Attitude Dynamics

We have come a long way on the coattails of Sir Isaac Newton. His second law enables us to calculate the trajectory of any aerospace vehicle, provided we can model the external forces. For many applications we have to predict only the movement of the c. m. and can neglect the details of the attitude motions. Three – and even five-DoF simulations can be built on Newton’s law only. If you were to stop here, what would you be missing?

It is like the difference between riding on a ferns wheel, which transports your c. m., vs the twists and turns you experience on the Kumba at Busch Gardens amusement park. Attitude dynamics brings excitement into the dullness of tra­jectory studies and three more dimensions to the modeling task. Do you see the affinity between geometry, kinematics, and dynamics? Chapter 3 dealt with ge­ometry, describing position in terms of location and orientation; Chapter 4 dealt with kinematics; and now we characterize dynamics, consisting of translation (Chapter 5) and attitude motions (this chapter).

Attitude dynamics are the domain of Leonhard Euler, a Swiss physicist of the 18th century, whose name we have used before, but who now competes with Newton head on. We will study his law of attitude motion in detail. It has a strong resemblance to Newton’s law. Newton’s building blocks are mass, linear velocity, and force, whereas Euler’s law uses moment of inertia, angular velocity, and moment. But it gets more complicated. Mass is a simple scalar, whereas the moment of inertia requires a second-order tensor as a descriptor.

To prepare the way, we start with the moment-of-inertia (MOI) tensor and derive some useful theorems that help us calculate its value for missiles and aircraft. The geometrical picture of a MOI ellipsoid will help us visualize the tensor character­istics. The concept of principal axes will be most useful in simplifying the attitude equations.

Combining the MOI tensor with the angular velocity vector will lead to the concept of angular momentum. We will learn how to calculate it for a collection of particles and a cluster of bodies. Again, we will see how important the c. m. is for simplified formulations.

From these elements we can formulate Euler’s law. As Newton’s law is often paraphrased as force = mass x linear acceleration, so can Euler’s law be regarded as moment = MOI x angular acceleration. For freely moving bodies like missiles and aircraft, we use the c. m. as a reference point. Some gyrodynamic applications with a fixed point—for instance the contact point of a top—will lead to an alternate formulation of Euler’s law.

Gyrodynamics is a fascinating study of rigid body motions, and we will devote some time to it, both in reverence to the giants of mechanics, like Euler, Poinsot, Klein, and Magnus, and because of its modem applications in INS and stabilization of spacecraft. I will introduce the kinetic energy of spinning bodies and the energy

ellipsoid. Two integrals of motion are particularly fertile for studying the motions of force-free rigid bodies.

If you persevere with me through this chapter, you will have mastered a mod­ern treatment of geometry, kinematics, and dynamics of Newtonian and Eulerian motions. The remaining chapters deal with a host of applications relevant for today’s aerospace engineer with particular emphasis on computer modeling and simulation. So, with verve let us tackle a new tensor concept.