Unformatted text preview: ASE 330M – Homework #4 Problem #1: 1.2‐2 (Textbook) Problem #2: Consider a differential equation in y t characterized by differential operators Q D D 2 D 3 ,
P D 1. Suppose the system is initially at rest ( y 0 0 , y 0 0 ). Find the complete solution if the forcing input signal is given x t 1 t 2 1 t 4 . Since the input signal is piecewise continuous, you must divide the solution process into three parts such that y1 t for 0 t 2 y t y2 t for 2 t 4 y t for t 4
3 Note that the zero initial state applies only for y1 t , the solution over the first segment. The initial condition for the next segment must equal the terminal condition for the previous segment. For example, y2 2 y1 2 , y2 2 y1 2 . Likewise, y3 4 y2 4 , y3 4 y2 4 . All four conditions above simply imply that, in a dynamical system, the response is always continuous, even when the input is not. The notation a denotes t a from the left, before the input signal switches value. Similarly, a implies that t a from the right. That is, the function is evaluated immediately after the input signal switches value. Problem #3 Consider a second order differential equation of the form 2 2 n y n 2 y n x t . y This equation is said to be in “standard second order form” when and n are constant positive scalars. The quantity n is denoted the “natural frequency” while represents the damping constant. Any second order ODE can be written in this form if all coefficients are positive. Find the step response of the system. That is, identify the complete solution to this differential equation when the system is initially at rest and x t 1 t . Write the solution in real form (i.e. no complex exponentials or coefficients. Use either real exponentials or sines/cosines). Consider the following cases: (a) 1 (underdamped) (b) 1 (overdamped) (c) 1 (critically damped) For each of the cases above, determine if the solution exhibits any oscillatory behavior or secular terms. Write your answers in terms of n and . Provide a MATLAB plot for the case when n 4 and 0, 0.25, 0.50, 0.75, 0.9,1.0,1.25 . Plot all solutions in one figure and include a legend. Discuss the general behavioral trend as is increased. Repeat the process for 0.5 and n 1,10,100 . Plot all three curves on one figure and include a legend. Discuss the general trend you observe as n is increased. Problem #4: 1.7‐1(a,c,e,h) Problem #5: 1.7‐9, 1.7‐10 ...
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- Spring '10
- input signal