4.10 PRACTICE PROBLEMS
1. Convert the following differential equations to state variable form.
2. a) Put the differential equations given below in state variable form.
b) Put the state equations in matrix form
3. Develop state equations for the mass-spring-damper system below.
4. The system below is comprised of two masses. There is viscous damping between the masses and between the bottom mass and the floor. The masses are also connected with a cable that is run over a massless and frictionless pulley. Write the differential equations for the system, and put them in state variable form.
5. Do a first order numerical integration of the derivative below from 0 to 10 seconds in one second steps. Assume the system starts undeflected. Write the equation for each time step.
6. Given the differential equation below integrate the values numerically for the first ten seconds with 1 second steps. Assume the initial value of x is 1. You may use first order or Runge Kutta integration.
7. Write a Scilab program to calculate the area under the function below using a numerical method, such as Simpson's rule. Test it by using the range from x=1 to x=1.2. Assume the sine function is in radians
8. Write a computer program in a language of your choice (C/C++, Java, Scilab script, etc.) that will numerically integrate the differential equation below.
9. Given the following differential equation and initial conditions, draw a sketch of the first 5 seconds of the output response. The input is a step function that turns on at t=0. Use at least two different methods, and compare the results.
10. a) For the mass-spring-damper system below solve the differential equation as a function of time. Assume the system starts at rest and undeflected. b) Also solve the problem using your Scilab (and state equations) to verify your solution. Sketch the results.
11. The mechanical system below is a mass-spring-damper system. A force 'F' of 100N is applied to the 10Kg cart at time t=0s. The motion is resisted by the spring and damper. The spring coefficient is 1000N/m, and the damper coefficient is to be determined. Follow the steps below to develop a solution to the problem. Assume the system always starts undeflected and at rest.
a) Develop the differential equation for the system.
b) Solve the differential equation using damper coefficients of 100Ns/m and 10000Ns/m. Draw a graph of the results.
c) Develop the state equations for the system.
d) Solve the system with a first order numerical analysis using Scilab for damper coefficients of 100Ns/m and 10000Ns/m. Draw a graph of the results.
e) Solve the system with a Runge Kutta numerical analysis using Scilab for damper coefficients of 100Ns/m and 10000Ns/m. Draw a graph of the results.
f) Write a computer program (in C, Java or Fortran) to do the Runge Kutta numerical integration in step e). Draw a graph of the results.
g) Compare all of the solutions found in the previous steps.
h) Select a damper coefficient to give an overall system damping factor of 1. Verify the results by numerically integrating.
12. For the mechanism illustrated in the figure below the values are K
s1=K
s2=100N/m, M
1=M
2=1kg, F=1N. Assume that the system starts at rest, and the springs are undeformed initially.
a. Derive the differential equations for the system.
b. Put the equations in state variable form.
c. Put the equations in state variable matrices.
d. Use Scilab to find values for x
1 and x
2 over the first 10 seconds. Provide the results in a table in 1 second intervals.
e. Use Scilab to plot the values for the first 10 seconds.
f. Use a Scilab program and the Runge-Kutta method to produce a graph of the first 10 seconds.
g. Repeat step g. using the first-order approximation method.
h. Use a C program to produce a graph of points for the first 10 seconds.
13. The second order response below was obtained for a unit step input (x) to a system. Develop the state equation and find the response to a ramp input.
14. Develop the state equations and numerically integrate the following differential equation for 20s for the given input function. Show the results in a table.
15. Find the response of the following differential equation to the given step input. Assume the initial conditions are all zero. Solve the problem using a numerical method and show the values from 0 to 10 seconds in 1 second intervals.
16. Write a Scilab program to implement the following equation to calculate the value of x.
17. Numerically integrate one time step of the differential equation below using a) first order integration and b) Runge Kutta integration.
18. For the mechanism shown in the figure below the values are K
s1=K
s2=100N/m, K
d1=10Nm/s, M
1=M
2=1kg, F=1N. Assume that the system starts at rest, and the springs are undeformed initially.
a. Derive the differential equations for the system.
b. Put the equations in state variable form.
c. Put the equations in state variable matrices.
d. Use Scilab and first order integration to find values for x
1 and x
2 over the first 10 seconds. Provide the results in a table in 1 second intervals.
e. Use Scilab to plot the values for the first 10 seconds using the values obtained in part d.
f. Use a Scilab program and the Runge-Kutta method to produce a graph of the first 10 seconds.
g. Use a C program to produce a list of points for the first 10 seconds.
h. Compare the results found in steps d, f and g in a table.
19. Use a numerical method to find the position of the mass below over the first 2s. Record these values in a table and sketch the curve. The mass starts at rest at x=0. The block experiences aerodynamic drag that opposes motion. Assume the surface is frictionless.
