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VIBRATION AND CONTROL OF VIBRATION 31.5 in Software
VIBRATION AND CONTROL OF VIBRATION 31.5 Recognizing GS1  13 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Encoding EAN13 Supplement 5 In None Using Barcode generation for Software Control to generate, create EAN13 image in Software applications. VIBRATION AND CONTROL OF VIBRATION
EAN13 Recognizer In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. EAN13 Creator In Visual C#.NET Using Barcode drawer for .NET framework Control to generate, create EAN13 image in VS .NET applications. FIGURE 31.3 Variation of the ratio of displacement maxima with damping.
Encode EAN13 In .NET Using Barcode creation for ASP.NET Control to generate, create EAN / UCC  13 image in ASP.NET applications. UPC  13 Maker In Visual Studio .NET Using Barcode drawer for VS .NET Control to generate, create EAN / UCC  13 image in .NET applications. The equivalent viscous damping in a system is measured experimentally by using this principle. The system at rest is given an impact which provides initial velocity to the system and sets it into free vibration. The successive maxima of the ensuing vibration are measured, and by using Eq. (31.17) the damping ratio can be evaluated. The variation of the decaying amplitudes of free vibration with the damping ratio is plotted in Fig. 31.3 for different values of n. Critically Damped System (z = 1). When the system is critically damped, the roots of the characteristic equation given by Eq. (31.3) are equal and negative real quantities. Hence, the system does not execute oscillatory motion.The solution is of the form x = (A + Bt) exp ( nt) and after substitution of initial conditions, x = [x0 + (v0 + x0 n)t] exp ( nt) (31.19) (31.18) Generate EAN / UCC  13 In VB.NET Using Barcode generator for .NET framework Control to generate, create EAN / UCC  13 image in .NET framework applications. EAN / UCC  13 Creator In None Using Barcode encoder for Software Control to generate, create European Article Number 13 image in Software applications. This motion is shown graphically in Fig. 31.4, which gives the shortest time to rest. Overdamped System (z > 1). When the damping ratio is greater than unity, there are two distinct negative real roots for the characteristic equation given by Eq. (31.3). The motion in this case is described by x = exp ( nt) [A exp nt 2 1 + B exp ( nt 2 1)] (31.20) Making UPC Symbol In None Using Barcode creator for Software Control to generate, create GS1  12 image in Software applications. Generating Code 128A In None Using Barcode printer for Software Control to generate, create Code 128 image in Software applications. Downloaded from Digital Engineering Library @ McGrawHill (www.digitalengineeringlibrary.com) Copyright 2004 The McGrawHill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. EAN128 Encoder In None Using Barcode encoder for Software Control to generate, create EAN / UCC  13 image in Software applications. Bar Code Maker In None Using Barcode generator for Software Control to generate, create barcode image in Software applications. VIBRATION AND CONTROL OF VIBRATION 31.6
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Scan Universal Product Code Version A In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Paint DataMatrix In VS .NET Using Barcode encoder for Reporting Service Control to generate, create Data Matrix 2d barcode image in Reporting Service applications. FIGURE 31.4 damping.
UCC  12 Encoder In .NET Using Barcode drawer for ASP.NET Control to generate, create UPCA Supplement 5 image in ASP.NET applications. Creating DataMatrix In None Using Barcode encoder for Font Control to generate, create Data Matrix image in Font applications. Free vibration of a singledegreeoffreedom system under different values of
Create GTIN  13 In None Using Barcode creator for Online Control to generate, create UPC  13 image in Online applications. Painting Matrix Barcode In Java Using Barcode generator for Java Control to generate, create 2D Barcode image in Java applications. where A= and 0 = n 2 1 All four types of motion are shown in Fig. 31.4. If the mass is suspended by a spring and damper as shown in Fig. 31.5, the spring will be stretched by an amount st, the static deflection in the equilibrium position. In such a case, the equation of motion is m + cx + k(x + st) = mg x (31.21) 1 v0 + nx0 x0 + n 2 B= 1 2 x0 + nx0 0 Downloaded from Digital Engineering Library @ McGrawHill (www.digitalengineeringlibrary.com) Copyright 2004 The McGrawHill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. VIBRATION AND CONTROL OF VIBRATION 31.7
VIBRATION AND CONTROL OF VIBRATION
FIGURE 31.5 Model of a singledegreeoffreedom system showing the static deflection due to weight.
Since the force in the spring due to the static equilibrium is equal to the weight, or k st = mg = W, the equation of motion reduces to m + cx + kx = 0 x (31.22) which is identical to Eq. (31.1). Hence the solution is also similar to that of Eq. (31.1). In view of Eq. (31.21) and since n = (k/m)1/2, the natural frequency can also be obtained by n = g st (31.23) An approximate value of the fundamental natural frequency of any complex mechanical system can be obtained by reducing it to a singledegreeoffreedom system. For example, a shaft supporting several disks (wheels) can be reduced to a singledegreeoffreedom 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. 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 gearmounting errors or tooth profile errors, which will result in torsional vibration of the geared shafts. A singledegreeoffreedom 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) Downloaded from Digital Engineering Library @ McGrawHill (www.digitalengineeringlibrary.com) Copyright 2004 The McGrawHill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

