Each impulse in Figure (3.3) will produce a vibration of this form. Because the system is linear, the effect of each impulse is independent of every other impulse and the total resulting motion can be obtained by the principle of super position.  

This integral is known as convolution or Duhamel integral. Explicit solution may be obtained for simple forms of forcing function such as rectangular and triangular.
From equations (3.8), (3.9), (3.10) and (3.15), the total response (given in equation (3.7)) of system can be given by

For the system with at rest condition (i.e. =0 and =0) the response is given by   

This is known as time domain solution because the response is calculated using time as a variable.
In order to obtain recurrence formulas for time domain analysis, consider a SDOF system with displacement and velocity defined at initial time, ti and the response is required at, ti+1 (refer Figure 3.4). Suppose and are the initial displacement and velocity of the system, respectively,