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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
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Chapter 9 Solutions
System Dynamics
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