Re: Non-linear springs

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Ed Schweber's question about a 1-D oscillator with a quadratic
restoring force spring (i.e. potential energy cubic in the displacement
from equilibrium: V(x) = (1/3)*k*|x|^3):
> ...
>   Since the amplitude is the only other factor that can effect the period
> this seems to suggest that the formula for the period is
>
>                       T = N*sqrt(m/k)*f(A)
>
> where N is a constant and f(A) is some function of the amplitude.
>
>  So my questions are:
>
>           -am I correct so far?

Yes.

>          -is N = 2*pi as for a linear spring?

No, N is given by:

N = sqrt(8*[pi]/3)*GAMMA(1/3)/GAMMA(5/6)   .

Here GAMMA(x) is the gamma function which is a continuous generalization
of the factorial function.  In particular, GAMMA(z) = (z-1)! when z is a
positive integer, but the gamma function is also defined for other
arguments z as well.  The gamma function is meromorphic in the complex
z plane being analytic everywhere except for its simple poles at each of
the non-positive integers.

I would quote you a numerical value for N but I'm posting from home and
don't have access to my copy (in my office) of Abramowitz & Stegun's
table of functions that would give values for GAMMA(1/3) and GAMMA(5/6).

>          - can f(A) by determined in closed form?

Yes.  f(A) = 1/sqrt(A)

> Thanks for any input

You're welcome.

David Bowman
David_Bowman@georgetowncollege.edu