Frequency of small oscillations in A cos(ωt - θ) + Bθ'' = 0

[yxgao]

What is the frequency of SMALL oscillations about è[t] = 0 of the following expression: Assume that w t is a constant.

A Cos[w t - è[t]] + B è''[t]==0, where A and B are arbitrary constants?

If you expand the Cosine term, you get A Cos[w t] Cos[è[t]] + A Sin[w t] Sin[è[t]] +B è''[t] ==0, which can be approximated as:

B è''[t] + A Sin[w t] è[t]== -A Cos[w t]

So is the frequency of small oscillations just Sqrt[(A Sin[w t])/B]?

 

[krab]

I cannot make head or tails of this post. Same is true of some of your posts on the College Help forum. If you use complicated formulas, please use the LaTeX typesetting described here .

 

[HallsofIvy]

Krab, this forum (like all of the internet) is very international. I consider myself fortuate in that the ONE language I speak well (English) is the standard language of the internet and am very tolerant of those whose English is not so good (but far better than my ability at any other language!).

My understanding of the problem is that you have
A Cos[w t] Cos[y[t]] + A Sin[w t] Sin[y[t]] +B y''[t] =0.

This is a badly non-linear equation but for "small oscillations", that is, when y(t) remains small we can approximate Sin(y[t]) by its tangent line approximation at 0: Sin(y[t])~ y[t]. Similarly the tangent line approximation of Cos(y[t]) at 0 is Cos(y[t])~ 1.

The differential equation itself is then approximated by the linear equation: A Cos[wt]+ A Sin[wt]y+ B y"[t]= 0.

This has, however, variable coefficients and so still cannot be given a elementary solution.