Rotational Dynamics

Virtually all games, but especially ones like SSX Blur for the Wii need to model the physics of rigid objects. Without good rotational dynamics, many of the motions would look cheap and unrealistic.

# Learning Objectives

1. Explain what the six degrees of freedom are for a rigid object.
2. Model the translation and rotation of a rigid object in 3D.

# Introduction

Any object in 3D space has at least three degrees of freedom. It can move up and down, left and right, and forward and back, or you can think of it as it being able to move in the x, y, and z directions. However for an extended rigid object, you can also have three additional degrees of freedom. Of course, the object can still move in the x, y and z directions, but it can also rotate about the x, y, and z axes. This brings the total degrees of freedom for an extended and rigid object up to a total of six. Modeling rotational motion is not that different that modeling linear motion. All of the same type of kinematics that we did with linear motion will be directly applicable to rotational motion.

Quantity Translation Rotation
Displacement $x$ $\theta$
Velocity $\vec v = \frac { \Delta x } { \Delta t }$ $\vec \omega = \frac { \Delta \theta } { \Delta t }$
Acceleration $\vec a = \frac { \Delta v } { \Delta t }$ $\vec \alpha = \frac { \Delta \omega } { \Delta t }$

If you know the angular acceleration, $\alpha$, then you can use the Forward Euler Method to determine the new angular velocity and angular displacement just as was done with linear motion.

(1)
\begin{align} \theta_{new} = \theta_{old} + \omega_{old} \Delta t \end{align}
(2)
\begin{align} \omega_{new} = \omega_{old} + \alpha_{old} \Delta t \end{align}

Finding the acceleration for linear motion involved first finding the sum of the forces acting on an object and using Newton’s 2nd Law, $\vec F = m \vec a$, to solve for the acceleration. For rotational motion, a similar thing is done. Just like a force causes a change in linear motion, a torque will cause a change in rotational motion. A torque is applied to an object when a force acts along a line not connected with the object’s center of mass. The distance between where a force acts on a body and the center of rotation for that body is called the lever arm. Torque is the cross-product of the lever arm and the force.

(3)
\begin{align} \vec \tau = \vec r \times \vec F \end{align}

or

(4)
\begin{align} \| \tau \| = r F \sin \left( \theta \right) \end{align}

The order of this, as with any cross-product, is important. The unit for torque is a Newton-meter (N-m) in the standard SI metric system and foot-pounds (ft-lbs) in the British system. By adding all of the torques produced by all of the different forces on an object, one gets the net torque and with the net torque one can find the angular acceleration. Just as with Newton’s 2nd Law, mass also has an effect on the acceleration. The greater the mass for a given net torque, the lower the angular acceleration. With rotational motion, though, where the mass is distributed is just as important as how much is there. The quantity that accounts for both the mass and its distribution in an object is called the moment of inertia, I. For a point mass, the moment of inertia is defined as

(5)
$$I = m r^2$$

where r is the distance from the mass to the center of rotation for the object. An extended object can be thought of as a collection of point masses, so the total moment of inertia is the sum of the moments of inertia for each of its individual parts. With the moment of inertia, we can now express the relationship between torque and angular acceleration.

(6)
\begin{align} \vec \tau_{net} = I \vec \alpha \end{align}

Notice that like Newton’s 2nd Law, this is a vector equation.

# Prelab Questions and Exercises

1. Fill in the angular velocity and position using the Forward Euler Method in the table below.
0.0 0.0 0.0 0.0
0.1 2.4
0.2 0.7
0.3 -0.3
0.4 3.1

2. Calculate the torque produced given the following force and lever arm:
$\vec F = \left( \begin{array}{cc} 14.0 \\ 9.3 \\ -2.1 \end{array} \right) N$ and $\vec r = \left( \begin{array}{cc} 0.4 \\ -3.8 \\ -2.7 \end{array} \right) m$.
3. Find the moment of inertia about the center of mass for the object shown below if $r = 1.20 m, m_1 = 0.50 kg$ and $m_2 = 1.80 kg$.

4. What amount of torque is needed to cause the object in Question 3 to have an angular acceleration of 7.42 rad/s2?

# Laboratory Procedures

Write a program that models the motion of a rigid object through 2D space after being accelerated by a user-supplied force for a user-supplied amount of time. The object is two massive balls connected by a thin light-weight and rigid rod. You can assume that the rod’s mass is negligible compared to the mass of the two balls. Details about the dimensions and masses for the object are shown in the figure below. Your program should output the coordinates for the center of mass of the system and the angular position.

Initial conditions:

• $r = 0.35 m, m_1 = 1.20 kg$ and $m_2 = 4.90 kg$
• The center of mass is located at the origin with the rod oriented along the x-axis and the larger ball to the right (+x).
• The force is applied to the small mass at an angle of 30 degrees.
• Use a timestep of 0.02 s.