Dynamical Model of a Quad Lutz

by George S. Rossano

The following is a simple dynamical model of a quad Lutz that illustrates the in-flight characteristics of a jump with 0.75 seconds of air time.

The parameters of this model are:

Air time: 0.75 sec
Takeoff rotation rate: 2.50 rotations per second
Pre-rotation at takeoff: 0.249 rotation
Moment of inertia in-flight: as shown in graph

Although the ISU tolerates any amount of pre-rotation on jumps, we have limited the model to a pre-rotation less than the one-quarter rotation that, by a strict application of the rules, should result in an under-rotation call. Even the best executed quads have at least this pre-rotation.

Normalized Moment of Inertia
This is an adjustable parameter of the model.

This model is derived from an aggregate of several competition quad Lutz examples, and illustrates the main features of a true quad Lutz jump - as opposed to pseudo quads that are often pre-rotated by one-half rotation on the takeoff and may also be missing rotation on the landing.

For this jump the skater spends about 200 msec pulling in and 75 msec opening up to land. About 475 msec is spent mid-flight near the minimum moment of inertia. Note that during mid-flight the skater typically makes small adjustments to their position and the minimum moment of inertia is generally not reached until late in mid-flight. We specify times for pulling in and opening up using the initial and final air times when the moment of inertia is one-half the initial moment of inertia.

The above moment of inertia sequence is an "ideal" example for a jump with 0.75 seconds of air time. Individual jump attempts with the same air time show variations in the time to pull in and check out, and show variations in the consistency with which the near minimum air position (and rotation rate) is held.

In our model for true quads we include a small pre-rotation at takeoff, and full rotation at the landing. This small pre-rotation seems to be necessary to develop the initial angular momentum needed to complete a quad, and is smaller than what would trigger an under-rotation call under a strict interpretation of ISU rules. Jumps that are missing more than this one-quarter pre-rotations we consider pseudo quads, from a dynamical point of view.

The in-flight characteristics of this jump are shown in the following graphs.

Flight of the Jump

Rotation Rate In-Flight

Rotations Completed

This model has a peak rotation rate of 5.8 rotations per second and a landing rotation speed of 2.8 rotations per second.

One could construct a family of models with slightly varying air times and initial rotation rates. All of such examples would qualitatively look similar to the graphs presented here.