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0593_C13_fm  Page 449  Monday, May 6, 2002  3:21 PM





                       Introduction to Vibrations                                                  449


                       where µ is:


                                                          µ= km                                (13.5.8)

                       In this case, the block moves to its static equilibrium position without oscillation. This
                       phenomenon is called critical damping.
                        Finally, suppose the damping constant is larger than critical damping, that is, larger
                       than 2 km . Then, from Eq. (13.5.4), we see that ω is imaginary and that the solution for
                       the displacement x of B becomes:

                                                             +
                                                     x =  Ae −(µν  t )  +  Be −(µ −ν  t )      (13.5.9)

                       where µ and ν are:

                                                                            /
                                                                   c  2  k   12
                                                µ = cm    and     ν =    2  −               (13.5.10)
                                                    2
                                                                   4 m  m 
                        Observe that µ is always larger than ν and we have a relatively rapidly decaying motion
                       of B. This phenomenon is called overdamping.







                       13.6 Forced Vibration of a Damped Linear Oscillator
                       Consider next the forced vibration of a damped linear oscillator as depicted in Figure
                       13.6.1. From the principles of dynamics, we readily find the governing equation of motion
                       to be:

                                                       mx cx kx = ()                           (13.6.1)
                                                          +
                                                             +
                                                            ˙
                                                        ˙˙
                                                                   F t
                       where as before m is the mass of the block B, c is the viscous damping coefficient, k is the
                       linear spring constant, and F(t) is the forcing function. Suppose that as in Section 13.4 F(t)
                       has the periodic form:
                                                        Ft () =  F cos  pt                     (13.6.2)
                                                              0


                                                                          c

                                                                     k      B
                                                                            m
                                                                                         F(t)
                       FIGURE 13.6.1
                       Forced motion of a damped mass–spring                      x
                       system.
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