MATH215 December 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) •
Question 03 (a)
Suppose the displacement x(t) of a damped mass-spring system subject to sinusoidal forcing of amplitude is modelled by:
(a) Find and sketch the solution x(t) when (no forcing).
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
If , the equation is homogeneous. Can you find the homogeneous solution?
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We have a homogeneous ODE with x(0) = 0, x'(0) = 1.
The roots of the corresponding characteristic polynomial are
The roots are complex, and we have a general solution of
We can use the initial conditions to find the constants. implies . Next, use the fact that the second term vanishes to calculate the first derivative of the solution.
Since x'(0) = 1 this implies . The final solution is therefore
A plot of the solution is given below. Note that the solution is bounded between so there is an envelope. The solution must fit between the upper and lower envelope functions. At , , and the solution is increasing at (the derivative is positive).