Homework2_ solutions

comparing to eq 216 we will have and ii

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Unformatted text preview: to Eq. 2.16 we will have: = / = and = (ii) The variable ωn denotes the natural frequency of the spring damper system. Show that the natural frequency ωn can be increased by increasing the value of the spring constant k or by decreasing the mass M. From = / we can see that increasing k increases ωn, or decreasing M. (iii) Determine whether the system is (a) linear 3 + = 1 + + 2 3 + 1 so =1 3 =[ + So y3(t)=αy1(t)+βy2(t) The system is linear. and + + 1 2 + 3 + + So y1(τ)=y(τ-t0) The system is time-invariant. (c) memoryless 1 τ+ 0 3 = 1+ [ (b) time-invariant For x(t-t0) applied as the input, the output y1(t) is: 1 1 + + 1 Substitute τ=t-t0 so dt=dτ 1 τ+ 0 + + + 1 2 + 2 + =2 2 2 = 1 τ+ 0 = +0 EE 341 Fall 2012 To get the output we need to integrate twice the second order diff. eqn. 1 = − − 1 It is clear that the output depends on past values of the input. System is not memoryless. (d) causal From part © we can deduce that the system is causal (e) invertible x(t) can be obtained from equation above so system is invertible. (f) stable. The system is BIBO. 2....
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