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0593_C13_fm  Page 446  Monday, May 6, 2002  3:21 PM





                       446                                                 Dynamics of Mechanical Systems


                        Equation (13.3.14) is identical in form to Eq. (13.3.2); therefore, the solution will have
                       the form of Eq. (13.3.7). That is,


                                                                ˙
                                                    =
                                                   θθ cos t     θ ω) sin ωt                   (13.3.15)
                                                       0   ω +( 0
                       where θ  and  θ ˙ 0  are the initial values of θ and  , respectively, and where ω is:
                                                                 θ
                                                                 ˙
                              0
                                                          ω= g l                              (13.3.16)

                        Suppose, for example, that the pendulum is released from rest at an angle  θ . The
                                                                                                0
                       subsequent motion of the pendulum is:

                                                         =
                                                        θθ cos g l  t                         (13.3.17)
                                                            0
                        Observe in Eq. (13.3.17) that the amplitude of the periodic motion is θ  and that the
                                                                                         0
                       frequency and period are:

                                               f = (12π )  g l    and     T = 2π l  g         (13.3.18)


                        Finally, observe that the frequency and period of the pendulum depend only upon the
                       length   of the pendulum. That is, the pendulum movement is independent of the mass m.
                       (This is the reason why pendulums have been used extensively in clocks and timing devices.)






                       13.4 Forced Vibration of an Undamped Oscillator

                       Consider again the undamped mass–spring system of the foregoing section and as
                       depicted again in Figure 13.4.1. This time, let the mass B be subjected to a time-varying
                       force F(t) as shown. Then, it is readily seen by using any of the principles of dynamics
                       discussed earlier, that the governing equation of motion for this system is:

                                                        mx kx = ()                             (13.4.1)
                                                           +
                                                          ˙˙
                                                                 F t
                        Suppose F(t) is itself a periodic function such as:

                                                        Ft () =  F cos pt                      (13.4.2)
                                                               0


                                                                          B
                                                                 k
                                                                         m            F(t)
                                                                                  frictionless
                       FIGURE 13.4.1
                       An undamped forced mass–spring                         x
                       system.
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