Pendulum DynamicsPendulum Dynamics

Introduction to the Project | Applications of Pendulums | Bifurcation Theory | Numerical Analysis for Nonlinear Systems
Modelling the Parametrically Forced Pendulum | Bifurcations occurring in the System
Stability of the Vertically Downward Position | The Effect of Varying Damping | Stability of the Vertically Upward Position
Summary of Findings | Extensions to the Project | Acknowledgements & References

The Applications of Pendulums

There are a number of real-life physical systems where the pendulum is the underlying mechanism. Lets examine some examples.

i) The human body. Walking is often likened to the motion of two coupled pendula where the supported leg is analogous to an inverted pendulum with the suspension point on the ground and the swinging leg analogous to a damped pendulum. As humans learn to walk, control of this system is developed and research suggests that individuals favour a step frequency and step length which minimises their expenditure of energy. The variables of step length and frequency are essentially parameters of the modelled system. Although such a model is a significant simplification of human motion, the pendulum remains the fundamental mechanism in walking.

ii) Rocket Science.  When a space rocket is launched, in order to control its trajectory, the rocket itself must be kept accurately balanced in line with the direction of thrust.  This is typically implemented using a thrust vector control system whereby corrective bursts are fired during the ascent from a set of gimballed control thrusters at the base of the rocket.  This system is analogous to a forced pendulum in three-dimensional space.  The explicit forcing, a physical external force exerted on the system, is provided by the set of control thrusters that stabilise the pendulum in the inverted position.  Again, this is a simplification of the entire dynamics of a space shuttle launch, but the classical engineering problem of balancing an inverted pendulum plays a key role in controlling the rocket.

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