a. Compute the required input to bring the system to rest in 10, 2, 0.5 and 0.00

Question

a. Compute the required input to bring the system to rest in 10, 2, 0.5 and 0.001 seconds.

b. In this problem,??1 and??2 represent the vertical displacement of the mass from rest. Let us

assume that the maximum the system can be stretched or compressed is 25cm, and that the

units of 1 and 2 are cm, that is max{|??1|} = max{|??2|} = 25. Can you estimate a lower

limit on the time for which we can drive the system from the initial state to rest without

reaching the physical limit of the system?

c. If perhaps the first damping coefficient in the system were 1 instead of 2,determine how this affects the controllability of the system?

Explanation / Answer

The undampednatural frequency(in rad/sec) is...

?0 = ?( k / m )

and the damping ratio (dimensionless) is...

? = c / ( 2 ?( k / m ) )

? = c / ( 2 ?0 )

Given a spring constant of 121 N/m and a mass of 5 kg, the undamped natural frequency is...

w0 = ?( 121 N/m / 5 kg )

w0 = 4.919 rad/sec

At critical damping, ? = 1, thus the damping coefficient c (in kg/sec) is...

c = ? / ( 2 ?0 )

c = 1 / ( 2 x 4.919 rad/sec )

c = 0.101 kg/sec

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