## PRACTICE PROBLEMS

1. The mass, M, illustrated below starts at rest. It can slide across a surface, but the motion is opposed by viscous friction (damping) with the coefficient B. Initially the system starts at rest, when a constant force, F, is applied. Write the differential equation for the mass, and solve the differential equation. Leave the results in variable form.

2. The following differential equation was derived for a mass suspended with a spring. At time 0s the system is released and allowed to drop. It then oscillates. Solve the differential equation to find the motion as a function of time.

3. Solve the following differential equation with the three given cases. All of the systems have a step input ’y’ and start undeflected and at rest.

4. Solve the following differential equation with the given initial conditions and draw a sketch of the first 5 seconds. The input is a step function that turns on at t=0.

5. Solve the following differential equation with the given initial conditions and draw a sketch of the first 5 seconds. The input is a step function that turns on at t=0.

6. a) Write the differential equations for the system below. Solve the equations for x assuming that the system is at rest and undeflected before t=0. Also assume that gravity is present.

b) State whether each system is first or second-order. If the system if first-order find the time constant. If it is second-order find the natural frequency and damping ratio.

7. Solve the following differential equation with the three given cases. All of the systems have a sinusoidal input ’y’ and start undeflected and at rest.

8. A spring mass system supports a mass of 34N. If it has a spring constant of 20.6N/cm, what is the systems natural frequency?

9. Using a standard lumped parameter model the weight is 36N, stiffness is 2.06*10^3 N/m and damping is 100Ns/m. What are the natural frequency (Hz) and damping ratio?

10. What is the differential equation for a second-order system that responds to a step input with an overshoot of 20%, with a delay of 0.4 seconds to the first peak?

11. A system is to be approximated with a mass-spring-damper model using the following parameters: weight 28N, viscous damping 6Ns/m, and stiffness 36N/m. Calculate the undamped natural frequency (Hz) of the system, the damping ratio and describe the type of response you would expect if the mass were displaced and released. What additional damping would be required to make the system critically damped?

12. Solve the differential equation below using homogeneous and particular solutions. Assume the system starts undeflected and at rest.

13. What would the displacement amplitude after 100ms for a second order system having a natural frequency of 13 rads/sec and a damping ratio of 0.20. Assume an initial displacement of 50mm, and a steady state displacement of 0mm. (Hint: Find the response as a function of time.)

14. Determine the first order differential equation given the graphical response shown below. Assume the input is a step function.

15. Explain with graphs how to develop first and second-order equations using experimental data.

16. The second order response below was obtained experimentally. Determine the parameters of the differential equation that resulted in the response assuming the input was a step function.

17. Develop equations (function of time) for the first and second order responses shown below.

18. Write a differential equation for a system that has a time constant of 2 s. For an input of 3, the steady state output is 6.

19. Find the explicit response of the following differential equation to the given step input. Assume the initial conditions are all zero.

20. A mass-spring-damper system has a mass of 10Kg and a spring coefficient of 1KN/m. Select a damper coefficient so that the system will have an overshoot of 20% for a step input.

21. Convert the following equation to phase-shift form.

22. Write the homogeneous differential equation for a second order system with the first peak at 1s and 10% overshoot. The system variable is ‘x’.

23. Write the differential equation for a first order system with a variable ‘x’. The system has the response shown in the graph below for an input of F=6.

24. A system is tested with a step input of F = 1N. The resulting output ‘y’ is shown in the graph below. a) Find the differential equation for the system. b) Find the explicit response (i.e., solve the differential equation) for an input of F=sin(t)N.