Short Notes: Effect of Damping | Short Notes for Mechanical Engineering PDF Download

 Page 1


Effect of Damping 
Damping
Any influence which tends to dissipate the energy of a system. Single degree of 
freedom system with viscous damping can be shown in the figure below.
mx cx kx
J J _1
]
1 Degree ol freedom system
m X + CX + X = 0
c k
XH------XH------- X = 0
m m
c k _ 
a * H — q H— = 0 
m m
Damping Factor or Damping Ratio
• It is the ratio of actual to critical damping coefficient.
c =
c
• Damping Coefficient of proportionality between the damping force and 
relative velocity
c= 2g-Jmk = 2£
k _
m
Page 2


Effect of Damping 
Damping
Any influence which tends to dissipate the energy of a system. Single degree of 
freedom system with viscous damping can be shown in the figure below.
mx cx kx
J J _1
]
1 Degree ol freedom system
m X + CX + X = 0
c k
XH------XH------- X = 0
m m
c k _ 
a * H — q H— = 0 
m m
Damping Factor or Damping Ratio
• It is the ratio of actual to critical damping coefficient.
c =
c
• Damping Coefficient of proportionality between the damping force and 
relative velocity
c= 2g-Jmk = 2£
k _
m
• £ = 1, the damping is known as critical under critical damping condition 
Critical damping coefficient
• £ > 1 i.e., the system is over damped
i-? ± Jte2-!))* ,
• Motion is a periodic (non-oscillatory). In practice, no mechanical systems 
have over damping.
• £ < 1 i.e., system is underdamped.
x= .-le + 2 te
• If the system oscillates with the frequency 
Damped frequency
(u )c/ is always less than cjn )
• The above equation can be written as 
X = constant
x = X e~' ' (sin xjt + o)
Linear frequency
Time period
where, C = actual damping coefficient
Page 3


Effect of Damping 
Damping
Any influence which tends to dissipate the energy of a system. Single degree of 
freedom system with viscous damping can be shown in the figure below.
mx cx kx
J J _1
]
1 Degree ol freedom system
m X + CX + X = 0
c k
XH------XH------- X = 0
m m
c k _ 
a * H — q H— = 0 
m m
Damping Factor or Damping Ratio
• It is the ratio of actual to critical damping coefficient.
c =
c
• Damping Coefficient of proportionality between the damping force and 
relative velocity
c= 2g-Jmk = 2£
k _
m
• £ = 1, the damping is known as critical under critical damping condition 
Critical damping coefficient
• £ > 1 i.e., the system is over damped
i-? ± Jte2-!))* ,
• Motion is a periodic (non-oscillatory). In practice, no mechanical systems 
have over damping.
• £ < 1 i.e., system is underdamped.
x= .-le + 2 te
• If the system oscillates with the frequency 
Damped frequency
(u )c/ is always less than cjn )
• The above equation can be written as 
X = constant
x = X e~' ' (sin xjt + o)
Linear frequency
Time period
where, C = actual damping coefficient
Cc = critical damping coefficient 
u)d = damped frequency 
£ = damping factor 
u)„ = natural frequency 
Logarithmic Decrement:
• In an underdamped system, the arithmetic ratio of two successive oscillations 
is called logarithmic decrement (constant).
Since,
X, X, X
Logarithmic decrement,
6 = In -—2 - 
X .
or
or
In
Forced Vibration
• Equation of forced vibration can be given as
m x+ kx = Fq sin xt
n .
mx + cx + fo e = F0 sin xt 
x=Xe~‘~ -r sin (x„t + Oj)
F r , sinter — o)
Page 4


Effect of Damping 
Damping
Any influence which tends to dissipate the energy of a system. Single degree of 
freedom system with viscous damping can be shown in the figure below.
mx cx kx
J J _1
]
1 Degree ol freedom system
m X + CX + X = 0
c k
XH------XH------- X = 0
m m
c k _ 
a * H — q H— = 0 
m m
Damping Factor or Damping Ratio
• It is the ratio of actual to critical damping coefficient.
c =
c
• Damping Coefficient of proportionality between the damping force and 
relative velocity
c= 2g-Jmk = 2£
k _
m
• £ = 1, the damping is known as critical under critical damping condition 
Critical damping coefficient
• £ > 1 i.e., the system is over damped
i-? ± Jte2-!))* ,
• Motion is a periodic (non-oscillatory). In practice, no mechanical systems 
have over damping.
• £ < 1 i.e., system is underdamped.
x= .-le + 2 te
• If the system oscillates with the frequency 
Damped frequency
(u )c/ is always less than cjn )
• The above equation can be written as 
X = constant
x = X e~' ' (sin xjt + o)
Linear frequency
Time period
where, C = actual damping coefficient
Cc = critical damping coefficient 
u)d = damped frequency 
£ = damping factor 
u)„ = natural frequency 
Logarithmic Decrement:
• In an underdamped system, the arithmetic ratio of two successive oscillations 
is called logarithmic decrement (constant).
Since,
X, X, X
Logarithmic decrement,
6 = In -—2 - 
X .
or
or
In
Forced Vibration
• Equation of forced vibration can be given as
m x+ kx = Fq sin xt
n .
mx + cx + fo e = F0 sin xt 
x=Xe~‘~ -r sin (x„t + Oj)
F r , sinter — o)
wvwwwwwwww
kx
I k
mx
J
- E d ~ *
I
F0 sin wf 
Forced vibration
Fq sin c o t F0 sin tot
• In the case of steady state response first term zero (e'“ = 0).
A
A
Fq
\J(k - w - : ) : + (c ^ f
• The amplitude of the steady-state response is given by 
Magnification Factor
• Ratio of the amplitude of the steady state response to the static deflection 
under the action force F0 is known as magnification factor.
X{F = *J(.k ~ mo1)2 +(C0? 
FJk
_______ k_______
J(k- mo1 )1 +(ca)'
• Let frequency ratio
r - —
MF [Function H(c j)]
________1 _______
7 (1 - r ) J+ (2 ' r f
For small values of damping, the peak can be assumed to be at which define 
the quality factor.
Page 5


Effect of Damping 
Damping
Any influence which tends to dissipate the energy of a system. Single degree of 
freedom system with viscous damping can be shown in the figure below.
mx cx kx
J J _1
]
1 Degree ol freedom system
m X + CX + X = 0
c k
XH------XH------- X = 0
m m
c k _ 
a * H — q H— = 0 
m m
Damping Factor or Damping Ratio
• It is the ratio of actual to critical damping coefficient.
c =
c
• Damping Coefficient of proportionality between the damping force and 
relative velocity
c= 2g-Jmk = 2£
k _
m
• £ = 1, the damping is known as critical under critical damping condition 
Critical damping coefficient
• £ > 1 i.e., the system is over damped
i-? ± Jte2-!))* ,
• Motion is a periodic (non-oscillatory). In practice, no mechanical systems 
have over damping.
• £ < 1 i.e., system is underdamped.
x= .-le + 2 te
• If the system oscillates with the frequency 
Damped frequency
(u )c/ is always less than cjn )
• The above equation can be written as 
X = constant
x = X e~' ' (sin xjt + o)
Linear frequency
Time period
where, C = actual damping coefficient
Cc = critical damping coefficient 
u)d = damped frequency 
£ = damping factor 
u)„ = natural frequency 
Logarithmic Decrement:
• In an underdamped system, the arithmetic ratio of two successive oscillations 
is called logarithmic decrement (constant).
Since,
X, X, X
Logarithmic decrement,
6 = In -—2 - 
X .
or
or
In
Forced Vibration
• Equation of forced vibration can be given as
m x+ kx = Fq sin xt
n .
mx + cx + fo e = F0 sin xt 
x=Xe~‘~ -r sin (x„t + Oj)
F r , sinter — o)
wvwwwwwwww
kx
I k
mx
J
- E d ~ *
I
F0 sin wf 
Forced vibration
Fq sin c o t F0 sin tot
• In the case of steady state response first term zero (e'“ = 0).
A
A
Fq
\J(k - w - : ) : + (c ^ f
• The amplitude of the steady-state response is given by 
Magnification Factor
• Ratio of the amplitude of the steady state response to the static deflection 
under the action force F0 is known as magnification factor.
X{F = *J(.k ~ mo1)2 +(C0? 
FJk
_______ k_______
J(k- mo1 )1 +(ca)'
• Let frequency ratio
r - —
MF [Function H(c j)]
________1 _______
7 (1 - r ) J+ (2 ' r f
For small values of damping, the peak can be assumed to be at which define 
the quality factor.
• For
the peak amplitudes occur at
A [(i_ r ) + (2 
dr'
C D
‘ A v