So we know that SHM can be described as: x(t) = Acos(ωt + ϕ) v(t) = -Aω sin(ωt + ϕ) a(t) = -Aω^2 cos(ωt + ϕ) it can be easily said that the max acceleration (in terms of magnitude) a SHM system can achieve is Aω^2 In Damped Harmonic Motion we know that: x(t) = (A)(e^(-bt/2m))cos(ωt + ϕ) given that: A' = (A)(e^(-bt/2m)) ω' = sqrt( (ω^2) - (b/2m)^2 ) Is it true that the max acceleration at any given time is (A')(ω')^2? My intuition tells me that the above statement is not true =/ because differentiating the function x(t) = (A)(e^(-bt/2m))cos(ωt + ϕ) gives me a complex function (which has sine & cosine in it) & it doesn't really give me anything close to the (A')(ω')^2 term
This is easier using complex numbers. For damped motion, ##x(t)## = the real part of ##Ae^{(-s + i\omega')t}## where ##s## is your ##b/2m##. Note, ##A## is a complex constant (to account for your phase angle ##\varphi##) and of course ##e^{i\theta} = \cos\theta + i\sin\theta##. So ##a(t)## = the real part of ##(-s + i\omega')^2 x(t)## Your intuition is right, but if the damping is small, ##(-s + i\omega')^2## is close to ##-\omega^2##.
I'm not really familiar with complex numbers (other than i^2 = -1) but your explanation does makes some sense to me. Thank You!! =)