123 Suppression of vibration and resonance

Consider the linear single degree-of-freedom oscillator shown in Figure 12.2(a): a mass m attached by a spring and a damper to a base. Assume that the base vibrates at a single frequency œ with input amplitude X, so that its displacement is x = Xelœt. The relative deflection of the mass is y = Yelœt is then given by the transfer function H(œ):

where w1 is the undamped natural frequency of the oscillator and 77 is the damping constant. The magnitude of H(w) is shown in Figure 12.2(b).

Figure 12.2 (a) Single degree of freedom oscillator subject to seismic input x at frequency w. (b) The transfer function for the relative displacement y

Single low-frequency undamped input

For small values of «/«1 and low damping

meaning that the response Y is minimized by making its lowest natural frequency «1 as large as possible. Real vibrating systems, of course, have many modes of vibration, but the requirement of maximum «1 is unaffected by this. Further, the same conclusion holds when the input is an oscillating force applied to the mass, rather than a displacement applied to the base. Thus the material index Mu

should be maximized to minimize response to a single low-frequency undamped input.

Consider, as an example, the task of maximizing «1 for a circular plate. We suppose that the plate has a radius R and a mass m1 per unit area, and that these are fixed. Its lowest natural frequency of flexural vibration is mXR4(X - v2)

where E is Young's modulus, v is Poisson's ratio and C2 is a constant (see Section 4.8). If, at constant mass, the plate is converted to a foam, its thickness, t, increases as (p/psVl and its modulus E decreases as (p/ps)2 (Section 4.2) giving the scaling law

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