Mechanical_Vibration..

Jiraphon Srisertpol, Ph.D
School of Mechanical Engineering
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.
1
2
Introduction to Vibration and The
Free Response
The Spring-Mass model
Single –degree of freedom
Simple harmonic motion
Relationship between Displacement, Velocity
and Acceleration
Representations of harmonic motion
3
School of Mechanical Engineering
4
The Vibration of a Fixed-Fixed String
School of Mechanical Engineering
5
The main mass and dynamic absorber at three
frequencies.
6
Vibration Characteristics of a
Spring
7
8
Fundamental Torsional Mode of a
Valve Support Stand
Deflected Elastomer Shock
Isolation
9
Structural Vibration
10
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Three degree of freedom system
Two degree of freedom system
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Infinite number of degree of freedom
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Dynamic System Modeling and Analysis, Hung V Vu and Ramin S. Esfandiari,
McGraw-Hill 1998
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4, 32
Mathematical Model of
Motorcycle
Vibration Analysis Procedure
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keq {kt , kr , ks }− equivalentstiffness.
ceq {cs , cr }− equivalentdamping constant.
meq {mr , mv , mw }− equivalent mass
r − rider ,t − tires, s −struts , v − vehicle body, w − wheels,
33
Mathematical Model of
Motorcycle
34
Mathematical Model of
Motorcycle
r − rider ,t − tires, s −struts ,v − vehicle body, w − wheels,
r − rider ,t − tires, s −struts ,v − vehicle body, w − wheels,
35
36
Mathematical Model of
Motorcycle
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Springs Elements
Mass or Inertia Elements
Damping Elements
r − rider ,t − tires, s −struts ,v − vehicle body, w − wheels,
37
38
Spring Elements
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 force
F = kx
k − spring contant or spring stiffness
x − displacement(deformation)
Potential energy in the spring :
39
U=
1 2
kx
2
40
Springs in Parallel
Springs in Series
1. Static of the system (δ st )
Equilibrium equation
W = k1δ st + k 2δ st
δ st = δ1 + δ 2
2. Equilibrium equation W = k1δ1
W = keqδ st
W = k2δ 2
where keq = k1 + k 2
3. keq for the same static deflection
W = keqδ st
k1δ1 = k 2δ 2 = keqδ st
or
Equivalent spring constant (keq ) in parallel
Equivalent spring constant (keq ) in series
keq = k1 + k 2 + k3 + ⋯ + k n
1
1 1 1
1
= + + +⋯+
keq k1 k 2 k3
kn
δ1 =
that is,
keqδ st
k1
, δ2 =
keqδ st
k2
1
1 1
= +
keq k1 k 2
41
42
Damping Elements
EX: Springs in Parallel
shear modulus G = 80 ×109 N m 2
mean coil diameter D = 20 cm
wire diameter d = 2 cm
The stiffness of helical spring is given by
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Friction or Coulomb Damping)
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d 4G (0.02 ) 80 ×109
=
40,000 N m
k=
3
8D 3n
8(0.2 ) 5
4
The equivalent spring constant of the suspension system is given by
keq = 3k = 3 × 40,000 = 120,000 N m
43
44
Viscous Damping
Damping
All 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.
Mostly a mathematically motivated form, allowing
a solution to the resulting equations of motion that predicts
reasonable (observed) amounts of energy dissipation.
Damper (c)
x
f c = −cv(t ) = −cxɺ (t )
fc
45
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Viscous Damping
Equivalentdampingconstant (ceq ) in series
1 1 1
1
= + +⋯
ceq c1 c2
cn
Damper
Damping coefficient
Critical damping coefficient
Damping ratio
Equivalentdampingconstant (ceq ) in parallel
ceq = c1 + c2 + ⋯ + c3
47
48
Ex: Horizontal milling machine
Underdamped Motion
Overdamped Motion
Critically Damped Motion
49
Ex: Horizontal milling machine
50
Ex: Horizontal milling machine
Fs = keq x
Fsi = ki x ; i = 1,2,3,4
Fdi = ci xɺ ; i = 1,2,3,4
Fd = ceq xɺ
Fs = Fs1 + Fs 2 + Fs 3 + Fs 4
Fd = Fd1 + Fd 2 + Fd 3 + Fd 4
where
keq = k1 + k2 + k3 + k4
ceq = c1 + c2 + c3 + c4
Fs + Fd = W − total vertical force
G − center of mass , Fsi − forces acting on the springs , Fdi − forces acting on the dampers
Fs − forces acting on all the springs , Fd − forces acting on all the dampers
51
52
Newton’s second law
Conservation of Energy
Potential Energy
Kinetic Energy
Natural frequency
53