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2. Let's go back to the same example from HW6, that is, the two mass-spring-damper system: ki •fi(+) kz ли M₂ •f₁₂(+) M₁ Da 00 00 1-7x2 1+x, From the Free-Body Diagrams of masses 1 and 2, we may write - m1x1 + c2(x1 − x2) + k2(x1 − x2) + C₁×1 + k₁x₁ = f1(t) = u1(t) m2x2 + c2(x2 − x1) + k2(x2 − x1) = ƒ2(t) = u2(t) = Using the same numerical value as HW06 m₁ 2, m2 = different outputs as yı = the position of mass 2, and Y2 the right) (note: it includes u2(t)) - = 1, C₁ = 3, C2 1, k₁ = 20, k2 10, but define two the total force felt by mass 2 (defined as positive to a) Derive the system TFs, find poles, and zeros. You CAN use MATLAB commands such as roots, ss2tf for this problem. Hint: you may reuse results from HW6. b) Use the initial value theorem to find the instantaneous change in the two outputs when each input is set to 1 while the other input remains as zero (note: there are a total of 4 responses here) c) Use the final value theorem to find the steady-state response of each output when each input is set to 1 while the other input remains at zero (note: there are a total of 4 responses here) d) Use step to verify your results from parts (b) and (c)

2. Let's go back to the same example from HW6, that is, the two mass-spring-damper system: ki •fi(+) kz ли M₂ •f₁₂(+) M₁ Da 00 00 1-7x2 1+x, From the Free-Body Diagrams of masses 1 and 2, we may write - m1x1 + c2(x1 − x2) + k2(x1 − x2) + C₁×1 + k₁x₁ = f1(t) = u1(t) m2x2 + c2(x2 − x1) + k2(x2 − x1) = ƒ2(t) = u2(t) = Using the same numerical value as HW06 m₁ 2, m2 = different outputs as yı = the position of mass 2, and Y2 the right) (note: it includes u2(t)) - = 1, C₁ = 3, C2 1, k₁ = 20, k2 10, but define two the total force felt by mass 2 (defined as positive to a) Derive the system TFs, find poles, and zeros. You CAN use MATLAB commands such as roots, ss2tf for this problem. Hint: you may reuse results from HW6. b) Use the initial value theorem to find the instantaneous change in the two outputs when each input is set to 1 while the other input remains as zero (note: there are a total of 4 responses here) c) Use the final value theorem to find the steady-state response of each output when each input is set to 1 while the other input remains at zero (note: there are a total of 4 responses here) d) Use step to verify your results from parts (b) and (c)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Let's go back to the same example from HW6, that is, the two mass-spring-damper system: ki •fi(+) kz ли M₂ •f₁₂(+) M₁ Da 00 00 1-7x2 1+x, From the Free-Body Diagrams of masses 1 and 2, we may write - m1x1 + c2(x1 − x2) + k2(x1 − x2) + C₁×1 + k₁x₁ = f1(t) = u1(t) m2x2 + c2(x2 − x1) + k2(x2 − x1) = ƒ2(t) = u2(t) = Using the same numerical value as HW06 m₁ 2, m2 = different outputs as yı = the position of mass 2, and Y2 the right) (note: it includes u2(t)) - = 1, C₁ = 3, C2 1, k₁ = 20, k2 10, but define two the total force felt by mass 2 (defined as positive to a) Derive the system TFs, find poles, and zeros. You CAN use MATLAB commands such as roots, ss2tf for this problem. Hint: you may reuse results from HW6. b) Use the initial value theorem to find the instantaneous change in the two outputs when each input is set to 1 while the other input remains as zero (note: there are a total of 4 responses here) c) Use the final value theorem to find the steady-state response of each output when each input is set to 1 while the other input remains at zero (note: there are a total of 4 responses here) d) Use step to verify your results from parts (b) and (c)
Transcribed Image Text:2. Let's go back to the same example from HW6, that is, the two mass-spring-damper system: ki •fi(+) kz ли M₂ •f₁₂(+) M₁ Da 00 00 1-7x2 1+x, From the Free-Body Diagrams of masses 1 and 2, we may write - m1x1 + c2(x1 − x2) + k2(x1 − x2) + C₁×1 + k₁x₁ = f1(t) = u1(t) m2x2 + c2(x2 − x1) + k2(x2 − x1) = ƒ2(t) = u2(t) = Using the same numerical value as HW06 m₁ 2, m2 = different outputs as yı = the position of mass 2, and Y2 the right) (note: it includes u2(t)) - = 1, C₁ = 3, C2 1, k₁ = 20, k2 10, but define two the total force felt by mass 2 (defined as positive to a) Derive the system TFs, find poles, and zeros. You CAN use MATLAB commands such as roots, ss2tf for this problem. Hint: you may reuse results from HW6. b) Use the initial value theorem to find the instantaneous change in the two outputs when each input is set to 1 while the other input remains as zero (note: there are a total of 4 responses here) c) Use the final value theorem to find the steady-state response of each output when each input is set to 1 while the other input remains at zero (note: there are a total of 4 responses here) d) Use step to verify your results from parts (b) and (c)
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