Figure 2 shows a roller rolling and slipping at the same time on a slippery surface. The center of the roller moves with a displacement r(t) and the roller rotates with angle (t). The roller has mass m, mass moment of inertia I, and radius r. In addition, a layer of fluid with viscous damping coefficient c is present to lubricate the roller. Moreover, the roller is pulled by a spring under a given (i.e., prescribed) displacement u(t), whereas the spring has spring constant k. Also, the roller is subjected to an applied torque M(t). Figure 3 shows the free-body diagram of the roller. Answer the following questions. (a) Apply F = ma to derive an equation of motion governing the translation z(t). (b) Apply M= Ia to derive an equation of motion governing the rotation 0(t). (c) Eliminate the variable 0(t) from the two equations of motion derived in parts (a) and (b). You should obtain a third-order differential equation in z(t). (d) Initially (i.e., at t = 0), the roller has no displacement but translates with a velocity v. The roller also has an initial spin rate wo. Derive the initial conditions for the third-order different equation governing r(t) in part (c).
Figure 2 shows a roller rolling and slipping at the same time on a slippery surface. The center of the roller moves with a displacement r(t) and the roller rotates with angle (t). The roller has mass m, mass moment of inertia I, and radius r. In addition, a layer of fluid with viscous damping coefficient c is present to lubricate the roller. Moreover, the roller is pulled by a spring under a given (i.e., prescribed) displacement u(t), whereas the spring has spring constant k. Also, the roller is subjected to an applied torque M(t). Figure 3 shows the free-body diagram of the roller. Answer the following questions. (a) Apply F = ma to derive an equation of motion governing the translation z(t). (b) Apply M= Ia to derive an equation of motion governing the rotation 0(t). (c) Eliminate the variable 0(t) from the two equations of motion derived in parts (a) and (b). You should obtain a third-order differential equation in z(t). (d) Initially (i.e., at t = 0), the roller has no displacement but translates with a velocity v. The roller also has an initial spin rate wo. Derive the initial conditions for the third-order different equation governing r(t) in part (c).
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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Transcribed Image Text:2. Figure 2 shows a roller rolling and slipping at the same time on a slippery surface. The
center of the roller moves with a displacement r(t) and the roller rotates with angle (t).
The roller has mass m, mass moment of inertia I, and radius r. In addition, a layer of fluid
with viscous damping coefficient c is present to lubricate the roller. Moreover, the roller is
pulled by a spring under a given (i.e., prescribed) displacement u(t), whereas the spring has
spring constant k. Also, the roller is subjected to an applied torque M(t). Figure 3 shows
the free-body diagram of the roller. Answer the following questions.
(a) Apply F = ma to derive an equation of motion governing the translation z(t).
(b) Apply M = Ia to derive an equation of motion governing the rotation 0(t).
(c) Eliminate the variable (t) from the two equations of motion derived in parts (a) and
(b). You should obtain a third-order differential equation in x(t).
(d) Initially (i.e., at t = 0), the roller has no displacement but translates with a velocity 0.
The roller also has an initial spin rate wo. Derive the initial conditions for the third-order
different equation governing r(t) in part (c).
m,
8
-x(t)
0 (t)
M
k
ww
1
u(t)
Figure 2: A slipping roller with a pre-
scribed displacement
c(x-ro)
mg
-M
ÎN
k(u - x)
Figure 3: Free-body diagram of the
slipping roller
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