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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(τt0)
The system is timeinvariant.
(c) memoryless 1 τ+ 0 3 = 1+ [ (b) timeinvariant
For x(tt0) applied as the input, the output y1(t) is:
1
1
+
+
1
Substitute τ=tt0 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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 Fall '09
 Volt

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