System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
Question
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Chapter 8, Problem 8.19P
To determine

(a)

The response x(t) for the model equation x¨+4x˙+8x=2us(t).

Expert Solution
Check Mark

Answer to Problem 8.19P

The response x(t) is x(t)=1414e2tcos2t14e2tsin2t.

Explanation of Solution

Given:

The given model equation is as:

x¨+4x˙+8x=2us(t)

With initial conditions as follows:

x(0)=0 and x˙(0)=0

Concept Used:

Laplace transform is used for obtaining the response of the model equation.

Calculation:

Equation to be solved is as:

x¨+4x˙+8x=2us(t)

By taking the Laplace of this equation that is,

x¨+4x˙+8x=2us(t)(s2X(s)sx(0)x˙(0))+4(sX(s)x(0))+8X(s)=2s(s2X(s)00)+4(sX(s)0)+8X(s)=2s x(0)=0,x˙(0)=0X(s).(s2+4s+8)=2sX(s)=2s(s2+4s+8)

On using partial fraction expansion, this expression could be expressed as follows:

X(s)=2s(s2+4s+8)=As+Bs+Cs2+4s+82s(s2+4s+8)=A(s2+4s+8)+s(Bs+C)s(s2+4s+8)2=s2(A+B)+s(4A+C)+8A A=14,B=14,C=1

Therefore, X(s)=2s(s2+4s+8)=14s14(s+4)(s+2)2+(2)2

Now, on taking the inverse Laplace of the above transfer function, we get

X(s)=14s14(s+4)(s+2)2+(2)2=14s14(s+2)(s+2)2+(2)2142(s+2)2+(2)2x(t)=1414e2tcos2t14e2tsin2t

Thus, this is the response for the given model equation.

Conclusion:

The total response x(t) for the model equation x(t)=1414e2tcos2t14e2tsin2t.

To determine

(b)

The response x(t) for the model equation x¨+8x˙+12x=2us(t).

Expert Solution
Check Mark

Answer to Problem 8.19P

The response x(t) is x(t)=1616e4tcosh2t13e4tsinh2t.

Explanation of Solution

Given:

The given model equation is as:

x¨+8x˙+12x=2us(t)

With initial conditions as follows:

x(0)=0 and x˙(0)=0

Concept Used:

Laplace transform is used for obtaining the response of the model equation.

Calculation:

Equation to be solved is as:

x¨+8x˙+12x=2us(t)

By taking the Laplace of this equation that is,

x¨+8x˙+12x=2us(t)(s2X(s)sx(0)x˙(0))+8(sX(s)x(0))+12X(s)=2s(s2X(s)00)+8(sX(s)0)+12X(s)=2s x(0)=0,x˙(0)=0

X(s).(s2+8s+12)=2sX(s)=2s(s2+8s+12)

On using partial fraction expansion, this expression could be expressed as follows:

X(s)=2s(s2+8s+12)=As+Bs+Cs2+8s+122s(s2+8s+12)=A(s2+8s+12)+s(Bs+C)s(s2+8s+12)2=s2(A+B)+s(8A+C)+12A A=16,B=16,C=43

Therefore, X(s)=2s(s2+4s+8)=16s16(s+8)(s+4)2(2)2

Now, on taking the inverse Laplace of the above transfer function, we get

X(s)=16s16(s+8)(s+4)2(2)2=16s16(s+4)(s+4)2(2)2132(s+4)2(2)2x(t)=1616e4tcosh2t13e4tsinh2t.

Conclusion:

The total response x(t) for the model equation x(t)=1616e4tcosh2t13e4tsinh2t.

To determine

(c)

The response x(t) for the model equation x¨+4x˙+4x=2us(t).

Expert Solution
Check Mark

Answer to Problem 8.19P

The response x(t) is x(t)=1212e2tte2t.

Explanation of Solution

Given:

The given model equation is as:

x¨+4x˙+4x=2us(t)

With initial conditions as follows:

x(0)=0 and x˙(0)=0

Concept Used:

Laplace transform is used for obtaining the response of the model equation.

Calculation:

Equation to be solved is as:

x¨+4x˙+4x=2us(t)

By taking the Laplace of this equation that is,

x¨+4x˙+4x=2us(t)(s2X(s)sx(0)x˙(0))+4(sX(s)x(0))+4X(s)=2s(s2X(s)00)+4(sX(s)0)+4X(s)=2s x(0)=0,x˙(0)=0X(s).(s2+4s+4)=2sX(s)=2s(s+2)2

On using partial fraction expansion, this expression could be expressed as follows:

X(s)=2s(s2+4s+4)=As+Bs+2+C(s+2)22s(s2+4s+4)=A(s+2)2+Bs(s+2)+Css(s+2)22=s2(A+B)+s(4A+2B+C)+4A A=12,B=12,C=1

Therefore, X(s)=2s(s2+4s+4)=12s121(s+2)1(s+2)2

Now, on taking the inverse Laplace of the above transfer function, we get

X(s)=12s121(s+2)1(s+2)2x(t)=1212e2tte2t

Thus, this is the response for the given model equation.

Conclusion:

The total response x(t) for the model equation x(t)=1212e2tte2t.

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