This document discusses forced vibrations, which occur when a system oscillates due to an external periodic driving force rather than its own internal restoring force. It provides the equation of motion for forced vibrations, which is a non-homogeneous differential equation. The document also notes that the general solution can be found by solving the particular solution and matching it to the homogeneous solution. It states that if the frequency of the driving force matches the natural frequency of the system, resonance will occur, causing the amplitude of oscillations to become very large or even infinite without damping. Two example problems on forced vibrations of a car suspension and pendulums driven by an oscillating beam are also included.

2. 6.5 Forced Vibrations and Effect of
Dissipative Forces
2
Abrar Hussain
Roll # 15081710-012

3. And those who believe
and do good work
Such are rightful of
the garden.
They will abide therein.
(Surah Al-Baqrah 2:28

4. VIBRATIONS
 Free Vibrations
 Forced Vibrations

5. FORCED VIBRATIONS
Free vibrations are those in which when a system is displaced
slightly from its mean position and then it vibrates with its own
force.
It is an ideal case. As mass spring system , simple pendulum
Forced vibrations are those vibrations in which a system
oscillates with the action of some external driving force or forces
that continue to act on after t = 0 . Force should be periodic in
nature.

6. EQUATION OF MOTION FOR FORCED VIBRATIONS
Forced vibrations are frequently seen in many mechanical systems .
In some cases these are called mechanical vibrations because there is mechanical
force behind it. Which is periodic
as example.
Running engine of car
Washing Machine
So equation of forced vibrations looks like
A
Where K → constant P → amplitude and w is frequency of force.
This is also known as general periodic function.
The above equation is a non-homogenous function.

7. SOLUTION:
 The general solution for this equation can be found by solving
a particular solution.
This general solution corresponds to a particular solution of a
homogenous equation
 We will guess a solution of this homogenous equation as:
→it’s a homogenous
deferential equation
1

8. by putting the respective derivatives of 1 in eq. A
 If then blow up (destroy). Here is
magnification factor

9. GRAPHICAL EXPLANATION
If the system is in resonance then amplitude becomes infinite.
In real case amplitude does not goes infinite because there are
some damping forces.

10. PROBLEM NO. 1
1. A 1000 KG car carrying four 82 KG of each person travels over washboader
dirt road with corrugation 4.o m apart. The car bounces with maximum
amplitude when its speed is 16 km/h. when car stops and peoples get out
of car, how much does the car rise on its suspensions.

12. PROBLEM# 2
.
 Hanging from a horizontal beam are 5 pendulums of following
lengths a)0.10 m b) 0.30 c)0.40 d)0.80 e)1.2m suppose beam
goes under horizontal oscillations with angular frequencies in
range of 2.0 rad/s to 4.0 rad/s. which pendulum have large
amplitude?

13. Thank You