Nonlinear Pendulum
Differential Equation of Oscillations
Pendulum is an ideal model in which the material point of mass
Dynamics of rotational motion is described by the differential equation
where
In our case, the torque is determined by the projection of the force of gravity on the tangential direction, that is
The minus sign indicates that at a positive angle of rotation
The moment of inertia of the pendulum is given by
Then the dynamics equation takes the form:
In the case of small oscillations, one can set
where
The period of small oscillations is described by the well-known formula
However, with increasing amplitude, the linear equation ceases to be valid. In this case, the correct description of the oscillating system implies solving the original nonlinear differential equation.
Period of Oscillation of a Nonlinear Pendulum
Suppose that the pendulum is described by the nonlinear second order differential equation
We consider the oscillations under the following initial conditions
The angle
The order of the equation can be reduced, if we find a suitable integrating factor. Multiply this equation by the integrating factor
After integration we obtain the first order differential equation:
Given the initial conditions, we find the constant
Then the equation becomes:
Next, we apply the double angle identity
which leads to the following differential equation:
Integrating this equation, we obtain
We denote
Then
It follows that
In the new notation, our equation can be written as
Next, we discuss the limits of integration. The passage of the arc from the lowest point
The integral on the right cannot be expressed in terms of elementary functions. It is the so-called complete elliptic integral of the
The function
The function
where the double factorials
Note that if we restrict ourselves to the zero term of the expansion, assuming that
Further terms of the series for