Oscillation
As vibrations or oscillations (Latin oscillare, rocking ') are called repeated temporal fluctuations of state variables of a system. As a variation, the deviation from a mean value will be referred to. Oscillations can occur in all feedback systems. Examples of oscillations in mechanics, electrical engineering, biology, economy ( pig cycle ) and can be found in many other areas.
We distinguish:
- Periodic and non-periodic ( quasi-periodic or chaotic ) vibrations
- Damped and undamped oscillations
- Free, forced ( separately excited ), self-excited and parameter- excited vibrations
- Linear and non-linear oscillations
- Vibration with one degree of freedom, with several degrees of freedom and with infinitely many degrees of freedom ( vibrations of a continuum )
- Continuous vibration and oscillation between discrete states
All these features may be combined.
A periodic mechanical vibration of bodies is called vibration. A vibration, which is used for transmission of information signal is sometimes called, for example, electrical signal. The spatial propagation of a disturbance or vibration is a wave.
- 4.1 Forced oscillations
- 4.2 Self-excited oscillations
- 4.3 Parameter excited vibrations
Harmonic oscillation
An oscillation is referred to as a harmonic, whose temporal profile may be described by a sine function.
The graph shows a harmonic oscillation with the deflection, the amplitude and the period.
The deflection current is at a time, the amplitude at the maximum possible value of the variable. The period or the period of oscillation is the time that elapses while precisely oscillating mechanical system undergoes an oscillation period, ie after it is back in the same vibrational state. The reciprocal of the period T, the frequency f, that is: . Instead of f also the Greek letter (pronounced " nu " ) will be used. The unit of frequency is the hertz ( 1 Hz = 1 s -1).
A sustained oscillation is harmonic, if the return value (eg the restoring force) is proportional to the deflection, for example, a spring pendulum. Here, one also speaks of a harmonic oscillator or a linear system, since the restoring force varies linearly with the deflection: Doubles these, also the restoring force doubled.
Such vibration can be described by
With
The times of the frequency, the angular frequency of the oscillation. By using the angular frequency results in a more compact notation:
Linear damped oscillation
Macroscopic physical systems are always damped. As they leave, for example, by friction energy to the environment, the amplitude of their oscillation over time decreases. If you leave such a system itself ( free vibration ), so this eventually leads to a "standstill", as is apparent from the second law of thermodynamics. So perpetual motion (see conservation of energy ) is not possible.
If you compare the equation of motion of a spring pendulum with a frequency proportional to speed damping on, we obtain the following differential equation:
It is
( For torsional vibration shall be replaced by the moment of inertia and by the deflection angle. )
This is a homogeneous linear ordinary differential equation of 2nd order, based on the general form
Can bring, if the (positive) abbreviations for the decay constant
And the undamped natural angular frequency
Introduces whose meanings are clear only in the interpretation of the solution.
In the classical way of solving such a linear homogeneous differential equation ( alternatively you can use methods of operational calculus ) can be measured using the
With possibly complex parameters two linearly independent solutions are found which form a fundamental system. Inserted into the differential equation yields:
In this equation can be equal to zero only the parenthetical expression. One obtains the so-called characteristic equation for determining the constant:
This is a quadratic equation whose discriminant
Determines whether it has two real solutions, two complex conjugate solutions or a so-called double root. Therefore a case distinction is required.
The theory of linear differential equations shows that the general solution of the homogeneous differential equation is a linear combination of the two determined solutions. Has the characteristic equation two solutions (that is, the discriminant equal to 0), then can the general solution of the equation of motion written as follows:
The two (generally complex ) constants and represent the two remaining degrees of freedom of the general solution. By establishing two initial conditions (eg and / or ) the two constants for a specific case must be clarified.
Resonant case
A vibration there can be only when the losses are small. Then the root expression is with the discriminant is negative, imaginary and we obtain two complex conjugate solutions:
With the damped natural angular frequency:
Results in shorter:
Thus one obtains
With the help of Euler's formulas can also specify in trigonometric form of the solution of the homogeneous differential equation. This is purely real and practically better interpretable:
Or
Here too in each case the two constants, respectively, and to determine the initial conditions. In particular, the last form is easy to interpret as " ringing ".
By setting the two initial conditions and the two constants can be eliminated. Starting from the first trigonometric form we obtain the concrete is dependent on two initial conditions solution
If the decay constant is zero, the amplitude remains constant. The oscillation is the undamped frequency circuit.
Aperiodic limiting case
The limit from the vibration is no longer possible forms of the aperiodic limiting case (respectively). The solution then contains no sine function. Since applies to a second independent solution must be constructed in other ways. The result is
Kriechfall
At high damping, ie for results in the Kriechfall whose solution is composed of two exponential functions with two real:
Frequency spectrum of vibration
A vibration can be considered instead of time-dependent change as a function in frequency space. Mathematical transformation called the Fourier transform. The information content remains intact and therefore can always be reconstructed from a frequency spectrum by inverse transformation the corresponding time-dependent oscillation. Background for this consideration is that each vibration can be by an additive superposition ( superposition ) represent harmonic oscillations of different frequency. The superposition of two harmonic oscillations is called the beat.
Excitation of an oscillation
Forced oscillations
Free oscillations resulting from an oscillatory system that left after a disturbance / displacement itself, oscillating (or in the case of critical or supercritical damping crawling ) returns to the equilibrium state; see above. The frequency of the free vibration, the natural frequency of the vibrator. Wherein oscillations at a plurality of degrees of freedom, there are a corresponding number of resonant frequencies.
Forced oscillations resulting from an oscillator which is excited by time varying external influence to swing ( forced ). Practical significance are mainly periodic excitations and including the harmonic sinusoidal excitation. The frequency of the periodic excitation is called the excitation frequency. There are also mehrfrequente excitations or excitations by random processes.
In the case of harmonic excitation linear system generally results in two oscillations simultaneously:
- The free oscillation ( with the natural frequency or more natural frequencies ), whose size depends on the initial conditions and which decays by the always present damping during the transient and
- The forced vibration to the excitation frequency at constant excitation intensity. The amplitude of this oscillation is constant after the end of the transient. The ratio between the amplitude and the strength of excitation is quantified by the magnification function.
In engineering mechanics, the main excitation mechanisms are the Weger impulse, the force excitation and the unbalance excitation ( see close function).
The amplitude of the forced oscillation takes in the case of resonance at a maximum. In the absence of damping and equality of ( a ) and excitation frequency ( a ) natural frequency, the amplitude becomes infinite. With increasing damping value, the resonance point shifts slightly and the resonance amplitude decreases.
Self-excited oscillations
Vibration systems where the power supply is controlled by a suitable control and the vibration process itself, perform self-excited vibrations, and are called oscillator. In the differential equations, this phenomenon affects so that the attenuation value is zero. A typical example in the field of mechanics are the vibrations of the strings of a violin. These are caused by the fact that the static friction between the sheet and the string is greater than the sliding friction and the sliding friction with increasing difference in speed or decrease. Other examples are the sounds of glasses by rubbing the rim and electronic clock ( oscillator circuit ).
Self-excited vibrations are increasing in amplitude up to the disproportionately increasing the amplitude attenuation compensates for the energy input or the oscillating system will be destroyed.
Parameters induced vibrations
A parameter- excited oscillation then occurs when parameters change of the vibration system ( inertia sizes, damping values, or spring constants) periodically, such as on the swings.
Linear and nonlinear vibrations
Linear oscillations are characterized in that they can be described by differential equations, in which all dependencies from the vibrating size and their time derivatives are linear. For non-linear vibration that is not the case. Nonlinear vibrations are therefore not strictly sinusoidal. Of greater practical importance, that when a driven oscillator the resonance behavior of forced vibrations and changes the amplitude of self-excited oscillations remain limited.
Nonlinear systems are often not integrable. This means that the differential equation (s ) have no analytical solution. The vibration behavior of such systems is therefore usually studied by numerical computer simulations. One of the first experiments was the Fermi pasta Ulam experiment, in which a string vibration was observed with a non-linear error term. The solution of such systems usually gives a quasi-periodic or chaotic oscillation, where the behavior ( quasi-periodic or chaotic ) often depends on the energy of the oscillation. A non-linear system which allows no chaotic behavior is of the van der Pol oscillator. Chaotic behavior can be observed for example in a double pendulum.
Oscillations at several degrees of freedom
Vibration with one degree of freedom are those which can be completely described with a vibrating size. An example of this are vibrations of the planar string pendulum. If we let the pendulum spatial movements like a Foucault's pendulum, as it is already an oscillator with two degrees of freedom. In the following we restrict ourselves to the consideration of small deflections.
On this example we can see that the term may depend as vibration of the variables considered, ie the choice of the generalized coordinates. Thus, for deflecting the pendulum, so that the oscillation occurs in a plane. Are you the pendulum additionally an initial velocity perpendicular to the deflection, as can be observed elliptical orbits or a circular motion with constant angular velocity.
Considering deflection angle of the pendulum from the side from two different directions, one obtains two harmonic oscillations of the same period. A superimposing two harmonic oscillations are called Lissajous figures. Another possibility is to consider the pendulum from the top and record distance from the rest position and the direction of the deflection as a continuous distance to the starting angle. In the case of a circular orbit are both no more vibrations.
The number of degrees of freedom of a mechanical system having a plurality of compositions which can move independently, the sum of all degrees of freedom. Other examples of vibration with multiple degrees of freedom are torsional vibrations of a crankshaft or the horizontal oscillations of a multi-storey building under seismic impact.
Some oscillations of a system with several degrees of freedom can be considered as several independent oscillations with a suitable choice of coordinates. For a vibration that can be described by differential equations, this means that the equation of the individual coordinates decouple. If the individual vibrations periodically, can then be decoupled from the differential equations, the natural frequencies of the system determine. Can be written as an integer multiple of a constant all natural frequencies, as well as the vibration of the whole system is periodic.
For nonlinear vibration systems, a decoupling of the differential equations in closed form is usually not possible. However, there are approximations to enable starting of a linearization of the differential equations, an iterative solution.
Vibrations of a continuum
Can be described by wave oscillations of a continuum. These waves can be reflected, if the medium changes. Within the vibrating body thus takes place an overlay, which leads to the standing waves. In one dimension, for example, a string vibration of a violin or two dimensions, the vibration of a diaphragm such as in a loudspeaker. Such a wave can be described mathematically by an infinite number of coupled oscillators, ie describe a system with infinitely many degrees of freedom. In contrast to a system with finitely many degrees of freedom, therefore, has a harmonic oscillator continuous no finite number of possible natural frequencies, but a fundamental frequency and an infinite number of harmonics. Such vibration will then be described by its frequency spectrum.
Of practical interest in the art are further the vibrations of rods, plates and bowls. A cantilever beam has many degrees of freedom of oscillation, which differ not only in their resonance frequencies, but also by the nature of its motion.
The cantilever bending vibrator vibrates in the first overtone
The same beam performs torsional oscillations of the lowest frequency
The same bar can also perform torsional vibrations of higher frequency
Other examples
In everyday life, we encounter vibrations, for example, musical instruments and the clock pendulum, but also in quartz crystal clocks or clock generation in other electronic devices.
The atoms or molecules in a crystal lattice able to oscillate about an equilibrium position and thus generate, for example, characteristic absorption spectra.
Oscillating reactions set the pace for breathing and the heartbeat.
In electron tubes microphonic is often observed. It is caused by externally applied to the components interfering mechanical oscillations about by close to it standing speakers.
As a regenerative effect is known in manufacturing technology oscillations that occur during the manufacturing process within a machine.
In geology and meteorology smaller and with a certain regularity of fluctuations in sea level, the Eisrandlagen, the Erdkrustenstücke, the earth's magnetic field or the climate is observed.
In economics, the Goodwin model is used to explain business cycles.
The Lotka -Volterra equations describe approximately the fluctuations of predator and prey populations.