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高瞻計畫_振動學課程

Lecture 10: Continuous

Vibration (II)

Prof. Kuo-Shen Chen

Department of Mechanical Engineering

National Cheng-Kung University

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Outline

Beam problem analysis

Membrane / Plate analysis

Approximation method

Simple problems

Finite Element Analysis

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Part I. Beam Problem Analysis

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Introduction Bending vibration of beam is the most seen

vibration in real applications

Tall buildings subject to earthquake

Motion induced vibrations of robotic arms

Wings of airplane

MEMS sensors

approximation methods provide effective ways to

estimate vibration properties

Natural frequencies

Rayleigh method, Rayleigh-Ritz method, Galerkin

method

Finite element analysis of continuous structures 4

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Equation of Motion (I)

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Equation of Motion (II)

If uniform/

homogeneous

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Natural Frequency / Mode Shape Separation of variables

where

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Natural Frequency / Mode Shape

where

Finally

And

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Typical Boundary Conditions

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Typical Boundary Conditions

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Flexural Bending Mode Shapes and

Boundary Conditions

Free-Free Beam

Clamped-Clamped Beam

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Flexural Bending Mode Shapes and

Boundary Conditions

Clamped-Free Beam

Simply Supported Beam

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Example Find the boundary conditions of the following system

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Mode Decomposition

Assume

Known mode

shapes

Unknown time-

dependent

weighting function

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Modal Decomposition

Convolution

solution

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Effect of Axial Tension

Beams with axial force: the natural frequencies will be changed

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Effect of Axial Tension

If homogeneous

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Effect of Axial Tension: Example

Natural frequency of a S-S beam subjected to

axial force

BCs

P >0 n ; P<0 n If P=2EI/l2 n =0

Critical load for buckling

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Timoshenko Beam

Thicker beam

Considering the shear

deformation and

rotary inertia effect

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Timoshenko Beam

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Timoshenko Beam

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Part II. Membrane/Plate Analysis

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Modeling of Membrane Vibration

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Modeling of Membrane Vibration

Membrane is the 2D

version of string

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Rectangular Membrane Vibration

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Vibrational Modes of a Rectangular

Membrane

(1,1) mode

(1,2) mode

(2,1) mode (2,2) mode 26

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Degenerate Modes for a Square Membrane

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Circular Membrane

(0,1) (0,2) (0,3)

(1,1) (1,2) (2,1)

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Plate Vibrations

29 Source Dr. Y-K Lee, HKUST

Boundary Conditions

30 Source Dr. Y-K Lee, HKUST

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Typical Plate Vibration Formula

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Steinberg, Vibration of Electronic Equipment, Wiley

Part III. Approximation Methods

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Rayleigh Method

Fundamental natural frequency estimation

Upper bound

Rayleigh Quotient

Equal max KE & PE X: assumed motion mode

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Rayleigh Method (Example)

Refer to the figure in previous page

assume

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Dunkerley Method

Fundamental natural frequency estimation

Lower bound

Starting from the characteristic equation

We have

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Dunkerley Method: Example mainly used for reconstruct the results from experimental data

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Rayleigh-Ritz Method

A basic extension of Rayleigh method

Motion is assumed to be composed by several

mode functions with different weighting

Minimize Rayleigh Quotient

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Rayleigh-Ritz Method

We have (n equations)

In matrix form

Solve 2 and mode shape weighting ratio

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Galerkin Method

Similar to RR method for finding approximated

solution of a continuous structure vibration

By minimizing the error in satisfying a differential

equation over the system range

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1

( ) ( )n

i i

j

z x c z x

( ) 2( ) ( ) ( ) 0ivL z EIz x z x

( ) ( ) 0 1,2,...L

i io

R z x L z dx i n

Assume

zi(x): assumed mode

ci: weighting

For beam L(z): operator

Weighted residual

N linear equations to find ci

Galerkin Method Example Find the first two natural frequencies of a fixed-fixed beam (length l,

mass/length , flexural rigidity EI)

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Trial modes

Weighted residual

Where

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Galerkin Method Example (Cont’d)

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2

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1

4

2

2

4

1

44

1

)8(

)2

(

ccR

ccR

0

82det

444

444

1 69.0 ,)(1.124

1 0.23 ,)(48.22

1

2/1

22

1

2/1

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cEI

l

cEI

l

l

x

l

xx

l

x

l

xx

4cos1)

2cos1(69.0)(

4cos1)

2cos1(23)(

2

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Formulas for Natural Frequency and

Mode Shape by R. D. Blevins

A handbook to list

natural frequency and

vibration modes of

elastic structures

Suitable for engineering

design

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Part IV. Simple Problems

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Problem 1. Natural Frequencies/Modes

of a Fixed-Pinned Beam (Rao. 8.7)

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Boundary Conditions:

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Problem 2. Forced Vibration of a

Simply Supported Beam (Rao 8.8)

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Problem 3. Vibration of a Railroad on

Elastic Foundation

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一個列車於軌道上運動 (速度 v0)所引發之振動問題. 我們可以將該題目簡化成 infinite beam on elastic foundation (with foundation (soil) modulus k)受到一個 moving loading的模型. 其運動方程式可以表示成

4 2

0 04 2( )

w wEI A kw F x v t

x t

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Part V. Flexible Beam-Hub Example

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Problem Statement

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Finite Element Frequency and Modal Dynamics

Analysis of the Hub-Beam Experiment

Modal Dynamics Analysis

Mode superposition method mentioned earlier

Hub

accelerometer (y2)

motor

accelerometer (y1) accelerometer (y

3)

4.5" 8.0"

Y1 Y

2Y

3

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FE MESHES

300 C3D20 elements; 45 C3D15 elements; 1500 nodes

modes

extration

modal

dynamics

linear dynamics w/

subspace projection

torque input

(pulse or

sinusoidal)

outputpower

spectrum

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Responses (II)

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Power Spectrum Density

PS

D

100

101

102

103

10-6

10-4

10-2

100

102

104

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POWER SPECTRUM DENSITY

Frequency Hz

Sensor Y3

Sensor Y2

Sensor Y1

PS

D

100

101

102

103

10-8

10-6

10-4

10-2

100

102

104

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POWER SPECTRUM DENSITY

Frequency Hz

Sensor Y3

Sensor Y2

Sensor Y1

Power spectrum density plots of system response. (a) without and (b) with input shaping

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Part VI. Youtube Demonstration

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