Forced vibration

Three identical pendulums at different excitation frequencies

The forced oscillation is the movement that an oscillating system ( oscillator ) executes due to a time-dependent external stimulus. If the excitation is periodic , the forced oscillation gradually changes into the steady-state forced oscillation after a settling process . In the case of stationary forced oscillation, the oscillator performs a periodic oscillation, the frequency of which, regardless of its natural frequency , is only given by the external excitation. The oscillator oscillates with an amplitude that is constant over time and which has particularly large values ​​when the oscillator is only weakly damped and the excitation frequency is close to its natural frequency (see resonance ).

Forced vibrations occur in many areas of everyday life. In physics and technology, forced harmonic oscillations in particular are often used as a model for the reaction of a system to external influences. In mechanics, the excitation is typically carried out by a periodic force on a body or a periodic shift of its rest position, in electrical engineering and electronics by an alternating voltage or an alternating current , in optics and quantum physics by an electromagnetic wave , in quantum physics also by a Matter wave .

A parameter-excited oscillation is not a forced oscillation, since the excitation does not occur through external influences, but rather through changes in system parameters such as the natural frequency or the position of the center of gravity .

Examples from mechanics

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

An air suspension seat dampens the vibrations of a forklift truck

If rotating machine parts are not carefully balanced , this always leads to a so-called critical speed , at which the forces stimulate the oscillating overall system (spring-mass system, consisting of rotor mass and shaft or total mass and suspension / foundation) to resonate. This is desirable with the vibratory plate , but must be avoided with the car engine or electric generator .

The drivers of earth-moving machines or forklifts - apart from the noise - are exposed to forced vibrations for hours , which can lead to occupational diseases .

If the sound waves did not force the eardrum of one ear to vibrate, there would be no hearing. The same applies to many of the animal's sense organs .

Unevenness in the roadway stimulates sprung cars driving over them to create forced vibrations which - if they are not dampened by shock absorbers in the shortest possible time - drastically reduce the ability to steer and brake.

High-rise buildings are stimulated to vibrate by earthquake waves , which can lead to collapse without a vibration absorber .

Examples from electrical engineering

Excitation of harmonics in a selective amplifier

In filter circuits , a combination of coils and capacitors, sometimes also resistors and crystals, is excited into forced oscillations by a mixture of electrical alternating voltages, which is generated, for example, by an antenna . The amplitude at the output of the filter depends heavily on the frequency . Radio devices such as televisions or radios would be impossible without filters , because otherwise the individual programs could not be separated from one another.

If the input (left in the picture) of the adjacent circuit is fed with alternating voltage with a frequency of 2 MHz and a sufficiently high amplitude, short current pulses of this frequency flow through the transistor . These contain many harmonics , the frequencies of which, according to the laws of the Fourier series, are always integer multiples of the fundamental frequency. In this example, the collector current contains components of 4 MHz, 6 MHz, 8 MHz etc. An oscillating circuit , the resonance frequency of which is matched to one of these frequencies by suitable selection of L and C , is stimulated to forced oscillations by the current pulses. The circuit is known as a frequency multiplier and allows the generation of the highest frequencies in a spectrum analyzer .

The alternating current generated by a transmission system excites the electrons in the wires of a transmission antenna into forced oscillations. With a suitable choice of antenna length, resonance is generated and the power is emitted particularly effectively.

When shielding electrotechnical devices, the electrons in the metal shell are excited to vibrate and emit electromagnetic waves with exactly the same frequency and amplitude, but out of phase. The fields compensate each other in the interior.

In every type of loudspeaker , the membrane is excited to create forced vibrations by alternating current (for electrodynamic loudspeakers ) or alternating voltage (for electrostatic loudspeakers ). In doing so, resonances should be avoided because they worsen the frequency response .

Examples from optics

The electrons of the irradiated surfaces are stimulated to oscillate by the sunlight and in turn emit light. Due to the surface properties of the body, the frequencies of some colors are preferred. The chlorophyll molecules of plants preferentially reflect green light because the electrons can not perform forced oscillations at other frequencies . Blue and red light is absorbed by chlorophyll for the purpose of photosynthesis . Quantum mechanics needs a more precise description .

The sunlight excites the electrons of the molecules of the earth's atmosphere to forced vibrations, which in turn emit light. The short-wave blue light spectrum is scattered around 16 times more than the red light. That is why the blue color predominates in the light of our atmosphere.

In the microwave oven , water molecules are forced to vibrate (more precisely: flipping over) by high frequency waves and heat up as a result of mutual friction. This does not work with frozen water because the molecules cannot fold over because of their mutual bonding in the crystal lattice.

generation

All oscillating systems are subject to damping . You therefore always need an external drive for permanent vibration . This compensates for the energy loss due to the damping. The continuous oscillation may be desirable, e.g. B. for sound generation, or undesirable. The amplitude of the system must then be kept low by means of vibration isolation .

Often there is no permanent drive. The system is only stimulated once (for example when a drum is beating) or for a limited period of time (for example when bowing with a violin bow). In this case, the oscillating system first travels through the so-called transient process in order to decay as a dampened oscillation after the end of the drive.

Forced oscillation on the harmonic oscillator

The easiest way to study the phenomena is on the harmonic oscillator , for example a mechanical mass-spring-damper system as shown on the right.

Mass, spring, damper system

In reality, most of the systems that can oscillate are only approximately harmonic, but they all show the phenomena of forced oscillation in at least a similar way (see anharmonic oscillator ).

Equation of motion

An external force is added to the homogeneous differential equation for a linearly damped harmonic oscillator , which acts on the mass. This makes the equation inhomogeneous. ${\displaystyle F(t)}$

${\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=F(t)}$

This describes the momentary deflection from the rest position, the mass of the body, the spring constant for the restoring force, and the damping constant (see fig.). ${\displaystyle x(t)}$ ${\displaystyle m}$ ${\displaystyle k}$ ${\displaystyle c}$

Without external force and damping, the system would oscillate freely with its natural angular frequency . In complex notation (with any real amplitude and phase ): ${\displaystyle \omega _{0}={\sqrt {\frac {k}{m}}}}$ ${\displaystyle A_{0}}$ ${\displaystyle \varphi _{0}}$

${\displaystyle x_{0}(t)=A_{0}e^{-\mathrm {i} (\omega _{0}t-\varphi _{0})}}$

If damping is added, the system can carry out free damped oscillations with the angular frequency , the amplitude of which decreases proportionally to where is and was assumed: ${\displaystyle \omega _{d}={\sqrt {\omega _{0}^{2}-\gamma ^{2}}}}$ ${\displaystyle e^{-\gamma t}}$ ${\displaystyle \gamma =c/(2m)}$ ${\displaystyle 0<\gamma <\omega _{0}}$

${\displaystyle x_{d}(t)=A_{0}e^{-\gamma t}e^{-\mathrm {i} (\omega _{d}t-\varphi _{0})}}$

A static constant force of the exciter would result in a shift in the rest position by . ${\displaystyle F(t)=F_{0}}$ ${\displaystyle A_{\text{E}}=F_{0}/k}$

Transient process, stationary oscillation, general solution

An arbitrary course of the force is given, i.e. not necessarily periodic or even sinusoidal. Depending on the initial conditions, the system will perform different movements. Let and be two such movements, i.e. solutions of the same equation of motion: ${\displaystyle F(t)}$ ${\displaystyle x_{1}(t)}$ ${\displaystyle x_{2}(t)}$

${\displaystyle m{\ddot {x_{1}}}+c{\dot {x_{1}}}+kx_{1}=F(t)}$

${\displaystyle m{\ddot {x_{2}}}+c{\dot {x_{2}}}+kx_{2}=F(t)}$.

If you subtract these equations from each other, because of the linearity in and , the difference between the two movements results in the equation of motion ${\displaystyle x}$ ${\displaystyle F}$ ${\displaystyle {\tilde {x}}(t)=x_{2}(t)-x_{1}(t)}$

${\displaystyle m{\ddot {\tilde {x}}}+c{\dot {\tilde {x}}}+k{\tilde {x}}=0}$

Fulfills. describes a dampened harmonic oscillation of the force-free oscillator. When damped , their amplitude approaches zero. Therefore (given the course of ) all different forced oscillations of the damped system change into a single one in the course of time. This process is called the transient process , its result is the steady, forced oscillation (referred to below as ). The transient process is a process that varies depending on the initial conditions , but is always irreversible. The steady, forced oscillation has no “memory” of the specific initial conditions from which it arose. ${\displaystyle {\tilde {x}}(t)}$ ${\displaystyle c>0}$ ${\displaystyle F(t)}$ ${\displaystyle X(t)}$ ${\displaystyle x(0),\ {\dot {x}}(0)}$ ${\displaystyle X(t)}$

The most general form of motion is given by a superposition of stationary solution and damped natural oscillation : ${\displaystyle X(t)}$ ${\displaystyle x_{d}(t)}$

${\displaystyle x(t)=X(t)+x_{d}(t)}$

Periodic stimulation

The system is stimulated by a sinusoidal periodic force that acts on the mass. If it was previously at rest, the amplitude initially increases and, if the excitation frequency is close to its natural frequency, it can reach higher values ​​than if the maximum force was constantly applied (see resonance ). Unless the oscillation system is overloaded ( resonance catastrophe ), the oscillation gradually changes into a harmonic oscillation with constant values ​​for amplitude, frequency and phase shift compared to the excitation oscillation. This behavior is shown to be perfectly consistent for each type of harmonic oscillator . In reality, most of the systems that can oscillate are only approximately harmonic oscillators, but they all show the resonance phenomena in at least a similar way.

If the force is sinusoidal with the amplitude and the excitation frequency , then applies ${\displaystyle F_{0}}$ ${\displaystyle \omega }$

${\displaystyle F(t)=F_{0}\cdot \sin \omega t}$.

A force with a different course, even if it is not periodic, can be represented by adding sine (or cosine) shaped forces of different excitation frequencies (see Fourier transformation ). Because of the linearity of the equation of motion, the resulting motion is then the corresponding sum of the forced vibrations for each of the occurring frequencies. The method of Green's function is mathematically equivalent , in which the response of the system to an impulse of any short duration (in the form of a delta function ) is determined, so to speak to a hammer blow. The responses to the force impulses are then added up or integrated with a corresponding time delay . ${\displaystyle F(t)\,dt}$

Steady state with sinusoidal excitation

Amplitude ratio as a function of Lehr's damping, plotted against the frequency ratio . The points of intersection of the dotted line with the resonance curves show the position of the resonance frequencies at . ${\displaystyle A(\omega )/A_{\text{E}}}$ ${\displaystyle \omega /\omega _{0}}$ ${\displaystyle \omega _{\mathrm {R} }}$ ${\displaystyle \omega _{\mathrm {R} }/\omega _{0}={\sqrt {1-2D^{2}}}}$

With periodic excitation, the steady state must show a constant amplitude . Therefore the exponential approach is sufficient for the calculation with complex numbers , from which and are determined. For the force is to be used instead of , so that here the imaginary part has the physical meaning. ${\displaystyle A}$ ${\displaystyle X(t)=Ae^{\lambda t}}$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle F(t)=F_{0}\cdot e^{\mathrm {i} \omega t}}$ ${\displaystyle F(t)=F_{0}\cdot \sin \omega t}$

It follows

${\displaystyle X(t)={\frac {\frac {F_{0}}{m}}{-\omega ^{2}+2\mathrm {i} \gamma \omega +\omega _{0}^{2}}}\,e^{\mathrm {i} \omega t}}$

or reshaped

${\displaystyle X(t)={\frac {\frac {F_{0}}{m}}{\left|-\omega ^{2}+2\mathrm {i} \gamma \omega +\omega _{0}^{2}\right|}}\,e^{\mathrm {i} (\omega t-\varphi )}}$

As with the formula for the complex force, only the imaginary part has direct physical meaning (the real part belongs to the force curve corresponding to the real part of the complex force):

${\displaystyle X(t)={\frac {\frac {F_{0}}{m}}{\left|-\omega ^{2}+2\mathrm {i} \gamma \omega +\omega _{0}^{2}\right|}}\,\sin(\omega t-\varphi )}$

This is a harmonic oscillation around the rest position with the circular frequency , the (real) amplitude ${\displaystyle x=0}$ ${\displaystyle \omega }$

${\displaystyle A(\omega )={\frac {\frac {F_{0}}{m}}{\sqrt {(\omega _{0}^{2}-\omega ^{2})^{2}+(2\gamma \omega )^{2}}}}\equiv {\frac {1}{\sqrt {(1-\eta ^{2})^{2}+(2\eta D)^{2}}}}\cdot A_{\text{E}}}$

and the constant phase shift with respect to the exciting force

${\displaystyle \varphi (\omega )=\arctan \left({\frac {2\omega \gamma }{\omega _{0}^{2}-\omega ^{2}}}\right)\equiv \arctan \left({\frac {2\eta D}{1-\eta ^{2}}}\right)}$

Inside is:

• ${\displaystyle A_{\text{E}}={\tfrac {F_{0}}{k}}}$: the amplitude of the exciter or the deflection when the force is static . ${\displaystyle F_{0}}$
• ${\displaystyle \eta ={\frac {\omega }{\omega _{0}}}}$: the excitation frequency related to the natural frequency,
• ${\displaystyle D={\frac {\gamma }{\omega _{0}}}}$: the related, dimensionless Lehr's damping , which is often also expressed by the quality factor . The quality factor means that it indicates the number of oscillations after which (in the absence of an external force) the amplitude has decayed to the initial value (after oscillations up ). ${\displaystyle \omega _{0}}$ ${\displaystyle Q={\tfrac {1}{2D}}}$ ${\displaystyle e^{-\pi }\!\approx 4\,\%}$ ${\displaystyle {\tfrac {Q}{\pi }}}$ ${\displaystyle {\tfrac {1}{e}}\approx 37\,\%}$

The dependence of the amplitude on the excitation frequency is shown in the figure. It is called the resonance curve of the amplitude or the amplitude response of the system. It has a maximum at the resonance frequency , if (dotted line in the figure). For a more detailed description of the phenomena near the maximum of the amplitude response, see the article resonance . ${\displaystyle A}$ ${\displaystyle \omega }$ ${\displaystyle \omega _{\mathrm {R} }={\sqrt {\omega _{0}^{2}-2\gamma ^{2}}}={\sqrt {1-2D^{2}}}\omega _{0}}$ ${\displaystyle \omega _{0}>{\sqrt {2}}\gamma }$

The phase shift is (with the sign convention used here) for low excitation frequencies between 0 and 90 °. In the quasi-static case , i.e. H. If the excitation varies very slowly, the oscillation of the system follows the oscillation of the exciting force with a slight delay. The expression for stationary oscillation can be transformed here (for ) to ${\displaystyle \omega <\omega _{0}}$ ${\displaystyle \omega \ll \omega _{0}}$

${\displaystyle X(t)\approx {\frac {1}{k}}\,F(t-\Delta t)}$,

where indicates the delay time. Accordingly, with a slowly varying force, the deflection is exactly as large at every moment as it would be with the force acting shortly before if it were acting constantly. ${\displaystyle \Delta t\approx {\frac {2\gamma }{\omega _{0}^{2}}}}$ ${\displaystyle X(t)}$ ${\displaystyle F(t-\Delta t)}$

At , the deceleration reaches exactly 90 °, so that force and speed always change their sign at the same time and thus energy constantly flows into the oscillating system. At this excitation frequency, the energy stored in the oscillation is maximal. ${\displaystyle \omega =\omega _{0}}$

The delay increases further at a higher excitation frequency. When excited far above the resonance frequency, the system vibrates almost in phase opposition to the exciting force.

Swinging in from the rest position

In order to find the movement that matches the initial condition “rest position” , in the general formula ${\displaystyle x(0)=0\;;\ {\dot {x}}(0)=0}$

${\displaystyle x(t)=X(t)+x_{d}(t)}$

the appropriate values ​​are used for the parameters and the damped free oscillation . In the simplest case, the zero point in time is based on the stationary oscillation and is set to a zero crossing of . Then: ${\displaystyle A_{0}}$ ${\displaystyle \varphi _{0}}$ ${\displaystyle x_{d}(t)}$ ${\displaystyle X(t)}$

${\displaystyle X(t)=A(\omega )\;\sin \omega t}$.

The exciting power is then given by. The initial condition "rest position" is just from ${\displaystyle F(t)=F_{0}\sin(\omega t+\varphi (\omega ))}$

${\displaystyle x(t)=A(\omega )\left(\sin \omega t-{\frac {\omega }{\omega _{d}}}e^{-\gamma t}\sin \omega _{d}t\right)}$

Fulfills. This reflects the complete sequence of movements. The second term in brackets represents the transient process. In the case of slow excitation, its contribution is small or even negligible. However, it becomes more and more important with increasing excitation frequency. In the case of high-frequency excitation, it accounts for the largest part of the movement for a certain period of time, until the pre-factor therein fulfills the condition due to the damping . ${\displaystyle \omega \ll \omega _{d}}$ ${\displaystyle \omega }$ ${\displaystyle \omega \gg \omega _{d}}$ ${\displaystyle {\frac {\omega }{\omega _{d}}}e^{-\gamma t}<1}$

In the case of low damping ( or ), there is a pronounced beat behavior at excitation frequencies in the resonance range : The oscillator oscillates at the center frequency , with the amplitude being modulated. It starts at zero and varies sinusoidally with half the difference frequency, rising and falling. First of all, the oscillation “sways” until the first amplitude maximum is reached. With weak damping ( ), double the resonance amplitude of the steady state can be reached. ${\displaystyle \gamma \ll \omega _{d}}$ ${\displaystyle D\ll 1}$ ${\displaystyle \omega }$ ${\displaystyle {\frac {1}{2}}(\omega +\omega _{d})}$ ${\displaystyle {\frac {1}{2}}(\omega -\omega _{d})}$ ${\displaystyle t_{A_{\text{max}}}={\frac {\pi }{|\omega -\omega _{d}|}}}$ ${\displaystyle A_{\text{max}}=A(\omega )\left(1+{\frac {\omega }{\omega _{d}}}e^{-\gamma t_{A_{\text{max}}}}\right)}$ ${\displaystyle \gamma \ll |\omega -\omega _{d}|}$ ${\displaystyle A_{\text{max}}}$ ${\displaystyle A_{\text{res}}\approx A(\omega _{0})}$

The closer the excitation frequency comes to the natural frequency , the longer the build-up takes ( ). In the case of the exact amplitude resonance , the transient process has the particularly simple form ${\displaystyle \omega }$ ${\displaystyle \omega _{d}}$ ${\displaystyle t_{A_{\text{max}}}\rightarrow \infty }$ ${\displaystyle (\omega =\omega _{d})}$

${\displaystyle x(t)=A(\omega _{d})\left(1-e^{-\gamma t}\right)\sin \omega _{d}t}$.

Here, the amplitude approaches the steady-state resonance amplitude asymptotically without overshooting.

Borderline case of vanishing attenuation

In the theoretical ideal case, the damping is vanishing . It is therefore no longer possible to speak of a transient process, as it originates in the damped case from the decaying natural oscillation of the free oscillator. ${\displaystyle \gamma =0}$

Stationary and general solution outside of resonance

However, for sinusoidal periodic excitation, there is also a well-defined stationary oscillation around the rest position with the angular frequency , as follows from the general formula (see above) for immediately: ${\displaystyle \omega \neq \omega _{0}}$ ${\displaystyle \omega }$ ${\displaystyle \gamma =0}$

${\displaystyle X(t)=\left({\frac {\omega _{0}^{2}}{\omega _{0}^{2}-\omega ^{2}}}A_{\text{E}}\right)\cdot \sin \omega t}$

With slow excitation, the amplitude of this stationary oscillation is as large as the deflection in the static case . When the resonance approaches , it grows beyond all limits and falls again towards a higher frequency, from below it is smaller than . Below the natural frequency, the stationary oscillation is in phase with the force (phase shift ); at an excitation frequency above the natural frequency, it is in opposite phase ( ). ${\displaystyle \omega \ll \omega _{0}}$ ${\displaystyle A_{\text{E}}}$ ${\displaystyle \omega \rightarrow \omega _{0}}$ ${\displaystyle \omega >{\sqrt {2}}\,\omega _{0}}$ ${\displaystyle A_{\text{E}}}$ ${\displaystyle \varphi (\omega )=0^{\circ }}$ ${\displaystyle \varphi (\omega )=180^{\circ }}$

The general solution of the equation of motion is called (always under the assumption ) ${\displaystyle \omega \neq \omega _{0}}$

${\displaystyle {x(t)=X(t)+x_{0}(t)=\left({\frac {\omega _{0}^{2}}{\omega _{0}^{2}-\omega ^{2}}}A_{\text{E}}\right)\cdot \sin \omega t+A_{0}\cdot \sin(\omega _{0}t-\varphi _{0})}}$.

The two free parameters are to be determined according to the initial conditions . Except in this case , there is a superposition of two harmonic oscillations, in the case of a beat that (theoretically) lasts for any length of time. ${\displaystyle A_{0}\ ,\varphi _{0}}$ ${\displaystyle x(0),\ {\dot {x}}(0)}$ ${\displaystyle A_{0}=0}$ ${\displaystyle \omega \approx \omega _{0}}$

Special and general solution in response

A special solution, suitable for the initial condition , is obtained as follows: In the above formula for the transient process from the rest position ${\displaystyle x(0)=0,\ {\dot {x}}(0)=0}$

${\displaystyle x(t)=A(\omega _{d})\left(1-e^{-\gamma t}\right)\sin \omega _{d}t}$.

can be substituted for the borderline case : ${\displaystyle \gamma \rightarrow 0}$

${\displaystyle \omega _{d}\rightarrow \omega _{0}\;;\quad A(\omega _{d})\rightarrow {\frac {F_{0}}{2m\omega _{0}\gamma }}\;;\quad 1-e^{-\gamma t}\rightarrow \gamma t}$.

It follows:

${\displaystyle x(t)={\frac {F_{0}}{2m\omega _{0}}}\cdot t\cdot \sin \omega _{0}t}$.

Accordingly, with resonant excitation from the rest position, the amplitude increases proportionally to the time, theoretically across all limits. For the general solution for any initial conditions, a free oscillation with suitable parameters must be added to this formula as above . ${\displaystyle x_{0}(t)=A_{0}\cdot \sin(\omega _{0}t-\varphi _{0})}$ ${\displaystyle A_{0}\ ,\varphi _{0}}$

Limit case free particle

Even without a restoring force, a body can perform a periodic movement if an external force acts on it accordingly. Examples of such "vibrations" are the sliding or rolling of an object on a surface when the friction is low and the surface does not remain horizontal with sufficient accuracy. Specifically: when a cup slips on the tray and you want to bring it to rest by inclining it in the opposite direction, or when the deck cargo has torn loose on a swaying ship, or when, in a patience game, you direct balls into a depression by tilting the playing surface are. The equation of motion (in one dimension, notation as above) is

${\displaystyle m{\ddot {x}}+c{\dot {x}}=F(t)}$.

It corresponds to that of the harmonic oscillator with natural frequency . ${\displaystyle \omega _{0}=0}$

A periodic excitation can e.g. B. can be realized by alternately tilting the surface. In the case of sinusoidal excitation, the statements made above for the forced oscillation apply, whereby this must be set and therefore always applies to the excitation frequency . The formula for the amplitude becomes: ${\displaystyle \omega _{0}=0}$ ${\displaystyle \omega >\omega _{0}}$

${\displaystyle A(\omega )={\frac {\frac {F_{0}}{m}}{\omega {\sqrt {\omega ^{2}+(2\gamma )^{2}}}}}}$.

The deflections become larger, the lower the excitation frequency. The “resonance catastrophe” will certainly occur if . It can not be prevented by damping . The movement is delayed compared to the force. The phase shift is given by ${\displaystyle A\rightarrow \infty }$ ${\displaystyle \omega \rightarrow 0}$ ${\displaystyle \gamma >0}$

${\displaystyle \varphi (\omega )=\arctan \left(-{\frac {2\gamma }{\omega }}\right)}$.

Accordingly, it is almost 180 ° with weak damping ( ) and decreases to 90 ° for strong damping ( ). ${\displaystyle \gamma \ll \omega }$ ${\displaystyle \gamma \gg \omega }$

literature

• Horst Stöcker: Pocket book of physics. 4th edition, Verlag Harry Deutsch, Frankfurt am Main, 2000, ISBN 3-8171-1628-4

Individual evidence

1. ↑ http://phys.org/news/2014-01-quantum-mechanics-efficiency-photosynthesis.html