This appendix derives some basic results from the field of physics, particularly mechanics and acoustics, which are referenced elsewhere in this book, or which are known to be needed in practical synthesis models.
Newton's Laws of Motion
Perhaps the most heavily used equation in physics is Newton's second law of motion:
In this formulation, the applied force is considered positive in the direction of positive mass-position . The force and acceleration are, in general, vectors in three-dimensional space . In other words, force and acceleration are generally vector-valued functions of time . The mass is a scalar quantity, and can be considered a measure of the inertia of the physical system (see §B.1.1 below).
Newton's three laws of motion may be stated as follows:
- Every object in a state of uniform motion will remain in that
state of motion unless an external force acts on it.
- Force equals mass times acceleration [
- For every action there is an equal and opposite reaction.
The first law, also called the law of inertia, was pioneered by Galileo. This was quite a conceptual leap because it was not possible in Galileo's time to observe a moving object without at least some frictional forces dragging against the motion. In fact, for over a thousand years before Galileo, educated individuals believed Aristotle's formulation that, wherever there is motion, there is an external force producing that motion.
The second law, , actually implies the first law, since when (no applied force), the acceleration is zero, implying a constant velocity . (The velocity is simply the integral with respect to time of .)
Newton's third law implies conservation of momentum . It can also be seen as following from the second law: When one object ``pushes'' a second object at some (massless) point of contact using an applied force, there must be an equal and opposite force from the second object that cancels the applied force. Otherwise, there would be a nonzero net force on a massless point which, by the second law, would accelerate the point of contact by an infinite amount.
Mass is an intrinsic property of matter. From Newton's second law, , we have that the amount of force required to accelerate an object, by a given amount, is proportional to its mass. Thus, the mass of an object quantifies its inertia--its resistance to a change in velocity.
We can measure the mass of an object by measuring the gravitational force between it and another known mass, as described in the next section. This is a special case of measuring its acceleration in response to a known force. Whatever the force , the mass is given by divided by the resulting acceleration , again by Newton's second law .
The usual mathematical model for an ideal mass is a dimensionless point at some location in space. While no real objects are dimensionless, they can often be treated mathematically as dimensionless points located at their center of mass, or centroid (§B.4.1).
The physical state of a mass at time consists of its position and velocity in 3D space. The amount of mass itself, , is regarded as a fixed parameter that does not change. In other words, the state of a physical system typically changes over time, while any parameters of the system, such as mass , remain fixed over time (unless otherwise specified).
where is the distance between the centroids of the masses and at time , and is the gravitation constant.B.2
The law of gravitation Eq.(B.2) can be accepted as an experimental fact which defines the concept of a force.B.3 The giant conceptual leap taken by Newton was that the law of gravitation is universal--applying to celestial bodies as well as objects on earth. When a mass is ``dropped'' and allowed to ``fall'' in a gravitational field, it is observed to experience a uniform acceleration proportional to its mass. Newton's second law of motion (§B.1) quantifies this result.
where is the displacement of the spring from its natural length. We call the spring constant, or stiffness of the spring. In terms of our previous notation, we have
Note that the force on the spring in Fig.B.1 is gravitational force. Equal and opposite to the force of gravity is the spring force exerted upward by the spring on the mass (which is not moving). We know that the spring force is equal and opposite to the gravitational force because the mass would otherwise be accelerated by the net force.B.4 Therefore, like gravity, a displaced spring can be regarded as a definition of an applied force. That is, whenever you have to think of an applied force, you can always consider it as being delivered by the end of some ideal spring attached to some external physical system.
Note, by the way, that normal interaction forces when objects touch arise from the Coulomb force (electrostatic force, or repulsion of like charges) between electron orbitals. This electrostatic force obeys an ``inverse square law'' like gravity, and therefore also behaves like an ideal spring for small displacements.B.5
The specific value of depends on the physical units adopted as well as the ``stiffness'' of the spring. What is most important in this definition of force is that a doubling of spring displacement doubles the force. That is, the spring force is a linear function of spring displacement (compression or stretching).
As a simple example, consider a mass driven along a frictionless surface by an ideal spring , as shown in Fig.B.2. Assume that the mass position corresponds to the spring at rest, i.e., not stretched or compressed. The force necessary to compress the spring by a distance is given by Hooke's law (§B.1.3):
where we have defined as the initial displacement of the mass along . This is a differential equation whose solution gives the equation of motion of the mass-spring junction for all time:B.7
where denotes the frequency of oscillation in radians per second. More generally, the complete space of solutions to Eq.(B.4), corresponding to all possible initial displacements and initial velocities , is the set of all sinusoidal oscillations at frequency :
Work = Force times Distance = Energy
Work is defined as force times distance. Work is a measure of the energy expended in applying a force to move an object.B.8
Work can also be negative. For example, when uncompressing an ideal spring, the (positive) work done by the spring on its moving end support can be interpreted also as saying that the end support performs negative work on the spring as it allows the spring to uncompress. When negative work is performed, the driving system is always accepting energy from the driven system. This is all simply accounting. Physically, one normally considers the driver as the agent performing the positive work, i.e., the one expending energy to move the driven object. Thus, when allowing a spring to uncompress, we consider the spring as performing (positive) work on whatever is attached to its moving end.
During a sinusoidal mass-spring oscillation, as derived in §B.1.4, each period of the oscillation can be divided into equal sections during which either the mass performs work on the spring, or vice versa.
Gravity, spring forces, and electrostatic forces are examples of conservative forces. Conservative forces have the property that the work required to move an object from point to point , either with or against the force, depends only on the locations of points and in space, not on the path taken from to .
Potential Energy in a Spring
When compressing an ideal spring, work is performed, and this work is stored in the spring in the form of what we call potential energy. Equation (B.6) above gives the quantitative formula for the potential energy stored in an ideal spring after it has been compressed meters from rest.
Kinetic Energy of a Mass
Kinetic energy is energy associated with motion. For example, when a spring uncompresses and accelerates a mass, as in the configuration of Fig.B.2, work is performed on the mass by the spring, and we say that the potential energy of the spring is converted to kinetic energy of the mass.
Suppose in Fig.B.2 we have an initial spring compression by meters at time , and the mass velocity is zero at . Then from the equation of motion Eq.(B.5), we can calculate when the spring returns to rest (). This first happens at the first zero of , which is time . At this time, the velocity, given by the time-derivative of Eq.(B.5),
Mass Kinetic Energy from Virtual Work
From Newton's second law, (introduced in Eq.(B.1)), we can use d'Alembert's idea of virtual work to derive the formula for the kinetic energy of a mass given its speed . Let denote a small (infinitesimal) displacement of the mass in the direction. Then we have, using the calculus of differentials,
Thus, by Newton's second law, a differential of work applied to a mass by force through distance boosts the kinetic energy of the mass by . The kinetic energy of a mass moving at speed is then given by the integral of all such differential boosts from 0 to :
The quantity is classically called the virtual work associated with force , and a virtual displacement .
Energy in the Mass-Spring Oscillator
Summarizing the previous sections, we say that a compressed spring holds a potential energy equal to the work required to compress the spring from rest to its current displacement. If a compressed spring is allowed to expand by pushing a mass, as in the system of Fig.B.2, the potential energy in the spring is converted to kinetic energy in the moving mass.
We can draw some inferences from the oscillatory motion of the mass-spring system written in Eq.(B.5):
- From a global point of view, we see that energy is conserved, since the oscillation never decays.
- At the peaks of the displacement (when is either or ), all energy is in the form of potential energy, i.e., the spring is either maximally compressed or stretched, and the mass is momentarily stopped as it is changing direction.
- At the zero-crossings of , the spring is momentarily relaxed, thereby holding no potential energy; at these instants, all energy is in the form of kinetic energy, stored in the motion of the mass.
- Since total energy is conserved (§B.2.5), the kinetic
energy of the mass at the displacement zero-crossings is exactly the
amount needed to stretch the spring to displacement (or compress
it to ) before the mass stops and changes direction. At all
times, the total energy is equal to the sum of the potential
energy stored in the spring, and the kinetic energy
stored in the mass:
It is a remarkable property of our universe that energy is conserved under all circumstances. There are no known exceptions to the conservation of energy, even when relativistic and quantum effects are considered.B.9
Energy Conservation in the Mass-Spring System
The momentum of a mass is usually defined by
Like energy, momentum is conserved in physical systems. For example, when two masses collide and recoil from each other, the total momentum after the collision equals that before the collision. Since the momentum is a three-dimensional vector in Euclidean space, momentum conservation provides three simultaneous equations, in general.B.10
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).
Young's modulus can be thought of as the spring constant for solids. Consider an ideal rod (or bar) of length and cross-sectional area . Suppose we apply a force to the face of area , causing a displacement along the axis of the rod. Then Young's modulus is given by
For wood, Young's modulus is on the order of N/m. For aluminum, it is around (a bit higher than glass which is near ), and structural steel has .
Recall (§B.1.3) that Hooke's Law defines a spring constant as the applied force divided by the spring displacement , or . An elastic solid can be viewed as a bundle of ideal springs. Consider, for example, an ideal bar (a rectangular solid in which one dimension, usually its longest, is designated its length ), and consider compression by along the length dimension. The length of each spring in the bundle is the length of the bar, so that each spring constant must be inversely proportional to ; in particular, each doubling of length doubles the length of each ``spring'' in the bundle, and therefore halves its stiffness. As a result, it is useful to normalize displacement by length and use relative displacement . We need displacement per unit length because we have a constant spring compliance per unit length.
The number of springs in parallel is proportional to the cross-sectional area of the bar. Therefore, the force applied to each spring is proportional to the total applied force divided by the cross-sectional area . Thus, Hooke's law for each spring in the bundle can be written
We may say that Young's modulus is the Hooke's-law spring constant for the spring made from a specifically cut section of the solid material, cut to length 1 and cross-sectional area 1. The shape of the cross-sectional area does not matter since all displacement is assumed to be longitudinal in this model.
The tension of a vibrating string is the force used to stretch it. It is therefore directed along the axis of the string. A force must be applied at the endpoint on the right, and a force is applied at the endpoint on the left. Each point interior to the string is pulled equally to the left and right, i.e., the net force on an interior point is . (A nonzero force on a massless point would produce an infinite acceleration.)
If the cross-sectional area of the string is , then the tension is given by the stress on the string times .
Wave Equation for the Vibrating String
Consider an elastic string under tension which is at rest along the dimension. Let , , and denote the unit vectors in the , , and directions, respectively. When a wave is present, a point originally at along the string is displaced to some point specified by the displacement vector
The displacement of a neighboring point originally at along the string can be specified as
Let denote string tension along when the string is at rest, and denote the vector tension at the point in the present displaced scenario under analysis. The net vector force acting on the infinitesimal string element between points and is given by the vector sum of the force at and the force at , that is, . If the string has stiffness, the two forces will in general not be tangent to the string at these points. The mass of the infinitesimal string element is , where denotes the mass per unit length of the string at rest. Applying Newton's second law gives
where has been canceled on both sides of the equation. Note that no approximations have been made so far.
The next step is to express the force
in terms of the tension
of the string at rest, the elastic constant of the string, and
geometrical factors. The displaced string element
Let's now assume the string is perfectly flexible (zero stiffness) so that the direction of the force vector is given by the unit vector tangent to the string. (To accommodate stiffness, it would be necessary to include a force component at right angles to the string which depends on the curvature and stiffness of the string.) The magnitude of at any position is the rest tension plus the incremental tension needed to stretch it the fractional amount
where no geometrical limitations have yet been placed on the magnitude of and , other than to prevent the string from being stretched beyond its elastic limit.
The four equations (B.31) through (B.35) can be combined into a single vector wave equation that expresses the propagation of waves on the string having three displacement components. This differential equation is nonlinear, so that superposition no longer holds. Furthermore, the three displacement components of the wave are coupled together at all points along the string, so that the wave equation is no longer separable into three independent 1D wave equations.
To obtain a linear, separable wave equation, it is necessary to assume that the strains , , and be small compared with unity. This is the same assumption ( ) necessary to derive the usual wave equation for transverse vibrations only in the - plane.
This is the linearized wave equation for the string, based only on the assumptions of elasticity of the string, and strain magnitudes much less than unity. Using this linearized equation for the force , it is found that (B.31) separates into the three wave equations
where is the longitudinal wave velocity, and is the transverse wave velocity.
In summary, the two transverse wave components and the longitudinal component may be considered independent (i.e., ``superposition'' holds with respect to vibrations in these three dimensions of vibration) provided powers higher than 1 of the strains (relative displacement) can be neglected, i.e.,
where and denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced, e.g.,
(See also Eq.(C.1).) We see that the term in the first equation above provides a mechanism for transverse waves to ``drive'' the generation of longitudinal waves. This coupling cannot be neglected if momentum effects are desired.
Physically, the rising edge of a transverse wave generates a longitudinal displacement in the direction of wave travel that propagates ahead at a much higher speed (typically an order of magnitude faster). The falling edge of the transverse wave then cancels this forward displacement as it passes by. See  for further details (including computer simulations).
Particle Velocity of a Gas
The particle velocity of a gas flow at any point can be defined as the average velocity (in meters per second, m/s) of the air molecules passing through a plane cutting orthogonal to the flow. The term ``velocity'' in this book, when referring to air, means ``particle velocity.''
It is common in acoustics to denote particle velocity by lower-case .
Volume Velocity of a Gas
When a flow is confined within an enclosed channel, as it is in an acoustic tube, volume velocity is conserved when the tube changes cross-sectional area, assuming the density remains constant. This follows directly from conservation of mass in a flow: The total mass passing a given point along the flow is given by the mass density times the integral of the volume volume velocity at that point, or
As a simple example, consider a constant flow through two cylindrical acoustic tube sections having cross-sectional areas and , respectively. If the particle velocity in cylinder 1 is , then the particle velocity in cylinder 2 may be found by solving
It is common in the field of acoustics to denote volume velocity by an upper-case . Thus, for the two-cylinder acoustic tube example above, we would define and , so that
Pressure is Confined Kinetic Energy
According the kinetic theory of ideal gases , air pressure can be defined as the average momentum transfer per unit area per unit time due to molecular collisions between a confined gas and its boundary. Using Newton's second law, this pressure can be shown to be given by one third of the average kinetic energy of molecules in the gas.
Proof: This is a classical result from the kinetic theory of gases . Let be the total mass of a gas confined to a rectangular volume , where is the area of one side and the distance to the opposite side. Let denote the average molecule velocity in the direction. Then the total net molecular momentum in the direction is given by . Suppose the momentum is directed against a face of area . A rigid-wall elastic collision by a mass traveling into the wall at velocity imparts a momentum of magnitude to the wall (because the momentum of the mass is changed from to , and momentum is conserved). The average momentum-transfer per unit area is therefore at any instant in time. To obtain the definition of pressure, we need only multiply by the average collision rate, which is given by . That is, the average -velocity divided by the round-trip distance along the dimension gives the collision rate at either wall bounding the dimension. Thus, we obtain
In an ideal inviscid, incompressible flow, we have, by conservation of energy,
This basic energy conservation law was published in 1738 by Daniel Bernoulli in his classic work Hydrodynamica.
From §B.7.3, we have that the pressure of a gas is proportional to the average kinetic energy of the molecules making up the gas. Therefore, when a gas flows at a constant height , some of its ``pressure kinetic energy'' must be given to the kinetic energy of the flow as a whole. If the mean height of the flow changes, then kinetic energy trades with potential energy as well.
The Bernoulli effect provides that, when a gas such as air flows, its pressure drops. This is the basis for how aircraft wings work: The cross-sectional shape of the wing, called an aerofoil (or airfoil), forces air to follow a longer path over the top of the wing, thereby speeding it up and creating a net upward force called lift.
Figure B.8 illustrates the Bernoulli effect for the case of a reservoir at constant pressure (``mouth pressure'') driving an acoustic tube. Any flow inside the ``mouth'' is neglected. Within the acoustic channel, there is a flow with constant particle velocity . To conserve energy, the pressure within the acoustic channel must drop down to . That is, the flow kinetic energy subtracts from the pressure kinetic energy within the channel.
Referring again to Fig.B.8, the gas flow exiting the acoustic tube is shown as forming a jet. The jet ``carries its own pressure'' until it dissipates in some form, such as any combination of the following:
- heat (now allowing for ``friction'' in the flow),
- vortices (angular momentum),
- radiation (sound waves), or
- pressure recovery.
Acoustic intensity may be defined by
For a plane traveling wave, we have
Therefore, in a plane wave,
The two forms of energy in a wave are kinetic and potential. Denoting them at a particular time and position by and , respectively, we can write them in terms of velocity and wave impedance as follows:
Since the acoustic energy density is the energy per unit volume in a 3D sound field, it follows that the total energy of the field is given by integrating over the volume:
Sabine's theory of acoustic energy decay in reverberant room impulse responses can be derived using this conservation relation as a starting point.
The ideal gas law can be written as
The alternate form comes from the statistical mechanics derivation in which is the number of gas molecules in the volume, and is Boltzmann's constant. In this formulation (the kinetic theory of ideal gases), the average kinetic energy of the gas molecules is given by . Thus, temperature is proportional to average kinetic energy of the gas molecules, where the kinetic energy of a molecule with translational speed is given by .
In an ideal gas, the molecules are like little rubber balls (or rubbery assemblies of rubber balls) in a weightless vacuum, colliding with each other and the walls elastically and losslessly (an ``ideal rubber''). Electromagnetic forces among the molecules are neglected, other than the electron-orbital repulsion producing the elastic collisions; in other words, the molecules are treated as electrically neutral far away. (Gases of ionized molecules are called plasmas.)
We normally do not need to consider the (nonlinear) ideal gas law in audio acoustics because it is usually linearized about some ambient pressure . The physical pressure is then , where is the usual acoustic pressure-wave variable. That is, we are only concerned with small pressure perturbations in typical audio acoustics situations, so that, for example, variations in volume and density can be neglected. Notable exceptions include brass instruments which can achieve nonlinear sound-pressure regions, especially near the mouthpiece [198,52]. Additionally, the aeroacoustics of air jets is nonlinear [196,530,531,532,102,101].
Isothermal versus Isentropic
If air compression/expansion were isothermal (constant temperature ), then, according to the ideal gas law , the pressure would simply be proportional to density . It turns out, however, that heat diffusion is much slower than audio acoustic vibrations. As a result, air compression/expansion is much closer to isentropic (constant entropy ) in normal acoustic situations. (An isentropic process is also called a reversible adiabatic process.) This means that when air is compressed by shrinking its volume , for example, not only does the pressure increase (§B.7.3), but the temperature increases as well (as quantified in the next section). In a constant-entropy compression/expansion, temperature changes are not given time to diffuse away to thermal equilibrium. Instead, they remain largely frozen in place. Compressing air heats it up, and relaxing the compression cools it back down.
In terms of , we have
where for dry air at normal temperatures. Thus, if a volume of ideal gas is changed from to , the pressure change is given by
The value is typical for any diatomic gas.B.31 Monatomic inert gases, on the other hand, such as Helium, Neon, and Argon, have . Carbon dioxide, which is triatomic, has a heat capacity ratio . We see that more complex molecules have lower values because they can store heat in more degrees of freedom.
An ideal monatomic gas molecule (negligible spin) has only three degrees of freedom: its kinetic energy in the three spatial dimensions. Therefore, . This means we expect
For an ideal diatomic gas molecule such as air, which can be pictured as a ``bar bell'' configuration of two rubber balls, two additional degrees of freedom are added, both associated with spinning the molecule about an axis orthogonal to the line connecting the atoms, and piercing its center of mass. There are two such axes. Spinning about the connecting axis is neglected because the moment of inertia is so much smaller in that case. Thus, for diatomic gases such as dry air, we expect
There is also a (weaker) dependence of air absorption on temperature .
Theoretical models of energy loss in a gas are developed in Morse and Ingard [318, pp. 270-285]. Energy loss is caused by viscosity, thermal diffusion, rotational relaxation, vibration relaxation, and boundary losses (losses due to heat conduction and viscosity at a wall or other acoustic boundary). Boundary losses normally dominate by several orders of magnitude, but in resonant modes, which have nodes along the boundaries, interior losses dominate, especially for polyatomic gases such as air.B.34 For air having moderate amounts of water vapor () and/or carbon dioxide (), the loss and dispersion due to and vibration relaxation hysteresis becomes the largest factor [318, p. 300]. The vibration here is that of the molecule itself, accumulated over the course of many collisions with other molecules. In this context, a diatomic molecule may be modeled as two masses connected by an ideal spring. Energy stored in molecular vibration typically dominates over that stored in molecular rotation, for polyatomic gas molecules [318, p. 300]. Thus, vibration relaxation hysteresis is a loss mechanism that converts wave energy into heat.
In a resonant mode, the attenuation per wavelength due to vibration relaxation is greatest when the sinusoidal period (of the resonance) is equal to times the time-constant for vibration-relaxation. The relaxation time-constant for oxygen is on the order of one millisecond. The presence of water vapor (or other impurities) decreases the vibration relaxation time, yielding loss maxima at frequencies above 1000 rad/sec. The energy loss approaches zero as the frequency goes to infinity (wavelength to zero).
Under these conditions, the speed of sound is approximately that of dry air below the maximum-loss frequency, and somewhat higher above. Thus, the humidity level changes the dispersion cross-over frequency of the air in a resonant mode.
Wave Equation in Higher Dimensions
The wave equation in 1D, 2D, or 3D may be written as
where, in 3D, denotes the amplitude of the wave at time and position , and
Plane Waves in Air
Figure B.10 depicts a more mathematical schematic of a sinusoidal plane wave traveling toward the upper-right of the figure. The dotted lines indicate the crests (peak amplitude location) along the wave.
The direction of travel and spatial frequency are indicated by the vector wavenumber , as discussed in in the following section.
where p(t,x) is the pressure at time (seconds) and position (3D Euclidean space). The amplitude , phase , and radian frequency are ordinary sinusoid parameters , and is the vector wavenumber:
- (unit) vector of direction cosines
- (scalar) wavenumber along travel direction
- wavenumber along the travel direction in its magnitude
- travel direction in its orientation
To see that the vector wavenumber has the claimed properties, consider that the orthogonal projection of any vector onto a vector collinear with is given by .B.35Thus, is the component of lying along the direction of wave propagation indicated by . The norm of this component is , since is unit-norm by construction. More generally, is the signed length (in meters) of the component of along . This length times wavenumber gives the spatial phase advance along the wave, or, .
For another point of view, consider the plane wave , which is the varying portion of the general plane-wave of Eq.(B.48) at time , with unit amplitude and zero phase . The spatial phase of this plane wave is given by
Solving the 2D Wave Equation
Since solving the wave equation in 2D has all the essential features of the 3D case, we will look at the 2D case in this section.
The sum of two such waves traveling in opposite directions with the same amplitude and frequency produces a standing wave. For example, if the waves are traveling parallel to the axis, we have
which is a standing wave along .
We often wish to find solutions of the 2D wave equation that obey certain known boundary conditions. An example is transverse waves on an ideal elastic membrane, rigidly clamped on its boundary to form a rectangle with dimensions meters.
Note that we can also use products of horizontal and vertical standing waves
To build solutions to the wave equation that obey all of the boundary conditions, we can form linear combinations of the above standing-wave products having zero displacement (``nodes'') along all four boundary lines:
The mathematics of 3D sound is quite elementary, as we will see below. The hard part of the theory of practical systems typically lies in the mathematical approximation to the ideal case. Examples include Ambisonics  and wave field synthesis .
Consider a point source at position . Then the acoustic complex amplitude at position is given by
The fundamental approximation problem in 3D sound is to approximate the complex acoustic field at one or more listening points using a finite set of loudspeakers, which are often modeled as a point source for each speaker.
Digital Waveguide Theory
A History of Enabling Ideas Leading to Virtual Musical Instruments