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Question 2 The diagram below shows a system for automatic control of pressure at a local chemical plant. y(t) is the displacement of the membrane, where y=0 corresponds to the initial equilibrium position due to a given constant pressure. Pressure p(t) M Y(s) = Membrane Spring y(t) : Displacement Q2 diagram The displacement, y(t) in mm, responding to the pressure change p=8u(t), in psi, is determined by the differential equation module as Equ1, d²y dy +7. dt² dt where t=0, y(0)=0, and y'(0) = 0; u(t) is unit step function. (a) show that the displacement Y(s) can be expressed as 1 8 1 2 3s +3 S+4 + 3 s - +12y = 8u(t) (Equ1) (b) by finding the inverse of Y(s), determine the displacement variation function f(t). (c) comment on the system response when there is such a pressure variation, i.e. outline, respectively, the transient and the steady state response of the displacement upon this pressure surge, and the time required for the system to reaches the steady state. (d) due to the fact that the elasticity of the membrane will change with time, the parameters of the system model will alter, i.e. the coefficient 7, and 12 (in model Equ1) will change after a period of usage, comment on how the change of coefficients of the module may affect the system response.

Question 2 The diagram below shows a system for automatic control of pressure at a local chemical plant. y(t) is the displacement of the membrane, where y=0 corresponds to the initial equilibrium position due to a given constant pressure. Pressure p(t) M Y(s) = Membrane Spring y(t) : Displacement Q2 diagram The displacement, y(t) in mm, responding to the pressure change p=8u(t), in psi, is determined by the differential equation module as Equ1, d²y dy +7. dt² dt where t=0, y(0)=0, and y'(0) = 0; u(t) is unit step function. (a) show that the displacement Y(s) can be expressed as 1 8 1 2 3s +3 S+4 + 3 s - +12y = 8u(t) (Equ1) (b) by finding the inverse of Y(s), determine the displacement variation function f(t). (c) comment on the system response when there is such a pressure variation, i.e. outline, respectively, the transient and the steady state response of the displacement upon this pressure surge, and the time required for the system to reaches the steady state. (d) due to the fact that the elasticity of the membrane will change with time, the parameters of the system model will alter, i.e. the coefficient 7, and 12 (in model Equ1) will change after a period of usage, comment on how the change of coefficients of the module may affect the system response.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Question 2 The diagram below shows a system for automatic control of pressure at a local chemical plant. y(t) is the displacement of the membrane, where y=0 corresponds to the initial equilibrium position due to a given constant pressure. Pressure p(t) M +7 dy dt Membrane Spring y(t) Displacement Q2 diagram The displacement, y(t) in mm, responding to the pressure change p=8u(t), in psi, is determined by the differential equation module as Equ1, +12y = 8u(t) d²y dt² where t=0, y(0)=0, and y'(0) = 0; u(t) is unit step function. (a) show that the displacement Y(s) can be expressed as 21 8 1 2 3s +3 S+4 Y(s) = + (Equ1) (b) by finding the inverse of Y(s), determine the displacement variation function f(t). (c) comment on the system response when there is such a pressure variation, i.e. outline, respectively, the transient and the steady state response of the displacement upon this pressure surge, and the time required for the system to reaches the steady state. (d) due to the fact that the elasticity of the membrane will change with time, the parameters of the system model will alter, i.e. the coefficient 7, and 12 (in model Equ1) will change after a period of usage, comment on how the change of coefficients of the module may affect the system response.
Transcribed Image Text:Question 2 The diagram below shows a system for automatic control of pressure at a local chemical plant. y(t) is the displacement of the membrane, where y=0 corresponds to the initial equilibrium position due to a given constant pressure. Pressure p(t) M +7 dy dt Membrane Spring y(t) Displacement Q2 diagram The displacement, y(t) in mm, responding to the pressure change p=8u(t), in psi, is determined by the differential equation module as Equ1, +12y = 8u(t) d²y dt² where t=0, y(0)=0, and y'(0) = 0; u(t) is unit step function. (a) show that the displacement Y(s) can be expressed as 21 8 1 2 3s +3 S+4 Y(s) = + (Equ1) (b) by finding the inverse of Y(s), determine the displacement variation function f(t). (c) comment on the system response when there is such a pressure variation, i.e. outline, respectively, the transient and the steady state response of the displacement upon this pressure surge, and the time required for the system to reaches the steady state. (d) due to the fact that the elasticity of the membrane will change with time, the parameters of the system model will alter, i.e. the coefficient 7, and 12 (in model Equ1) will change after a period of usage, comment on how the change of coefficients of the module may affect the system response.
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Step 1: Summarize the given information.

"Since you have posted a question with multiple sub parts, we will provide the solution only to the first three sub parts as per our Q&A guidelines. Please repost the remaining sub parts separately."

The given differential equation is shown below.

fraction numerator d squared y over denominator d t squared end fraction plus 7 fraction numerator d y over denominator d t end fraction plus 12 y equals 8 u open parentheses t close parentheses ...(i)

The initial conditions are written below.

y open parentheses 0 close parentheses equals 0 comma space y apostrophe open parentheses 0 close parentheses equals 0 at t equals 0.

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