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Home GATE Study MaterialCivil Engineering � Vibrations of Structures

Vibrations of Structures

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Vibrations of Structures

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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