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0593_C13_fm  Page 447  Monday, May 6, 2002  3:21 PM





                       Introduction to Vibrations                                                  447


                       where F  and p are constants. The governing equation then becomes:
                              0
                                                          +
                                                       mx kx =  Acos pt                        (13.4.3)
                                                        ˙˙
                        From Eqs. (13.2.32) through (13.2.35) we see that the general solution of Eq. (13.4.3) is:
                                                                 F (
                                                      +
                                            x =  Acosω t Bsinω t +  [ 0  k mp )] cos  pt       (13.4.4)
                                                                     −
                                                                         2
                       where as before A and B are constants to be determined from auxiliary conditions (such
                       as initial conditions) and where ω is defined as:


                                                            D
                                                          ω =  km                              (13.4.5)

                                                                                           2
                        Consider the last term, [F /(k – mp )]cospt of Eq. (13.4.4). Observe that if p  has values
                                                        2
                                               0
                       nearly equal to k/m (that is, if p is nearly equal to ω) the denominator becomes very small,
                       producing large-amplitude oscillation. Indeed, if p is equal to ω, the oscillation amplitude
                       becomes unbounded. This means that by stimulating the mass B with a periodic force
                       having a frequency nearly equal to ω (that is,  km ), the amplitude of the resulting oscillation
                       becomes unbounded.
                        The quantity  km  is called the natural frequency of the system. When the frequency of
                       the loading function is equal to the natural frequency, giving rise to a large-amplitude
                       response, we have the phenomenon commonly referred to as resonance.






                       13.5 Damped Linear Oscillator

                       Consider next the damped linear oscillator as depicted in Figure 13.5.1. This is the same
                       system we considered in the previous sections, but here the movement of the mass B is
                       restricted by a “damper” in the form of a dashpot. For simplicity of illustration, we will
                       assume viscous damping, where the force exerted by the dashpot on B is proportional to
                       the speed of  B and directed opposite to the motion of  B with  c being the constant of
                       proportionality. That is, the damping force F  on B is:
                                                              D

                                                          F =− ˙ n                             (13.5.1)
                                                               cx
                                                           D
                       where n is a unit vector in the positive X direction as shown in Figure 13.5.1.

                                                                         c


                                                                             B
                                                                     k
                                                                            m
                                                                                         n

                       FIGURE 13.5.1                                             x
                       A damped mass–spring system.
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