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passive isolation system. (t)

passive isolation system. (t)

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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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. of 2 y(t) FIGURE 4.P.1 Equipment supported by flexible mountings. Project 1 PROBLEMS 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- 3. Plot the graphs of TR versus the dimensionless ratio w/on for ( = 0.1, 0.5, 1, 2. For what values of w/an is Ta = 1? Explain why the graphs imply that it is desirable that the mountings have a low natural frequency in order to isolate the vibration source from the equipment platform, and that using low-stiffness isolators is one way to achieve this. () 4. The vibrations in the floor of an industrial plant lie in the range 16-75 Hz. The combined mass of the equipment and platform is 38 kg, and the viscous damping ratio of the suspension is 0.2. Find the maximum value of the spring stiffness if the ampli- tude of the transmitted vibration is to be less than erence 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" +25omy' +why = 2 conl + wiu, where a = k/m and = y/2/mk is a dimen- sionless parameter known as the viscous damping ratio. 2. Denote by G(ia) the frequency response of Eq. (i), that is, the ratio of the steady-state response to the harmonic input u(t) = Ae The transmis- sibility TR of the platform mounting system is then defined to be the corresponding gain function, TR = |G(iw)|. Show that TR can be expressed as 10% of the amplitude of the floor vibration over the given frequency range. 5. Test the results of your design strategy for the situ- ation described in Problem 4 by performing several numerical simulations of Eq. (i) in Problem 1 using various values of k as well as various input frequen- cies while holding the value of fixed. For each simulation, plot the input u(t) and the response y(1) on the same set of coordinate axes. Do the numerical 2 1+(2 TR = + 20 simulations support your theoretical results? wwww
Transcribed Image Text: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. of 2 y(t) FIGURE 4.P.1 Equipment supported by flexible mountings. Project 1 PROBLEMS 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- 3. Plot the graphs of TR versus the dimensionless ratio w/on for ( = 0.1, 0.5, 1, 2. For what values of w/an is Ta = 1? Explain why the graphs imply that it is desirable that the mountings have a low natural frequency in order to isolate the vibration source from the equipment platform, and that using low-stiffness isolators is one way to achieve this. () 4. The vibrations in the floor of an industrial plant lie in the range 16-75 Hz. The combined mass of the equipment and platform is 38 kg, and the viscous damping ratio of the suspension is 0.2. Find the maximum value of the spring stiffness if the ampli- tude of the transmitted vibration is to be less than erence 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" +25omy' +why = 2 conl + wiu, where a = k/m and = y/2/mk is a dimen- sionless parameter known as the viscous damping ratio. 2. Denote by G(ia) the frequency response of Eq. (i), that is, the ratio of the steady-state response to the harmonic input u(t) = Ae The transmis- sibility TR of the platform mounting system is then defined to be the corresponding gain function, TR = |G(iw)|. Show that TR can be expressed as 10% of the amplitude of the floor vibration over the given frequency range. 5. Test the results of your design strategy for the situ- ation described in Problem 4 by performing several numerical simulations of Eq. (i) in Problem 1 using various values of k as well as various input frequen- cies while holding the value of fixed. For each simulation, plot the input u(t) and the response y(1) on the same set of coordinate axes. Do the numerical 2 1+(2 TR = + 20 simulations support your theoretical results? wwww
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