Solutions to Differential Equations of Motion for Vibrating Systems Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Solutions to Differential Equations of Motion for Vibrating Systems Civil Engineering (CE) Notes | EduRev

 Page 1


                     
 
 
  
    
 
 
   
 
 
 
Solutions to Differential Equations of Motion for Vibrating Systems 
 
 
Here, we summarize the solutions to the most important differential equations of motion that we 
encounter when analyzing single degree of freedom linear systems. 
 
 
CASE I:                   
2
22
1
n
dx
xC
dt ?
?? 
 
CASE II                    
2
22
1 dx
xC
dt ?
? ? ? 
 
CASE III                  
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
 
CASE IV                 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
CASE V                    
2
22
1 2 2
nn
n
d x dx dy
x C K y
dt dt
dt
??
??
?
??
? ? ? ? ?
??
??
  with 
0
sin y Y t ? ? 
 
CASE VI                  
22
2 2 2 2
12
n
nn
d x dx K d y
xC
dt
dt dt
?
?
??
? ? ? ? with 
0
sin y Y t ? ? 
 
 
 
 
 
 
 
 
 
 
Page 2


                     
 
 
  
    
 
 
   
 
 
 
Solutions to Differential Equations of Motion for Vibrating Systems 
 
 
Here, we summarize the solutions to the most important differential equations of motion that we 
encounter when analyzing single degree of freedom linear systems. 
 
 
CASE I:                   
2
22
1
n
dx
xC
dt ?
?? 
 
CASE II                    
2
22
1 dx
xC
dt ?
? ? ? 
 
CASE III                  
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
 
CASE IV                 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
CASE V                    
2
22
1 2 2
nn
n
d x dx dy
x C K y
dt dt
dt
??
??
?
??
? ? ? ? ?
??
??
  with 
0
sin y Y t ? ? 
 
CASE VI                  
22
2 2 2 2
12
n
nn
d x dx K d y
xC
dt
dt dt
?
?
??
? ? ? ? with 
0
sin y Y t ? ? 
 
 
 
 
 
 
 
 
 
 
SOLUTION 1: 
 
The equation 
2
22
1
n
dx
xC
dt ?
?? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ?
0
2 2 2 1 0
0 0 0
0
sin
()
( ) / tan
n
n
n
x C X t
xC
X x C v
v
??
?
??
?
? ? ?
?? ?
? ? ? ?
??
??
 
or, equivalently 
0
0
( ) ( )cos sin
nn
n
v
x t C x C t t ??
?
? ? ? ? 
 
 
 
 
 
SOLUTION 2 
 
The equation 
 
2
22
1 dx
xC
dt ?
? ? ? 
with initial conditions 
 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ? ? ?
00
00
11
( ) ( ) exp ( ) exp
22
vv
x t C x C t x C t ??
??
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


                     
 
 
  
    
 
 
   
 
 
 
Solutions to Differential Equations of Motion for Vibrating Systems 
 
 
Here, we summarize the solutions to the most important differential equations of motion that we 
encounter when analyzing single degree of freedom linear systems. 
 
 
CASE I:                   
2
22
1
n
dx
xC
dt ?
?? 
 
CASE II                    
2
22
1 dx
xC
dt ?
? ? ? 
 
CASE III                  
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
 
CASE IV                 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
CASE V                    
2
22
1 2 2
nn
n
d x dx dy
x C K y
dt dt
dt
??
??
?
??
? ? ? ? ?
??
??
  with 
0
sin y Y t ? ? 
 
CASE VI                  
22
2 2 2 2
12
n
nn
d x dx K d y
xC
dt
dt dt
?
?
??
? ? ? ? with 
0
sin y Y t ? ? 
 
 
 
 
 
 
 
 
 
 
SOLUTION 1: 
 
The equation 
2
22
1
n
dx
xC
dt ?
?? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ?
0
2 2 2 1 0
0 0 0
0
sin
()
( ) / tan
n
n
n
x C X t
xC
X x C v
v
??
?
??
?
? ? ?
?? ?
? ? ? ?
??
??
 
or, equivalently 
0
0
( ) ( )cos sin
nn
n
v
x t C x C t t ??
?
? ? ? ? 
 
 
 
 
 
SOLUTION 2 
 
The equation 
 
2
22
1 dx
xC
dt ?
? ? ? 
with initial conditions 
 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ? ? ?
00
00
11
( ) ( ) exp ( ) exp
22
vv
x t C x C t x C t ??
??
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
SOLUTION 3 
 
The equation 
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has the following solutions: 
 
Case I: Overdamped System 1 ? ? 
0 0 0 0
( )( ) ( )( )
( ) exp( ) exp( ) exp( )
22
n d n d
n d d
dd
v x C v x C
x t C t t t
? ? ? ? ? ?
? ? ? ?
??
?? ? ? ? ? ? ?
? ? ? ? ?
??
??
 
where 
2
1
dn
? ? ? ?? 
 
Case II: Critically Damped System  1 ? ? 
? ? ? ?
0 0 0
( ) ( ) ( ) exp( )
nn
x t C x C v x C t t ?? ? ? ? ? ? ? ? 
 
Case III: Underdamped System 1 ? ? 
00
0
()
( ) exp( ) ( )cos sin
n
n d d
d
v x C
x t C t x C t t
??
? ? ? ?
?
?? ??
? ? ? ? ?
??
??
 
where 
2
1
dn
? ? ? ??  
 
The graphs below show x(t) for two types of initial condition: the first graph shows results with 
0
0 v ? , while the second graph shows results with 
0
0 x ? .  Both results are for C=0. 
 
Graphs of solutions to ODE governing free vibration of a damped spring-mass system 
 
 
 
Page 4


                     
 
 
  
    
 
 
   
 
 
 
Solutions to Differential Equations of Motion for Vibrating Systems 
 
 
Here, we summarize the solutions to the most important differential equations of motion that we 
encounter when analyzing single degree of freedom linear systems. 
 
 
CASE I:                   
2
22
1
n
dx
xC
dt ?
?? 
 
CASE II                    
2
22
1 dx
xC
dt ?
? ? ? 
 
CASE III                  
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
 
CASE IV                 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
CASE V                    
2
22
1 2 2
nn
n
d x dx dy
x C K y
dt dt
dt
??
??
?
??
? ? ? ? ?
??
??
  with 
0
sin y Y t ? ? 
 
CASE VI                  
22
2 2 2 2
12
n
nn
d x dx K d y
xC
dt
dt dt
?
?
??
? ? ? ? with 
0
sin y Y t ? ? 
 
 
 
 
 
 
 
 
 
 
SOLUTION 1: 
 
The equation 
2
22
1
n
dx
xC
dt ?
?? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ?
0
2 2 2 1 0
0 0 0
0
sin
()
( ) / tan
n
n
n
x C X t
xC
X x C v
v
??
?
??
?
? ? ?
?? ?
? ? ? ?
??
??
 
or, equivalently 
0
0
( ) ( )cos sin
nn
n
v
x t C x C t t ??
?
? ? ? ? 
 
 
 
 
 
SOLUTION 2 
 
The equation 
 
2
22
1 dx
xC
dt ?
? ? ? 
with initial conditions 
 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ? ? ?
00
00
11
( ) ( ) exp ( ) exp
22
vv
x t C x C t x C t ??
??
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
SOLUTION 3 
 
The equation 
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has the following solutions: 
 
Case I: Overdamped System 1 ? ? 
0 0 0 0
( )( ) ( )( )
( ) exp( ) exp( ) exp( )
22
n d n d
n d d
dd
v x C v x C
x t C t t t
? ? ? ? ? ?
? ? ? ?
??
?? ? ? ? ? ? ?
? ? ? ? ?
??
??
 
where 
2
1
dn
? ? ? ?? 
 
Case II: Critically Damped System  1 ? ? 
? ? ? ?
0 0 0
( ) ( ) ( ) exp( )
nn
x t C x C v x C t t ?? ? ? ? ? ? ? ? 
 
Case III: Underdamped System 1 ? ? 
00
0
()
( ) exp( ) ( )cos sin
n
n d d
d
v x C
x t C t x C t t
??
? ? ? ?
?
?? ??
? ? ? ? ?
??
??
 
where 
2
1
dn
? ? ? ??  
 
The graphs below show x(t) for two types of initial condition: the first graph shows results with 
0
0 v ? , while the second graph shows results with 
0
0 x ? .  Both results are for C=0. 
 
Graphs of solutions to ODE governing free vibration of a damped spring-mass system 
 
 
 
 
SOLUTION 4: 
 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
and initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution of the form 
( ) ( ) ( )
hp
x t C x t x t ? ? ? 
where the steady state solution (or particular integral) is 
? ?
? ?
? ?
0
1 0
0
1/2 2 2
2
2
22
( ) sin
2/
tan
1/
1 / 2 /
p
n
n
nn
x t X t
KF
X
??
? ? ?
?
??
? ? ? ? ?
?
??
?
??
?
??
??
??
??
 
 
while the transient solution (or homogeneous solution, or complementary solution) is: 
 
 
Case I: Overdamped System 1 ? ? 
0 0 0 0
( ) ( )
( ) exp( ) exp( ) exp( )
22
h h h h
n d n d
h n d d
dd
v x v x
x t t t t
? ? ? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
? ? ? ?
??
??
??
 
where 
2
1
dn
? ? ? ?? 
 
 
Case II: Critically Damped System  1 ? ? 
? ? 0 0 0
( ) exp( )
h h h
h n n
x t x v x t t ??
??
? ? ? ?
??
 
 
 
Case III: Underdamped System 1 ? ? 
00
0
( ) exp( ) cos sin
hh
h n
h n d d
d
vx
x t t x t t
??
? ? ? ?
?
??
? ??
? ? ?
??
??
??
 
where 
2
1
dn
? ? ? ?? 
 
In all three preceding cases, we have set 
0 0 0 0
0 0 0 0
0
(0) sin
cos
h
p
p
h
t
x x C x x C X
dx
v v v X
dt
?
??
?
? ? ? ? ? ?
? ? ? ?
 
Observe that for large time, the transient solution always decays to zero. 
 
Page 5


                     
 
 
  
    
 
 
   
 
 
 
Solutions to Differential Equations of Motion for Vibrating Systems 
 
 
Here, we summarize the solutions to the most important differential equations of motion that we 
encounter when analyzing single degree of freedom linear systems. 
 
 
CASE I:                   
2
22
1
n
dx
xC
dt ?
?? 
 
CASE II                    
2
22
1 dx
xC
dt ?
? ? ? 
 
CASE III                  
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
 
CASE IV                 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
CASE V                    
2
22
1 2 2
nn
n
d x dx dy
x C K y
dt dt
dt
??
??
?
??
? ? ? ? ?
??
??
  with 
0
sin y Y t ? ? 
 
CASE VI                  
22
2 2 2 2
12
n
nn
d x dx K d y
xC
dt
dt dt
?
?
??
? ? ? ? with 
0
sin y Y t ? ? 
 
 
 
 
 
 
 
 
 
 
SOLUTION 1: 
 
The equation 
2
22
1
n
dx
xC
dt ?
?? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ?
0
2 2 2 1 0
0 0 0
0
sin
()
( ) / tan
n
n
n
x C X t
xC
X x C v
v
??
?
??
?
? ? ?
?? ?
? ? ? ?
??
??
 
or, equivalently 
0
0
( ) ( )cos sin
nn
n
v
x t C x C t t ??
?
? ? ? ? 
 
 
 
 
 
SOLUTION 2 
 
The equation 
 
2
22
1 dx
xC
dt ?
? ? ? 
with initial conditions 
 
00
0
dx
x x v t
dt
? ? ? 
has solution 
? ? ? ?
00
00
11
( ) ( ) exp ( ) exp
22
vv
x t C x C t x C t ??
??
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
SOLUTION 3 
 
The equation 
2
22
12
n
n
d x dx
xC
dt
dt
?
?
?
? ? ? 
with initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has the following solutions: 
 
Case I: Overdamped System 1 ? ? 
0 0 0 0
( )( ) ( )( )
( ) exp( ) exp( ) exp( )
22
n d n d
n d d
dd
v x C v x C
x t C t t t
? ? ? ? ? ?
? ? ? ?
??
?? ? ? ? ? ? ?
? ? ? ? ?
??
??
 
where 
2
1
dn
? ? ? ?? 
 
Case II: Critically Damped System  1 ? ? 
? ? ? ?
0 0 0
( ) ( ) ( ) exp( )
nn
x t C x C v x C t t ?? ? ? ? ? ? ? ? 
 
Case III: Underdamped System 1 ? ? 
00
0
()
( ) exp( ) ( )cos sin
n
n d d
d
v x C
x t C t x C t t
??
? ? ? ?
?
?? ??
? ? ? ? ?
??
??
 
where 
2
1
dn
? ? ? ??  
 
The graphs below show x(t) for two types of initial condition: the first graph shows results with 
0
0 v ? , while the second graph shows results with 
0
0 x ? .  Both results are for C=0. 
 
Graphs of solutions to ODE governing free vibration of a damped spring-mass system 
 
 
 
 
SOLUTION 4: 
 
2
22
12
()
n
n
d x dx
x C KF t
dt
dt
?
?
?
? ? ? ? with 
0
( ) sin F t F t ? ? 
 
and initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution of the form 
( ) ( ) ( )
hp
x t C x t x t ? ? ? 
where the steady state solution (or particular integral) is 
? ?
? ?
? ?
0
1 0
0
1/2 2 2
2
2
22
( ) sin
2/
tan
1/
1 / 2 /
p
n
n
nn
x t X t
KF
X
??
? ? ?
?
??
? ? ? ? ?
?
??
?
??
?
??
??
??
??
 
 
while the transient solution (or homogeneous solution, or complementary solution) is: 
 
 
Case I: Overdamped System 1 ? ? 
0 0 0 0
( ) ( )
( ) exp( ) exp( ) exp( )
22
h h h h
n d n d
h n d d
dd
v x v x
x t t t t
? ? ? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
? ? ? ?
??
??
??
 
where 
2
1
dn
? ? ? ?? 
 
 
Case II: Critically Damped System  1 ? ? 
? ? 0 0 0
( ) exp( )
h h h
h n n
x t x v x t t ??
??
? ? ? ?
??
 
 
 
Case III: Underdamped System 1 ? ? 
00
0
( ) exp( ) cos sin
hh
h n
h n d d
d
vx
x t t x t t
??
? ? ? ?
?
??
? ??
? ? ?
??
??
??
 
where 
2
1
dn
? ? ? ?? 
 
In all three preceding cases, we have set 
0 0 0 0
0 0 0 0
0
(0) sin
cos
h
p
p
h
t
x x C x x C X
dx
v v v X
dt
?
??
?
? ? ? ? ? ?
? ? ? ?
 
Observe that for large time, the transient solution always decays to zero. 
 
 
The graphs below plot the amplitude of the steady state vibration and the steady state phase lead. 
 
(a)                                                              (b) 
Steady state response of a forced spring —mass system (a) amplitude and (b) phase 
 
 
SOLUTION 5 
 
The equation 
2
22
1 2 2
nn
n
d x dx dy
x C K y
dt dt
dt
??
??
?
??
? ? ? ? ?
??
??
 with 
0
( ) sin y t Y t ? ? 
and initial conditions 
00
0
dx
x x v t
dt
? ? ? 
has solution of the form 
( ) ( ) ( )
hp
x t x t x t ?? 
 
where the steady state solution (or particular integral) is 
? ?
? ?
? ?
? ?
? ?
0
1/2
2
33
0
1
0
1/2 2 2 2
2
2
22
( ) sin
1 2 /
2/
tan
1 (1 4 ) /
1 / 2 /
p
n
n
n
nn
x t X t
KY
X
??
? ? ?
? ? ?
?
? ? ?
? ? ? ? ?
?
??
?
?
??
??
??
??
??
??
 
 
 
while the transient solution (or homogeneous solution) is: 
 
Case I: Overdamped System 1 ? ? 
0 0 0 0
( ) ( )
( ) exp( ) exp( ) exp( )
22
h h h h
n d n d
h n d d
dd
v x v x
x t t t t
? ? ? ? ? ?
? ? ? ?
??
??
? ? ? ? ??
? ? ? ?
??
??
??
 
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