please help me in number 1 Project 1 A Vibration Insulation ProblemPassive isolation systems are sometimes used to insulate delicate equipment from unwantedvibrations. For example, in order to insulate electrical monitoring equipment from vibrationspresent in the floor of an industrial plant, the equipment may be placed on a platformsupported by flexible mountings resting on the floor. A simple physical model for such asystem is shown in Figure 4.P.1, where the mountings are modeled as an equivalent linearspring with spring constant k, the combined mass of the platform and equipment is m, andviscous damping with damping coefficient y is assumed. Assume also that only verticalmotion occurs. In this project, we use this model to illustrate an important fundamentalprinciple in the design of a passive isolation system.of 2y(t)FIGURE 4.P.1 Equipment supported by flexible mountings.Project 1 PROBLEMS1. Denote by y(t) the displacement of the platformfrom its equilibrium position relative to a fixedframe of reference. Relative to the same fixed ref-3. Plot the graphs of TR versus the dimensionlessratio w/on for ( = 0.1, 0.5, 1, 2. For what valuesof w/an is Ta = 1? Explain why the graphs implythat it is desirable that the mountings have a lownatural frequency in order to isolate the vibrationsource from the equipment platform, and that usinglow-stiffness isolators is one way to achieve this.() 4. The vibrations in the floor of an industrial plant liein the range 16-75 Hz. The combined mass of theequipment and platform is 38 kg, and the viscousdamping ratio of the suspension is 0.2. Find themaximum value of the spring stiffness if the ampli-tude of the transmitted vibration is to be less thanerence frame, let u(t) represent the displacement ofthe plant floor from its equilibrium position. Showthat the differential equation for the motion of theplatform isy" +25omy' +why = 2 conl + wiu,where a = k/m and = y/2/mk is a dimen-sionless parameter known as the viscous dampingratio.2. Denote by G(ia) the frequency response of Eq. (i),that is, the ratio of the steady-state response tothe harmonic input u(t) = Ae The transmis-sibility TR of the platform mounting system isthen defined to be the corresponding gain function,TR = |G(iw)|. Show that TR can be expressed as10% of the amplitude of the floor vibration over thegiven frequency range.5. Test the results of your design strategy for the situ-ation described in Problem 4 by performing severalnumerical simulations of Eq. (i) in Problem 1 usingvarious values of k as well as various input frequen-cies while holding the value of fixed. For eachsimulation, plot the input u(t) and the response y(1)on the same set of coordinate axes. Do the numerical21+(2TR =+ 20simulations support your theoretical results?wwww

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please help me in number 1 Project 1 A Vibration Insulation ProblemPassive isolation systems are sometimes used to insulate delicate equipment from unwantedvibrations. For example, in order to insulate electrical monitoring equipment from vibrationspresent in the floor of an industrial plant, the equipment may be placed on a platformsupported by flexible mountings resting on the floor. A simple physical model for such asystem is shown in Figure 4.P.1, where the mountings are modeled as an equivalent linearspring with spring constant k, the combined mass of the platform and equipment is m, andviscous damping with damping coefficient y is assumed. Assume also that only verticalmotion occurs. In this project, we use this model to illustrate an important fundamentalprinciple in the design of a passive isolation system.of 2y(t)FIGURE 4.P.1 Equipment supported by flexible mountings.Project 1 PROBLEMS1. Denote by y(t) the displacement of the platformfrom its equilibrium position relative to a fixedframe of reference. Relative to the same fixed ref-3. Plot the graphs of TR versus the dimensionlessratio w/on for ( = 0.1, 0.5, 1, 2. For what valuesof w/an is Ta = 1? Explain why the graphs implythat it is desirable that the mountings have a lownatural frequency in order to isolate the vibrationsource from the equipment platform, and that usinglow-stiffness isolators is one way to achieve this.() 4. The vibrations in the floor of an industrial plant liein the range 16-75 Hz. The combined mass of theequipment and platform is 38 kg, and the viscousdamping ratio of the suspension is 0.2. Find themaximum value of the spring stiffness if the ampli-tude of the transmitted vibration is to be less thanerence frame, let u(t) represent the displacement ofthe plant floor from its equilibrium position. Showthat the differential equation for the motion of theplatform isy&#34; +25omy' +why = 2 conl + wiu,where a = k/m and = y/2/mk is a dimen-sionless parameter known as the viscous dampingratio.2. Denote by G(ia) the frequency response of Eq. (i),that is, the ratio of the steady-state response tothe harmonic input u(t) = Ae The transmis-sibility TR of the platform mounting system isthen defined to be the corresponding gain function,TR = |G(iw)|. Show that TR can be expressed as10% of the amplitude of the floor vibration over thegiven frequency range.5. Test the results of your design strategy for the situ-ation described in Problem 4 by performing severalnumerical simulations of Eq. (i) in Problem 1 usingvarious values of k as well as various input frequen-cies while holding the value of fixed. For eachsimulation, plot the input u(t) and the response y(1)on the same set of coordinate axes. Do the numerical21+(2TR =+ 20simulations support your theoretical results?wwww

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passive isolation system. (t)