 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
barcode generator vb.net source code VIBRATION AND CONTROL OF VIBRATION 31.15 in Software
VIBRATION AND CONTROL OF VIBRATION 31.15 EAN13 Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Paint GTIN  13 In None Using Barcode creator for Software Control to generate, create EAN13 Supplement 5 image in Software applications. VIBRATION AND CONTROL OF VIBRATION
Recognize GS1  13 In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. EAN13 Printer In C# Using Barcode creation for VS .NET Control to generate, create EAN13 Supplement 5 image in Visual Studio .NET applications. FIGURE 31.13 Phase angle between transmitted and applied forces.
Print GTIN  13 In VS .NET Using Barcode creator for ASP.NET Control to generate, create EAN13 image in ASP.NET applications. EAN / UCC  13 Generator In Visual Studio .NET Using Barcode creator for .NET framework Control to generate, create UPC  13 image in VS .NET applications. FIGURE 31.14 Dynamic system subject to unbalanced excitation.
Encode EAN13 In VB.NET Using Barcode creation for VS .NET Control to generate, create EAN13 image in .NET framework applications. Make Code128 In None Using Barcode creator for Software Control to generate, create Code128 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. GS1 128 Encoder In None Using Barcode drawer for Software Control to generate, create GS1 128 image in Software applications. Bar Code Creation In None Using Barcode printer for Software Control to generate, create bar code image in Software applications. VIBRATION AND CONTROL OF VIBRATION 31.16
Creating EAN13 Supplement 5 In None Using Barcode maker for Software Control to generate, create EAN13 Supplement 5 image in Software applications. Code39 Generation In None Using Barcode maker for Software Control to generate, create Code 3/9 image in Software applications. LOAD CAPABILITY CONSIDERATIONS
Code 11 Generation In None Using Barcode creator for Software Control to generate, create USD  8 image in Software applications. Painting Code 128 Code Set A In None Using Barcode creator for Word Control to generate, create Code 128B image in Microsoft Word applications. FIGURE 31.15 A base excited system.
EAN / UCC  13 Reader In Visual C# Using Barcode reader for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Make Linear Barcode In .NET Framework Using Barcode maker for ASP.NET Control to generate, create Linear Barcode image in ASP.NET applications. where F0 = u0 (k2 + c2 2)1/2 and = tan 1 k c (31.49) (31.48) Create EAN / UCC  13 In VS .NET Using Barcode creator for Reporting Service Control to generate, create UCC128 image in Reporting Service applications. Encoding GS1  12 In Java Using Barcode generation for Android Control to generate, create UCC  12 image in Android applications. Equation (31.47) is identical to Eq. (31.25) except for the phase . Hence the solution is similar to that of Eq. (31.25). If the ratio of the system response to the base displacement is defined as the motion transmissibility, it will have the same form as the force transmissibility given in Eq. (31.38). Resonance, System Bandwidth, and Q Factor. A vibrating system is said to be in resonance when the response is maximum. The displacement and acceleration responses are maximum when = n(1 2 2)1/2 whereas velocity response is maximum when = n (31.51) (31.50) Code128 Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. ECC200 Creation In None Using Barcode encoder for Microsoft Excel Control to generate, create Data Matrix 2d barcode image in Excel applications. In the case of an undamped system, the response is maximum when = n, where n is the frequency of free vibration of the system. For a damped system, the frequency of free oscillations or the damped natural frequency is given by d = n (1 2 )1/2 (31.52) In many mechanical systems, the damping is small and the resonant frequency and the damped natural frequency are approximately the same. When the system has negligible damping, the frequency response has a sharp peak at resonance; but when the damping is large, the frequency response near resonance will be broad, as shown in Fig. 31.8. A section of the plot for a specific damping value is given in Fig. 31.16. The Q factor is defined as Q= 1 = Rmax 2 (31.53) 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 31.17
VIBRATION AND CONTROL OF VIBRATION
FIGURE 31.16 Resonance, bandwidth, and Q factor.
which is equal to the maximum response in physical systems with low damping. The bandwidth is defined as the width of the response curve measured at the halfpower points, where the response is Rmax/ 2. For physical systems with < 0.1, the bandwidth can be approximated by = 2 n = n Q (31.54) Forced Vibration of Torsional Systems. In the torsional system of Fig. 31.3, if the disk is subjected to a sinusoidal external torque, the equation of motion can be written as (31.55) J + c + k = T0 sin t Equation (31.55) has the same form as Eq. (31.25). Hence the solution can be obtained by replacing m by J and F0 by T0 and by using torsional stiffness and torsional damping coefficients for k and c, respectively, in the solution of Eq. (31.25). 31.2.4 Numerical Integration of Differential Equations of Motion: RungeKutta Method When the differential equation cannot be integrated in closed form, numerical methods can be employed. If the system is nonlinear or if the system excitation can 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 31.18
LOAD CAPABILITY CONSIDERATIONS
not be expressed as a simple analytical function, then the numerical method is the only recourse to obtain the system response. The differential equation of motion of a system can be expressed in the form = f(x,x ,t) x or x = y = F1(x,y,t) y = f(x, ,t) = F2(x,y,t) x x0 = x(0) x 0 = x (0) (31.56) where x0 and x 0 are the initial displacement and velocity of the system, respectively. The form of the equation is the same whether the system is linear or nonlinear. Choose a small time interval h such that tj = jh for j = 0, 1, 2, . . . Let wij denote an approximation to xi (tj) for each j = 0, 1, 2, . . . and i = 1, 2. For the initial conditions, set w1,0 = x0 and w2,0 = x 0. Obtain the approximation wij + 1, given all the values of the previous steps wij, as [31.1] 1 wi,j + 1 = wi,j + (k1,i + 2k2,i + 2k3,i + k4,i) i = 1, 2 (31.57) 6 where k1,i = hFi(tj + w1,j, w2,j) k2,i = hFi tj + k3,i = hFi tj + h 1 1 , w1,j + k1,1, w2i,j + k1,2 2 2 2 1 h 1 , w1,j + k2,1, w2,j + k2,2 2 2 2 i = 1, 2 (31.58) k4,i = hFi(tj + h, w1,i + k3,1, w2, i + k3,2 ) Note that k1,1 and k1,2 must be computed before we can obtain k2,1. Example 1. Obtain the response of a generator rotor to a shortcircuit disturbance given in Fig. 31.17. The generator shaft may be idealized as a singledegreeoffreedom system in torsion with the following values: 1 = 1737 cpm = 28.95(2 ) rad/s = 182 rad/s J = 8.5428 lb in s2 (25 kg m2) k = 7.329 106 lb in/rad (828 100 N m/rad) Solution Hence, J + k = f(t) = f(t) k = + J J k + f(t) = J J (31.59) where f(t) is tabulated. Since 1 = 182 rad/s, the period = 2 /182 = 0.00345 s and the time interval h must be chosen to be around 0.005 s. Hence, tabulated values of f(t) must be available for t intervals of 0.005 s, or it has to be interpolated from Fig. 31.17. 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.

