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FIGURE 3.4 Spring-mass and viscous damping model.

Damping MX force

H: Damping force

FIGURE 3.4 Spring-mass and viscous damping model.

FIGURE 3.5 Motion with underdamping. (Reprinted from Ref. [1, p. 11] by courtesy of Marcel Dekker, Inc.)

FIGURE 3.5 Motion with underdamping. (Reprinted from Ref. [1, p. 11] by courtesy of Marcel Dekker, Inc.)

Note that the oscillatory motion has the frequency, damped frequency, and the associated period, damped period, which may be expressed as p* = pV1^

Actually, the difference between p*, T * and p, T is slight. The terms p and T may be used instead of p* and T* in damped vibrations without introducing a serious error.

If the given initial conditions of x0 = 0 and x = p at t = 0 are inserted into Equation 3.12b, we obtain x = ffiffiffi e-ppt cos(V1 - P2 pt - P) (3.15)

which is plotted in Figure 3.5 for various damping factors of 0.05, 0.10, 0.25, and 0.35. It can be seen that the amplitudes of successive cycles are different and the periods of successive cycles are the same; strictly speaking, the motion is not regarded as being periodic but as time-periodic. In most engineering structures, p may vary from 0.02 to 0.08. Of course, the damping factor for some buildings may be as high as 0.15, depending on the nature of the material used in their construction and the degree of looseness in their connections.

3.2.3 Undamped and Damped Forced Vibration

3.2.3.1 Undamped Forced Vibration with Harmonic Force (Steady-State Response)

Consider the spring-mass model shown in Figure 3.3 where the mass is subjected to a harmonic force F sin ot with forced frequency o. Let F sin ot be considered positive to the right of the equilibrium position from which displacement, x, is measured. The differential equation of motion is

in which xst = F/Kas static displacement. The homogeneous solution is xh = A sin pt + B cos pt; the term, p = \J(K/M), is independent of the forced frequency, o. The particular solution may be obtained by trying the following:

Substituting xp and xp in Equation 3.17 for C1 and C2, we then obtain the following complete solution:

0 0

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