The three, counter-rotating wheels will
slip against neighboring wheel(s), thus experiencing friction, if the angular
speeds are not commensurate with the wheels rolling together. The rotation
of the wheels is governed by the following system of ODE’s:
x’1 = -x1 – 2×2
x’2 = -3×1 – 12×2 – 2×3
x’3 = -3×2 – x3 ;
where xj is the frequency of the j-th wheel (in cycles per second, with
clockwise positive and counter-clockwise negative) and the over-dot indicates
the derivative with respect to time.
(a) Find the solution of the system subject to the initial values
x1(0) = 5 ; x2(0) = 0 ; x3(0) = 0 ;
and use a plotting device to illustrate x1(t), x2(t), and x3(t) together.
(b) Find the solution of the system subject to the initial values
x1(0) = 0 ; x2(0) = 0 ; x3(0) = 5 :
and use a plotting device to illustrate x1(t), x2(t), and x3(t) together.
(c) Describe, briey and in your own words, all the possible behaviors of the
system (for different initial conditions) as t goes to +infinity.
Show all work.
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