Mass-spring question (frequency and amplitude)

[pacgcrosss]

1. I am confused by this question and was hoping someone could clear this up for me, I know this is simple.
   
   A mass-spring system vibrates at 10 Hz. Ideally, i.e., without friction, it would continue forever at the same amplitude. In practice, its amplitude is found to decay such that it decreases by 1/e every 10 minutes. Estimate the width of its frequency spectrum.
   
   So I know the frequency is 10Hz and how much the amplitude is decreasing. How am I supposed to determine the width of the frequency spectrum from this? Isn't there not a relationship between amplitude and frequency?
   

 

[kuruman]

You might find this useful.
https://en.wikipedia.org/wiki/Q_factor

 

[BvU]

Hi pac,

This question is put to you in the context of some teaching material. Control engineering, Laplace ? You have to use the template and list some relevant equations plus an attempt at solution before we are allowed to assist you

 

[pacgcrosss]

@BvU, this is for a sensors course in robotics. And @kuruman, thanks very much for that link that really helps. I didn't really know where to start. Reading that wiki, the width of the frequency spectrum is equivalent to the bandwidth. I simply use the formula Δf = f_o / Q where f_o is the resonant frequency (10Hz) in this case. I can calculate the Q factor using Q = π*τ*f_o.

τ = 1/λ where lambda is the decay rate. I know that the amplitude is decaying at a rate of 1/e every 10 minutes.

Therefore Δf = f_o / Q → Δf = f_o / (π τ f_o) → Δf = 1/(πτ) → Δf = λ/π

Which leads me to two questions.

1. λ looks like it must have units of 1/s so would I just need to convert the decay rate from what it is every 10 minutes to what it is every second?

2. Does the resonant frequency really not matter here? It canceled out of my final equation.

Thanks a lot.

 

[kuruman]

Yes and yes.