Below are selected topics from rigid-body dynamics, a subtopic of classical mechanics involving the use of Newton's laws of motion to solve for the motion of rigid bodies moving in 1D, 2D, or 3D space.B.11 We may think of a rigid body as a distributed mass, that is, a mass that has length, area, and/or volume rather than occupying only a single point in space. Rigid body models have application in stiff strings (modeling them as disks of mass interconnect by ideal springs), rigid bridges, resonator braces, and so on.
We have already used Newton's to formulate mathematical dynamic models for the ideal point-mass (§B.1.1), spring (§B.1.3), and a simple mass-spring system (§B.1.4). Since many physical systems can be modeled as assemblies of masses and (normally damped) springs, we are pretty far along already. However, when the springs interconnecting our point-masses are very stiff, we may approximate them as rigid to simplify our simulations. Thus, rigid bodies can be considered mass-spring systems in which the springs are so stiff that they can be treated as rigid massless rods (infinite spring-constants , in the notation of §B.1.3).
So, what is new about distributed masses, as opposed to the point-masses considered previously? As we will see, the main new ingredient is rotational dynamics. The total momentum of a rigid body (distributed mass) moving through space will be described as the sum of the linear momentum of its center of mass (§B.4.1 below) plus the angular momentum about its center of mass (§B.4.13 below).
A nice property of the center of mass is that gravity acts on a far-away object as if all its mass were concentrated at its center of mass. For this reason, the center of mass is often called the center of gravity.
Linear Momentum of the Center of Mass
Thus, the momentum of any collection of masses (including rigid bodies) equals the total mass times the velocity of the center-of-mass.
Whoops, No Angular Momentum!
The previous result might be surprising since we said at the outset that we were going to decompose the total momentum into a sum of linear plus angular momentum. Instead, we found that the total momentum is simply that of the center of mass, which means any angular momentum that might have been present just went away. (The center of mass is just a point that cannot rotate in a measurable way.) Angular momentum does not contribute to linear momentum, so it provides three new ``degrees of freedom'' (three new energy storage dimensions, in 3D space) that are ``missed'' when considering only linear momentum.
To obtain the desired decomposition of momentum into linear plus angular momentum, we will choose a fixed reference point in space (usually the center of mass) and then, with respect to that reference point, decompose an arbitrary mass-particle travel direction into the sum of two mutually orthogonal vector components: one will be the vector component pointing radially with respect to the fixed point (for the ``linear momentum'' component), and the other will be the vector component pointing tangentially with respect to the fixed point (for the ``angular momentum''), as shown in Fig.B.3. When the reference point is the center of mass, the resultant radial force component gives us the force on the center of mass, which creates linear momentum, while the net tangential component (times distance from the center-of-mass) give us a resultant torque about the reference point, which creates angular momentum. As we saw above, because the tangential force component does not contribute to linear momentum, we can simply sum the external force vectors and get the same result as summing their radial components. These topics will be discussed further below, after some elementary preliminaries.
Translational Kinetic Energy
The translational kinetic energy of a collection of masses is given by
More generally, the total energy of a collection of masses (including distributed and/or rigidly interconnected point-masses) can be expressed as the sum of the translational and rotational kinetic energies [270, p. 98].
Rotational Kinetic Energy
The rotational kinetic energy of a rigid assembly of masses (or mass distribution) is the sum of the rotational kinetic energies of the component masses. Therefore, consider a point-mass rotatingB.13 in a circular orbit of radius and angular velocity (radians per second), as shown in Fig.B.4. To make it a closed system, we can imagine an effectively infinite mass at the origin . Then the speed of the mass along the circle is , and its kinetic energy is . Since this is what we want for the rotational kinetic energy of the system, it is convenient to define it in terms of angular velocity in radians per second. Thus, we write
is called the mass moment of inertia.
The mass moment of inertia (or simply moment of inertia), plays the role of mass in rotational dynamics, as we saw in Eq.(B.7) above.
The mass moment of inertia of a rigid body, relative to a given axis of rotation, is given by a weighted sum over its mass, with each mass-point weighted by the square of its distance from the rotation axis. Compare this with the center of mass (§B.4.1) in which each mass-point is weighted by its vector location in space (and divided by the total mass).
Equation (B.8) above gives the moment of inertia for a single point-mass rotating a distance from the axis to be . Therefore, for a rigid collection of point-masses , ,B.14 the moment of inertia about a given axis of rotation is obtained by adding the component moments of inertia:
where is the distance from the axis of rotation to the th mass.
For a continuous mass distribution, the moment of inertia is given by integrating the contribution of each differential mass element:
where is the distance from the axis of rotation to the mass element . In terms of the density of a continuous mass distribution, we can write
The moment of inertia for the same circular disk rotating about an axis in the plane of the disk, passing through its center, is given by
Perpendicular Axis Theorem
In general, for any 2D distribution of mass, the moment of inertia about an axis orthogonal to the plane of the mass equals the sum of the moments of inertia about any two mutually orthogonal axes in the plane of the mass intersecting the first axis. To see this, consider an arbitrary mass element having rectilinear coordinates in the plane of the mass. (All three coordinate axes intersect at a point in the mass-distribution plane.) Then its moment of inertia about the axis orthogonal to the mass plane is while its moment of inertia about coordinate axes within the mass-plane are respectively and . This, the perpendicular axis theorem is an immediate consequence of the Pythagorean theorem for right triangles.
Let denote the moment of inertia for a rotation axis passing through the center of mass, and let denote the moment of inertia for a rotation axis parallel to the first but a distance away from it. Then the parallel axis theorem says that
Note that the moment of inertia does not change when masses are moved along a vector parallel to the axis of rotation (see, e.g., Eq.(B.9)). Thus, any rigid body may be ``stretched'' or ``squeezed'' parallel to the rotation axis without changing its moment of inertia. This is known as the stretch rule, and it can be used to simplify geometry when finding the moment of inertia.
For example, we saw in §B.4.4 that the moment of inertia of a point-mass a distance from the axis of rotation is given by . By the stretch rule, the same applies to an ideal rod of mass parallel to and distance from the axis of rotation.
Note that mass can be also be ``stretched'' along the circle of rotation without changing the moment of inertia for the mass about that axis. Thus, the point mass can be stretched out to form a mass ring at radius about the axis of rotation without changing its moment of inertia about that axis. Similarly, the ideal rod of the previous paragraph can be stretched tangentially to form a cylinder of radius and mass , with its axis of symmetry coincident with the axis of rotation. In all of these examples, the moment of inertia is about the axis of rotation.
The area moment of inertia is the second moment of an area around a given axis:
In a planar mass distribution with total mass uniformly distributed over an area (i.e., a constant mass density of ), the mass moment of inertia is given by the area moment of inertia times mass-density :
For a planar distribution of mass rotating about some axis in the plane of the mass, the radius of gyration is the distance from the axis that all mass can be concentrated to obtain the same mass moment of inertia. Thus, the radius of gyration is the ``equivalent distance'' of the mass from the axis of rotation. In this context, gyration can be defined as rotation of a planar region about some axis lying in the plane.
For a bar cross-section with area , the radius of gyration is given by
where is the area moment of inertia (§B.4.8) of the cross-section about a given axis of rotation lying in the plane of the cross-section (usually passing through its centroid):
For a rectangular cross-section of height and width , area , the area moment of inertia about the horizontal midline is given by
The radius of gyration can be thought of as the ``effective radius'' of the mass distribution with respect to its inertial response to rotation (``gyration'') about the chosen axis.
Most cross-sectional shapes (e.g., rectangular), have at least two radii of gyration. A circular cross-section has only one, and its radius of gyration is equal to half its radius, as shown in the next section.
Using the elementrary trig identity , we readily derive
For a circular tube in which the mass of the cross-section lies within a circular annulus having inner radius and outer radius , the radius of gyration is given by
Two Masses Connected by a Rod
As an introduction to the decomposition of rigid-body motion into translational and rotational components, consider the simple system shown in Fig.B.5. The excitation force densityB.15 can be applied anywhere between and along the connecting rod. We will deliver a vertical impulse of momentum to the mass on the right, and show, among other observations, that the total kinetic energy is split equally into (1) the rotational kinetic energy about the center of mass, and (2) the translational kinetic energy of the total mass, treated as being located at the center of mass. This is accomplished by defining a new frame of reference (i.e., a moving coordinate system) that has its origin at the center of mass.
First, note that the driving-point impedance (§7.1) ``seen'' by the driving force varies as a function of . At , The excitation sees a ``point mass'' , and no rotation is excited by the force (by symmetry). At , on the other hand, the excitation only sees mass at time 0, because the vertical motion of either point-mass initially only rotates the other point-mass via the massless connecting rod. Thus, an observation we can make right away is that the driving point impedance seen by depends on the striking point and, away from , it depends on time as well.
To avoid dealing with a time-varying driving-point impedance, we will use an impulsive force input at time . Since momentum is the time-integral of force ( ), our excitation will be a unit momentum transferred to the two-mass system at time 0.
First, consider . That is, we apply an upward unit-force impulse at time 0 in the middle of the rod. The total momentum delivered in the neighborhood of and is obtained by integrating the applied force density with respect to time and position:
The kinetic energy of the system after time zero is
In this case, the unit of vertical momentum is transferred entirely to the mass on the right, so that
Note that the velocity of the center-of-mass is the same as it was when we hit the midpoint of the rod. This is an important general equivalence: The sum of all external force vectors acting on a rigid body can be applied as a single resultant force vector to the total mass concentrated at the center of mass to find the linear (translational) motion produced. (Recall from §B.4.1 that such a sum is the same as the sum of all radially acting external force components, since the tangential components contribute only to rotation and not to translation.)
All of the kinetic energy is in the mass on the right just after time zero:
However, after time zero, things get more complicated, because the mass on the left gets dragged into a rotation about the center of mass.
To simplify ongoing analysis, we can define a body-fixed frame of referenceB.16 having its origin at the center of mass. Let denote a velocity in this frame. Since the velocity of the center of mass is , we can convert any velocity in the body-fixed frame to a velocity in the original frame by adding to it, viz.,
In summary, we defined a moving body-fixed frame having its origin at the center-of-mass, and the total kinetic energy was computed to be
It is important to note that, after time zero, both the linear momentum of the center-of-mass ( ), and the angular momentum in the body-fixed frame ( ) remain constant over time.B.17 In the original space-fixed frame, on the other hand, there is a complex transfer of momentum back and forth between the masses after time zero.
Similarly, the translational kinetic energy of the total mass, treated as being concentrated at its center-of-mass, and the rotational kinetic energy in the body-fixed frame, are both constant after time zero, while in the space-fixed frame, kinetic energy transfers back and forth between the two masses. At all times, however, the total kinetic energy is the same in both formulations.
When working with rotations, it is convenient to define the angular-velocity vector as a vector pointing along the axis of rotation. There are two directions we could choose from, so we pick the one corresponding to the right-hand rule, i.e., when the fingers of the right hand curl in the direction of the rotation, the thumb points in the direction of the angular velocity vector.B.18 The length should obviously equal the angular velocity . It is convenient also to work with a unit-length variant .
As introduced in Eq.(B.8) above, the mass moment of inertia is
given by where is the distance from the (instantaneous)
axis of rotation to the mass located at
terms of the angular-velocity vector
, we can write this as
The vector cross product (or simply vector product, as
opposed to the scalar product (which is also called the
dot product, or inner product)) is commonly used in
vector calculus--a basic mathematical toolset used in
acoustics , electromagnetism , quantum
mechanics, and more. It can be defined symbolically in the form of
a matrix determinant:B.19
where denote the unit vectors in . The cross-product is a vector in 3D that is orthogonal to the plane spanned by and , and is oriented positively according to the right-hand rule.B.20
The second and third lines of Eq.(B.15) make it clear that . This is one example of a host of identities that one learns in vector calculus and its applications.
where denotes the identity matrix in , denotes the orthogonal-projection matrix onto , denotes the projection matrix onto the orthogonal complement of , denotes the component of orthogonal to , and we used the fact that orthogonal projection matrices are idempotent (i.e., ) and symmetric (when real, as we have here) when we replaced by above. Finally, note that the length of is , where is the angle between the 1D subspaces spanned by and in the plane including both vectors. Thus,
The direction of the cross-product vector is then taken to be orthogonal to both and according to the right-hand rule. This orthogonality can be checked by verifying that . The right-hand-rule parity can be checked by rotating the space so that and in which case . Thus, the cross product points ``up'' relative to the plane for and ``down'' for .
To see this, let's first check its direction and then its magnitude. By the right-hand rule, points up out of the page in Fig.B.4. Crossing that with , again by the right-hand rule, produces a tangential velocity vector pointing as shown in the figure. So, the direction is correct. Now, the magnitude: Since and are mutually orthogonal, the angle between them is , so that, by Eq.(B.16),
Relation of Angular to Linear Momentum
Thus, the angular momentum is times the linear momentum .
Linear momentum can be viewed as a renormalized special case of angular momentum in which the radius of rotation goes to infinity.
Angular Momentum Vector
Like linear momentum, angular momentum is fundamentally a vector in . The definition of the previous section suffices when the direction does not change, in which case we can focus only on its magnitude .
More generally, let denote the 3-space coordinates of a point-mass , and let denote its velocity in . Then the instantaneous angular momentum vector of the mass relative to the origin (not necessarily rotating about a fixed axis) is given by
where denotes the vector cross product, discussed in §B.4.12 above. The identity was discussed at Eq.(B.17).
For the special case in which is orthogonal to , as in Fig.B.4, we have that points, by the right-hand rule, in the direction of the angular velocity vector (up out of the page), which is satisfying. Furthermore, its magnitude is given by
In the more general case of an arbitrary mass velocity vector , we know from §B.4.12 that the magnitude of equals the product of the distance from the axis of rotation to the mass, i.e., , times the length of the component of that is orthogonal to , i.e., , as needed.
It can be shown that vector angular momentum, as defined, is conserved.B.22 For example, in an orbit, such as that of the moon around the earth, or that of Halley's comet around the sun, the orbiting object speeds up as it comes closer to the object it is orbiting. (See Kepler's laws of planetary motion.) Similarly, a spinning ice-skater spins faster when pulling in arms to reduce the moment of inertia about the spin axis. The conservation of angular momentum can be shown to result from the principle of least action and the isotrophy of space [270, p. 18].
The matrix is the Cartesian representation of the mass moment of inertia tensor, which will be explored further in §B.4.15 below.
In summary, the angular momentum vector is given by the mass moment of inertia tensor times the angular-velocity vector representing the axis of rotation.
Note that the angular momentum vector does not in general point in the same direction as the angular-velocity vector . We saw above that it does in the special case of a point mass traveling orthogonal to its position vector. In general, and point in the same direction whenever is an eigenvector of , as will be discussed further below (§B.4.16). In this case, the rigid body is said to be dynamically balanced.B.24
As derived in the previous section, the moment of inertia tensor, in 3D Cartesian coordinates, is a three-by-three matrix that can be multiplied by any angular-velocity vector to produce the corresponding angular momentum vector for either a point mass or a rigid mass distribution. Note that the origin of the angular-velocity vector is always fixed at in the space (typically located at the center of mass). Therefore, the moment of inertia tensor is defined relative to that origin.
The moment of inertia tensor can similarly be used to compute the mass moment of inertia for any normalized angular velocity vector as
Since rotational energy is defined as (see Eq.(B.7)), multiplying Eq.(B.22) by gives the following expression for the rotational kinetic energy in terms of the moment of inertia tensor:
We can show Eq.(B.22) starting from Eq.(B.14). For a point-mass located at , we have
where again denotes the three-by-three identity matrix, and
which agrees with Eq.(B.20). Thus we have derived the moment of inertia in terms of the moment of inertia tensor and the normalized angular velocity for a point-mass at .
For a collection of masses located at , we simply sum over their masses to add up the moments of inertia:
Now let the mass be located at so that
For , the result is
A principal axis of rotation (or principal direction) is an eigenvector of the mass moment of inertia tensor (introduced in the previous section) defined relative to some point (typically the center of mass). The corresponding eigenvalues are called the principal moments of inertia. Because the moment of inertia tensor is defined relative to the point in the space, the principal axes all pass through that point (usually the center of mass).
since is unit length, and projecting it onto any other vector can only shorten it or leave it unchanged. That is, , with equality occurring for for any nonzero . Zooming out, of course we expect any moment of inertia for a positive mass to be nonnegative. Thus, is symmetric nonnegative definite. If furthermore and are not collinear, i.e., if there is any nonzero angle between them, then is positive definite (and ). As is well known in linear algebra , real, symmetric, positive-definite matrices have orthogonal eigenvectors and real, positive eigenvalues. In this context, the orthogonal eigenvectors are called the principal axes of rotation. Each corresponding eigenvalue is the moment of inertia about that principal axis--the corresponding principal moment of inertia. When angular velocity vectors are expressed as a linear combination of the principal axes, there are no cross-terms in the moment of inertia tensor--no so-called products of inertia.
The three principal axes are unique when the eigenvalues of (principal moments of inertia) are distinct. They are not unique when there are repeated eigenvalues, as in the example above of a disk rotating about any of its diameters (§B.4.4). In that example, one principal axis, the one corresponding to eigenvalue , was (i.e., orthogonal to the disk and passing through its center), while any two orthogonal diameters in the plane of the disk may be chosen as the other two principal axes (corresponding to the repeated eigenvalue ).
Symmetry of the rigid body about any axis (passing through the origin) means that is a principal direction. Such a symmetric body may be constructed, for example, as a solid of revolution.B.26In rotational dynamics, this case is known as the symmetric top . Note that the center of mass will lie somewhere along an axis of symmetry. The other two principal axes can be arbitrarily chosen as a mutually orthogonal pair in the (circular) plane orthogonal to the axis, intersecting at the axis. Because of the circular symmetry about , the two principal moments of inertia in that plane are equal. Thus the moment of inertia tensor can be diagonalized to look like
Rotational Kinetic Energy Revisited
If a point-mass is located at and is rotating about an axis-of-rotation with angular velocity , then the distance from the rotation axis to the mass is , or, in terms of the vector cross product, . The tangential velocity of the mass is then , so that the kinetic energy can be expressed as (cf. Eq.(B.23))
In a collection of masses having velocities , we of course sum the individual kinetic energies to get the total kinetic energy.
When twisting things, the rotational force we apply about the center is called a torque (or moment, or moment of force). Informally, we think of the torque as the tangential applied force times the moment arm (length of the lever arm)
as depicted in Fig.B.7. The moment arm is the distance from the applied force to the point being twisted. For example, in the case of a wrench turning a bolt, is the force applied at the end of the wrench by one's hand, orthogonal to the wrench, while the moment arm is the length of the wrench. Doubling the length of the wrench doubles the torque. This is an example of leverage. When is increased, a given twisting angle is spread out over a larger arc length , thereby reducing the tangential force required to assert a given torque .
For more general applied forces , we may compute the tangential component by projecting onto the tangent direction. More precisely, the torque about the origin applied at a point may be defined by
where is the applied force (at ) and denotes the cross product, introduced above in §B.4.12.
Note that the torque vector is orthogonal to both the lever arm and the tangential-force direction. It thus points in the direction of the angular velocity vector (along the axis of rotation).
The torque magnitude is
Newton's Second Law for Rotations
The rotational version of Newton's law is
where denotes the angular acceleration. As in the previous section, is torque (tangential force times a moment arm ), and is the mass moment of inertia. Thus, the net applied torque equals the time derivative of angular momentum , just as force equals the time-derivative of linear momentum :
To show that Eq.(B.28) results from Newton's second law , consider again a mass rotating at a distance from an axis of rotation, as in §B.4.3 above, and let denote a tangential force on the mass, and the corresponding tangential acceleration. Then we have, by Newton's second law,
In summary, force equals the time-derivative of linear momentum, and torque equals the time-derivative of angular momentum. By Newton's laws, the time-derivative of linear momentum is mass times acceleration, and the time-derivative of angular momentum is the mass moment of inertia times angular acceleration:
As discussed above, it is useful to decompose the motion of a rigid body into
- the linear velocity of its center of mass, and
- its angular velocity about its center of mass.
The linear motion is governed by Newton's second law , where is the total mass, is the velocity of the center-of-mass, and is the sum of all external forces on the rigid body. (Equivalently, is the sum of the radial force components pointing toward or away from the center of mass.) Since this is so straightforward, essentially no harder than dealing with a point mass, we will not consider it further.
The angular motion is governed the rotational version of Newton's second law introduced in §B.4.19:
where is the vector torque defined in Eq.(B.27), is the angular momentum, is the mass moment of inertia tensor, and is the angular velocity of the rigid body about its center of mass. Note that if the center of mass is moving, we are in a moving coordinate system moving with the center of mass (see next section). We may call the intrinsic momentum of the rigid body, i.e., that in a coordinate system moving with the center of the mass. We will translate this to the non-moving coordinate system in §B.4.20 below.
The driving torque is given by the resultant moment of the external forces, using Eq.(B.27) for each external force to obtain its contribution to the total moment. In other words, the external moments (tangential forces times moment arms) sum up for the net torque just like the radial force components summed to produce the net driving force on the center of mass.
Rotation is always about some (instantaneous) axis of rotation that is free to change over time. It is convenient to express rotations in a coordinate system having its origin ( ) located at the center-of-mass of the rigid body (§B.4.1), and its coordinate axes aligned along the principal directions for the body (§B.4.16). This body-fixed frame then moves within a stationary space-fixed frame (or ``star frame'').
In Eq.(B.29) above, we wrote down Newton's second law for angular motion in the body-fixed frame, i.e., the coordinate system having its origin at the center of mass. Furthermore, it is simplest ( is diagonal) when its axes lie along principal directions (§B.4.16).
As an example of a local body-fixed coordinate system, consider a spinning top. In the body-fixed frame, the ``vertical'' axis coincides with the top's axis of rotation (spin). As the top loses rotational kinetic energy due to friction, the top's rotation-axis precesses around a circle, as observed in the space-fixed frame. The other two body-fixed axes can be chosen as any two mutually orthogonal axes intersecting each other (and the spin axis) at the center of mass, and lying in the plane orthogonal to the spin axis. The space-fixed frame is of course that of the outside observer's inertial frameB.28in which the top is spinning.
Similarly, the total external forces on the center of mass become
Substituting this result into Eq.(B.30), we obtain the following equations of angular motion for an object rotating in the body-fixed frame defined by its three principal axes of rotation:
For a uniform sphere, the cross-terms disappear and the moments of inertia are all the same, leaving , for . Since any three orthogonal vectors can serve as eigenvectors of the moment of inertia tensor, we have that, for a uniform sphere, any three orthogonal axes can be chosen as principal axes.
For a cylinder that is not spinning about its axis, we similarly obtain two uncoupled equations , for , given (no spin). Note, however, that if we replace the circular cross-section of the cylinder by an ellipse, then and there is a coupling term that drives (unless happens to cancel it).
Properties of Elastic Solids