Anharmonicity

Anharmonic the oscillator is a physical system capable of oscillation, in which the restoring force is not proportional to the displacement from the rest position. As a result, the oscillation does not extend strictly sinusoidal. Mechanical examples include gravity pendulum ( anharmonicity noticeable in larger deflection), wagging ( the tipping of an upright object), Bouncing a ball on a flat surface.

On closer inspection, almost all real oscillatory systems are anharmonic. But most come close to a harmonic oscillator, the smaller the displacements from the equilibrium position are, because then the approximation of a linear restoring force always better to apply ( see Hooke's law for the mechanics, Mathematical pendulum). In such anharmonic oscillators run small oscillations approximation sinusoidal and at a certain natural frequency, the fundamental frequency of the oscillator.

When anharmonic oscillator occur compared to the harmonic oscillator fundamentally new phenomena:

• The deviation from the sinusoidal shape means that the vibration and harmonics ( acoustic: overtones ) contains the fundamental frequency.
• For unbalanced force law, the center of oscillation compared to the rest position shifts. This occurs, for example in the vibrations between the atoms of the solid body and is the cause of the thermal expansion.
• If an external excitation with a periodic force, the resulting forced vibration components at the difference frequency of the excitation frequency and the fundamental frequency and other integral combinations thereof. Technical application of this is, for example, in non-linear optics in the frequency doubling with laser light.
• If an external excitation with a periodic force of the anharmonic oscillator can also react with a chaotic motion when the initial conditions are chosen according to its parameters or outside certain limits ( eg in case of low damping).

Equation of motion

There are a number of different anharmonic oscillators and the corresponding number of equations of motion. Their common characteristic is that its restoring force is not as according to the harmonic oscillator only linearly with the deflection, but also by higher powers of ( " non-linear power law "). In the adjacent figure, the simplest examples are shown, along with the approximate linear power law (blue line):

The equations of motion with the damping term are then obtained from the Newtonian law

Wherein the first and the second derivative of the function with respect to time referred to:

For one again obtains in both cases, the linear force law as well as the differential equation of the damped harmonic oscillator. Even for small deflections the solution of the anharmonic oscillator is again almost harmonious as the first trajectories show in the picture on the right.

Due to the nonlinearity of the differential equations, the superposition principle is overridden. This means that no longer is any multiple of a solution is also a solution of the differential equation, and more generally that two solutions are not any linear combination is also a solution (where any solid numbers). The solution of the equation of motion is usually an elliptic integral and therefore in closed form using elementary functions can not be represented. This article focuses on periodic movements of the anharmonic oscillator. The damping is locally neglected, ie set. Only in this way arise if no external force is applied, periodic movements in the strict sense. The chaotic motion forms see chaos theory and literature given there.

Approximate solution

In the case of a weak anharmonic disorder, that is, respectively, one can obtain the solution by perturbation theory. To this is added in the form of a power series of a Störparameters to:

Here, the first member, the ( initial conditions adapted ) solution for the harmonic case, for example, when the oscillator is enabled at the time when the deflection at speed and the fundamental frequency has.

After substituting in the equation of motion, which is expressed by, a power series yields in whose coefficients are all equal to zero set. One obtains equations for the various approximation functions, which are recursively solved. Concretely, the differential equation of the form of equation of motion of a harmonic oscillator which is excited by an external force which is given by the previous one and forced vibrations.

In case A, follows in the first step

This already occurs in the doubled fundamental frequency. The mathematical reason for this can be traced back to the appearance of the quadratic term in the equation of motion, which is linearized by the trigonometric identity. In the further approximations to summands arise with correspondingly higher multiples of the fundamental frequency, ie a total of a whole spectrum of harmonics.

In the case B results in the first step an unstable solution, because tains a proportional Term This can be eliminated, however, if is set. This results already in the first approximation the dependence of the oscillation frequency of the amplitude ( as in the case A only in the 2nd step).

Applications

Real built- in technical equipment springs generally have, sometimes intended by the design, within certain limits, a linear relationship betwee restoring force and deflection on. The dynamics of a system with such a feather then follows the non-linear equations of motion, as introduced above.

Important applications for anharmonic vibrations can be found approximately in molecular physics in the vibration of diatomic molecules or in solid state physics in heat-induced vibrations of atoms. The anharmonicity is formed by the different effects upon approach ( electrostatic repulsion, partially shielded by the electrons of the atomic shell, but by the Pauli principle reinforced) and distance ( restoring force by the covalent bonding of atoms) of the ions from. As shown in the figure on the right, such oscillations can be calculated approximately in a Morse potential.

Forced anharmonic vibrations

In the motion under the influence of a time-dependent external force is anharmonic and harmonic oscillator are fundamentally different from each other. For example, even after a transient, the anharmonic oscillator at different frequencies vibrate as represented in the exciting force. There may be a sharp change in the amplitude even in slow variation of the excitation frequency. These phenomena are also of great interest in practice, since real oscillators only as long behave harmonically, are respected as certain limits for deflection and / or frequency.

These phenomena can be generally attributed to the non-linear form of the equation of motion, which the superposition principle can not be applied here. Some of the consequences:

• There is no universal solution procedures as of the forced oscillations of the harmonic oscillator. You have to integrate the equation of motion numerically or using analytical solutions that are win only approximations.
• The transient (in the case of periodic excitation ) is given by the superposition of the steady oscillation of a free vibration.
• The steady- stationary vibration is not always independent of the initial conditions.
• After Fourier transformation, the equations of motion for different frequencies remain coupled.

Amplitude jumps

As an example, the symmetric power law ( the above case B) was examined: The equation of motion is ( through after division, as well as with ):

Starting from the assumption of a stationary harmonic

Results from the force causing

The one with oscillating portion of the force is due to the transformation ago. This component is disregarded in the following. The force can then be approximated to

Summarize, so here's a harmony with the frequency of vibrating energy creates a harmonic oscillation of the same frequency. Taking into account a phase shift and the amplitude of the force by

Given. This equation can be not change in the manner customary for the resonance curve shape. However, you can resolve it by and gets the relationship between excitation frequency and stationary oscillation amplitude (for the given force by amplitude) in the form:

The two solutions that result from the solution of a quadratic equation here express that in general two excitation frequencies lead to equally large amplitude of the stationary oscillation, as is the case already with the harmonic oscillator on the left and right of the resonance peak. For the harmonic case, is not this formula consistent with the resonance curve of the harmonic forced vibrations, which has its amplitude maximum at and on both sides drops down symmetric. What is new is the anharmonic oscillator, the resonance frequency with increasing amplitude shifts (term in the formula). This can be in the graph of the resonance curve the whole resonance peak curve in such a way that it assumes an S -shape in specific frequency ranges, so despite having the same force amplitude and excitation frequency up to display three different possible values ​​for the stationary amplitude. When such a range is reached at a slow, continuous variation of the excitation frequency, the amplitude jumps from one branch to another of the resonance curve: The oscillation " tilts ".

Subharmonic excitation

In the previous section, a contribution to the force which oscillates with the frequency in the equation (** ) simply omitted. This is not always justified, because this contribution can also play the lead role in certain conditions. If it is true that

Then disappear from equation (**), all with periodic terms. It remains:

Is a solution to the external force

Example: The mathematical pendulum with fundamental frequency ( acceleration due to gravity, pendulum length ), the law of force approached by selecting the parameter will, driven by an external force

It oscillates with amplitude and a frequency three times as squat. To observe this behavior, however, you must either make the right initial conditions, or wait for the decay of additional self-oscillations, which can take a long time because of the assumption of a negligible attenuation.

Intermodulation

With intermodulation is the phenomenon that the oscillator with excitation at two frequencies respond with a vibration, are also in the combination frequencies ( and integer) represented. During the transient, but usually does not take long because of damping, such frequency multiples and combination frequencies are also present in relation to the fundamental frequency. In acoustics, it can occur as audible tones, which therefore have their origin in the fact that the eardrum or a loudspeaker diaphragm is excited over that deflection addition to which applies a linear force law for the restoring force.