# Troublshooting your pc for dummies - Bourg D.M.

ISBN 0-596-00006-5

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dG/dt — ma

and

7" F - dG/dt ma

So far, we have considered only translation of the body without rotation. In generalized 3D motion, you must account for the rotational motion of the body and will therefore need some additional equations to fully describe the body’s motion. Specifically, you will require analogous formulas relating the sum of all moments (torque) on a body

16 I Chapter 1: Basic Concepts

to the rate of change in its angular momentum over time or the derivative of angular momentum with respect to time. Thus,

^Mcg = d/dt(Hcg)

where ^2 MCg is the sum of all moments about the body center of gravity, and H is the angular momentum of the body Mcg can be expressed as

Mcg - rxF

where F is a force acting on the body, r is the distance vector from F, perpendicular to the line of action of F, to the center of gravity of the body, and x is the vector cross product operator.

The angular momentum of the body is the sum of the moments of the momentum of all particles in the body abqut the axis of rotation, which in this case we assume passes through the center of gravity of the body This can be expressed as

Hcg = ^ri x mi(o} x ri)

where i represents the i th particle making up the body, to is the angular velocity of the body about the axis under consideration, and (cox r;) is the angular momentum of the i th particle, which has a magnitude of tori. For rotation about a given axis this equation can be rewritten in the form

Hcg = J cor2 dm

Given that the angular velocity is the same for all particles making up the rigid body, we have

Hcg = co j rzdm

and recalling chat moment of inertia, I, equals /r2 dm, we get

Hcg — I co

Taking the derivative with respect to time, we obtain

dHcg/dt = d/dt(Ico) Idco/dt — la where a is the angular acceleration of the body about a given axis.

Finally we can write

As I stated in our discussion on mass moment of inertia, we will have to further generalize our formulas for moment of inertia and angular moment to account for general rotation about any body axis. Generally, M and a will be vector quantities, while I will be a tensor,* since the magnitude of moment of inertia for a body may vary depending on the axis of rotation.

* In this case, I will be a second rank tensor, which is essentially a 3 x 3 matrix. A vector is actually a tensor of rank 1, and a scalar is actually a tensor of rank zero.

Newton's Second Law of Motion 17

Tensors

A tensor is a mathematical expression that has magnitude and dife'ction, but its magnitude might not be unique, depending on the direction. Tensors are usually used to represent properties of materials when these properties have different magnitudes in different directions. Materials with properties that vary depending on direction are called anisotropic (isotropic implies the same magnitude in all directions). For example, consider the elasticity (or strength) of two common materials, a sheet of plain paper and a piece of woven or knitted cloth. Take the sheet of paper and, holding it flat, pull on it softly from opposing ends. Try this lengthwise, widthwise, and then along a diagonal. You should observe that the paper seems just as strong, or stretches about the same, in all directions. It is isotropic; therefore, only a single scalar constant is required to represent its strength for all directions.

Now try to find a piece of cloth with a simple, relatively loose weave in which the threads in one direction are perpendicular to the threads in the other direction. Most neckties will do. Try the same pull test that you conducted with the sheet of paper, pulling the cloth along each thread direction and then at a diagonal to the threads. You should observe that the clorh stretches more when you pull it along a diagonal to the threads than when you pull it along the direction of the run of the threads. The cloth is anisotropic in that it exhibits different elastic (or strength) properties depending on the direction of pull; thus, a collection of vector quantities (a tensor) is required to represent its strength for all directions. .

In the context of the subject of this book, the property under consideration is a body’s moment of inertia, which in 3D requires nine components to fully describe it for any arbitrary rotation. Moment of inertia is not a strength property as in the paper and cloth example, but it is a property of the body that varies with the axis of rotation. Since nine components are required, moment of inertia will be generalized in the form of a 3 x 3 matrix (second-rank tensor) later in this book. .

I need to mention a few things at this point regarding coordinates, which will become important when you’re writing your real-time simulator. Both the equations of motion have, so far, been written in terms of global coordinates and not body-fixed coordinates. That’s okayfor the linear equation of motion, in which you can track the body’s location and velocity in the global coordinate system. However, from a computational point of view, you don’t want to do that for the angular equation of motion for bodies that rotate in three dimensions.* The reason why is because the moment of inertia term, when calculated with respect to global coordinates, actually changes depending on the body’s position and orientation. This means that during your simulation you’ll have to recalculate the inertia matrix (and its inverse) a lot, which is computationally inefficient. It’s better to rewrite the equations of motion in terms of local (attached to the body) coordinates so that you have to calculate the inertia matrix (and its inverse) only once, at the start of your simulation.

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