Question

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 | xe) WI Tut) FIGURE 4.P.1 Equipment supported by flexible mountings.

Answer 1

Given! Denote by Gliw) the frequency response of eqr , that is the ratio of the steady-state response to the hormonic input ult= Aeint the transmissibility TR of the plate form mounting system is then defined to be the corresponding gain function, TR = | Gliw i show that Trcan expressed as be TR = ('-(0))+ es ff = max. tronsmitted force ALLL FO C & mic FO 'Ft transmissibility, t= A ( performance of vibration isolation) E-0 (Best) Fo OLELI IT = S(SA)² + (CWAjr = PA 1+ (EWA)? SA fr = SA 1+(24w12 Fo = ton J () +25je Dividing both sides- where on external force freqerencey E = wor notural frequency of the system ç - compingfact e = 6 che (critical damping Vai J + (29.4