FORCE DAMPRD VIBRATION

1. Forced Damped Vibrations
 If the system vibrates under the
influence of external periodic force,
the vibrations are known as forced
vibration.
 In forced vibrations, there is an
impressed force acting on the system
which keeps it vibrating.
 The vibration of air compressors, I.C.
engines, turbines, pumps, etc. are the
example of forced vibrations.

2.  All the rotating machines like
compressors, pumps, turbines, etc. are
carefully designed and manufactured,
still there are some unbalanced forces
acting on such machines which act as
harmonic excitation forces.
 The problem becomes more severe
when frequency of external excitation
force coincides with the natural
frequency of a system, which result in
resonance and the system starts
vibrating dangerously with large
amplitude.

3.  In order avoid the resonance either the
operational speed is changed or
structure altered to change its natural
frequency.
 In addition, sufficient amount of damping
may be provided to keep the vibrating
amplitude as low as possible and
components are designed to withstand
the dynamic loads.
 In general, the external periodic forces
acting on the system are of two types :
Harmonic Forces and Non-harmonic
forces.

4. Forced Damped Vibrations with
Constant Harmonic Excitation
 The harmonic excitation forces which
are commonly produced by the
unbalanced in rotating machines, are
often encountered in engineering
systems.
 Consider the system having
spring-mass-damper system excited
by a harmonic force Fo sin t, as
shown in figure.

5. where, Fo = magnitude of external excitation
harmonic force, N
= circular frequency of external
excitation force, rad/s

6.  At any instant, when the mass is
displaced from the equilibrium position
the forces acting on the system are :
(i) External Harmonic Force, Fo sin t,
(downward)
(ii) Inertia force, m x (upward)
(iii) Damping force, x (upward)
(iv) Spring force, Kx (upward)

7.  Consider F.B.D. of mass shown in
figure.
∑ [Inertia force + External force] = 0
m x + c x + Kx – Fo sin t = 0
…..(1)
 The Equation (1) is a linear, second
order differential equation.

8.  The solution of this equation consist of
two parts :
1. Complementary Function (xc)
2. Particular Integral (xp)
 The complete solution to the linear,
second order differential Equation(1)
is
x = xc + xp …….(2)

9. Complementary Function(xc)
 The Complementary Function is obtain by
considering no forcing function. i.e.
considering Equation (1) as,
This Equation is same as the equation
obtained for damped free vibration system.
The solution of Equation is given by,

10. Complete solution for differential
Equation
 The complete solution for differential
equation is,
x = xc + xp

12. Transient Vibrations
 This is the complete solution to an
under damped system subjected to
sinusoidal excitation.
 The first part of the complete solution
that is complementary function, is
seen to decay with time and vanishes
ultimately. The part is called as
Transient Vibrations.

13. Steady state Vibrations
 The second part of complete solution
that is particular solution is seen to be
sinusoidal vibration with a constant
amplitude and is called as Steady
state Vibrations.

14. Complete Solution
 The complete solution is obtained by
superposition of transient and steady
state vibration.
 After the transient vibration die out,
the complete solution consist of
steady state vibration only.
 The two constant X1 and Ø1
determined by applying the initial
conditions to the complete solution.

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