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The mathematical model representing the equation of motion of a pendulum isnonlinear. Solutions that satisfy the equation cannot be represented byelementary functions, such as trigonometric functions. To solve such problems,it is common to linearize the nonlinear equations and derive approximatenumerical solutions and exact solutions. Applying such linearization is limitedto cases in which the angle of the pendulum is relatively small. In cases wherethe angle of the pendulum is large, various methods have been presented thatrely on numerical solutions and the exact solutions based on Jacobian ellipticfunctions, to name one example. On the other hand, the author has been studyinga certain type differential equation. The second derivative of a function isequal to the term multiplied by $-n$(or $n$) for terms whose function is raisedto $2n-1$. Curves based on solving this differential equation are constructedas regular waves with periods. The author has termed the function satisfyingthis differential equation the leaf function (or the hyperbolic leaf function).In this paper, we attempt to apply this leaf function (or hyperbolic leaffunction) to undamped pendulums and overdamped pendulums for large angles.

Pendulum Analysis by Leaf Functions and Hyperbolic Leaf Functions