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An approximate value of the fundamental natural frequency of any complex mechanical system can be obtained by reducing it to a single-degree-of-freedom system. For example, a shaft supporting several disks (wheels) can be reduced to a single-degree-of-freedom system by lumping the masses of all the disks at the center and obtaining the equivalent stiffness of the shaft by using simple flexure theory.

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31.2.2 Torsional Systems Rotating shafts transmitting torque will experience torsional vibrations if the torque is nonuniform, as in the case of an automobile crankshaft. In rotating shafts involving gears, the transmitted torque will fluctuate because of gear-mounting errors or tooth profile errors, which will result in torsional vibration of the geared shafts. A single-degree-of-freedom torsional system is shown in Fig. 31.6. It has a massless shaft of torsional stiffness k, a damper with damping coefficient c, and a disk with polar mass moment of inertia J. The torsional stiffness is defined as the resisting torque of the shaft per unit of angular twist, and the damping coefficient is the resisting torque of the damper per unit of angular velocity. Either the damping can be externally applied, or it can be inherent structural damping. The equation of motion of the system in torsion is given J + c + k = 0 (31.24)

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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

-(void)setParticle:(NSString*)atlasKey { . . . mesh.colors = colors; mesh.colorSize = 4; }

Equation (31.24) is in the same form as Eq. (31.1), except that the former deals with moments whereas the latter deals with forces. The solution of Eq. (31.24) will be of the same form as that of Eq. (31.1), except that J replaces m and k and c refer to torsional stiffness and torsional damping coefficient.

31.2.3 Forced Vibration System Excited at the Mass. A vibrating system with a sinusoidal force acting on the mass is shown in Fig. 31.7. The equation of motion is m + cx + kx = F0 sin t x (31.25)

Listing 4-1

Assuming that the steady-state response lags behind the force by an angle , we see that the solution can be written in the form xs = X sin ( t ) (31.26)

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

Now you have a place to put your colors, and the mesh is all hooked up. Next you just need to set the color in the particles and generate the color array during your update loop. Let s start at the beginning of the particle life cycle: emitParticles. You need to set the initial RGBA values on the particle as well as the new startingLife value:

Substituting in Eq. (31.26), we find that the steady-state solution can be obtained: xs = (F0/k) sin ( t ) [(1 2/ 2 )2 + (2 / n)2]1/2 n (31.27)

Using the complementary part of the solution from Eq. (31.19), we see that the complete solution is x = xs + exp ( nt) [A exp ( nt 2 1) + B exp ( nt 2 1)] (31.28)

CONTENTS AT A GLANCE Conserving Power Basic Components Construction of the Communicator IrDA Communicator (Low-Power Version) Parts List IrDA Communicator (High-Power Version) Parts List

If the system is undamped, the response is obtained by substituting c = 0 in Eq. (31.25) or = 0 in Eq. (31.28). When the system is undamped, if the exciting frequency coincides with the system natural frequency, say / n = 1.0, the system response will be infinite. If the system is damped, the complementary part of the solution decays exponentially and will be nonexistent after a few cycles of oscillation; subsequently the system response is the steady-state response. At steady state, the nondimensional response amplitude is obtained from Eq. (31.27) as X = F0/k 1 2 2 n

-(void)emitNewParticles:(NSTimeInterval)deltaTime { . . . NSInteger index; for (index = 0; index < newParticleCount; index++) { . . . p.r = startR; // set the colors p.g = startG; p.b = startB; p.a = startA; p.life = BBRandomFloat(lifeRange); p.startingLife = p.life; // set so you can do color animation . . . } }

2 n

2 1/2

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