Question

A critically damped, driven oscillator’s displacement x(t) satisfies the equation of motion

x¨ + 2ω0x˙ + ω

2

0x = f0 cos ωt (1)

where ω0 is the natural frequency, and ω is the “driving frequency”.

(i) Find the particular solution to the above equation, in the form xp(t) = A cos(ωt + φ).

Your answer should clearly give the expressions for A and φ.

(ii) The homogeneous equation ¨x + 2ω0x˙ + ω

2

0x = 0 has e

−ω0t as one solution. Show, by

substitution, that the function te−βt can be the second solution. Find β in terms of ω0.

(iii) Use the above results to construct the complete solution to Eq. 1, subject to the initial

conditions x(0) = 0 = ˙x(0). (3 MARKS)

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Answer 1

Answer:

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