Short Notes: Free and Forced Vibration | Short Notes for Mechanical Engineering PDF Download

 Page 1


Free and Forced Vibration 
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. System with a finite number of 
degrees of freedom are called discrete or lumped parameter system, and those 
with an infinite number of degrees of freedom are called continuous or distributed 
systems.
Single degree of freedom: The number of degree of freedom of a mechanical 
system is equal to the minimum number of independent co-ordinates required to 
define completely the positions of all parts of the system at any instance of time.
Multi degree of freedom: A multi degree of freedom system is one for which 2 or 3 
co-ordinates are required to define completely the positions of the system at any 
instance of time.
Free vibration: When there is no external force acts on the body after giving an 
initial displacement, then the body is said to be under free or natural vibration.
Forced vibration: When the body vibrates under the influence of external force the 
body is said to be under forced vibration. The frequency of forced vibration is 
called forced frequency.
Forced Vibrations with Damping
In this section, we will restrict our discussion to the case where the forcing 
function is a sinusoid. Thus, we can make some general statements about the 
solution:
The equation of motion with damping will be given by:
m u" — ~u' — k t t = F r cos ( )
Page 2


Free and Forced Vibration 
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. System with a finite number of 
degrees of freedom are called discrete or lumped parameter system, and those 
with an infinite number of degrees of freedom are called continuous or distributed 
systems.
Single degree of freedom: The number of degree of freedom of a mechanical 
system is equal to the minimum number of independent co-ordinates required to 
define completely the positions of all parts of the system at any instance of time.
Multi degree of freedom: A multi degree of freedom system is one for which 2 or 3 
co-ordinates are required to define completely the positions of the system at any 
instance of time.
Free vibration: When there is no external force acts on the body after giving an 
initial displacement, then the body is said to be under free or natural vibration.
Forced vibration: When the body vibrates under the influence of external force the 
body is said to be under forced vibration. The frequency of forced vibration is 
called forced frequency.
Forced Vibrations with Damping
In this section, we will restrict our discussion to the case where the forcing 
function is a sinusoid. Thus, we can make some general statements about the 
solution:
The equation of motion with damping will be given by:
m u" — ~u' — k t t = F r cos ( )
Its solution will be of the form:
u (r) = C j M j (r)+ c2 u2 (r) + A cos(^-r) 4- B sin (u*)
fcofflogtatous solution : 
’tnr.si«r.: sobtion'
pimr^tir jobtioe u 
't i b y ttu t soluaon'
Note:
• The homogeneous solution Uh(t) -» 0 as t -» °°, which is why it is called the 
"transient solution."
• The constants cl and c2 of the transient solution are used to satisfy given 
initial conditions.
• The particular solution up(t) is all that remains after the transient solution dies 
away, and is a steady oscillation at the same frequency of the driving 
function. That is why it is called the "steady state solution," or the "forced 
response."
• The coefficients A and B must be determined by substitution into the 
differential equation.
• If we replace up(t) = U(t) = Acos(ajt) + Bsin (cut) with up(t) = U(t) = Rcos(a)t-5), 
then
F0 U)
k
m
Note that as c j -* 0
cos(5)— + 1 and sin(f)— » 0 =»[ < 5 — > 0
• Note that when
• Note that as
(mass is out of phase with drive).
The amplitude of the steady state solution can be written as a function of all 
the parameters of the system:
Page 3


Free and Forced Vibration 
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. System with a finite number of 
degrees of freedom are called discrete or lumped parameter system, and those 
with an infinite number of degrees of freedom are called continuous or distributed 
systems.
Single degree of freedom: The number of degree of freedom of a mechanical 
system is equal to the minimum number of independent co-ordinates required to 
define completely the positions of all parts of the system at any instance of time.
Multi degree of freedom: A multi degree of freedom system is one for which 2 or 3 
co-ordinates are required to define completely the positions of the system at any 
instance of time.
Free vibration: When there is no external force acts on the body after giving an 
initial displacement, then the body is said to be under free or natural vibration.
Forced vibration: When the body vibrates under the influence of external force the 
body is said to be under forced vibration. The frequency of forced vibration is 
called forced frequency.
Forced Vibrations with Damping
In this section, we will restrict our discussion to the case where the forcing 
function is a sinusoid. Thus, we can make some general statements about the 
solution:
The equation of motion with damping will be given by:
m u" — ~u' — k t t = F r cos ( )
Its solution will be of the form:
u (r) = C j M j (r)+ c2 u2 (r) + A cos(^-r) 4- B sin (u*)
fcofflogtatous solution : 
’tnr.si«r.: sobtion'
pimr^tir jobtioe u 
't i b y ttu t soluaon'
Note:
• The homogeneous solution Uh(t) -» 0 as t -» °°, which is why it is called the 
"transient solution."
• The constants cl and c2 of the transient solution are used to satisfy given 
initial conditions.
• The particular solution up(t) is all that remains after the transient solution dies 
away, and is a steady oscillation at the same frequency of the driving 
function. That is why it is called the "steady state solution," or the "forced 
response."
• The coefficients A and B must be determined by substitution into the 
differential equation.
• If we replace up(t) = U(t) = Acos(ajt) + Bsin (cut) with up(t) = U(t) = Rcos(a)t-5), 
then
F0 U)
k
m
Note that as c j -* 0
cos(5)— + 1 and sin(f)— » 0 =»[ < 5 — > 0
• Note that when
• Note that as
(mass is out of phase with drive).
The amplitude of the steady state solution can be written as a function of all 
the parameters of the system:
• Notice that
is dimensionless (but proportional to the amplitude of the motion), since
k
is the distance a force of F0 would stretch a spring with spring constant k. 
• Notice that
mk
is dimensionless...
m ass
. time
m ass
mass-------
time'
• Note that as
x — * 0. R - v l R
• Note that as c j -* R -* 0 (i.e., the drive is so fast that the system cannot 
respond to it and so it remains stationary).
• The frequency that generates the largest amplitude response is:
Page 4


Free and Forced Vibration 
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. System with a finite number of 
degrees of freedom are called discrete or lumped parameter system, and those 
with an infinite number of degrees of freedom are called continuous or distributed 
systems.
Single degree of freedom: The number of degree of freedom of a mechanical 
system is equal to the minimum number of independent co-ordinates required to 
define completely the positions of all parts of the system at any instance of time.
Multi degree of freedom: A multi degree of freedom system is one for which 2 or 3 
co-ordinates are required to define completely the positions of the system at any 
instance of time.
Free vibration: When there is no external force acts on the body after giving an 
initial displacement, then the body is said to be under free or natural vibration.
Forced vibration: When the body vibrates under the influence of external force the 
body is said to be under forced vibration. The frequency of forced vibration is 
called forced frequency.
Forced Vibrations with Damping
In this section, we will restrict our discussion to the case where the forcing 
function is a sinusoid. Thus, we can make some general statements about the 
solution:
The equation of motion with damping will be given by:
m u" — ~u' — k t t = F r cos ( )
Its solution will be of the form:
u (r) = C j M j (r)+ c2 u2 (r) + A cos(^-r) 4- B sin (u*)
fcofflogtatous solution : 
’tnr.si«r.: sobtion'
pimr^tir jobtioe u 
't i b y ttu t soluaon'
Note:
• The homogeneous solution Uh(t) -» 0 as t -» °°, which is why it is called the 
"transient solution."
• The constants cl and c2 of the transient solution are used to satisfy given 
initial conditions.
• The particular solution up(t) is all that remains after the transient solution dies 
away, and is a steady oscillation at the same frequency of the driving 
function. That is why it is called the "steady state solution," or the "forced 
response."
• The coefficients A and B must be determined by substitution into the 
differential equation.
• If we replace up(t) = U(t) = Acos(ajt) + Bsin (cut) with up(t) = U(t) = Rcos(a)t-5), 
then
F0 U)
k
m
Note that as c j -* 0
cos(5)— + 1 and sin(f)— » 0 =»[ < 5 — > 0
• Note that when
• Note that as
(mass is out of phase with drive).
The amplitude of the steady state solution can be written as a function of all 
the parameters of the system:
• Notice that
is dimensionless (but proportional to the amplitude of the motion), since
k
is the distance a force of F0 would stretch a spring with spring constant k. 
• Notice that
mk
is dimensionless...
m ass
. time
m ass
mass-------
time'
• Note that as
x — * 0. R - v l R
• Note that as c j -* R -* 0 (i.e., the drive is so fast that the system cannot 
respond to it and so it remains stationary).
• The frequency that generates the largest amplitude response is:
Plugging this value of the frequency into the amplitude formula gives us:
r j x j ^
if
- — > i 
4 mk
, then the maximum value of R occurs for c j = 0.
Resonance is the name for the phenomenon when the amplitude grows very large 
because the damping is relatively small and the drive frequency is close to the 
undriven frequency of oscillation of the system.
Forced Vibrations without Damping
The equation of motion of an undamped forced oscillator is:
mu —ku = F t cos (jjt I 
When
a * < v 0
(non-resonant case), the solution is of the form:
When a) = u)0 (resonant case), the solution is of the form:
Page 5


Free and Forced Vibration 
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. System with a finite number of 
degrees of freedom are called discrete or lumped parameter system, and those 
with an infinite number of degrees of freedom are called continuous or distributed 
systems.
Single degree of freedom: The number of degree of freedom of a mechanical 
system is equal to the minimum number of independent co-ordinates required to 
define completely the positions of all parts of the system at any instance of time.
Multi degree of freedom: A multi degree of freedom system is one for which 2 or 3 
co-ordinates are required to define completely the positions of the system at any 
instance of time.
Free vibration: When there is no external force acts on the body after giving an 
initial displacement, then the body is said to be under free or natural vibration.
Forced vibration: When the body vibrates under the influence of external force the 
body is said to be under forced vibration. The frequency of forced vibration is 
called forced frequency.
Forced Vibrations with Damping
In this section, we will restrict our discussion to the case where the forcing 
function is a sinusoid. Thus, we can make some general statements about the 
solution:
The equation of motion with damping will be given by:
m u" — ~u' — k t t = F r cos ( )
Its solution will be of the form:
u (r) = C j M j (r)+ c2 u2 (r) + A cos(^-r) 4- B sin (u*)
fcofflogtatous solution : 
’tnr.si«r.: sobtion'
pimr^tir jobtioe u 
't i b y ttu t soluaon'
Note:
• The homogeneous solution Uh(t) -» 0 as t -» °°, which is why it is called the 
"transient solution."
• The constants cl and c2 of the transient solution are used to satisfy given 
initial conditions.
• The particular solution up(t) is all that remains after the transient solution dies 
away, and is a steady oscillation at the same frequency of the driving 
function. That is why it is called the "steady state solution," or the "forced 
response."
• The coefficients A and B must be determined by substitution into the 
differential equation.
• If we replace up(t) = U(t) = Acos(ajt) + Bsin (cut) with up(t) = U(t) = Rcos(a)t-5), 
then
F0 U)
k
m
Note that as c j -* 0
cos(5)— + 1 and sin(f)— » 0 =»[ < 5 — > 0
• Note that when
• Note that as
(mass is out of phase with drive).
The amplitude of the steady state solution can be written as a function of all 
the parameters of the system:
• Notice that
is dimensionless (but proportional to the amplitude of the motion), since
k
is the distance a force of F0 would stretch a spring with spring constant k. 
• Notice that
mk
is dimensionless...
m ass
. time
m ass
mass-------
time'
• Note that as
x — * 0. R - v l R
• Note that as c j -* R -* 0 (i.e., the drive is so fast that the system cannot 
respond to it and so it remains stationary).
• The frequency that generates the largest amplitude response is:
Plugging this value of the frequency into the amplitude formula gives us:
r j x j ^
if
- — > i 
4 mk
, then the maximum value of R occurs for c j = 0.
Resonance is the name for the phenomenon when the amplitude grows very large 
because the damping is relatively small and the drive frequency is close to the 
undriven frequency of oscillation of the system.
Forced Vibrations without Damping
The equation of motion of an undamped forced oscillator is:
mu —ku = F t cos (jjt I 
When
a * < v 0
(non-resonant case), the solution is of the form:
When a) = u)0 (resonant case), the solution is of the form:
p
u (r) = C j cos( ^ 0 r) + c, sin (x j) + r sin (^0 t)
Free Vibration of Undamped One Degree-of-Freedom Systems 
Translation:
mx+ k x = 0
(a mass-spring system)
Solution:
x(r) = A sin u j + B cos x H t = A' sinU',/ + o)
Frequency:
Rotation:
J J - K 9 = 0 
W-g8 = 0
(a pendulum)
Free Vibration of Damped One Degree-of-Freedom Systems
m x+cx+kx= 0 
x + 2 C ^ „x + ^ x = 0 
Damping factor:
2m u:a 2-Jbti
c is the damping coefficient in the units of lbs per in/sec. 
Solution:
x(r) = exp(-rcJlir)(Asin«( S r+ B cos&i/)
= X exp(-Ayn r)sin(«s r+ < p )
The damped natural frequency is:
-‘-'a — —n C
The system response when under-damped: £ < 1