MATH215 December 2011
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) •
Question 04 (a)
Suppose the motion of a spring-mass system is described by the following differential equation
(a) Find all values of the parameter for which the system is overdamped or critically damped (that is the mass cannot pass its equilibrium position more than once, so there are no oscillatory solutions).
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What are the roots of the characteristic equation as a function of .
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Given the equation
we can obtain the characteristic equation by substituting a solution
and looking at the resulting polynomial. Doing this we get
which has roots
Now we study the behaviour of the roots of the equation. Note that is necessary for physical damping.
- When , there are complex roots to the system, and there are oscillations. Thus the system is underdamped.
- When , the square root always yields a real value. Since (assuming ), it follows that both roots are negative (so the amplitude decays) and real (so there are no oscillations). Thus, the system is overdamped.
Thus, there is overdamping for and underdamping for (and therefore critical damping if ).