Introduction

A structure is said to be under free vibration when it is disturbed from its static equilibrium position and then allowed to vibrate without any external force. If it is not damped it will vibrate continually. Actually due to energy dissipating mechanism acting simultaneously vibration will be stopped like frictional forces. Here we are considering vibrations of single degrees of freedom structure a portal frame two columns and one beam.

Undamped Free Vibration

The differential equation of a SDF system of a one-story frame subjected to external force P(t) is

Md2s / dt2 + cds / dt + ks=0 if the system is not damping c=0 then the equation becomes

Md2s / dt2 + ks = 0.

Where M = mass, s = displacement, c = viscous damping coefficient, k = lateral stiffness

Free vibration is given by imparting the mass some displacement s= s (0) velocity ds / dt = d s/ dt (0) applying these initial conditions, the solution to the differential equation is

s (0) = s (0) coswt + ds / dt (0) sinwt / w where m = square root of k / M

If the above equation is plotted we get a sin wave, the vibration is simple harmonic motion or SHM. S (0) means un deformed

The period of vibration is T = 2pi/w that is time taken to complete one vibration. Frequency f = 1 / T.

 w = Angular Velocity Radian per second