Topic 3: Rotations and Rigid Body Motion

Describing a Rigid Body

A macroscopic object can often be approximated by a "particle", which has a mass and position in space. A particle has one physical parameter, its mass, and three translational degress of freedom because it can move in 3-dimensional space.

The equations of motion of a particle can be generalized to a system of    particles. Such a system is defined by    mass parameters, and has    translational degrees of freedom. Its configuration at any time can be represented by    points in 3-dimensional space, or by a single point in    dimensional configuration space.

When the size and shape of a macroscopic object matters, it can often be approximated by a "rigid body". A rigid body is a system of particles in which every pair of particles has fixed relative displacement. This is an approximation because the smallest parts of objects are atoms which do not have definite positions according to quantum theory. It is also an approximation because a change in position of one particle cannot affect the position of another particle instantaneously according to the theory of relativity.

Suppose that the rigid body is made of    particles or "atoms", how many degrees of freedom does it have, and how many physical parameters are needed to describe it? Its location and orientation are completely fixed by specifying the positions in space of any three non-collinear particles. A rigid triatomic molecule, which can translate and rotate but not vibrate, has 6 degrees of freedom, 3 translational and 3 rotational. Therefore a rigid body also has 6 degrees of freedom. The configuration space of a rigid body is the product space of a 3-dimensional Euclidean space of translational motion with a 3-dimensional closed ball of radius    with antipodal points identified. The rotations of a rigid body belong to the rotation group SO(3), which is an extremely important concept in physics.

The number of physical parameters required to describe a rigid body approximated by    particles is    masses plus    parameters to specify the fixed relative locations of all the particles.

Rigid Body Motion

The mechanics of rigid bodies was developed in great detail by Leonhard Euler in his book "Theoria motus corporum solidorum seu rigidorum". He introduced many important concepts including moment of inertia and Euler angles. His student Lagrange later applied the Lagrangian method to rigid body motion.

Space-Fixed and Body-Fixed Coordinate Systems

A rigid body can move in space like a free particle, and can rotate, tumble, spin and twist in three indpendent directions at the same time! To simplify this complex motion it is very useful to introduce two coordinate systems.

The space-fixed coordinate system is chosen in an inertial reference frame in which Newton's equations of motion are valid. These equations are exactly the   -body equations of Topic 1.

The body-fixed coordinate system is an origin and three orthogonal axes (unit vectors) that are fixed to the body and rotate, tumble, spin and twist along with it. This is definitely not an inertial coordinate system, so Newton's equations must be modified by non-inertial accelerations, centripetal, Coriolis, etc.

Dr. Butikov's Gyroscope Precession Applet.

The velocity of particle    in the space-fixed system is related to the velocity in the body-fixed system by

where    is the angular velocity about the instantaneous rotation axis passing through the origin of the body-fixed system.

Kinetic Energy

The equations of motion can be derived from the Lagrangian of the system   . The kinetic energy is given by

The middle term is zero if we choose the body-fixed origin at the center of mass of the rigid body

The third term can simplified using

to obtain

Scalars, Vectors, and Tensors

A Tensor is a geometric object associated with a space. Tensors of different "ranks" can be defined in spaces of any dimension. A scalar is a rank-0 tensor and a vector is a rank-1 tensor.

In mechanics we deal with 3-dimensional Euclidean space. Displacements, velocities, and accelerations are vectors, i.e., rank-1 tensors. The dot product of two vectors is a scalar, i.e., a rank-0 tensor.

Scalars, vectors, and tensors of any rank are objects in the space that do not depend on the choice of coordinate system. It is possible to do calculations directly with tensors without choosing a coordinate system. However, it is often convenient, especially in numerical calculations to introduce a coordinate system defined by an origin and a set of basis vectors, e.g. a Cartesian basis   , relative to which a tensor of rank    has    components.

Mathematically, tensors are defined by how their components transform under a change of basis or coordinate system. A scalar has one component, its value, which does not change. A vector in 3-dimensional space has 3 components. It is convenient to write a vector    in column format and express the relation between components in different bases using a    transformation matrix:

The    transformation is an "orthogonal" matrix with the property that its matrix inverse    its matrix transpose with rows and columns interchanged.

A scalar product of two vectors can be expressed by the product of a "transposed" row vector    with the column vector   :

A rank-2 tensor can be constructed as the "tensor product" of two rank-1 tensors. A rank-2 tensor in 3-dimensional space can be represented by a    matrix whose components are constructed from the direct product of a column vector    with the "transposed" row vector   :

In this notation, the cross product of two vectors in 3-dimensions is an antisymmetric rank-2 tensor or "pseudovector":

Moment of Inertia Tensor

The symmetric rank-2 tensor

where    is the unit    matrix, represents the moment of inertia tensor of the rigid body relative to the body-fixed coordinate system. The kinetic energy of the rigid body, which is a scalar, is compactly represented in tensor notation:

An important theorem of linear algebra states that a real symmetric matrix can be diagonalized by an orthogonal transformation:

where the orthogonal matrix    transforms from the body-fixed coordinate system to a "principal axes" coordinate system. The constants    are called the "principal moments of inertia" of the rigid body.

The moment of inertia tensor is defined relative to a point in space. A very simple and useful formula relates the moment of inertia tensor    about the origin of coordinates defined above to the moment of inertia tensor    defined relative to the center of mass of the rigid body.

where

is the position of the center of mass relative to the body-fixed coordinate system.

To prove this result write

where    is the position of    relative to the center of mass. Then

because

and

Parallel Axes and Perpendicular Axes Theorems

The relation

implies

The is the parallel axes theorem: the moment of inertia    about an axis is related to the momentum of inertia about a parallel axis through the center of mass by

where    is the perpendicular distance between the two axes. This theorem applies to any rigid body.

Chapter 8 of Fowles/Cassiday discusses the Perpendicular-Axis Theorem:

The moment of inertia of any plane lamina about an axis normal to the plane of the lamina is equal to the sum of the moments of inertia about any two mutually perpendicular axes passing through the given axis and lying in the plane of lamina.

Carefully note the conditions for this theorem to be applicable!!

In general, the moments of inertia of any rigid body obey an inequality

For a rigid body in general, the sum of moments of inertia about any two perpendicular axes through a point is greater than the moment of inertia about the third perpendicular axis through that point!

The equality holds if and only if the body is laminar and the first two axes lie in the plane of lamina so that    for each particle. The perpendicular-axis theorem breaks down if any part of the body lies off the lamina.

Lagrange's Equations of Motion

The Lagrangian    is the difference between kinetic and potential energies. If there are no external forces, the potential energy of the rigid body is the sum of interaction potential energies between the constituent atoms. Because the separations of pairs of atoms do not change, this internal potential energy does not affect the motion of the rigid body. Internal forces are not zero, in fact they must be infinitely strong for the body to be perfectly rigid, but they cancel in pairs according to Newton's Third Law.

The final expression for the kinetic energy is

where    are the angular velocity components with respect to the three principal axes. This expression shows that the kinetic energy of a rigid body depends on 4 constant parameters and 6 velocity variables.

The Lagrangian is a function of 6 position variables, called generalized coordinates   , and the 6 corresponding generalized velocities   . There is one Lagrange equation for each generalized coordinate:

With no external forces acting, the Lagrangian depends only on the generalized velocities   , so Lagrange's equations are

Angular Momentum

The angular momentum of the system of particles comprising the rigid body about the origin of the inertial space-fixed coordinate system is

where the location of the body-fixed origin at the center of mass, and the vector triple product identity

have been used. The angular momentum of the rigid body is the sum of an "orbital" angular momentum of a equivalent particle of mass   , and an internal "spin" angular momentum about its center of mass

Using Lagrange's equations of motion we see that orbital and spin angular momentum of a rigid body are separately conserved in the absence of external forces:

Gravitational Potential Energy

External forces can act on the rigid body and contribute to its acceleration. Fundamental gravitational and electromagnetic forces are conservative and can be derived from a potential energy function of the coordinates of the rigid body and the other bodies with which it interacts.

If the rigid body comes in contact with another macroscopic object, contact forces such as a normal force of constraint or tangential frictional forces can also act. A rigid body moving through a medium can experience pressure and drag forces.

Some contact forces that do not perform work on the rigid body can be eliminated from Lagrange equations by a suitable choice of generalized coordinates, or by explicitly adding them to Lagrange's equations using Lagrange multipliers.

Frictional and drag forces generally do not conserve energy and require the use of thermodynamics and atomic physics to account for energy losses.

Consider the simple case of gravitational external forces on the rigid body due to a point mass at the origin of the space-fixed coordinate frame. The force on a constituent particle    of the rigid body is given by

The potential energy function

depends in a complicated way on the shape of the rigid body and distribution of particles within it. If the gravitational mass is far away from the rigid body then the potential energy function can be evaluated approximately using a multipole expansion

where

The first term in the expansion, which is called the monople term, is

which is just the gravitational potential energy of a point particle with the rigid body mass located at its the center of mass.

The second term in the expansion is

The first sum on the right hand side gives the dipole term, which vanishes because the origin of the body-fixed system is at the center of mass of the rigid body.

The second term on the right hand side is of order    and for consistency, we must add to it the contribution from the    term in the Taylor series expansion:

The gravitational potential energy of the rigid body in the external field in this approximation is

The second term on the right hand side is called the quadrupole energy of the rigid body. It is determined by the moments and products of inertia of the rigid body.

Forces derived from the quadrupole energy cause torques about its center of mass. These tidal forces cause tides in Earth's oceans, the tidal locking that causes the Man in the moon to keep his face towards us, and are responsible for the peculiar motion of other moons and satellites, for example the chaotic tumbling motion of Saturn's moon Hyperion.

Free Rotation of a Rigid Body

Consider a rigid body that can rotate freely about its center of mass. This can be achieved by mounting the rigid body on Gimbals so its center of mass can be located at any point in space without interfering with its rotation. It can also be achieved in outer space, or during free fall in a constant gravitational field.

Spherical, Symmetrical, and Asymmetrical Tops

Four constant parameters, the mass    and three principal moments of inertia   , characterize the free rotational motion of a Top.

A rigid body with    is called a spherical top. It need not actually be a sphere! A uniform cube is a spherical top. Baseballs, soccer and tennis balls are approximately spherical tops.

A rigid body with an axis of symmetry has    and is called a symmetrical top. A football has a smaller moment of inertial about its long axis and approximates a prolate spheroid. Prolate symmetrical tops have   . A frisbee approximates half of a hollow oblate spheroid, which has a larger moment of inertia about its axis of symmetry. Oblate symmetrical tops have   .

Poinsot's Ellipsoid

Conservation of spin angular momentum and rotational kinetic energy constrain the motion of freely rotating rigid bodies.

If the top is not spherical, then the direction of the instantaneous angular velocity    is different from the direction of the conserved angular momentum. Most of the interesting properties of top motion arise from this fundamental property.

The conservation of angular momentum implies that

This is the equation of an ellipsoid in 3-D Euclidean space of the components of    along the principal axis directions. With    fixed, the tip of the vector    is constrained to lie on the surface of this ellipsoid, if it moves.

Conservation of rotational kinetic energy

constrains the tip of    to the surface of Poinsot's ellipsoid.

To satisfy angular momentum and energy conservation simultaneous, the tip of    must is confined to the curve of intersection of the two ellipsoide, which is called the polhode or path of the pole, see Polhode motion for more.

Poinsot's construction can be used to understand the The Tennis Racket Theorem. If    then the racket set spinning around    or    is in stable rotational motion, but unstable if set spinning around   .

Euler's Equations of Motion for Free Rotation

To find the trajectory of the tip of the angular velocity vector we need to solve Newton's (or Lagrange's) equations of motion for the rigid body.

The direction of the conserved angular momentum is fixed in an inertial reference frame. The angular velocity vector    is most conveniently defined in terms of its components in the principal axes coordinate system, which is rotating relative to the inertial reference frame.

The transformation of the free body equation of motion from fixed to rotating frame is given by

Expressed in principal axes components

Symmetric Top: Space and Body Cones

For a symmetrical top    so the third Euler equation is

Defining a constant with dimensions of angular speed

the first two equations can be written

which can be solved to give

In the inertial reference frame, the angular velocity vector    describes a circle of radius    around    with angular frequency    which is called the angular frequency of precession. The cone of half angle

traced out by    is called the space cone.

In the body-fixed reference frame    describes a cone of half angle

called the body cone with angular frequency

For a nice animation, see Dr. Butikov's Free Rotation of an Axially Symmetrical Body Java Applet.

Eulerian Angles and Euler's Equations

The description of a rigid body is simplest in the body-fixed reference frame which uses the principal axes coordinate system. The moment of inertia tensor is diagonal and constant. The equations of motion are easily expressed in terms of the angular velocity components    along the principal axes directions.

Rigid bodies are usually observed from a space-fixed inertial reference frame. The moment of inertia tensor is not diagonal in general, and its components change with time. We would like to write the equations of motion in terms of vector components in the inertial reference frame.

Euler introduced a very convenient notation for relating quantities in the two frames in terms of Euler angles.

To focus on rotational motion, suppose that the origin of coordinates in the inertial frame is chosen to coincide with the origin in the body-fixed frame at a particular instant of time   , and that the inertial frame is moving with the same instantaneous velocity as the rigid body at this time   . Of course this will change with time if the body is accelerating, but we just want to obtain the form of the equations in the fixed frame at this instant: by Galilean invariance, this form will hold in all inertial frames.

Figure 9.6.1 from Fowles/Cassiday above shows a standard definition of the Euler angles   . The intersection of the inertial and body-fixed   -   planes is called the line of nodes. The coordinate systems are both right-handed,    is a polar angle in the range   , and    are azimuthal angles in the range   .

The figure also shows the instantaneous angular velocity    of the rigid body about the origin. As the body rotates, the Euler angles will change with rates    about the space-fixed    axis, the line of nodes, and the body-fixed    axis, respectively:

where    are principal axes unit vectors, and    is the unit vector along the line of nodes.

The dot products above are most easily evaluated by noting that the    axis direction has polar angle    and azimuthal angle    with respect to the principal axes

and that

The Symmetric Top

Consider a symmetric top spinning about a tip of its symmetric axis as shown in Figure 9.7.1 of Fowles/Cassiday:

Note that its center of mass is a distance    from the tip. The moments of inertia about the tip are

The rotational kinetic energy of a rigid body with axis of symmetry    in terms of Euler angles is

The gravitational potential energy relative to the level of the tip is

and the Lagrangian function is

Note that the Lagrange function does not depend on    and    The Lagrange equations of motion for    and   

show that the angular momentum components along the vertical and symmetric directions are conserved

These equations can be solved for

The equation of motion for    is

Steady Precession

In the bicycle wheel lecture demo the angle    of the symmetry axis with the vertical is initially constant around    while the wheel precesses around the vertical and then slow increases towards    as the wheel spins down.

Look for a solution with    and   :

If the conserved spin    is very large, there will be a slow precession about the vertical with constant angular speed   . In reality, friction in the bearings and atmospheric drag will cause the spin    to decrease, causing    and    to increase.

In fact steady precession is also possible at    values other than   . Assume again that    and solve

for two precession angular speeds

If the wheel is set spinning fast with   , then the plus sign gives a fast precession and the minus sign a slow precession.

Note that steady precession implies a minimum angular momentum given by

Energy Equation and Nutation

A constant external force acting on a symmetric top can cause a "bobbing" motion called nutation, in addition to regular precession. These effects can be seen in the Earth's rotation as the Precession of the equinoxes, and the Chandler wobble which was originally predicted by Newton and by Euler.

The Lagrangian function for the symmetric top can be written in terms of the conserved angular momentum components    and   

and the conserved energy of the top is

The motion of the top now reduces to a one-dimensional problem in the variable   , which can be solved with the help of the effective potential energy

The angular velocities are determined by

Given    there are two turning points    between which    oscillates periodically. This periodic "bobbing" motion is called nutation.

There are two possibilities for the azimuthal motion in the variable   . The velocity    can change sign periodically resulting in a spiral path for the symmetry axis direction, or maintain the same sign resulting in a wave like path. The spiral and wave like solutions are separated by the vanishing of    at a turning point in   , resulting in a cusp in the path of the axis direction. To see this let

f[u_, a_, b_, c_, d_] := (1 - u^2) (a - b u) - (c - d u)^2

phidot[u_, c_, d_] := (c - d u) / (1 - u^2)

Manipulate[Plot[{f[u, a, b, c, d], phidot[u, c, d]}, {u,-1.5,2.5}], 
                {a, 0.1, 10}, {b, 0.1, 10}, {c, 0.1, 10}, {d, 0.1, 10}]

Applications of Rigid Body Motion

The procedure described above to obtain the motion of symmetric top can be used to study a variety of applications of rigid body motion.

• Find the center of mass and principal moments of inertia of the rigid body. If the object has an axis of symmetry, as is does in many applications, choose this direction as the body-fixed    axis.
• Choose a convenient inertial reference frame. The inertial    coordinate system can be chosen arbitrarily at any instant of time to simplify the derivation of the equations of motion.
• Choose the primed frame used in defining the Euler angles and the primed coordinate system    to simplify the equations. The line of nodes direction    is the positive    axis direction. The    axis is usually chosen along the symmetry axis so   .
• The Euler angles are defined relative to the primed frame, which will generally not be inertial. The equations of motion in the primed frame will involve noninertial accelerations that
  

  where    is the torque of the net external force about the origin of the primed system.

Gryroscopes and The Gyrocompass

A Gyroscope is a symmetrical top mounted so it can rotate freely about its center of mass, see YouTube Gyroscope video. This can be achieved using Gimballs. In the absence of external torques its angular momentum is constant in any inertial reference frame, and can be used for navigation.

A Gyrocompass is a symmetrical top mounted so that it can rotate freely in the horizontal plane on Earth's surface. Gyrocompasses point true North, unlike magnetic compasses which align with Earth's magnetic field.

The angular velocity of the body-fixed frame due to Earth's rotation is

where    is the latitude of the center of mass on Earth's surface.

The angular velocity of the gyrocompass in the body-fixed frame is got by adding its spin angular velocity to its orbital angular velocity:

The gyrocompass has principal moments of inertia    and   . Its rotational kinetic energy is

and its angular momentum is

The rate of change of angular momentum in the inertial frame is determined by the torque on the gyrocompass that constrains it to the horizontal.

The    component

shows that the spin of the gyrocompass

The    component gives the acceleration of the gyroscope in the horizontal plane

The second term on the right hand side is small because the gyrocompass is set spinning at several Hz, while Earth rotates much more slowly. The equation reduces to that of a physical pendulum in the variable    which measures deviation from the North-South direction:

The gyrocompass is mounted in oil so that its motion is damped, and its spin is maintained by an electric motor. If the motion of the ship causes its symmetry axis to deviate from the North-South direction, it reverts back to pointing North in a damped harmonic motion. The compass works best at the Equator and becomes less effective near the Poles.

Stability of Bicycle Motion

A circular laminar object, like a coin or wheel, that rolls without slipping on a surface is stabilized by its spin angular momentum if it spins sufficiently fast. This is called gyroscopic stabilization. The stability of bicycle motion is not primarily due to gyroscopic stabilization of the spinning wheels, although this principle can be used to improve stability as in the Gyrobike or the Gyro Monorail.

Consider a wheel of mass   , radius   , and principal moments of inertia    and   , that rolls without slipping on a horizontal surface. The symmetry axis with direction    makes an angle    with the vertical direction   , the    direction is taken opposite to the point of contact    with the surface, and the    (line of nodes) direction is horizontal.

The wheel can wobble in addition to spinning about its axle:

so that

The kinetic energy of rotation about the center of mass is

The condition of rolling without slipping is

where    is the position vector of the point of contact relative to the center of mass. The cross product

The translational kinetic energy of the center of mass is

To investigate the stability of a vertical fast rolling wheel, we can assume that    is small so    and   , and    and    are small compared with   .

The equations of motion in this approximation reduce to

The first equation implies that the spin is conserved   . The second equation can then be solved to give

Assuming    and there is initially no precession   , the third equation becomes

This is the equation for simple harmonic motion provided that