Single degree freedom free vibration

Single Degree of Freedom Systems
(Free Vibrations )
© Dr.V.R Deulgaonkar, 2018

Mechanical Vibrations
Essentials

What is Vibration?????
• Vibration is the physical movement or oscillation of
a mechanical part about a reference position.
• Why do we care for vibration???
Wasted Energy
Major cause of premature component faliure

Vibration Terminology
1) Amplitude: Amplitude is an indicator of the
severity of a vibration. Amplitude can be
expressed as one of the following engineering
units:
a)Velocity b) Acceleration c)Displacement

Velocity
• It is defined as the rate of change of
displacement or rate of change in position.
• It is measured in mm/s, m/s or IPS(Inches Per
Second).
• As it is not frequency related it is the most
accurate measure of vibration.s

Acceleration
• Acceleration is the rate of change of velocity
and is the measurement of the force being
produced.
• Acceleration is expressed in gravitational
forces or “G’s”.
• Acceleration is frequency related.

Displacement
• Displacement is a measure of the actual
distance an object is moving from a reference
point.
• Measured in mm, m inches, etc..
• It is also frequency related.

Elements of a Vibrating System
• Springs: Potential energy storage.
• Mass: Inertial element .
• Damper : Energy dissipation.

Energy Methods
9
• Kinetic Energy
– Discrete masses:
• Point mass: Has translation only, therefore kinetic energy is

Rigid body: Has both translation and rotation, therefore kinetic energy is
 The so called “Energy Methods” draw on the interplay between kinetic and
potential energies associated with a conservative system

Potential Energy Component
• U – change in potential energy of the system from its value in the static-
equilibrium configuration
10
 Conservation of Energy
 Can use with conservative systems only
 At any two different moments of time t1 and t2 we have that

Derivation of EOM
• For the free vibration of an undamped system the energy is partly kinetic and
partly potential.
• Kinetic energy T is stored in the mass by virtue of its velocity
• Potential energy U is stored in the form of strain energy in elastic deformation
or work done in a force field such as gravity.
11
 The expression obtained above after taking the time derivative is (after some
massaging) precisely the EOM

Single Degree of Freedom
Vibrating Systems

One Degree of Freedom Systems
• Single degree Degree of Freedom Un-damped Free
Vibration
k m
x
K (δ+x)
mg
m
δ x
m x’’
F = kδ F = mg
Static Equilibrium δ= static deflection
K = spring rate = force/deflection
X = displacement

• Single degree Degree of Freedom Un-damped Free Vibration
m
K (δ+x)
mg
x
m x’’
0=Fx∑
0)(2
2
=−++ mgxk
dt
xd
m δ
0'' =−++ mgkkxmx δ
mgk =δsince 0'' =+ kxmx

0'' =+ kxmx
0'' =





+ x
m
k
x
m
k
=2
ω
m
k
=ω
0'' 2
=+ xx ω
Solution is of the form of
st
ex = st
sex ='
st
esx 2
'' =
Therefore, by substitution
0'' 2
=+ xx ω 022
=+ stst
ees ω

By substitution
022
=+ stst
ees ω
or 022
=+ ωs ωω is −+=−= /2
from Euler identity,
titi
eCeCx ωω −
+= 21
( ) ( )titCtitCx ωωωω sincossincos 21 +++=
Grouping terms
( ) ( ) tiCCtCCx ωω sincos 2121 −++=

( ) ( ) tiCCtCCx ωω sincos 2121 −++=
tBtAx ωω sincos +=
( ) ( ) BiCCACCLet =−=+ 2121
A & B are now found from initial conditions
nt)displacemeinitial()0(0@ 0XAxt ===
)0()1()0( BAx +=
tBtXx ωωωω cossin' 0 +−=
00 )1()0()0(' VBXx =+−= ωω

00 )1()0()0(' VBXx =+−= ωω
ω
0V
B =
therefore, for a free vibrating single DOF system
t
V
tXtx ω
ω
ω sincos)( 0
0 





+=
m
k
=ω
m
k
fn
π2
1
=

Single Degree of Freedom Damped
Systems

• Single degree Degree of Freedom Damped Free Vibration
k m
x
K (x)
m
x
m x’’
C (x’)
C
0=Fx∑ 0)(2
2
=++
dt
dx
Cxk
dt
xd
m

• Single degree Degree of Freedom damped system
0=Fx∑ 0)(2
2
=++
dt
dx
Cxk
dt
xd
m
Utilizing the quadratic equation
02
2
=





+





+ x
m
k
dt
dx
m
C
dt
xd
m
k
m
C
m
C
or
m
k
m
C
m
C
−





−+





−
−





−+





−
=
2
2
2
/
22
4/
λ

m
k
m
C
m
C
ss −





−+





−=
2
21
2
/
2
,
imaginaryand-real,and0,becan
2
2
+=−





m
k
m
C
The solution is of the form
tsts
eCeCtx 21
21)( +=
m
k
m
C
m
C
ss −





−+





−=
2
21
2
/
2
,

• Expanding
t
m
k
m
C
m
C
t
m
k
m
C
m
C
eCeCtx








−





−





−








−





+





−
+=
22
22
2
22
1)(
or










+=








−





−








−





+






−
t
m
k
m
C
t
m
k
m
C
t
m
C
eCeCetx
22
2
2
2
1
2
)(
imaginaryand-real,and0,becan
2
2
+=−





m
k
m
C
Note that three conditions can occur

– Condition 1, CRITICAL DAMPING
0
2
2
=−





m
k
m
C
kmCor
m
k
m
C
24
2
==





damping)critical(2 cCkmC ==
{ })()1()( 21
2
tCCetx
t
m
C
+=






−
recall
m
k
n =ω
m
k
n =
2
ω
By substitution nc mkmC ω22 ==

– Condition 2, UNDER DAMPING
}{imaginary0
2
2
<−





m
k
m
C
















−





+








−





=






−
t
m
k
m
C
SinCt
m
k
m
C
CosCetx
t
m
C 2
2
2
1
2
22
)(
Note: The system now has damped oscillatory behavior

– Condition 3, OVER DAMPING
Real}{All0
2
2
>−





m
k
m
C
Note: The system now has no oscillatory behavior










+=








−





−








−





+






−
t
m
k
m
C
t
m
k
m
C
t
m
C
eCeCetx
22
2
2
2
1
2
)(

damping)critical(2 cCkmC ==
By definition
ratio)damping(== ξ
cC
C
kmC
km
C
C
C
c
2
2
ξξ ===
therefore
n
m
k
m
km
m
C
ωξξξ ===
2
2
2

also
therefore
n
m
k
m
km
m
C
ωξξξ ===
2
2
2
1
2
2222
2
−=−=−





ξωωωξ nnn
m
k
m
C
12
−= ξωω nd

– Critical Damping
– Over Damped Condition
– Under damping
This is the condition of greatest interest as it represents normal
damping on vehicles. Solve for the constants in the system.
1=ξ
{ })()( 21 tCCetx tn
+= − ωξ
( )
( ) ( ){ }tSinCtCosCetx dd
tn
ωωωξ
21)( += −
1<ξ
( )






+=



 



 −−


 



 −
−
tt
t nn
n
eCeCetx
ωξωξ
ξω
1
2
1
1
22
)(
1>ξ

Under Damped System Behavior
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3
Time (sec)
Displacement(cm)
Forcing function
Displace (cm)
ξ = 0.3 ωn = 7.0 V0 = 0

– Under Damped condition
• @ t = 0
• @ t = 0
( )
( ) ( ){ }tSinCtCosCetx dd
tn
ωωωξ
21)( += −
( )
( ) ( ){ }
( )
( ) ( ){ }tCosCtSinCe
tSinCtCosCe
dt
dx
dddd
t
dd
t
n
n
n
ωωωω
ωωωξ
ωξ
ωξ
21
21
+−+
+−=
−
−
{ } 101 )1()1()0( CXCx ==
{ } { })1()1()1()1( 200 CXV
dt
dx
dn ωωξ +−==

– Under Damped condition
• @ t = 0
{ } { })1()1()1()1( 200 CXV
dt
dx
dn ωωξ +−==
d
n XV
C
ω
ξω 00
2
+
=
( )





 +
+= −
tSin
XV
tCosXetx d
d
n
d
tn
ω
ω
ξω
ωωξ 00
0)(
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3
Time (sec)
Displacement(cm)
Forcing function
Displace (cm)

Single degree freedom free vibration

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