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SYSTEM MODELING AND ANALYSIS
BASIC ELEMENTS OF MECHANICAL SYSTEMS
In general, vibrating systems have potential and kinetic energy
storage elements. If the system includes damping, there is also energy
absorbing elements. The basic relations for these elements are given
below.
a) Elastic Elements (Spring): Springs connect the masses in mechanical
systems and allows the relative motion between them. Springs have
linear or nonlinear characteristics. Linear springs behaves in
accordance with the Hooke’s law and the force accumulated in the
spring is proportional with the elastic deformation. Springs may have
nonlinear charactersitics depending on their geometry and/or material.
Different spring behaviors are shown in Figure 1.
Figure 1. Spring characteristics
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SYSTEM MODELING AND ANALYSIS
b) Inertia Elements: Inertia elements store the kinetic energy. Inertia
elements can make translational and rotational motion. They also make
translational and rotational motion at the same time. Basic relations
for inertia elements are given below.
Translational
Rotational
ω
I: Mass moment of inertia
m, I
1 2
Ek  I 
2
(Moment) T
d
TI
 I
dt
2
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SYSTEM MODELING AND ANALYSIS
c) Damping Elements: Damping elements absorb the energy in damped
systems. Dashpot type elements provide damping with fluid friction and
decrease the vibration amplitudes in an exponential manner. In this
element, mechanical energy is converted to heat energy. The element
equation is given below.
Orifice
Energy loss
Energy dissipation occurs during motion. So,
reaction force is propotional with the velocity
(not displacement)
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SYSTEM MODELING AND ANALYSIS
Homogeneous slender bars are freqeuntly used in vibrating
mechanical sytems. This element can simply rotate about an axis
placed on it or both rotates and translates in a plane.
In rotational motion, the kinetic energy stored in the element
is directly proportional with the inertia of the body.
(Angular velocity)
(mass)
(length)
(Mass per unit length)
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SYSTEM MODELING AND ANALYSIS
If the homogeneous slender beam rotates about the axis which passes
through the point B, the mass moment of inertia about point B can be
calculated using the parallel axis theorem (Steiner’s theorem).
Mass moment of inertia about center of gravity
Distance between the center of gravity and rotation axis
Kinetic energy of the bar rotating about point B can be written as
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SYSTEM MODELING AND ANALYSIS
LINEARIZATION OF VIBRATION PROBLEMS
(SMALL DISPLACEMENTS)
Vibration problems can be analysed by the solution of linear differential
equations for small displacements and rotations. If the displacements and
rotations are large, then the nonlinear forms of governing differential
equations must be considered.

k
x
tan  
x
R

R
sin  x

cos  R
sin 
xR
cos 
Taylor-series expansion of sin is
For small angular displacements
 <<1  sin   
Taylor-series expansion of cos is
For small angular displacements
 <<1  cos   1
1 3 5
sin      
1! 3! 5!
2 4
cos   1 


2! 4!

7
x  R  R
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SYSTEM MODELING AND ANALYSIS
For small motions, the same relation between the angular and translational
displacements is valid for a beam which rotates about an axis.
A
O

xA
xA
sin  
OA
x A  OA sin   OA 
F(t)
OBTAINING THE GOVERNING EQUATIONS OF
MOTION FOR A MECHANICAL SYSTEM
m
x(t)
g
The governing equations of motion of a mechanical
system determines the dynamic response of the system
under an external excitation (force or displacements)
or predefined initial conditions. The equations of
motion can be obtained using different methods.
Commonly used methods are described below.
1.
1642-1726
k
c
Newton’s 2nd Law: The mechanical system shown in the figure
has one degree of freedom and the motion of mass m can be
represented by coordinate x. As per the Newton’s 2nd law of
motion, the sum of the forces acting on the mass m is equal
to mass * acceleration of the mass.
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SYSTEM MODELING AND ANALYSIS
mg
k
xs
m
k
k
x(t)
Static equilibrium
xd(t)
x(t)  x s  x d (t)
x ( t )  x d
x( t )  x d
Free Body Diagram
F(t)
xs: static displacement of m
xd: dynamic displacement of m (about xs)
Newton’s
2nd
mg
m
x(t)=xs+xd(t)
law for translational systems
 F  m x
k(xs+xd)
cx d
Newton’s 2nd law for rotational systems

M

I

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SYSTEM MODELING AND ANALYSIS
F(t)
mg
F(t )  mg  k x s  x d   cx d  mx
m
x(t)=xs+xd(t)
k(xs+xd)
cx d
xd  x
mg
F( t )  mg  k
 kx d  cx d  mx d
k
mx d  cx d  kx d  F( t )
Displacement of m is the displacement about static equilibrium
m x  cx  kx  F( t )
d2x
dx
m 2  c  kx  f ( t )
dt
dt
2. Dynamic Equilibrium Method (d’Alembert Principle):
In this method, inertia forces are placed in the FBD and the
static equilibrium equations are applied.
1717-1783
F  0
or
M  0
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SYSTEM MODELING AND ANALYSIS
d’Alembert or inertia force
F(t)
mg
mx d
m
x(t)=xs+xd(t)
k(xs+xd)
cx d
mg
F( t )  mg  k
 kx d  cx d  mx d  0
k
use x=xd
m x  cx  kx  F( t )
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SYSTEM MODELING AND ANALYSIS
3. Energy Method :
In this method, principle of conservation of energy is applied. The rate of
increase in the total energy of a system is equal to the power supplied to
the system.
dE t
 Pnet
dt
Where, Et denotes the sum of potential and kinetic energies of the system, Pnet
denotes the supplied net power; the sign of the power supplied by the
external forces and moments is +, the power given to the external systems and
heat power in the damping elements has negative (–) sign.
Pnet   Pg   Pv   Pd
Sum of the
mechanical power
given to the
system
Sum of the power
given to the
external systems
Sum of the thermal
power discharged
from the damping
element
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SYSTEM MODELING AND ANALYSIS
1
E k  mx 2
2
Ep 
1 2
kx
2
1
1 2
2

E t  mx  kx
2
2
Pnet  F(t )x  cx x
d 1
1 2
2

 mx  kx   F( t ) x  cx x
dt  2
2

m x x  kxx  F( t ) x  cx x
mx  cx  kx  F( t )
4. Lagrange’s Method:
1736-1813
In this method, potential and kinetic energy expressions of the
system are written. Then, generalized forces are derived from the
virtual work of external forces and damping forces. Finally, the
Lagrange’s equation is used to obtain the equations of motion of the
system.
The Lagrangian of the system is defined as the difference between
kinetic and potential energies of the system
L  Ek  Ep
Lagrange’s
equation
d  L  L

 
 Qi
dt  q i  q i
Generalized
Force
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SYSTEM MODELING AND ANALYSIS
d  E k E p  E k E p

 


 Qi
dt  q i q i  q i q i
Where qi denotes the ith generalized cordinate of the system. Qi denotes the
generalized force acting on the ith generalized coordinate. Generalized force
is obtained from the Wirtual Work expression.
In general, kinetic energy is related with the velocity of the generalized
coordinate and potential energy is related with the generalized coordinate,
so the simple form of the Lagrange’s equation can be written as
d  E k  d  E p  E k E p

  
 

 Qi
dt  q i  dt  q i  q i q i
d  E k

dt  q i
 E p
 
 Qi
 q i

On the other hand, in some mechanical
systems,
kinetic
energy
may
be
a
function of generalized coordinate. For
such systems, 3rd term of the Lagrange’s
equation should be taken into account.
1
E k  m r 2    r 2  2
2

O
l
g
Lagrange’s equation is a force equilibrium or a moment equilibrium
for translational and rotational systems, respectively.
θ
r
m
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SYSTEM MODELING AND ANALYSIS
The virtual works of external forces and damping forces on the generalized
coordinates are considered in order to obtain the generalized forces. For this
purpose, a small displacement () independent of time is applied to each
generalized coordinate and the virtual works done by these forces are written as
W  F(t) q i  cq i q i
W  Qi q i
1
E k  mx 2
2
Ep 
1
k x2
2
W  F(t)x  cx x  F(t)  cx x
d   1
 1

2 
  mx     kx 2   F( t )  cx
dt  x  2
  x  2

Qx
d
mx   kx  F(t )  cx
dt
 x  mx  kx  F( t )  cx
m
mx  cx  kx  F( t )
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SYSTEM MODELING AND ANALYSIS
Example: Obtain the equation of motion for the single degree of freedom
system shown in the figure.
Equation of motion is :
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SYSTEM MODELING AND ANALYSIS
Example: Obtain the equations of motion for two-degree of freedom system
shown in the figure.
1
1
2

E k  mx1  2mx 22
2
2
Ep 
1 2 1
1
kx1  2k x 2  x1 2  kx 22
2
2
2
W  f1x1  cx 2  x 1  x 2  x1 
For multi degree of freedom systems, Lagrange’s equation is written for every
generalized coordinate.
Lagrange’s equation for x1 ,
d  E k

dt  x 1
 E p
 
 Qx1
 x1
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SYSTEM MODELING AND ANALYSIS
mx1  kx1  2k(x 2  x1 )  f1  c(x 2  x 1 )
mx1  cx 1  cx 2  3kx1  2kx 2  f1
Lagrange’s equation for x2,
d  E k

dt  x 2
 E p
 
 Qx 2
 x 2
2mx 2  2k(x 2  x1 )  kx 2  c(x 2  x 1 )
2mx 2  cx 1  cx 2  2kx1  3kx 2  0
If we write the equations of motion in matrix form
For lineer systems, Mass, Damping and Stifness matrices are symmetric.
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SYSTEM MODELING AND ANALYSIS
Example: Obtain the equations of motion for two-degree of freedom system
shown in the figure.
h
h  L  L cos 1  L1  cos 1 
 


1
1
2
2
m1 L 1  m 2 L 2
2
2
2
1 L
L 
E p  m1gL 1  cos 1   m 2 gL 1  cos  2   k  2  1 
2 2
2 
Ek 
W  0
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SYSTEM MODELING AND ANALYSIS
Lagrange’s equation for θ1,
m1L2 1  m1gL sin 1  k
m1L21
LL
L 
  2  1   0
22
2 
L2
L2
k
1  k
 2  m1gL 1  0
4
4
Lagrange’s equation for θ2,
m 2 L2 2  m 2 gL sin  2  k
LL
L 


1   0

2
22
2 
L2
L2
m 2L 2  k
1  k
 2  m 2 gL  2  0
4
4
2 
m1L2

 0
 L2

L2
k
  1  0
0  1  k 4  m1gL
4
    

2
2
2   
L
L
m 2 L   2  
  2  0

k
k

m
gL
2


4
4
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