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Queries on Damped Harmonic Motion 
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#1
Dec513, 08:56 PM

P: 6

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 


#2
Dec513, 10:18 PM

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P: 6,950

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##. 


#3
Dec513, 10:55 PM

P: 6

I'm not really familiar with complex numbers (other than i^2 = 1) but your explanation does makes some sense to me. Thank You!! =)



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