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HW1_Solution

# HW1_Solution - MAE 171A Introduction to Feedback and...

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MAE 171A Introduction to Feedback and Control Systems Fall 2010 HW#1 Solutions Exercises from Feedback Control of Dynamic S ystems (Sixth Edition) by Gene F. Franklin, J.David Powell and Abbas Emami-Naeini 2.3 Write the equations of motion for the double-pendulum system shown in the figure below. Assume that the displacement angles of the pendulums are small enough to ensure that the spring is always horizontal. The pendulum rods are taken to be massless, of length l , and the springs are attached 3/4 of the way down. Double pendulum From the FBD we proceed by summing the moments about the pivot point of the left pendulum. ( 29 1 1 2 1 1 3 3 sin sin sin cos 4 4 M I mgl k l l θ = = - + - 1 3 cos 4 l ( 29 2 1 3 sin sin 4 k l - ( 29 1 2 3 sin sin 4 k l - mg mg We begin by developing the FBD.

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Modify the equation of motion for the cruise control in derived in Example 2.1, so that it has a control law; that is, let where ( ), r u K ν = - , b u m m + = This is a “proportional” control law in which the difference between v r and the actual speed is used as a signal to speed the engine up or slow it down. Revise the equations of motion with v r as the input and v as the output and find the transfer function. Assume that m = 1000 kg and b = 500 N ·
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HW1_Solution - MAE 171A Introduction to Feedback and...

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