Suppose a spring-mass system is on a table retarded by a Coulomb frictional force (equation 10.3): dx Y if dt d²x m- + kx = Ff, where F; dx ny if > 0. dt (a) If dx/dt > 0, determine the energy equation. Sketch the resulting phase plane curves. [Hint: By completing the square show that the phase plane curves are ellipses centered at v = =0.] 0, x = -y/k; not centered at (b) If dx/dt < 0, repeat the calculation of part (a). [Hint: The ellipses now are centered at v 0, x = = ylk.] (c) Using the results of (a) and (b), sketch the solution in the phase plane. Show that the mass stops in a finite time!!! (d) Consider a problem in which the mass is initially at x = vo. Determine how many times the mass passes x = 0 as a function of O with velocity vo.

Suppose a spring-mass system is on a table retarded by a Coulomb frictional force (equation 10.3): dx Y if dt d²x m- + kx = Ff, where F; dx ny if > 0. dt (a) If dx/dt > 0, determine the energy equation. Sketch the resulting phase plane curves. [Hint: By completing the square show that the phase plane curves are ellipses centered at v = =0.] 0, x = -y/k; not centered at (b) If dx/dt < 0, repeat the calculation of part (a). [Hint: The ellipses now are centered at v 0, x = = ylk.] (c) Using the results of (a) and (b), sketch the solution in the phase plane. Show that the mass stops in a finite time!!! (d) Consider a problem in which the mass is initially at x = vo. Determine how many times the mass passes x = 0 as a function of O with velocity vo.

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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Forced undamped vibrations

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Question

Suppose a spring-mass system is on a table retarded by a Coulomb frictional
force (equation 10.3):
dx
у if
dt
d?x
+ kx = Ff, where F,
dx
y if
> 0.
dt
(a) If dx/dt > 0, determine the energy equation. Sketch the resulting phase
plame curves. [Hint: By completing the square show that the phase
plane curves are ellipses centered at v =
0.)
0, x = -y/k; not centered at
(b) If dx/dt < 0, repeat the calculation of part (a). [Hint: The ellipses now
are centered at v =
0, x =
vlk.]
(c) Using the results of (a) and (b), sketch the solution in the phase plane.
Show that the mass stops in a finite time!!!
(d) Consider a problem in which the mass is initially at x =
vo. Determine how many times the mass passes x = 0 as a function of
O with velocity
vo.

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