Download MA 2051 A B `06 Spring-Mass Problems 1. Consider the following

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MA 2051 A B ’06
Spring-Mass Problems
1. Consider the following IVP for an undamped spring-mass model.
s00 + 4s = 0, s(0) = −2, s0 (0) = 3
(a) Find the solution, and put it in phase-amplitude form.
(b) Find the first positive time t∗ such that s(t∗ ) = 0.
(c) What is the maximum velocity of the mass?
2. Consider the following differential equation for a damped spring-mass model.
s00 + 4s0 + 3s = 0
(a) Find the roots of the characteristic equation and verify that the model is
overdamped.
(b) Find the solution satisfying s(0) = 2, s0 (0) = −8. Is there a time t such that
s(t) = 0? If so, find the time and explain why there is only one such time.
(c) (Challenge Problem) Find the solution satisfying the initial conditions s(0) =
s0 , s0 (0) = v0 , with s0 6= 0. Show that if
0<
3s0 + v0
<1
s0 + v 0
then there is a unique value of t such that s(t) = 0.
3. Consider the following IVP for a damped spring-mass model.
s00 + 4s0 + 8s = 0, s(0) = 1, s0 (0) = 2
(a) Find the solution.
(b) Show that this model is underdamped and write the solution in phase-amplitude
form.
(c) Estimate the time t∗ which guarantees that | s(t) |< 0.02 for t ≥ t∗ .
4. Consider the differential equation for a damped, forced spring-mass system given
below. You may assume that the mass is 1.
s00 +
1 0
s + 9s = sin(ωt)
10
(a) Find the roots of the characteristic equation and show that resonance will
occur.
(b) Using the formula from class for R, the amplitude of the forced response, find
the value of ω that maximizes R and determine the maximum value of R.
(c) Determine the values of ω that will guarantee that the amplitude R is less
than 2.
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