Harmonicaly Excited Forced Vibration of a Single DOF System

2.1 Objective

To analyze the forced vibration response of a harmonically excited single degree-of-freedom system at different damping ratios and frequency ratios.


2.2 Basic Terminology

2.2.1 Periodic Motion:

A motion which repeats itself after equal intervals of time.


2.2.2 Frequency:

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


2.2.3 Amplitude:

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


2.2.4 Natural Frequency:

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


2.2.5 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.


2.2.6 Fundamental Mode of Vibration: 

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


2.2.7 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.


2.2.8 Mechanical System: 

The system consisting of mass, stiffness and damping are known as mechanical system.


2.2.9 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 machinery, 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). 


2.2.10 Resonance:

If the frequency of excitation coincides with one of the natural frequencies of the system, the amplitude of vibration becomes excessively large. This concept is known as resonance.