introduction to vibration and the free...

14
1 Jiraphon Srisertpol, Ph.D School of Mechanical Engineering 2 Recommended reading : ... ก : ก, Pearson Education Indochina 2545 Singiresu S.Rao : Mechanical Vibration (Fourth Edition) ,Prentice Hall 2004. SI Edition Leonard Meirovitch : Fundamentals of Vibrations , Mc-Graw Hill 2001. Kelly S. Graham : Fundamentals of Mechanical Vibrations, Mc-Graw Hill 2000. 3 Introduction to Vibration and The Introduction to Vibration and The Introduction to Vibration and The Introduction to Vibration and The Free Response Free Response Free Response Free Response The Spring-Mass model Single –degree of freedom Simple harmonic motion Relationship between Displacement, Velocity and Acceleration Representations of harmonic motion 4 School of Mechanical Engineering 4

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Page 1: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

1

Jiraphon Srisertpol, Ph.D

School of Mechanical Engineering

22

Recommended reading :� ��.��.��. ก����� �������������: ก���� �!�"ก#, Pearson Education

Indochina 2545� Singiresu S.Rao : Mechanical Vibration (Fourth Edition) ,Prentice Hall

2004. SI Edition� Leonard Meirovitch : Fundamentals of Vibrations , Mc-Graw Hill 2001.� Kelly S. Graham : Fundamentals of Mechanical Vibrations,

Mc-Graw Hill 2000.

3

Introduction to Vibration and The Introduction to Vibration and The Introduction to Vibration and The Introduction to Vibration and The Free ResponseFree ResponseFree ResponseFree Response� The Spring-Mass model

� Single –degree of freedom

� Simple harmonic motion

� Relationship between Displacement, Velocity and Acceleration

� Representations of harmonic motion

4School of Mechanical Engineering 4

Page 2: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

5School of Mechanical Engineering 5 6

The Vibration of a Fixed-Fixed String

7

The main mass and dynamic absorber at three

frequencies.

8

Vibration Characteristics of a Vibration Characteristics of a Vibration Characteristics of a Vibration Characteristics of a SpringSpringSpringSpring

Page 3: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

9

Fundamental Fundamental Fundamental Fundamental TorsionalTorsionalTorsionalTorsional Mode of a Mode of a Mode of a Mode of a Valve Support Stand Valve Support Stand Valve Support Stand Valve Support Stand

10

Deflected Deflected Deflected Deflected ElastomerElastomerElastomerElastomer Shock Shock Shock Shock Isolation Isolation Isolation Isolation

11

Structural VibrationStructural VibrationStructural VibrationStructural Vibration

12

����� �!"#$%&'ก�)*+ก,�!)-.&'%&'ก�)/0.$1�'�2*�ก))�

� ���������ก ���������ก�������������ก���� ��������� !��ก���ก�����������"" #�$�%�������� &����'�()���������*+#�+� �,��)-�.�������ก����ก.""ก����"���ก������ .�ก���/��"ก������� ก��.ก�)01#�ก�������*��ก�/��2�ก�"%�������� #�$���""

Page 4: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

13

���������� ก��������ก��ก������������ก���� #����BC�D�EF�GH�I������ (Degree of Freedom, DOF) - L���F�

M�ก��(Coordinate) !O �P�Q!O �R�!O SGTCUD��T�ก��VU "L��H�I��P�"W�P���B�Q���XY�T"�T�"Z C�"!Rก�TF�W���BB!O C[�HF#�Y�U "

14

���������� ก��������ก��ก������������ก���

� Single degree of freedom system

15

Two degree of freedom systemTwo degree of freedom systemTwo degree of freedom systemTwo degree of freedom system

16

Three degree of freedom systemThree degree of freedom systemThree degree of freedom systemThree degree of freedom system

Page 5: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

17

Infinite number of degree of freedom Infinite number of degree of freedom Infinite number of degree of freedom Infinite number of degree of freedom systemsystemsystemsystem

18

���������� ก��������ก��ก������������ก���

� Discrete System (Lumped System)-��""�*�������ก'�#�/ก����$���(#�(/�/��+'�/�"��2�����������*��'�ก�/���#����

� Continuous System (Distributed System)- ��""�*��*'�/�"��2������)-������(���'�ก�/

19

ก��������ก���ก������

� ก������.""����� (Free Vibration)

� ก������."""����" (Forced Vibration)

� ก������.""(���*����#���� (Undamped Vibration)

� ก������.""�*����#���� (Damped Vibration)

� ก������.""�,������ (Linear Vibration)

� ก������.""(���,������ (Nonlinear Vibration)

� ก������.""ก'�#�/(/� (Deterministic Vibration)

� ก������.""���� (Random Vibration)

20

ก�������������� (Free Vibration)� �$�ก�����������""���ก ��*�#����ก�*ก���"ก����""�*�#+�/�����+���*���/��/����2�.�ก� $���'��#��ก�/ก��������2�.�� ก��������2�/'��������()%/+(���*.����ก3�+��ก��ก���'�ก�"��""�*ก�+

� ก���"ก����""������)-�ก���'��#��ก�/ก�����/ #�$��'��#��ก�/������4���������#�$���2����.""���ก��

Page 6: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

21

ก���� �XBBB�"E�B (Forced Vibration)� �$�ก�����������""3�+���.��ก���'���ก3�+��ก &���.��ก���'���ก3�+��ก�*2

������)-�.�����ก �&2'�#�$�(��&2'�������ก4(/�� ก���������ก ��*2 �,�� ก��������$�����ก����(����/�������$�����ก��*��ก�/ก��

#���� ��������*����ก �.""�*2() ���ก�"�����*�5���,��������"" ก�������*2���*

�ก ��*��*,���ก���� (amplitude) ก�������*������ก �����*+กก�������ก ��*2��� ก������ ��� (Resonance)

22

ก���� �XBBSGTGOEF�GY�TF" (Undamped Vibration)

� #��+���ก�������*�(���*ก����1��*+ ������#�ก�"����.�/��������"" (��������+������).����*+/��� #�$�.�������$���/

� ��$����""��$����*�.""(���*����#�������'��#� �������������""����#����ก����$����*��*2�*������*�

� ก�������*�(���*����#���������""�������ก�/��2�(/�����ก��������2�� ก������.""(���*����#����.�ก������.""����������*������""����*+ก���

�����*�5���,��� (Natural Frequency)

23

ก���� �XBBGOEF�GY�TF" (Damped Vibration)

� #��+���ก�������*��*ก����1��*+ ���������#�����ก�/ก����$����*������"" (�����/��+���#���/ก4���

� %/+����().��ก����������3� �����)-�������2����)-�ก������.""�*����#����.�"��2���2�

24

ก��������� ��� ��� (Linear Vibration)� ��""���ก������&���)��ก�"/��+ �� �)��� .����#���� �* !��ก����)*�+�.)�������.��ก���'�()�+����,������ก�"��+����#�$�������4���� �ก�/�*��,�

� ก������.""�,����������������,� #�ก���ก������'�.#���(Principle of Superposition)

Page 7: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

25

ก���� �XBBSGTH�I�H��"H�P� (Nonlinear Vibration)

� ��""���ก������&���)��ก�"/��+ �� �)��� .����#���� �* !��ก����)*�+�.)�������.��ก���'�()�+���(���)-��,������ก�"��+����#�$�������4���� �ก�/�*��,�

26

ก���� �XBBก��Y��S�P (Deterministic Vibration)

� ��""�*��ก�/ก�������+��3�+���.��ก���'���ก3�+��ก�ก ��/ก4��� ���.���*�ก���'��+����2�������ก'�#�/���/���.��(/�#�$����"���������� ��56���.�� &����)-�70�ก6,����������*�.����2�ก���'�

27

ก���� �XBB�RTG (Random Vibration)

� ��""�*��ก�/ก�������+��3�+���.��ก���'���ก3�+��ก�ก ��/ก4��� ���.���*�ก���'������""(��������ก'�#�/���/���.��(/�

28

C�D����ก��F�HE���Y�ก���� �!�"F��Fก��G (Vibration Analysis Procedure)� ก�������.""�'��������������6 (Mathematical Modeling)

� ก��#���ก��ก����$����*� (Derivation of Governing Equations)

� ก��#�8�9+��ก��ก����$����*� (Solution of Governing Equations)

� ก���������#68�*�(/� (Interpretation of the Results)

Page 8: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

29

ก����P�"XBBL��#�"!�"E[��������(Mathematical Modeling)� ก�������������:���"$2����������""�*��ก�/ก������ %/+.�������*��*����/��+.""�'������ก�+3� �,��

� .��.#������ �������ก+6��ก.""����""/��+�)���(spring )

� .�������*��'��#��ก�/ก����1��*+ �����/��+���#����(damped)

� .�������*��)-�.#������ �������6 /��+ ��(mass) , �����9$��+

30

ก��Y��Gก��ก��HE#̀ ��!O (Derivation of Governing Equations)� ก��.ก���ก����� ��56� ก���)*�+���)����)�& (Laplace)s transform)

� ��5*�,������� (Numerical method)

Dynamic System Modeling and Analysis, Hung V Vu and Ramin S. Esfandiari,McGraw-Hill 1998

31

ก��Y�b#Hc#Q�Gก��ก��HE#̀ ��!O (Solution of Governing Equations)

� .����ก��ก����$����*������""����ก�".""�'����*��������2���&��������(/���#�+��5* �,��

� ก;ก����$����*����������� ก;ก�������ก 6 �����

32

ก��F�HE���Y�b#!O S�P(Interpretation of the Results)

� �������5�"�+.����)8� 56�*�(/� * ���/���.���*�ก���'�* ��+�ก����$���(#�* ������4�, ��������

Page 9: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

33

Vibration Analysis ProcedureVibration Analysis ProcedureVibration Analysis ProcedureVibration Analysis Procedure

34

Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle� ����������ก."�'����*����+�*���/ • Single-degree of freedom model �*�.�/�����) b.

{ } stiffness. equivalent,, −srteq kkkk

{ } constant. damping equivalent, −rseq ccc

{ } mass equivalent,, −wvreq mmmm

wheels, body, vehicle, struts tires,,rider −−−−− wvstr

35

Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle

wheels, body, vehicle, struts tires,,rider −−−−− wvstr

36

Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle

wheels, body, vehicle, struts tires,,rider −−−−− wvstr

Page 10: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

37

Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle

wheels, body, vehicle, struts tires,,rider −−−−− wvstr

38

����������ก������ ������ ��ก�����!�"#����$

� Springs Elements

� Mass or Inertia Elements

� Damping Elements

39

� Stiffness (N/m)

� Young’s modulus (N/m²)

� Density (kg/m³)

� Shear modulus G(N/m²)

� Springs in series

� Springs in parallel

40

Spring Elements

� �)���.����)ก�6�/� ����""�*��*��/����#��+�*��,�� $��ก������ �������ก+6�����""

� Spring force

kxF =

stiffness springor contant spring−k

tion)nt(deformadisplaceme−x

2

2

1 :spring in theenergy Potential kxU =

Page 11: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

41

Springs in Parallel

where

equation mEquilibriu

21

21

kkk

kW

kkW

eq

steq

stst

+=

=

+=

δ

δδ

( )

neq

eq

kkkkk

k

++++= ⋯321

parallelin constant spring Equivalent

42

Springs in Series( ) 21 system theof Static 1. δδδδ +=stst

22

11

equation mEquilibriu 2.

δ

δ

kW

kW

=

=

steqeq kWk δ= deflection static same for the .3

( )

neq

eq

kkkkk

k

11111

seriesin constant spring Equivalent

321

++++= ⋯

, or

22

11

2211

k

k

k

k

kkk

steqsteq

steq

δδ

δδ

δδδ

==

==

21

111 is, that

kkkeq

+=

43

EX: Springs in Parallel

cm 2ddiameter wire

cm 20Ddiameter coilmean

mN 1080G modulusshear 29

=

=

×=

The stiffness of helical spring is given by

( ) ( )( )

mN 000,4052.08

108002.0

8 3

94

3

4 ×==

nD

Gdk

The equivalent spring constant of the suspension system is given by

mN 120,000 000,4033 =×== kkeq

44

Damping Elements

� ���#������$�����ก����#�$/ (Viscous Damping)

� ���#������$�����ก.����*+/�����#�������.�4�ก�"���.�4� (Dry

Friction or Coulomb Damping)

� ���#������$�����ก����(��+$/#+���������/� (Hysteretic Damping or

Structural Damping)

Page 12: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

45

DampingDampingDampingDamping

46

Viscous DampingViscous DampingViscous DampingViscous DampingAll real systems dissipate energy when they vibrate. To account for this we must consider damping. The most simple form of damping (from a mathematical point of view) is called viscous damping. A viscous damper (or dashpot) produces a force that is proportional to velocity.

Damper (c)

( ) ( )cf cv t cx t= − = − ɺ x

fc

Mostly a mathematically motivated form, allowinga solution to the resulting equations of motion that predictsreasonable (observed) amounts of energy dissipation.

47

Viscous Damping

( )

neq

eq

cccc

c

1111

seriesin constant damping Equivalent

21

⋯++=

( )

321

parallelin constant damping Equivalent

cccc

c

eq

eq

+++= ⋯

48

� Damper

� Damping coefficient

� Critical damping coefficient

� Damping ratio

Page 13: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

49

� Underdamped Motion

� Overdamped Motion

� Critically Damped Motion

50

Ex: Horizontal milling machine

51

Ex: Horizontal milling machine

dampers theallon acting forces , springs theallon acting forces

dampers on the acting forces , springs on the acting forces , mass ofcenter

−−

−−−

ds

disi

FF

FFG

4,3,2,1 ;

4,3,2,1 ;

==

==

ixcF

ixkF

idi

isi

ɺ

4321

4321

ddddd

sssss

FFFFF

FFFFF

+++=

+++=

force verticaltotal where −=+ WFF ds

52

Ex: Horizontal milling machine

xcF

xkF

eqd

eqs

ɺ=

=

4321

4321

ccccc

kkkkk

eq

eq

+++=

+++=

Page 14: Introduction to Vibration and The Free Responseeng.sut.ac.th/me/meold/1_2552/425304/Mechanical_Vibrations-1[1].pdf · SingiresuS.Rao: Mechanical Vibration (Fourth Edition) ,Prentice

53

� Newton’s second law

� Conservation of Energy

� Potential Energy

� Kinetic Energy

� Natural frequency