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 x₁ (t) and the horizontal displacement of the right mass is denoted by x2(t). Both masses are initially at rest (the initial displacement and speed are zero). A force f₁ (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. f2(t) = 2 N. a) Show that the displacements x₁ (t) and x2 (t) are given by the solutions of the equations mä₁ (t) + cx₁ (t) + k(x₁(t) − x₂(t)) = f₁(t), mä₂(t)+ci₂(t) + k(x₂(t) = x₁(t)) = f₂(t). b) Introducing the change of variable 9₁ (t) = x₁(t) + x₂(t), 92 (t) = x₁(t) = x₂(t), and combining the two equations established in Q1-a, show that q₁ (t) and 92 (t) are given by the solutions of the equations 91 mäi(t) + cġ₁ (t) = f₁(t) + f₂(t), mä2 (t) + cġ₂ (t) + 2kq2 (t) = f₁(t) - f2(t). c) Calculate q₁ (t) and q2 (t) (hint for q₁ (t): introduce the change of variable z(t) = ġ₁ (t)) 91 d) Using 91 (t) and q2 (t) calculated in Q1-c and equations (1.3) and (1.4), calculate x₁ (t) and x₂(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 x₁ (t) and the horizontal displacement of the right mass is denoted by x2(t). Both masses are initially at rest (the initial displacement and speed are zero). A force f₁ (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. f2(t) = 2 N. a) Show that the displacements x₁ (t) and x2 (t) are given by the solutions of the equations mä₁ (t) + cx₁ (t) + k(x₁(t) − x₂(t)) = f₁(t), mä₂(t)+ci₂(t) + k(x₂(t) = x₁(t)) = f₂(t). b) Introducing the change of variable 9₁ (t) = x₁(t) + x₂(t), 92 (t) = x₁(t) = x₂(t), and combining the two equations established in Q1-a, show that q₁ (t) and 92 (t) are given by the solutions of the equations 91 mäi(t) + cġ₁ (t) = f₁(t) + f₂(t), mä2 (t) + cġ₂ (t) + 2kq2 (t) = f₁(t) - f2(t). c) Calculate q₁ (t) and q2 (t) (hint for q₁ (t): introduce the change of variable z(t) = ġ₁ (t)) 91 d) Using 91 (t) and q2 (t) calculated in Q1-c and equations (1.3) and (1.4), calculate x₁ (t) and x₂(t).
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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Question
I am struggling with a, b,c,d
Transcribed Image Text:C
Data:
m = 2 kg.
k = 1 N/m.
m
www.
b) Introducing the change of variable
☐
C
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 ₁ (t) and the horizontal displacement of the right mass
is denoted by x2(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.
m
c = 4 Ns/m.
fi(t) = -1 N.
f2(t) = 2 N.
a) Show that the displacements x₁ (t) and x2 (t) are given by the solutions of the equations
mä₁ (t) + cx₁ (t) + k(x₁(t) − x₂(t)) = f₁(t),
mä₂ (t)+ci₂(t) + k(x₂(t) − x₁(t)) = f₂(t).
91 (t) = x₁(t) + x₂ (t),
92 (t) = x₁(t) = x₂(t),
91
and combining the two equations established in Q1-a, show that q₁ (t) and q2 (t) are given by the
solutions of the equations
mä₁ (t) + cġ₁ (t) = f₁(t) + f₂(t),
mä2 (t) + cġ₂ (t) + 2kq2 (t) = f1(t) - f2(t).
c) Calculate q₁ (t) and q2 (t) (hint for q₁ (t): introduce the change of variable z(t) = ġ₁ (t))
91
d) Using q₁ (t) and 92 (t) calculated in Q1-c and equations (1.3) and (1.4), calculate x₁ (t) and x₂(t).
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