analytical, numerical and experimental study of the ......tight compressors and helicoidal springs....

Analytical, Numerical and Experimental Study of the

Dynamical Response of Helicoidal Springs and Period ic

Bars

Hugo Filipe Dinis Policarpo

Dissertação para obtenção do Grau de Mestre em

Engenharia Mecânica

Júri

Presidente: Prof. Nuno Manuel Mendes Maia

Orientador: Prof. Miguel António Lopes de Matos Neves

Co-orientador: Prof. António Manuel Relógio Ribeiro

Vogais: Prof. Arlindo José de Pinho Figueiredo e Silva

Prof. Mihail Fontul

Setembro 2008

Aos meus pais e avós,

à Rita e a todos os que me ajudaram.

"If I have seen further it is by standing on the shoulders of giants."

Isaac Newton

i

Agradecimentos

Quero expressar os meus sinceros agradecimentos:

Ao meu orientador, o Professor Miguel Matos Neves pela orientação, disponibilidade, apoio e

incentivo que me facultou durante a realização deste meu trabalho.

Ao meu Co-orientador, o Professor António Relógio Ribeiro pela orientação prestada na área de

vibrações, com particular ênfase na componente experimental.

Ao Professor Mihail Fontul por todo o auxílio prestado no Laboratório de Vibrações do DEM-IST.

Ao Professor Arlindo Silva pela disponibilização de blocos de aglomerado de cortiça e cedência de

informação acerca dos mesmos.

Ao Engenheiro Olavo Silva da UFSC (Brasil) pela disponibilidade e informação fornecida sobre

compressores herméticos e molas helicoidais.

Ao Engenheiro Diogo Montalvão e Silva pela ajuda prestada no Laboratório de Vibrações do DEM-

IST, pela cedência da versão do programa informático BETAlab por si desenvolvido, assim como,

uma formação de ambientação ao mesmo.

Aos técnicos do Laboratório de Técnicas Oficinais do DEM-IST, nomeadamente ao Sr. Nelson

Fernandes, ao Sr. Carlos Faria e especialmente ao Sr. Pedro Alves por todo o apoio, disponibilidade

e tempo despendido na maquinagem dos materiais utilizados na construção dos provetes de ensaio

desenvolvidos.

À empresa Valdemar Ribeiro Representação de Comércio e Ferramentas, Lda. pela oferta das

molas helicoidais utilizadas, assim como, a disponibilidade do Sr. Valdemar Ribeiro em me receber.

À minha família, de um modo muito especial aos meus pais, avós e à Rita pelo apoio prestado em

todas as situações.

Aos meus amigos: Bruno Gomes por toda a ajuda prestada na realização dos ensaios

experimentais os quais sem ele teriam sido mais morosos e de difícil execução; Jorge Fonseca,

Pedro Lagos e Tiago Massano pela sua disponibilidade, apoio e incentivo prestado.

Ao meu amigo Peter Muelhan, amigo da família, residente nos Estados Unidos da América, pelo

envio de materiais alternativos susceptíveis a futuras análises e construção de provetes de ensaio.

Aos meus amigos Max e Flor pela sua companhia durante o decorrer deste trabalho.

Ao IDMEC e à fundação para a ciência e tecnologia no âmbito do projecto FCT POCTI

44728/EME/2002, designado por Técnicas de Análise de Ondas de Bloch Aplicadas ao Projecto

Óptimo de Estruturas Periódicas, pelos apoios financeiros concedidos.

ii

Acknowledgments

I want to express my sincere acknowledgments:

To my thesis orientator, Professor Miguel Matos Neves for the orientation, availability, support and

incentive provided which made possible the accomplishment of this work.

To my Co-orientator, Professor António Relógio Ribeiro for the orientation given in the area of

vibrations, with particular emphasis in the experimental component.

To Professor Mihail Fontul for all the support provided in the Laboratory of Vibrations of DEM-IST.

To Professor Arlindo Silva for supplying the agglomerated blocks of cork as well as some of their

properties.

To Engineer Olavo Silva of the UFSC (Brasil) for the availability and supplied information on air-

tight compressors and helicoidal springs.

To Engineer Diogo Montalvão e Silva for the support given in the Laboratory of Vibrations of DEM-

IST, for the availability of the software version of BETAlab, developed by him, as well as, the an

essential formation to get familiarized with the software.

To the technicians of the Laboratory of Machining Techniques of DEM-IST, namely to Mr. Nelson

Fernandes, Mr. Carlos Faria and especially to Mr. Pedro Alves for all the support, availability and time

dispended in the machining of the materials used in the construction of the experimental specimens.

To the company Valdemar Ribeiro Representação de Comércio e Ferramentas, Lda. for offering

the helicoidal springs used in this work, as well as, the availability of Mr. Valdemar Ribeiro in receiving

me.

To my family, and in a very special way to my parents, grandparents and Rita for the support given

in all situations.

To my friends: Bruno Gomes for all the support given in the accomplishment of the experimental

tasks which without him would have been time consuming and of difficult execution; Jorge Fonseca,

Pedro Lagos and Tiago Massano for their availability, support and incentive.

To my friend Peter Muelhan, a family friend, resident in the United States of America, for mailing

some alternative materials susceptive of future analysis and construction of experimental specimens.

To my friends Max and Flor for their company during this work.

To IDMEC and to the Foundation for Science and Technology regarding the project FCT POCTI

44728/EME/2002, designated by Analyzing Techniques of Bloch Waves Applied to the Optimum

Project of Periodic Structures, for the financial support.

iii

Resumo

Este documento apresenta uma caracterização da atenuação de vibrações em sistemas com

molas helicoidais e/ou estruturas periódicas multilaminadas. Consideram-se três abordagens

complementares: analítica ou teórica, numérica e experimental. O objectivo consiste no controlo

passivo da transmissibilidade longitudinal de vibração em determinadas gamas de frequência, através

do projecto de estruturas alternativas de suspensão com custo monetário semelhante ao das molas.

Este trabalho é motivado pelo facto das suspensões que utilizam molas helicoidais não

proporcionarem um desempenho satisfatório para todas as gamas de frequências a que estão

sujeitas, dado que geralmente apresentam uma distribuição de frequências naturais regularmente

espaçadas no espectro da frequência, originando vibração indesejável para frequências próximas das

de ressonância.

Estruturas periódicas multilaminadas são introduzidas como alternativas, apresentando em

determinadas gamas de frequência um afastamento significativo entre frequências naturais

adjacentes, relativamente ao das molas. Estas estruturas são posteriormente combinadas com molas

helicoidais tentando tirar partido das vantagens inerentes a cada estrutura.

Os modelos analíticos e numéricos são verificados e validados experimentalmente. É apresentada

uma relação interessante entre a análise modal (meio finito) e a teoria de ondas de Bloch (meio

infinito), a qual não se encontrou referenciada na literatura pesquisada.

Dois problemas de optimização estrutural são apresentados para estruturas multilaminadas

(periódicas e não-periódicas).

Uma breve análise relativamente à selecção de materiais é apresentada. Aço, latão, PMMA e

aglomerado de cortiça são utilizados no fabrico dos provetes experimentais.

Como trabalho futuro sugere-se o estudo com outros materiais (e.g. materiais visco elásticos) que

permitam alcançar regiões de atenuação para frequências mais baixas.

Palavras-chave: Atenuação Passiva, Estruturas Periódicas, Hiato de Ressonâncias, Optimização

Estrutural, Vibração.

iv

Abstract

This document presents a characterization on the attenuation of vibrations of systems with

helicoidal springs and/or multilaminated periodic bars. Three complementary approaches are

considered: analytical or theoretical, numerical and experimental. The objective is to control, in a

passive form, longitudinal vibration transmissibility in specific frequency ranges of interest, where the

possibility to design alternative suspension structures with similar monetary cost, relatively to the

spring, is of interest.

This work is motivated by the fact that suspensions systems using helicoidal springs do not always

provide a satisfactory performance in the service frequency range, since they generally present a

regularly spaced distribution of the natural frequencies, originating excessive vibration levels for

frequencies nearby these resonances.

Multilaminated periodic structures are introduced alternatively, presenting a significant wider gap

between adjacent frequencies, relatively to the spring, in specific frequency ranges. These structures

are posteriorly combined with the helicoidal spring trying to take advantage of the inherent

characteristics of each structure.

The analytical and numerical models are verified and validated experimentally. An interesting

relation between the modal analysis (finite medium) and the Bloch wave theory (infinite medium) is

proposed for which no reference to it was found in the researched literature.

Two structural optimization problems are presented for the multilaminated periodic and non-

periodic structures.

A brief material selection analysis is performed. Steel, brass, PMMA and agglomerated cork are

used in the construction of the experimental specimens.

Future work involving other materials (e.g. viscoelastic materials, foam materials) should be

explored to achieve even lower frequency attenuation regions.

Keywords: Passive Attenuation, Periodic Structure, Frequency Bandgap, Structural Optimization,

Vibration.

v

Table of Contents AGRADECIMENTOS ...................................................................................................................................... I

ACKNOWLEDGMENTS .................................................................................................................................. II

RESUMO .................................................................................................................................................... III

ABSTRACT ................................................................................................................................................ IV

TABLE OF CONTENTS .................................................................................................................................. V

LIST OF FIGURES ..................................................................................................................................... VIII

LIST OF TABLES ....................................................................................................................................... XII

NOTATION ............................................................................................................................................... XIII

1. INTRODUCTION .................................................................................................................................... 1

2. ANALYTICAL MODELS .......................................................................................................................... 6

2.1. Elastic Characterization ............................................................................................................ 6

2.2. Finite Element Method .............................................................................................................. 9

2.3. Types of Structural Analysis ..................................................................................................... 9

2.3.1. Static Analysis ...................................................................................................................... 10

2.3.2. Modal Analysis ..................................................................................................................... 10

2.3.3. Harmonic Analysis ............................................................................................................... 11

2.4. Dynamic Characterization of Structures ................................................................................. 12

2.4.1. Resonances and Anti-Resonances ...................................................................................... 14

2.4.2. Residuals .............................................................................................................................. 14

2.4.3. Modal Identification – The Characteristic Response Function (CRF) .................................. 15

2.5. Modal Identification Technique ............................................................................................... 17

2.6. Transmissibility – Overview .................................................................................................... 17

2.7. Verification and Validation of Models – Overview ................................................................... 18

2.8. Springs .................................................................................................................................... 19

2.8.1. Helicoidal Spring .................................................................................................................. 19

2.8.2. Static Deflection of a Helicoidal Spring ................................................................................ 21

2.8.3. Nonlinearities........................................................................................................................ 22

2.8.4. Natural Frequencies of a Helicoidal Spring .......................................................................... 22

2.9. Multilaminated Periodic Structures ......................................................................................... 23

2.9.1. Material Properties and Parameters Influence ..................................................................... 24

2.9.2. Static Deflection of a Multilaminated Bar ............................................................................. 25

2.9.3. Nonlinearities........................................................................................................................ 26

2.9.4. Bloch Wave Analysis ............................................................................................................ 26

2.10. Optimization ......................................................................................................................... 30

2.10.1. Structural Optimization of the Proportion of the Material in Each Repetitive Cell (Periodic

Case) .............................................................................................................................................. 31

2.10.2. Structural Optimization of the Distribution of Two Materials in Each Design Variable

(General Case) ............................................................................................................................... 32

vi

3. NUMERICAL MODELS .........................................................................................................................33

3.1. Basic Steps in Finite Element Method .................................................................................... 34

3.1.1. Preprocessing ...................................................................................................................... 34

3.1.2. Solution ................................................................................................................................ 34

3.1.3. Postprocessing ..................................................................................................................... 34

3.2. 2D and 3D Numerical Models ................................................................................................. 34


3.2.2. Multilaminated Bar ............................................................................................................... 36

3.3. Analysis: Static / Modal / Harmonic / Bloch ............................................................................ 36

3.4. Geometry and Mesh................................................................................................................ 37

3.5. Finite Element Convergence Study for the Necessary Number of F.E. per Wave Length ..... 38


3.7. Optimization Technique .......................................................................................................... 39

4. EXPERIMENTAL MODELS AND METHODOLOGY ....................................................................................40

4.1. Hardware Equipment .............................................................................................................. 41

4.1.1. Excitation of the Structure .................................................................................................... 41

4.1.2. Sensing Transducers ........................................................................................................... 41

4.1.2.1. Piezoelectric Accelerometer ......................................................................................... 42

4.1.2.2. Force Transducer ......................................................................................................... 42

4.1.3. Data Acquisition and Processing Components .................................................................... 42

4.2. Equipment Used ...................................................................................................................... 42

4.3. Experimental Setup ................................................................................................................. 43

4.4. Calibration and Finite Element Model Updating ..................................................................... 44

4.5. Experimental Specimens and Constructive Aspects .............................................................. 45

4.6. Adhesive ................................................................................................................................. 45


4.8. Methodology ............................................................................................................................ 48

5. RESULTS AND DISCUSSION ................................................................................................................49

5.1. Numerical vs. Analytical Verification ....................................................................................... 49


5.1.2. Homogeneous Bar ............................................................................................................... 52

5.1.3. Periodic Bar .......................................................................................................................... 52

5.2. Helicoidal Spring: Parameters and Properties Influence ........................................................ 53

5.2.1. Spring’s Parameters Influence in the Natural Frequencies ................................................. 54

5.2.2. Spring’s Parameters Influence in the Attenuation Region ................................................... 54

5.3. Bar: Parameters and Properties Influence .............................................................................. 56

5.3.1. Convergence Study for the Necessary Number of Finite Elements per Wave Length ........ 56

5.3.2. Homogeneous Bar: Parameters and Properties Influence .................................................. 57

5.3.3. Periodic Bar: Parameters and Properties Influence ............................................................. 57

5.3.4. A Material Selection on the Frequency Range of Interest ................................................... 59

vii

5.4. Experimental vs. Numerical and Analytical ............................................................................. 60

5.4.1. Calibration and Finite Element Model Updating ................................................................... 60

5.4.1.1. Homogeneous Bar ....................................................................................................... 60

5.4.2. Adhesive Influence ............................................................................................................... 61

5.4.2.1. Heterogeneous Steel-PMMA Specimen: 1 Cell ........................................................... 62

5.4.3. Heterogeneous Steel-PMMA Specimen: 5.5 Cells .............................................................. 62

5.5. In the Search for Lower Frequency Ranges: Agglomerated Cork .......................................... 64

5.5.1. Heterogeneous Steel-Commercial Agglomerated Cork: 3.5 Cells ....................................... 64

5.5.2. Heterogeneous Brass-Commercial Agglomerated Cork: 3.5 Cells ...................................... 65

5.5.3. Heterogeneous Steel-Agglomerated Cork Ref.8123 bar with Rectangular Section .......... 65

5.6. Helicoidal Spring ..................................................................................................................... 67

5.6.1. Helicoidal Spring Combined with Heterogeneous Periodic Bar ........................................... 67

5.6.1.1. Heterogeneous Steel-PMMA Specimen Combined with Helicoidal Spring ................. 68

5.6.1.2. Heterogeneous Steel-Commercial Agglomerated Cork Specimen Combined with

Helicoidal Spring ......................................................................................................................... 68

5.6.1.3. Heterogeneous Brass-Commercial Agglomerated Cork Specimen Combined with

Helicoidal Spring ......................................................................................................................... 69

5.6.1.4. Heterogeneous Steel-Agglomerated Cork Ref.8123 bar with Rectangular Section

Combined with Helicoidal Spring ................................................................................................ 70

5.6.2. Compressor Motor Base Support Simulating Structure ....................................................... 72

5.7. Optimization Results ............................................................................................................... 73

5.7.1. Results from the Maximization of the Separation of Two Adjacent Eigenfrequencies ωi+1

and ωi for a Periodic Material Distribution (Periodic Case) ............................................................ 73

5.7.2. Results from the Maximization of the Separation of Two Adjacent Eigenfrequencies ωi+1

and ωi by a Distribution of Two Different Materials in Each Design Variable (General Case) ...... 74

6. CONCLUSIONS ...................................................................................................................................75

REFERENCES .......................................................................................................................................76

APPENDIX .............................................................................................................................................80

A.1 – TYPE OF SPRING ENDS ....................................................................................................................81 A.2 – SPRINGS END CONDITIONS ..............................................................................................................82 A.3 – COLUMN EFFECTIVE LENGTH ...........................................................................................................82 A.4 – OPTIMIZATION TECHNIQUE ...............................................................................................................82 A.5 – CALIBRATION CHART: BRÜEL & KJAER TYPE 4508-B ......................................................................83 A.6 – CALIBRATION CHART: BRÜEL & KJAER TYPE 8200 ..........................................................................83 A.7 – FEA RESULT COMPARISON WITH [15] .............................................. ................................................84 A.8 – FEA RESULT COMPARISON WITH [22] .............................................. ................................................84 A.9 – EFFECT OF THE HELIX ANGLE ..........................................................................................................86 A.10 – EFFECT OF THE NUMBER OF ACTIVE COILS ....................................................................................86 A.11 – EFFECT OF THE DIAMETER RELATION .............................................................................................87 A.12 – CONVERGENCE STUDY FOR THE NECESSARY NUMBER OR F.E. PER WAVE LENGTH.........................87 A.13 – CONTINUATION OF EXPERIENCE ADAPTATION .................................................................................88 A.14 – ACCELEROMETER MOUNTING CLIP ................................................................................................90

viii

List of Figures Figure 2.1 – Body and surface forces acting on arbitrary portion of a continuum. .................................. 7

Figure 2.2 – Example of a model with N degrees of freedom. ................................................................ 9

Figure 2.3 – Natural frequencies plot based on modal analysis. .......................................................... 10

Figure 2.4 – Frequency response of a structure subject to a harmonic excitation force for N degrees of

freedom. ......................................................................................................................................... 11

Figure 2.5 – Dynamic models interrelation (undamped case) [61]........................................................ 13

Figure 2.6 – Resonance and anti-resonance of a 2 degree of freedom system [68]. ........................... 14

Figure 2.7 – CRF’s terms graphical representation obtained from BETAlab [62]: (a) Real term; (b)

Imaginary term. .............................................................................................................................. 16

Figure 2.8 – Theoretical transmissibility for a system with a natural frequency of 20 Hz. .................... 17

Figure 2.9 – Verification and validation procedure between the theoretical, numerical and experimental

models. ........................................................................................................................................... 18

Figure 2.10 – Circular helix.................................................................................................................... 20

Figure 2.11 – Type of ends for helicoidal springs: (a) Open, plain end; (b) Open, squared; (c)

Closed, plain end; (d) Closed, squared [75]. .................................................................................. 21

Figure 2.12 – Periodic structures: (a) Pure crystals; (b) Pin joint bars; (c) Beam or bar. ..................... 23

Figure 2.13 – Bar with a static force applied: (a) Heterogeneous multilaminated bar; (b) Homogeneous

bar. ................................................................................................................................................. 25

Figure 2.14 – Schematic drawing of two material periodic structure. ................................................... 28

Figure 2.15 – Dispersion curves obtained from Eq.(93). ....................................................................... 30

Figure 2.16 – Multilaminated bar: (a) Periodic bar; (b) Non-periodic bar. ............................................. 31

Figure 3.1 – Finite elements, nodes and mesh of a sub-structure Ω. ................................................... 33

Figure 3.2 – Order of degrees of freedom in the beam element [58]. ................................................... 35

Figure 3.3 – Finite element geometry used for the bar [58]. ................................................................. 36

Figure 3.4 – Helicoidal spring’s geometry: (a) Continuous model; (b) FEM model............................... 37

Figure 3.5 – Multilaminated periodic bar: (a) Continuous model; (b) FEM model. ................................ 38

Figure 3.6 – Multilaminated non-periodic bar: (a) Continuous model; (b) FEM model. ........................ 38

Figure 3.7 – Combined structure: (a) Continuous model (b) FEM model. ............................................ 38

Figure 4.1 – Transmissibility between the shaker and the structure through a push-rod [61]. ............. 41

Figure 4.2 – Sensing transducers [61]: (a) Cross-section of a piezoelectric accelerometer; (b) Cross-

section of a piezoelectric force transducer. .................................................................................... 41

Figure 4.3 – Hardware components and auxiliary support structures: (a) Workstation: PC, amplifiers,

data acquisition components, etc.; (b) Fixation and support structure. ......................................... 42

Figure 4.4 – Experimental setup. .......................................................................................................... 43

Figure 4.5 – Amplified photographs of some hardware equipment: (a) Vibration exciter B&K Type

4809; (b)Force transducer B&K Type 8200; (c) Accelerometer B& K Type 4508-B. .................... 44

Figure 4.6 – Finite element analysis updating process of the material Young’s modulus. ................... 44

ix

Figure 4.7 – Steel and PMMA half-cells with a diameter of 0.020m and a length of 0.135m: (a) Top

view; (b) Six steel half-cell front view; (c) Five PMMA half-cells front view. .................................. 45

Figure 4.8 – L shaped beam to assist the bonding process: (a) Half-cell order; (b) L shaped beam

(side view). ..................................................................................................................................... 45

Figure 4.9 – Adhesive layer of material in a unit cell: (a) Via SolidWorks®; (b) Via experimental. ....... 45

Figure 4.10 – Adhesive X 60 Schnellklebstoff: (a) Exterior package; (b) Interior of the Package. ....... 46

Figure 4.11 – BETAlab software environment screen shot. .................................................................. 47

Figure 4.12 – Diagram of the methodology adopted in this work. ......................................................... 48

Figure 5.1 – Helicoidal spring parameters identification. ...................................................................... 50

Figure 5.2 – Relative error obtained between the analytical and finite element results. ....................... 50

Figure 5.3 – Relative error obtained between the finite element method applied and those of [15]. .... 51

Figure 5.4 – Finite elements, transference matrix and the dynamic stiffness methods [22] results

comparison. .................................................................................................................................... 51

Figure 5.5 – Periodic bar: (a) Modal analysis, finite structure; (b) Bloch wave analysis, infinite

structure. ........................................................................................................................................ 52

Figure 5.6 – Limit curves of the bandgap and frequency band versus number of cells: (a) First

frequency band, (b) Second frequency band. ............................................................................... 53

Figure 5.7 – Helicoidal spring’s parameters identification. .................................................................... 53

Figure 5.8 – Parameters influence in the natural frequencies: (a) Helix angle α; (b) Number of active

coils Na; (c) Diameter relation D/d. ............................................................................. 54

Figure 5.9 – Influence of the helix angle. .............................................................................................. 55

Figure 5.10 – Effect of the helix angle. .................................................................................................. 55

Figure 5.11 – Influence of the active number of coils. ........................................................................... 55

Figure 5.12 – Effect of the active number of coils. ................................................................................ 55

Figure 5.13 – Influence of the diameter relation D/d. ............................................................................ 55

Figure 5.14 – Effect of the diameter relation D/d. ................................................................................. 55

Figure 5.15 – Homogeneous bar with diameter of 0.01m and Lt=0.825m: (a) Natural frequencies; (b)

Frequency response....................................................................................................................... 57

Figure 5.16 – Periodic bar with diameter of 0.01m, 2.5≤n≤5.5, L1=L2=0.075m, Lt=0.825m: Natural

frequencies; .................................................................................................................................... 58

Figure 5.17 – Periodic bar with diameter of 0.01m, n=5.5, L1=L2=0.075m, Lt=0.825m. ....................... 58

Figure 5.18 – Periodic bar with diameter of 0.01m, n=5.5, L1=L2=0.075m, Lt=0.825m: (a) Natural

frequencies; (b) Frequency response. ........................................................................................... 58

Figure 5.19 – Results obtained for the periodic model presented in section 3.2.2 with different

materials. ........................................................................................................................................ 59

Figure 5.20 – Experimental setup of the homogeneous steel bar specimen: (a) Vibration exciter; (b)

Specimen; (c) Accelerometer. ........................................................................................................ 60

Figure 5.21 – Frequency response curves for the homogeneous steel bar specimen. ........................ 61

Figure 5.22 – Experimental setup of the homogeneous PMMA bar specimen. .................................... 61

Figure 5.23 – Frequency response curves for the homogeneous steel bar specimen. ........................ 61

x

Figure 5.24 – Experimental setup of the one cell heterogeneous steel-PMMA bar specimen. ............ 62

Figure 5.25 – Frequency response curves for the one cell heterogeneous one cell steel-PMMA bar

specimen. ....................................................................................................................................... 62

Figure 5.26 – Heterogeneous specimen: (a) Experimental setup of the 5.5 cells steel-PMMA bar

specimen; (b) Accelerometer positioning in the end extremity. ..................................................... 62

Figure 5.27 – Frequency response curves for the 5.5 cells heterogeneous steel-PMMA bar specimen.

........................................................................................................................................................ 63

Figure 5.28 – Frequency response curves for the 5.5 cells heterogeneous steel-PMMA bar specimen,

including the regenerated curve. .................................................................................................... 63

Figure 5.29 – Experimental setup of the 3.5 cells heterogeneous steel-commercial agglomerated cork

bar specimen. ................................................................................................................................. 64

Figure 5.30 – Frequency response curves for the 3.5 cells heterogeneous steel- commercial

agglomerated cork bar specimen. .................................................................................................. 64

Figure 5.31 – Experimental setup of the 3.5 cells heterogeneous brass-commercial agglomerated cork

bar specimen. ................................................................................................................................. 65

Figure 5.32 – Frequency response curves for the 3.5 cells heterogeneous brass- commercial

agglomerated cork bar specimen. .................................................................................................. 65

Figure 5.33 – Experimental setup of 3.5 cells heterogeneous steel-agglomerated cork Ref.8123 [87]

(with 0.05m cell length) square cross-section bar specimen. ........................................................ 65

Figure 5.34 – Frequency response curves for the 3.5 cells heterogeneous steel-agglomerated cork

Ref.8123 [87] (with 0.05m cell length) square cross-section bar specimen. ................................. 66

Figure 5.35 – Experimental setup of for the 3.5 cells heterogeneous steel-agglomerated cork Ref.8123

[87] (with 0.10m cell length) square cross-section bar specimen. ................................................. 66


Ref.8123 [87] (with 0.10m cell length) square cross-section bar specimen. ................................. 66

Figure 5.37 – Experimental helicoidal spring: (a) Parameters; (b) Frequency response curves of four

identical springs. ............................................................................................................................. 67

Figure 5.38 – Numerical frequency response curves of the combined structure (example). ................ 67

Figure 5.39 – Steel-PMMA specimen combined with the helicoidal spring: (a) Experimental setup; (b)

Extremity close-up. ......................................................................................................................... 68

Figure 5.40 – Frequency response curves for the 3.5 cells heterogeneous steel-PMMA bar specimen

combined with the helicoidal spring. .............................................................................................. 68

Figure 5.41 – Steel-Commercial Agglomerated cork specimen combined with the helicoidal spring: (a)

Experimental setup; (b) Extremity close-up. .................................................................................. 68

Figure 5.42 – Frequency response curves for the 3.5 cells heterogeneous steel- commercial

agglomerated cork bar specimen combined with the helicoidal spring. ......................................... 69

Figure 5.43 – Brass-Commercial Agglomerated cork specimen combined with the helicoidal spring: (a)

Experimental setup; (b) Extremity close-up. .................................................................................. 69

Figure 5.44 – Frequency response curves for the 3.5 cells heterogeneous brass- commercial

agglomerated cork bar specimen combined with the helicoidal spring. ......................................... 69

xi

Figure 5.45 – Steel- Agglomerated cork ref.8123 [87] specimen (0.05m cell length), with a square

cross-section, combined with the helicoidal spring: (a) Experimental setup; (b) Extremity close-up.

........................................................................................................................................................ 70


Ref.8123 [87] (with 0.05m cell length) square cross-section bar specimen combined with the

helicoidal spring. ............................................................................................................................. 70

Figure 5.47 – Steel-Agglomerated cork ref.8123 [87] specimen (0.10m cell length), with a square

cross-section, combined with the helicoidal spring: (a) Experimental setup; (b) Extremity close-up.

........................................................................................................................................................ 70


Ref.8123 [87] (with 0.10m cell length) square cross-section bar specimen combined with the

helicoidal spring. ............................................................................................................................. 71

Figure 5.49 – Experimental setup compressor motor base simulating structure with specimens: (a)

Helicoidal springs; (b) Multilaminated bars; (c) Springs + bars. ..................................................... 72

Figure 5.50 – Frequency response curves of the compressor motor base simulating structure with

different specimens. ....................................................................................................................... 72

Figure 5.51 – Results for n=3 cells: a) Eigenfrequencies curve plot; b) Dispersion curves and c)

Frequency response plot. ............................................................................................................... 73

Figure 5.52 – Results for i=1 : a) Eigenfrequencies curve plot ; b) Frequency response plot and c)

Material distribution in the bar. ....................................................................................................... 74

Figure 5.53 – Results for i=5 : a) Eigenfrequencies curve plot ; b) Frequency response plot and c)

Material distribution in the bar. ....................................................................................................... 74

Figure A.1 – Reproduction of Figure 6.1: Type of ends for helicoidal springs: (a) Open, plain end; (b)

Open, squared; (c) Closed, plain end; (d) Closed, squared [75]. ................................................. 81

Figure A. 4 – Diagram of the implemented optimization technique. ......................................................82 Figure A. 5 – Calibration chart supplied with the accelerometer: Brüel & Kjaer Type 4508 [92]. ......... 83 Figure A. 6 – Calibration chart supplied with the force transducer: Brüel & Kjaer Type 8200 [93]. ...... 83 Figure A.12.1 – Frequency response of homogeneous steel experimental specimen with 8, 16 and 32

finite elements per wave length. ..................................................................................................... 87 Figure A.13.1 – Experimental setup of heterogeneous specimen of the 4.5 cells steel-PMMA bar. .... 88 Figure A.13.2 – Frequency response curves for the 4.5 cells heterogeneous steel-PMMA bar specimen………………………………………………………………………………………………………...88 Figure A.13.3 – Frequency response curves for the 3.5 cells heterogeneous steel-PMMA bar specimen………………………………………………………………………………………………..……….88 Figure A.13.4 – Frequency response curves for the 3.5 cells heterogeneous steel-PMMA bar specimen…………………………………………………………………………………………………..…….89 Figure A.13.5 – Frequency response curves for the 1.5 cells heterogeneous steel-PMMA bar specimen……………………………………………………………………………………………….………..89 Figure A.13.6. – Frequency response curves for the 1.5 cells heterogeneous steel-PMMA bar specimen………………………………………………………………………………………………….……..89

xii

List of Tables

Table 5.1 – Helicoidal spring: Theoretical results compared with those obtained via FEM. ................. 50

Table 5.2 – Standard material properties: E- longitudinal modulus of elasticity; ρ- mass density. ....... 52

Table 5.3 – Homogenous bar: Theoretical results compared with those obtained via FEM. ................ 52

Table 5.4 – Spring’s parameters influence in the attenuation region. ................................................... 56

Table 5.5 – Wave length of the studied materials. ................................................................................ 56

Table 5.6 – Steel and aluminum properties (E, ρ and c). ...................................................................... 57

Table 5.7 – Experimental mass density of the studied materials. ......................................................... 60

Table 5.8 – Specimen type, length and initial frequency of the attenuation region. ............................. 71

Table 5.9 – Output results from the optimization cases with i = 3. ........................................................ 73

Table 5.10 – Output results from the optimization cases with i = 1 and 5............................................. 74

Table A.1 – Expressions for helicoidal spring dimensions [75].…………………………………………...81

Table A.2 – End condition constants α for helicoidal compression springs [75]. .................................. 82

Table A.3 – Effective length of column for various end conditions [75]. ............................................... 82

Table A.7– Results obtained via FEM, for the first eight eigenmodes, compared with those of [15]. ... 84

Table A.8.1 – Natural frequencies in Hz of a spring with fixed – fixed boundary conditions; E=2.09·1011

N/m2; υ=0.28; ρ=7800kg/m3; r=6mm; R=65mm; α=7.44º; N=6; and L0 = 320mm; all modes below

100 Hz are presented. .................................................................................................................... 84

Table A.9– Effect of the helix angle. ...................................................................................................... 86

Table A.10– Effect of the number of active coils. .................................................................................. 86

Table A.11– Effect of the diameter relation D/d. ................................................................................... 87

xiii

Notation

A Area α Helix angle; Phase angle b Number of helix revolutions β Phase angle c Viscous damping coefficient; Γ Boundary Wave speed ε Strain C Spring index; Constants θ Angle of revolution D, d Diameter λ Wave length E Longitudinal modulus of elasticity μ Wave propagation coefficient or Young’s modulus μa Anti-resonance frequency F Force ρ Mass density f , ω Frequency σ Stress G Shear modulus or modulus of rigidity τ Torsion of the helix; Torsional stress h Pitch of the helix υ Poisson’s coefficient i, j, k, l Complex unity (i2=-1; j2=-1); Φ, φ Phase angle J Polar moment of inertia Ω Domain KB Bergstrasser’s correction factor k Spring stiffness or spring constant; Stiffness; Wave number κ Curvature of the helix L Length L0 Free length of the spring Ls Solid length of the spring Le Effective length M, m Mass N, Na Number of active coils R, r - Radius S Surface T Surface Force; Torsional moment t Time variable; Traction vector u Displacement U Strain energy; Displacement V Volume; Potential energy X Maximum amplitude x, y, z zz Rectangular coordinates; Displacement; Deflection z Impedance; Dynamic stiffness matrix

1

1. Introduction

This document presents a study on the analytical and numerical modulation and experimental

validation of elastic structures, namely helicoidal springs and multilaminate structures. The objective is

to control in a passive form longitudinal vibration transmissibility in specific frequency ranges, where

the possibility to design alternative suspension structures with similar monetary cost, relatively to the

spring, is of interest.

The present work is motivated by the fact that traditional suspension systems using helicoidal

springs, being both inexpensive and critical to reliability of many other systems do not always provide

a satisfactory performance in the service frequency range [1]-[3], since they present a regular spaced

distribution of the natural frequencies in the frequency spectrum, originating unwanted vibration and

possible acoustic discomfort, for frequencies nearby the resonances. Multilaminated structures are

introduced as alternative structures because for certain frequency ranges they can be designed to

present significant wider gaps between adjacent frequencies, relatively to the spring [4]. A structure

that combines both the helicoidal spring and multilaminated structure is analyzed trying to take

advantage of the inherent characteristics of each structure.

A brief literature review concerning the helicoidal spring (also known as helical coil spring) and its

optimization, the transmissibility concept, the periodic structures and its optimization as well as wave

propagation in mediums is presented next.

From a historical perspective, the first coil spring was produced in the 15th century [5]. The first

differential motion equations applicable to the vibration of helices and rings dates from 1890 when

Michell [6] obtained three motion equations using the first form of Lagrange’s fundamental equation.

Based on Michell’s work, in 1899, Love [7] enunciates six 12th order differential motion equations for a

helical wire. These motion equations revealed to be of great importance on future developments in the

field.

In 1936, Timoshenko [8] derives the spring’s stiffness and the longitudinal and transversal

deflections defining the stability limits of the helicoidal spring. Ancker and Goodier [9] study helicoidal

springs with circular wire cross-section considering tension, deflection, the coupling effects and the

curvature effect. Further derivations of the motion equations were obtained which contemplate the

effects of a pre-static axial load [10] and torsional movements of the helicoidal spring [11] and [12].

In 1980, Mottershead [13] develops a finite element method to obtain the displacement functions by

integration of the differential equations. Swanobori et al. [14] study the static and dynamic tensions of

helicoidal springs recurring to the finite element method. Other methods, such as the dynamic stiffness

method [15]-[17], variational methods [18] and [19] and differential geometry [20] were used to obtain

exact and/or approximate solutions for the natural frequencies of helicoidal springs with circular wire

cross-section, variable pitch angle, variable helix radius and the curvature effects based on the Euler-

Bernoulli beam theory. This is generalized for arbitrary shape helicoidal springs in [21] and [22].

2

In [23], Yildirim determines the natural frequencies of helicoidal springs using the distributed mass

model and Timoshenko beam theory together with the axial deformation. For the circular wire cross-

section, the effects of the helix pitch angle, the number of active turns, the ratio of diameters of the

minimum cylinder to the maximum cylinder, and the ratio of diameters of maximum cylinder to the

diameter of wire on the free vibration frequencies of all types of helices are studied. It is shown that

the numerical results agree well with previous theoretical and experimental published results.

Regarding optimization of springs and spring structures, in [24] Lin et al. develop and improve

optimization routines, which include the spring dynamic model, with the objective of minimizing the

amplitude of helicoidal spring in resonance. In [25], Paredes et al. present an advanced sizing tool for

compression spring design, making use of the optimization capabilities currently available, going

beyond the existing industrial software based on comprehensive validation tools. Pearce in [26] and

[27] carries out speculative research work, presenting a study on ceramic PZT (lead zirconate titanate)

springs and arrives at some exciting potential applications for these devices e.g., audio speakers. A

“new” methodology for the optimal design of composite helicoidal springs considering a multi-objective

evolutionary algorithm is implemented in [28] to optimize two conflicting goals: minimize mass and

maximize stiffness. A range of composite springs is analyzed, among which an optimal spring was

selected for an automotive application, namely to replace the metallic spring of the suspension of a

sport utility vehicle. Royston and Singh [29] improve an analytical strategy for the design optimization

of complex, active or passive, non-linear mounting systems. In [30] Ashrafiuon optimizes a base of an

aircraft engine applying an enhanced artificial life algorithm to minimize the vibration transmissibility.

To better understand the concept of vibration transmissibility, since it is an important concept in

most areas of vibration analysis and has only recently been addressed for multi degree of freedom

(MDOF) systems, a suitable definition of the concept can be found in [31]. Some practical applications

are referred in [32], respecting the predicted responses at points where the transducers are not

allowed in field service or in damage detection. More recently, in [33] Ribeiro et al. continue to develop

the transmissibility concept for MDOF systems, which include an alternative formulation that allow the

concept’s expansion to other research areas namely, systems identification and structural

modification.

Multilaminated structures have received a lot of attention and extensive efforts were made to

analyze the propagation of waves in periodic structures, which consist of a number of identical

elements joined together in an identical manner. Distinguished among these efforts is the unified

approach of Brillouin for the analysis of the dynamics of a wide variety of periodic structures which is

documented in his classical book [4]. The theory of periodic structures was originally developed for

solid state applications [34] and extended, in the early seventies, to the design of mechanical

structures [35]. Since then, the theory has been extensively applied to a wide variety of structures

such as spring-mass systems [36] periodic beams [35]-[38], stiffened plates [39] and space structures.

Apart from their unique filtering characteristics, the ability of periodic structures to transmit waves, from

one location to another, within the pass-bands can be greatly reduced when the ideal periodicity is

disrupted or disordered [39] and [40]. Among these pioneering efforts is also the work of Mead and his

3

co-workers which extends over a period of thirty five years and includes many of the original

contributions in the analysis and characterization of the wave propagation in periodic structures [41]-

[45].

To study structures with infinite periodicity repetition, Bloch's theorem [34] also related with

Lyapunov-Floquet’s theorem [46] and [47] can be used to obtain a characterization of longitudinal

waves leading to the corresponding dispersion relation that present a band structure analogue to

those of electrons in a crystal and light in supper-lattices [34]. In mediums with periodic heterogeneity,

there are ranges of frequencies, known as pass-bands and stop-bands, over which all incident waves

are allowed to propagate and effectively stopped, respectively. This frequency-banded dynamic

response has interest across multiple disciplines. For structures with finite periodic repetition,

researchers are still reviewing and trying to improve existing models for vibration problems. A

description of the basics can be found in [4] and [48] while for structural optimization of longitudinal

vibrations it can be found, among others, in [49]-[51]. In [52] Kosevich present a one dimension model

that describes the frequency spectrum of acoustic and electromagnetic waves in periodic elastic

structures.

In respect to the optimization of periodic or near-periodic structures, Barbaroise and Neves [53]

describe Floquet–Bloch’s finite element analysis and optimization techniques to deal with the periodic

structural design problems. The main advantage of periodic structures is simple fabrication. The

results presented illustrate how a material distribution inside a cell can create bandgaps in the

dispersion curves of wave propagation. This way it is possible to systematically design simple

frequency filters that maximize specific bandgaps within the dispersion curves. In [54], Jensen and

Pederson present a method to maximize the separation of two adjacent eigenfrequencies in structures

with two material components. The method is based on finite element analysis and topology

optimization in which an iterative algorithm is used to find the optimal distribution of the materials. In

[55], Hussein et al. study the electrodynamics of 1D periodic materials and finite structures comprising

these materials with particular emphasis on correlating their frequency-dependent characteristics and

on elucidating their pass-band and stop-band behaviors. It is shown that the maximum localized

response under stop-band conditions could be significantly less than in an equivalent homogenous

structure and the converse is true for pass-band conditions. In [56], Hussein et al. uses a multi-

objective genetic algorithm to design the topologies of one-dimensional periodic unit cells for target

frequency band structures characterizing longitudinal wave motion. It is shown that the widths and

locations of these stop and pass-bands in the frequency domain depend on the layout of contrasting

materials and the ratio of their properties. In [57], Silva et al. review and investigate a numerical

technique of combining finite element analysis and optimization techniques to deal with passive

vibration attenuation problems at structural components. It is also investigated a procedure to

minimize the frequency response displacement in a frequency range of interest and a modified

methodology to get a wider attenuation region in the referred frequency range is proposed. In [58]

Yilmaz and Kikuchi design a stiff and lightweight passive vibration isolator that presents a wide stop-

4

band at low frequencies. In [59], Lopes et al. extend the models of cellular/composite material design

to nonlinear material behavior and apply them for design of materials for passive vibration control.

Having exposed the bibliography review, the methodology adopted in this work is now presented.

The methodology used is based on analytical methods, finite element numerical methods,

experimental analysis and structural optimization techniques. To simplify the problem it is assumed

that each material is isotropic with linear elastic behavior. The structural models are limited to the

elastic bar (i.e., link or rod elements) and to the Euler-Bernoulli beam model [57]. No damping effects

are considered. The helicoidal springs and the multilaminated structures are modeled using finite

elements and verified with published work: [16] and [23] for helicoidal springs and [54] and [55] for

multilaminated structures. The helicoidal spring is studied with circular wire cross-section for which the

effects of the helix angle, the number of active coils and the ratio of diameters (wire and coil) on the

vibration frequencies and on the attenuation regions are studied.

As previously mentioned, traditional suspension systems using helicoidal springs do not always

provide a satisfactory performance in the service frequency range. So in the search for an alternative

solution the periodic structures are studied. The methodology proposed reveals a new perspective on

the problem in which is observed the relation between the eigenfrequency results (modal problem),

the dispersion relations (Bloch wave problem) and frequency response (harmonic problem). This can

be combined with a selection of materials taken for a frequency range of interest for practical

applications. To illustrate the new perspective, two structural optimization problems are introduced

which reveal significant attenuations for the case of longitudinal vibrations. The numerical

implementation technique used combines a commercial finite element analysis code (Ansys®) [59] for

the finite element analysis with optimization algorithms running on a problem solving environment

(MatLab®) [61].

Nevertheless, these multilaminated types of structures are not advantageous for low frequencies,

due to the mechanical properties required by the composing materials [4]. A structure that combines

both the helicoidal spring and multilaminated structure is analyzed trying to that take advantage of the

inherent characteristics of each structure.

Having all the numerical models successfully verified against published work, the next step

consists in the experimental analysis, trying to reproduce the numerical results obtained and

consequently validate the developed models. After routine hardware and software calibration tests and

simplified numerical model updating procedures, the experimental specimens, composed of two

different materials bonded together by a thin adhesive layer, are built. The experimental results

obtained and the modal parameters from modal identification procedures [62], described in section

2.4.3, are compared with the numerical ones. Bode diagrams for the numerical and the regenerated

transfer functions are represented in the same plot for easier comparison.

This study supports the possibility to use this design methodology for the development of vibration

and shock isolation structures, sound isolation pads/partitions, and multiple band frequency filters,

among other applications.

5

To conclude this introductory chapter, it is now presented the layout of this work.

This work is composed by six main chapters. The first is an introductory chapter where the theme

is presented justifying the importance and motivation of this work in terms of engineering and

particularly in the structural vibrations field. This is followed by a bibliography review where some of

the most relevant work in this field is referred. The methodology adopted in this work, where the main

guidelines are presented, terminates this introductory chapter.

The second chapter introduces the analytical or theoretical structural models for the helicoidal

spring and for the multilaminated periodic structures. An elastic characterization of the solid body,

types of analysis, definitions, properties and characteristics, used is presented in this chapter. A

general dynamic characterization of structures is followed by an overview on modal identification,

transmissibility concept, verification and validation of models and the helicoidal spring and

multilaminated periodic analytical models themselves are presented. Two optimization problems

regarding the periodic and non-periodic cases, respectively, are presented concluding this chapter.

In the third chapter, the basic steps of the finite element method are introduced allowing a

numerical approach to evaluate the analytical structural models. The numerical models for both

structures, helicoidal spring and multilaminated periodic bar, are presented. A study on the necessary

number of finite elements per wave length is performed which is followed by the modal identification

and optimizations techniques used in this work.

In the fourth chapter, the experimental models as well as the equipment (hardware and software)

necessary for the analysis are presented which includes brief references to the excitation

transmissibility, sensing transducers and data acquisition and processing components. This is

followed by a description of the equipment used, the experimental setup, constructive aspects of the

specimens, an introduction to the modal identification software used and the experimental

methodology adopted concludes this chapter.

In the fifth chapter the results obtained are presented and discussed. It starts by the verification

process of the analytical and numerical results obtained. This is followed by the presentation of the

results obtained for the helicoidal spring referring the spring’s parameters and properties influence in

the natural frequencies and in the attenuation regions. A brief similar analysis that illustrates how the

material properties of the periodic bar (the modulus of elasticity and the mass density) affect the

natural frequencies and the attenuation regions is also presented. To demonstrate the applicability of

these structures at frequencies of practical interest, a plot of results for some different combinations of

materials is presented. The experimental results obtained for the specimens constructed and tested

are here presented and validated with the numerical ones. To conclude, the optimization results for

the two proposed optimization problems are presented.

In the sixth and final chapter, entitled, Conclusions, the main conclusions obtained throughout this

work are referred and some suggestions for future work are presented.

6

2. Analytical Models

The present chapter introduces the analytical, mathematical or theoretical structural models,

definitions, properties and characteristics, used in this work. It begins by introducing the elastic

characterization of a solid body, based on the theory of elasticity, in which the stress-strain relations

and the variational (weak) and differential (strong) formulations of the equilibrium equations are

presented. This is followed by an insight on the mathematical formulation (variational formulation and

approximations) of the finite element method which is the theme of Chapter 3. From the equilibrium

equation three different analysis: Static; Modal; and Harmonic are derived and presented for the

undamped problem. This is followed by a general dynamic characterization of structures which

includes resonances and anti-resonances, residuals, modal identification and an overview of the

transmissibility concept. A brief introduction on verification and validation of models which consists in

analyzing and providing a direct comparison between two or more sets of results, e.g., experimental

and numerical, is presented.

The analytical structural models are presented in which the spring element model is particularized

for the case of the cylindrical helicoidal spring. The static and dynamic analysis of this specific spring

is boarded and its behavior analyzed, including: the maximum stress; the curvature effect; the type of

ends; the static deflection; the maximum static force; the stability; the natural frequencies; and the

harmonic analysis. A general overview of the periodic structures is then presented including the static

and dynamic behavior for the particular case of the periodic bar referring: the influence of the

materials’ properties and parameters of the structures; the static deflection; the maximum static force;

the harmonic response; and the natural frequencies. A Bloch wave analysis is also presented for

these types of periodic structures particularizing for a two material multilaminated periodic bar.

To conclude this chapter, an overview on optimization concepts and two particular problems

regarding the periodic and non-periodic case are presented.

2.1. Elastic Characterization

Recurring to the theory of elasticity, the stress analysis of elastic bodies or structures is commonly

used to predict mechanical material behavior [64]. The mechanical behavior of solids is normally

defined by constitutive stress-strain relations expressing the stress as a function of the strain, strain

rate, strain history, temperature, and material properties. In this work, an introduction to this theory is

presented, boarding the necessary concepts and formulation for many engineering materials and

design scenarios, therefore, used extensively in structural analysis and engineering design, often

through the aid of finite element analysis. A material model called the elastic solid [64] that does not

include or rate history effects is described as a deformable continuum that recovers its original

configuration when the loadings causing the deformation are removed [65]. Furthermore, the

constitutive stress-strain law is restricted to be linear. These assumptions simplify the model and linear

elasticity predictions have shown good agreement with experimental data and have provided useful

methods to conduct stress analysis [65]. Many structural materials including metals, plastics,

ceramics, wood, rock, concrete, and so forth exhibit linear elastic behavior under small deformations.

7

Consider a closed sub-domain Ω with volume V and surface S within a body in equilibrium. The

region has a general distribution of surface forces T on the boundary surface Γt and body forces F as

illustrated in Figure 2.1.

Figure 2.1 – Body and surface forces acting on arbitrary portion of a continuum.

After applying the load, the body (or a material point of the body) changes its position from the initial or

undeformed configuration to the deformed configuration. The displacement of the body is defined by

the difference of these two positions vectors [65]. The infinitesimal strain tensor ε : ε;<e;e< introduced

by Cauchy neglects the nonlinear terms of the generalized strain tensor and is given by

ε;< : 12 ?∂u;∂x< A ∂u<∂x;B, (1)

where u is the displacement, x is the orientation coordinate and i and j are notation indices. In the

above, and in the sequel, the convention that repeated indices in a term are summed over the range

of the index is adopted. The stress tensor σ : σ<;e<e; in the deformed configuration is defined by the

traction vector t; on the boundary surface Γt characterized by a unit vector, n<, normal to it given by t; : σ<;n< . (2)

Based on the general concept of conservation of linear momentum, the governing equations of the

distribution of the stress tensor within a body may be given by

E T dA A E F dVF : E ρuGF dV ,HI (3)

where V is the volume, Γt is the boundary surface of a closed sub-domain Ω, T is the surface forces, F

is the body forces, ρ is the mass density and uG is the displacement’s second order derivative with

respect to time (a superposed dot denotes partial differentiation in respect to time). Eq.(3) may be

rewritten using Eq.(2) and the index notation as

E σ<;n< dA A E F; dVF : E ρuG ;F dV.HI (4)

Using the Divergence theorem of Gauss Eq.(4) becomes

E σ<;,<n< dV A E F; dVF J EKσ<;,<n< A F;LF dV : E ρuG ;F dV,F (5)

where σ<;,< : ∂σ<; ∂x<⁄ (a partial derivative with respect to the coordinate xj is indicated by a comma).

Since Ω is of arbitrary volume, Eq.(5) may be rewritten as

8

σ<;,< A F; : ρuG ; . (6)

A variational (weak) form of the equations may be written using the procedures described in

Chapter 3 of [66] and yield the virtual work equations given by

E δu;ρuG ;dΩ A E δε;<σ;<dΩΩ O E δu;F;Ω dΩ O E δu;t; HP dΓ : 0 .Ω (7)

In the above Cartesian tensor form, virtual strains are related to virtual displacements as

δε;< : 12 Kδu;,< A δu<,;L . (8)

The equilibrium equations and displacement-strain relations previously presented do not

contemplate any mechanical characteristic of the solid body. For this purpose the elasticity tensor Eijkl is introduced to define the linear material behavior by the constitutive equations [65]

σ;< : E;<QRεQR . (9)

The elasticity tensor does not depend on stress or strain and assumes its simplest form for the

isotropic materials where it is expressed by

E;<QR : λ δ;<δQR A μKδ;Qδ<R A δ;Rδ<QL , (10)

where λ and μ are the Lamé constants related to Young’s modulus E and Poisson’s ratio υ by

λ : E υS1 O 2υTS1 A υT , μ : E 2S1 A υT . (11)

Applying these concepts to the equilibrium equations and displacement-strain relations previously

presented, Eq.(6) may be rewritten as

UE;<QR 12 KuQ,R A uR,QLV,< A F; : ρuG ; . (12)

To obtain a unique solution to this equation it is necessary to define the initial conditions for a given

instant t0 and the boundary conditions. The problem discreterization follows the path of the parabolic

problem [67] obtaining a system of second order differential equations that in matrix notation may be

given by WMX YuG Z A WKX YuZ : YFZ , (13)

which can be generalized to include viscous damping and hysteretic damping, leading to

WMXYuG Z A WCXYu[ Z A SWKX A iWDXTYuZ : YFZ , (14)

where WMX is the mass matrix, WCX and WDX are the viscous and hysteretic damping matrices, WKX is the

stiffness matrix and YFZ is the applied load vector. This system of equations can be numerically solved

by the usual methods, e.g., Euler method.

9

2.2. Finite Element Method

The mathematical formulation of the finite element method [68] can be presented as a variational

problem with an element-wise Rayleigh-Ritz treatment and shape function discretization. Alternatively

the finite element equations may be obtained directly from the differential equations using a Galerkin

approach weighted by the element’s shape function.

The variational (weak) form of Eq.(6) presented in the virtual work equation, Eq.(7), may be written

in the matrix form as

EWδuX\ρYuG ZdΩ A EWδεX\YσZdΩΩ O EWδuX\YFZΩ dΩ O EWδuX\YtZ HPdΓ : 0.Ω (15)

Finite element approximations to displacements and virtual displacements may be defined as

uSx, tT : NSxTu]StT and δuSxT : NSxTδu] , (16)

where N are shape functions. A detailed formulation can be found in [68] where the multidimensional

form of Newton’s second law (F:ma), for a multi-element structure may be represented by Eq.(14).

2.3. Types of Structural Analysis

Consider the multi degree of freedom (MDOF) system illustrated by Figure 2.2. Degrees of freedom

indicate how many variables are required to express the systems geometrical position at any instant.

Figure 2.2 – Example of a model with N degrees of freedom.

In this work no damping effects are considered (c=0; d=0) and three different analysis, derived from

the system of equations Eq.(13), are presented in the following sections by

Static: WKXYUZ : YFZ, (17)

Modal: KWKX O ω^ WMXL YΦZ : Y0Z and (18)

Harmonic: KWKX O ω_`^ WMXL YUZ : aFbcde . (19)

where ωap is the applied force excitation frequency and YFωapZ is the applied force vector.

10

2.3.1. Static Analysis

The static analysis consists in solving Eq.(14) in which the inertia and damping effects, such as those

caused by time-varying loads are not considered, i.e.: YuG Z=0; Yu[ Z=0; WCX=0; WDX=0; and YFStTZ=0,

obtaining a linear force-displacement law expressed by Eq. (17) also known as Hook’s law, which is

rewritten here for demonstration purposes, as

YFZ : WKX YUZ. (20)

2.3.2. Modal Analysis

Modal analysis consists in calculating the natural vibration mode shapes and frequencies of a

system by solving the free undamped problem Eq.(18) which analyzes the possible existence of

harmonic body motion when no forces are applied [68]. A harmonic motion may be described as

YUZ : gUhe;SbIijTk , (21)

where ω is the natural frequency and Uh is the amplitude and α is the phase angle of the harmonic

motion, t is the time variable, ex is the exponential function and i is the imaginary unity (i2=-1). By

substituting Eq.(21) and its second order derivative in respect to time in Eq.(13), after some symbolic

mathematical manipulation, Eq.(18) is derived and rewritten here for demonstration purposes as

SWKX O ω^WMXTYΦZ : 0 , (22)

which is a homogeneous system of equations with non-zero solution when

det SWKX O ω^WMXT : 0 . (23)

Solving this equation leads to the characteristic equation, and the left-hand side is called the

characteristic polynomial. When expanded, this gives a polynomial equation for λ S:ω2T.

Figure 2.3 – Natural frequencies plot based on modal analysis.

Figure 2.3 illustrates the results obtained through modal analysis, i.e., the order of the natural

frequencies and the respective value in frequency. It is a simple plot that reveals to be of interest in

the characterization of the periodic structures presented in this work.

11

2.3.3. Harmonic Analysis

Whenever external energy is supplied, through either dynamic applied forces or displacements, to

a structural system it is said to undergo forced vibration. The dynamical applied forces and

displacements may be classified as: periodic harmonic or periodic non-harmonic; non-periodic; with

long or short duration; or random in nature. The dynamical response of a system to a harmonic

excitation is called harmonic response. In this work it is considered a harmonic excitation presented in

the complex form as

YFStTZ : gFhe;SbIilTk , (24)

where F0 is the force amplitude, ω is the excitation frequency and β is the phase angle of the harmonic

excitation, t is the time variable, ex is the exponential function and i is the imaginary unity (i2:-1). For

an excitation frequency coincident to the natural frequency, the response of the system will be large,

tending to infinite if there is no damping. This is known as resonance (see section 2.4.1) and should

be avoided to prevent system failure.

Applying a similar methodology as presented in the previous section (section 2.3.2), i.e., substitute

Eq.(21) and its second order derivative in respect to time in Eq.(13) obtaining, Eq.(19) which is

rewritten here for demonstration purposes as

KWKX O ω_`^ WMX LYUZ : gFb_`k , (25)

where ωap is the applied force excitation frequency, YUZ and YFωapZ are the displacement and applied

force vectors, respectively.

This work is restricted to the steady state motion region, thus the solution of Eq.(19) may be

represented in a frequency response plot as illustrated by Figure 2.4.

Figure 2.4 – Frequency response of a structure subject to a harmonic excitation force for N degrees of freedom.

The amplitude may refer to displacement, velocity or acceleration and is usually represented in

logarithm to the base ten (log) or in decibel (dB). The peaks observed are called resonances (see

section 2.4.1) which physically represent the tendency of a structure to oscillate at maximum

amplitude at certain frequencies due to the overcome of the inertia effects over the structures

stiffness.

12

2.4. Dynamic Characterization of Structures

In [62] it is mentioned that the dynamical properties of a system with N degrees of freedom may be

described by three different types of models: the Spatial model; the Modal model; and the Response

model. It is worth recalling that degrees of freedom are the number of independent generalized

coordinates necessary to completely describe the motion of that system.

In the first case, the dynamic’s system characteristics are contained in the spatial distribution of its

mass, stiffness and damping properties, described by the mass matrix WMX, the stiffness matrix WKX and

the viscous and hysteretic damping matrices WCX and WDX, respectively. Considering that the

displacement of each degree of freedom can be described by YuZ and to it is associated an excitation

force YfZ, the system can be modeled by Eq. (13), rewritten here for demonstration purpose as

The spatial model given by these matrices leads to an eigenproblem which, having been solved,

yields the modal model described by the modal properties (N natural frequencies and N mode shape

vectors) contained in eigenvalues matrix [\λ2\] and eigenvectors [Φ]. Each element of the diagonal of

the eigenvalue matrix contains information about the eigenfrequencies for each eigeinmode and the

eigenvectors matrix contains the modal vectors (eigenmodes) normalized to the mass, in which the

orthogonal properties of the modal model are defined as

where WIX is the identity matrix. The modal vectors, due to the orthogonal properties, are linearly

independent, thus WΦX is a non singular and invertible matrix. So being, through Eq.(27) and Eq.(28) it

should be possible to obtain a spatial model from the knowledge of the modal model and vice-versa if

the complete model is available.

However, if a system is too complex and therefore cannot be modeled analytically, one must revert

to experimental analysis described by a response model WHSωTX. In a steady state, the vibration of a

structure may be characterized by

where Xo and Fp are the structure’s response and solicitation complex amplitudes, or phasors,

respectively, (the phasor [62] includes the phase angle φ of the response displacement relatively to

the force: Xo : Xe;q). The function HSωT is frequency dependent and is designated as the Frequency

Response Function (FRF).

The FRF may be defined in terms of displacement (as in Eq. (29)), velocity or acceleration which

are mathematically interrelated quantities: through integration and derivation, knowledge of an FRF in

terms of any one of the motion parameters allows obtaining any of the other FRF forms.

It is possible to relate the modal model with the experimental response model, as shown in [62], by

WMXYuG Z A WKXYuZ : YFZ . (26)

WΦX\WMXWΦX : WIX , (27)

WΦX\WKXWΦX : r λ^ \\ t , (28)

YXoZ : WHSωTXYFpZ , (29)

13

where ω2 is the squared frequency. The receptance matrix WαSωTX is obtained experimentally using

specific equipment, where the signals are acquired by force and response transducers.

The receptance αSωT is used to describe the structure’s response relation between the

displacement and the excitation FSωT, being a particular case of the generalized representation of the

FRFs. Other particular representations of the FRFs are the mobility YSωT (which relates the velocity

with the solicitation) and the accelerance ASωT (which relates the acceleration with the excitation).

In Eq.(30), each term of the receptance matrix may be written as

where Xo< is the response complex amplitude in coordinate j, FQ is the force amplitude in coordinate k, N is the total number of modes, r is the vibration mode, ωr is the natural frequency of mode r and

vAo<Q is the Modal Constant.

There are many techniques that allow derivation of the modal characteristics of a given system

from the experimentally obtained response model. The procedure is called Modal Identification or

System Identification. From the modal model thus derived, one may apply Eq.(27) and Eq.(28) and

obtain a spatial model. In section 2.4.3, a variant of a frequency domain method called the inverse

method [62] applied to the Characteristic Response Function (CRF) defined by Ribeiro in [69] is

presented. Figure 2.5 summarizes the presented model interrelation for the undamped case.

Figure 2.5 – Dynamic models interrelation (undamped case) [62].

Through Eq.(31) and all other mentioned considerations, it is possible to obtain the FRF’s graphical

representation, that present peculiar graphic characteristics emphasis by the presence of maximums

(resonance) and minimums (anti-resonance), which are discussed in the next section.

WαSωTX : WΦX r λ^ O ω^\\ twx WΦX\ , (30)

α<QSωT : Xo<FQ : y vAo<Qωv O ω^z

vx , (31)

14

2.4.1. Resonances and Anti-Resonances

By definition [70], for undamped systems, the resonance frequency of a FRF, considering the

receptance αjk, is the vibration frequency for which the amplitude of the displacement at coordinates j increases progressively to infinity when an applied force at coordinate k decreases to zero, i.e.

where ωr is the resonance frequency. On the other hand, at an anti-resonance frequency of αjk, the

amplitude of the displacement at coordinate j is infinitesimal for an applied force at coordinates k

increasing to infinity, i.e.

where ωa is the anti-resonance frequency.

Figure 2.6 – Resonance and anti-resonance of a 2 degree of freedom system [69].

In the case of a direct FRF (when the response and solicitation are measured in the same

coordinate, j:k) the anti-resonances appear intercalated with the resonances (see Figure 2.6) in the

frequency spectrum of the system [70].

2.4.2. Residuals

One of the parameters that is necessary to define in modal analysis is the frequency range, that is

inevitability limited due to equipment limitations or other practical reasons. This means that it is not

possible to identify the properties of the vibration modes out of the frequency range of analysis, even

though their influence is present in an important and significant way on the experimental FRF

obtained, thus its influence should be considered.

The regeneration of an FRF from the obtained modal parameters, e.g., through modal identification

(see section 2.4.3) of the experimental FRF, may be obtained by [71]

where the number of resonances is limited ranging from m1 to m2. Generally, an experimental analysis

begins with the first mode of vibration (r=1) and it is never able to reach the highest mode (r:N), since N→∞ for real systems. Limiting the experimental data acquisition to a frequency range between m1 and m2 does not mean that the FRF is not affected by the modes out of this range. Recalling Eq.(31) it

is possible, without losing any generality, to rewrite it as [71]

ωv : ωKx< → ∞, fQ → 0L; f : 0, m k , (32)

ω_ : ωKx< → 0, fQ → ∞L; f : 0, m k , (33)

α<QSωT : y vAo<Qωv O ω^ A iηvωv v

, (34)

15

In this equation, the first term refers to the “low frequency” vibration modes, the third term refers to the

“high frequency” vibration modes and the second terms refers to the identified vibration modes in the

analyzed frequency range (only term to appear in Eq.(34)). Within the frequency range of interest, the

first term (“low frequency modes”), tends to have an approximately “mass contribution” while the third

term (“high frequency modes”) tends to have an approximately “stiffness contribution” [71]. Based on

this, it is now possible to rewrite Eq.(35) introducing the Residual terms that substitute the “low” and

“high” frequency terms as [71]

where M<Q and K<Q are the respective mass and stiffness residuals.

Sometimes it is convenient to treat the residual terms as vibration modes, i.e., instead of

representing the effect of each residual by a constant, each one of them may be represented by a

“pseudo vibration mode”, this is the same as saying that each residual term has a mass and stiffness.

In order to translate the effects of the “low”/ “high” frequency modes, there will exist a “pseudo

vibration mode” with a natural frequency below/above the inferior/superior limit of the analyzed

frequency range.

2.4.3. Modal Identification – The Characteristic Re sponse Function (CRF)

In this section hysteretic damping is considered in order to present the CRF as defined by Ribeiro

in [69]. Eq.(31) and Eq.(36) show that nearby each resonance, the contribution of the adjacent

vibration modes is more important (contributive) than the total contribution of all the other vibration

modes. In this neighborhood, the respective vibration mode may be referred to as a dominant vibration

mode.

Considering that the contribution of all the other vibration modes of the structure, nearby each

dominant vibration frequency r may be approximated by a constant, the receptance function may be

approximated by

where i is the complex unity (i2=-1) and ηr is the hysteretic damping factor. The constant represents

the contribution of a residual term as described in section 2.4.2, since it represents the difference

between the value of the receptance for a specific frequency and the contribution for the same

dominant vibration mode. Note that if there are very close vibration modes their contribution would be

better approximated by a frequency dependent term. After some symbolical mathematical

manipulation, Ribeiro [69] defined a function Characteristic Response Function (CRF) as

α<QSωT : y vAo<Qωv O ω^ A y vAo<Qωv O ω^

vA y vAo<Qωv O ω^ z

vixwx

vx . (35)

α<QSωT O 1ωv M<Q A y vAo<Qωv O ω^

vA 1 K<Q , (36)

vα<QSωT vAo<Qωv O ω^ A iηvωv A cte , (37)

16

which is similar to a term of the modal receptance (nearby each resonance frequency ωr) with a

unitary modal constant. It depends only of the global parameters, structural characteristics, and not on

local parameters (complex modal constants) which depend of the coordinates chosen for the

experimental setup.

The CRF presents the form of a one degree of freedom receptance (nearby each resonance

frequency ωr). Since the numerator is unitary, a modal identification method such as the inverse

method [62] can be simply applied. Rearranging Eq.(38) as

it becomes clear that the real term (see Figure 2.7 (a)) is represented against ω2 by a straight line with

negative unitary slope and the imaginary term (see Figure 2.7 (b)) is a constant that depends of the

damping factor and of the natural frequency

Noting that

so, nearby each vibration mode, this graphic representation should allow the calculation of the

resonance frequency and the damping factor of the respective vibration mode. Note that these lines

evidenciate the fact that the resonance out of the neighborhood of the CRF are meaningless.

(a)

(b)

Figure 2.7 – CRF’s terms graphical representation obtained from BETAlab [63]: (a) Real term; (b) Imaginary term.

βSω^T : 1ωv O ω^ A iηvωv , (38)

1βSω^T : ωv O ω^ A iηvωv ^ , (39)

Real 1βSω^T : ωv O ω^ , (40)

Imag 1βSω^T : iηvωv . (41)

∂∂ω^ UReal 1βSω^TV : O1 , (42)

∂∂ω^ UImag 1βSω^TV : 0 , (43)

17

2.5. Modal Identification Technique

When the measured data indicates heavily coupled modes or noise contamination, or when high

accuracy is required for the estimation, a modal identification technique can then be used to improve

the modal parameter estimation. Diverse modal identification algorithms are available for modal

analysis, but independently of the parameter estimation technique used, the estimation must always

be based on sufficient data that correctly represents the dynamics of the structure. The most important

part of modal testing may be the mobility measurements since no identification algorithm can make

reliable parameter estimates from poor measurements.

Modal identification is where the mathematical theory and practical measurements meet. The

theory provides the mathematical parametric model for the theoretical FRF of a structure and the

measurements provide the real FRF. In this work a specific software, entitled BETAlab, developed by

Diogo Montalvão e Silva [63], based on the CRF defined by Ribeiro, is used for this purpose.

2.6. Transmissibility – Overview

Transmissibility [31] is an important concept, especially in forced vibrations. Transmissibility is

defined as being the ratio between the output and input entities. If the ratio is greater than one, the

entity is amplified, and if the ratio is less than one, the entity is reduced, isolated or attenuated (see

Figure 2.8).

Figure 2.8 – Theoretical transmissibility for a system with a natural frequency of 20 Hz.

If one thinks of the transmissibility function, as previously defined and usually conceived in a single

degree of freedom systems (SDOF), it can be assumed that an equivalent matrix should exist for a

multi degree of freedom system (MDOF) [32]. This matrix should be the transmissibility matrix of the

structure.

In any case, displacement transmissibility relates two sets of forces allowing to obtain one set from

the knowledge of the other and the number of known displacements must be at least equal to the

number of applied forces. For MDOF systems, there are multiple possibilities for the number and

location of applied forces and/or moments. In this case the number of generalized forces and their

locations must be equal. This is a minimum mathematical requirement, as can be appreciated from the

formulation presented in [32]. The relationships among the responses at various co-ordinates will

depend on the number and co-ordinates of the applied forces.

18

2.7. Verification and Validation of Models – Overvi ew

One, if not the most used, application of experimental testing is to provide a direct comparison

between predictions (obtained analytically or numerically) of the dynamic behavior of a structure and

those actually observed in practice. This process is usually referred to as verification and validation of

models which have received (more-or-less) consensus definitions and are not yet established

concepts. The American Institute of Aeronautics and Astronautics (AAIA) and the American Society of

Mechanical Engineers (ASME) define:

• Verification as the process of determining that a model implementation accurately represents

the developer’s conceptual description of the model and the solution to the model and;

• Validation as the process of determining the degree to which a model is an accurate

representation of the real world from the perspective of its intended uses.

In a synthetic way, it is common to state that verification is about mathematics and validation is about

physics.

Figure 2.9 – Verification and validation procedure between the theoretical, numerical and experimental models.

To do this effectively several steps must be taken [72], see Figure 2.9. The first of these is to make

a direct and objective comparison of specific static and dynamic properties analytically obtained vs.

predicted vs. the experimental. The second, or perhaps still part of the first, is to quantify the extent of

the differences/similarities between the sets of data. Then third is to make adjustments or

modifications, also referred to as updating [73] and [74] to one or more sets of results, in order to bring

them closer into line with each other. When this is achieved, the theoretical and/or numerical models

can be said to have been verified and validated and are fit to be used for further analysis.

It should be noted that no model is ever 100% verified or validated. Thus, verification and validation

are not an absolute. Any model is a representation of a system, and the model’s behavior is at best an

approximation to the system’s behavior. When it is stated that a model has been verified and

validated, it is meant that it explicitly carried out a series of tasks to verify and validate the model to the

degree necessary for its purposes.

19

2.8. Springs

Springs may be classified according to the type of load solicitation: tension/extension and

compression springs for axial loading; and torsional spring for torsional loading. One of the most

common springs consists of wire, with circular constant cross-section, wounded into a circular helix. A

tension/extension spring is a coiled spring whose coils normally touch each other under tension; as a

force is applied to stretch the spring, the coils separate. In contrast, a compression spring is a coiled

spring with space between successive coils; as a force is applied to shorten the spring, the coils are

pushed closer together. Among the compression helicoidal spring, these may present cylindrical or

non-cylindrical shapes, such as: barrel; conical; and hyperboloidal. A third type of coiled spring is the

torsion spring, designed so that the applied force twists the coil into a tighter spiral. Another type of

coiled springs is the watch spring, which is coiled into a flat spiral, rather than a cylinder.

Regarding the non-coil springs, the most common is the leaf spring, which is shaped like a shallow

arch; it is commonly used for the heavy automobiles suspension systems. Another type is a disc

spring, a washer-like device also known as Belleville spring which has a bi-stable load-deflection

characteristic. Open-core cylinders of solid, elastic material can also act as springs. Generally, springs

are made from high carbon steels, alloy steels, stainless steel, copper alloy types, however the most

common are made of high carbon steels ranging from 0.50% to 1.20%. Other materials may be used

such us, ceramic, cement and more recently piezoelectric ceramic material such as PZT (lead

zirconate titanate) is used to form a helicoidal spring [26] and [27].

In this work, the spring analysis is restricted to the cylindrical helicoidal compression springs with

constant steel wire cross-section. They are commonly used in the motor suspension of air-tight

compressors which is a practical application field of interest.

By definition, a spring is a flexible mechanical link between two points in a mechanical system [5].

In reality a spring itself is a continuous system [8]. However, the inertia of the spring is usually small

compared to the other elements in the mechanical system and may be neglected. Under this

assumption the force applied to each end of the spring is the same.

The length of the spring when it is not subject to external forces is called unstretched length [1].

Since the spring is made of a flexible material, the applied force F that must be applied to a linear

spring to change its length by x may be expressed by Hook’s law Eq.(20) which is rewritten here for

demonstration purpose

F : k x , (44)

where k is called the spring stiffness or spring constant and has dimensions of force per length.

2.8.1. Helicoidal Spring

If a point (x, y, z) revolves around the z-axis at a constant distance r from it and simultaneously

moves parallel to the z-axis in such a way that its z-component is proportional to the angle of

revolution, the resulting path is called a circular helix [74]. An example is illustrated in Figure 2.10

20

Figure 2.10 – Circular helix.

where r is the average radius and h is the helical pitch. If θ denotes the angle of revolution, the

coordinates of each point are given by

x : r cos θ , y : r sin θ , z : b θ , (45)

where r>0, and b≠0. When θ varies from 0 to 2π r, the x and y coordinates return to their original

values while z changes from 0 to 2πb. The number h S:2πbT is often referred to as the pitch of the

helix.

The curvature κ of the helix is given by

κ : rr^ A b^ . (46)

The torsion τ of the helix is given by

τ : br^ A b^ , (47)

thus the ratio of curvature to torsion κτ : rb , (48)

is a constant. In fact, Lancret's theorem [74] states that a necessary and sufficient condition for a

curve to be a helix is that the ratio of curvature to torsion be constant.

Helices can be either right-handed or left-handed defined by the line of sight being the helical axis.

If clockwise movement of the helix corresponds to axial movement away from the observer, then it is a

right-handed helix. If counter-clockwise movement corresponds to axial movement away from the

observer, it is a left-handed helix.

The curvature of the wire increases the stress on the inside of the spring but decreases it only

slightly on the outside. This curvature stress is primarily important in fatigue because the loads are

lower and there is no opportunity for localized yielding [9]. For static loading, these can be normally

neglected because of strain-strengthening with the first application of load. For dynamical analysis, or

when intended, a spring index which is a measure of coil curvature defined by

C : Dd , (49)

should be considered. The suggested range of spring index is 4≤ C ≤12, [76], with the lower index

being more difficult to form, due to the danger of surface cracking, and springs with higher index

tending to tangle often enough to require individual packing.

21

Assuming elastic behavior, the maximum stress in the wire is computed by superposition of the

direct shear and torsional stresses [76] given by

τ : K 8 F Dπ d , (50)

where F is the axial force, D is the mean coil diameter, d is the wire diameter and KB is a correction

factor, known as Bergstrasser’s factor, which considers transverse shear and curvature effects,

defined by

K : 2C A 12C . (51)

Eq.(50) is quite general and applies for both static and dynamic loads.

The four types of ends generally used in helicoidal springs are illustrated in Figure 2.11.

A spring with plain ends has non-interrupted helicoids while a spring with squared or closed ends is

obtained by deforming the ends to a zero-degree helix angle. For a better load transfer springs should

always be both closed and squared. Table A.1 in appendix shows how the type of end used affects

the number of coils and the spring length.

2.8.2. Static Deflection of a Helicoidal Spring

The deflection-force relations are obtained using Castigliano’s theorem [74]. The total strain energy

for a helicoidal spring is composed of a torsional and shear component given by

U : T^ L2 G J A F^ L2 A G , (52)

where G is the shear modulus of the wire. Substituting T:FD/2, L:πDN, J:πd4/32, A: πd2/4 gives

U : 4 F^ D Nd G A 2 F^ D Nd^ G , (53)

where N is the number of active coils. Then using Castigliano’s theorem to find the total deflection x

gives

x : ∂U∂F : 8 F D Nd G A 4 F D Nd^ G . (54)

Since C:D/d, Eq.(54) can be rearranged to yield

(a) (b) (c) (d)

Figure 2.11 – Type of ends for helicoidal springs: (a) Open, plain end; (b) Open, squared; (c) Closed, plain

end; (d) Closed, squared [76].

22

x : 8 F D Nd G 1 A 12 C^ 8 F D Nd G . (55)

Comparing Eq.(55) with Eq.(44) leads to the conclusion that under the assumptions stated, a

helicoidal spring can be modeled as a linear spring of stiffness

2.8.3. Nonlinearities

Three types of nonlinearities, namely geometric, material and contact, may be considered.

Generally, the geometric and/or material nonlinearities are often related to problems subject to large

displacements, material plasticity and material creep while contact is to time-variant boundary

conditions.

In this work to maintain linearity it is considered that: all boundary conditions are applied for a

steady-state; the spring is subject to small displacements; no contact between coils occur, i.e., when a

spring is about to close, it is necessary to avoid the gradual touching of coils, as the recommend range

for the number of active coils is 3 ≤ N ≤ 15, [76]. Thus the maximum force that can be applied to a

helicoidal spring may be given by F_ : kSLh O LT , (57)

where L0 is the free length and LS is the solid length of the spring; to avoid buckling of a compression

helicoidal spring, which may occur when the deflection becomes critical, Samanóv [77] and [78] states

that the critical deflection equation cited by Wahl [1] and verified experimentally by Haringx [79]

resume the condition for absolute stability given by

Lh π Dα 2SE O GT2 G A E x, (58)

where α is the end condition constant given by Table A.2 in appendix. The symbol E is for the

longitudinal modulus of elasticity and G is for the shear modulus of the wire’s material; and the material

is assumed homogeneous, isotropic with a linearized elastic behavior.

2.8.4. Natural Frequencies of a Helicoidal Spring

The longitudinal (axial) natural frequencies of helicoidal springs may be obtained by imposing the

boundary conditions, e.g., the end conditions of the spring, into the homogeneous solution of the

undamped system’s motion equation, Eq.(13). The harmonic, natural, frequencies ω for a spring

clamped between two flat and parallel plates is expressed by

ω : m πL kM , m : 1, 2, 3, … (59)

where the first natural frequency is given for m=1, the second for m=2, and so on. M is the weight of

the active part of the helicoidal spring given by

k d G8 D N . (56)

23

M : A Lh ρ : π^d^D N ρ4 cos θ , (60)

where ρ is the mass density of the wire. The harmonic analysis Eq.(19) of the helicoidal spring may be

identically obtained, i.e., by imposing the initial and boundary conditions into the homogeneous

solution of the undamped system’s motion equation, Eq.(13).

2.9. Multilaminated Periodic Structures

Multilaminated structures may be defined as structures composed of two or more layers of firmly

united material, made by bonding or, for e.g., impregnating superposed layers with resin and

compressing under pressure and heat. The layer disposition, in the multilaminated structure, can be

periodic or non-periodic. In this work, the analysis are limited to the periodic layer disposition

structures due to their pass-band filter like behavior [4], i.e., if the energy dissipation is omitted, a high

distinction between frequency bands exhibiting wave propagation, also known as pass-bands, and

those showing no propagation, also known as stop-bands, is observed.

A periodic or repetitive structure consists of a number of identical structural elements ("periodic

elements") that are joined together through the extremities and/or side by side in order to form the

structure [4]. The atomic lattices of pure crystals [34] (see Figure 2.12 (a)) constitute perfect periodic

structures, but these are particular structures with discrete mass (the atoms) interconnected by inter-

atomic elastic forces. In the theory of elasticity [8] as well as in structural engineering the mass and

elasticity of the structural members may be considered continuous and constitute periodic structures

when arranged in regular meshes.

Periodic structures may be categorized in one, two or three dimensional and consist of beam, bars

flat plates or curved shells in various combinations [38] and with different support conditions (see

Figure 2.12). Their applied time dependent loadings may be localized or widely distributed, harmonic

or random, impulsive, short term or longer term transient. While the simplest structures transmit

vibrational energy by just one type of wave motion (ex: flexural waves) others transmit it in

simultaneous and particular combinations of longitudinal torsional and bi-directional flexure [38]. When

these different wave types encounter a discontinuity in the periodicity they interact and are converted

from one type into another.

(a)

(b)

(c)

Figure 2.12 – Periodic structures: (a) Pure crystals; (b) Pin joint bars; (c) Beam or bar.

24

By controlling the wave propagation in the periodic structure, controlling the materials layout and

the ratio of their properties within a unit cell, a periodic structure can be designed to have a desired

frequency band filter structure.

2.9.1. Material Properties and Parameters Influence

The wave propagation in a material depend on the materials properties, namely on the modulus of

elasticity and mass density and on geometrical parameters such as the length of the material. Solid

mechanics and continuum mechanics consider three main types of material models to describe the

behavior of solids response to external solicitations, e.g., forces, temperature changes, displacements,

etc. The three main types are:

1. Elastic – Linearly elastic materials can be described by linear elastic equations such as Hooke's

law, Eq.(20): when the applied stress is removed, the material returns to its previous state.

2. Viscoelastic – These are materials that behave in an intermediate way between elastic solid and

viscous liquid and, as such, exhibit time dependent strain. Elasticity is usually the result of bond

stretching along crystallographic planes in an ordered solid, whereas viscoelasticity is the result

of the diffusion of atoms or molecules inside of an amorphous material.

3. Plastic – Materials that behave elastically when the applied stress is less than a yielding stress

and undergo non-reversible changes of shape when the stress is greater than the yield stress.

Essentially, this work is restricted to linear elastic material. The linear elastic behavior of materials

have been extensively studied [64], thus no extended formulation will be presented. In order to work

materials with low Young’s modulus, like agglomerated cork it is necessary to understand its complex

modulus nature. The Dynamic Mechanical Analysis (DMA) is a suitable technique that enables the

individualization and characterization of the elastic and viscoelastic components of solids [80]-[82].

The characterization of the viscoelastic behavior of materials may be done through dynamic

analysis in which a sinusoidal time variant stress σ is applied with a certain frequency ω

σStT : σhe;bI , (61)

where σ0 is the maximum stress amplitude. The resultant strain ε is also time variant with the same

frequency ω, but with an angle δ between the in-phase and out-phase components in the cyclic

motion, being εStT : εhe;SbIwT , (62)

where ε0 is the maximum strain amplitude. Since σ/ε is time dependant, when δ0, the elasticity

modules must be redefined, using complex quantities, to include this time dependence.

The complex longitudinal modulus of elasticity, E* is defined as

E : σStTεStT e; → E : σhεh cosδE : σhεh sinδ ; R¡¢£¤¤¥ E : E A iE , (63)

25

where E’ is the real term of E*, designated by storage modulus and is a measure of the energy stored

elastically and E’’ is the imaginary term of E* designated by loss modulus and is a measure of the

energy lost as heat or sound.

2.9.2. Static Deflection of a Multilaminated Bar

Consider a cylindrical multilaminated or homogeneous elastic bar as illustrated in Figure 2.13 (a)

and Figure 2.13 (b), respectively, with a cross-sectional area A, a total length L and an applied force F.

An analogy can be drawn between the bar and the spring since both objects are continuously

distributed elements, in that their stiffness and mass are spread uniformly throughout their interiors.

(a)

(b)

Figure 2.13 – Bar with a static force applied: (a) Heterogeneous multilaminated bar; (b) Homogeneous bar.

As the spring, the bar obeys a force-displacement law under a static load, Eq.(17), which is

generalized here for demonstration convenience

F : k § x , (64)

where keq is the equivalent stiffness of the bar and x is the change in length of the bar. The force

developed in each layer of material is the same and equal to the force acting on the bar. The

displacement of the bar is the sum of the changes in length of the layers of materials in the series

combination. Denoting xi as the change in length of the ith layer of material, then

x : xx A x^ A x A … A x;wx A x; : y x;©

;h . (65)

Since the force is the same in each layer of material (xi:F/ki), Eq.(65) becomes

x : y Fk;©

;h . (66)

Substituting in Eq.(64), lead to the equivalent stiffness of a bar, keq, with i material layers, given by

k § : 1∑ 1k;©;h , (67)

where for each layer of material, the particular case of the homogeneous bar, see Figure 2.13 (b), may

be applied in which the stiffness of the material layer klayer is given by

kR_ v : E ALR_ v , (68)

where E is the modulus of elasticity of the material, A is the cross-sectional area and Llayer is the length

or thickness of the layer.

26

2.9.3. Nonlinearities

As previously presented in section 2.8.3, three types of nonlinearities may be considered. In this

work to maintain linearity it is considered that: all boundary conditions are applied for a steady-state;

the bar is subject to small displacements; the two material layers considered are assumed perfectly

bonded along plane interfaces across which there is continuity of displacements and equilibrium of

stresses; to avoid buckling of the bar subject to a compressive load, sometimes called the critical load,

the Euler’s formula for columns under compression is

σ_ π^ E SL /rT^ , (69)

where E is the longitudinal modulus of elasticity, Le/r the slenderness ratio of the bar, Le is the effective

length factor of the bar, whose value depends on the conditions of end support of the column, given by

Table A.3 in appendix, and r is the radius of the bar; and the material is assumed homogeneous,

isotropic with a linear elastic behavior. To assure stability it is common, in linear elastic materials, to

use a value inferior to ten for the slenderness L r 10 . (70)

2.9.4. Bloch Wave Analysis

Recalling, a mechanical or structural system it is said to undergo forced vibration when external

energy is supplied, through dynamical force or displacement. The dynamic response of a

homogeneous system under harmonic excitations is presented in Eq.(24). The equation of motion for

the forced longitudinal vibration of a uniform homogeneous bar may be expressed [52] by

EA ∂ûSx, tT∂x^ O ρA ∂ûSx, tT∂t^ : FSx, tT , (71)

where E is the modulus of elasticity, A is the cross sectional area, ρ is the mass density, uSx,tT is the

displacement at the longitudinal coordinate x and at time t and F is the applied force. Since this

equation is non-homogeneous, its general solution uSx,tT is given by the sum of the homogeneous

solution, uhSx,tT, and the particular solution, upSx,tT.

The homogeneous solution, uhSx,tT, can be obtained from Eq.(71), considering FSx,tT:0

∂ûSx, tT∂x^ O 1c^ ∂ûSx, tT∂t^ : 0 , (72)

where c is the longitudinal wave velocity (of a bar with a uniform cross sectional area) given by

c : Eρ , (73)

can be written as

u«Sx, tT : USxT TStT J ¬Bxcos ωc x A B^ sin ωc x SBcos ωt A B sin ω tT , (74)

27

where the function USxT represents the normal mode and depends only on x and the function TStT

depends only on t. B1, B2, B3 and B4 are constants determined by the initial conditions imposed on the

system. Since the excitation force, FSx,tT, is harmonic, the particular solution, upSx,tT, is also harmonic

with the same frequency ω. The complete solution of Eq.(71) can be sought as

uSx, tT : uSx, tT A u`Sx, tT J USxT e<bI. (75)

In this work the study is restricted to the steady state motion, thus, it does not consider the transient

period.

The longitudinal (axial) natural frequencies for a homogeneous bar may be obtained by imposing

the initial conditions, e.g., the end conditions of the bar, into the homogeneous solution, Eq.(72). The

harmonic, natural, frequencies ω for a homogenous bar under free-free boundary conditions may be

expressed by

where the first natural frequency is given for n=1, the second for n=2, and so on. E is the modulus of

elasticity, ρ is the mass density and L is the length of the bar.

Generalizing the harmonic response for the case of a periodic elastic bar in the form of alternating

plane parallel homogeneous layers of materials, which are assumed to be isotropic, differing in elastic

properties. The longitudinal wave propagation in the bar in absence of body forces is governed by the

following one dimensional equation [4]

∂∂x ESxT ∂uSx, tT∂x O ρSxT ∂^uSx, tT∂t^ : 0 , (77)

where ρSxT and ESxT are respectively the mass density and the Young’s modulus of the bar at position x. In Eq.(77) uSx,tT is the displacement at the longitudinal coordinate x and at time t. For harmonic

motion of excitation frequency ω, a solution can be sought as

uSx, tT : USxT e<bI . (78)

Under this condition, Eq.(77) is reduced to the following ordinary differential equation

ddx ESxT dUSxTdx A ω^ ρSxT USxT : 0 . (79)

The material layers’ are assumed perfectly bonded along plane interfaces across which there is

continuity of displacements and equilibrium of stresses. The periodic bar is considered to be infinite so

that the dynamics of wave propagation in its periodic layering can be studied by analyzing the

behavior of its smallest physical unit, denoted as cell, whose repeated translation generates the

periodic structure [4]. It is known that, by virtue of the periodicity of the structure with period d,

eigenmodes can be characterized by a quasi-wave number k, considering that the displacement field

in the ith unit cell takes the form

ω© : n ® °±ρ

, n : 1, 2, 3, … , (76)

28

u;SxT : uhSx O idT e<SQ;¡T, i : 1, 2, j : √O1 . (80)

In this work two material layers are considered, see Figure 2.14,

Figure 2.14 – Schematic drawing of two material periodic structure.

Thus the material properties can be described by the following periodic functions of period d

Within each material, which is assumed to be homogenous and isotropic, the wave propagation is

described by the solution of the one dimensional wave equation

¡³ST¡ A ¬bµ^ USxT : 0, i : 1, 2 , (84)

where ci is the longitudinal wave velocity given by Eq.(73), in the ith layer, with i=1 and 2 denoting the

layers defined over L1<x<0 and 0<x<L2, respectively.

By applying the Lyapunov-Floquet’s theorem [46] and [47], it can be shown that the solution of

Eq.(79) when ESxT and ρSxT are described by periodic functions such as those given in Eq.(82) and

Eq.(83), can be expressed as a wave characterized by a quasi-wave number k, and can be sought as

u;Sx, tT : U;SxT e<SQ ib IT, i : 1, 2, j : √O1 , (85)

where UiSxT is defined as

The physical meaning of the quasi-wave number k can be extracted from the application of the

Lyapunov-Floquet’s theorem as shown by Brillouin in [4]. The wave propagation at location x of the

periodic bar can be expressed as

uSx, tT : ASxT e<SQ ib IT, (87)

where ASxT is a periodic function of period d. According to Eq.(87) the wave propagation at location SxAdT can be expressed as

d : Lx A L^ , (81)

ESxT : ¶Ex, OLx · 0 E^, 0 · L^ , (82)

ρSxT : ¶ρx, OLx · 0 ρ^, 0 · L^ . (83)

U;SxT : A; e<bµwQ A B; ew<bµiQ, i : 1, 2, j : √O1 . (86)

29

uSx A d, tT : ASx A dT e< QSi¡Ti< b I : ASxT e< Q i< b I e< Q ¡ : uSx, tT e< Q ¡. (88)

Therefore, k acts as an effective wave number that relates the displacements of any two points

located at a distance d apart inside material i. Hence, the nature of wave propagation inside the

material depends on the quasi-wave number k. If k is real, then the wave at uiS0,tT will propagate

whereas if k is complex, the wave will be attenuated. The coefficients A1, A2, B1 and B2 in Eq.(86) can

be found by imposing the continuity at the interface x=0, where both the displacement and the normal

stress have to join smoothly. The other conditions required for determining the constants can be

obtained by imposing the periodicity of the problem. The Lyapunov-Floquet’s theorem states that each

function UiSxT, as it has to be continuous across the layer interfaces and does not depend on time,

must also be a periodic function of x of period equal to the period d of the periodic bar. Therefore,

UxSOLxT : U^SL^T . (89)

A similar requirement is imposed on the function describing the stresses, that is

Imposing these four continuity conditions yields to a homogeneous system of four linear equations

with A1, A2, B1 and B2 as unknowns. The system can be conveniently expressed in matrix form [52] as

¹¹¹º 1 1 O1 O1zx Ozx Oz^ z^ew¬ bwQ¯ e¬ biQ¯ Oew¬ bwQ¯ Oew¬biQ¯zxew¬ bwQ¯ Ozxew¬ biQ¯ Oz^ ew¬ bwQ¯ z^ ew¬ biQ¯»¼¼

¼½ AxA^BxB^¾ : 0000¾ (91)

where zi is the impedance of the ith layer given by

z; : ω ¿E; ρ; . (92)

The homogeneous system in Eq.(91) has a non-trivial solution if the determinant of the matrix of

the coefficients is zero. Setting the determinant to zero results in the following characteristic equation

whose roots give, for any frequency ω, the quasi-wave number k. The constant k defines an effective

wave number that quantifies the nature of the wave propagation along the periodic bar. Eq.(93) which

relates the effective wave number k to the frequency ω, represents the ‘‘dispersion characteristics’’ for

the steady-state wave propagation [52]. The solution of Eq.(93) may be represented in a frequency vs.

wave number k plot as illustrated in Figure 2.15.

Ex dKUxSxTLdx Àw¯ : E^ dKU^SxTLdx À¯ . (90)

cosSkdT : cosSkx LxT cosSk^ L^T O 12 kxk^ A k^kx sinSkx LxT sinSk^ L^T , (93)

30

Figure 2.15 – Dispersion curves obtained from Eq.(93).

For a multilaminated bar, Eq.(93) yields propagating or decaying solutions depending on whether

the value of k is real or complex. Therefore, k is expressed in terms of its real and imaginary parts k:kR-jkI. Accordingly, Eq.(86) can be rewritten as

U;SxT : ewQÁ A;e<bµwQÂ A B; ew<bµiQÂ , i : 1, 2 . (94)

When the wave propagates at the values of frequency making the wave number complex, the

amplitude of the displacement are attenuated exponentially with x. Under such conditions, the

frequency bands where k is complex are called stop-bands, while the bands within which k is real are

called pass-bands, see Figure 2.15.

A step can be taken from the infinite to finite repetition mediums of interest as used in structural

engineer applications. Doing so, the stop-band regions, where all incident waves are not allowed to

propagate, which are a characteristic of the infinite medium has in reality for finite mediums only

attenuation regions, where all incident waves are significantly attenuated. These attenuation regions

tend to the stop-band regions as the finite medium tends to infinite, i.e., for a large number of unit

cells. This becomes clear, when analyzing this medium transition through a graphical relation between

the Bloch wave theory and the harmonic analysis [52]-[57]. In this work, this relation is explored and a

new relation, considered to be more intuitive, at least for engineers, based on modal analysis and the

previous two is proposed.

2.10. Optimization

Optimization is an important tool in decision science and in the analysis of physical systems. To

make use of this tool, it is necessary to first identify some objective, a quantitative measure of the

performance of the system under study. The objective depends on certain characteristics of the

system, called variables or unknowns. The goal is to find values of the variables that optimize the

objective. Often the variables are restricted, or constrained, in some way. The process of identifying

objective, variables, and constraints for a given problem is known as modeling.

31

Construction of an appropriate model is the first step, sometimes the most important step, in the

optimization process. Once the model has been formulated, an optimization algorithm can be used to

find its solution. There is no universal optimization algorithm but rather a collection of algorithms, each

of which is conceived for a particular type of optimization problem. If the optimality conditions are not

satisfied, they may give useful information on how the current estimate of the solution can be

improved. The model may be improved by applying techniques such as sensitivity analysis revealing

the sensitivity of the solution to changes in the model and data. Interpretation of the solution in terms

of the application may also suggest ways in which the model can be refined or improved. If any

changes are made to the model, a new the optimization problem is solved, and the process repeats.

Mathematically speaking, optimization is the minimization or maximization of a function subject to

constraints on its variables, for a more detailed description see for e.g., Arora [83]. The optimization

problem can be formulated as max ÃÄÅ fSxT

Subject to:

c;SxT : 0 , i Ã Æ,

c;SxT Ç 0 , i Ã È,

(95)

where f is the objective function, x is the vector of variables, also referred to as unknowns or

parameters, ci are constraint functions, which are scalar functions of x that define certain equations

and inequalities that the unknown vector x must satisfy and È and Æ are sets of indices for equality and

inequality constraints, respectively.

In this work, two optimization problems are considered: the first, very simple, regards the periodic

case in which the design variable is the proportion of each material (in axial length) of each repetitive

cell (see Figure 2.16 (a)); and the second regards the non-periodic case in which the design variable

is the two-material distribution in the structure (see Figure 2.16 (b)).

(a)

(b)

Figure 2.16 – Multilaminated bar: (a) Periodic bar; (b) Non-periodic bar.

2.10.1. Structural Optimization of the Proportion o f the Material in Each Repetitive Cell

(Periodic Case)

Consider a periodic bar made of two different materials, where the objective is to optimize the

portion of each material, in axial length, of each repetitive cell, as illustrated by Figure 2.16 (a). Doing

so, the separation of two specified adjacent eigenfrequencies ωiA1 and ωi, i.e. for a given i (i=1, 2, 3, 4,

5,…) is maximized.

The size optimization problem is formulated as

32

max ¡ Sω;ix O ω;T

Subject to:

SWKX O ω; WMXT YÉ;Z : Y0Z , d : dx A d^ , n Ê d Ë L .

(96)

The design variable in the repetitive cell is d2 where the total number of cells is n and the total length

of the bar, Lt, are known.

2.10.2. Structural Optimization of the Distribution of Two Materials in Each Design Variable

(General Case)

The objective is to find an optimized distribution of two different material components along a given

bar, as the distribution illustrated by Figure 2.16 (b). The distribution should maximize the separation

of two specified adjacent eigenfrequencies ωiA1 and ωi, i.e. for a given i (i=1, 2, 3, 4, 5,…).

A topology optimization method is formulated as max IÌ Sω;ix O ω;T

Subject to:

SWKX O ω; WMXT YÉ;Z : Y0Z , 0 Ë t< Ë 1 j : 1,2, … , NDes .

(97)

The formulation follows the presented in Jensen and Pedersen [84] where the material interpolation

proposed for this problem is different from the usual SIMP technique described by BendsØe and

Sigmund [85]. It is expressed by the following relations

The design variable is tj and it has a value in the interval between tj=0 (where 0 means only material 1

is present) and tj=1 (where 1 means only material 2). To relax the problem for a continuum variation of

the parameter tj, mixtures between these materials are allowed and represented by intermediate

values of tj. Material properties are then computed by Eq.(98). Each finite element is associated with

one value of tj although each design variable can be assigned to several consecutive finite elements.

E : Ex1 A t<SEx/E^ O 1T and ρ : ρx A t<Sρ^OρxT. (98)

33

3. Numerical Models

In a numerical simulation of a physical system one can apply a numerical approach to evaluate its

mathematical model. The finite elements method (FEM) is a technique of numerical analysis of

differential equations. Nowadays, the FEM has been established and is universally accepted as an

analysis method in structural design. The method leads to a discrete system of matrix equations to

represent the mass and stiffness effects of a continuous structure. No restriction is placed upon the

geometrical complexity of the structure since the mass and stiffness matrices are assembled from the

contributions of the individual finite elements with simple shapes and a mathematical formulation

associated to the element’s geometrical description, independent of the overall geometry of the

structure. Accordingly, the structure is divided into discrete lines, areas or volumes known as

elements. Element boundaries are defined when nodal points are connected by a polynomial curve or

surface. In the most popular isoparametric displacement type elements identical polynomial

description is used to relate the internal element displacements to the displacements of the nodes,

which is a process generally known as shape function interpolation. The boundary nodes are shared

between neighboring elements, thus the displacement field is usually continuous across the elements

boundaries. Figure 3.1 illustrates the geometric assembly of finite elements to form part of a mesh of a

modeled structure.

Figure 3.1 – Finite elements, nodes and mesh of a sub-structure Ω.

The mathematical formulation of the finite element method can be presented as a variational

problem with an element-wise Rayleigh-Ritz [68] treatment and shape function discretisation.

Alternatively the finite element equations may be obtained directly from the differential equations using

a Galerkin [68] approach weighted by the elements shape function.

Because finite element methods can be adapted to problems of great complexity and unusual

geometry, they are an extremely powerful tool in the solution of important problems in heat transfer,

fluid mechanics, and mechanical systems. Furthermore, the availability of fast and inexpensive

computers allows problems which are intractable using analytic or mechanical methods to be solved in

a straightforward manner using finite element methods.

In this work the finite element method implemented recurs to two different types of finite elements

to model the structures: a three-dimensional Euler-Bernoulli beam element, which is capable of axial

and torsional deflections as well as two-plane bending, is used to model the helicoidal spring; and a

two-dimensional elastic bar element, subjected to axial forces only, is used to model the

multilaminated bar. The two and three dimension models are presented followed by the developed

analysis, namely: the static; modal; harmonic; and Bloch’s analysis, the models geometry and mesh. A

convergence study for the necessary number of finite elements per length wave is performed. To

conclude, the modal identification and optimization techniques implemented are described.

34

3.1. Basic Steps in Finite Element Method

The basic steps involved in most finite element analysis are: Preprocessing; Solution; and

Postprocessing. For a more detailed description see Cook [68].

3.1.1. Preprocessing

The preprocessing step is, quite generally, described as defining the model and includes:

• Define the geometric domain of the problem;

• Define the element types to be used (bar, beam);

• Define the material properties of the elements (E, υ);

• Define the geometric properties of the elements (length, section properties);

• Define the element connectivities (mesh the model);

• Define the physical constraints (boundary conditions and loading).

The preprocessing step is critical since a perfectly computed finite element solution is of absolutely no

value if it corresponds to a different problem.

3.1.2. Solution

Initiating the solution step, the analysis type is chosen:

• Static;

• Modal;

• Harmonic.

The governing algebraic equations are assembled the in matrix form and the unknown values (usually

displacements) computed. The computed values are then used by back substitution to compute

additional, derived variables, such as reaction forces, element stresses, etc.

3.1.3. Postprocessing

Analysis and evaluation of the solution results is commonly referred to as postprocessing.

• Generic postprocessing (Static and Modal: deformed structure, plot ωi vs. i); • Time-History postprocessing (Harmonic: plot ui/u0 vs. ω);

The objective is to apply knowledge and judgment in determining whether the solution results are

reasonable.

3.2. 2D and 3D Numerical Models

The formulation of the finite element characteristics is based on the following main assumptions:

the material presents a linear elastic behavior, is isotropic, and homogeneous; the displacement

and/or deflection of the elements are small in comparison to the characteristic dimensions of the

element; the element is of length L and has two nodes, one at each end; the element is connected to

other elements only at the nodes; element loading occurs only at the nodes; the bar element supports

axial loading only; bending, torsion, and shear are not transmitted to the element via the nature of its

connections to other elements; and the beam element is prismatic where the cross-section has an axis

of symmetry in the plane of bending.

35


The helicoidal spring is modeled using three-dimensional straight beam elements considering

tension, compression, torsion capabilities and four degrees of freedom at each node; translation in the z direction and rotations about the nodal x, y, and z axes (see Figure 3.2), obtaining the following

elementar stiffness WKeX, mass WMeX and force YFeZ matrixes of an element with length L [59],

Figure 3.2 – Order of degrees of freedom in the beam element [59].

where A is the cross sectional area, E is the longitudinal modulus of elasticity, L is the element length, G is the shear modulus, J is the Saint-Venant’s torsional constant given by

where Ix is the 2nd moment of area (inertia) normal to the x direction, Jx is the Saint-Venant’s torsional

constant and

where Ii is the 2nd moment of area (inertia) normal to the i direction, A;¢ is the shear area normal to

direction i:A/F;¢ and F;¢ is the shear coefficient. The elementar mass matrix may be given by

(99)

J : ¶ J if I : 0 I if I 0 , (100)

e; : S4 A θ;TE I;S1 A θ;TL ; f; : S2 O θ;TE I;S1 A θ;TL ; θ : 12 E IÍG AÍ¢L^ ; θÍ : 12 E IG A¢ L^ , (101)

(102)

36

where A is the cross sectional area, ρ is the mass density, L is the element length, G is the shear

modulus, Jx is the Saint-Venant’s torsional constant given by Eq.(100) and

In the above expressions, θi has been previously defined in Eq.(101) and ri is the radius of gyration.

Finally, the elementar force vector is represented by

F :ÎÏÏÏÐÏÏÏÑ FMÒMÒMÒÍFMÒMÒMÒÍÓÏÏ

ÏÔÏÏÏÕ

e< b I . (104)

3.2.2. Multilaminated Bar

The multilaminated bar is modeled by a two-dimensional elastic bar element with uniaxial tension -

compression and two degrees of freedom at each of the two nodes: translations in the nodal x and y

directions.

Figure 3.3 – Finite element geometry used for the bar [59].

For the multilaminated structure, the nodes to displace only in the axial direction obtaining the

following elementar stiffness WKeX, mass WMeX and force YFeZ matrixes of an element with length L,

K : ESxTAL r 1 O1O1 1 t , M : ρSxTA L6 r2 11 2t , F : ¶FxF^× e< b I , (105)

where A is the cross-sectional area and ESxT and ρSxT are the longitudinal modulus of elasticity and

mass density of each material layer, respectively.

3.3. Analysis: Static / Modal / Harmonic / Bloch

The structural analysis presented in section 2.3, are performed via finite element method [68]

and after the respective matrices assemblage, are expressed by the respective equations which

are rewritten here for easier visualization. The static analysis Eq.(17) to obtain the static

displacement,

E; : 1105 A 160 θ; A 124 θ; A ¬ 215 A 16 θ; A 13 θ; ¬rL^ L^S1 A θ;T^ ;

F; : 1105 A 160 θ; A 124 θ; A ¬ 130 A 16 θ; O 16 θ; ¬rL^ L^S1 A θ;T^ ; r; : I;;A .

(103)

37

by the modal analysis Eq.(18) to obtain the eigenfrequencies and eigenmodes

and by the harmonic analysis Eq.(19) to obtain the frequency response

In the above equations, ωi is the eigenfrequency, Фi is the eigenmode, ωap is the applied force

excitation frequency, U is the respective maximum displacement and Fωap the magnitude of the

applied force.

All finite element analysis are programmed in APDL files and then executed. APDL stands for

Ansys® Parametric Design Language, a scripting language that can be used to automate common

tasks or even build models in terms of selected parameters. APDL also encompasses a wide range of

other features such as repeating a command, macros, if-then-else branching, do-loops, and scalar,

vector and matrix operations.

The Bloch wave analysis, previously mentioned in section 2.9.4 consists in solving the

characteristic equation, Eq.(93), whose roots give for each given frequency ω, the corresponding wave

number k. In this work a commercial numerical code is used to solve this equation. A graphical relation

between the Bloch wave theory and the harmonic analysis is explored and a new relation, considered

to be more intuitive, based on modal analysis between and the previous two is proposed and

presented in section 5.7.

3.4. Geometry and Mesh

The spring in analysis is a right handed cylindrical helicoidal spring with open and plane ends (see

Figure 2.11) with wire of cylindrical section modeled in finite elements having present the concept of

cylindrical helix (see section 2.8.1).

The figures bellow illustrate the geometry of the spring, created using (4Na+ 1) keypoints and (4Na) lines and the respective mesh defined by 5 divisions per line and 100 finite elements per division.

(a)

(b)

Figure 3.4 – Helicoidal spring’s geometry: (a) Continuous model; (b) FEM model.

The multilaminated periodic, Figure 3.5, or non-periodic, Figure 3.6, elastic bar in the form of two

homogeneous alternating plane parallel layers of materials, which are assumed to be isotropic,

differing in elastic properties. The layers are assumed to be perfectly bonded along plane interfaces

across which there is continuity of displacements and equilibrium of stresses.

WKX YUZ : YFZ, (106)

SWKX O ω; WMXT YФ;Z : Y0Z , (107)

KWKX Oω_`^ WMXL YUZ : aFωcde. (108)

38

(a)

(b)

Figure 3.5 – Multilaminated periodic bar: (a) Continuous model; (b) FEM model.

(a)

(b)

Figure 3.6 – Multilaminated non-periodic bar: (a) Continuous model; (b) FEM model.

The number of finite elements used is adjusted taking in account the highest frequency in analysis

and the finite element length. Finite elements analysis are performed considering no transversal

displacements, i.e., the nodes are allowed to displace only in the axial direction.

The structure combining the multilaminated bar and the helicoidal spring is modeled considering

the combination of both structures connected through one point, Figure 3.7.

(a)

(b)

Figure 3.7 – Combined structure: (a) Continuous model (b) FEM model.

The individual characteristics and assumptions of each finite element used is maintained in the

combined structure, e.g., no bending is considered on the two-dimensional elastic bar element.

The APDL files created to model and analyze the structures allows three different types of

analyses: static; modal; and harmonic (steady-state), for three different boundary conditions: fixed-

fixed; free-free; and free-fixed and the following input variables: number of finite elements per line;

applied force; spring’s medium radius; wire’s radius; helix angle; helical pitch; active number of coils.

For the modal and harmonic analysis it is possible to choose the number of modes to extract and

the range of frequencies intended for the analysis, respectively.

3.5. Finite Element Convergence Study for the Neces sary Number of F.E. per Wave Length

In the numerical models it is taken into account the necessary number of finite elements per wave

length, λ. To do so, it is necessary to determine the wave length in each material which is given by

λ : cω : °Eρω , (109)

39

where c is the longitudinal wave’s velocity, E is the modulus of elasticity, ρ is the mass density and ω

the frequency.


The commercially available modal parameter identification algorithms tend to be used as "black

boxes", where the calculation process is more or less transparent to the user. For investigators it is a

frequent practice to use algorithms where the process remains under the user’s control.

In this work a specific software, entitled BETAlab [63], previously referred in section 2.5, running in

LabVIEW® environment is used. The software was developed using the concepts described in section

2.4. Conceived to simultaneously identify, accept and to process a set of transference functions, this

software uses the experimental results obtained by a spectral analyzer. The software uses the CRF, in

accordance with the presented in section 2.4.3 to identify the modal, global and local parameters.

Once the CRF is obtained, the calculation of the global parameters of the structure is performed using

the inverse method [62]. The inversion of Eq.(39) leads to a line with theoretical unitary negative slope

(relatively to the real part) which eases the mode identification. In particularly complex cases, it is

possible to adjust the theoretical model to the experimental curves with higher precision with the

possibility to manually modify the results obtained.

3.7. Optimization Technique

The implemented optimization technique combines the commercial software Ansys®, for the finite

element analysis, with medium scale Square Quadratic Programming (SQP) [86] optimization

algorithms, that run in MatLab® environment (see Figure A.4 in appendix) in order to maximize the

separation of two adjacent eigenfrequencies as presented in the optimization problems of section

2.10. The implemented method uses the “fmincon” function, available in MatLab’s optimization

toolbox, due to the fact that in the optimization group of Prof. M. M. Neves when one recurs to

MatLab® for optimization, the optimization toolbox or the Method of the Moving Asymptotes (MMA) is

used. The function of optimization toolbox, in this case “fmincon”, reveals to be effective for the

proposed problems, reason why it is implemented.

40

4. Experimental Models and Methodology

One can say that there is no single right way to perform a vibration test. In almost every case, the

support, the excitation equipment or the transducers will influence the dynamic behavior of the

mechanical component under test. The science in vibration testing is in realizing that these influences

exist, understanding them and then designing the test to minimize their effects on the dynamic

behavior of the structure. The art and science of vibration testing, as in general for mechanical

engineering, is to obtain results that are as close to the correct answer as required, at a cost that is

within budget, and to achieve all this in useful time. In many aspects the practice of vibration testing

can be considered as more an art than a science.

The object of the performed vibration analysis was to acquire sets of FRFs that are sufficiently

extensive and accurate, in both the frequency and spatial domains, to enable analysis and extraction

of the properties for all the required modes of the structure. The advantage of this approach is that the

actual structure is tested. However, for practical reasons, the extent of a model obtained in this way is

more limited in spatial and frequency domains than a Finite Element model. The frequency domain

extent of the model obtained from experimental measurements will be dictated by the capabilities of

the transducers and data processing equipment used in the measurements.

Remembering, one of the main objectives of this work is to design a multilaminated structure that,

by itself or when combined with the helicoidal spring, is able to attenuate structural vibrations in

specified frequency ranges. In this chapter the experimental (test) models, often referred to as

specimens, used in the analysis of vibration attenuation, are presented. In the initial stage of the

experimental work, routine hardware and software calibration tests were performed. The first

specimens built were simple but of great importance, since they reveal the materials experimental

modulus of elasticity that was useful in the updating of the finite element model [73]. For this purpose

the first specimens built were one-material structures, either steel or PMMA (Poly methyl

methacrylate). Afterwards, multilaminated structures were built in which the materials’ half-cells (a

half-cell is composed of one material while a cell is composed of two half-cells and consequently of

two different materials) were bonded with an adhesive, thus a bonds adhesive influence analysis was

performed.

Searching for the experimental modal validation some specimens were built. The first consisted in

an adaptation of the one used in the experience carried out by Jensen, Sigmund and Thomsen [84]

where the specimen is constituted by a total of 5.5 cells, six and five steel and PMMA half-cells,

respectively. To demonstrate the applicability of these structures at frequencies of practical interest,

some other materials were tested, namely brass and agglomerated cork, and new specimens were

built and tested. This was followed by the test of the cylindrical helicoidal spring and its combination

with the previously built specimens. A complex specimen intending to simulate the base of an air-tight

compressor motor was built and tested for three different suspension situations using: helicoidal

springs; multilaminated specimens; and both combined. To conclude this experimental journey an

overview on modal identification and experimental methodology is presented.

41

4.1. Hardware Equipment

In experimental vibration analysis, various hardware components are necessary to test specimen

that is usually suspended from an auxiliary structure. The measurement chain can be divided into

three main sub-sets [62]: the excitation components; the sensing components; and the data

acquisition and processing components.

4.1.1. Excitation of the Structure

The excitation component, also known as exciter or shaker, provides the input solicitation,

generally in the form of a driving force applied to the structure under analysis. The majority of

structural excitation techniques require some physical contact with the structure, see Figure 4.1. The

objective is to transmit controlled excitation to the structure in a given direction and, simultaneously, to

impose as small restrain on the structure as possible in all other directions. A push-rod is usually used

to form a link between the shaker and the structure under test.

Figure 4.1 – Transmissibility between the shaker and the structure through a push-rod [62].

One may state that, without prior knowledge of the dynamic characteristics of a structure, the

location of the excitation and response measurement points is a matter of trial and error coupled with

experience and engineering judgment.

4.1.2. Sensing Transducers

The technology developed in measurement equipment allows the direct measurement of three

physical quantities that characterize movement: displacement, velocity and acceleration. So, for

measuring displacements the most common instruments are potentiometers (of simple conception),

while for measuring velocities, Laser equipment (using the Doppler’s effect) is probably the most

popular option. Relatively to acceleration measurements transducers, usually designated as

accelerometers, they can be piezoelectric, piezoresistive, capacitive or force balanced.

(a)

(b)

Figure 4.2 – Sensing transducers [62]: (a) Cross-section of a piezoelectric accelerometer; (b) Cross-section of a

piezoelectric force transducer.

42

4.1.2.1. Piezoelectric Accelerometer

The cross-section of a typical piezoelectric accelerometer is illustrated in Figure 4.2 (a). There are

four basic components: a base and case, a center post, an annular section of piezoelectric ceramic

and an annular seismic mass element. The base of the accelerometer moves with the motion structure

to which is attached, and to cause equivalent motion of the seismic mass, a force must be applied.

This force is transmitted through the piezoelectric crystal that deforms slightly as a consequence. The

deformation produces a charge in the piezoelectric crystal that is proportional to the deformation and

hence, ultimately, to the acceleration of the seismic mass and structure. These devices operate well

over a fairly wide frequency range, but they are not generally well suited to low frequency applications

[62], see the calibration chart in Figure A.5 in appendix.

4.1.2.2. Force Transducer

The most common type of force transducers (see Figure 4.2 (b)), works on the principle that the

deformation of a piezoelectric crystal produces a charge output proportional to the force acting on that

crystal (same principle as described for the piezoelectric accelerometer in the previous section). In

Figure 4.2 (b) it is illustrated the cross-section of a typical piezoelectric force transducer from which is

possible to identify two constitutive masses: one below the piezoelectric element (“base-side” mass)

with 3 grams; and the other above the piezoelectric element (“live-side” mass) with 18g [62]. The “live-

side” mass of the force transducer is kept small minimizing the modification to the structure. For a

typical force transducer, the “base-side” is only 3 grams. In reference [62] it is discussed the

importance and influence of the active and total mass of the force transducers in experimental work.

4.1.3. Data Acquisition and Processing Components

These components are used to measure and process the signals developed by the sensing

components, e.g., magnitudes and phases of the excitation forces and its response. Signal

conditioning and power amplifiers, signal generators and analyzers are an example of types of these

components. Nowadays, this information is transmitted to a digital computer, in which specific

software processes the acquired data.

4.2. Equipment Used

The experimental tests were performed in the Laboratory of Vibrations of Mechanical Department

(DEM) in Instituto Superior Técnico.

(a)

(b)

Figure 4.3 – Hardware components and auxiliary support structures: (a) Workstation: PC, amplifiers, data

acquisition components, etc.; (b) Fixation and support structure.

43

In the laboratory there is an adequate isolated ground area available as well as hardware components

(eg., shakers, amplifiers, transducers, etc.) and auxiliary support structures for these types of tests

(see Figure 4.3). The hardware equipment used to conduct the tests were the following:

• Data acquisition equipment Brüel & Kjaer Type 3560-C [87];

• Power supply unit Brüel & Kjaer Type 2827 [87];

• Lan interface module Brüel & Kjaer Type 7533 [89];

• Generator module, 4/2 ch. Input/Output Brüel & Kjaer Type 3109 [87];

• Vibration exciter Brüel & Kjaer Type 4809 [90];

• Power amplifier Brüel & Kjaer Type 2706 [91];

• Charge amplifier Brüel & Kjaer Type 2635 [92];

• Accelerometer Brüel & Kjaer Type 4508-B [92], calibration chart in appendix: Figure A.5;

• Force transducer Brüel & Kjaer Type 8200 [94], calibration chart in appendix: Figure A.6;

• Mounting Clips ref. UA1407 [93];

• Cables and connectors.

4.3. Experimental Setup

The experimental setup is illustrated in Figure 4.4. In order to obtain the experimental results of the

specimens under test a signal is generated and transmitted to a shaker (see Figure 4.5 (a)) which

transmits a longitudinal force, through a force transducer along the specimen (see Figure 4.5 (b)). The

dynamical deformation propagates throughout the specimen, and in the opposite extremity the

longitudinal acceleration is measured by an accelerometer (see Figure 4.5 (c)). The signal is then

treated and saved for posterior analysis and processing. The specimen is suspended from a fix

support by two thin lines while the shaker is suspended from a mobile support by metallic chains.

Figure 4.4 – Experimental setup.

44

(a)

(b)

(c)

Figure 4.5 – Amplified photographs of some hardware equipment: (a) Vibration exciter B&K Type 4809;

(b)Force transducer B&K Type 8200; (c) Accelerometer B& K Type 4508-B.

4.4. Calibration and Finite Element Model Updating

In the beginning of the experimental analysis, routine hardware and software calibrations tests

were performed. The sensibility of the force transducer had to be set in the respective amplifier, the

hardware data had to be introduced into the software analysis program, which is Bruel & Kjaer

PULSE® Labshop (vers.6.1.5.65).

In the numerical finite element models (see section 3.2) the material properties used were standard

properties for the respective material (see Table 5.6). The first structures tested were one-material

structures from which was calculated the experimental value for the modulus of elasticity, E, of the

material used, useful in the finite element model updating (see Figure 4.6), by

that is a particular solution of Eq.(77) when solved for free-free boundary condition of a bar. In

Eq.(110), L is the length of the bar, f:ω/2π is the first natural frequency, obtained from the

experimental FRF curve. The mass density, ρ may be given by

where m and V are the mass and the volume of the specimen, respectively. Thus, the experimental

mass density of a material was obtained through the direct weight and dimension measurements of a

specimen of the respective material.

Figure 4.6 – Finite element analysis updating process of the material Young’s modulus.

E : 4 L^f ^ ρ, (110)

ρ : mV , (111)

45

4.5. Experimental Specimens and Constructive Aspect s

The method applied to build the specimens start with cutting the material half-cells as illustrated in

Figure 4.7. The extremities of the half-cells were polished minimizing the rugosity originated during the

cutting process.

(a)

(b)

(c)

Figure 4.7 – Steel and PMMA half-cells with a diameter of 0.020m and a length of 0.135m: (a) Top view; (b) Six

steel half-cell front view; (c) Five PMMA half-cells front view.

To minimize eventual centering deviations an L shaped beam was used to correctly align the half-

cell units. The adhesive was applied to the extremities of the half-cells which were placed on the L

shaped beam and compressed against each other and against the beam while the adhesive cures.

(a)

(b)

Figure 4.8 – L shaped beam to assist the bonding process: (a) Half-cell order; (b) L shaped beam (side view).

4.6. Adhesive

Adhesive was used to bond the different material half-cells along the specimens. Even though it

introduces a relatively thin layer of material (see Figure 4.9 (a) and Figure 4.9 (b)), this is a different

material for which the necessary properties, modulus of elasticity and mass density, are unknown. A

brief analysis on how this affects the experimental results is presented.

(a)

(b)

Figure 4.9 – Adhesive layer of material in a unit cell: (a) Via SolidWorks®; (b) Via experimental.

A methyl methacrylate based structural adhesive was used in the construction of the experimental

specimens involving PMMA material, minimizing the PMMA material transition. The adhesive is

46

named X 60 Schnellklebstoff®, and is composed by two components, a solid component in the form of

powder and a liquid component, identified as Komponente A and Komponente B, respectively, see

Figure 4.10.

(a)

(b)

Figure 4.10 – Adhesive X 60 Schnellklebstoff: (a) Exterior package; (b) Interior of the Package.

Legend of Figure 4.10 (b):

A - X 60 Schnellklebstoff®, Komponente A, (solid component);

B – X 60 Schnellklebstoff®, Komponente B, (liquid component);

C – Spoon for the solid component;

D – Mixture recipients;

E – Mixing stick.

The adhesive’s preparation consists of inserting a portion of the solid component, with the spoon,

in the mixture recipient. Some drops of the liquid component are then added to the recipient and the

components mixture is homogenized with the stick. The adhesive is ready to be applied. During the

cure time, approximately 90 seconds, an exothermic reaction occurs and some heat is transferred to

the mixture recipient.

For different materials (e.g., cork) a different type of adhesive was used to bond the different

material half-cells along the specimens. The adhesive used was an Araldite® commercial high

performance, two component epoxy adhesive, with a cure time of 5 minutes. The adhesive’s

preparation consists in mixing equal portions of each of the components in a mixture recipient until a

homogenized mixture is obtained. When so, the adhesive is ready to be applied.


As presented in sections 2.5 and 3.6, when the measured data indicates heavily coupled modes or

noise contamination, or when high accuracy is required for the estimation, an identification technique

can be used to obtain the modal parameters estimation. In this work, a recent version of the software

BETAlab [63] was used. This software uses the CRF (Characteristic Response function) to identify the

modal, global and local parameters, in consistent form as described in section 2.4. The figure bellow

illustrates BETAlab’s environment.

47

Figure 4.11 – BETAlab software environment screen shot.

The identification procedure used with this software is easy to learn and follows the steps described

below:

1. Identification of the source of the input result data file is the first option to be introduce;

2. The software reads the data file, calculates and presents the ALFA-FRF and BETA-FRC curves;

3. The user makes use of the FRF and FRC curves visualization and selects the region of points

intended to identify each mode;

4. The step size may be increased to ease the mode visualization in the FRC function;

5. The user can, optionally, isolate a mode to deduct the effect from the previously identified modes,

or to decrease the influence of these in future identified modes;

6. Identification of additional modes is supported;

7. Residuals may be included in other to contemplate the contribution of the modes that out of the

frequency range of analysis;

8. The input data curve may be overlapped to the regenerated curves allowing a direct comparison

between them;

9. In case the user is not satisfied with some of the calculated parameters, he can always manually

modify it, immediately observing the produced effect;

10. Result and project files may be saved and/or loaded.

48

4.8. Methodology

To conclude this chapter the methodology adopted is presented. In the figure bellow are illustrated

the paths and main steps taken in this work.

Figure 4.12 – Diagram of the methodology adopted in this work.

1. Identification of the physical problem;

2. Characterize the problem and evaluate it to determine relevant data such as displacements,

stresses, natural frequencies, etc.;

2.1. Analytically;

2.2. Experimentally;

2.3. Numerically (FEM was used);

3. Compare the sets of the results obtained to verify and/or validate the models (see section 2.7);

4. If the results are not satisfactory, i.e., verification and/or validation was not successfully achieved;

4.1. Refine FE model;

4.2. Improve analytical model;

4.3. Improve structural model;

5. Once the models are verified and validated they are then fit to be used for further analysis.

49

5. Results and Discussion

This chapter consists on the presentation and discussion of the results obtained throughout this

work. The verification of the analytical and numerical models is performed checking the results against

those of others studies carried out, which are found in the bibliography. The helicoidal spring, the

homogeneous and the heterogeneous cylindrical periodic bar results are presented and verified.

The results obtained for the helicoidal spring, showing the influence of the spring’s parameters,

namely the helix angle, the number of active coils and the relation of diameters D/d, on the natural

frequencies and on the attenuation regions are presented. This is followed by a similar analysis that

shows how the material properties of the periodic bar (modulus of elasticity and the mass density)

affect the natural frequencies and the attenuation regions.

In order to demonstrate the applicability of the multilaminated periodic bar at frequencies of

practical interest, a plot of results for some different combination of materials is presented.

Regarding the experimental analysis, the first experimental specimen’s results reveal their

importance in the updating of the finite element model [73], which was initially modeled using standard

material properties. This is followed by the presentation of the adhesive bonds results regarding its

influence in the FRF curves.

The results obtained from the adaptation of the experience carried out by Jensen, Sigmund and

Thomsen [84] as well as from identical structural analysis, intended to verify the influence of the non-

longitudinal vibration modes and the attenuation’s amplitude, are presented. This is followed by the

analysis regarding the applicability of these specimens for frequencies of practical interest involving

specimens built from other materials, e.g. agglomerated cork.

Remembering, one of the main objectives of this work is to design a multilaminated structure that,

by itself or when combined with the helicoidal spring, is able to attenuate structural vibrations in

specified frequency ranges. The results for the combined structures are presented as well as the

results obtained for the complex specimen intending to simulate the base of an air-tight compressor

motor tested with three different suspension configurations. To conclude this chapter, the numerical

results of the previously introduced optimization problems are presented and discussed by the relation

between frequencies’ plots (modal analysis) associated with a finite repetition and dispersion curves

(Bloch wave analysis) associated with an infinite repetition for a periodic bar.

5.1. Numerical vs. Analytical Verification

The first step in verifying (see section 2.7) the finite element models presented consisted in

comparing the numerical results obtained with the analytical ones.


The spring modeled, see Figure 5.1, was a right handed cylindrical helicoidal spring, with open and

plane ends, with wire of cylindrical section and with the following properties: Young’s modulus E=2.09·1011 N/m2; Poisson’s coefficient υ=0.28; mass density ρ=7800kg/m3; average wire radius

50

r=6mm; average coil radius R=65mm; helix angle α=7.44º; number of active coils N=6; and total free

length L0 = 320mm.

Figure 5.1 – Helicoidal spring parameters identification.

The analytical results are obtained solving the respective equation, whereas the numerical results,

see section 3.2.1, are obtained using the following parameters to define the geometry of the spring:

(4N+1) keypoints; (4N) lines; and the respective mesh defined by 5 divisions per line and 100 finite

elements per division.

Table 5.1 – Helicoidal spring: Theoretical results compared with those obtained via FEM.

Analytical FEM er

Eq.(50) with KB=1 τmax =759.29 MPa Eq.(17) τmax = 744.26 MPa 1.99 %

Eq.(50) KB from Eq.(51) τmax =815.78 Mpa Eq.(17) τmax = 799.63 Mpa 1.98 %

Eq.(56) k=16.053·103 N/m Eq.(17) k=15.736·103 N/m 1.97 %

Eq.(57) Fmax=3.789·103 N Eq.(17) Fmax=3.714·103 N 1.98 %

Eq.(59) f = 42.9 Hz Eq.(18) f = 41.040 Hz 4.34 %

The FEM results obtained are presented and compared with the analytical ones, where er is the

relative error between them illustrated in Figure 5.2. Comparing the analytical and numerical results

obtained a relative error interval can be defined within 1.97% and 4.34%.

Figure 5.2 – Relative error obtained between the analytical and finite element results.

51

In Table A.7 (in appendix), the FEM results obtained solving Eq.(18), for the first eight eigenmodes,

for three different boundary conditions (B.C.), are presented and compared with those of [16], where er is the relative error between them illustrated in Figure 5.2.

Figure 5.3 – Relative error obtained between the finite element method applied and those of [15].

The results comparison, between both FEM’s, define a relative error interval within 0.02% and 0.27%.

In [23], others numerical methods are applied; the transference matrix method and the dynamic

stiffness method. These results are presented in appendix (Table A.8.1, Table A.8.2 and Table A.8.3)

for three different boundary conditions and successfully compared with the ones obtain by the

implemented FEM model, where er is the relative error between them illustrated in Figure 5.4

(a)

(b)

(c)

Figure 5.4 – Finite elements, transference matrix and the dynamic stiffness methods [23] results comparison.

The results obtained are coherent and can be considered acceptable within all the approximations

inherent to the FEM, thus the finite element model for the helicoidal spring can be considered as

verified [72], see section 2.7.

52

5.1.2. Homogeneous Bar

Three different material homogenous bars were considered whose standard properties are

presented in the following table.

Table 5.2 – Standard material properties: E- longitudinal modulus of elasticity; ρ- mass density.

Steel PMMA Brass EEEE [Pa] 200·109 4.1·109 115·109

ρρρρ [kg/m 3] 7860 1140 8750

The results were obtained with a cylindrical bar with a uniform diameter of 0.020m and a total

length of 1m under free-fixed boundary conditions for the stiffness calculation, Eq.(68) and Eq.(17),

and under free-free boundary condition for the frequency calculation, Eq.(77) and Eq.(18).

Table 5.3 – Homogenous bar: Theoretical results compared with those obtained via FEM.

Analytical FEM

Steel PMMA Brass Steel e r PMMA er Brass e r kkkk [N/m] 62.83·106 1.29·106 36.13·106 62.89·106 0.09% 1.29·106 0% 36.10·106 0.08% ffff [Hz] 2522.17 948.22 1812.65 2522.50 0.01% 948.35 0.01% 1812.91 0.01%

Comparing the theoretical and numerical results obtained, shown in Table 5.3, a relative error, er, interval can be defined within 0% and 0.09%.

5.1.3. Periodic Bar

To verify the finite element model of the periodic bar, two different approaches were considered.

The first, a modal analysis Eq.(18), with n cells for a given Lt (see Figure 3.5), identifies the lower and

upper bounds of the first and second attenuation bands. The second, a Bloch wave analysis which

consists on solving Eq.(93), allow to compute and plot the frequency as an implicit function of the

quasi-wave number, where the stop-bands are identified for complex wave number kimag.

(a)

(b)

Figure 5.5 – Periodic bar: (a) Modal analysis, finite structure; (b) Bloch wave analysis, infinite structure.

A relation between the modal analysis (see Figure 5.5 (a)) associated with a finite repetition and

Bloch wave analysis associated with an infinite repetition (see Figure 5.5 (b)), is presented in Figure

53

5.6. Consider a periodic bar as described in section 3.2.2 and 3.4 and shown in Figure 3.5, where the

two materials considered are: material 1 (MAT.1) and material 2 (MAT.2) with the following properties:

Young’s modulus EMAT.1=0.003GPa, EMAT.2=200.0GPa and mass density ρMAT.1=1142kg/m3, ρMAT.2=7860kg/m3. The bar was analyzed for different cell numbers (n= 2,…,9) where L2:Lt/2n with a

uniform diameter of 0.01m and a total length of Lt=0.036m.

Figure 5.6 – Limit curves of the bandgap and frequency band versus number of cells: (a) First frequency band,

(b) Second frequency band.

This way, the lower and upper bounds of the first and second attenuation bands were identified.

For the second attenuation band, see Figure 5.6 (b), the results obtained via modal and Bloch wave

analysis present a slight difference in the results (maximum relative error of 5.2%) for the upper bound

(ωiA1) that is associated to the number of finite elements used and to the move from the infinite to the

finite repetition case.

As illustrated in Figure 5.6 (a) and Figure 5.6 (b), for the first and second attenuation bands,

respectively, one can state that such a relation exists. The relation obtained by these two different

approaches is not well known, not mentioned in the literature, but it is an important relation since in

Bloch wave analysis equations get larger and more complicated leading to longer and heavier

computational time and requirements.

5.2. Helicoidal Spring: Parameters and Properties I nfluence

In this section a brief analysis that illustrates how the parameters of the spring affect the natural

frequencies and the attenuation regions is presented. The parameters considered are the helix angle α, the number of active coils N and the relation of diameters D/d as illustrated in Figure 5.7.

Figure 5.7 – Helicoidal spring’s parameters identification.

54

5.2.1. Spring’s Parameters Influence in the Natural Frequencies

The results for the analyzed parameters are illustrated by Figure 5.8: the helix angle α (see Figure

5.8(a)), the number of active coils Na (see Figure 5.8(b)) and the relation and diameters D/d (see

Figure 5.8(c)).

(a) (b)

(c)

Figure 5.8 – Parameters influence in the natural frequencies: (a) Helix angle α; (b) Number of active coils Na;

(c) Diameter relation D/d. As the helix angle (α), the active number of coils (Na) and the diameter relation D/d increase, the

total length of the spring increases and consequently the stiffness of the system and its natural

frequencies decrease. With the increase of these parameters the coupled modes begin to appear

(see Table A.9, Table A.10 and Table A.11 in appendix).

5.2.2. Spring’s Parameters Influence in the Attenua tion Region

In this section, an analysis on how the parameters of the spring affect the attenuation regions is

presented. The figures presented in this section the attenuation region is normalized by a frequency of

4000 Hz.

55

Figure 5.9 – Influence of the helix angle.

Figure 5.10 – Effect of the helix angle.

Figure 5.11 – Influence of the active number of coils.

Figure 5.12 – Effect of the active number of coils.

Figure 5.13 – Influence of the diameter relation D/d.

Figure 5.14 – Effect of the diameter relation D/d.

As the helix angle α (see Figure 5.9 and Figure 5.10), the active number of coils Na (see Figure

5.11 and Figure 5.12) and the diameter relation D/d (see Figure 5.13 and Figure 5.14) increase, the

amplitude of the attenuation regions (A.R.) decreases. A summary on how the parameters of the

spring affect the attenuation regions is presented in the table below.

56

Table 5.4 – Spring’s parameters influence in the attenuation region.

Variable Constant

NNNN αααα C:D/dC:D/dC:D/dC:D/d αααα ↓ N → ↑ A.R. * - ↓ C → ↑ A.R. ** NNNNaaaa - ↓ α → ↑ A.R. * ↓ C → ↑ A.R. ** D/D/D/D/dddd ↓ N → ↑ A.R. ** ↓ α → ↑ A.R. * -

* First option – The variation can be approximated by a linear function such as: y:m xAb;

** Second option - The variation can be approximated by a power function such as: y:a x-b. The first and second options should be applied, if acceptable, in this order if it is intended to

minimize the decrease of the attenuation regions or in the reverse order if it is intended to maximize

the decrease of the attenuation regions. Generally, i.e., considering normalized springs, decreasing

the parameters D/d, Na or α in this order of decrease influence, increases the amplitude of the

attenuation regions.

Helicoidal springs have been and continue to be widely studied leading to today’s known

performances. It terms of optimization some work has been done [24]-[29] and in respect to the use of

common materials (excluding piezoelectric [26] and [27] and smart materials) to build these types of

springs there is not much that can be done to optimize their performance (shown by the previous

parametric study). A new approach that intends to substitute the helicoidal springs with a periodic

structure is presented. The main idea consists in positioning the attenuation region of the periodic

structure in the desired frequency range of interest.

5.3. Bar: Parameters and Properties Influence

In this section a brief analysis that illustrates how the parameters and material properties of the bar

affect the natural frequencies and the attenuation regions is presented. The material properties

considered are the modulus of elasticity E and the mass density ρ.

5.3.1. Convergence Study for the Necessary Number o f Finite Elements per Wave Length

When updating the numerical model it was taken into account the necessary number of finite

elements per wave length, λ. Applying Eq.(109) the results obtained are presented in Table 5.5.

Table 5.5 – Wave length of the studied materials.

Steel PMMA Brass Aluminum EEEE [Pa] 200·109 4.1·106 115·106 70·109 ρρρρ [kg/m 3] 7860 1142 8750 2780 cccc [m/s ] 5086.81 1938.87 3625.31 5017.9 ffff [Hz] 5000 5000 5000 5000 λλλλ [m] 1.0174 0.3878 0.7250 1.0036

It was verified (see Figure A.12 in appendix) that in general 8 finite elements per wave length is not

sufficient to obtain reasonable results for these types of analysis at high frequencies. The use of 16 or

32 finite elements per wave length revealed more accurate results at high frequencies but the

computational effort is greater. In this study 16 finite elements per wave length are used in all

numerical analysis.

57

5.3.2. Homogeneous Bar: Parameters and Properties I nfluence

Consider a cylindrical bar as ilustrated in Figure 3.5, with a diameter of 0.01m, a total length (Lt) of

0.825m and a total of 5.5 unit cells, made of steel or aluminum with the properties presented in the

following table.

Table 5.6 – Steel and aluminum properties (E, ρ and c).

Solving Eq.(17) and Eq.(18) for the homogenous steel and aluminum bars, the results obtained are

presented in Figure 5.15.

Figure 5.15 – Homogeneous bar with diameter of 0.01m and Lt=0.825m: (a) Natural frequencies; (b) Frequency

response.

Figure 5.15 illustrates that both steel and aluminum bars (black and blue lines, respectively)

present identical natural frequencies, Figure 5.15 (a), and steel present a “slightly” grater attenuation

in displacement than aluminum, Figure 5.15 (b). Thus, one can conclude that materials with identical

wave propagation speed Eq.(73) have identical natural frequencies but denser materials present

higher attenuation displacements. The periodic bar is analyzed in the next section.

5.3.3. Periodic Bar: Parameters and Properties Infl uence

Consider a cylindrical bar as described in sections 3.2.2 and 3.4 and shown in Figure 3.5, with a

uniform diameter of 0.01m, a total length (Lt) of 0.825m and a total of unit cells (n) varying unitarily

from 2.5 to 5.5, made of steel or aluminum and Poly methyl methacrylate (PMMA) with the properties

represented in Table 5.6. Solving Eq.(17) for the periodic steel or aluminum and PMMA bars, the

results obtained are presented in Figure 5.16.

Material EEEE [GPa] ρρρρ [kg/m 3] cccc [m/s]

Steel 200 7860 5044.3

Aluminum 70 2780 5017.9

PMMA 4.1 1140 1896.4

58

Figure 5.16 – Natural frequencies of periodic bar with diameter of 0.01m, 2.5≤n≤5.5, L1=L2=0.075m, Lt=0.825m.

On Figure 5.16 the attenuation regions may be identified by the lines that present higher slopes,

which reflect the distance between the natural frequencies. Figure 5.16 illustrates that as the number

of unit cells increase the attenuation regions also increase leading to the Bloch wave theory, section

2.9.4, which states that for an infinite medium stop-band regions (where waves are not allow to

propagate) substitute the attenuation regions. The steel and PMMA periodic bars (filled and doted

lines, respectively) present grater attenuation regions than aluminum and PMMA bars due to the

higher mass density of steel (as presented in Table 5.6).

Particularizing for the case of a periodic bar with 5.5 unit cells as illustrated in Figure 5.17.

Figure 5.17 – Periodic bar with diameter of 0.01m, n=5.5, L1=L2=0.075m, Lt=0.825m.

Solving Eq.(17) and Eq.(18) periodic steel or aluminum and PMMA bars for 5.5 unit cells, the results

obtained are presented in Figure 5.18.

Figure 5.18 – Periodic bar with diameter of 0.01m, n=5.5, L1=L2=0.075m, Lt=0.825m: (a) Natural frequencies; (b)

Frequency response.

59

Figure 5.18 illustrates that the steel-PMMA and aluminum-PMMA bars (black and blue lines,

respectively) the attenuation regions increase in frequency and attenuation of the displacement also

increases with the increase difference between the materials properties of the periodic bar. In

conclusion, one can state that the dissimilarity in the properties of the materials control the amplitude

of the attenuation regions, and displacements, in the frequency domain.

One other parameter that influences the natural frequencies and the attenuation regions is the cell

size. It is obvious that natural frequencies decrease as the total length of the bar increases, Eq.(76),

so adding additional unit cells, that make the bar longer, will consequently decrease the frequency

attenuation region and increase the attenuation of the displacement in that region, and vice-versa. If

the length of the bar is fixed and the length of the cells is decreased, meaning that the number of units

cells in the structure increase, then as presented in section 2.9.4, the structure’s attenuation region is

approximately maintained but the attenuation of the displacement in that region increases. This can be

seen in the next section where a parametric study involving the variation of the total number of cells,

the length of the bar and the materials properties is presented.

5.3.4. A Material Selection on the Frequency Range of Interest

For a structure in the conditions of the implemented model one can recur to Figure 5.19 and

select the suited for a particular frequency range of interest. The limit values presented were

obtained considering the model presented in section 3.4, with a total number of cells n, 6≤n≤12, a

total length of the bar Lt, 0.036m≤ Lt ≤0.1m and a relative material density ρv, 0.5≤ ρv≤1. The left

points (lower bounds) of each horizontal line are calculated for n=6; Lt =0.1m ; ρv=0.5 and the right

points (upper bounds) of each horizontal line are calculated for n=12 ; Lt =0.036m ; ρv=1. The

shaded area represents the use of foam to model the material with lowest modulus of elasticity in

the structure. The foam model used was: Efoam:ρv3 E and ρfoam:ρv ρ [95].

Figure 5.19 – Results obtained for the periodic model presented in section 3.2.2 with different materials.

From a simple parametric study, which originated Figure 5.19, it is possible to choose an appropriate

combination of material that best suits the frequency range of interest.

60

5.4. Experimental vs. Numerical and Analytical

All experimental results were subject to modal identification (see sections 2.4.3, 2.5, 3.6 and 4.7)

with the specific software BETAlab [63]. The identified parameters were used to regenerate the FRF

curves; these curves fit the experimental data. The obtained FRF curves are presented where the

blue, red and green lines represent the numerical, experimental and regenerated curves, respectively.

5.4.1. Calibration and Finite Element Model Updatin g

In the beginning of the experimental analysis routine hardware and software calibrations and tests

were performed as described in section 4.4. In the numerical finite element models the material

properties used, were standard properties for the respective material (see Table 5.5). The first

specimens analyzed were one material specimens that supply the experimental value for the modulus

of elasticity, Eq.(110), of the material used. The experimental mass density of a material was obtained

as described in section 4.4 through the direct weight and dimension measurements of the specimen of

the respective material. The data obtained for the materials experimental mass density is presented in

Table 5.7.

Table 5.7 – Experimental mass density of the studied materials.

The updating of the finite element models was performed, taking into account all the considerations

stated above, based on the experimental results. With this purpose, the first specimens tested were

cylindrical homogeneous steel and PMMA bars each with a uniform diameter of 0.020m and a total

length of 1m. The mass density initially used in the FE models present a relative error of 0.48% and

4.8% for steel and PMMA, respectively.

5.4.1.1. Homogeneous Bar

This specimen is a cylindrical homogeneous steel bar with a uniform diameter of 0.020m and a

total length of 1m. The experimental setup is illustrated by Figure 5.20.

(a) (b)

(c)

Figure 5.20 – Experimental setup of the homogeneous steel bar specimen: (a) Vibration exciter; (b) Specimen; (c)

Accelerometer.

Steel PMMA Brass Cork Cork 8123 [87]

Mass [kg] 0.3395 0.1147 0.0476 0.0048 0.0369

Length [m] 0.1364 0.3050 0.0206 0.0379 0.050

Diameter [m] 0.0200 0.0200 0.0191 0.0228 -

Volume [m3] 4.285·10-5 9.582·10-5 5.947·10-6 1.551·10-5 1.445·10-4

Density [kg/m3] 7922.51 1197.05 8011.74 312.15 255.85

61

Figure 5.21 – Frequency response curves for the homogeneous steel bar

specimen.

First natural frequency:

f1 =2544 Hz

Experimental E:

ESteel=205 GPa

The next specimen is a cylindrical homogeneous PMMA bar with a uniform diameter of 0.020m and

a total length of 1m. The experimental setup is illustrated by Figure 5.22

Figure 5.22 – Experimental setup of the homogeneous PMMA bar specimen.

Figure 5.23 – Frequency response curves for the homogeneous steel bar

specimen.

First natural frequency:

f1 =992 Hz

Experimental E:

EPMMA=4.7 GPa

Based on the experimental results, the modulus of elasticity initially used in the FE models presented

a relative error of 2.5% and 14.6% for steel and PMMA, respectively.

5.4.2. Adhesive Influence

As presented in section 4.6, a methyl methacrylate based structural adhesive, X 60

Schnellklebstoff® (see Figure 4.10), is used to bond the different material cells along the structures. A

brief analysis on how it can affect the experimental results is presented and a heterogeneous one cell

Steel-PMMA specimen was built.

62

5.4.2.1. Heterogeneous Steel-PMMA Specimen: 1 Cell

This specimen is a 1 cell periodic bar, composed of one steel and one PMMA half-cells each with a

total length of 1m and a uniform diameter of 0.020m. The half-cells are bonded with a structural

adhesive creating a specimen with 2m in length. The experimental setup is illustrated by Figure 5.24.

Figure 5.24 – Experimental setup of the one cell heterogeneous steel-PMMA bar specimen.

Figure 5.25 – Frequency response curves for the one cell heterogeneous one cell steel-PMMA bar specimen.

For the first adhesive bond, the experimental results (red curve) do not allow for the identification of

the second natural frequency but “clearly” identify the other three. Suspecting that this may be due to

deficient adhesive bond, a new adhesive bond containing a “very thin” layer of adhesive, referred to as

bond 2 is made (green line) from which it is possible to identify the second natural frequency at

approximately 1580Hz. The FRF curves are presented in Figure 5.25 where the blue line represents

the numerical results the red and green lines represent the experimental results of the first and second

adhesives’ bonds, respectively.

Having seen the influence of the adhesive, namely its layer thickness that must be carefully

controlled, in the bonding of the material that composes the structures, the following step consisted in

adapting the experience carried out in [84], by Jensen, Sigmund and Thomsen.

5.4.3. Heterogeneous Steel-PMMA Specimen: 5.5 Cells

This experience adaptation consisted in testing a 5.5 unit cell periodic cylindrical bar with a uniform

diameter of 0.020m, composed of six steel and five PMMA half-cells each with a length of 0.135m.

The half-cells are bonded with a structural adhesive creating a specimen with a total length of 1.485m.

The experimental setup is illustrated by Figure 5.26 (a).

(a)

(b)

Figure 5.26 – Heterogeneous specimen: (a) Experimental setup of the 5.5 cells steel-PMMA bar specimen; (b)

Accelerometer positioning in the end extremity.

63

The positioning of the accelerometers is illustrated in Figure 5.26 (b) where the F.V. and F.H.

accelerometers capture the flexural modes in the vertical and horizontal directions, respectively. This

positioning is applied due to the acquired experimental experience from the previous tests where the

presence of non-longitudinal vibration modes was detected and their influence significantly noted,

especially when analyzing long bars as is this case.

Figure 5.27 – Frequency response curves for the 5.5 cells heterogeneous steel-PMMA bar specimen.

The flexural modes in the vertical and horizontal directions are illustrated in Figure 5.27, purple and

green lines, respectively, which significantly affect the longitudinal vibration modes after 1800 Hz.

In Figure 5.28 are presented the numerical, experimental and regenerated curves for the

longitudinal vibrations, where the blue, red and green lines represent the numerical, experimental and

regenerated curves, respectively.

Figure 5.28 – Frequency response curves for the 5.5 cells heterogeneous steel-

PMMA bar specimen, including the regenerated curve.

Attenuation Region:

After 1840 Hz

Figure 5.27 / Figure 5.28 illustrate the presence of an attenuation region after 1840 Hz.

In order to verify the decrease influence of the non-longitudinal vibration modes as well as the

decrease in attenuation’s amplitude, three identical tests were performed for the same structure,

64

involving the same materials but for different number of cells ( 4.5, 3.5 and 1.5 cells), see appendix

A.13. From these tests it was confirmed that the decrease in the number of cells correspond to a

decrease in the attenuation, as stated in section 2.9 and section 5.3.3. The experimental results agree

well with the FEM predictions, i.e., the filtering characteristics of these type of structures is

experimentally verified.

5.5. In the Search for Lower Frequency Ranges: Aggl omerated Cork

The next step consisted in developing specimens for a lower and more practical frequency range of

interest (see sections 5.3.3 and 5.3.4) which were tested here using other material. In this work, the

other materials used were brass and agglomerated cork. Since PMMA is no further used, a different

type of adhesive, Araldite® (see section 4.6) was used to bond the different material half-cells along

these new specimens.

5.5.1. Heterogeneous Steel-Commercial Agglomerated Cork: 3.5 Cells

This specimen is a 3.5 unit cell periodic cylindrical bar with a uniform diameter of 0.020m,

composed of four steel and three standard commercial agglomerated cork half-cells each with a length

of 0.020m. The half-cells are bonded with a structural adhesive creating a specimen with a total length

of 0.140m. The experimental setup is illustrated by Figure 5.29.

Figure 5.29 – Experimental setup of the 3.5 cells heterogeneous steel-commercial agglomerated cork bar

specimen.

The FRF curves are presented in Figure 5.30 where the blue, red, green and purple lines represent

the numerical experimental, regenerated and experimental F.H. (see Figure 5.26 (b)) results,

respectively.


commercial agglomerated cork bar specimen.

First natural frequency: f1 =351 Hz

Experimental E:

EAgl. cork=35 MPa

Attenuation Region:

After 980 Hz

65

5.5.2. Heterogeneous Brass-Commercial Agglomerated Cork: 3.5 Cells

This specimen is similar to the previous and is a 3.5 unit cell periodic cylindrical bar with a uniform

diameter of 0.020m, composed of four brass and three standard commercial agglomerated cork half-

cells each with a length of 0.020m. The half-cells are bonded with a structural adhesive creating a

specimen with a total length of 0.140m. The experimental setup is illustrated by Figure 5.31.

Figure 5.31 – Experimental setup of the 3.5 cells heterogeneous brass-commercial agglomerated cork bar

specimen.

The FRF curves are presented in Figure 5.32 where the blue, red and green lines represent the

numerical experimental and regenerated results, respectively.

Figure 5.32 – Frequency response curves for the 3.5 cells heterogeneous brass-

commercial agglomerated cork bar specimen.


Experimental E: EAgl. cork =30 MPa

Attenuation Region:

After 908 Hz

The advantage of using materials highly different in their properties was experimentally verified and

attenuation regions were obtained in a lower frequency range. Two new specimens were built with

agglomerated cork, referenced as 8123 in [87], with experimentally verified properties, namely the

modulus of elasticity, E, and the mass density, ρ.

5.5.3. Heterogeneous Steel-Agglomerated Cork Ref. 8123 bar with Rectangular Section

This specimen is a 3.5 unit cell periodic bar with square cross-section with a side length of 0.050m,

composed of four steel and three cork ref.8123 [87] half-cells each with a length of 0.010m and

0.050m, respectively. The half-cells are bonded with a structural adhesive creating a specimen with a

total length of 0.190m. The experimental setup is illustrated by Figure 5.33.

Figure 5.33 – Experimental setup of 3.5 cells heterogeneous steel-agglomerated cork Ref.8123 [87] (with 0.05m

cell length) square cross-section bar specimen.

66




agglomerated cork Ref.8123 [87] (with 0.05m cell length) square cross-section bar

specimen.


Experimental E:

EAgl. cork=43 MPa

Attenuation Region:

After 896 Hz

Trying to lower the attenuation frequency range (see sections 5.3.3 and 5.3.4) the size of the

agglomerated cork cells was increased.

This specimen is a 3.5 unit cell periodic bar a with square cross-section with a side length of

0.050m, composed of four steel and three cork ref.8123 [87] half-cells each with a length of 0.010m

and 0.100m, respectively. The half-cells are bonded with a structural adhesive creating a specimen

with a total length of 0.340m. The experimental setup is illustrated by Figure 5.35.

Figure 5.35 – Experimental setup of for the 3.5 cells heterogeneous steel-agglomerated cork Ref.8123 [87] (with

0.10m cell length) square cross-section bar specimen.





specimen.


Experimental E:

EAgl. cork=43 MPa

Attenuation Region:

After 665 Hz

67

5.6. Helicoidal Spring

Considerer a helicoidal spring with the following properties (see, Figure 5.37(a)): wire radius r=0.7mm; average coil radius R=6.1mm; helix angle α=9,23º; number of active coils N=5.8; and total

length L0 = 35.8mm. Four of these springs, presenting similar properties, are tested and their FRF

curves are presented in Figure 5.37(b) where the regenerated curve is an average result of the four

FRF curves.

(a)

(b)

Figure 5.37 – Experimental helicoidal spring: (a) Parameters; (b) Frequency response curves of four identical

springs.

5.6.1. Helicoidal Spring Combined with Heterogeneou s Periodic Bar

The next step consisted in combining the periodic bar and the helicoidal spring to observe the

advantage of combined structures, for which a numerical example illustrating the frequency response

curves of the combined structures is presented. The combination is assured by a structural adhesive.

The numerical frequency response curves are presented in Figure 5.38 where the red, blue and green

lines represent the helicoidal spring, the multilaminated periodic bar and the combined structures

numerical results, respectively.

Figure 5.38 – Numerical frequency response curves of the combined structure (example).

68

5.6.1.1. Heterogeneous Steel-PMMA Specimen Combined with Helicoidal Spring

This specimen is a combination of a similar experimental specimen analyzed in section 5.4.3 and

the helicoidal spring, see Figure 5.39, where the specimen is a 3.5 unit cell cylindrical periodic bar with

a uniform diameter of 0.020m, composed of five steel and four PMMA half-cells each with a length of

0.135m.

(a)

(b)

Figure 5.39 – Steel-PMMA specimen combined with the helicoidal spring: (a) Experimental setup; (b) Extremity

close-up.

The FRF curves are presented in Figure 5.40 where the red, blue and green lines represent the

helicoidal spring, the multilaminated periodic bar and the combined specimens’ experimental results,

respectively.


PMMA bar specimen combined with the helicoidal spring.

Attenuation Region:

After 1750 Hz

5.6.1.2. Heterogeneous Steel-Commercial Agglomerate d Cork Specimen Combined with

Helicoidal Spring

This specimen is a combination of the experimental specimen analyzed in section 5.5.1 and the

helicoidal spring, see Figure 5.41, where the specimen is a 3.5 unit cell cylindrical periodic bar with a

uniform diameter of 0.020m, composed of four steel and three standard commercial agglomerated

cork half-cells each with a length of 0.020m.

(a)

(b)

Figure 5.41 – Steel-Commercial Agglomerated cork specimen combined with the helicoidal spring: (a)

Experimental setup; (b) Extremity close-up.

69



respectively.


commercial agglomerated cork bar specimen combined with the helicoidal spring.

Attenuation Region:

After 980 Hz

5.6.1.3. Heterogeneous Brass-Commercial Agglomerate d Cork Specimen Combined with

Helicoidal Spring


helicoidal spring, see Figure 5.43, where the specimen is a 3.5 unit cell cylindrical periodic bar with a

uniform diameter of 0.020m, composed of four brass and three standard commercial agglomerated

cork half-cells each with a length of 0.020m.

(a)

(b)

Figure 5.43 – Brass-Commercial Agglomerated cork specimen combined with the helicoidal spring: (a)

Experimental setup; (b) Extremity close-up.



respectively.

Figure 5.44 – Frequency response curves for the 3.5 cells heterogeneous brass-

commercial agglomerated cork bar specimen combined with the helicoidal spring.

Attenuation Region:

After 908 Hz

70

5.6.1.4. Heterogeneous Steel-Agglomerated Cork Ref. 8123 bar with Rectangular Section

Combined with Helicoidal Spring


helicoidal spring, see Figure 5.45, where the specimen is a 3.5 unit cell periodic bar with a square

cross-section with a side length of 0.050m, composed of four steel and three cork ref.8123 [87] half-

cells each with a length of 0.010m and 0.050m, respectively.

(a)

(b)

Figure 5.45 – Steel- Agglomerated cork ref.8123 [87] specimen (0.05m cell length), with a square cross-section,

combined with the helicoidal spring: (a) Experimental setup; (b) Extremity close-up.



respectively.



specimen combined with the helicoidal spring.

Attenuation Region:

After 896 Hz

This structure identical to the previous and is a combination of the specimen tested in section 5.5.3

and the helicoidal spring, see Figure 5.47, where the specimen is a 3.5 unit cell periodic bar with a

square cross-section with a side length of 0.050m, composed of four steel and three agglomerated

cork ref.8123 [87] half-cells each with a length of 0.010m and 0.100m, respectively.

(a)

(b)

Figure 5.47 – Steel-Agglomerated cork ref.8123 [87] specimen (0.10m cell length), with a square cross-section,

combined with the helicoidal spring: (a) Experimental setup; (b) Extremity close-up.

71



respectively.



specimen combined with the helicoidal spring.

Attenuation Region:

After 665 Hz

A result summary of the initial frequency in the attenuation regions of each specimen, including the

combined specimens, is presented in the following table.

Table 5.8 – Specimen type, length and initial frequency of the attenuation region.

Specimen length [m] Attenuation region: initial frequency [Hz]

Steel-PMMA 1.485 1840

Steel-Com. Agglomerated Cork 0.140 980

Brass-Com. Agglomerated Cork 0.140 908

Steel-Agl. Cork ref.8123 (0.050m) 0.190 896

Steel- Agl. Cork ref.8123(0.100m) 0.340 665

The filtering characteristics of these type of structures was experimentally verified. Relatively to the

displacement amplitude, some discrepancy is noted at higher frequencies, above 2.5 kHz for the

homogeneous steel specimen and 2 kHz for the homogeneous PMMA specimen for which case above

4 kHz the experimental results were not suitable for comparison. This is probably due to the fact that

the accelerometers sensibility is affected by the use of mounting clips (see, A.14 in appendix) and

consequently the acceptable frequency range is reduced from 5 kHz to 2 kHz.

The increased dissimilarity between the properties of the materials used (Steel or brass-Com.

Agglomerated Cork) and/or the increase of the cell’s length (Steel-Agl. Cork ref.8123), show a

tendency to decrease the initial frequency of the attenuation regions to lower and more practical

frequency ranges of interest. From the experimental results obtained (see Figure 5.40, Figure 5.42,

Figure 5.44, Figure 5.46 and Figure 5.48) it was possible to verify that the filtering characteristic was

present in the combined specimens.

72

5.6.2. Compressor Motor Base Support Simulating Str ucture

Intending to generalize the developed specimens for industrial application, a more complex

specimen was built and tested. This structure intends to simulate the base of an air-tight compressor

motor. The response of the structure is tested for three different suspension situations using: helicoidal

springs; multilaminated specimens (steel-commercial agglomerated cork, see section 5.5.1; and both

combined.

(a)

(b)

(c)

Figure 5.49 – Experimental setup compressor motor base simulating structure with specimens: (a) Helicoidal

springs; (b) Multilaminated bars; (c) Springs + bars.

Figure 5.50 – Frequency response curves of the compressor motor base simulating structure with different

specimens.

From the experimental results obtained, see Figure 5.50, it is possible to verify that the helicoidal

spring structure by itself (red curve) is advantageous relatively the multilaminated structure (green

curve) presenting lower displacements amplitude due to the higher stiffness of the bars. However,

when the springs or bars are substituted with the combined structure (purple curve) it is verified that

the attenuation amplitude increases and in general present higher amplitude attenuation relatively to

the other configurations. Future development on this structure is needed, e.g. structural changes such

as thicker plates, to achieve the full model verification and validation [72] and [74].

73

5.7. Optimization Results

Here, a given bar was optimized for the formulation of sections 2.10 and 3.7 (to maximize the

separation of two adjacent eigenfrequencies). For all the presented results, it was assumed that

the bar has a uniform diameter of 0.01m, a total length of Lt=0.036m and the two material

properties are: EMat.1= 205.0GPa, ρMat.1=7860kg/m3 and EMat.2= 0.003GPa, ρMat.2=1140kg/m3. Bar

finite elements (element LINK1 in Ansys®) were used, observing that the number of finite

elements should be adjusted taking in account the highest frequency in analysis and the finite

element length. Finite element analysis were performed at Ansys® with no transversal

displacements, Uy=0 for all nodes, and Ux=0 at x:Lt (see Figure 5.17).

5.7.1. Results from the Maximization of the Separat ion of Two Adjacent Eigenfrequencies ωi+1

and ωi for a Periodic Material Distribution (Periodic Cas e)

The initial design consisted in the design variables d2 being Lt/2n. Improvement in the

separation of two adjacent frequencies (i and iA1), relatively to the initial design, i.e. from initial

separation ωini to optimized final value ωfin were obtained as shown by the values in Figure 5.51a),

for n=3. The dispersion curves (see section 2.9.4) obtained are presented in Figure 5.51 b). The

frequency response of displacement at node x:SL-SL/nTT the node that separates left variables

from the last node is presented at Figure 5.51 c) for an applied axial harmonic force with 200N

magnitude at x=0 m. The attenuation bands are identified (by the gray areas) to relate it with the

stop-bands given by the dispersion curves.

a) b) c)

Figure 5.51 – Results for n=3 cells: a) Eigenfrequencies curve plot; b) Dispersion curves and c) Frequency response plot.

Table 5.9 – Output results from the optimization cases with i = 3.

dddd2222 [m] ΔΔΔΔ ωωωωiiii [Hz]

Initial 6.00 × 10-3 3.44× 103

Optimized 1.20 × 10-3 20.16× 103

These separations of adjacent eigenvalues are evident when comparing frequency response

curves from initial design (see Figure 5.51 c)) with corresponding curves of optimized designs for i=3.

74

5.7.2. Results from the Maximization of the Separat ion of Two Adjacent Eigenfrequencies ωi+1

and ωi by a Distribution of Two Different Materials in Ea ch Design Variable (General Case)

The number of design variables tj used was eleven, all with the same length. The initial design

consisted in all design variables tj being 0.5, as indicated by a blue line at the Figure 5.52 c) and

Figure 5.53 c). Using the optimization procedure described in Section 3.7, the optimized designs

were obtained for a specific i value and presented at mentioned figures, the red line in Figure 5.52

c) and Figure 5.53 c), where in the first it indicates a sequence of materials: Mat.2/Mat.1/

Mat.2/Mat.1. Improvement in the separation of two adjacent frequencies (ωi and ωiA1), relatively to

the initial design were obtained as shown by the values in Figure 5.52 a) and Figure 5.53 a), for i=1 and 5, respectively. The frequency response of displacement at node x=(L-(L/11)) at Figure

5.52 b) and Figure 5.53 b) for an applied axial harmonic force with 200N magnitude at x=0 m.

a) b) c)

Figure 5.52 – Results for i=1 : a) Eigenfrequencies curve plot ; b) Frequency response plot and c) Material

distribution in the bar.

a) b) c)

Figure 5.53 – Results for i=5 : a) Eigenfrequencies curve plot ; b) Frequency response plot and c) Material

distribution in the bar.

Table 5.10 – Output results from the optimization cases with i = 1 and 5.

These separations of adjacent eigenvalues are evident when comparing frequency response

curves from initial design (see Figure 5.52 b) and Figure 5.53 b)) with corresponding curves of

optimized designs for i=1 and 5, respectively.

iiii 1 5 ΔΔΔΔ initial [Hz] 0.507× 103 0.512× 103 Δ Δ Δ Δ optimized [Hz] 2.40× 103 6.32× 103

75

6. Conclusions

A study on the dynamical characterization of passive attenuation of longitudinal vibrations through

helicoidal springs and periodic bars is presented in this thesis. For it, analytical, numerical and

experimental work was done to verify and in a certain sense to validate the models used. It required

the use of finite element method and the construction of experimental specimens. The main

conclusions achieved are:

• It is possible to design, develop and build specimens for reasonable low frequency ranges with

interest for mechanical engineering applications at cost of lowering the stiffness. One material

explored is agglomerated cork for which is showed that a linear analysis is enough to

characterize its dynamical behavior in the tested conditions.

• It is possible to have a good characterization of the wide attenuation regions in frequency from

Modal analysis for the first natural frequencies (not mentioned in the researched literature).

• An intuitive relation (not mentioned in the researched literature) to analyze wave propagation in

simple finite structures, similar to the ones presented is achieved based on the modal analysis

(finite repetitive structure) and the Bloch wave analysis (infinite repetitive structure).

• Good correlation for the applications of interest is obtained from the analytical Bloch wave

analysis applied to infinite structures which present stop-band regions (described in section

5.1.3), from finite element modal analysis and from finite element harmonic analysis applied to

finite structures which present attenuation regions (described in sections 5.3.2 and 5.3.3).

Some future work involving these types of structures contemplating damping and the use of

viscoelastic materials such as urethane is of immediate benefit in the increase of the amplitude of the

attenuation and displacement regions. The effect of the temperature and humidity in experimental

analysis should be accounted for when susceptible materials are used (as in this work with the use of

agglomerated cork). Experimental testes of these types of structures in real application are needed.

76

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80

APPENDIX

81

A.1 – Type of Spring Ends

The four types of ends generally used in helicoidal springs are illustrated in Figure 2.11, which is

here reproduced for easier reference. Table A.1 shows how the type of end used affects the number

of coils and the spring length.

Table A.1 – Expressions for helicoidal spring dimensions [76].

Type of Spring Ends

Plain Plain and Ground

Squared or Closed

Squared and Ground

End coils, NNNNeeee 0 1 2 2

Total coils, NNNNtttt NNNNaaaa NNNNaaaa+1 NNNNaaaa+2 NNNNaaaa+2

Free lenght, LLLL0000 p.Np.Np.Np.Naaaa. )θcos( +d pSNpSNpSNpSNaaaa. )θcos( +1) p.Np.Np.Np.Naaaa. )θcos( +3dddd p.Np.Np.Np.Naaaa.. )θcos( +2dddd

Solid lenght, LLLLssss dSNdSNdSNdSNtttt+1) d.Nd.Nd.Nd.Ntttt dSNdSNdSNdSNtttt+1) d.Nd.Nd.Nd.Ntttt Pitch, pppp

)θcos( N

d-L

a

0 1)cos((N

L

a

0

)+ θ

cos(N

3d-L

a

0

)θ

cos(N

2d-L

a

0

)θ

Figure A. 2 –

Figure A. 3 –

(a) (b) (c) (d)

Figure A.1 – Reproduction of Figure 6.1: Type of ends for helicoidal springs: (a) Open, plain end; (b) Open,

squared; (c) Closed, plain end; (d) Closed, squared [76].

82

A.2 – Springs End Conditions

α is the end condition constant used to describe the condition for absolute stability given by

Eq.(66).

Table A.2 – End condition constants α for helicoidal compression springs [76].

End Condition Constant αααα

Spring supported between flat parallel surfaces (fixed ends) 0.5

One end supported by flat surface perpendicular to spring axis (fixed);

other end pivoted (hinged) 0.707

Both ends pivoted (hinged) 1

One end clamped; other end free 2

A.3 – Column Effective Length

Le is the effective length factor of the column, whose value depends on the conditions of end support

of the column. Used in Eq.(73) for the stability of columns.

Table A.3 – Effective length of column for various end conditions [76].

Fixed-Free Pined-Pined Fixed-Pined Fixed-Fixed

LLLLeeee 2 L L 0.7 L 0.5 L

A.4 – Optimization Technique

The implemented optimization technique combines the commercial software Ansys®, for the finite

element analysis, optimization algorithms that run in MatLab® environment. See section 3.7.

Figure A. 4 – Diagram of the implemented optimization technique.

83

A.5 – Calibration Chart: Brüel & Kjaer Type 4508-B

Calibration chart of the accelerometer used in the experimental analysis (see section 4.1.2.1) in a

frequency range between 0 and 3 kHz.

Figure A. 5 – Calibration chart supplied with the accelerometer: Brüel & Kjaer Type 4508 [93].

Table A.4

A.6 – Calibration Chart: Brüel & Kjaer Type 8200

Calibration chart of the force transducer used in the experimental analysis (see section 4.1.2.2) in a

frequency range between 0 and 3 kHz.

Figure A. 6 – Calibration chart supplied with the force transducer: Brüel & Kjaer Type 8200 [94].

Table A.5

Table A.6

84

A.7 – FEA Result Comparison with [15]

In Table A.7, the FEM results obtained, solving Eq.(18), for the first eight eigenmodes, for three

different boundary conditions (B.C.), are presented and compared with those of [16],

Table A.7– Results obtained via FEM, for the first eight eigenmodes, compared with those of [16].

B.C. Fixed - Fixed Free - Free Fixed - Free

Modes f [Hz] f [Hz] [16] er f [Hz] f [Hz] [16] er f [Hz] f [Hz] [16] er

1 41.040 41.105 0.16% 42.012 42.087 0.18% 9.4744 9.4777 0.03%

2 45.160 45.212 0.12% 43.301 43.386 0.20% 9.5032 9.5054 0.02%

3 46.986 47.049 0.13% 44.086 44.184 0.22% 21.381 21.429 0.22%

4 47.713 47.785 0.15% 49.357 49.445 0.18% 24.207 24.194 0.05%

5 81.187 81.325 0.17% 81.275 81.415 0.17% 42.152 42.182 0.07%

6 89.537 89.787 0.27% 86.723 86.936 0.25% 42.908 42.942 0.08%

7 91.627 91.774 0.16% 88.309 88.499 0.21% 63.186 63.310 0.20%

8 93.297 93.352 0.06% 94.891 95.072 0.19% 71.317 71.281 0.05%

A.8 – FEA Result Comparison with [22]

In [27], others numerical methods are applied; the transference matrix method, the dynamic

stiffness method and Banerjee and Williams method for purely extensional/torsional modes. These

results are presented in the following tables (Table A.8.1, Table A.8.2 and Table A.8.3) for three

different boundary conditions and compared with the ones obtain by the implemented FEM model.

Table A.8.1 – Natural frequencies in Hz of a spring with fixed – fixed boundary conditions; E=2.09·1011 N/m2;

υ=0.28; ρ=7800kg/m3; r=6mm; R=65mm; α=7.44º; N=6; and L0 = 320mm; all modes below 100 Hz are presented.

Modes

FEM FEM

Transfer

stifness

method

Dynamic stiffness

method

Pure

extension/torsion

Principal

Motion

f [Hz] f [Hz]

[22] er

f [Hz]

[22] er

f [Hz]

[22] er

f [Hz]

[22] er

1 41.040 41.105 0.16% 40.99 0.12% 40.994 0.11% 42.725 3.94% Axial

2 45.160 45.212 0.12% 45.13 0.07% 45.135 0.06% - - Lateral

3 46.986 47.049 0.13% 46.95 0.08% 46.951 0.07% - - Lateral

4 47.713 47.785 0.15% 47.72 0.01% 47.726 0.03% 48.348 1.31% Torsional

5 81.187 81.325 0.17% 81.09 0.12% 81.091 0.12% 85.449 4.99% Axial

6 89.537 89.787 0.27% 88.97 0.08% 88.976 0.07% - - Lateral

7 91.627 91.774 0.16% 91.59 0.04% 91.586 0.04% - - Lateral

8 93.297 93.352 0.06% 93.18 0.13% 93.173 0.13% 96.676 3.50% Torsional

85

Table A.8.2 – Natural frequencies in Hz of a spring with free – free boundary conditions; E=2.09·1011 N/m2; υ=0.28; ρ=7800kg/m3; r=6mm; R=65mm; α=7.44º; N=6; and L0 = 320mm; all modes below 100 Hz are presented.

Modes

FEM FEM

Transfer

stifness

method

Dynamic stiffness

method

Pure

extension/torsion

Principal

Motion

f [Hz] f [Hz]

[22] er

f [Hz]

[22] er

f [Hz]

[22] f [Hz]

f [Hz]

[22] er

1 42.012 42.087 0.18% 41.96 0.12% 41.962 0.12% 42.725 1.67% Axial

2 43.301 43.386 0.20% 43.32 0.04% 43.309 0.02% - - Lateral

3 44.086 44.184 0.22% 44.11 0.05% 44.106 0.05% - - Lateral

4 49.357 49.445 0.18% 49.38 0.05% 49.384 0.05% 48.348 2.09% Torsional

5 81.275 81.415 0.17% 81.18 0.12% 81.178 0.12% 85.449 4.88% Axial

6 89.537 89.787 0.27% 86.72 0.00% 86.721 0.00% - - Lateral

7 88.309 88.499 0.21% 88.30 0.01% 88.303 0.01% - - Lateral

8 94.891 95.072 0.19% 94.93 0.04% 94.927 0.04% 96.676 1.85% Torsional

Table A.8.3 – Natural frequencies in Hz of a spring with fixed – free boundary conditions; E=2.09·1011 N/m2; υ=0.28; ρ=7800kg/m3; r=6mm; R=65mm; α=7.44º; N=6; and L0 = 320mm; all modes below 100 Hz are presented.

Modes

FEM FEM

Transfer

stifness

method

Dynamic stiffness

method

Pure

extension/torsion

Principal

Motion

f [Hz] f [Hz]

[22] er

f [Hz]

[22] er

f [Hz]

[22] f [Hz]

f [Hz]

[22]

1 9.4744 9.4777 0.03% 9.472 0.03% 9.4719 0.03% - - Lateral

2 9.5032 9.5054 0.02% 9.50 0.03% 9.4998 0.04% - - Lateral

3 21.381 21.429 0.22% 21.36 0.10% 21.359 0.10% 21.362 0.09% Axial

4 24.207 24.194 0.05% 24.17 0.15% 24.170 0.15% 24.169 0.16% Torsional

5 42.152 42.182 0.07% 42.10 0.12% 42.101 0.12% - - Lateral

6 42.908 42.942 0.08% 42.86 0.11% 42.857 0.12% - - Lateral

7 63.186 63.310 0.20% 63.11 0.12% 63.109 0.12% 64.087 1.41% Axial

8 71.317 71.281 0.05% 71.20 0.16% 71.205 0.16% 72.507 1.64% Torsional

9 88.344 88.435 0.10% 88.22 0.14% 88.227 0.13% - - Lateral

10 90.015 90.095 0.09% 89.89 0.14% 89.893 0.14% - - Lateral

86

A.9 – Effect of the Helix Angle

Results obtained from the analysis on how the parameters of the spring affect the natural

frequencies, presented in section 5.2.1.

Table A.9– Effect of the helix angle.

ωωωω[Hz] 1 2 3 4 5 6 7 8 9 10

5º 472.19 520.66 550.66 565.78 928.63 992.73 1052.40 1096.70 1342.00 1459.60

8.57º 403.06 405.51 468.66 533.10 881.68 898.24 921.53 1051.50 1336.60 1374.90

10º 360.45 362.35 466.35 530.26 819.68 831.25 915.69 1043.60 1310.50 1329.50

15º 245.21 255.56 454.92 519.46 627.19 636.68 887.76 1021.50 1074.70 1099.10

20º 189.15 190.23 434.21 486.18 495.39 512.72 827.06 869.09 918.89 998.06

25º 145.48 146.32 375.32 382.59 438.58 490.02 696.00 702.83 845.40 963.00

30º 113.93 114.59 302.10 303.84 412.45 468.53 567.22 568.58 794.06 872.03

Figure A. 7

A.10 – Effect of the Number of Active Coils



Table A.10– Effect of the number of active coils.

ω [Hz] 1 2 3 4 5 6 7 8 9 10

n=4 844.33 975.56 1030.70 1082.50 1590.30 1772.40 1812.60 1936.60 2088.30 2268.40

n=6 547.50 578.05 620.96 660.58 1105.40 1188.70 1242.50 1279.30 1591.50 1760.70

n=7.6 403.06 405.51 468.66 533.10 881.68 898.24 921.53 1051.50 1336.60 1374.90

n=8 359.53 363.03 445.42 497.90 800.60 814.87 886.30 976.12 1234.70 1299.50

n=10 246.16 246.87 353.84 399.30 581.23 584.08 702.38 787.69 964.95 975.96

n=12 177.64 177.82 294.26 333.18 434.86 435.79 584.97 659.18 748.85 752.23

Figure A. 8

Figure A. 9

Figure A. 10

Figure A. 11

87

A.11 – Effect of the Diameter Relation



Table A.11– Effect of the diameter relation D/d.

ω [Hz] 1 2 3 4 5 6 7 8 9 10

D/d=4 2512.00 2527.40 2928.20 3310.70 5482.00 5585.10 5756.30 6532.10 8343.30 8542.80

D/d=6 1118.50 1125.40 1301.70 1477.60 2444.80 2490.70 2559.30 2914.80 3711.30 3811.50

D/d=8 629.59 633.43 732.26 832.40 1376.90 1402.70 1439.80 1641.90 2088.20 2146.90

D/d=10 403.06 405.51 468.66 533.10 881.68 898.24 921.53 1051.50 1336.60 1374.90

D/d=12 279.95 281.65 325.47 370.35 612.46 623.97 639.97 730.47 928.26 955.14

D/d=14 205.70 206.95 239.12 272.15 450.06 458.51 470.19 539.79 682.01 701.89

D/d=16 157.50 158.46 183.08 208.40 344.51 351.09 360.00 411.04 522.18 537.46

A.12 – Convergence Study for the Necessary Number o r F.E. per Wave Length

Numerical analysis were performed for the homogeneous specimens presented in section 5.1.2 ,

obtaining the figures bellow, and results compared with those obtained experimentally. The red curve

represents the experimental results while the blue, green and purple curves represent the numerical

results obtained using 8, 16 and 32 finite elements per wave length, respectively.

Figure A.12.1 – Frequency response of homogeneous steel experimental specimen with 8, 16 and 32 finite

elements per wave length.

Figure A.12.2 – Frequency response of homogeneous PMMA experimental specimen with 8, 16 and 32 finite

elements per wave length.

88

A.13 – Continuation of Experience Adaptation

In order to verify the decrease influence of the non-longitudinal vibration modes as well as the

decrease in attenuation’s amplitude, three similar tests to those presented in section 5.4.3 were

performed for a similar structure, involving the same materials but for different number of cells, n, (n=4.5, 3.5 and 1.5 cells).


composed of five steel and four PMMA half-cells each with a length of 0.135m. The half-cells are

bonded with a structural adhesive creating a specimen with a total length of 1.215m. The experimental

setup is illustrated by Figure A.13.1.

Figure A.13.1 – Experimental setup of heterogeneous specimen of the 4.5 cells steel-PMMA bar.

The FRF curves are presented in Figure A.13.2 where the blue, red and green lines represent the


Figure A.13.2 – Frequency response curves for the 4.5 cells heterogeneous steel-

PMMA bar specimen.

Attenuation Region:

After 1800 Hz

The experimental results present identical longitudinal natural frequencies to those obtained

numerically. The filtering characteristics of these type of specimens are verified experimentally,

meaning that after the fourth natural frequency at approximately 1800 Hz the longitudinal waves are

attenuated.


composed of four steel and three PMMA half-cells each with a length of 0.135m. The half-cells are



Figure A.13.3 – Frequency response curves for the 3.5 cells heterogeneous steel-PMMA bar specimen.

89

The FRF curves are presented in Figure A.13.4. where the blue, red and green lines represent the



PMMA bar specimen.

Attenuation Region:

After 1750 Hz



meaning that after the third natural frequency at approximately 1750 Hz the longitudinal waves are

attenuated.


composed of two steel and one PMMA half-cells each with a length of 0.135m. The half-cells are



Figure A.13.5 – Frequency response curves for the 1.5 cells heterogeneous steel-PMMA bar specimen.

The FRF curves are presented in Figure A.13.6 where the blue, red and green lines represent the



PMMA bar specimen.

Attenuation Region:

After 1300 Hz

90



meaning that after the first natural frequency at approximately 1300 Hz the longitudinal waves are

attenuated.

A.14 – Accelerometer Mounting Clip

This information was taken from the product data sheet of Brüel & Kjaer Type 4508-B accelerometer.

analytical, numerical and experimental study of the ......tight compressors and helicoidal springs....

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