# Harmonically Excited Support Motion of a Single DOF System

4.1 Objective

To analyze the response for forced vibration of a system excited by motion of support point at different damping ratios and frequency ratios.

4.2 Basic Terminology

4.2.1 Periodic Motion:

A motion which repeats itself after equal intervals of time.

4.2.2 Frequency:

The number of oscillations completed per unit time is known as frequency of the system.

4.2.3 Amplitude:

The maximum displacement of a vibrating body from its equilibrium position.

4.2.4 Natural Frequency:

The frequency of free vibration of a system is called Natural Frequency of that particular system.

4.2.5 Damping:

The resistance to the motion of a vibrating body is called Damping. In actual practice, there is always some damping (e.g., the internal molecular friction, viscous damping, aero dynamical damping, etc.) present in the system which causes the gradual dissipation of vibration energy and results in gradual decay of amplitude of the free vibration. Damping has very little effect on natural frequency of the system, and hence, the calculations for natural frequencies are generally made on the basis of no damping. Damping is of great importance in limiting the amplitude of oscillation at resonance.

4.2.6 Fundamental Mode of Vibration:

The fundamental mode of vibration of a system is the mode having the lowest natural frequency.

4.2.7 Degrees Of Freedom:

The minimum number of independent coordinates needed to describe the motion of a system completely, is called the degree-of-freedom of the system. If only one coordinate is required, then the system is called as single degree-of-freedom system.

4.2.8 Mechanical System:

A system consisting of mass, stiffness and damping is known as a mechanical system.

4.2.9 Forced vibration:

When a dynamic system is subjected to a steady-state harmonic excitation, it is forced to vibrate at the same frequency as that of the excitation. Harmonic excitation is often encountered in engineering systems. It is commonly produced by the unbalance in rotating machineries, forces produced by the reciprocating machines, or the motion of machine itself. Although pure harmonic excitation is less likely to occur than the periodic or other types of excitation, understanding the behavior of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system. The harmonic excitation can be given in many ways like with constant frequency and variable frequency or a swept-sine frequency, in which the frequency changes from the initial to final values of frequencies with a given time-rate (i.e., ramp).

4.2.10 Support Motion:

In case of locomotives or vehicles, the wheels act as base or support for the system. The wheels can move vertically up and down on the road surface during the motion of the vehicle. At the same time there is relative motion between the wheels and the chassis. So chassis is having motion relative to the wheels and the wheels are having motion relative to the road surface. The amplitude of vibration in case of support motion depends on the speed of the vehicle and the nature of road surface. The vibration measuring instruments are designed on the support motion approach.

4.2.11 Absolute Motion:

Absolute motion of mass means its motion with respect to the co-ordinate system attached to the earth.