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An engineer is interested in the horizontal displacement of the dynamic system depicted in Fig. Q1. The two solids have the same mass m, are connected by a spring with stiffness k, and are both connected to the ground by dampers (viscous damping) with a damping coefficient c. The horizontal displacement of the left mass is denoted by 2₁ (t) and the horizontal displacement of the right mass is denoted by 22(t). Both masses are initially at rest (the initial displacement and speed are zero). A force fi(t) is applied on the left mass and a force f2(t) is applied on the right mass. Data: m = 2 kg. k = 1 N/m. c = 4 Ns/m. fi(t)= -1 N. fa(t)=2 N. a) Show that the displacements ₁ (t) and 2 (t) are given by the solutions of the equations mä(t) + ci(t) + k(x₁ (t) - 1₂(t)) = f(t), mä(t)+ci₂(t) + k(r₂(t)-1(t)) = f(t). b) Introducing the change of variable 9₁ (t) = x₁(t) + x₂(1), 92(t)=1(t)- 22(1), and combining the two equations established in Q1-a, show that q₁ (t) and q2 (t) are given by the solutions of the equations mãi(t) + chi(t) = fi(l) + b(t), mia(t) + c(t) + 2kga(t) = f(t)-fa(t). c) Calculate qi(t) and ga(1) (hint for q (t); introduce the change of variable z(t) = 9₁ (t))

An engineer is interested in the horizontal displacement of the dynamic system depicted in Fig. Q1. The two solids have the same mass m, are connected by a spring with stiffness k, and are both connected to the ground by dampers (viscous damping) with a damping coefficient c. The horizontal displacement of the left mass is denoted by 2₁ (t) and the horizontal displacement of the right mass is denoted by 22(t). Both masses are initially at rest (the initial displacement and speed are zero). A force fi(t) is applied on the left mass and a force f2(t) is applied on the right mass. Data: m = 2 kg. k = 1 N/m. c = 4 Ns/m. fi(t)= -1 N. fa(t)=2 N. a) Show that the displacements ₁ (t) and 2 (t) are given by the solutions of the equations mä(t) + ci(t) + k(x₁ (t) - 1₂(t)) = f(t), mä(t)+ci₂(t) + k(r₂(t)-1(t)) = f(t). b) Introducing the change of variable 9₁ (t) = x₁(t) + x₂(1), 92(t)=1(t)- 22(1), and combining the two equations established in Q1-a, show that q₁ (t) and q2 (t) are given by the solutions of the equations mãi(t) + chi(t) = fi(l) + b(t), mia(t) + c(t) + 2kga(t) = f(t)-fa(t). c) Calculate qi(t) and ga(1) (hint for q (t); introduce the change of variable z(t) = 9₁ (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
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Data: m = 2 kg. k = 1 N/m. x₁ (t) c = 4 Ns/m. fi(t) = -1 N. f2(t)=2 N. C m ww Fig. Q1 Dynamic system An engineer is interested in the horizontal displacement of the dynamic system depicted in Fig. Q1. The two solids have the same mass m, are connected by a spring with stiffness k, and are both connected to the ground by dampers (viscous damping) with a damping coefficient c. The horizontal displacement of the left mass is denoted by 2₁ (t) and the horizontal displacement of the right mass is denoted by r2(t). Both masses are initially at rest (the initial displacement and speed are zero). A force fi(t) is applied on the left mass and a force f2(t) is applied on the right mass. b) Introducing the change of variable r₂(t) C 9₁ (t) = x₁(t) + x₂(1), 92 (t)=1 (t)-T2(t), m a) Show that the displacements r(t) and r2(t) are given by the solutions of the equations mä(t) + ci(t) +k(ri(t)- x₂(t)) = fi(t), m₂(t)+ci₂(t)+k(r2(t)-1(t)) = f(t). and combining the two equations established in Q1-a, show that q₁ (t) and q2 (t) are given by the solutions of the equations mãi(t) + ch1(t) = fill + b(1) mia(t) + cáz(t) + 2kga(t) = f(t)-fa(t). c) Calculate q (t) and qz (1) (hint for q (t); introduce the change of variable z(t) = d (t))
Transcribed Image Text:Data: m = 2 kg. k = 1 N/m. x₁ (t) c = 4 Ns/m. fi(t) = -1 N. f2(t)=2 N. C m ww Fig. Q1 Dynamic system An engineer is interested in the horizontal displacement of the dynamic system depicted in Fig. Q1. The two solids have the same mass m, are connected by a spring with stiffness k, and are both connected to the ground by dampers (viscous damping) with a damping coefficient c. The horizontal displacement of the left mass is denoted by 2₁ (t) and the horizontal displacement of the right mass is denoted by r2(t). Both masses are initially at rest (the initial displacement and speed are zero). A force fi(t) is applied on the left mass and a force f2(t) is applied on the right mass. b) Introducing the change of variable r₂(t) C 9₁ (t) = x₁(t) + x₂(1), 92 (t)=1 (t)-T2(t), m a) Show that the displacements r(t) and r2(t) are given by the solutions of the equations mä(t) + ci(t) +k(ri(t)- x₂(t)) = fi(t), m₂(t)+ci₂(t)+k(r2(t)-1(t)) = f(t). and combining the two equations established in Q1-a, show that q₁ (t) and q2 (t) are given by the solutions of the equations mãi(t) + ch1(t) = fill + b(1) mia(t) + cáz(t) + 2kga(t) = f(t)-fa(t). c) Calculate q (t) and qz (1) (hint for q (t); introduce the change of variable z(t) = d (t))
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