hwk2 - y ( t ) is in this case the input force, while x 1 (...

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MAE 143 A: Signals and Systems. Homework #2. Assigned Jan 13. Due Jan 20. 1. The circuit in the left-hand graph of Figure 1 is modeled by means of the following equations: ± LC d 2 V ( t ) dt + CR dV ( t ) dt + V ( t ) = U ( t ) , i ( t ) = C dV ( t ) dt . Here, L, C and R are the constants associated with the inductor, capacitor and resistor elements, respectively. The signal U ( t ) represents an input voltage, V ( t ) is the voltage at the capacitor, and i ( t ) is the output current going through it. Is this system linear? (why?) Rewrite the system equations in state space form using matrices. 2. Using Newton’s laws, one can model the multibody system in the right-hand graph of Figure 1 by the following set of ODEs: m 1 ¨ x 1 = - k 1 x 1 - b 1 ˙ x 1 - k 2 ( x 1 - x 2 ) , m 2 ¨ x 2 = - k 2 ( x 2 - x 1 ) - k 3 ( x 2 - y ) - b 2 ˙ x 2 . Here, m 1 , m 2 are the masses of the connected bodies, b 1 , b 2 are friction coeﬃcients, and k 1 , k 2 , and k 3 , are the spring constants. The signal
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Unformatted text preview: y ( t ) is in this case the input force, while x 1 ( t ) and x 2 ( t ) are the unknown output displacements. Rewrite the system equations in state-space form using matrices. Figure 1: Left-hand ﬁgure corresponds to Problem 1. Right-hand ﬁgure corresponds to Problem 2. 3. A system is described by the following ODE plus output equation: 7 d 3 y ( t ) dt 3-21 cos( y ( t )) d 2 y ( t ) dt 2 + 7 dy ( t ) dt + 14 y ( t ) = U ( t ) , z ( t ) = sin( y ( t )-π ) . Here U ( t ) is the input signal, y ( t ) is a system unknown, and z ( t ) is the system output. (a) Is the system linear? Write the system in state space form. (b) Consider the point ( y, ˙ y, ¨ y ) = ( π, , 0). Linearize the system around this point. 1...
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This note was uploaded on 04/26/2010 for the course MAE 101B 101B taught by Professor Rohr during the Summer '09 term at UCSD.

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