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0593_C13_fm  Page 471  Monday, May 6, 2002  3:21 PM





                       Introduction to Vibrations                                                  471



                                                                           B
                                                                                    F  (t)
                                                                                     1
                                                                           m
                       FIGURE P13.4.4                                               F  (t)
                                                                                     2
                       An undamped mass–spring system
                       with two forcing functions.           frictionless
                       P13.4.4: Suppose an undamped mass–spring system is subjected to two forcing functions
                       as represented in Figure P13.4.4. Determine the governing equation of motion and its
                       general solution. Let the forcing functions F (t) and F (t) have the forms:
                                                                      2
                                                              1
                                                        Ft () =  F cos p t
                                                        1     10    1

                       and

                                                       Ft () =  F cos p t
                                                        2     20    2
                       P13.4.5: Solve Eq. (13.4.1) if F(t) has the form:


                                                      Ft () =  F cos ( pt + ) φ
                                                            0
                       where F , p, and φ are constants.
                              0


                       Section 13.5 Damped Linear Oscillator
                       P13.5.1: Consider the damped mass–spring system depicted in Figure P13.5.1. Let the
                       block B have a mass of 0.25 slug, let the spring modulus be 4 lb/ft, and let the damping
                       coefficient c be 0.5 lb⋅sec/ft. Let B be displaced to the right 1.5 ft from its equilibrium
                       position and then released from rest. Determine the subsequent motion x(t) of the system.

                                                                       c

                                                                           B
                                                                   k
                                                                          m


                       FIGURE P13.5.1                                           x
                       A damped mass–spring system.

                       P13.5.2: See Problem P13.5.1. Show that the system is underdamped (see Eq. (13.5.5)).
                       P13.5.3: See Problem P13.5.1. The logarithm  δ of the ratio of amplitudes of successive
                       cycles of vibration is called the logarithmic decrement. Compute δ for the system of Problem
                       P13.5.1.
                       P13.5.4: See Problem P13.5.1. Determine the value of the damping coefficient c so that the
                       system is critically damped.
                       P13.5.4: Repeat Problems P13.5.1 through P13.5.4 if m, k, and c have the values 3.5 kg,
                       60 N/m, and 6.5 N⋅sec/m, respectively.
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