1. Vibrations FreeForced UndampedDampedUndampedDamped

2. 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.

3. Equation of Motion: Solution:

4. Solution for a General Force F(t):

5. Example:

7. Excitations Changing at Discrete Times: A step force f(t) starting at t = 0:

8. A step force f(t) starting at t = t 0 : In general, we have the convolution integral:

9. Some Excitations Given in Figure 4.5:

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

11. Transient Motion due to Base Excitation: x: Displacement of system y: Displacement of base Equation of Motion : Defining z = x  y, we get Or,

12. Solution with Convolution Integral :