Oscillating Systems

Type of physical science: Classical physics

Field of study: Mechanics

A system (from a simple pendulum to electronic circuitry), the amplitude of which changes in a repetitive way with time, is said to be oscillating or in a state of oscillatory motion. This motion not only explains familiar mechanical and electrical phenomena, such as clocks, tuning forks, and electrical circuits, but also is at the basis of many quantum field theories.

Overview

Oscillatory motion is the repetitive change of the amplitude of a mechanical or electrical system with time. In its simplest form, the motion repeats itself at regular time intervals, maintaining a constant frequency of motion. Any such motion is called "periodic motion" or "harmonic motion"; any motion that repeats itself over the same path is called "vibratory" or "oscillatory."

Oscillatory motion is characterized by a number of attributes, including its "period," or the amount of time required to complete one cycle of motion; its "amplitude," which is the maximum distance, or displacement, from the rest point that the mass (or other variable, such as a current) achieves; and its "frequency," or the number of complete oscillations or cycles that occur within a given period of time. Most media, whether a mass in a mechanical system or a current in an electrical system, possess a "natural" frequency, a characteristic frequency at which the medium will oscillate as a result of the properties of its component parts.

Simple oscillatory motion can be understood most easily when it occurs in a simple "linear system." In mechanics, a mass driven by a force (such as gravity) that moves it toward its point of rest (known also as its position of equilibrium or attracting point), with a magnitude proportional to the displacement, is referred to as a "linear" system, and the force is known as a linear force. There are many examples in nature of linear systems. One such mechanical system is the pendulum. In this situation, a displacement raises the mass (the bob) against gravity; when the bob is released, it falls, but it is constrained by its shaft or string to move along a circular path toward its original resting point. It overshoots its original point and rises against gravity; when the bob reaches the highest point of its swing, it again returns toward its resting point, and the motion repeats itself. Here, the pendulum's period is the amount of time required to make one complete cycle to and from the resting point; the frequency is the number of cycles (oscillations) it makes in a given length of time (such as one second); and the amplitude is the height (distance from resting point) that it reaches at the highest point of its swing.

The underlying beauty of a linear system is that it can be analyzed mathematically in a relatively simple, straightforward way. In a simple system, using given values in an equation for such properties of the system as the nature of the force, the mass, and the system's characteristic properties, one can determine what the system's natural frequency is. Determining a system's frequency is made only slightly more difficult when one must allow for other forces that may act on the system: More variables are simply added to the equation, bringing it closer to providing the frequencies for real (instead of theoretical) oscillating systems.

An important example of such a force is that which causes damping. Taking the earlier example of the pendulum, damping occurs as the bob continues to oscillate about its original rest position: Unless energy is added to this system, it will eventually slow down. Any such force that reduces the amplitude of an oscillatory motion is called a "damping force." In mechanical systems, the damping force is usually friction. The nature of such frictional forces can be quite complicated, but in many circumstances they can be described as a force that is proportional to the velocity (speed) of the mass in the oscillating system. Motion can be underdamped (the system continues to oscillate, but the amplitude of the oscillation decays at a rate proportional to the damping), "critically" damped (the damping force is proportional to the natural frequency of the system), and "overdamped" (the magnitude of the damping is still greater). Critical damping and overdamping result in the system's coming to rest after one or two cycles. In the case of the pendulum, that might happen if the bob were moving through a thick liquid such as molasses.

The quality factor, or Q, of either a mechanical system or an electrical circuit is a measure of how quickly its motion decays--high Q implying a small amount of damping. Q is not measured in time but in terms of the number of oscillation cycles needed for the motion to decay. Q is proportional to the natural frequency of oscillation divided by the damping. For mechanical systems, Q can be as high as thousands; for electrical systems, Q can be up to tens of thousands; for superconducting electrical cavities, Q can be billions.

Another type of force that will influence a linear system's natural frequency is the restoring force--that force which works to return the mass to its equilibrium point. In some linear systems, the mass influences this restoring force, and thus the system's natural frequency; in other systems, the mass has no effect. In the case of a pendulum, the restoring force is the result of gravity and is proportional to the mass of the pendulum's bob. Both the restoring (gravitational) force and the mass times acceleration term (ma) in Newton's second law (Force = ma) contain the mass as a common factor. Since mass occurs in both terms, in determining the frequency of a system, the mass cancels itself in the equation. With restoring forces at work and in the absence of damping, therefore, the frequency of a system does not depend on the mass. Thus, a pendulum 1 meter long has a frequency of 2 Hertz, no matter how heavy the bob is. This independence of oscillation frequency from the mass of the pendulum was first noticed by Galileo as he idly watched the movement of chandeliers during a church service.

Similarly, most solids, when deformed slightly, have a restoring force (in this case, elasticity) that opposes the deformation and is proportional to it (Hooke's law). This property forms the basis for springs, which are designed to produce a linear restoring force over a large range of stretching or compression. The restoring force is provided by the elasticity of the spring: A displacement of the mass compresses or stretches the spring, and the force then exerted by the spring on the mass attempts to return the mass to its original position. In this case, the natural frequency of the system does depend on the mass; heavier masses or weaker springs produce lower frequencies, and stiffer springs or lighter masses produce higher frequencies. The constant of proportionality between the force and the displacement is called the spring constant. There are many other examples of systems in which the force that results when the mass is displaced a small amount is linear, although larger displacements produce more complicated forces. A pendulum is just such a case: The force is linear for small displacements (as when the bob of a grandfather clock is swinging only a few inches), but not for large displacements, which will bring other forces into play (such as when the bob swings 180 degrees or more from its resting point).

The current in electrical circuits depends on quantities analogous to the mass, the damping force, and the spring constant in mechanical systems. These are, respectively, the inductance (L), the resistance (R), and the reciprocal of the capacitance (C). Thus, one can create exactly the same kind of oscillatory behavior for the current in electrical circuits as for the displacement in mechanical systems. In fact, this analogy is exploited in using analogue computers to mimic the behavior of mechanical systems.

Linear systems can be quite complicated, consisting of a number of interacting linear forces. No matter how complicated these linear systems are, however, their motion can be reduced mathematically to a set of noninteracting linear motions, or "normal modes," where each normal mode can be described as if it were an isolated, one-mass linear system. If a system is started in a normal mode, it will continue to execute that motion; only some sort of outside force can couple one normal mode to another one. Once a system's normal modes have been determined, the system can be treated as a set of uncoupled simple linear oscillators.

A mass-spring system can provide a good example of normal modes. Imagine three identical springs that are separated by two identical masses, all laid out along a straight line. The free ends of the outside springs are then attached to fixed points. One normal mode occurs when the two masses move in tandem in the same direction--that is, in such a way that they do not compress the spring between them. Here, only the two masses and the two outside springs are involved, giving a natural frequency for this mode that is exactly equal to that for a one-mass, one-spring system. In the same system, however, another normal mode occurs where the two masses move exactly opposite to each other, compressing the center spring symmetrically and stretching the two outside springs symmetrically. All three springs are stretched or compressed; that is, the masses are harder to displace. The system acts as if it had a stiffer spring, and thus this mode has a higher natural frequency than the first.

The normal coordinate of a particular mode is equivalent to the displacement in a single oscillator with that natural frequency. In the first normal mode of the system, in which the two masses move in the same direction, the normal coordinate of the mode is the sum of the displacements of the two masses; in the second, in which the masses move in opposite directions, thus stretching and compressing all three springs, the normal coordinate is the difference of the displacement of the two masses. Scientists and engineers are very interested in determining the normal coordinates of different modes, because each mode acts as if it were a simple, independent oscillator.

Applications

Oscillatory motion is the basis for many familiar devices. The fact that the natural frequency of oscillation depends only on the mechanical properties of a linear system has been exploited for centuries in timekeeping instruments. Being able to break time periods up into well-determined, stable, fixed intervals is equivalent to measuring distance by means of a ruler, which has broken space up into well-determined, stable intervals. In spring-driven wristwatches, the fundamental frequency is the oscillation of a spring-driven balance wheel; in spring-driven or weight-driven clocks, the frequency standard is the pendulum (swinging or torsional). Even when electric wristwatches first appeared, the frequency was controlled by a small tuning fork, which acts exactly like a single mass-spring system. In these devices, external energy fed into the system by the spring or weight compensates for the friction (damping force) that would otherwise eventually make the watch or clock stop.

It is important to understand how the oscillatory motion in a linear system will respond to an external force that varies with time. Suppose that a simple damped spring system is shaken (or driven) with another oscillator that has an adjustable frequency. The driven system then oscillates with the shaking frequency. If the damping is small, the amplitude of the driven system can become very large when the shaking frequency approaches the natural frequency of the system. This condition is called "resonance"; that is, the system is "resonant" at a shaking frequency that is approximately equal to its natural frequency. The effect of combining these two near-identical frequencies (that of the driving force and the natural frequency of the system being driven) is to amplify the overall oscillation of the system exponentiallly. The destruction of many of the buildings during the Mexico City earthquake of 1985 is an example of unexpected resonance effects; another example is the collapse of the Tacoma Narrows Bridge during high winds in 1940. Once scientists and engineers understand the natural frequencies at which a system resonates, they can introduce suitable damping mechanisms that will reduce the amplitude of the resonant motion or shift the natural frequency to another value that is less likely to be excited.

On the other hand, sometimes one wants just the opposite: sensitivity to motion or a signal in a narrow frequency range. Remote-control garage door openers and television channel selectors, for example, must be sensitive to the control device's frequency but very insensitive to noise or other signals. The ability to separate radio stations operating at slightly different frequencies requires that the receiver have a high Q for the detection of the station desired.

Simple linear systems are easily dealt with analytically, by solving the equations of motion after finding the normal modes and natural frequencies of the system. For complex systems, however, this is possible only in principle, and unless the system has a high degree of symmetry (many identical subsystems), scientists and engineers must rely on models or numerical methods. The need to test the limits of each component of a complicated system makes the use of computer simulations not only helpful but also necessary: These programs can be used to identify possible weak points in a system (such as a building or an airplane) while it is still on the drawing board. A particularly important modern application of these ideas is to deal with an entire structure as if it were separate pieces that are coupled by springs and that have associated damping/frictional forces. This mathematical approach is called "finite-element analysis" and has allowed the mechanical behavior of complex structures to be estimated before they are built. Such finite-element modeling has become standard in the automobile and aerospace industries and has allowed engineers to anticipate responses of buildings and highway structures to storms and earthquakes.

Similarly, in the response of linear electrical circuits, computer simulation has become a guide to creating prototype ("breadboard") circuits. This tendency is driven by the growing complexity of computer systems and by the resultant need to investigate performance over a large range of circuit parameters. Designers of electronic control systems for aircraft, nuclear reactors, and similar complex devices clearly must anticipate potential difficulties before the system is constructed. Likewise, hybrid electromechanical systems, in which one can use the electrical circuit to shift the resonant frequency, are often studied to provide damping that adapts to changing conditions or to broaden the frequency range possible in the mechanical system alone. Such methods have been used to design stereophonic loudspeakers that will obtain the highest fidelity in reproducing the originally recorded sound--the broadest, smoothest frequency response.

Context

The modern, detailed understanding of mechanical and dynamical phenomena began with Isaac Newton in the eighteenth century. His invention of the calculus and its applications to solving the equations that arise from his three laws of motion form the foundation of all that has followed. Oscillating systems are only one fruit of his pioneering discoveries and inventions.

With the tools he created, it became possible to understand at the most basic level why these systems behave as they do, and thus to exploit their properties more completely. Before Newton's time, scientists had only a catalog of the observed properties of these systems; nothing that has been done with the classical behavior of oscillating systems in the three centuries since Newton's death would be much of a surprise to him.

With the invention of quantum mechanics, however, Newton's work on these systems assumed even greater significance. The underlying mathematical simplicity of complex linear systems has allowed the mathematics of linear systems to form the basis of most classical and quantum field theories. This is possible because even if the force field (for example, the electromagnetic field) is thought of as having a large number (even an infinite number) of pieces, it can be separated into its normal modes. Each mode can then be dealt with as if it had only one component, and the rules of quantum mechanics are then applied to that simple piece. This approach is very similar to that used by physicists to deal with complex classical systems. In making the transition to quantum mechanical systems, the decomposition of a classical system--that is, the ability to break it down into normal modes--is indispensable. This was first done in the case of the vibrational spectrum of excited states of simple molecules--that is, when they act as if they were made of masses separated by springs (the way water molecules or carbon dioxide molecules absorb light, for example). Later, the method became the cornerstone of the shell model of atomic nuclei, which provides physicists with an intuitive understanding of the systematic properties of the nuclei of all elements.

Physicists have also applied the principles of oscillatory motion to an area of the quantum field theory of solids known as quantized normal modes (phonons) of the crystalline lattice of the material. Simply put, they are applying the normal mode idea, familiar from classical mechanical systems, to the atomic structure of a crystalline material in order to reduce the enormous number of possible equations to a manageable number. These equations may then be put in a form suitable for the quantum mechanical description of each normal mode. This field theory of solids has many promising practical applications, for it enables scientists to predict the thermal and mechanical properties of materials. Similarly, the theoretical treatment of superconductivity by John Bardeen, Leon N Cooper, and John Robert Schrieffer (BCS theory) exploits the same trick.

Being able to understand a system as commonplace as a pendulum has thus led physicists not only to a wide range of practical applications in seemingly complex systems but also to powerful techniques that allow a similar understanding of the quantum nature of the microscopic world.

Principal terms

AMPLITUDE: the maximum displacement of an oscillating medium (either a mass in a mechanical system or electric current in an electrical system) from its attracting point (its point of equilibrium or resting position)

DAMPING FORCE: any force that causes a loss of energy (slowing or stopping) in a mechanical or electrical system--generally friction and electrical resistance, respectively

DISPLACEMENT: the distance, at any given time, that an oscillating medium is from its attracting point

FREQUENCY: the number of oscillations occurring during a unit of time; the usual unit of frequency is the Hertz (Hz), and one Hertz is one oscillation per second

LINEAR RESTORING FORCE: a force that drives an oscillating medium back toward its attracting point and is in proportion to its displacement, thus "restoring" the medium to its resting position

NATURAL FREQUENCY: the preferred frequency of oscillation of a given medium, as dictated by its nature and properties

NORMAL MODE: a simple type of motion into which a complex system (a system composed of several joined masses or circuits) may be separated; there may be many normal modes in a complex system, each behaving like an independent oscillating system with its own natural frequency

RESONANCE: the response of a system to being stimulated by a time-dependent force at a frequency near one of the system's natural frequencies, which causes the system to oscillate with a relatively large amplitude

Bibliography

Benade, Arthur. HORNS, STRINGS, AND HARMONY. New York: Doubleday, 1960. This is a superbly written discussion which presents, in simple terms and without recourse to mathematics, the oscillatory systems identified in the title; the reader comes away with an excellent sense of the physical ideas and processes. The author is a physicist who is also a musician.

Feynman, Richard P., R. B. Leighton, and M. Sands. THE FEYNMAN LECTURES ON PHYSICS. Vol. 1. Reading, Mass.: Addison-Wesley, 1963. This remarkable set of lectures on mechanics was originally presented to freshmen at the California Institute of Technology in 1961-1962. It contains many examples of linear systems and where they occur in physical processes. The approach is entirely original and characterized by the author's well-known wit and charm. Excellent diagrams.

Frautschi, S. C., R. P. Olenick, T. M. Apostol, and D. L. Goodstein. THE MECHANICAL UNIVERSE: MECHANICS AND HEAT. New York: Cambridge University Press, 1986. This is the text version of the material presented in Goodstein's television series, THE MECHANICAL UNIVERSE, which was aired by the Public Broadcasting System. A clearly written introduction to the subject for college freshmen, with considerable attention to placing the phenomena and principles in their historical context. Excellent illustrations, references, and diagrams.

French, A. P. NEWTONIAN MECHANICS. New York: W. W. Norton, 1971.

French, A. P. VIBRATION AND WAVES. New York: W. W. Norton, 1971.

French, A. P. AN INTRODUCTION TO QUANTUM MECHANICS. New York: W. W. Norton, 1978. These three volumes form the introductory physics series for the Massachusetts Institute of Technology. The material on linear systems includes examples taken from other areas of physics: atomic physics and planetary motion, for example. These books are exceptionally well written, with clear explanations, appropriate diagrams and examples, and an excellent list of references.

Sutton, O. G. MATHEMATICS IN ACTION. New York: Harper Torchbooks, 1960. An informal introduction to the use of mathematics in physical problems. The chapter entitled "An Essay on Waves" provides an excellent discussion of the mathematics of oscillatory systems.

Linear oscillating system

Nonlinear Maps and Chaos