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Geared Systems. When a shaft transmits torque to another through a gear drive of speed ratio n, it is necessary to reduce the geared torsional system to an equivalent single-shaft system to find its natural frequency.The moments of inertia and the stiffness of the equivalent system are obtained through a consideration of the kinetic and potential energies of the system. Consider the geared torsional system in Fig. 31.22a. The speed of the second shaft is 2 = n 1. Assuming massless gears, we see that the kinetic energy of the system is T= 1 2 1 1 J1 1 + J2 n2 2 2 2 (31.77)

The stem and the head of the flower, because they re more rounded and solid, were rendered with standard diffuse and ambient reflection Both ambient and diffuse reflection are standard lighting models in OpenGL ES 11 With ambient reflection, a surface is uniformly shaded depending on the light color and intensity Diffuse reflection, on the other hand, changes the shading on the surface depending on the angle between the light direction and the surface normal The following equation describes the color of a surface with diffuse reflection: Cd = max(0, L N) Cl Cd is the color of the diffuse reflection, L is the vector from the surface to the light, N is the surface normal, and Cl is the color of the light.

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Thus the equivalent mass moment of inertia of disk 2 referred to shaft 1 is n2J2. If disks 1 and 2 are clamped and a torque is applied to gear 1, rotating it through an angle 1, there will be deformations in both shafts 1 and 2. Gear 2 will rotate through an angle 2 = n 1. The potential energy stored in the two shafts is U= 1 1 k 1 2 + k2 n2 2 1 1 2 2 (31.78)

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Hence the equivalent stiffness of shaft 2 referred to shaft 1 is n2k2. The equivalent torsional system is shown in Fig. 31.22b, where the stiffness and inertia of one side of the system are multiplied by the square of the speed ratio to obtain the corresponding equivalent values for Holzer calculations.

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31.3.3 Continuous Systems Engineering structures, in general, have distributed mass and elasticity. Such structures can be treated as multidegree-of-freedom systems by lumping their masses at certain locations and connecting them by representative spring elements. However, it is necessary to consider several such lumped masses and springs to get sufficiently accurate values for the natural frequencies. If only the fundamental natural frequency or the first few natural frequencies are of interest, it is convenient to use some approximate methods based on energy formulations discussed here. Rayleigh Method. This method can give the natural frequency of a structure of any specific mode of vibration. A deflection shape satisfying the geometric boundary conditions has to be assumed initially. If the natural frequency of the fundamental mode of vibration is of interest, then a good approximation would be the static deflection shape. For a harmonic motion, the maximum kinetic energy of a structure can be written in the form Tmax = 2 C1 (31.79)

Notice that the color specifically depends on the dot product between L and N, which means it s directly related to the cosine of the angle between those two vectors When the dot product is negative (L and N are pointing away from each other), there is no diffuse reflection The following code creates an OpenGL light with diffuse properties and a global ambient light:.

0.1- F 35-V tantalum capacitor 20-pF disk capacitor 4.7- F 35-V electrolytic capacitor 0.22- F 35-V disk capacitor Red LED IR LED IR detector transistor Normally open momentary pushbutton SPST toggle power switch PC board, 4-pin male header, 4-pin female header, DB-9 connector wire, enclosure, etc.

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.

where C1 depends on the assumed deflection shape. The maximum potential energy is of the form Umax = C2 (31.80)

Neglecting damping, we see that the maximum kinetic energy must be equal to the maximum potential energy. Hence, the natural frequency is 2 = C2 C1 (31.81)

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