En mecánica clásica , un oscilador armónico es un sistema que, cuando se desplaza de su posición de equilibrio , experimenta una fuerza restauradora F proporcional al desplazamiento x :
donde k es una constante positiva .
Si F es la única fuerza que actúa sobre el sistema, el sistema se denomina oscilador armónico simple , y experimenta un movimiento armónico simple : oscilaciones sinusoidales alrededor del punto de equilibrio, con una amplitud constante y una frecuencia constante (que no depende de la amplitud ).
Si también está presente una fuerza de fricción ( amortiguación ) proporcional a la velocidad , el oscilador armónico se describe como un oscilador amortiguado . Dependiendo del coeficiente de fricción, el sistema puede:
- Oscila con una frecuencia más baja que en el caso no amortiguado y una amplitud que disminuye con el tiempo ( oscilador subamortiguado ).
- Decaimiento a la posición de equilibrio, sin oscilaciones ( oscilador sobreamortiguado ).
La solución límite entre un oscilador subamortiguado y un oscilador sobreamortiguado se produce en un valor particular del coeficiente de fricción y se denomina críticamente amortiguado .
Si está presente una fuerza externa dependiente del tiempo, el oscilador armónico se describe como un oscilador accionado .
Los ejemplos mecánicos incluyen péndulos (con pequeños ángulos de desplazamiento ), masas conectadas a resortes y sistemas acústicos . Otros sistemas análogos incluyen osciladores armónicos eléctricos como circuitos RLC . El modelo de oscilador armónico es muy importante en física, porque cualquier masa sujeta a una fuerza en equilibrio estable actúa como un oscilador armónico para pequeñas vibraciones. Los osciladores armónicos ocurren ampliamente en la naturaleza y se explotan en muchos dispositivos artificiales, como relojes y circuitos de radio. Son la fuente de prácticamente todas las vibraciones y ondas sinusoidales.
Oscilador armónico simple
Un oscilador armónico simple es un oscilador que no es impulsado ni amortiguado . Consiste en una masa m , que experimenta una sola fuerza F , que tira de la masa en la dirección del punto x = 0 y depende solo de la posición x de la masa y una constante k . El equilibrio de fuerzas ( segunda ley de Newton ) para el sistema es
Resolviendo esta ecuación diferencial , encontramos que el movimiento es descrito por la función
dónde
El movimiento es periódica , se repite en una sinusoidal de la manera con amplitud constante A . Además de su amplitud, el movimiento de un oscilador armónico simple se caracteriza por su período , el tiempo para una sola oscilación o su frecuencia , el número de ciclos por unidad de tiempo. La posición en un tiempo determinado t también depende de la fase φ , que determina el punto de inicio de la onda sinusoidal. El período y la frecuencia están determinados por el tamaño de la masa my la constante de fuerza k , mientras que la amplitud y la fase están determinadas por la posición inicial y la velocidad .
La velocidad y la aceleración de un oscilador armónico simple oscilan con la misma frecuencia que la posición, pero con fases desplazadas. La velocidad es máxima para el desplazamiento cero, mientras que la aceleración es en la dirección opuesta al desplazamiento.
La energía potencial almacenada en un oscilador armónico simple en la posición x es
Oscilador armónico amortiguado
En osciladores reales, la fricción o la amortiguación ralentizan el movimiento del sistema. Debido a la fuerza de fricción, la velocidad disminuye en proporción a la fuerza de fricción que actúa. Mientras que en un oscilador armónico simple no impulsado la única fuerza que actúa sobre la masa es la fuerza de restauración, en un oscilador armónico amortiguado hay además una fuerza de fricción que siempre está en una dirección opuesta al movimiento. En muchos sistemas vibratorios, la fuerza de fricción F f puede modelarse como proporcional a la velocidad v del objeto: F f = - cv , donde c se denomina coeficiente de amortiguación viscoso .
El equilibrio de fuerzas ( segunda ley de Newton ) para osciladores armónicos amortiguados es entonces
which can be rewritten into the form
where
- is called the "undamped angular frequency of the oscillator",
- is called the "damping ratio".
The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
- Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly.
- Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero). This is often desired for the damping of systems such as doors.
- Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by the exponential decay of the underdamped harmonic oscillator is given by
The Q factor of a damped oscillator is defined as
Q is related to the damping ratio by the equation
Osciladores armónicos impulsados
Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).
Newton's second law takes the form
It is usually rewritten into the form
This equation can be solved exactly for any driving force, using the solutions z(t) that satisfy the unforced equation
and which can be expressed as damped sinusoidal oscillations:
in the case where ζ ≤ 1. The amplitude A and phase φ determine the behavior needed to match the initial conditions.
Step input
In the case ζ < 1 and a unit step input with x(0) = 0:
the solution is
with phase φ given by
The time an oscillator needs to adapt to changed external conditions is of the order τ = 1/(ζω0). In physics, the adaptation is called relaxation, and τ is called the relaxation time.
In electrical engineering, a multiple of τ is called the settling time, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term overshoot refers to the extent the response maximum exceeds final value, and undershoot refers to the extent the response falls below final value for times following the response maximum.
Sinusoidal driving force
In the case of a sinusoidal driving force:
where is the driving amplitude, and is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC-driven RLC circuits (resistor–inductor–capacitor) and driven spring systems having internal mechanical resistance or external air resistance.
The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude , driving frequency , undamped angular frequency , and the damping ratio .
The steady-state solution is proportional to the driving force with an induced phase change :
where
is the absolute value of the impedance or linear response function, and
is the phase of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).
For a particular driving frequency called the resonance, or resonant frequency , the amplitude (for a given ) is maximal. This resonance effect only occurs when , i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.
The transient solutions are the same as the unforced () damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored.
Osciladores paramétricos
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing.[4][5][6] A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency and damping .
Parametric oscillators are used in many applications. The classical varactor parametric oscillator oscillates when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, waveguide/YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.
Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For example, the Optical parametric oscillator converts an input laser wave into two output waves of lower frequency ().
Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability phenomenon.
Ecuación del oscilador universal
The equation
is known as the universal oscillator equation, since all second-order linear oscillatory systems can be reduced to this form.[citation needed] This is done through nondimensionalization.
If the forcing function is f(t) = cos(ωt) = cos(ωtcτ) = cos(ωτ), where ω = ωtc, the equation becomes
The solution to this differential equation contains two parts: the "transient" and the "steady-state".
Transient solution
The solution based on solving the ordinary differential equation is for arbitrary constants c1 and c2
The transient solution is independent of the forcing function.
Steady-state solution
Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:
Supposing the solution is of the form
Its derivatives from zeroth to second order are
Substituting these quantities into the differential equation gives
Dividing by the exponential term on the left results in
Equating the real and imaginary parts results in two independent equations
Amplitude part
Squaring both equations and adding them together gives
Therefore,
Compare this result with the theory section on resonance, as well as the "magnitude part" of the RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Phase part
To solve for φ, divide both equations to get
This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Full solution
Combining the amplitude and phase portions results in the steady-state solution
The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions:
For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.
Sistemas equivalentes
Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see universal oscillator equation above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators – their output waveform, resonant frequency, damping factor, etc. – are the same.
Translational mechanical | Rotational mechanical | Series RLC circuit | Parallel RLC circuit |
---|---|---|---|
Position | Angle | Charge | Flux linkage |
Velocity | Angular velocity | Current | Voltage |
Mass | Moment of inertia | Inductance | Capacitance |
Momentum | Angular momentum | Flux linkage | Charge |
Spring constant | Torsion constant | Elastance | Magnetic reluctance |
Damping | Rotational friction | Resistance | Conductance |
Drive force | Drive torque | Voltage | Current |
Undamped resonant frequency : | |||
Damping ratio : | |||
Differential equation: | |||
Aplicación a una fuerza conservadora
The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator.
A conservative force is one that is associated with a potential energy. The potential-energy function of a harmonic oscillator is
Given an arbitrary potential-energy function , one can do a Taylor expansion in terms of around an energy minimum () to model the behavior of small perturbations from equilibrium.
Because is a minimum, the first derivative evaluated at must be zero, so the linear term drops out:
The constant term V(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
Thus, given an arbitrary potential-energy function with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.
Ejemplos de
Simple pendulum
Assuming no damping, the differential equation governing a simple pendulum of length , where is the local acceleration of gravity, is
If the maximal displacement of the pendulum is small, we can use the approximation and instead consider the equation
The general solution to this differential equation is
where and are constants that depend on the initial conditions. Using as initial conditions and , the solution is given by
where is the largest angle attained by the pendulum (that is, is the amplitude of the pendulum). The period, the time for one complete oscillation, is given by the expression
which is a good approximation of the actual period when is small. Notice that in this approximation the period is independent of the amplitude . In the above equation, represents the angular frequency.
Spring/mass system
When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.
By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
the latter being Newton's second law of motion.
If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by
Given an ideal massless spring, is the mass on the end of the spring. If the spring itself has mass, its effective mass must be included in .
Energy variation in the spring–damping system
In terms of energy, all systems have two types of energy: potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which is then transferred into kinetic energy. The potential energy within a spring is determined by the equation
When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.
Definición de términos
Symbol | Definition | Dimensions | SI units |
---|---|---|---|
Acceleration of mass | m/s2 | ||
Peak amplitude of oscillation | m | ||
Viscous damping coefficient | N·s/m | ||
Frequency | Hz | ||
Drive force | N | ||
Acceleration of gravity at the Earth's surface | m/s2 | ||
Imaginary unit, | — | — | |
Spring constant | N/m | ||
Mass | kg | ||
Quality factor | — | — | |
Period of oscillation | s | ||
Time | s | ||
Potential energy stored in oscillator | J | ||
Position of mass | m | ||
Damping ratio | — | — | |
Phase shift | — | rad | |
Angular frequency | rad/s | ||
Natural resonant angular frequency | rad/s |
Ver también
- Anharmonic oscillator
- Critical speed
- Effective mass (spring-mass system)
- Normal mode
- Parametric oscillator
- Phasor
- Q factor
- Quantum harmonic oscillator
- Radial harmonic oscillator
- Elastic pendulum
Notas
- ^ Fowles & Cassiday (1986, p. 86)
- ^ Kreyszig (1972, p. 65)
- ^ Tipler (1998, pp. 369,389)
- ^ Case, William. "Two ways of driving a child's swing". Archived from the original on 9 December 2011. Retrieved 27 November 2011.
- ^ Case, W. B. (1996). "The pumping of a swing from the standing position". American Journal of Physics. 64 (3): 215–220. Bibcode:1996AmJPh..64..215C. doi:10.1119/1.18209.
- ^ Roura, P.; Gonzalez, J.A. (2010). "Towards a more realistic description of swing pumping due to the exchange of angular momentum". European Journal of Physics. 31 (5): 1195–1207. Bibcode:2010EJPh...31.1195R. doi:10.1088/0143-0807/31/5/020.
Referencias
- Fowles, Grant R.; Cassiday, George L. (1986), Analytic Mechanics (5th ed.), Fort Worth: Saunders College Publishing, ISBN 0-03-96746-5, LCCN 93085193CS1 maint: ignored ISBN errors (link)
- Hayek, Sabih I. (15 Apr 2003). "Mechanical Vibration and Damping". Encyclopedia of Applied Physics. WILEY-VCH Verlag GmbH & Co KGaA. doi:10.1002/3527600434.eap231. ISBN 9783527600434.
- Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8
- Serway, Raymond A.; Jewett, John W. (2003). Physics for Scientists and Engineers. Brooks/Cole. ISBN 0-534-40842-7.
- Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1 (4th ed.). W. H. Freeman. ISBN 1-57259-492-6.
- Wylie, C. R. (1975). Advanced Engineering Mathematics (4th ed.). McGraw-Hill. ISBN 0-07-072180-7.
enlaces externos
- The Harmonic Oscillator from The Feynman Lectures on Physics
- "Oscillator, harmonic", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Harmonic Oscillator from The Chaos Hypertextbook
- A Java applet of harmonic oscillator with damping proportional to velocity or damping caused by dry friction
- Damped Harmonic Oszillator Detailed solution from beltoforion.de