Page 1 - Combined_52_OCR
P. 1

                                     usually results. It can be shown 2 that the frequency that is found by using any shape
                                     except the correct shape always is higher than the actual frequency. Therefore, if more
                                      than one calculation is made, using different assumed shapes, the lowest computed fre­
                                     quency is closest to the actual frequency of the system.                      -ofm-ti::
                                       In many problems for which a classical solution would be possible, the work involved
                                     is excessive. Often a satisfactory answer to such a problem can be obtained by the ap­  ORTH
                                     plication of Rayleigh’s method. In this chapter several examples are worked using both   L
                                     the classical method and Rayleigh’s method. In all, Rayleigh’s method gives a good   system -.
                                     approximation to the correct result with relatively little work. Many other examples of   .-«-h-cted -
                                     solutions to problems by Rayleigh’s method are in the literature.3-4-5        that of ! :
                                       Ritz’s method is a refinement of Rayleigh’s method. A better approximation to tie­  tions as '
                                     fundamental natural frequency can be obtained by its use, and approximations to higher
                                     natural frequencies can be found. In using Ritz’s method, the deflections which are as­
                                     sumed in computing the energies are expressed as functions with one or more undeter­
                                     mined parameters; these parameters are adjusted to make the computed frequencj- a   •a here
                                     minimum. Ritz’s method has been used extensively for the determination of the natural   mode,
                                     frequencies of plates of various shapes, and is discussed in the section on the lateral   For a b
                                     vibrations of plates.                                                         i.c., in th-
                                       Lumped Parameters. A procedure that is useful in many problems for finding ap­
                                     proximations to both the natural frequencies and the mode shapes is to reduce the system
                                     with distributed parameters to one having a finite number of degrees-of-freedom. This
                                     is done by lumping the parameters for each small region into an equivalent mass and
                                     elastic element. Several formalized procedures for doing this and for analyzing the re­  •a here q J
                                                                                                                   fcth norm;
                                                                                                                    For a s;
                                     Table 7.2. Approximate Formulas for Natural Frequencies of Systems Having     dimension
                                                      Both Concentrated and Distributed Mass

                                                TYPE OF SYSTEM      FREQUENCY       STIFFNESS
                                                k’m A                                                               Equatio:
                                                                                D= COIL DIA                       tudinal, t<
                                                                                d= WIRE DIA                       equations
                                           SPRING WITH MASS ATTACHED            n=NUMBER OF TURNS                 the soluti
                                                                                                                  cro-vs sec’i
                                                       I                                                            In anaiy
                                               k..I. n
                                                                                                                  tudinal d:
                                                          o                                                       stresses in:
                                                                                 D = ROD DIAMETER                 this rnotio:
                                           CIRCULAR ROD, WITH DISC               I = ROD LENGTH                     (onside:
                                           ATTACHED, IN TORSION
                                                                                                                  by passing
                                                                                                                  K’. 7.1/A
                                              m/2         m/2
                                                                                    L 48EI
                                                                               1=BEAM LENGTH
                                           UNIFORM SIMPLY SUPPORTED
                                                                               1= MOMENT OF INERTIA
                                           BEAM WITH MASS IN CENTER
                                                                               1=BEAM LENGTH
                                           UNIFORM CANTILEVER BEAM
                                                                               I = MOMENT OF INERTIA
                                           WITH MASS ON END                                                      * ’r‘ ’-1. (.4
   1   2