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Problem A - RLC circuit analysis L R + + vi vo C - - (s) 1. Wr ite the transfer function, VVout , for the circui t shown above. in ( s ) 2. Gi ven C = 1 m F , find the values of R and L such that x = 0 .707 and the undamped natural frequency is 5 kHz. ( Don’t forget to convert to rad/sec!!! ) 3. Using the same values of L and C from part(2), and the locations of any system pole(s) and zero(s) given R = 1000•.W Sketch the unit step response, clearl y i ndicating the time and magnitude scaling. (Hi nt: use a dominant pole approximation.) Use the IVT and FVT to show that your response starts and ends at the appropriate values. 4. Sketch the log Magnitude vs log frequency, and linear phase vs log frequency (Bo de plot) for t his system based on your calculated poles from part 3. 5. Again using C = 1 x 10 -6 F and R = 1000 W now let L = 0 H . (i.e. Remove the inductor from the circuit) Calculate the location of the pole and compare this to the dominant pole found in part 2. 6. Wr ite the transfer function, V out ( s ) V in ( s ) , for the two circuits shown below. R1 R1 L - + + vi + + vi L C R2 vo - - C vo R2 - 1 Problem B - Blo ck Diagram 1. Reduce the block diagram below to derive the transfer function for the system. Find the value of K that will result i n a critically damped response. X(s) + K s(s+20) - Y(s) 2. Deriv e the transfer functions VVout1 ( s( s) ) and VVout2 ( s( s) ) for the block diagram shown below. (Hin t, you can always label individual positions along the block diagram with variable names and write out and solve the relevant algebraic equations.) V1 + - s+3 + + V2 + + 6s+1 14s+8 Vout 3. Wr ite the complete di•eren tial equation for Vout ( t) i n terms of V1 ( t) a nd V2 ( t). 3 Problem C - Step resp onse x m w k c f r You h ave been given the system illustrated ab ove. The system consist of a cylinder with a mass (m) with a r adius (r=0. 5 m) which spin s about an axle. The cylinder rolls without slip on the ground. Attac hed to the axle housing are a d amp er (c), a spring (k=200 N/m), and a force source (f ). You measure th e following resp onse x(t) to a step input of the force source. Step Response 0.8 0.7 0.6 Amplitude 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Time (sec) 3 3.5 4 4.5 1. Using the provide parameters and the step resp onse, determine the damping constan t ( c) and the equivalent mass (m eq ), where m eq is mass equivalent o f the comb ined inertia and mass. 2. If the cylinder has a m=3 kg, determine th e inerti a of the cylinder. 4 Problem D θ(t) k φ(t) c J Bode Diagram Magnitude (dB) 5 0 5 10 15 20 Phase (deg) 0 45 90 135 180 0 1 10 10 Frequency (rad/sec) Yo u perform a frequency analysis of the system shown above and obtain the bode plot shown above. The inertia J=15 N/ m 2 . 1. Using the data in the bode plot determine damping ratio and natural frequency for this system. 2. Using J and the values determined in part 1, determine c and k for the system. 3. Determine an expression for the system output for an input sin (wt) where w=1.1, 10, and 20 rad/s. 5