Undamped Free Vibration

INTRODUCTION
• If the external force is removed after giving the initial
displacement to the system, such vibrations are known as free
vibrations, if there is no external resistance(damping) to the
vibrations then such vibrations are known as Undamped free
vibrations.
• When frequency of external exciting force is equal to natural
frequency of vibrating body, the amplitude of vibration becomes
excessively large. Such state is known as Resonance.
• Resonance is dangerous and it may lead to the failure of part.
• Free vibration means that no time varying external forces act on
the system.
• The pendulum will continue to oscillate with the same time
period and amplitude for any length of time.

• The natural frequency of any body or a system depend
upon the geometrical parameters and mass property of the
body.
• It is independent of the forces acting on the body or a
system.
• There are various method to obtain the equation of a
vibrating systems, which can be used to find the natural
frequency of the given vibratory system.
1. Equilibrium Method(D’Alemberts’s Principle)
2. Energy Method
3. Reyleigh’s Method
INTRODUCTION

• The simplest mechanical vibration equation
occurs when γ = 0, F(t) = 0. This is the undamped
free vibration. The motion equation is
mu″ + ku = 0.
• The characteristic equation is mr² + k = 0. Its
solutions are
r = + √(K/m)i or - √(K/m)i
GENERAL EQUATION

• The general solution is then
U(t) = C1 cos ω˳ t + C2 sin ω˳ t
Where ω˳ = √(K/m) is known as natural
frequency of system.
• frequency at which the system tends to oscillate
in the absence of any damping. A motion of this
type is called simple harmonic motion. It is a
perpetual, sinusoidal, motion.
GENERAL EQUATION

Example of Simple harmonic Motion

NATURAL FREQUENCY OF
UMDAMPE FREE VIBRATION BY
EQUILIBRIUM METHOD
• A body or structure which is not in static
equilibrium due to acceleration it possesses can be
brought to static equilibrium by introducing the
inertia force on it.
• The inertia force is equal to the mass times the
acceleration direction is opposite to that of
acceleration.
• The principle is used for developing the equation
of motion for vibrating system which is further
used to find the natural frequency of the vibrating
system.

NATURAL FREQUENCY OF UMDAMPED FREE
VIBRATION BY EQUILIBRIUM METHOD

• The gravitational force must be equal to zero.
mg=kδ
------- (1)
The force acting on the mass are :
1. inertia force : mẍ (upwards)
2. spring force : K(x+δ) (upwards)
3. gravitational force : mg
VIBRATION BY EQUILIBRIUM METHOD

• We know that the fundamental equation of SHM
ẍ + ω²x = 0 ------- (3)
• Comparing equation 2 & 3 ,
ω² = K̲ rad/s -------(4)
m
• According to D’Alembert’s principle ,
mẍ + K(x+δ) – mg = 0
mẍ + Kδ +Kx – mg = 0
mẍ + Kx = 0
ẍ + K̲ x = 0 ------ (2)
m

• The natural frequency f of vibration is ,
f = ω /2∏ or f = ½∏ √(K/m) Hz
----- (5)
• also from eq. (1),
mg = Kδ → K/m = g/δ ------ (6)
• substituting eq (5) in eq (2) , we get
f = ½∏ √(g/δ) H ------ (7)
• the time period t is ,
t = 1/f = 1/ (1/2∏)(√(K/m)) or
t = 2∏ √(m/K) s. ------ (8)

NATURAL FREQUENCY OF UMDAMPED
FREE VIBRATION BY ENERGY METHOD
• According to lao of conservation of energy , the
energy can neither be created nor be destroyed, it
can be converted from one form to another form.
• In free undamped vibrations, no energy is
transferred to the system or from the system.
• Therefore the total mechanical energy i.e. the sum
of kinetic energy and potential energy remains
constant.

• Kinetic energy is due to motion of the body or
system.
• Potential energy consist of two parts
I. Gravitational Potential Energy : Due to
position of body or system with respect to
equilibrium or mean position.
II. Strain Energy : Due to elastic deformation of
body or system

• At mean position, the kinetic energy is maximum
and potential energy is zero; whereas at extreme
positions. The kinetic energy is zero and
potential energy is maximum.
• E = K.E + P.E = Constant _______(9)

E = ½MV² + ½Kx ² ______(10)
E = ½M ẍ² + ½Kx ² ______(11)

(dE/dt) = ½M.2ẍ.ẋ + ½K.2x.ẋ _______(12)
0 = ½.2ẋ(M ẍ + K ẋ) _______(13)
M ẍ + K ẋ = 0 _______(14)
• We know that the fundamental equation of
SHM
ẍ + ω²x = 0 _______(15)
ẍ + (k/m) ẋ = 0 _______(16)
• Comparing equation 2 & 3 ,
ω n = √(K/M) _______(17)
f = ½∏ √(K/m) Hz

M ẍ + K ẋ = 0 _______(18)
ẍ + (k/m) ẋ = 0 _______(19)
• We know that the fundamental equation of SHM
ẍ + ω²x = 0 _______(20)
• By comparing above Equation with equation of
natural frequency
ωn = √(K/M) _______(21)

f = ½∏ √(K/m) Hz _______(22)
• The (natural) period of the oscillation is given by
T = 2∏/ ωn (seconds).

• This is extension of energy method.
• According to principle of conservation of energy,
(Total Energy)mean position = (Total Energy)extreme position
(K.E + P.E)1 = (K.E + P.E)2
(K.E)1+ (P.E)1 = (K.E) 2 + (P.E)2
• At Position 1 P.E is zero and K.E is Maximum.
• At position 2 K.E is zero and P.E is Maximum
FREE VIBRATION BY RAYLEIGH METHOD

• But at mean position K.E is maximum and at
extreme position P.E. is maximum.
(K.E)MAX = (P.E)MAX
• Therefore, According to Lord Reyleigh’s, the
maximum energy is at mean position is equal to
maximum potential energy which is at extreme
position.
FREE VIBRATION BY RAYLEIGH METHOD

(P.E)max = (K.E) max
P.E = ½Kx ² ; x=Xsin(ωn t)
_______(23)
(P.E)max = ½KX² ; ωn t = 90°
_______(24)
VIBRATION BY RAYLEIGH METHOD

K.E = ½M ẋ ² ; ẋ = X ωn Cos(ωn t)
_______(25)
(K.E) max = ½M ωn² X²
_______(26)
(P.E)max = (K.E)max
_______(27)
½KX² = ½M ωn² X² _______(28)
- By simulating the above equation we get next
equation,

ωn = √(K/M) _______(29)
f = ½∏ √(K/m) Hz
• The (natural) period of the oscillation is given
by
T = 2∏/ ωn (seconds).

• Due to gravitation force ‘mg’, the cantilever
beam is deflected by ‘δ’.
• At Equilibrium position mg = Kδ.
• Let the system is subjected to one time external
force due to which it will displaced by ‘x’ from
equilibrium position.
Undamped Free Transverse Vibration

• Forces acting on mass beyond mean position
are,
1. Inertia Force, mẍ (upward) _________(30)
2. Resisting Force, Kx (upward)
• According to D’amberte’s principle,
Ʃ(Inertia Force + External Force) = 0
mẍ + Kx = 0
ẍ + (K/M)x = 0 _____(31)

• Comparing Eq. 31 with Eq. of S.H.M.,
ωn² = (K/M) rad/s
ωn = √(K/M) rad/s or f = ½∏ √(K/m) Hz
• From Eq. 30,
(K/M) = (g/ δ)
• Substituting above values,
Fn = (0.4985/ √ δ) Hz

• Consider a disc having mass moment of inertia
‘I’ suspended on shaft with negligible mass, as
shown in fig.
• If the disc is given a angular displacement
about a axis of shaft, it oscillates about that
axis, such vibrations are known as Torsional
vibrations.

Undamped Free Torsional Vibration

• For angular displacement of disc ‘Ɵ’ in clockwise
direction, the torques acting on the disc are:
• According to D’amberte’s principle,
Ʃ(Inertia Force + External Force) = 0
I Ɵˊˊ + Kt. Ɵ = 0
Ɵˊˊ + (Kt/I).Ɵ = 0 _______(32)

• The fundamental Eq. of S.H.M.
Ɵˊˊ + ωn Ɵ = 0
_____(33)
• By Comparison of above Eq. 32 & 33
ωn = √(Kt/I) rad/s _____(34)
f = ½∏ √(Kt/I) Hz ______(35)
Undamped Free Torsional Vibration

Undamped Free Vibration

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