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*3.24. Use the $p$ operator to obtain the input-output equation relating $x_1$ and $x_2(t)$ for the system shown in Figure 3.7(a), starting with a. The state-variable equations for $x_1$ and $v_1$ given in (18) b. The state-variable equations for $x_1$ and $q$ given in (22) Figure 3.7 (a) Translational system with displacement input. (b), (c) Free-body diagrams. For Part A: $x_1 = v_1$ (18a) $\dot{v_1} = \frac{1}{M_1}[-(K_1 + K_2)x_1 - Bv_1 + Bx_2 + K_2x_2(t)]$ (18b) For Part B: $x_1 = q + \frac{B}{M_1}x_2(t)$ $\dot{q} = \frac{1}{M_1}[-(K_1 + K_2)x_1 - Bq + (K_2 - \frac{B^2}{M_1})x_2(t)]$ (22)

          *3.24. Use the $p$ operator to obtain the input-output equation relating $x_1$ and $x_2(t)$ for the
system shown in Figure 3.7(a), starting with
a. The state-variable equations for $x_1$ and $v_1$ given in (18)
b. The state-variable equations for $x_1$ and $q$ given in (22)

Figure 3.7 (a) Translational system with displacement input. (b), (c) Free-body diagrams.
For Part A:
$x_1 = v_1$ (18a)
$\dot{v_1} = \frac{1}{M_1}[-(K_1 + K_2)x_1 - Bv_1 + Bx_2 + K_2x_2(t)]$ (18b)
For Part B:
$x_1 = q + \frac{B}{M_1}x_2(t)$
$\dot{q} = \frac{1}{M_1}[-(K_1 + K_2)x_1 - Bq + (K_2 - \frac{B^2}{M_1})x_2(t)]$ (22)

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*3.24. Use the p operator to obtain the input-output equation relating x1 and x2(t) for the
system shown in Figure 3.7(a), starting with
a. The state-variable equations for x1 and v1 given in (18)
b. The state-variable equations for x1 and q given in (22)

Figure 3.7 (a) Translational system with displacement input. (b), (c) Free-body diagrams.
For Part A:
x1 = v1 (18a)
v̇1̇ = (1)/(M1)[-(K1 + K2)x1 - Bv1 + Bx2 + K2x2(t)] (18b)
For Part B:
x1 = q + (B)/(M1)x2(t)
q̇ = (1)/(M1)[-(K1 + K2)x1 - Bq + (K2 - (B^2)/(M1))x2(t)] (22)

Added by Gregory R.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Madhur L

07/12/2023

Step 1

Step 1: To use the "P" operator to find the input-output equation relating x1 and x2(t), we need to first express the state-variable equations in terms of the "P" operator. Show more…

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Thanks for your feedback!

I understand how to find state variable equations, but I'm lost when it comes to this. Can someone teach me how to use the "P" operator to find the input-output equation relating x1 and x2(t)? You don't even need to answer this question, I just need to know how to use and apply it on problems like this. (I just need to know how to find the input-output equation given what I have). Also, if you do this problem, I'm gonna be honest, I do not know how to set up my "Q" variables here. I'm used to doing q1 = x1, q1' = x1', q2 = x2, q2' = found from EOM. I couldn't tell you how to set it up, any help would be appreciated. *3.24. Use the p operator to obtain the input-output equation relating x and x2 for the system shown in Figure 3.7(a), starting with a. The state-variable equations for x1 and x2 given in (18). b. The state-variable equations for x and q given in (22). Figure 3.7(a): Translational system with displacement input. Figure 3.7(b), (c): Free-body diagrams. For Part A: x1 = v1 1 = [-K + K2x1 - Bv1 + Bx2 + K2x2t] / M (18a) (18b) For Part B: x2t = a(K1 + K2x1 - Bq + Mi) (22)

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