Chapter 9, Problem 9.33P

To determine

The expression for the steady state displacement x(t)for a cam.

The steady stat response for the given Fourier series is given by

Given:

The displacement produced by the cam is given by the following figure

Figure 1                     Figure 2

Where m = 1 Kg, and k = 4900N/m.

Fourier series approximation =

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 =

From Figure 1 equation of motion is given by,

(1)

Substituting for m = 1 Kg, and k = 4900N/m in equation (1)

Applying Laplace transform for the above equation

The transfer function for the given system

T(s) (2)

Therefore magnitude of above transfer function is given by

(3)

The phase angle is given by

(4)

From the equation (3) we can observe that attains a peak of 1 when 0. Hence = 0 and is calculated as follows

Squaring on both sides we get,

On further simplification we get

The roots of the equation are

Taking only the positive value

143.42 rad/s

Therefore, the bandwidth lies between 0 and 143.42 rad/s that is

As is greater than 143.42 rad/s which is outside the required bandwidth, we consider only 0, 10 20 30 and 40 for frequency values.

From equation (3)

Substituting for 0, 10 20 30 and 40 respectively we get

.... (5)

=

=1.015

1.263

1.038

0.806

From equation (4) we have

Now calculating the phase angles for corresponding frequencies 0, 10 20 30 and 40 respectively we get,

rad

2.247 rad

2.038 rad

The steady state response for the given Fourier series

Substituting the values of and for corresponding frequencies 0, 10 20 30 and 40 respectively we get,

Conclusion:

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

Answer to Problem 9.33P

The steady stat response for the given Fourier series is given by

Explanation of Solution

Given:

The displacement produced by the cam is given by the following figure

Figure 1                     Figure 2

Where m = 1 Kg, and k = 4900N/m.

Fourier series approximation =

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 =

From Figure 1 equation of motion is given by,

(1)

Substituting for m = 1 Kg, and k = 4900N/m in equation (1)

Applying Laplace transform for the above equation

The transfer function for the given system

T(s) (2)

Therefore magnitude of above transfer function is given by

(3)

The phase angle is given by

(4)

From the equation (3) we can observe that attains a peak of 1 when 0. Hence = 0 and is calculated as follows

Squaring on both sides we get,

On further simplification we get

The roots of the equation are

Taking only the positive value

143.42 rad/s

Therefore, the bandwidth lies between 0 and 143.42 rad/s that is

As is greater than 143.42 rad/s which is outside the required bandwidth, we consider only 0, 10 20 30 and 40 for frequency values.

From equation (3)

Substituting for 0, 10 20 30 and 40 respectively we get

.... (5)

=

=1.015

1.263

1.038

0.806

From equation (4) we have

Now calculating the phase angles for corresponding frequencies 0, 10 20 30 and 40 respectively we get,

rad

2.247 rad

2.038 rad

The steady state response for the given Fourier series

Substituting the values of and for corresponding frequencies 0, 10 20 30 and 40 respectively we get,

Conclusion:

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