Chapter 9, Problem 9.34P

To determine

The expression for the steady state response

The steady state response for the given function is given by

Given:

Fourier series approximation and is given by

Fourier series approximation =

The model equation is

Concept Used:

We first derive an expression for transfer function of the given system and then calculate magnitude of the given transfer function, phase angle, bandwidth, and magnitude of phase angles of respective frequencies and finally calculate the expression for steady state response.

Calculation:

Fourier series approximation

The model equation is

Applying Laplace transform for the above equation we get,

Therefore, the transfer function of the given model is given by

(1)

Comparing the above equation with standard transfer function we get

Time constant 0.1s

The bandwidth is given by

Substituting for 0.1s we get

10 rad/s

Considering the transfer equation in equation (1), the magnitude of transfer function is given by

(2)

The phase angle for the given model is given by

(3)

The bandwidth for the given system should lie between 0 and 10 as 10 rad/s

So, we consider only 4 and 8

The phase angle for the given model is given by

The steady state response for the given Fourier series,

Conclusion:

Therefore, the steady stat response for the given function is given by