Harmonic oscillator#Driven harmonic oscillators

The forced vibration is the movement which an oscillatory system ( oscillator ) due to a time-dependent external excitation. If the excitation periodically forced oscillations goes after a transient gradually into the stationary forced oscillations. In the stationary forced oscillation, the oscillator performs a periodic vibration, the frequency, regardless of the natural frequency, is given only by the external stimulus. Here, the oscillator oscillates at a constant amplitude in time, which has particularly high values ​​, when the oscillator is only slightly attenuated, and the exciting frequency in the vicinity of its natural frequency (see resonance).

Forced oscillations occur in many areas of daily life. In physics and engineering in particular forced harmonic oscillations are widely used as a model for the response of a system to external influences. In mechanics, the excitation is typically done by a periodic force on a body or a periodic shift in its rest position, in electrical engineering and electronics by an alternating voltage or alternating current, in optics and quantum physics by an electromagnetic wave in quantum physics by a matter wave.

A parameter- excited oscillation is not a forced oscillation, since the excitation is not done by the external force, but by changing the native parameters such as natural frequency, or center of gravity.

Examples of the mechanical

The tides excite the masses of water in bays to forced oscillations. With a suitable total length of the bay, the amplitude of the tides can be particularly high, as in the Bay of Fundy.

When rotating parts are not carefully balanced, which always leads to a so-called critical speed at which the forces ( consisting spring -mass system of rotor mass and wave or total mass and suspension / foundation ) stimulate the oscillatory overall system response. This is desirable in the vibrating plate, but must be avoided in the car's engine or electric generator.

The driver of earth-moving equipment or forklifts are - apart from the noise - for hours exposed to forced oscillations, which can lead to occupational disease.

When the tympanic membrane of an ear will not be excited by the sound waves to forced vibrations, there is no hearing. The same applies to many sensory organs of the animals.

Bumps in the road about rain propelled, spring loaded cars to be forced oscillations, which - if they are not damped by shock absorbers in no time - reduce steering and braking ability drastically.

Skyscrapers are excited by seismic waves to forced oscillations, which can lead to collapse without mass dampers.

Examples from electrical engineering

Filter circuits in a combination of inductors and capacitors, resistors and sometimes crystals excited by a mixture of an alternating electric voltage generated, for example, through an antenna, to forced vibrations. The amplitude at the output of the filter is strongly dependent on the frequency. Without filter radios were like TV or radio impossible because otherwise the individual programs could be separated from each other.

If the input ( left) fed the adjacent circuit with AC voltage of frequency 2 MHz and sufficiently high amplitude flow through the transistor short current pulses that frequency. These contain many harmonics whose frequencies under the laws of the Fourier series are always integer multiples of the fundamental frequency. In this example, the collector current levels of 4 MHz, 6 MHz, 8 MHz and so on, an oscillation circuit, whose resonant frequency is tuned by appropriate selection of L and C on one of these frequencies has, is excited by current pulses to forced vibrations. The circuit is referred to as a frequency in a spectrum analyzer and allows the generation of high frequencies.

The alternating current generated by a transmitter excites the electrons in the wires of a transmitting antenna to forced oscillations. By suitable choice of length of the antenna resonance is generated and the power radiated particularly effective.

For shielding of electrical equipment, the electrons of the metal shell are excited to forced vibrations and in turn gives off electromagnetic waves with exactly the same frequency and amplitude but opposite in phase from. Inside, the fields compensate.

For each type of speaker, the membrane is excited by alternating current ( the electrodynamic loudspeaker ) or alternating current ( in electrostatic speakers) to forced oscillations. In this case, resonance is to be avoided, because these deteriorate the frequency response.

Examples of the optical

From the sunlight, the electrons the illuminated surfaces are encouraged to forced vibrations and in turn gives off light. Due to the surface properties of the body, the frequencies are some colors are preferred. The chlorophyll molecules of plant preferred reflect green light because the electrons at other frequencies can not perform forced oscillations. Blue and red light is absorbed by the chlorophyll, for the purpose of photosynthesis. A more detailed description needs quantum mechanics.

The sunlight excites the electrons of the molecules of the atmosphere to forced oscillations, which in turn emit light. Here, the short-wave blue light spectrum is about 16 times more scattered than the red light. Therefore, the blue color predominates in the light of our atmosphere.

In the microwave oven, water molecules are high frequency waves to vibrations (more precisely, folding ) and forced heat up due to mutual friction. With frozen water does not work, because the molecules can not fold because of their mutual bond in the crystal lattice.

Generation

All vibrating systems are subject to attenuation. You always need for a permanent oscillation therefore an external drive. This compensates for the loss of energy by the damping. The duration of the vibration may be desirable, for example, tone generation, or undesirable. By vibration isolation then the amplitude of the system must be kept low.

Often done no permanent drive. The system is so excited only once ( for example, when striking a drum ) or for a limited period of time ( for example, when painting with a violin bow ). In this case, the oscillating system, first traverses the so-called transient response to decay after the end of the drive as a damped oscillation.

Forced oscillation at the harmonic oscillator

In the harmonic oscillator, for example, a mechanical mass-spring -damper system as shown on the left, the phenomena can be studied most easily.

In reality though most systems that can perform oscillations are only approximately harmonic, but they all show the phenomena of forced vibration in a manner at least similar (see anharmonic oscillator ).

Equation of motion

Of the homogeneous differential equation for a damped harmonic oscillator is linearly an external force added to act on the mass. The equation is thus inhomogeneous.

This refers to the instantaneous displacement from the equilibrium position, the mass of the body, the spring constant for the restoring force and the damping constant (see figure).

Without external force and damping, the system would oscillate freely at its natural angular frequency. In complex notation ( with arbitrary real amplitude and phase):

If added damping, the system can run free damped oscillations with angular frequency whose amplitude decreases proportionally, which is and has been adopted:

A static constant force would result in a shift in the position of rest to the result.

Transient, steady vibration, general solution

Given an arbitrary course of the force, so not necessary periodically or even sinusoidal. Depending on the initial conditions, the system will perform various movements. Let and be two such movements, ie solutions of the same equation of motion:

Subtracting these equations each other, is obtained due to the linearity, and that the difference between the two movements, the motion equation

Met. thus describes a damped harmonic oscillation of the force-free oscillator. In damping the amplitude approaches zero. Therefore go (for a given course of ) all the different forced vibrations of the damped system over time in a single. This process is called transient, its result is the steady-state forced vibration (hereinafter referred to ). The transient is different depending on the initial conditions, but always irreversible process. The steady-state forced vibration has no "memory" from which concrete initial conditions out it was made.

The most common form of motion is given by a superposition of stationary solution and steamed Eigenschschwingung:

Periodic excitation

The system is excited by a periodic force, acting on the mass. It was previously at rest, the first amplitude increases and, when the excitation frequency is close to its natural frequency, can reach values ​​larger than at a constant exposure to the maximum force (see resonance). Provided that the vibration system is not overloaded (resonance disaster ), the oscillation gradually changes into a harmonic oscillation with constant values ​​for amplitude, frequency and phase shift from the exciting vibration. This behavior is completely consistent for each type of harmonic oscillator. In reality though, most systems that can perform oscillations only approximately harmonic oscillators, but they all show the resonance phenomena in at least a similar manner.

When the force extends sinusoidal with the amplitude and the excitation frequency, applies

A force with another course, even if it is not periodic, can be represented by adding sine (or cosine ) shaped forces of different excitation frequencies (see Fourier transform). Because of the linearity of the equation of motion, the resulting motion is an appropriate sum of the forced vibrations to each of the frequencies present. Mathematically, the method of Green's function, in which first the response of the system is determined to an arbitrarily short acting force impulse ( in the form of a delta function ), so to speak, a hammer blow. The responses to the impulses of the starch are then summed or integrated in accordance with a time lag.

Steady state with sinusoidal excitation

With periodic excitation the steady state must show a constant amplitude. Therefore, sufficient for the complex number calculation of Exponentialansatz, from and determined. Is to use for the force.

It follows

Or transformed

As with the formula for the complex force here, only the imaginary part has a direct physical meaning:

This is a harmonic oscillation around the rest position with the angular frequency, the (real) amplitude

And the constant phase shift with respect to the exciting force

Therein:

• : The deflection under static action of the force,
• The related to the natural frequency of the excitation frequency,
• The Lehr to related dimensionless damping, which is often expressed by the quality factor. The quality factor has the meaning that it indicates the number of vibrations, by which (in the absence of an external force ), the amplitude has decayed to the initial value (after oscillations ).

The dependence of the amplitude of the excitation frequency is illustrated in the figure. It is referred to as a resonance curve of the amplitude or the amplitude response of the system. She's at a maximum, if. For a more detailed description of the phenomena in the vicinity of the maximum amplitude of the resonance passage, see the article.

The phase shift is ( when used here sign convention ) for low excitation frequencies between 0 and 90 °. In the quasi-static case, that is very slowly varying excitation, the system follows in its oscillation with a slight delay of the oscillation of the exciting force. The expression for stationary oscillation can be here formed ( for ) to

Wherein the delay time indicates. Thus, for slowly varying force, the deflection at any moment is just as great as they would in a short time before acting force when this act would constant.

When the delay reaches exactly 90 °, so that power and speed always simultaneously change sign and thus constantly flowing into energy in the oscillating system. At this excitation frequency in the stored energy of the vibration becomes a maximum.

At higher excitation frequency the delay continues to increase. Upon excitation far above the resonance frequency of the system oscillates almost in antiphase to the exciting force.

Transient from rest

To find the initial condition for the " rest position " matching exercise, in need of general formula

For the parameters and the damped free vibration, the appropriate values ​​are used. In the simplest case, the time zero point is oriented towards the stationary vibration and straight down to a zero crossing. Then:

The exciting force is then given by. The initial condition " rest position " is in use by

Met. This gives the full movement again. The second term in the brackets represents the transient dar. His contribution is at low excitation due to small or even negligible. It is with increasing excitation frequency but still significant. With high frequency excitation it makes for a certain amount of time for the largest portion of the movement until the prefactor due to the damping is fulfilled the condition.

In the case of a low damping (or ) is at excitation frequencies in the range of the resonance behavior of a strong beat: the oscillator oscillates at the center frequency, wherein the amplitude is modulated. It starts at zero and will vary with the half of the difference frequency sinusoidally rising and falling. First, " Rocks vibration on " until around the time of the first amplitude maximum is reached. For weak damping () twice the resonance amplitude of the steady state can be reached.

The closer the excitation frequency to the natural frequency, the longer runs towards the building-up (). In the case of exact resonance amplitude of the transient has the particularly simple form

Here, the amplitude approaches to no overshoot asymptotically stationary resonance amplitude.

Limit of vanishing damping

Is the theoretical ideal case of vanishing damping. From a transient, as it arises in the case of the decaying damped natural frequency of the free oscillator, one can no longer speak thus.

Stationary and general solution outside the resonance

However, there exists for sinusoidal periodic excitation here, provided a well-defined stationary oscillation around the rest position with the angular frequency, how ( as ) follows from the general formula for immediately:

The amplitude of this oscillation is stationary as great as the deflection in the static case at low excitation. When approaching the resonance it increases beyond all limits, and falls down again to a higher frequency, as it is less than. Below the natural frequency of oscillation is stationary with the force in the phase ( phase shift), when the excitation frequency is above the resonant frequency in anti-phase ().

The general solution of the equation of motion is called ( always provided )

The two free parameters are appropriately set to the initial conditions. Except in the case of a superposition of two harmonic oscillations, in the case thus yields a beat that (theoretically) continues indefinitely.

Special and general solution in the resonance

A special solution to match the initial condition, one gets this: In the above formula for the transient from rest

Can be substituted for the limiting case:

It follows:

Accordingly, the amplitude increases with resonant excitation from the equilibrium position proportional to time to, theoretically, beyond all limits. For the general solution for arbitrary initial conditions must be added to this formula, as above, a free vibration with matching parameters.

Limiting case Free particles

Without restoring force, a body to perform a periodic movement when an external force is applied in accordance with on it. Examples of such " vibrations" slide back and forth or roll an object on a surface when the friction is low and the area is not sufficiently remains exactly horizontal. Specifically: if a cup on the tray into the slides is coming and you will bring them through opposite inclination to rest, or if on a rocking ship deck cargo has broken loose, or if to conduct with a headache just by tilting the playing balls into a depression are. The equation of motion ( in one dimension, such as numbers above)

It corresponds to the harmonic oscillator with natural frequency.

A periodic excitation may be implemented, for example, by alternately tilting the surface. With sinusoidal excitation for the statements made for the forced oscillation above, it being set and therefore always applies to the excitation frequency apply. From the formula for the amplitude is as follows:

The rashes are greater, the lower the excitation frequency. The " resonance catastrophe" occurs with certainty when. You can not be prevented by damping. The movement is retarded with respect to the force. The phase shift is between 90 ° ( for ) and 180 ° ( for ). The phase shift is given by