which can be written as follows:
x二 e (J12m)t[A cos w生t + B sin w生t](5-16)
Forced Vibrations
Sofar， the study of vibrating systems has been limited to free vibrations where there is no external input into the system. A free vibration system vibrates at its natural resonant frequency until the vibration dies down due to energy dissipation in the damping.
Now the influence of external excitation will be considered. Inpractice，dynamic systems are excited by external forces， which are themselves periodic in nature. Consider the system shown in Figure 5-10.
The externally applied periodic force has a frequency w， which can vary independently of the system parameters. The motion equation for this system may be obtained by any of the previously stated methods. The Newtonian approach will be used here because of its conceptual simplicity. The freebody diagram of the mass m is shown in Figure 5-11.
The motion equation for the mass m is given by: mx
二F sin wt王x Jxi(5-17) and can be rewritten as mx
+Jxi+王x二F sin wt Assuming that the steady-state oscillation of this system is represented by
the following relationship: x二D sin (wt .)(5-18) where: D二amplitude of the steady-state oscillation
.二phase angle by which the motion lags the impressed force The velocity and acceleration for the system are given by the following relationships:
()
V二xi二Dw cos (wt .)二Dw sin wt . + Z 2 (5-19)
G二x
二Dw 2 sin (wt .)二Dw 2 sin (wt . + Z 2) (5-20)
Substituting the previous relationships into motion equation(5-17)， the following relationship is obtained:
()
Z
mDw 2 sin (wt .) JDw sin wt . + D sin (wt .)+F sin wt二02 (5-21)
Inertia force +Damping force +Spring force +Impressed force二0From the previous equation， the displacement lags the impressed force by the phase angle .， and the spring force acts opposite in direction to
displacement. The damping force lags the displacement by 900 and is there-fore in the opposite direction to the velocity. The inertia force is in phase with the displacement and acts in the opposite direction to the acceleration. This information is in agreement with the physical interpretation of harmo-nic motion. The vector diagram as seen in Figure 5-12 shows the various forces acting onthe body， which is undergoing a forced vibration withviscous damping. Thus， from the vectordiagram， it is possible to obtain the value of the phase angle and the amplitude of steady oscillation
D二 J (王 mF w2)2 + Jw2 (5-22)
tan .二 王Jwmw2 (5-23)
The nondimensional form of D and . can be written as
F1王
D二 . w2 w 2 (5-24)
1 w2 n + 2〈 wn
w
2〈
tan .二 wn 2 (5-25)
w
1
wn
where:
J
wn二王1m 二 natural frequency
〈二 JJJ二 damping factor
JJ二 2 mwn二 critical damping coefficient.
From these equations， the effect on the magnification factor ( D1F1王)and the phase angle (.) is mainly a function of the frequency ratio w1wn and the damping factor〈. Figures 5-13a and 5-13b show these relationships. The damping factor has great influence on the amplitude and phase angle in the region of resonance. For small values of w1wn < 1.0， the inertia and damping force terms small and result in small phase angle. For a value of w1wn二1.0，thearephaseangleis900.Theaamplitude at resonance approaches infinity as the damping factor approaches zero. The phase angle undergoes nearly a 1800 shift for light damping as it passes through the critical frequency ratio. For large values of w1wn > 1.0， the phase angle approaches 180 0，and the impressed force is expended mostly in overcoming the large inertia force.
Design .onsiderations
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