System motion fundamentals
Tossing stuff into the air helps us understand moments of inertia and principal axes that are essential for design.
By Kevin C. Craig, PhD -- Test & Measurement World, 8/1/2010 12:00:00 AM
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Take any book and wrap a few rubber bands around it. Toss the book in the air three times, each time giving it a pure rotation, as best you can, about one of the three axes perpendicular to its sides. What do you observe? This simple experiment demonstrates fundamentals essential to the design of rotating machines, space satellites, and much more.
The motion of any system depends on the forces acting on it and its constitution—that is, the manner in which its mass is distributed, usually in response to strength, weight, space, and stiffness requirements. To predict dynamic behavior, you need to know the mass, the location of the mass center, and six quantities called the inertia scalars. The concept of mass center is well known, and its location is used to determine the translational motion of a body. But inertia scalars are not well understood. At any point in a body, you can determine six independent quantities called the three mass moments of inertia and the three products of inertia. Together, they quantify how mass is distributed with respect to three perpendicular axes fixed in the body at that point. The mass moments of inertia quantify the resistance of the body to angular acceleration about each axis, and the products of inertia quantify the symmetry of the mass distribution with respect to each plane. In addition, there is always a particular orientation of those axes such that the products of inertia are all zero. The remaining three quantities—the principal mass moments of inertia—play an important role in dynamic analysis.
In the tossed book experiment, the only force acting on the book is gravity, and that force goes through the mass center. The book then is moment-free, spinning freely in space. Since the book is moment-free, the magnitude of its angular momentum vector, H, must be constant (conserved), and if you neglect any translation, the rotational kinetic energy, T, must be constant (conserved). Plotting constancy of T and H using the absolute angular velocities ω1, ω2, and ω3 as ordinates gives two ellipsoids.
The only allowable spinning states are at the intersections of these two ellipsoids. The lines on the figure are the intersections for a fixed value of T and various values of H, where I1 > I2 > I3. The three intersections are circles at the greatest and least axes and a saddle at the intermediate axis. This indicates that rotation about the axes with the greatest and least moments of inertia is stable to small oscillations, while rotation with respect to the intermediate axis is unstable to small oscillations.
Another way to arrive at this conclusion is by considering Euler's Equations for this situation, where the 1, 2, 3 axes are body-fixed principal axes through the mass center.
If the body is given a constant spin rate, Ω, exactly about any one of its principal axes, it will continue to spin about that axis. But what happens if that motion is perturbed by an angular velocity ωp? Assume ω1 = Ω + ωp. Analysis of Euler's Equations with linearization shows the resulting equation. If the coefficient of ω2 is negative, the solution for ω2 grows with time. This happens if the 2-axis is the intermediate principal axis.
You can apply the topic of principal axes to everyday practice. Modern machines have high-speed rotors fastened to shafts. If the principal axis of the mounted object does not coincide with the axis of the shaft, making the system dynamically balanced, then dynamic bearing reactions result that could lead to premature bearing failure.
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