5. A damped mass-spring system satisfies the differential equation and initial conditions d²x dx + + 10x = F(t). dt2 dt (a) When y = 0 and F(t) = 0, find the natural frequency of the system. %3D (b) When F(t) = 0, find the values of y for which the system is over-damped, critically damped and under-damped. (c) When y = 6 and F(t) = 0, find the general solution r(t).
5. A damped mass-spring system satisfies the differential equation and initial conditions d²x dx + + 10x = F(t). dt2 dt (a) When y = 0 and F(t) = 0, find the natural frequency of the system. %3D (b) When F(t) = 0, find the values of y for which the system is over-damped, critically damped and under-damped. (c) When y = 6 and F(t) = 0, find the general solution r(t).
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Transcribed Image Text:5. A damped mass-spring system satisfies the differential equation and initial conditions
da
+y
+ 10x = F(t).
dt2
dt
(a) When y = 0 and F(t) = 0, find the natural frequency of the system.
(b) When F(t) = 0, find the values of y for which the system is over-damped, critically
damped and under-damped.
(c) When y = 6 and F(t) = 0, find the general solution x(t).
(d) When y = 6 and F(t) = 25 cos 4t, use the method of undetermined coefficients to show
that a particular solution xp(t) is
Xp(t) =
25
cos 4t +
102
50
sin 4t.
51
(e) What is the amplitude of æp(t)? HINT: use the method employed in question 3 of this
problem set.
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