## 1.8 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 damper 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*103 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 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. A mass-spring-damper system has a mass of 10Kg and a spring coefficient of 1KN/m. Select a damping coefficient so that the system will have an overshoot of 20% for a step input.