Introduction

Numerical solutions to ODEs are approximate solutions given at discrete points in time instead of at all points in time as are analytical solutions. On the other hand, if done properly, numerical solutions are extremely accurate and can be used to greatly enhance or predictive capabilities as engineers. As a matter of fact, modern engineering and science would not exist were it not for numerical approximations to engineering problems. See:

• Numerical Methods at Penn State.
• Guide to Available Mathematical Software.

In Part A of today's mini-project, we will derive the equations of motion of a swinging pendulum bob attached to an elastic rod. You will find that the equations of motion are not solvable and therefore you must resort to computer approximation. In Part B of this mini-project you will look at the dynamics of a mass on the end of an elastic rod sliding on a hydrodynamic layer. Again, the equations will not be solvable and you will have to numerically solve them.

Part A - Motion of an Elastic Pendulum

Figure 1. Pendulum consisting of a mass attached to an elastic rod.

The 0.25-kg mass, which is attached to the elastic rod of stiffness 10 N/m and undeformed length 0.5 m, is free to move in the vertical plane under the influence of gravity. The mass is released from rest when the angle q = 0 with the rod stretched 0.25 m. Assume that the rod can only undergo tension and compression and that it always remains straight as the pendulum swings in the vertical plane. For this system:

1. Derive the equations of motion and state the initial conditions.
2. Solve the equations numerically from the time of release t until t = 10 seconds.
3. Plot the speed of the mass during this period of integration.
4. Plot the tension in the rod as a function of time for this period of integration. Is this what you expected to see?
5. On the same figure, plot the kinetic energy, potential energy, and total energy of the mass as a function of time for the period of integration. Comment on what you see.
6. Plot the actual trajectory of the mass as you would see it for t = 0 until t = 10 seconds.
7. Now, add aerodynamic drag that is linearly proportional to the speed of the mass, solve the equations for 15 secs and re-do the plots in Parts 3, 5 and 6. Use two different drag coefficients: c = 0.05 kg/s and c = 0.15 kg/s. Try others if you like. Comment on your results.

Finally, here is a very cool QuickTime movie of the motion of the elastic pendulum for 10 seconds.

Part B - Whirling Motion With Combined Elasticity and Friction

Introduction

Figure 2. Material point sliding on the xy-plane while attached at the end of an elastic rod.

The surface is covered by a film of lubricant intended to facilitate the sliding motion. The action of the lubricant on the moving mass turns out to be equivalent to a viscous resistance force proportional to the mass' velocity with viscosity coefficient c = 0.3 Ns/m. The mass is connected to the (fixed) origin of the xy-plane via an elastic rod with a free length L = 0.5 m and elasticity constant k = 100 N/m. The rod can elastically extend but cannot bend. The mass is acted upon by a force F = (5.0/R) N oriented always in a direction perpendicular to the rod, where R is the length of the rod. From a physical viewpoint, the force F results from the application of a constant moment of magnitude 5.0 N m to the elastic rod. At time t = 0, the mass is at rest with an initial position characterized by R = 0.1 m and y = 0.

Analysis

1. Derive the equations of motion and state the corresponding initial conditions.
2. As it turns out, after some time the system at hand will be characterized by a circular motion with constant angular velocity. For convenience, this part of the motion will be referred to as the steady state solution. Analytically (i.e., non-numerically) determine the radius of the circular trajectory and the corresponding value of the angular velocity for the steady state solution.
3. Numerically integrate the equations of motion to compute and then plot the trajectory of the mass during the interval of time [0, 10 s]. Verify that the trajectory will, at some point, coincide with the circle determined at point 2.
4. Finally, repeat the operations done in Part 3 for other two sets of arbitrarily assigned initial conditions and verify that, regardless of initial conditions the motion of the mass will converge to the steady state solution. Provide a physical explanation for this behavior.

Here is a very cool QuickTime movie of the whirling motion of the mass for the first 5 seconds.

Mini-Project Report

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[ESM] [ESM Faculty] [PSU Engineering] [Penn State]

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