## PRACTICE PROBLEMS

1. Draw the FBDs and write the differential equations for the mechanism below. The right most shaft is fixed in a wall. 2. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 3. Draw the FBDs and write the differential equations for the mechanism below. 4. The system below consists of two masses hanging by a cable over mass ‘J’. There is a spring in the cable near M2. The cable doesn’t slip on ‘J’.

a) Derive the differential equations for the following system.

b) Convert the differential equations to state variable equations 5. Write the state equations for the system to relate the applied force ’F’ to the displacement ’x’. Note that the rotating mass also experiences a rotational damping force indicated with Kd1 6. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 7. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 8. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 9. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 10. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 11. For the system pictured below a) write the differential equations (assume small angular deflections) and b) put the equations in state variable form. 12. Find the polar moments of inertia of area and mass for a round cross section with known radius and mass per unit area. How are they related?

13. The rotational spring is connected between a mass ‘J’, and the wall where it is rigidly held. The mass has an applied torque ‘T’, and also experiences damping ‘B’.

a) Derive the differential equation for the rotational system shown.

b) Put the equation in state variable form (using variables) and then plot the position (not velocity) as a function of time for the first 5 seconds with your calculator using the parameters below. Assume the system starts at rest. c) A differential equation for the rotating mass with a spring and damper is given below. Solve the differential equation to get a function of time. Assume the system starts at rest. 14. Find the response as a function of time (i.e. solve the differential equation to get a function of time.). Assume the system starts undeflected and at rest. 15. For the system pictured below a) write the differential equation for the system with theta2 as the output (assume small angular deflections) and b) put the equations in state variable form. 16. Analyze the system pictured below assuming the rope remains tight. a) Draw FBDs and write the differential equations for the individual masses.

b) Write the equations in state variable matrix form.

c) Use Runge-Kutta integration to find the system state after 1 second.

17. Analyze the system pictured below assuming the rope remains tight and gravity acts downwards. a) Draw FBDs and write the differential equations for the individual masses.

b) Combine the equations and simplify the equations as much as possible.

c) Write the equations in state variable matrix form.

d) Use Runge-Kutta to find the system state after 1 second.