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Vibration analysis and control of vibrations are important and integral aspects of every machine design procedure. Establishing an appropriate mathematical model, its analysis, interpretation of the solutions, and incorporation of these results in the design, testing, evaluation, maintenance, and troubleshooting require a sound understanding of the principles of vibration. All the essential materials dealing with various aspects of machine vibrations are presented here in a form suitable for most design applications. Readers are encouraged to consult the references for more details.

(15,000 rows)

31.2.1 Free Vibration A single-degree-of-freedom system is shown in Fig. 31.1. It consists of a mass m constrained by a spring of stiffness k, and a damper with viscous damping coefficient c. The stiffness coefficient k is defined as the spring force per unit deflection. The coef-

2

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31.1 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.

.. ...

ficient of viscous damping c is the force provided by the damper opposing the motion per unit velocity. If the mass is given an initial displacement, it will start vibrating about its equilibrium position. The equation of motion is given by m + cx + kx = 0 x (31.1)

. .. . .

where x is measured from the equilibrium position and dots above variables represent differentiation with respect to time. By substituting a solution of the form x = est into Eq. (31.1), the characteristic equation is obtained: ms 2 + cs + k = 0 The two roots of the characteristic equation are s = n i n(1 2)1/2 where n = (k/m)1/2 is undamped natural frequency = c/cc is damping ratio cc = 2m n is critical damping coefficient i= 1 (31.3) (31.2)

Rights Management is a five-stage process: packaging, distribution, establishing a license server, license acquisition, and playback (Figure 2.16). To package a file, the Rights Manager is used to encrypt it and lock it with a key. The key is stored in an encrypted license, which is distributed separately. Other information is added to the media file, such as the URL where the license can be acquired. The packaged digital media file is saved in Windows Media Format. To distribute the rights managed package, it can be placed on a Web site for download, placed on a media server for streaming, distributed on a CD, or e-mailed to consumers. Consumers are free to redistribute the rights managed package to their friends. In order for consumers to be able to unlock the rights managed packages, the content owner chooses a license clearinghouse to store the specific rights of the license and implements the Windows Media Rights Manager license services. That s all they need do to establish a license server. The role of the clearinghouse is to authenticate the consumer s request for a license. Because the media files and licenses are stored and distributed separately, it is easier to manage the entire system. To play a packaged media file, the consumer must acquire a license key to unlock the file. The process of acquiring a license begins automatically when the consumer attempts to acquire the protected content, acquires a pre-delivered license, or plays the file for the first time. Windows Media Rights Manager either sends the consumer to a registration page where information is requested or payment

.. ...

Depending on the value of , four cases arise. Undamped System (z = 0). In this case, the two roots of the characteristic equation are s = i n = i(k/m)1/2 and the corresponding solution is x = A cos nt + B sin nt (31.5) (31.4)

. .. . .

where A and B are arbitrary constants depending on the initial conditions of the motion. If the initial displacement is x0 and the initial velocity is v0, by substituting these values in Eq. (31.5) it is possible to solve for constants A and B. Accordingly, the solution is

... ..... ...... .. .

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.