What is the formula for periodic force

B.1. Vibrations

[2] periodic oscillation

Periodic oscillations are time-periodic changes in the state of a system in same Time intervals.

A system that is able to vibrate is called oscillator.

At unequal There is a time interval non-periodic oscillation in front.

Under Changes of state is to be understood as an alternating increase and decrease in mechanical, electrical or thermal states. Changes of state of a mechanical and electrical nature are primarily important for music.

[Example] The rotation of the earth, heartbeat and breathing are also periodic vibrations - albeit very slowly. The most vivid example of periodic oscillations is a clock pendulum. With the so-called seconds pendulum, a complete oscillation lasts two seconds. The period of oscillation of clock pendulums depends solely on their length and the force of gravity, but not on their mass.

In order for vibrations to arise, the system has to move out of its equilibrium state (Rest position) and there must be forces that move the system back towards the rest position (Restoring forces).

Under Rest position one understands the state of an oscillator before an external disturbance takes place.

Restoring forces are the forces present in the oscillator that are directed towards the rest position.

An oscillator that has been brought out of the rest position tends to return to the rest position.

Every mechanical vibration is a constant conversion of kinetic into potential energy and vice versa.

[Example] The pendulum movement makes this clear: When passing through the rest position, the pendulum has its highest speed and therefore only kinetic energy. It turns into potential energy when the pendulum moves towards the highest point of its swing and thereby slows down. At the turning point of the movement, the speed has dropped to zero and the kinetic energy has thus been completely converted into potential energy. This process is repeated as long as the pendulum is swinging.

One differentiates:

freeVibrations:

when the oscillator is excited once, it oscillates at its natural frequency. It is not connected to other oscillators that can affect the oscillation.

[Example] When the piano strings are struck, they vibrate with their natural frequencies.

forcedVibrations (→ [7]):

in which the oscillator is constantly excited by an external periodic force

[Example] Vibrations of the resonance body of a violin with a sustained tone

mutedVibrations (→ [4]):

in which the oscillator constantly releases energy to the environment through friction

[Example] A struck bell releases its vibrational energy through friction to the surrounding air, which makes it increasingly quieter.

self-excitedVibration:

when the energy lost during a period is fed back to an oscillator from an internal energy source (feedback → [13])

[Example] Clock pendulum that is constantly stimulated to vibrate by the steering wheel and armature. The reed of a clarinet, which opens and closes periodically, also provides for the replacement of the lost energy and thus maintains the oscillation.

Fig. [2] -1: Relationship between free, forced, damped and undamped oscillation

The deflection of the oscillator at a certain point in time (i.e. its distance from the rest position) is called Elongation.

Formula symbols: y Unit: m [meter or derived unit]

In the rest position, the amount of elongation is zero. The elongation is an instantaneous value that depends on the time of observation.

The amplitude is the amount of the greatest elongation (= maximum value of the elongation).

Formula symbols: A. Unit: m [meter or derived unit]

[2-1]    A. = yMax.

A period is a complete vibration.

If successive oscillations are the same in their course, one speaks of purely periodic oscillation.

Under Period duration one understands the time interval between two identical successive oscillation states.

Formula symbols: T Unit: s [second]

The frequency is the number of periods per second. It indicates how often a periodic process is repeated per second.

Formula symbols: f Unit: Hz [Hertz = s-1]

For particularly high frequencies, the unit used is kHz [kilohertz = 1,000 Hz], MHz [megahertz = 1 million Hz] or GHz [gigahertz = 1 billion hertz].

Frequency and period can be related:

Frequency = reciprocal of the period

Period duration = reciprocal value of the frequency

[2-2]                     

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[3] Sinusoidal oscillation

The simplest form of vibration is that Sine wave. The course of a sinusoidal oscillation can be described mathematically with a uniform circular movement (rotation). The circular movement is projected onto a plane perpendicular to the circular path. A point P performs a circular movement, with the perpendicular distance to the diameter of the circle constantly changing. This distance corresponds to the elongation.

[Example] The circling of the pedals while cycling - looking in the direction of travel - shows a regular up and down movement.

The elongation y is at any point in time:

[3-1]                 

Fig. [3] -1: Projection of the circular movement into the plane (sine function)

A is the radius of the circle (amplitude) and at the same time the largest perpendicular distance from the diameter of the circle.

With Phase angle the angle between the radius and the diameter of the circle is called.

Formula symbols: Unit: rad [radian]

A complete oscillation (i.e. a complete revolution) corresponds to an angle of 360 ° (in radians: 2π)

There is a vibration T Takes seconds, gets in any time t the angle

[3-2]                 

run through.

The Angular frequency is the quotient of a complete rotation (2π) and the period.

Formula symbols:ω Unit: rad / s [radians per second]

[3-3]                 

Under Phase shift one understands the value of the phase angle at the point in time t0 (Beginning of observation of the oscillation):

[3-4]                 

Depending on the value of the phase shift, the phase is called leading ( > 0) or lagging ( < 0).

The elongation with phase shift can be expressed as a function of time:

[3-5]                 

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[4] damped oscillation

When an oscillation is excited, energy is supplied to the oscillator. As soon as the excitation ceases, the oscillator loses the energy supplied, essentially through friction. The mechanical energy of the vibration is converted into thermal energy. The result is that the amplitude of the oscillation steadily decreases and the oscillator finally stops oscillating. This process is called damping.

Fig. [4] -1: damped oscillation

Most often, the decrease in amplitude takes place in the form of a geometric series. The quotient of two successive amplitudes represents a constant that one Damping ratio is called.

Formula symbols:k Unit 1

[4-1]                 

The natural logarithm of the damping ratio becomes logarithmic decrement called.

Formula symbols:Λ Unit: 1 Dimension: Np [Neper]

[4-2]                 

The following applies to small attenuation:         d : Loss factor

The Loss factor is defined:

Formula symbols:dUnit 1

Loss factor = double the degree of damping

[4-3]                 

The quality is the reciprocal of the loss factor:

Formula symbols:Q Unit 1

[4-4]                 

If the Brigg logarithm of the damping ratio is given, one speaks of Decay measure.

Formula symbols:Λ Unit: 1 Measure: dB [decibel]

[4-5]                 k : Damping ratio

The period of a damped oscillation is less than the period of the corresponding undamped oscillation.

For the elongation of a damped oscillation at the point in time t applies:

[4-6]                 

The Degree of damping is the parameter for the damping of a vibration system.

Formula symbols:D. Unit: s [second]

[4-7]                 

Depending on the value of the degree of damping, one speaks of

Vibration case

D. < 1

Creep fall

D. > 1

there is no oscillation, the amplitude decreases monotonically

aperiodic borderline case

D. = 1

there is no longer any vibration

If the oscillation comes to rest after half a period due to very large damping, one speaks of aperiodic oscillation.

Fig. [4] -2

The Cooldown of a vibration system is the time in which the amplitude of a vibration has dropped to 1 / e (= 0.37; e: Euler's number) of the initial value. In acoustics, the reverberation time is usually specified.

Formula symbols:τ Unit: s [second]

[4-8]                 

The Decay coefficient is the quotient of the damping coefficient and double the mass:

Formula symbols:δ Unit: 1 / s [per second]

[4-9]                 

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[5] composite vibration

Compound vibrations arise through Superposition (additive superposition) of two or more sinusoidal oscillations (Partial vibrations). The result is a resulting oscillation (Resultant).

The superposition of sinusoidal oscillations of the same frequency is called interference. The resultant is also sinusoidal.

Two oscillations with the same phase are called in phase, with different phase than out of phase to the Phase difference.

[5-1]       

Fig. [5] -1: Superposition of two vibrations with
same frequency and phase but different amplitude

If two sine waves of the same frequency with the same phase and the same amplitude are superimposed, the resultant has twice the amplitudeReinforcement).

[5-2]                 

With the superposition of two sine waves of the same frequency with the same amplitude but a phase difference from π, 3π, 5π, ... (Phase opposition), the amplitude of the resultant is zero, i.e. H. the two vibrations cancel each other out (Extinction).

[5-3]                 

[Experiment] A tuning fork consists of two bars that vibrate against each other. The sound waves emitted by the two rods are superimposed.If a struck tuning fork is rotated around its axis in front of the ear, the sound waves amplify or cancel each other out, which is noticeable in a permanent change in volume during the rotation.

When superimposed sinusoidal oscillations of unequal frequency, the resultant is in most cases not sinusoidal.

Become two vibrations, their frequencies f1 and f2 differ only slightly, superimposed, the amplitude of the resultant changes periodically (Beat). The number of changes per second is called Beat frequency (fs). The beat frequency is the amount of the difference between the two partial oscillations and occurs as Envelope curve, which envelops the amplitudes of the resultants in appearance. The following applies:

[5-4]       

The frequency of the resultant fr is the mean value of the frequencies of the two partial oscillations.

[5-5]       

Fig. [5] -2: beat

The resultant shows in the beating minima Phase jumps (Rotation of the phase by 180 °).

The Beat duration is the time interval between two minimum amplitudes of the beat (reciprocal of the beat frequency).

Formula symbols:TsUnit: s [second]

[5-6]                 

If the amplitudes of the two partial oscillations are the same, they cancel each other out at phase opposition. With in-phase, the amplitude of the resultant reaches twice the amplitude of one of the two partial oscillations (perfect beat).

A beat can be heard at a beat frequency of 1 - 8 Hz as a constantly fluctuating change in volume. The two partial vibrations are not perceived separately, but as only one. If the number of phase jumps per second increases over 10, the ear feels an increasing "roughness" until two different tones are finally perceived.

If the amplitudes of the partial oscillations are unequal, the amplitude of the resultant goes back to the minimum of the difference in the amplitudes of the two partial oscillations.

The phenomenon of beating can be used when tuning musical instruments: if both frequencies are brought closer together until the beating frequency is 0 Hz, they are of the same frequency (Beat zero).

Compound vibrations can be represented graphically in two ways:

  • as Oscillogram: in a coordinate system the time is shown on the x-axis and the elongation of the partial oscillations and the resulting oscillation on the y-axis as a curve
  • as Frequency spectrum: in a coordinate system the frequencies of the partial oscillations are on the x-axis and their amplitudes on the y-axis as Spectral lines shown

At a Sonogram both the time function (i.e. the time-dependent change) and the composition of a sound to be examined are shown: the different thicknesses of the spectral lines show the amplitude curve of the corresponding frequencies.

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[6] harmonic analysis

With the help of harmonic analysis (Fourier analysis) any periodic, non-sinusoidal oscillation is clearly broken down into a sum of (mostly infinitely many) harmonic oscillations.

It is assumed that the vibration to be considered consists of a Fundamental oscillation with a certain basic frequency and Harmonics is composed of frequencies that are integral multiples of the fundamental frequency.

The fundamental is also referred to as the 1st harmonic, the 1st, 2nd, 3rd, ..., nth harmonic as the 2nd, 3rd, 4th, ..., nth harmonic.

The elongation of a movement is at a given point in time t (Fourier series):

[6-1]       

with the Fourier coefficients:

[6-2]       

[6-3]       

The phases of the partial oscillations are not taken into account, which anyway have no practical significance for the acoustic-physiological perception of composite oscillations. The human ear generally does not perceive phase shifts between partial oscillations: one and the same sound is perceived.

Special composite vibrations are sawtooth, square and triangular vibrations:

Fig. [6] -1: Sawtooth, square and triangular oscillation

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[7] forced oscillation

Becomes a body through the action of an external force F.ext. stimulated to vibrate and then left to its own devices, he performs Natural vibrations a certain Natural frequencyfe, which depends on the oscillating mass and the amount of the restoring force. Every oscillatory system has its own natural frequency. The lower the damping of the system, the stronger the effect of the natural frequency.

The amplitude of the forced oscillation is proportional to the amplitude of the external force.

If the vibration is excited periodically with a Excitation frequencyfa, the oscillator assumes this frequency and the natural frequency is suppressed. An oscillator excited in this way is called Resonator. The vibrations that it performs are forced vibrations. The time that elapses until the resonator has assumed the excitation frequency is called Settling process. As long as the forced oscillation does not change, one speaks of stationary phase.

Fig. [7] -1: Vibration curve of a forced vibration

The transient process plays a decisive role in recognizing instrument tones.

Under Phase shift one understands the phase difference between excitation and forced oscillation.

If the stimulating force disappears, the body continues to oscillate at its natural frequency and is normally dampened. This process is called Decay process.

If the excitation frequency increases, the amplitude of the resonator initially increases. When the excitation frequency corresponds to the natural frequency, the maximum amplitude has been reached. This point is called resonance between natural oscillation and forced oscillation. If the excitation frequency is increased further, the amplitude of the forced oscillation decreases again.

With a damped system, the amplitude maximum is not only reached when the natural frequency is reached, but rather earlier. (→ Figure [2] -1, bottom right)

Matter can be destroyed by resonance (Resonance disaster).

[Example] Kidney stone shattered - bridge collapse due to wind

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© 2005 Everard Sigal