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FREE ROTATION OF AN AXIALLY SYMMETRICAL BODY
Simulations in Physics


A small simulation program (Java-applet) presented here visualizes torque-free rotation (or "inertial rotation") of an axially symmetrical body, that is, a body whose two of three principal moments of inertia are equal. Kinematically, such inertial rotation is called precession. Being applied to a gyroscope (see "Forced precession of a gyroscope") such a precession is called nutation.

Rotation of a rigid body about a fixed point is characterized by a vector of momentary angular velocity. Any point of the rotating body has a (linear) velocity, which at every moment of time is exactly the same as if the body were rotating around an (immovable) axis directed along the angular velocity vector. However, for a general case of free rotation, the vector of angular velocity and hence the momentary axis of rotation change continuously their direction in space. This mysterious motion of the axis of rotation sometimes is called "wobbling." Even in the absence of external torques, that is, during the inertial rotation, behavior of the momentary axis of rotation seems to be very complicated. And the trajectories of different points of the rotating body seem even more complicated.

For a rotating rigid body, the vector of angular momentum L is proportional to the momentary angular velocity ω, but generally the spatial direction of L differs from the direction of ω. Their directions coincide only if the angular velocity is directed along one of the three mutually orthogonal axes called the principal axes of inertia of the body. For symmetric bodies manufactured of a uniform material, these principal axes coincide with the axes of symmetry. For example, the principal axes of inertia of a rectangular box pass through its center of mass parallel to the edges. Moments of inertia, calculated with respect to the principal axes of inertia that pass through the center of mass, are called the central principal moments of inertia.

Inertial rotation of a rigid body around the principal axes of inertia actually is a very simple motion. Indeed, for such cases, vectors L and ω have a common direction. In the absence of external torques, vector L of the angular momentum remains constant, and therefore both the direction and magnitude of the angular velocity vector ω also remain constant. For this reason the principal axes of inertia are called also the axes of free rotation. If we set a rigid body into rotation around one of these axes and then release it, the body simply continues spinning uniformly about the axis whose direction in space and in the body does not change. All points of the body synchronously trace circles whose centers are located on this axis.

If the direction of the initial angular velocity deviates from a principal axis of inertia, the inertial rotation is relatively simple for any body which can be called a symmetrical top. Symmetrical top is a body whose two of three principal moments of inertia are equal. Examples of such bodies are given by a rod or bar with a circular or square cross section, as well as by any prism or pyramid whose base is a regular polygon (including triangle), manufactured of a homogeneous material; a circular disc, cylinder or a cone; an ellipsoid of rotation (oblate or prolate spheroid), and so on. (Simulation of inertial rotation of a cylinder (or circular disc) can be found on the web page "Torque-free rotation of a cylinder.")

When a symmetrical top is spinning around the axis of symmetry, the angular momentum is also directed along this axis. If the vector of angular velocity deviates from the axis of symmetry by some angle, the directions of vector L of the angular momentum and vector ω does not coincide, but they both lie in the same plane with the axis of symmetry of the body.

Taking into account the mutual arrangement of the angular momentum, angular velocity and the axis of the body, we can easily show that in the absence of external torques the axis of the body and the vector of momentary angular velocity both perform precession about the fixed direction of the angular momentum L. In other words, the axis of the body and the angular velocity move synchronously in space describing circular cones whose common apex is located at the center of mass. The angles at vertexes of these cones remain constant during the rotation of the body. Click here in order to observe the simulation of such motion (in the left-hand part of the applet window). (Click also here to put the applet to the upper part of the screen.)


You can make a pause in the simulation and resume it by clicking on the button "Start/Pause" on the control panel located on the left-hand side of the applet window. This control panel allows you also to vary parameters of the system and conditions of the simulation. By dragging the pointer of the mouse inside the applet window (moving the pointer with the left button pressed) you can rotate the image around the vertical and horizontal axes for a more convenient point of view.

The vector of angular velocity (the yellow arrow on the black background) shows in space the direction of momentary axis of rotation. The set of these momentary axes at different moments of time forms in space a circular cone whose vertex is located at the center of mass and whose axis is directed along vector L of the angular momentum (vertically on the screen). This space cone is also called the immovable axoid (see the right-hand side of the applet window).

Next we imagine one more circular cone, this time attached firmly to the body. The vertex of this cone is also located at the center of mass, and its axis is directed along the axis of symmetry of the body. Let the generator of this cone be the vector of angular velocity ω (the yellow arrow on the screen), that is, the momentary axis of rotation. In other words, the lateral surface of this cone associated with the body is formed by the set of momentary axes of rotation at different moments of time, and shows how these axes are located inside the body (relative to the body). For this reason, this imaginary cone, associated with the rotating body, is called the moving axoid (or the body cone).

The moving and immovable cones touch one another (outwardly for a prolate body, whose transverse moment of inertia is greater than longitudinal) by their lateral surfaces along vector ω that shows the momentary axis of rotation. All the points of the body, which are located at a given moment of time on the momentary axis of rotation, have zero linear velocities. This means that the moving cone (attached to the body) is just rolling without slipping over the surface of the immovable cone. This clear geometrical interpretation of kinematics of the free precession (of the inertial rotation) is shown in the right-hand side of the applet. (Click here to put the applet on the upper part of the screen.)

We can associate also this obvious geometrical interpretation of the free precession (as rolling of the moving cone without slipping over the surface of the immovable cone) with the decomposition of vector ω of momentary angular velocity onto the vector sum of two components shown by red arrows in the applet. One of these components corresponds to the axial rotation (spinning) of the body around its axis of symmetry. This component of the angular velocity is directed along the axis of symmetry, that is, its direction inside the body does not change. In space, this component generates a circular cone together with the axis of the body. The other component does not change its direction in space: this component corresponds to precession of the axis of symmetry about angular momentum L, whose direction in space (vertical on the screen) is preserved in the absence of external torques.

Points of the body located on its axis of symmetry trace circular paths whose centers lie on the axis of the immovable cone, that is, on the axis of precession (vertical on the screen). We can consider complicated motions of points that doesn't lie on the axis of symmetry as a superposition of two relatively simple motions, namely of a rotation around the axis of symmetry with simultaneous motion of this axis in space (precession) along the surface of a vertical cone. The simulation program can show trajectories of such points. To make the program do this, mark the check box on the control panel labelled "Trace of a point", and indicate position of the desired point by entering the angle between the axis of the body and direction to the point in question. For more convenient observation, the program shows the trajectory traced by the end point of an imaginary long arrow fixed to the body. The end point of this arrow lies beyond the surface bounding the body. All points of this arrow (that starts at the center of mass) trace similar trajectories, but the path of its end point shows their peculiarities in a larger scale. Click here in order to observe the trajectory of a point located on the surface of a moving cone. (Initially this point lies on the momentary axis). Click also here to put the applet on the upper part of the screen.

The geometrical interpretation of inertial rotation of a symmetrical body considered above is also applicable to a special case of a body whose longitudinal and transverse moments of inertia are equal. For such a body called a spherical top moments of inertia for all possible axes passing through the center of mass are equal. The body itself should not necessarily be spherically symmetrical: a cube, tetrahedron, or any regular polyhedron, in respect to rotation, are all dynamically equivalent and give examples of spherical tops. For such bodies, the angular velocity and the angular momentum always have a common direction, and an arbitrary axis is the axis of free rotation. This means that for a spherical top vector w of the angular velocity and the momentary axis of rotation keep their directions in space fixed (they do not precess), and the immovable cone degenerates into a ray directed along the angular momentum L (vertically on the screen). Rolling of the moving cone attached to the body is reduced in this case to a uniform rotation of this cone around the vertical ray. This ray (the degenerate immovable cone) lies on the lateral surface of the moving cone and is directed along ω. Any point of the body (say, the endpoint of an arrow sticked into the body) generates a circular path whose center is located on the axis of rotation. Click here to simulate the inertial rotation of a spherically symmetrical body. Click also here to put the applet on the upper part of the screen.

For an oblate symmetrical body the imaginary cone fixed to the body is touching the immovable space cone internally, by its interior surface. In this case the geometrical interpretation of inertial rotation is less obvious and even counterintuitive in some way. We note that when the symmetry axis of the body performs precession counterclockwise and the wide moving cone is rolling by its interior surface without slipping over the enveloped immovable cone, the body itself is spinning (rotating around its axis) clockwise – in the opposite sense with respect to the precession. This is clearly seen from the decomposition of the angular velocity vector (yellow arrow on the screen) onto components (red arrows) corresponding to precession and axial rotation: the component that corresponds to spin makes an obtuse angle with L (is directed downward on the screen). Click here in order to simulate the inertial rotation of an oblate body. Click also here to put the applet on the upper part of the screen.

A detailed theoretical description of an inertial rotation of an axially symmetrical body can be found in the paper «Inertial Rotation of a Rigid Body».

An understanding about the inertial (torque-free) rotation of an axially symmetrical body is an important prerequisite for the study of a counterintuitive behavior of gyroscopes, whose torque-induced precession is generally complicated by a nutation (see «Precession and Nutation of a Gyroscope»).

Controlling the program.
The simulation program allows you to vary parameters of the system and conditions of the simulation. You can rotate the cube with the image of the body around the vertical and horizontal axes (for a more convenient point of view) simply by dragging the pointer of the mouse inside the applet window (moving the pointer with the left button pressed). If in addition you will hold the "Control" key on the keyboard, the image will move in the desired direction. If you will hold the "Shift" key, dragging the mouse pointer will change the scale – the image will move closer or farther.

Rotation of the body can be represented in a convenient time scale. To change the time scale, you can vary the delay value (in milliseconds) by dragging a slider in the lower part of the control panel. The functions of other controls are rather obvious. The upper button starts the simulation and makes a pause. The second button allows you to execute the simulation by steps. The third button ("Initialize") retrieves the initial conditions. The fourth ("Reset") restores the default values.

At first acquaintance with the program, instead of entering the values of parameters, you can open the list of predefined examples and choose from it a suitable example. Performing experiments on your own, you can vary the values of parameters either by dragging the sliders, of by typing the desired values with the keyboard. Before changing parameters, you should pause the simulation by using the "Start/Pause" button. When you type new values into the window, the colour of its background changes to bright yellow. You should finish the input by pressing the "Enter" key. If the value is admissible, the window assumes its usual colour.

Inertial properties that characterize rotation of a symmetrical body are given by two central moments of inertia: one about the axis of symmetry, the other about any transverse axis passing through the center of mass. Only the ratio of these moments of inertia is essential for the simulation. In the program, the ratio of the transverse and longitudinal moments of inertia is defined by the parameter "Prolateness". Admissible values of this parameter lie in the interval from 0.5 up to 5.0. If this parameter equals 1, the transverse and longitudinal moments of inertia are equal, which is characteristic of a spherically symmetric body or a regular polyhedron. For a prolate body (stretched along the axis of symmetry) this parameter is greater than unity; for an oblate (squeezed) body – smaller than unity.

Another parameter that you can vary in the simulation program is the angle between the axis of symmetry of the body and direction of the angular velocity vector (direction of the momentary axis of rotation). On the control panel of the program, this angle is labelled "Deviation." The value of this angle should be entered in degrees. Admissible values lie in the interval from 0 up to 90 degrees. The magnitude of the angular velocity can be varied in the interval from 0.5 up to 10 arbitrary units. Variation of this parameter influences the speed of axial rotation, but doesn't change characteristic features of the body motion.

We have already mentioned earlier how you can switch on drawing of the trajectory of an arbitrary point of the body by marking the check box "Show point trace" and indicating the angular position of the desired point ("TraceAngle" parameter). The background color of the applet window depends on the state of the check box "Dark background".


Simulation of inertial rotation of a cylinder (or circular disc) can be found on the web page "Torque-free rotation of a cylinder."

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The applet presented here was created with the help of Easy Java Simulations tool developed by Francisco Esquembre, professor at University of Murcia, Spain.



Last revision – 20 June 2012.

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Simulations in Physics