MECN 4100
History of Vibration
Strings (Music)
Egyptians
Pythagoras: Monochord
Aristotle, Aristoxenus & Euclid
Vitruvius – acoustic properties of theater
Seismograph
Zhang Heng
History of Vibration
Laws of Vibrating String
Galileo – simple pendulum, resonance
Mersenne – “father of acoustics”
Hooke – relation between pitch and frequency
Sauveur – modes shapes and nodes, harmonics
Equation of Motions- Vibrating Body
Newton - Newton‟s Second Law
Taylor – Taylor‟s Theorem
History of Vibration
Principle of Superposition
Bernoulli
Thin Beam Theory
Euler-Bernoulli
Analytical Solution of Vibrating String
Lagrange
Torsional Oscillations
Coulomb
History of Vibration
Theory of Vibrating Plates
E.F.F. Chladni
Sophie Germain
Vibration of Flexible Membrane
Poisson
Clebsh
Thick Beam Theory
Timoshenko
History of Vibration
Thick Plates
Mindlin
Nonlinear
Poincare – pertubation
Lyapunov – stability
Random
Lin and Rice
Crandall and Mark
Finite Element Method
Importance of the Study of
Vibration
Importance of the Study of
Vibration
Importance of the Study of
Vibration
Importance of the Study of
Vibration – on the good side!
Basic Concepts
• Vibration
– Any motion that repeats itself after an interval of time
• Theory of Vibration
– Deals with the study of oscillatory motions of bodies and the forces associated with them
Basic Concepts
• Vibratory System
– Storing potential energy (spring)
– Storing kinetic energy (mass)
– Energy dissipation (damper)
Basic Concepts
• Degree of Freedom
– The minimum number of independent coordinates required to determine completely the position of all parts of a system at any instant of time defines the degree of freedom of the system
• Generalized Coordinates
– Coordinates necessary to describe the motion of a system
Basic Concepts
• Discrete (Lumped) System
– A system that can be describe using a finite number of degree of freedom
• Continuous (distributed)System
– A system that can be describe using a infinite number of degree of freedom
Classification
• Free Vibration
– A system which after an initial disturbance is left to vibrate on its own. No external force acts on the system
• Forced Vibration
– A system subjected to an external force resulting in a vibrating system
Classification
• Undamped
• Damped
• Linear
• Nonlinear
• Deterministic
• Random
Vibration Analysis Procedure
• A vibratory system is a dynamic system for which the variables such as the excitations (inputs) and response (outputs) are time-dependent. The response of a vibrating system generally depends on the initial as well as the external excitations.
• Predict the behavior under specified input conditions
• Consider a simple model of the complex physical model
Vibration Analysis Procedure
• Procedure
–Mathematical Modeling
–Derivation of the governing
equations
–Solution of the equations
–Interpretation of the results
Vibration Analysis Procedure
• Mathematical Modeling– Represent all important features for the
purpose of deriving the mathematical equations governing the system behavior
– Simple as possible
– Linear or Nonlinear
– Great deal of “engineering judgment”
– Sequential: First a crude or elementary model and then a refined model including more components and/or details
Vibration Analysis Procedure
• Derivation of Governing Equations
– Use principle of dynamics and derive the
descriptive equations of a vibration system
– The equation of motion is usually in the form of a
set of ordinary differential equations for a discrete
system and partial differential equations for a
continuous system
– Linear or Nonlinear
– Approaches: Newton‟s second law, D‟Alambert‟s
principle, and principle of conservation of energy
Vibration Analysis Procedure
• Solution of the governing equations– Standard methods of solving DFQs
• Ordinary
• Partial
– Laplace transform methods
– Matrix methods
– Numerical methods
Vibration Analysis Procedure
• Interpretation of the Results
– Displacements
– Velocities
– Accelerations
Vibration Analysis Procedure
• Example– The following figure shows a motorcycle with
a rider. Develop a sequence of three
mathematical models of the system for
investigating vibration in the vertical direction.
Consider the elasticity of the tires, elasticity
and damping of the strut, masses of the
wheels, and elasticity , damping, and mass of
the rider
Vibration Analysis Procedure
Vibration Analysis Procedure
Vibration Analysis Procedure
Vibration Analysis Procedure
• Example– A reciprocating engine is mounted on a
foundation as shown in the following figure. The
unbalanced forces and moments developed in
the engine are transmitted to the frame and the
foundation. An elastic pad is placed between the
engine and the foundation block to reduce the
transmission of vibration. Develop two
mathematical models of the system using gradual
refinement of the modeling process.
Vibration Analysis Procedure
Vibration Analysis Procedure
Elements
• Springs
• Mass/Inertial
• Damping
Spring Elements
• A linear spring is a type of mechanical link with
negligible mass and damping
• The spring force is proportional to the amount of
deformation
• The work done in deforming a spring is stored
as strain or potential energy in the spring
kxF
2
21 kxU
Spring Elements
• Actual spring are nonlinear
kxF
Spring Elements
• Linearization process
2
2
2*
** 2
1x
dx
Fdx
dx
dFxFFF
xx
xkF
Spring Elements
• Elastic elements like beams also behave like
spring
EI
Wlst
3
3
3
3
l
EIWk
st
Spring Elements
• Combination of Springs
– Spring in Parallel
stst kkW 21
steqkW
neq kkkk 21
Spring Elements
• Combination of Springs
– Spring in Series
21st
11kW
neq kkkk
1111
21
22kW
steqkW
eqeqkkk 2211
1
1k
k eqeq
2
2k
k eqeq
st
steqsteq
k
k
k
k
21
Spring Elements
• Example– The figure shown the suspension system of a
freight truck with a parallel-spring arrangement.
Find the equivalent spring constant of the
suspension if each of the three helical springs is
made of steel with a shear modulus G = 80 x 109
N/m2 and has five effective turns, mean coil
diameter D = 20 cm, and wire d = 2 cm
nD
Gdk
3
4
8mN /000,40
52.08
10802.03
94
mNkkeq /000,1203
Spring Elements
• Example– Determine the torsional spring constant of the
steel propeller shaft
Spring Elements
12
12
12 l
GJkt
12
4
12
4
12
32l
dDG
232
2.03.01080 449
23
23
23 l
GJkt
23
4
23
4
23
32l
dDG
332
15.025.01080 449
radmN /105255.25 6
radmN /109012.8 6
2312
2312
tt
tt
tkk
kkk
eq
radmN /105997.6 6
Spring Elements
• Find the equivalent
spring constant of the
system shown in the
figure. Assume that
k1, k2, k3 and k4 are
torsional and k5 ,k6
are linear spring
constant.
Spring Elements
;1111
321123 kkkk
Series of spring,
133221
321123
kkkkkk
kkkk
Using energy equivalence,
2
6212
5212
123212
4212
21 RkRkkkkeq
2
6
2
51234 RkRkkkkeq
65
2
133221
3214 kkR
kkkkkk
kkkkkeq
Spring Elements
• Consider two helical springs with the
following characteristic:
– Determine the equivalent spring constant
when (a) spring 2 is placed inside 1 (b) spring
2 is placed on top of 1
material # turns Mean
coil dia.
Wire dia. Free
length
Shear
modulus
Spring 1 steel 10 12 in. 2 in. 15 in. 12 x 106
psi
Spring 2 Aluminum 10 10 in. 1 in. 15 in. 4 x 106
psi
Spring Elements
3
4
64nR
GdkFor helical spring,
inlbk /89.388,161064
210123
25
1
inlbkkka eq /89.438,1: 21
inlbk /00.5051064
11043
25
2
inlbkkk
beq
/26.48111
:21
Mass or Inertia Elements
Assumed to be a rigid
body
Gain or lose kinetic
energy whenever the
velocity changes
The work done on the
mass is stored for in the
form of mass „s kinetic
energy
Mass or Inertia Elements
Combination of masses
Translational Masses Connected by a Rigid
Bar
1
1
22 x
l
lx
1
1
33 x
l
lx
Mass or Inertia Elements
Translational Masses Connected by a Rigid
Bar
Equating the KE of the three-mass system to that of the
equivalent mass system
1xxeq
2
212
33212
22212
1121
eqeqxmxmxmxm
3
2
1
32
2
1
21 m
l
lm
l
lmmeq
Mass or Inertia Elements
Combination of masses
Translational Masses and Rotational Masses
Coupled Together
○ Equivalent translational mass 2
212
21
oJxmT
xxeq
eqeqeq xmT 21
Rx /2
212
212
21
R
xJxmxm oeq
R
Jmm o
eq
Mass Elements
Example
A cam-follower mechanism is used to convert
the rotary motion of a shaft into the oscillating
or reciprocating motion valve. The follower
system consist of a pushrod of mass mp, a
rocker arm of mass mr, and mass moment of
inertia Jr , a valve of mass mv , and a valve
spring of negligible mass. Find the equivalent
mass of this cam-follower system by assuming
the location as (i) point A and (ii) point C
1/ lxr
122 // lxllx rv
133 / lxllx rr
Mass Elements
2
212
212
212
21
rrrrvvpp xmJxmxmT
eqeqeq xmT 21
1
2
l
lxxv
xxp
1l
xr
1
3
l
lxxr
2
1
2
3
2
1
2
2
2
1 l
lm
l
lm
l
Jmm rv
rpeq
Mass Elements
• In the figure find the equivalent mass of the rocker arm assembly with respect to the x coordinate
Mass Elements
• In the figure find the equivalent mass of the rocker arm assembly with respect to the x coordinate
2
0212
2212
11212
21 Jxmxmxmeq
b
x
b
xax1
2
02
2
1
1
bJm
b
ammeq
Damping Elements
The mechanism by which the
vibrational energy is gradually
converted into heat or sound
A damper is assume to have
neither mass nor elasticity, and
exist only if there is relative
velocity between the two ends of
the damper
Damping Elements
Types;
Viscous
○ Resistance offered by the fluid to the moving
body that causes the energy to be dissipated
○ Factors affecting the damping;
Size
Shape
Viscosity
Frequency
Velocity
Damping Elements
○ The damping force is proportional to the velocity
of the vibrating body
○ Typical example;
Fluid film between sliding surfaces
Fluid around a piston in a cylinder
Fluid flow through an orifice
Fluid film around a journal in a bearing
Coulomb/Dry Friction
○ Constant in magnitude but opposite in direction
of motion to the vibrating body
○ Caused by friction between rubbing surfaces
Damping Elements
Material or Solid or Hysteretic
○ Energy is dissipated by the material when
deform. This is due to friction between the
internal planes, which slip or slide
Damping Elements
Construction
dy
du
hdy
du
ch
AAF
h
Ac
Example 1
Develop an expression for the damping
constant of the dashpot shown
Example 1
Using the shear stress and rate of fluid flow,
dydy
dDlDldF
dy
dv
2
2
dy
vdDldyF
2
4
D
Pp
2
4
D
PDdyp
The pressure,
The pressure force,
Example 1
Integrating this equation twice and using the boundary condition v=0 at y=0 and v=0 at y=d
2
24
dy
vdDldydy
P
lD
P
dy
vd22
2 4
d
yvyyd
lD
Pv 1)(
20
2
2
4
002
3
2
1
6
2dv
lD
PdDDdyvQ
The rate of flow,
Example 1
The volume of the liquid flowing through
03
3
4
213
vd
D
dlD
P
D
d
d
lDc
21
4
33
3
2
4
0 DQ
v
Substituting,
Writing P=cv,
Example 2
The force (F) –velocity (x) relationship of
a nonlinear damper is given by
where a and b are constant. Find the
equivalent linear linear damping constant
when the relative velocity is 5 m/s with
a=5 N s/m and b=0.2 N s2/m2
2xbxaF
Example 222 2.05 xxxbxaF
00
0
)( xxxd
dFxFxF
x
,/50 smxat
30252.055)( 0xF
74.055
0
xxd
dF
x
57)5(730)( xxxF
xcxxF eq 7)( msNceq /7
Example 3
The damping constant (c) due to skin friction drag of a rectangular plate moving in a fluid of a viscosity μ is given by
Design a plate-like damper that provide an identical damping constant for the same fluid
dlc 2100
Example 3
dlc 2100
h
Ac
h
lddl 2100
lc
100
1
Harmonic Motion
tAAx sinsin
tAdt
dxcos
xtAdt
xdsin
2
2
Harmonic
Vectorial
Representation
tAy sin
tAx cos
OPvector
Amagnitude
Harmonic Motion
Complex
Number
2/122 baA
a
b1tan
ibaX
sincos iAAX
iAeiAX sincos
Harmonic
Using complex
number
tiAeX
XiAeiAedt
d
dt
Xd titi
XAeAeidt
d
dt
Xd titi
22
2
2
Harmonic Motion
The displacement, velocity, and
acceleration;
tAAe ti cos]Re[ntdisplaceme
tAAei ti sin]Re[velocity
tAAe ti cos]Re[onaccelerati 22
90cos tA
180cos tA
Harmonic Motion
magnitude,2
2
2
21 sincos AAAA
cos
sintan
21
21
AA
Athe angle,
Check -Example 1.11
Harmonic Motion
2
2
1f
Definition and terminology
Cycle
Amplitude
Period of oscillation
Frequency of oscillation
Harmonic Motion
tAx sin11
Definition and terminology
Phase angle tAx sin22
Harmonic Motion
tt
Xtx2
cos2
cos2
Definition and terminology
Natural frequency
Beats
Octave
Harmonic Motion
0
2
0
log20log10X
X
X
XdB
Definition and terminology
Decibel
Harmonic Analysis
In many cases the vibrations of a system are
periodic. Any periodic function of time can be
represented by Fourier series as an infinite
sum of sine and cosine terms
Harmonic Analysis
)sincos(2
2sinsin
2coscos2
)(
1
0
21
210
tnbtnaa
tbtb
tataa
tx
n
n
n
0
/2
00
2txdttxa
tdtntxdttntxan cos2
cos0
/2
0
tdtntxdttntxbn sin2
sin0
/2
0
Harmonic Analysis
1
000
1
0
2222
222)(
n
nntinnntinti
n
tintin
n
tintin
n
i
ibae
ibae
ibae
i
eeb
eea
atx
tin
n
nectx
dtetxc tin
n
0
1
Complex Fourier Series
Harmonic Analysis
Frequency Spectrum
Harmonic Analysis Time and Frequency Domain Representation
Check: Example 1.12/13