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Project 1 A Vibration Insulation Problem Passive isolation systems are sometimes used to insulate delicate equipment from unwanted vibrations. For example, in order to insulate electrical monitoring equipment from vibrations present in the floor of an industrial plant, the equipment may be placed on a platform supported by flexible mountings resting on the floor. A simple physical model for such a system is shown in Figure 4.P.1, where the mountings are modeled as an equivalent linear spring with spring constant k, the combined mass of the platform and equipment is m, and viscous damping with damping coefficient y is assumed. Assume also that only vertical motion occurs. In this project, we use this model to illustrate an important fundamental principle in the design of a passive isolation system. m Tyce ) WI Tut) FIGURE 4.P.1 Equipment supported by flexible mountings.
1. Denote by y(t) the displacement of the platform from its equilibrium position relative to a fixed frame of reference. Relative to the same fixed ref- crence frame, let u(t) represent the displacement of the plant floor from its equilibrium position. Show that the differential equation for the motion of the platform is y" +2500y' + way = 25onu' + wiu, (i) where we = /k/m and 5 = x/2/mk is a dimen- sionless parameter known as the viscous damping ratio.