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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.