F

x = xh + xp = A sin pt + B cos pt H--- sin ot (3.18a)

in which A and B should be determined by using the initial conditions of free vibration. When the force Fsin ot is applied from a position of rest, ignore xh (free vibration due to xo and xo). The response of forced vibration corresponding to the third term of Equation 3.18a is

which indicates that the motion is periodic with the same frequency as that of the force and may endure as long as the force remains on the mass; this is called steady-state vibration. Note that F/K is static displacement xst; when sin ot = 1, the displacement x is the amplitude X. In general, the application of a disturbing force can produce an additional motion superimposed on steady-state motion. This additional motion is from the homogeneous solution of the free vibration. Consider the initial conditions of x0 = 0 and xo = 0 at t = 0 in Equation 3.18a. The result is

Comparing Equation 3.18b with Equation 3.19 reveals that there is another term associated with sin pt. This is due to the fact that the application of a disturbing force produces some free vibrations of the system. Thus, the actual motion is a superposition of two harmonic motions with different frequencies, amplitudes, and phase angles. In practical engineering, there is always some damping. So, free vibration is eventually damped out and only forced vibration remains. The early part of a motion consisting of a forced vibration and a few cycles of free vibration is called transient vibration, which can be important in aircraft design for landing and for gust loading.

3.2.3.2 Undamped Forced Vibration with Impulses (Shock Spectra)

When the structure is subjected to impulses of duration Z, the maximum response can be defined in terms of amplification factor, Am, expressed in terms of amplitude (X) and static displacement (xst) as

The variation of amplitude in terms of force duration and structural natural period (Z/T) is expressed in the shock spectrum shown in Figure 3.6a. The shock spectra for three other types of impulses are similarly sketched in Figure 3.6b. Note that maximum amplification factor is twice the static displacement and that the amplification factor can be higher after the impulse than that during the impulse (see the case of triangular impulse in sine function shape).

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