A pendulum is properly modelled by a harmonic oscillator
only for small angles of elongation. Maybe you have
observed in the lab that the period of oscillation increases with increasing amplitude of oscillation. Starting
near the upside-down position, you will find that the period becomes much larger than for small-angle oscillations.
In fact, the period approaches infinity in the limit max180°.
In the virtual lab (and in reality too) you will never reach this limit.
Even though the equation of motion of an undamped and undriven pendulum is nonlinear, one can calculate
its frequency as a function of the amplitude max. In order to solve this nonlinear differential equation
notice that the total energy
is a constant during the motion of the pendulum. In the maximum elongation max, the kinetic energy is zero. Thus,
E = -02cosmax.
Solving (2) for d/dt
d/dt = ±[2(E+02cos)]1/2.
This first-order differential equation can be solved by separating the independent variable t from the dependent
one . That is, deal with the
differential quotient as if it would be a real quotient d divided by dt. Then, put all terms with on the left-hand side of the equation and all terms with t on the right-hand
d/[2(E + 02cos)]1/2 = ±dt.
To get the period of oscillations, integrate over a half cycle
Because the integrand on the left-hand side is an even function in , you will get
Note that E has been replaced by (3).
This integral can not be expressed by elementary functions like polynomials or trigonometric functions. This
is possible only in the limit of max0, where the cosine function
can be approximated by a Taylor series. Taking only the first and the second term, you will get an elementary integral
leading to the well-known result of T = 2/0.
For an arbitrary value of max,
the integral of (4) defines a so-called complete elliptic integral.
It sounds like a naïve trick to give an unsolvable integral a name.
Indeed, it is a trick but a fruitful one because elliptic integrals
appear quit often in mathematics and physics. Not as often as trigonometric functions of course,
but often enough to give
them a name, to find out their mathematical properties, to tabulate their values, and to include them into standard
mathematical subroutine packages on computers. One distinguishes three different types of elliptic integrals. The
second one is related to the circumference of an ellipse, that's why the name. In our case, the first one is relevant.
The substitution x = sin(/2)/sin(max/2)
turns (4) into the canonical form of the complete elliptic integral of the first kind
|QUESTIONS worth to think about:
- What is the reason for a diverging period of oscillations? Can you calculate
analytically the solution of (1) in that limit? Can you find
other nonlinear oscillators with diverging periods?
- What happens if the energy is larger than -02cosmax?
© 1998 Franz-Josef Elmer,
last modified Monday, July 20, 1998.