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 Page 1


 
FUNDAMENTAL EQUATIONS OF VIBRATIONS 
 
Free Vibration of Single-Degree-of-Freedom Systems 
Equation of Motion:  0 ? ? ? kx x c x m ? ? ? 
m
k
n
? ? ; 
c
c
c
? ? ; 
n c
m c ? 2 ? ; 
2
1 ? ? ? ? ?
n d
 
Underdamped Response 
? ? ? ? ? ?
??
? ?
?
t Ae t x
d
t
n
sin ; 
? ? ? ?
2
2
0
2
0 0
d
d n
x x v
A
?
? ?? ? ?
? ; 
0 0
0 1
tan
x v
x
n
d
??
?
?
?
?
?
 
Critically Damped Response 
? ? ? ?
t
n
e t a a t x
? ?
? ?
2 1
 where 
0 1
x a ? ; 
0 0 2
x v a
n
? ? ? 
Overdamped Response 
? ?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
t t
t
n n
n
e a e a e t x
1
2
1
1
2 2
? ? ? ?
??
; 
? ?
1 2
1
2
0
2
0
1
?
? ? ?
?
? ?
? ? ?
n
n
x v
a ; 
? ?
1 2
1
2
0
2
0
2
?
? ? ? ? ?
?
? ?
? ? ?
n
n
x v
a 
Logarithmic Decrement 
1
1
ln
1
?
?
n
X
X
n
? where n is the number of complete cycles and 
2 2
4 ? ?
?
?
?
? 
 
Harmonically Excited Vibration 
Equation of Motion:  ? ? t F kx x c x m ? ? ? ? ? ? 
Fixed-Base Harmonic Excitation ? ? t F t F ? cos
0
? 
? ? ? ? ? ? ? ? t X t x cos where 
? ? ? ?
2
2
2
0
? ? c m k
F
X
? ?
? ; 
?
?
?
?
?
?
?
?
?
2
1
tan
?
?
?
m k
c
 
Force Transmissibility:   
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
0
2 1
2 1
?
?
?
?
?
?
?
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?
? ?
?
?
n n
n
T
c m k
c k
F
F
?
?
?
?
?
?
?
?
? ?
?
 
Page 2


 
FUNDAMENTAL EQUATIONS OF VIBRATIONS 
 
Free Vibration of Single-Degree-of-Freedom Systems 
Equation of Motion:  0 ? ? ? kx x c x m ? ? ? 
m
k
n
? ? ; 
c
c
c
? ? ; 
n c
m c ? 2 ? ; 
2
1 ? ? ? ? ?
n d
 
Underdamped Response 
? ? ? ? ? ?
??
? ?
?
t Ae t x
d
t
n
sin ; 
? ? ? ?
2
2
0
2
0 0
d
d n
x x v
A
?
? ?? ? ?
? ; 
0 0
0 1
tan
x v
x
n
d
??
?
?
?
?
?
 
Critically Damped Response 
? ? ? ?
t
n
e t a a t x
? ?
? ?
2 1
 where 
0 1
x a ? ; 
0 0 2
x v a
n
? ? ? 
Overdamped Response 
? ?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
t t
t
n n
n
e a e a e t x
1
2
1
1
2 2
? ? ? ?
??
; 
? ?
1 2
1
2
0
2
0
1
?
? ? ?
?
? ?
? ? ?
n
n
x v
a ; 
? ?
1 2
1
2
0
2
0
2
?
? ? ? ? ?
?
? ?
? ? ?
n
n
x v
a 
Logarithmic Decrement 
1
1
ln
1
?
?
n
X
X
n
? where n is the number of complete cycles and 
2 2
4 ? ?
?
?
?
? 
 
Harmonically Excited Vibration 
Equation of Motion:  ? ? t F kx x c x m ? ? ? ? ? ? 
Fixed-Base Harmonic Excitation ? ? t F t F ? cos
0
? 
? ? ? ? ? ? ? ? t X t x cos where 
? ? ? ?
2
2
2
0
? ? c m k
F
X
? ?
? ; 
?
?
?
?
?
?
?
?
?
2
1
tan
?
?
?
m k
c
 
Force Transmissibility:   
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
0
2 1
2 1
?
?
?
?
?
?
?
?
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? ?
?
?
n n
n
T
c m k
c k
F
F
?
?
?
?
?
?
?
?
? ?
?
 
 
Harmonic Excitation due to Base Motion 
Displacement Transmissibility: 
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
2 1
2 1
?
?
?
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? ?
?
?
n n
n
c m k
c k
Y
X
?
?
?
?
?
?
?
?
? ?
?
 
Force Transmissibility: 
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2
2 2
2
2 1
2 1
?
?
?
?
?
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?
? ?
?
?
?
?
?
?
?
?
?
?
n n
n
n n
T
c m k
c k
kY
F
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
 
Harmonic Excitation due to Rotating Unbalance 
? ? ? ? ? ? ? ? t X t x cos where 
? ? ? ?
2
2
2
2
? ?
?
c M k
me
X
? ?
? ; 
?
?
?
?
?
?
?
?
?
2
1
tan
?
?
?
M k
c
 
Force Transmissibility: 
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
2
2 1
2 1
?
?
?
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n n
n
T
c M k
c k
me
F
?
?
?
?
?
?
?
?
? ?
?
?
 
 
Lagrange’s Equation 
j
j j j j
Q
q
V
q
D
q
T
q
T
dt
d
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
 where T is the kinetic energy, D the Rayleigh’s 
dissipation function, V the potential energy, 
j
q the generalized coordinates and 
j
Q 
the generalized forces. 
 
Vibrations of Continuous Systems 
Wave Equation Solution:  ? ? ? ? t D t C
c
x
B
c
x
A t x y ? ?
? ?
cos sin cos sin , ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
? 
Page 3


 
FUNDAMENTAL EQUATIONS OF VIBRATIONS 
 
Free Vibration of Single-Degree-of-Freedom Systems 
Equation of Motion:  0 ? ? ? kx x c x m ? ? ? 
m
k
n
? ? ; 
c
c
c
? ? ; 
n c
m c ? 2 ? ; 
2
1 ? ? ? ? ?
n d
 
Underdamped Response 
? ? ? ? ? ?
??
? ?
?
t Ae t x
d
t
n
sin ; 
? ? ? ?
2
2
0
2
0 0
d
d n
x x v
A
?
? ?? ? ?
? ; 
0 0
0 1
tan
x v
x
n
d
??
?
?
?
?
?
 
Critically Damped Response 
? ? ? ?
t
n
e t a a t x
? ?
? ?
2 1
 where 
0 1
x a ? ; 
0 0 2
x v a
n
? ? ? 
Overdamped Response 
? ?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ?
?
t t
t
n n
n
e a e a e t x
1
2
1
1
2 2
? ? ? ?
??
; 
? ?
1 2
1
2
0
2
0
1
?
? ? ?
?
? ?
? ? ?
n
n
x v
a ; 
? ?
1 2
1
2
0
2
0
2
?
? ? ? ? ?
?
? ?
? ? ?
n
n
x v
a 
Logarithmic Decrement 
1
1
ln
1
?
?
n
X
X
n
? where n is the number of complete cycles and 
2 2
4 ? ?
?
?
?
? 
 
Harmonically Excited Vibration 
Equation of Motion:  ? ? t F kx x c x m ? ? ? ? ? ? 
Fixed-Base Harmonic Excitation ? ? t F t F ? cos
0
? 
? ? ? ? ? ? ? ? t X t x cos where 
? ? ? ?
2
2
2
0
? ? c m k
F
X
? ?
? ; 
?
?
?
?
?
?
?
?
?
2
1
tan
?
?
?
m k
c
 
Force Transmissibility:   
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
0
2 1
2 1
?
?
?
?
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? ?
?
?
n n
n
T
c m k
c k
F
F
?
?
?
?
?
?
?
?
? ?
?
 
 
Harmonic Excitation due to Base Motion 
Displacement Transmissibility: 
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
2 1
2 1
?
?
?
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?
?
n n
n
c m k
c k
Y
X
?
?
?
?
?
?
?
?
? ?
?
 
Force Transmissibility: 
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2
2 2
2
2 1
2 1
?
?
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?
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?
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?
?
?
?
n n
n
n n
T
c m k
c k
kY
F
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
 
Harmonic Excitation due to Rotating Unbalance 
? ? ? ? ? ? ? ? t X t x cos where 
? ? ? ?
2
2
2
2
? ?
?
c M k
me
X
? ?
? ; 
?
?
?
?
?
?
?
?
?
2
1
tan
?
?
?
M k
c
 
Force Transmissibility: 
? ?
? ? ? ?
2
1
2
1
2
2
2
2
2
2
2
2 2
2
2 1
2 1
?
?
?
?
?
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? ?
?
?
n n
n
T
c M k
c k
me
F
?
?
?
?
?
?
?
?
? ?
?
?
 
 
Lagrange’s Equation 
j
j j j j
Q
q
V
q
D
q
T
q
T
dt
d
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
 where T is the kinetic energy, D the Rayleigh’s 
dissipation function, V the potential energy, 
j
q the generalized coordinates and 
j
Q 
the generalized forces. 
 
Vibrations of Continuous Systems 
Wave Equation Solution:  ? ? ? ? t D t C
c
x
B
c
x
A t x y ? ?
? ?
cos sin cos sin , ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
? 
 
Transverse Vibration of Strings or Cables 
Boundary Conditions:  Fixed ? ? 0 , ? t x y ; 
?
T
c ? where ? is mass per unit length 
Longitudinal Vibration of Bars 
Boundary Conditions:  Fixed ? ? 0 , ? t x u ; Free 
? ?
0
,
?
?
?
x
t x u
; 
?
E
c ? where ? is 
mass per unit volume 
Torsional Vibration of Shafts 
Boundary Conditions:  Fixed ? ? 0 , ? t x ? ;  Free 
? ?
0
,
?
?
?
x
t x ?
; 
?
G
c ? where ? is 
mass per unit volume 
Lateral Vibration of Beams 
General Solution:  ? ? x D x C x B x A t x y ? ? ? ? sin cos sinh cosh , ? ? ? ? 
Undamped Natural Frequencies:  ? ?
4
2
l
EI
l
n n
?
? ? ? where ? is mass per unit length 
Boundary Conditions:  Simply Supported / Pinned  ? ? 0 , ? t x y and 
? ?
0
,
2
2
?
?
?
x
t x y
; 
Free 
? ?
0
,
2
2
?
?
?
x
t x y
 and 
? ?
0
,
3
3
?
?
?
x
t x y
; Fixed ? ? 0 , ? t x y and 
? ?
0
,
?
?
?
x
t x y
. 
 
Vibration Control 
Undamped Dynamic Absorbers 
Amplitude of Primary Mass:  
? ?
? ? ? ?
2 2 2
0
2
a a a a
a a
k m k m k k
F m k
X
? ? ? ?
?
?
? ?
?
  
Amplitude of Absorber Mass:  
? ? ? ?
2 2 2
0
a a a a
a
a
k m k m k k
F k
X
? ? ? ?
?
? ?
  
Natural Frequencies:  
? ?
? ? ? ? 1 1 2 1
2
1
2
1 1
2 2 4
2 2
2
2
? ? ? ? ?
? ?
?
?
?
?
?
?
?
?
?
? ? ? ?
? ?
? ?
?
?
a
n
 
 where
m
m
a
? ? , 
p
a
?
?
? ? , 
a
a
a
m
k
? ? and
m
k
p
? ?

	Formula Sheets of Mechanical Engineering 13 docs

Formula Sheets of Mechanical Engineering

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FAQs on Formula Sheet: Vibrations - Formula Sheets of Mechanical Engineering

1. What are the fundamental principles of vibrations in mechanical engineering?

Ans. Vibrations in mechanical engineering are governed by principles such as mass-spring-damper systems, natural frequency, damping ratio, resonance, and modes of vibration.

2. How do engineers analyze vibrations in mechanical systems?

Ans. Engineers analyze vibrations by studying the system's mass, stiffness, and damping characteristics, using tools like modal analysis, frequency response analysis, and time domain analysis.

3. What are some common sources of vibrations in mechanical systems?

Ans. Common sources of vibrations in mechanical systems include unbalanced rotating machinery, impact forces, turbulent air flow, and structural instabilities.

4. How can engineers mitigate vibrations in mechanical systems?

Ans. Engineers can mitigate vibrations by adding damping materials, adjusting the system's mass distribution, tuning the stiffness of components, and implementing vibration isolation techniques.

5. What are some real-world applications of vibration analysis in mechanical engineering?

Ans. Vibration analysis is used in various applications such as automotive suspension systems, aerospace structures, earthquake engineering, machinery health monitoring, and civil infrastructure design.

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Formula Sheet: Vibrations Mechanical Engineering Questions

The "Formula Sheet: Vibrations Mechanical Engineering Questions" guide is a valuable resource for all aspiring students preparing for the Mechanical Engineering exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

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Formula Sheet: Vibrations | Formula Sheets of Mechanical Engineering PDF Download

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Formula Sheet: Vibrations Notes

Formula Sheet: Vibrations Mechanical Engineering Questions

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