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PRACTICE PROBLEMS

1. Develop the input-output equation and transfer function for the mechanical system below. There is viscous damping between the block and the ground. A force is applied to cause the mass to accelerate.

2. Find the input-output form for the following equations.

3. Find the transfer function for the systems below. Here the input is a torque, and the output is the angle of the second mass.

4. Find the input-output form for the following equations.

5. The following differential equations were converted to the matrix form shown. Use Cramer’s rule to find an input-output equation for ‘y’.

6. Find the input output equation for y2. Ignore the effects of gravity.

7. Find the input-output equations for the systems below. Here the input is the torque on the left hand side.

8. Write the input-output equations for the mechanical system below. The input is force ‘F’, and the output is ‘y’ or the angle theta (give both equations). Include the inertia of both masses, and gravity for mass ‘M’.

9. The applied force ‘F’ is the input to the system, and the output is the displacement ‘x’.

a) Find x(t), given F(t) = 10N for t >= 0 seconds.

b) Using numerical methods, find the steady-state response for an applied force of F(t) = 10cos(t + 1) N ?

c) Solve the differential equation to find the explicit response for an applied force of F(t) = 10cos(t + 1) N ?

d) Set the acceleration to zero and find an approximate solution for an applied force of F(t) = 10cos(t + 1) N. Compare the solution to the previous solutions.

10. Find the transfer function for the system below.

11. For the system below find the a) state and b) input-output equations. The cable always remains tight, and all deflections are small. Assume that the value of J2 is negligible. The input is the force F and the output is the angle ‘theta’.

12. Find the input-output equations for the differential equations below if both ’x’ and ’y’ are outputs.

13. For the system pictured below find the input-output equation for y2.