Harmonic Oscillator

FIELDS OF STUDY: Harmonics; Classical Mechanics; Electronics

ABSTRACT: Harmonic oscillations are a fundamental feature of all mass-spring systems. They are modeled by the motion of pendulums. They can be mathematically described using the sine and cosine functions. Newton’s laws of motion describe the physical behaviors of pendulums. Hooke’s law describes the force that restores such systems to their equilibrium positions. Communication, power generation, motor control, and all digital electronic devices depend on harmonic oscillation to function.

PRINCIPAL TERMS

• damped oscillator: an oscillator that is subject to friction or other braking forces.
• Hooke’s law: the law stating that the deformation of an elastic object, such as a spring, is directly proportional to the force acting on the object, as long as the object’s elastic limit is not exceeded.
• mass-spring system: a system consisting of an elastic object connected to an object with mass.
• net force: the overall force acting on a system, calculated as the vector sum of all forces acting on and within the system.
• pendulum: a suspended mass that can undergo regular oscillations.
• resonance: the response of an elastic body to a force acting on the body at its natural frequency.
• spring constant: a characteristic factor of a particular spring that determines the expansion or contraction of the spring when displaced by a specific force.
• torque: a turning force acting radially about an axis or point.

Oscillations

An oscillation is the variance of a physical property or its magnitude from one value to another and back again in a regular or cyclic manner. A harmonic oscillator is simply an object or system whose oscillation is caused by displacement that results in a restoring force. For example, imagine a marble sitting in the bottom of a circular bowl. The center of the bowl represents the marble’s equilibrium point, because it is the lowest point to which the marble can roll. If the marble is held at the edge of the bowl, it has been displaced from its equilibrium point. When the marble is released, gravity acts as a restoring force, causing the marble to roll back down toward its equilibrium point. However, the marble will not stop at the bottom of the bowl. Instead, it will roll partway up the opposite side of the bowl, then back down to the bottom again and partway up the other side. This motion will repeat over and over until the marble eventually comes to rest at the bottom of the bowl. This repeated motion is oscillation, and the marble is a harmonic oscillator.

Oscillations can be described mathematically using sine and cosine waves. A sine wave begins at a value of zero (or some middle point represented by zero), increases to its maximum positive value, decreases below zero to its minimum negative value, and then increases to zero again. This is one cycle of the oscillation of a sine wave. The phrase "sine wave" describes a smooth, repetitive oscillation, such as the swing of a pendulum or the vibration of a guitar string. A cosine wave is similar to a sine wave, except that its cycle begins and ends at its maximum positive value rather than at zero. Thus, a cosine wave is also a type of sine wave. Electromagnetic waves are also sinusoidal in nature, as are waves rippling across a pond.

Oscillations are characterized by their amplitude, frequency, period, and sometimes wavelength. All oscillations have an equilibrium or neutral point in each cycle that is exactly midway between the extremes of their values. Displacement is the extent to which the value of the oscillating property varies from the neutral point at any given stage in its cycle. The amplitude of an oscillation is its maximum displacement from the neutral point. For example, the amplitude of a sine wave would be the distance of the maximum positive (or minimum negative) value from zero. The frequency of an oscillation is the number of cycles that occur in a given time. The period of the oscillation is the time required for one complete cycle. "Wavelength" is a term used to describe oscillations that travel in a linear fashion, such as electromagnetic waves. The wavelength is the distance traveled by the wave during one complete cycle of the oscillation.

Strings, Springs, and Resonance

A string on a musical instrument, such as a guitar or a piano, produces a sound when it vibrates. The frequency of the sound depends on the length of the vibrating string, and the intensity or loudness of the sound depends on the displacement of the string from its neutral position. The physical displacement of the string translates directly to the amplitude of the sound waves that are produced by the oscillation of the string as it vibrates. The sound fades and disappears as the string gradually ceases to vibrate and the amplitude decreases to zero. The frequency and the wavelength, of the sound remain the same, however, since these characteristics are not dependent on the amplitude or displacement of the oscillating string. In contrast, an ideal harmonic oscillator would produce the same values of frequency, wavelength, and amplitude over time.

A classic type of harmonic oscillator is the spring. Every spring has its own characteristic spring constant, which describes its stiffness and strength. The spring constant shows how much energy is required to displace the spring from equilibrium. It also shows how much restoring force the spring exerts to return to equilibrium. A coil spring can be either expanded or compressed. After either action, when the spring is released, it exerts a force that returns it to its neutral resting position, about which it then oscillates. This restoring force (F) is described by Hooke’s law as the product of the spring’s displacement (x) and its particular spring constant (k):

F = −kx

The negative sign means that the restoring force is acting against the force that displaced the spring from its equilibrium position.

A spring does not have to be coiled in shape. Any structure that exerts a restoring force when displaced from its characteristic equilibrium shape can act as a spring system that obeys Hooke’s law. The displacement can be lateral. It can also be caused by torque, as in some kinds of clocks or when a cable twists and untwists while supporting a suspended mass.

Hooke’s law applies to all types of mass-spring systems. Every mass-spring system has its own characteristic resonance frequency (ω), which is related to its mass (m) and its particular spring constant (k), according to the following equation:

When a system encounters another system or force oscillating at its resonance frequency, the first system resonates and vibrates at this frequency as well.

When resonant waveforms combine, their amplitudes also combine and increase, occasionally with disastrous results. Large groups of soldiers marching in lockstep are advised to break step when crossing a bridge so that they do not cause the bridge to resonate and collapse. On November 7, 1940, wind caused a bridge over the Tacoma Narrows strait in Washington State to oscillate at its resonance frequency. The amplitude of the vibration exceeded the limitations of the bridge’s structural integrity, causing it to collapse. Similarly, Canadian truck drivers in the far north often take heavy loads over the "ice highway" on the frozen surfaces of lakes. More than one has been lost when the resonance frequency of the ice layer was met, causing the ice to break beneath the weight of the truck.

Pendulums

According to Isaac Newton’s (1642–1727) second law of motion, a net force (F_net) acting on a body will cause the body to accelerate in the direction of the force. The acceleration (a) of the body will be directly proportional to the net force and inversely proportional to the mass (m) of the body:

When an initial net force displaces a pendulum from its equilibrium position, it provides the pendulum with potential energy due to gravity. When the pendulum is released, gravity draws the pendulum back toward its equilibrium position. As it passes through equilibrium, the pendulum has kinetic energy. The momentum that the pendulum has acquired due to its motion, in accord with Newton’s first law, requires it to continue on its path until gravity, acting as a damping force, stops the motion at a certain distance and makes the pendulum once again fall back toward its equilibrium position. This oscillating motion continues until the kinetic energy and momentum of the pendulum have been lost to friction, air resistance, and other energy-consuming factors. The net force acting on the pendulum at any point is the vector sum of the force of gravity (its weight) and any frictional or other braking forces acting on it. The force of gravity imparts the motion, while braking forces detract from it.

A harmonic oscillator that is not subject to any braking forces is called a simple harmonic oscillator. An oscillator whose amplitude is decreased toward its equilibrium value by a braking force is called a damped oscillator. Almost all real physical oscillators are damped oscillators, because in the real world, braking forces are almost always present. To maintain the amplitude of a damped oscillation requires the constant input of energy into the system from an external source.

Torsional pendulums obey the same physical laws as linear pendulums. With a torsional pendulum, the displacement from the equilibrium position is brought about by torque about the central axis of the system. The restoring force of a torsional pendulum is not the force of gravity but the torque applied by the material that makes up the axis of the pendulum. A simple example of such a system is a metal wire or rod fixed to a weight. The radial length of the weight about the central axis determines the period and frequency at which the system can oscillate, while the radial displacement from its equilibrium position determines the spring force that acts on the system.

Sample Problem

The displacement of an oscillating string is described by the equation

x = Acos(ωt)

where x is the displacement at time t, A is the initial displacement of the string, and ω is the frequency of the oscillation. Calculate the displacement of a string oscillating at 440 hertz (Hz), or cycles per second (1/s), at a time of 100 milliseconds (ms) after release from an initial displacement of 5 millimeters (mm).

Answer:

Assign the variable values as

A = 5 mm

ω = 440 1/s

t = 100 ms = 0.1 s

Calculate:

x = Acos(ωt)

x = 5 mm × cos(440 1/s × 0.1 s)

x = 5 mm × cos(44)

x = 3.6 mm

The displacement of the string 100 milliseconds after release is 3.6 millimeters.

Oscillations in Other Systems

Electromagnetic waves and electronic signals are oscillating systems that obey the same mathematical principles as pendulums and vibrating strings. Describing such waves requires more complex calculus involving sine and cosine functions than is required for the description of simple harmonic oscillators and pendulums. Electromagnetic waves are characterized by their frequency, period, and amplitude. These are also related to circular motion through the sine and cosine functions. Electromagnetic waves combine as the vector sum of their sinusoidal waveforms and resonate at specific frequency combinations to produce purely sinusoidal waveforms. This is an essential feature of "phase shift" in electrical power generation and motor control. Digital electronic devices also depend on harmonic oscillation. They use the specific vibrational frequency of quartz or other materials to control the timing of transistor switches.

Bibliography

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Matthews, Michael R., Colin F. Gauld, and Arthur Stinner, eds. The Pendulum: Scientific, Historical, Philosophical & Educational Perspectives. Dordrecht: Springer, 2005. Print.