(a)
The transfer functions
Answer to Problem 2.36P
The transfer functions for the model equations are as:
Explanation of Solution
Given:
The given model equations are as:
Concept Used:
Laplace transform is employed for obtaining the transfer functions for the model conditions keeping the initial conditions zero. In addition, for the system having multiple inputs, the superposition principle is used for finding the impact of one input at a single time by disconnecting the system from other inputs.
Calculation:
Model equations to be solved are as:
For
On taking Laplace for both the model equations, that is
Similarly,
Using equations (1) and (2), we get
Similarly, for
On taking Laplace for both the model equations, that is
Similarly,
Using equations (3) and (4), we get
Conclusion:
The obtained transfer functions are as:
(b)
The values for
Answer to Problem 2.36P
The parameter values are:
Explanation of Solution
Given:
The given system model equations are as:
Concept Used:
Characteristic equation is obtained from transfer function obtained in part a, that is
Since,
Therefore, the characteristic equation is as
This equation is compared with
And
Calculation:
Characteristic equation of the system is as:
On comparing this with
Conclusion:
The obtained values for the parameters are as shown:
(c)
The time taken for the responses
Answer to Problem 2.36P
At
Explanation of Solution
Given:
The obtained transfer functions in the first sub-part ‘a’ are:
Also,
Concept Used:
Since
Therefore,
Using these modified model equations, expressions for
Using these responses, the time taken by oscillations to decay is observed.
Calculation:
Since,
On taking
On taking Laplace transform, we have
Using equations (1) and (2), we have
On taking inverse Laplace of this, we get
On observing the response
Here, at
Similarly, we have
That is the response
Therefore, at
Conclusion:
The obtained responses
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Chapter 2 Solutions
System Dynamics
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