Vibrations of Structures
Introduction
Many a times, we use the terms vibration and oscillation without knowing the
difference between them.
The term oscillation refers strictly to the repeating
motion of a point mass or that of a rigid body while the term vibration refers
to the repeating motion or deformations of an elastic structure. Thus any
oscillation is a term used only in cases like the motion of a pendulum or that
of a ship as a rigid body moving on the wavy seas while vibration is a term used
for phenomenon exhibited by structures such as rotating fans or motors etc.
Vibration involves deformation by definition while an oscillating structure does
not deform.
Dynamic Load and D'Alembert's Principle
We have studied how we find the response of a structure to static load. When
we come to study vibrations, what we have is not static load but dynamic load.
The very nature of the load means that the response of the structure i.e the
displacement, stress, reactions etc. also varies with time.
The fundamental difference between static and dynamic loads is that
acceleration of the body must be taken into account. From Newton's Law, we know
that any force produces acceleration, however in statics we assumed that the
load had been applied slowly and whatever effects were to take place are over
and the body is now in equilibrium. This cannot be assumed under dynamic load.
The solution to this problem was proposed by D'Alembert who said that the
acceleration can be treated as another force acting opposite to the acceleration
and of magnitude the mass times the acceleration. This force is called the
inertial force. The forces that act on the body can now be treated as a set of
forces in equilibrium.
Forces Involved
In any vibrating system, there are a total of four types of forces that need
to be taken into account. These are listed here.
- Inertial Force
- Spring Force
- Damping Force
- Total External Force
The inertial force has been described already. It occurs because acceleration
is present.
The Spring force arises due to the elasticity of the material. It follows the
Hooke's Law and is proportional to the displacement.
While these forces are enough to set up harmonic vibrations in a system, most
systems also have intrinsic damping even if an external damper is not used. The
damping force tends to oppose motion and acts against the velocity in most cases
(the exception is negative damping). It generally varies with some power of the
velocity though the most useful is viscous damping which varies linearly with
velocity.
The total External force is the remaining force that acts on the system. It
is the force that causes the excitation in the first place and may or may not be
present while the system vibrates.
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