Spring-Mass Systems

Aims

• To understand how to use numerical integration for 2nd order differential equations.
• To compare the Euler and Improved Euler integration methods.
• To investigate linear spring-mass systems.

Exercises

1. Modify the Java projectile motion program which uses the analytical solution to use numerial integration via the Euler approach as discussed in Acheson (equation 4.35). Remember, this is a two step process: use acceleration to update velocity and velocity to update position.
2. Plot the trajectory of both the analytical and the numerical approaches (use different colours) in the same window and compare.
3. Add the improved Euler approach - including using the variables as suggested in equation 4.35 in Acheson.
4. Compile and run the single spring mass sample program. Modify it to use the improved Euler method.
5. Modify it further to use both the Euler and improved Euler methods at the same time (two masses) and draws the mass using each approach.
6. Modify the program so that it draws the position curve (position versus time) Just a series of dots will do in the first instance.
7. Adjust the parameters so that they match those in Figure 4.7 and then Figure 4.9 of the textbook. Run the program and compare results.
8. Further modify the program so that the position of the mass is plotted against its velocity, that is, a phase diagram. What shape do you observe?