The document discusses undamped free vibration in machinery. It defines undamped free vibration as vibration of a system with no external damping forces after an initial displacement. It describes methods to determine the natural frequency of vibrating systems including the equilibrium method, energy method, and Rayleigh's method. The equilibrium method uses D'Alembert's principle. The energy method equates kinetic and potential energy. Rayleigh's method equates maximum kinetic and potential energy. Examples of undamped free transverse and torsional vibration are also presented and the equations for their natural frequencies are derived.

1. Dynamics of machinery
Undamped free vibration
Prepared by:-
Dungarani Urvesh (140050119506)
Lab Faculty:-
Chetan K Gohil

2. Content
 Introduction
 Determination of natural frequency
 Undamped free transverse vibration
 Undamped free torsional vibration

3. Introduction
 If the external forces is removed after giving an
initial displacement to the system, then the
system vibrates on its own due to internal
elastic forces. Such vibrations are known as free
vibration.
 And if there is no external artificial resistance
to the vibrations then such vibrations are
known as undamped free vibration.

4. Introduction
 In most of the free vibration there is always
certain amount of damping associated with the
system.
 However damping is very small, for all
practical purpose it can be neglected and the
vibrations considered as undamped vibration.

5. Resonance condition
 When the frequency of external excitation fore
acting on a body is equal to the natural
frequency of a vibrating body, the amplitude of
vibrations becomes excessively large. Such
state is known as resonance.
 The resonance is dangerous and it may lead to
the failure of the part.

6. Determination of natural frequency
 The natural frequency of any body or a system
is depends upon the geometrical parameters
and mass properties of the body.
 There are various methods to obtained the
equation of vibrating system,
1. Equilibrium method
2. Energy method
3. Rayleigh’s method

7. Equilibrium method
 According to a D’Alembert’s principal, a body
or a system which is not in static equilibrium
due to acceleration it possese, can be brought to
static equilibrium by introducing the inertia
force on it.
 This principal is used for developing the
equation of the motion for vibrating system
which is further used to find the natural
frequency of the vibrating system.

8. Equilibrium method
 Consider a spring-mass system as shown in fig.

9. Equilibrium method
 A spring has a negligible mass.
 The forces acting on the mass are:
1. Inertia force, mẍ
2. Spring force, K(x+δ)
3. Gravitational force, mg
 According to D’Alembert’s principal,
∑(Inertia force + External force) = 0
m + K(x+δ)+ mg = 0ẍ
+(K/m)x = 0ẍ

10. Equilibrium method
 Comparing the above equation with the
fundamental equation of the simple
harmonic motion equation
 We get,
●
+ (ωẍ n)2 x = 0
 The natural frequency ,
●
(ωn) = (K/m)1/2
●
fn = ωn/2π

11. Energy method
 According to law of conservation of energy, the
energy can neither be created nor be destroyed
but it can be transfer from the one from of
energy to another form of energy.
 In free damped vibration, no energy is
transferred to the system or from the system,
therefore total mechanical energy ramains
constant.

12. Energy method
 The kinetic energy due to motion of body.
 The potential energy due to
1. Gravitational potential energy
2. Strain energy
 At equilibrium position the kinetic energy is maximum
and the potential energy is zero and vice versa.
 According to law of energy conservation,
Total energy = Constant
KE + PE = Constant

13. Energy method
 Differentiating equation,
d/dt(KE + PE) = 0
 Kinetic energy = (1/2)m 2ẋ
 Potential energy = (1/2) Kx2
 Substituting the all equations we get,
+(K/m)x = 0ẍ
 Comparing it with fundamental equation of the S.H.M.
we get,
(ωn) = (K/m)1/2
fn = ωn/2π

14. Rayleigh’s method
 This is the extension of energy method, which is developed
by the Lord Rayleigh.
Total energy = Constant
(KE)1 + (PE)1 = (KE)2 + (PE)2
 The subscripts 1 and 2 denotes two different positions. Let
subscript 1 denote the mean position where potential
energy is zero.
 And subscript 2 denotes the extreme position where kinetic
energy is zero.
 The above equation will be,
(KE)1 = (PE)2

15. Rayleigh’s method
 But at mean position the kinetic energy is
maximum and at extream position the potential
energy is maximum.
(KE)max = (PE)max
 Therefore according to Lord Rayleigh’s the
maximum kinetic energy which is at the mean
position is equal to maximum potential energy
which is the extreme position.

16. Rayleigh’s method
 Let body is moving with simple harmonic
motion, therefore the displacement of the body
is given by,
x = Xsinωnt
 Differentiating above equation,
= dx/dt = ωẋ nXcosωnt
ẋmax = ωnt (t=0, at mean position)

17. Rayleigh’s method
 Maximum kinetic energy at mean position,
(KE)max = (1/2)m ẋmax2
(KE)max = (1/2)m(ωnX)2
 Maximum potential energy at extreme
position,
(PE)max = (1/2)KX2

18. Rayleigh’s method
 Comparing the both equation
– (1/2)m(ωnX)2 = (1/2)KX2
 The natural frequency will be,
(ωn) = (K/m)1/2
fn = ωn/2π

19. Undamped free transverse
vibration
 Consider a
cantilever
beam of
negligible
mass carrying
a
concentrated
mass “m” at
the free end,
as shown in
fig,

20. Undamped free transverse
vibration
 Considering the force acting on the mass beyond
equilibrium position, the forces acting on the
masses are,
 Inertia force, Kẍ
 Resisting force, Kx
 By applying the D’Alembert’s principal,
∑ (Inertia force + External force)= 0
m + Kx = 0ẍ
+ (K/m)x = 0ẍ

21. Undamped free transverse
vibration
 Comparing the above equation with the
fundametal equation of the S.H.M. we get,
(ωn) = (K/m)1/2
 For the frequency,
fn = ωn/2π
 But K/m = g/δ substituting in above equation,
fn = 1/ 2π (K/m)(1/2)
fn = 0.4985/ (δ)(1/2)

22. Torsional stiffness
 Trosional stiffness is define as the torque
required to produce unit angular deflection in
the direction of applied force.
– Kt = T/θ
–
Kt = GJ/θ
– Where, G = modulus of rigidity
– J = Polar moment of inertia
–

23. Parameters for linear & torsional
vibration
Parameters Linear vibration Torsional vibration
Symbol Unit Symbol Unit
Displacement x m θ rad
Velocity ẋ m/s dθ/dt Rad/s
Acceleration ẍ m/s2 d2θ/dt Rad/s2
Inertia force mẍ N Id2θ/dt N-m
Stiffness K N/m Kt N-m/rad

24. Undamped free torasional
vobration
 Consider a disc
having mass
moment of inertia ‘I’
suspended on a
shaft with negligible
mass, as shown in
fig,

25. Undamped free torasional
vobration
 For angular displacement of disc ‘θ’ in
clockwise direction, the torques acting on the
disc are :
 Inertia torque
 Restoring troque
 Therefor according to D’Alembert’s priciple,
∑ (Inertia force + External force)= 0
Id2θ/dt + (Kt/I) θ = 0

26. Undamped free torasional
vobration
 Comparing the above equation ith the
fundamental equation of the SHM. We get.
ωn = (Kt/I)1/2
 The natural frequency will be,
fn = ωn/2π
fn = (1/2π)(GJ/Il)1/2

27. Thank You