

Exam #1 Problem #1 Consider the system with the transfer function 30/(s^{2} + 8s+7) (5 pts) Is the system underdamped, overdamped, or critically damped? We need to find z. The value of z tells us whether the system is overdamped or critically damped or underdamped. Rearrange the polynomial into a more general form: t^{2}s^{2} + 2tzs + 1:
So, t^{2} = 1/7, meaning t = 0.37796. 2tz = 8/7, so: This is greater than 1, so the system is overdamped.
(20 pts) If x(t) = e^{6t}, find an expression for y(t). Show all work. First invert x(t): So: Do partial fraction expansion: Use Heaviside expansion to find the constants. Multiply by s+7, let s = 7: Multiply by s+6, let s = 6: Multiply by s+1, let s = 1: So, Invert: 
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