introduction to vibration and the free...
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Jiraphon Srisertpol, Ph.D
School of Mechanical Engineering
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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.
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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
5School of Mechanical Engineering 5 6
The Vibration of a Fixed-Fixed String
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The main mass and dynamic absorber at three
frequencies.
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Vibration Characteristics of a Vibration Characteristics of a Vibration Characteristics of a Vibration Characteristics of a SpringSpringSpringSpring
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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
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Deflected Deflected Deflected Deflected ElastomerElastomerElastomerElastomer Shock Shock Shock Shock Isolation Isolation Isolation Isolation
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Structural VibrationStructural VibrationStructural VibrationStructural Vibration
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����� �!"#$%&'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�"%�������� #�$���""
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���������� ก��������ก��ก������������ก���� #����BC�D�EF�GH�I������ (Degree of Freedom, DOF) - L���F�
M�ก��(Coordinate) !O �P�Q!O �R�!O SGTCUD��T�ก��VU "L��H�I��P�"W�P���B�Q���XY�T"�T�"Z C�"!Rก�TF�W���BB!O C[�HF#�Y�U "
14
���������� ก��������ก��ก������������ก���
� Single degree of freedom system
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Two degree of freedom systemTwo degree of freedom systemTwo degree of freedom systemTwo degree of freedom system
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Three degree of freedom systemThree degree of freedom systemThree degree of freedom systemThree degree of freedom system
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Infinite number of degree of freedom Infinite number of degree of freedom Infinite number of degree of freedom Infinite number of degree of freedom systemsystemsystemsystem
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���������� ก��������ก��ก������������ก���
� Discrete System (Lumped System)-��""�*�������ก'�#�/ก����$���(#�(/�/��+'�/�"��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���������#�$���2����.""���ก��
21
ก���� �XBBB�"E�B (Forced Vibration)� �$�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)
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)
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����$���(#�* ������4�, ��������
33
Vibration Analysis ProcedureVibration Analysis ProcedureVibration Analysis ProcedureVibration Analysis Procedure
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Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle� ����������ก."�'����*����+�*���/ • Single-degree of freedom model �*�.�/�����) b.
{ } stiffness. equivalent,, −srteq kkkk
{ } constant. damping equivalent, −rseq ccc
{ } mass equivalent,, −wvreq mmmm
wheels, body, vehicle, struts tires,,rider −−−−− wvstr
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Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle
wheels, body, vehicle, struts tires,,rider −−−−− wvstr
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Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle
wheels, body, vehicle, struts tires,,rider −−−−− wvstr
37
Mathematical Model of Mathematical Model of Mathematical Model of Mathematical Model of MotorcycleMotorcycleMotorcycleMotorcycle
wheels, body, vehicle, struts tires,,rider −−−−− wvstr
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����������ก������ ������ ��ก�����!�"#����$
� Springs Elements
� Mass or Inertia Elements
� Damping Elements
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� Stiffness (N/m)
� Young’s modulus (N/m²)
� Density (kg/m³)
� Shear modulus G(N/m²)
� Springs in series
� Springs in parallel
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Spring Elements
� �)���.����)ก�6�/� ����""�*��*��/����#��+�*��,�� $��ก������ �������ก+6�����""
� Spring force
kxF =
stiffness springor contant spring−k
tion)nt(deformadisplaceme−x
2
2
1 :spring in theenergy Potential kxU =
41
Springs in Parallel
where
equation mEquilibriu
21
21
kkk
kW
kkW
eq
steq
stst
+=
=
+=
δ
δδ
( )
neq
eq
kkkkk
k
++++= ⋯321
parallelin constant spring Equivalent
42
Springs in Series( ) 21 system theof Static 1. δδδδ +=stst
22
11
equation mEquilibriu 2.
δ
δ
kW
kW
=
=
steqeq kWk δ= deflection static same for the .3
( )
neq
eq
kkkkk
k
11111
seriesin constant spring Equivalent
321
++++= ⋯
, or
22
11
2211
k
k
k
k
kkk
steqsteq
steq
δδ
δδ
δδδ
==
==
21
111 is, that
kkkeq
+=
43
EX: Springs in Parallel
cm 2ddiameter wire
cm 20Ddiameter coilmean
mN 1080G modulusshear 29
=
=
×=
The stiffness of helical spring is given by
( ) ( )( )
mN 000,4052.08
108002.0
8 3
94
3
4 ×==
nD
Gdk
The equivalent spring constant of the suspension system is given by
mN 120,000 000,4033 =×== kkeq
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Damping Elements
� ���#������$�����ก����#�$/ (Viscous Damping)
� ���#������$�����ก.����*+/�����#�������.�4�ก�"���.�4� (Dry
Friction or Coulomb Damping)
� ���#������$�����ก����(��+$/#+���������/� (Hysteretic Damping or
Structural Damping)
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DampingDampingDampingDamping
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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
+++= ⋯
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� Damper
� Damping coefficient
� Critical damping coefficient
� Damping ratio
49
� Underdamped Motion
� Overdamped Motion
� Critically Damped Motion
50
Ex: Horizontal milling machine
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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
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Ex: Horizontal milling machine
xcF
xkF
eqd
eqs
ɺ=
=
4321
4321
ccccc
kkkkk
eq
eq
+++=
+++=
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� Newton’s second law
� Conservation of Energy
� Potential Energy
� Kinetic Energy
� Natural frequency
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