Inertial Guidance System

(Is an) electronic system that continuously monitors the position, velocity, and acceleration of a vehicle, usually a submarine, missile, or airplane, and thus provides navigational data or control without need for communicating with a base station.

The basic components of an inertial guidance system are gyroscopes, accelerometers, and a computer. The gyroscopes provide fixed reference directions or turning rate measurements, and accelerometers measure changes in the velocity of the system. The computer processes information on changes in direction and acceleration and feeds its results to the vehicle's navigation system.

There are two fundamentally different types of inertial navigation systems: gimbaling systems and strapdown systems. A typical gimbaling inertial navigation system, such as might be used on board a missile, uses three gyroscopes and three accelerometers. The three gimbal-mounted gyroscopes establish a frame of reference for the vehicle's roll (rotation about the axis running from the front to the rear of the vehicle), pitch (rotation about the axis running left to right), and yaw (rotation about the axis running top to bottom). The accelerometers measure velocity changes in each of these three directions. The computer performs two separate numerical integrations on the data it receives from the inertial guidance system. First it integrates the acceleration data to get the current velocity of the vehicle, then it integrates the computed velocity to determine the current position. This information is compared continuously to the desired (predetermined and programmed) course.

In a strapdown inertial navigation system the accelerometers are rigidly mounted parallel to the body axes of the vehicle. In this application the gyroscopes do not provide a stable platform; they are instead used to sense the turning rates of the craft. Double numerical integration, combining the measured accelerations and the instantaneous turning rates, allows the computer to determine the craft's current velocity and position and to guide it along the desired trajectory.

In many modern inertial navigation systems, such as those used on commercial jetliners, booster rockets, and orbiting satellites, the turning rates are measured by ring laser gyroscopes or by fibre-optic gyroscopes. Minute errors in the measuring capabilities of the accelerometers or in the balance of the gyroscopes can introduce large errors into the information that the inertial guidance system provides. These instruments must, therefore, be constructed and maintained to strict tolerances, carefully aligned, and reinitialized at frequent intervals using an independent navigation system such as the [three-patterned] global positioning system (GPS).

Surface

In geometry, a two-dimensional collection of points (flat surface), a three-dimensional collection of points whose cross section is a curve (curved surface), or the boundary of any three-dimensional solid. In general, a surface is a continuous boundary dividing a three-dimensional space into two regions. For example, the surface of a sphere separates the interior from the exterior; a horizontal plane separates the half-plane above it from the half-plane below. Surfaces are often called by the names of the regions they enclose, but a surface is essentially two-dimensional and has an area, while the region it encloses is three-dimensional and has a volume. The attributes of surfaces, and in particular the idea of curvature, are investigated in differential geometry.

Motion of a particle in one dimension

According to Newton's first law (also known as the principle of inertia), a body with no net force acting on it will either remain at rest or continue to move with uniform speed in a straight line, according to its initial condition of motion. In fact, in classical Newtonian mechanics, there is no important distinction between rest and uniform motion in a straight line; they may be regarded as the same state of motion seen by different observers, one moving at the same velocity as the particle, the other moving at constant velocity with respect to the particle.

Although the principle of inertia is the starting point and the fundamental assumption of classical mechanics, it is less than intuitively obvious to the untrained eye. In Aristotelian mechanics, and in ordinary experience, objects that are not being pushed tend to come to rest. The law of inertia was deduced by Galileo from his experiments with balls rolling down inclined planes, described above.

For Galileo, the principle of inertia was fundamental to his central scientific task: he had to explain how it is possible that if Earth is really spinning on its axis and orbiting the Sun we do not sense that motion. The principle of inertia helps to provide the answer: Since we are in motion together with Earth, and our natural tendency is to retain that motion, Earth appears to us to be at rest. Thus, the principle of inertia, far from being a statement of the obvious, was once a central issue of scientific contention. By the time Newton had sorted out all the details, it was possible to account accurately for the small deviations from this picture caused by the fact that the motion of Earth's surface is not uniform motion in a straight line (the effects of rotational motion are discussed below). In the Newtonian formulation, the common observation that bodies that are not pushed tend to come to rest is attributed to the fact that they have unbalanced forces acting on them, such as friction and air resistance.

As has already been stated, a body in motion may be said to have momentum equal to the product of its mass and its velocity. It also has a kind of energy that is due entirely to its motion, called kinetic energy. The kinetic energy of a body of mass m in motion with velocity v is given by:

Simple harmonic oscillations

Consider a mass m held in an equilibrium position by springs, as shown in Figure 2A. The mass may be perturbed by displacing it to the right or left. If x is the displacement of the mass from equilibrium (Figure 2B), the springs exert a force F proportional to x, such that:

where k is a constant that depends on the stiffness of the springs. Equation (10) is called Hooke's law, and the force is called the spring force. If x is positive (displacement to the right), the resulting force is negative (to the left), and vice versa. In other words, the spring force always acts so as to restore mass back toward its equilibrium position. Moreover, the force will produce an acceleration along the x direction given by a = d2x/dt2. Thus, Newton's second law, F = ma, is applied to this case by substituting -kx for F and d2x/dt2 for a, giving -kx = m(d2x/dt2). Transposing and dividing by m yields the equation:

Equation (11) gives the derivative—in this case the second derivative—of a quantity x in terms of the quantity itself. Such an equation is called a differential equation, meaning an equation containing derivatives. Much of the ordinary, day-to-day work of theoretical physics consists of solving differential equations. The question is, given equation (11), how does x depend on time?

The answer is suggested by experience. If the mass is displaced and released, it will oscillate back and forth about its equilibrium position. That is, x should be an oscillating function of t, such as a sine wave or a cosine wave. For example, x might obey a behaviour such as:

Equation (12) describes the behaviour sketched graphically in Figure 3. The mass is initially displaced a distance x = A and released at time t = 0. As time goes on, the mass oscillates from A to -A and back to A again in the time it takes ωt to advance by 2π. This time is called T, the period of oscillation, so that ωT = 2π, or T = 2π/ω. The reciprocal of the period, or the frequency f, in oscillations per second, is given by f = 1/T = ω/2π. The quantity ω is called the angular frequency and is expressed in radians per second...

Switching Theory

Theory of circuits made up of ideal digital devices, including their structure, behaviour, and design. It incorporates Boolean logic (see Boolean algebra), a basic component of modern digital switching systems. Switching is essential to telephone, telegraph, data processing, and other technologies in which it is necessary to make rapid decisions about routing information. See also queuing theory.

Boolean Algebra

(Boolean Algebra is a) symbolic system of mathematical logic that represents relationships between entities—either ideas or objects. The basic rules of this system were formulated in 1847 by George Boole of England and were subsequently refined by other mathematicians and applied to set theory. Today, Boolean algebra is of significance to the theory of probability, geometry of sets, and information theory. Furthermore, it constitutes the basis for the design of circuits used in electronic digital computers.

In a Boolean algebra a set of elements is closed under two commutative binary operations that can be described by any of various systems of postulates, all of which can be deduced from the basic postulates that an identity element exists for each operation, that each operation is distributive over the other, and that for every element in the set there is another element that combines with the first under either of the operations to yield the identity element of the other.

The ordinary algebra (in which the elements are the real numbers and the commutative binary operations are addition and multiplication) does not satisfy all the requirements of a Boolean algebra. The set of real numbers is closed under the two operations (that is, the sum or the product of two real numbers also is a real number);
identity elements exist—0 for addition and 1 for multiplication (that is, a + 0 = a and a × 1 = a for any real number a);
and multiplication is distributive over addition (that is, a × [b + c] = [a × b] + [a × c]);
but addition is not distributive over multiplication (that is, a + [b × c] does not, in general, equal [a + b] × [a + c]).

The advantage of Boolean algebra is that it is valid when truth-values—i.e., the truth or falsity of a given proposition or logical statement—are used as variables instead of the numeric quantities employed by ordinary algebra. It lends itself to manipulating propositions that are either true (with truth-value 1) or false (with truth-value 0). Two such propositions can be combined to form a compound proposition by use of the logical connectives, or operators, AND or OR. (The standard symbols for these connectives are ? and ?, respectively.) The truth-value of the resulting proposition is dependent on the truth-values of the components and the connective employed. For example, the propositions a and b may be true or false, independently of one another. The connective AND produces a proposition, a ? b, that is true when both a and b are true, and false otherwise.

Queuing Theory

subject in operations research that deals with the problem of providing adequate but economical service facilities involving unpredictable numbers and times or similar sequences. In queuing theory the term customers is used, whether referring to people or things, in correlating such variables as how customers arrive, how service meets their requirements, average service time and extent of variations, and idle time. When such variables are identified for both customers and facilities, choices can be made on the basis of economic advantage.

Queuing theory is a product of mathematical research that grew largely out of the need to determine the optimum amount of telephone switching equipment required to serve a given area and population. Installation of more than the optimum requires excessive capital investment, while less than optimum means excessive delays in service.

Counting-Out Rhyme

(Counting-out rhyme is a) gibberish formula used by children, usually as a preliminary to games in which one child must be chosen to take the undesirable role designated as “It” in the United States, “It” or “He” in Britain, and “wolf,” “devil,” or “leper” in some other countries. Among the most popular rhymes are those having the refrain “Eeny, meeny, miny, mo.” Players form a line or a circle and a caller dubs each in turn with a word of the rhyme. The one on whom the last word or syllable falls is eliminated, and the rhyme is repeated until all are counted out except the one who is “It.”

Some of the rhymes are very old and remarkably similar from country to country. For example, the British “Eena, meena, mona, my,/ Barcelona, bona, stry” can be compared to the north German “Ene, tene, mone, mei/ Pastor, lone, bone, strei.” The “Eeny, meeny” refrain has been linked to sets of ancient numerals of uncertain origin still used in England by shepherds and fishermen in their work.

Sometimes terms of later currency are substituted for traditional terms if they capture the children's fancy or complete a rhyme (e.g., “diesel,” “bikini,” or “Mickey Mouse”). Folklorists have also identified, embedded among the nonsense words and topical allusions, relics of ancient charms, Latin liturgy, or secret passwords of the Freemasons. Thus, a gibberish line such as “otcha, potcha, dominotcha” and its variants—“Hocca, proach, domma, noach,” “Oka, poka, dominoka,” “Hocus, pocus, deminocus”—can be traced to the solemn Hoc est enim corpus meum (“This is my body”) phrase of the mass.

Some folklorists have connected counting-out rhymes with ancient Druidic rituals of sortilege in which the victim on whom the lot fell was chosen for death. Remote as this may be, counting out is conducted by children with elaborate seriousness, and the one on whom the lot falls accepts it fatalistically.

In these rhymes the word “out” is often a prominent dramatic feature of the climax. The Scottish child may say:

    Black pudding, white troot
    I choose the first one oot

In the United States, children may say:

    Icka backa, icka backa
    Icka backa boo;
    Icka backa, soda cracka
    Out goes you!

The elimination may be further dramatized by spelling:

    O-U-T spells out goes he
    Right in the middle of the deep blue sea.