Vibrations FreeForced UndampedDampedUndampedDamped.

Vibrations

Free Forced

Undamped Damped Undamped Damped

Transient Vibrations of

Single Degree of Freedom Systems

Single Degree of Freedom System with an applied force f(t):

where f(t) can be of any form.

Equation of Motion:

Solution:

Solution for a General Force F(t):

Example:

Excitations Changing at Discrete Times:

A step force f(t) starting at t = 0:

A step force f(t) starting at t = t0:

In general, we have the convolution integral:

Some Excitations Given in Figure 4.5:

Response of an Undamped Single-Degree-of-Freedom System:

Transient Motion due to Base Excitation:

x: Displacement of system

y: Displacement of base

Equation of Motion :

Defining z = x y, we get

Or,

Solution with Convolution Integral :