Given: X (t) = Ae^(-αt)* cos(ωt) and m(d2x/dt2) = -kx – β(dx/dt), prove: ω^2 = k/m – (β/2m)^2 and α = β/2m?

Question

anyone who has:

BSc Physics or Mechanical Engineering notes or in a Mechanics Book.

please write the proof out for me since I do not have the book. I would really appreciate it.

Thanks in advance

Answer 1

Dear Great One,
This is a standard question on damped SHM.
The solution can be found in your BSc Physics or Mechanical Engineering notes or in a Mechanics Book.

I suggest that you go through and do this substitution yourself - rather than copy it from a book. This will give you good practice for the rest of your course!

These are the steps:
1 Write down the equation for x(t) - given in the Question
2 Differentiate it with respect to (wrt) t - gives dx/dt
using the differential of a Product formula
3 Differentiate again wrt t - gives d2x/dt2
using the differential of a Product formula
4 Substitute for d2x/dt2; dx/dt and x(t) in the dynamic equation - given in the Question (ie the one with m, k and β in it)

5 You get an equation in which Ae(-αt) will cancel through which shows you that the Amplitude of the oscillation does not affect the frequency of the system
6 The equation now is in the form with cos (ωt) and sin (ωt)

Cos and sin are known as Orthogonal Functions: when Cos = 1 then Sin = 0 and conversely.
This means we can Seperately Equate:
The coefficients of Cos on both sides of our final equation
AND
The coefficients of Sin on both sides of our final equation

When we do this, we get:
m(α^2 - ω^2) = -αβ -k (check this)
and we get:
2mαω = βω

Now manipulate them to get the Answers that you need.
Notice that the lossy term in β reduces the frequency of the system from Root(k/m)

α β ω CopyLeft RC

Answer 2

Depends on what is meant by prove. All I can think to do is to
substitute values for w and alpha in X, put X(t) into equation , take derivatives and show it to be true.

Answer 3

*33(x3)m - w^5 < (9x)

Answer 4

you prove it...lol