Addendum:

Today, I am certainly in a position to calculate the correct periodic times and to state the error concerning my approximate formula.

It applies: mgh_{0} = mgh + ½m(ds/dt)^{2}.

With x = x(t) and y = y(t) applies:

T = (8/g)∫(0, s_{0}, (1/(h_{0}-h))^{½}ds) = (8/g)∫(0, x_{0}, ((1+y'^{2})/(y_{0}-y))^{½}dx) = (8/g)∫(0, t_{0}, ((x^{.2}+y^{.2})/(y_{0}-y))^{½}dt).

The approximate formula reads:

T = 2π((2lh)^{½}/a)^{½} = 2π((2(1+y'(0)^{2})^{3/2}y_{0}(1+y'_{0}^{2}))^{½}/(gy'_{0}y''(0)^{½}))^{½} = 2π((2(x^{.}(0)^{2}+y^{.}(0)^{2})^{3/2}y_{0}(x^{.}_{0}^{2}+y^{.}_{0}^{2}))^{½}/(gy^{.}_{0}(x^{.}(0)y^{..}(0) - y^{.}(0)x^{..}(0))^{½}))^{½}.

With x = a sin φ, y = b - b cos φ, ds = (dx^{2}+dy^{2})^{½} = (a^{2}cos^{2}φ + b^{2}sin^{2}φ)^{½}dφ, k = sin(φ_{0}/2) and ε^{2} = (a^{2}-b^{2})/a^{2} exactly applies for the ellipse pendulum:

T = ((8a^{2})/(bg))^{½}∫(0, φ_{0}/2, (1 - ε^{2}sin^{2}φ)/(cos φ - cos φ_{0})dφ).

With ε^{2} < 1 the calculation of the integral yields the series expansion:

T = (2πa/(bg)^{½}){1 + 1/4 k^{2} - ε^{2}k^{2} + 9/64 k^{4} + 3/8 ε^{2}k^{4} - 3/4 ε^{4}k^{4} + O(k^{6})}.

The series expansion of the approximate formula T = 2πa(1 - ε^{2}sin^{2}φ_{0})^{¼}/(bg cos(φ_{0}/2))^{½} reads:

T = (2πa/(bg)^{½}){1 + 1/4 k^{2} - ε^{2}k^{2} + 5/32 k^{4} + 3/4 ε^{2}k^{4} - 3/2 ε^{4}k^{4} + O(k^{6})}.

Both formulas are identical up to the quadratic term as for the mathematical pendulum and confirm the quality of the approximation procedure. To obtain a better approximate formula, y(x) is to develop into a Taylor series around x = 0. For |y'| < 1, then numerator and denominator of the integral can be developed into series and easily be integrated after multiplying them out. Otherwise, the integral can be computed by numerical quadrature or by development of a Taylor series of the integrand with arbitrary precision. The latter is always possible by suitable substitution of x, and sufficient differentiability of the integrand. Interval arithmetic secures the results.

My actual aim was less the calculation of concrete periodic times, but the mathematical reduction of the difficult integration to a relatively easy differentiation (with simple integration). Precisely this yields the just mentioned development of a Taylor series, which is possibly to repeat for separate interval limits.

© 13.03.2010 by Boris Haase

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