Dynamics of a Spring-Pendulum System

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
A string of fixed length is free to move over two frictionless rollers fixed at the top. A spring is at one end of the string and a bob is at the other end, forming a pendulum of variable length.
[more]
Contributed by: Erik Mahieu (October 2014)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The spring and pendulum system has two degrees of freedom: , the angular displacement of the pendulum bob, and
, the vertical displacement of the spring.
Lagrangian mechanics can be used to simulate this system.
The potential energy of the system at time is:
.
The kinetic energy of the system at time is:
.
The Lagrangian is given by
.
The algebraic constraint expresses the constant length of the string:
.
The resulting equations of motion are
,
.
Permanent Citation