Nonlinear finite elements/Homework 1

Problem 1: Mathematical Model Development

Consider the motion of a pendulum (see Figure 1).

Figure 1. The motion of a pendulum.

Assume that:

1. The rod and the mass at the end are rigid.
2. The mass of the rod is negligible compared to the mass of the bob.
3. There is no friction at the pivot.

Find the following:

1. The equation of motion of the bob (angular displacement as a function of time). State all other assumptions.
2. Why is the equation of motion "nonlinear"?
3. Assume that $\theta$ is "small". Derive the equation of motion for this situation.
4. Why is the small $\theta$ equation of motion "linear"?
5. Derive a finite element formulation for the linear equation of motion.
6. Use ANSYS or some other tool of your choice (including your own code) to solve the linear equation of motion via finite elements. Compare with the exact solution. You can use any reasonable values for time ($t$), mass ($m$), and length ($l$).
7. What happens when you try to solve the nonlinear equation of motion with finite elements? (You will need to formulate the FE formulation to see the difference.)

Problem 2: Numerical Exercise

In the learning resources section, you have read about the process of formulating a FE model for an axial bar under a distributed body load. Ansys 9 shows you how to solve the same problem using ANSYS. Work the example out on your own.

Next, consider a mini centrifuge of the type manufactured by Kisker Biotech (see Figure 2).

Figure 2(b). Plan view of mini centrifuge.

This type of mini centrifuge is widely used in the fields of genetic research and microbiology. The centrifuge is rate to run at 15,000 RPM. The allowable temperature range is -20 C to 40 C. The nominal radius is 140 mm. Assume other geometric quantities on the basis of the figure.

We want to extend the temperature range to -80 C - 100 C and the allowable RPM to 25,000.

Assume that the current material is a light metal (any other material of your choice will also do). The only requirement is that it should be cheap and stiff. Ignore bending and the weight of the sample tubes.

Find the thinnest beam of the material of your choice that can stand the axial load that is generated due to rotation at the new speed. Use ANSYS or your own code to solve the problem. State all simplifying assumptions.

We just need a ballpark value. A 1-D simulation will do.

Nonlinear finite elements