A critically damped, driven oscillator’s displacement x(t) satisfies the equation of motion
x¨ + 2ω0x˙ + ω
0^2x = f0 cos ωt
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˙ +ω0^2x = 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).
1) driving frequency = f0 cos ωt
ii) x¨ + 2ω0x˙ + ω
iii) A cos(ωt + φ).