# Nonlinear finite elements/Homework 1

## Problem 1: Mathematical Model Development

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

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).