Elastic pendulum


In physics and mathematics, in the area of dynamical systems, an elastic pendulum is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The system exhibits chaotic behaviour and is sensitive to initial conditions. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations.

Analysis and interpretation

The system is much more complex than a simple pendulum, as the properties of the spring add an extra dimension of freedom to the system. For example, when the spring compresses, the shorter radius causes the spring to move faster due to the conservation of angular momentum. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum.

Lagrangian

The spring has the rest length and can be stretched to length. The angle of oscillation of the pendulum is.
The Lagrangian is:
where is the kinetic energy and is the potential energy.
See. Hooke's law is the potential energy of the spring itself:
where is the spring constant.
The potential energy from gravity, on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is:
where is the gravitational acceleration.
The kinetic energy is given by:
where is the velocity of the mass. To relate to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:
So the Lagrangian becomes:

Equations of motion

With two degrees of freedom, for and, the equations of motion can be found using two Euler-Lagrange equations:
For :
isolated:
And for :
isolated:
The elastic pendulum is now described with two coupled ordinary differential equations. These can be solved numerically. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order in this system.