Equation for a Spring in Damped Harmonic Motion

Question

So I got this question on a project for a class, and I've never seen anything like it before. I'm pretty confused on how to solve this, so I would really appreciate it if someone could explain to me how to do it.

The question is: "One equation modeling the displacement of a spring in damped harmonic motion is d = A e^(at) sin(2πt). Find the times when the spring is at it's equilibrium point, when d=0."

I know A is the amplitude, e is just a number, and t is time. I've also done sine graphs so I'm assuming this is something close to that. I guess I just don't understand how I'm supposed to solve when d=0. Any help is appreciated.

Answer 1

for d to equal zero, we either need A, e^(at), or sin(2πt) to be zero.

A=0 would be a trivial solution with no motion ever, so we will skip that.

e^(at) can never be zero, so we skip that.

This leaves us with 0 = sin(2πt)

replace 2πt with x to make it easier, and plug it back in after:

sin(x) is zero for x = nπ for all integers n

now we can plug back in for x:

2πt = nπ

solve for t:

t = n/2 for all integers n

We can check by plugging a few values of t back into the equation:

d = Ae^(a*0/2)*sin(2π*0/2) --> d = 0

d = Ae^(a*1/2)*sin(2π*1/2) --> d = 0

d = Ae^(a*2/2)*sin(2π*2/2) --> d = 0

etc