Step forcing function second order system

A forcing function, whose transform is a constant K is applied to an under-damped second-order system having a time constant of 0.5 min and a damping coefficient of 0.5. Show that the decay ratio for the resulting response is the same as that due to the application of a unit step function to the same system. 

Example 6il2. A general underdamped second-order system is forced by a unit step function ... 

Fig. 7.28. Response of second-order system to step forcing function...

The normal function of any control system is to ensure that the controlled variable attains its desired value as rapidly as possible after a disturbance has occurred, with the minimum of oscillation. Determination of the response of a system to a given forcing function will show what final value the controlled variable will attain and the manner in which it will arrive at that value. This latter is a function of the stability of the response. For example, in considering the response of a second order system to a step change, it can be seen that oscillation increases... 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

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