This article is about detailed information about the harmonic oscillator in classical mechanics. For an introduction to the simple harmonic oscillator, see simple harmonic motion. For an introduction to damped harmonic motion, see damping. For its uses in quantum mechanics, see quantum harmonic oscillator.
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An undamped spring-mass system is a simple harmonic oscillator.In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).
Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator).
If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
Contents [hide]
1 Simple harmonic oscillator
2 Damped harmonic oscillator
3 Driven harmonic oscillators
3.1 Step input
3.2 Sinusoidal driving force
4 Parametric oscillators
5 Universal oscillator equation
5.1 Transient solution
5.2 Steady-state solution
5.2.1 Amplitude part
5.2.2 Phase part
5.3 Full solution
6 Equivalent systems
7 Applications
8 Examples
8.1 Simple pendulum
8.2 Pendulum swinging over turntable
8.3 Spring–mass system
8.3.1 Energy variation in the spring–damping system
9 In-line notes and references
10 Further reading
11 See also
12 External links
[edit] Simple harmonic oscillator
Main article: Simple harmonic motion
Simple harmonic motion.A simple harmonic oscillator is an oscillator that is neither driven nor damped. Its motion is periodic— repeating itself in a sinusoidal fashion with constant amplitude, A. Simple harmonic motion SHM can serve as a mathematical model of a variety of motions, such as a pendulum with small amplitudes and a mass on a spring. It also provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.
In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation, its frequency, f, the reciprocal of the period f = 1⁄T (i.e. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system. Overall then, the equation describing simple harmonic motion is
.
Alternatively a cosine can be used in place of the sine with the phase shifted by π⁄2.
The general differential force equation for an object of mass m experiencing SHM is:
,
where k is the spring constant which relates the displacement of the object to the force applied to the object. The general solution for this equation is given above with the frequency of the oscillations given by:
.
The velocity (green arrow) and acceleration (red arrow) oscillate with a quarter and half a period delay from the displacement (black arrow)The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.
The potential energy of SHM is:
.
[edit] Damped harmonic oscillator
Main article: Damping
Dependence of the system behavior on the value of the damping ratio ζ.In real oscillators friction, or damping, slows the motion of the system. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is the viscous damping coefficient.
Similar damped oscillator behavior occurs for a diverse range of disciplines that include control engineering, mechanical engineering and electrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, the speed of an electric motor, or the current through an RLC circuit. Generally, damped harmonic oscillators satisfy:
,
where ω0 is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio. For a mass on a spring having a spring constant k and a damping coefficient c, ω0 = √k/m and ζ = c/2mω0.
Step-response of a damped harmonic oscillator; curves are plotted for three values of μ = ω1 = ω0√1−ζ2. Time is in units of the decay time τ = 1/(ζω0).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 equilibrium without oscillating. Larger values of the damping ratio ζ return to equilibrium slower.
Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. 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
[edit] Driven harmonic oscillators
Driven harmonic oscillators are damped oscillators driven by an externally applied force F(t), for example, a step function, an impulse function or a continuous sinusoidal force. In general, driven harmonic oscillators satisfy the nonhomogeneous second order linear differential equation:
This equation can be solved exactly for any driving force using the solutions z(t) to the unforced equation, which satisfy:
and which can be expressed as a damped sinusoidal oscillation:
in the case where ζ ≤ 1. The amplitude A and phase φ determine the behavior needed to match the initial conditions.
Classical mechanics
Newton's Second Law
History of ... [show]Branches
Statics · Dynamics / Kinetics · Kinematics · Applied mechanics · Celestial mechanics · Continuum mechanics · Statistical mechanics
[show]Formulations
Newtonian mechanics
Lagrangian mechanics
Hamiltonian mechanics
[show]Fundamental concepts
Space · Time · Velocity · Speed · Mass · Acceleration · Gravity · Force · Torque / Moment / Couple · Momentum · Angular momentum · Inertia · Moment of inertia · Reference frame · Energy · Mechanical work · Virtual work · D'Alembert's principle
[show]Core topics
Rigid body · Rigid body dynamics · Motion · Newton's laws of motion · Newton's law of universal gravitation · Equations of motion · Inertial frame of reference · Non-inertial reference frame · Rotating reference frame · Fictitious force · Displacement (vector) · Relative velocity · Friction · Simple harmonic motion · Harmonic oscillator · Vibration · Damping · Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Non-uniform circular motion · Centripetal force · Centrifugal force · Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement
[show]Scientists
Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut · Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson
This box: view • talk • edit
An undamped spring-mass system is a simple harmonic oscillator.In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).
Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator).
If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
Contents [hide]
1 Simple harmonic oscillator
2 Damped harmonic oscillator
3 Driven harmonic oscillators
3.1 Step input
3.2 Sinusoidal driving force
4 Parametric oscillators
5 Universal oscillator equation
5.1 Transient solution
5.2 Steady-state solution
5.2.1 Amplitude part
5.2.2 Phase part
5.3 Full solution
6 Equivalent systems
7 Applications
8 Examples
8.1 Simple pendulum
8.2 Pendulum swinging over turntable
8.3 Spring–mass system
8.3.1 Energy variation in the spring–damping system
9 In-line notes and references
10 Further reading
11 See also
12 External links
[edit] Simple harmonic oscillator
Main article: Simple harmonic motion
Simple harmonic motion.A simple harmonic oscillator is an oscillator that is neither driven nor damped. Its motion is periodic— repeating itself in a sinusoidal fashion with constant amplitude, A. Simple harmonic motion SHM can serve as a mathematical model of a variety of motions, such as a pendulum with small amplitudes and a mass on a spring. It also provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.
In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation, its frequency, f, the reciprocal of the period f = 1⁄T (i.e. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system. Overall then, the equation describing simple harmonic motion is
.
Alternatively a cosine can be used in place of the sine with the phase shifted by π⁄2.
The general differential force equation for an object of mass m experiencing SHM is:
,
where k is the spring constant which relates the displacement of the object to the force applied to the object. The general solution for this equation is given above with the frequency of the oscillations given by:
.
The velocity (green arrow) and acceleration (red arrow) oscillate with a quarter and half a period delay from the displacement (black arrow)The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.
The potential energy of SHM is:
.
[edit] Damped harmonic oscillator
Main article: Damping
Dependence of the system behavior on the value of the damping ratio ζ.In real oscillators friction, or damping, slows the motion of the system. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is the viscous damping coefficient.
Similar damped oscillator behavior occurs for a diverse range of disciplines that include control engineering, mechanical engineering and electrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, the speed of an electric motor, or the current through an RLC circuit. Generally, damped harmonic oscillators satisfy:
,
where ω0 is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio. For a mass on a spring having a spring constant k and a damping coefficient c, ω0 = √k/m and ζ = c/2mω0.
Step-response of a damped harmonic oscillator; curves are plotted for three values of μ = ω1 = ω0√1−ζ2. Time is in units of the decay time τ = 1/(ζω0).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 equilibrium without oscillating. Larger values of the damping ratio ζ return to equilibrium slower.
Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. 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
[edit] Driven harmonic oscillators
Driven harmonic oscillators are damped oscillators driven by an externally applied force F(t), for example, a step function, an impulse function or a continuous sinusoidal force. In general, driven harmonic oscillators satisfy the nonhomogeneous second order linear differential equation:
This equation can be solved exactly for any driving force using the solutions z(t) to the unforced equation, which satisfy:
and which can be expressed as a damped sinusoidal oscillation:
in the case where ζ ≤ 1. The amplitude A and phase φ determine the behavior needed to match the initial conditions.