ࡱ>  q` RΉbjbjqPqP8::4HHHHHHHdL܇5,xxxxx$hHgxxggHHxx'''gVHxHx'g''nHHx Ըe5. 0 2 @ Hs0"'ң gggg555.d!555d\DHHHHHH Simple harmonic motion-  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" http://en.wikipedia.org/wiki/Simple_harmonic_motion, From Wikipedia, the free encyclopedia  HYPERLINK "http://en.wikipedia.org/wiki/Image:Simple_harmonic_motion_animation.gif" \o "Simple harmonic motion."  INCLUDEPICTURE "http://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Simple_harmonic_motion_animation.gif/180px-Simple_harmonic_motion_animation.gif" \* MERGEFORMATINET   HYPERLINK "http://en.wikipedia.org/wiki/Image:Simple_harmonic_motion_animation.gif" \o "Enlarge"  Animation file: Simple harmonic motion. Simple harmonic motion is the motion of a  HYPERLINK "http://en.wikipedia.org/wiki/Harmonic_oscillator" \l "Simple_harmonic_oscillator" \o "Harmonic oscillator" simple harmonic oscillator, a motion that is neither driven nor  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \o "Damping" damped. The motion is  HYPERLINK "http://en.wikipedia.org/wiki/Periodic_function" \o "Periodic function" periodic, as it repeats itself at standard intervals in a specific manner - described as being  HYPERLINK "http://en.wikipedia.org/wiki/Sine_wave" \o "Sine wave" sinusoidal, with constant amplitude. It is characterized by its  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude (which is always positive), its  HYPERLINK "http://en.wikipedia.org/wiki/Periodicity" \o "Periodicity" period which is the time for a single oscillation, its  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency which is the number of cycles per unit time, and its  HYPERLINK "http://en.wikipedia.org/wiki/Phase_%28waves%29" \o "Phase (waves)" phase, which determines the starting point on the sine wave. The period, and its inverse the frequency, are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and  HYPERLINK "http://en.wikipedia.org/wiki/Velocity" \o "Velocity" velocity) of that system. Contents  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \l "Overview#Overview" 1 Overview  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \l "Mathematics#Mathematics" 2 Mathematics  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \l "Examples#Examples" 3 Examples  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \l "See_also#See_also" 4 See also  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \l "External_links#External_links" 5 External links 1. Overview Simple harmonic motion is defined by the  HYPERLINK "http://en.wikipedia.org/wiki/Differential_equation" \o "Differential equation" differential equation, , where k is a positive constant. In words simple harmonic motion is "motion where the acceleration of a body is proportional to, and opposite in direction to the displacement from its equilibrium position". The above differential equation has a  HYPERLINK "http://en.wikipedia.org/wiki/Sinusoid" \o "Sinusoid" sinusoidal solution, pictured below.  HYPERLINK "http://en.wikipedia.org/wiki/Image:Periodampwave.svg" \o "Periodampwave.svg"  A general equation describing simple harmonic motion is , where x is the  HYPERLINK "http://en.wikipedia.org/wiki/Displacement" \o "Displacement" displacement, A is the  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude of oscillation, f is the  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency, t is the elapsed time, and  is the phase of oscillation. If there is no displacement at time t = 0, the phase  = 0. A motion with frequency f has  HYPERLINK "http://en.wikipedia.org/wiki/Period" \o "Period" period . Simple harmonic motion can serve as a  HYPERLINK "http://en.wikipedia.org/wiki/Mathematical_model" \o "Mathematical model" mathematical model of a variety of motions and provides the basis of the characterization of more complicated motions through the techniques of  HYPERLINK "http://en.wikipedia.org/wiki/Fourier_analysis" \o "Fourier analysis" Fourier analysis. Mathematics It can be shown, by  HYPERLINK "http://en.wikipedia.org/wiki/Derivative" \o "Derivative" differentiating, exactly how the acceleration varies with time. Using the  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" angular frequency , defined as the displacement is given by the function Differentiating once gives an expression for the velocity at any time. And once again to get the acceleration at a given time. These results can of course be simplified, giving us an expression for acceleration in terms of displacement. a(t) = " 2x(t). Examples  HYPERLINK "http://en.wikipedia.org/wiki/Image:Simple_harmonic_oscillator.gif" \o "An undamped spring-mass system undergoes simple harmonic motion."  INCLUDEPICTURE "http://upload.wikimedia.org/wikipedia/commons/9/9d/Simple_harmonic_oscillator.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://upload.wikimedia.org/wikipedia/commons/9/9d/Simple_pendulum_height.png" \* MERGEFORMATINET  An undamped  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \o "Spring-mass system" spring-mass system undergoes simple harmonic motion. Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples: Mass on a spring: A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with Alternately, if the other factors are known and the period is to be found, this equation can be used: The total energy is constant, and given by where E is the total energy. Uniform circular motion: Simple harmonic motion can in some cases be considered to be the one-dimensional  HYPERLINK "http://en.wikipedia.org/wiki/Projection" \o "Projection" projection of  HYPERLINK "http://en.wikipedia.org/wiki/Uniform_circular_motion" \o "Uniform circular motion" uniform circular motion. If an object moves with angular frequency  around a circle of radius R centered at the  HYPERLINK "http://en.wikipedia.org/wiki/Origin" \o "Origin" origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude R and angular speed . Mass on a pendulum: In the  HYPERLINK "http://en.wikipedia.org/wiki/Small-angle_approximation" \o "Small-angle approximation" small-angle approximation, the motion of a pendulum is approximated by simple harmonic motion. The period of a mass attached to a string of length with gravitational acceleration g is given by This approximation is accurate only in small angles because of the expression for  HYPERLINK "http://en.wikipedia.org/wiki/Angular_acceleration" \o "Angular acceleration" angular acceleration being proportional to the sine of position: where I is the  HYPERLINK "http://en.wikipedia.org/wiki/Moment_of_inertia" \o "Moment of inertia" moment of inertia; in this case . When  is small, and therefore the expression becomes which makes angular acceleration directly proportional to , satisfying the definition of Simple Harmonic Motion. For a solution not relying on a small-angle approximation, see  HYPERLINK "http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29" \o "Pendulum (mathematics)" pendulum (mathematics). See also  HYPERLINK "http://en.wikipedia.org/wiki/Isochronous" \o "Isochronous" Isochronous  HYPERLINK "http://en.wikipedia.org/wiki/Uniform_circular_motion" \o "Uniform circular motion" Uniform circular motion  HYPERLINK "http://en.wikipedia.org/wiki/Complex_harmonic_motion" \o "Complex harmonic motion" Complex harmonic motion  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \o "Damping" Damping External links  HYPERLINK "http://www.phy.hk/wiki/englishhtm/SpringSHM.htm" \o "http://www.phy.hk/wiki/englishhtm/SpringSHM.htm" Java simulation of spring-mass oscillator  HYPERLINK "http://www.acoustics.salford.ac.uk/feschools/index.htm" \o "http://www.acoustics.salford.ac.uk/feschools/index.htm" Explanations and animations of Simple Harmonic Motion Harmonic oscillator From Wikipedia, the free encyclopedia (Redirected from  HYPERLINK "http://en.wikipedia.org/w/index.php?title=Spring-mass_system&redirect=no" \o "Spring-mass system" Spring-mass system) Jump to:  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "column-one#column-one" navigation,  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "searchInput#searchInput" search This article is about the harmonic oscillator in classical mechanics. For its use in  HYPERLINK "http://en.wikipedia.org/wiki/Quantum_mechanics" \o "Quantum mechanics" quantum mechanics, see  HYPERLINK "http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" \o "Quantum harmonic oscillator" quantum harmonic oscillator.  HYPERLINK "http://en.wikipedia.org/wiki/Image:Simple_harmonic_oscillator.gif" \o "An undamped spring-mass system is a simple harmonic oscillator."  INCLUDEPICTURE "http://upload.wikimedia.org/wikipedia/commons/9/9d/Simple_harmonic_oscillator.gif" \* MERGEFORMATINET  An undamped spring-mass system is a simple harmonic oscillator. In  HYPERLINK "http://en.wikipedia.org/wiki/Classical_mechanics" \o "Classical mechanics" classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring  HYPERLINK "http://en.wikipedia.org/wiki/Force" \o "Force" force F proportional to the displacement x according to  HYPERLINK "http://en.wikipedia.org/wiki/Hooke%27s_law" \o "Hooke's law" Hooke's law: where k is a positive  HYPERLINK "http://en.wikipedia.org/wiki/Constant" \o "Constant" constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \o "Simple harmonic motion" simple harmonic motion:  HYPERLINK "http://en.wikipedia.org/wiki/Sinusoidal" \o "Sinusoidal" sinusoidal  HYPERLINK "http://en.wikipedia.org/wiki/Oscillations" \o "Oscillations" oscillations about the equilibrium point, with a constant  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude and a constant  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency (which does not depend on the  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude). If a frictional  HYPERLINK "http://en.wikipedia.org/wiki/Force" \o "Force" force ( HYPERLINK "http://en.wikipedia.org/wiki/Damping" \o "Damping" damping) proportional to the  HYPERLINK "http://en.wikipedia.org/wiki/Velocity" \o "Velocity" velocity is also present, the harmonic oscillator is described as a damped oscillator. In such situation, the  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency of the  HYPERLINK "http://en.wikipedia.org/wiki/Oscillations" \o "Oscillations" oscillations is smaller than in the non-damped case, and the  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude of the  HYPERLINK "http://en.wikipedia.org/wiki/Oscillations" \o "Oscillations" oscillations decreases with time. If an external time-dependent  HYPERLINK "http://en.wikipedia.org/wiki/Force" \o "Force" force is present, the harmonic oscillator is described as a driven oscillator. Mechanical examples include  HYPERLINK "http://en.wikipedia.org/wiki/Pendulum" \o "Pendulum" pendula (with small angles of displacement), masses connected to  HYPERLINK "http://en.wikipedia.org/wiki/Spring_%28device%29" \o "Spring (device)" springs, and  HYPERLINK "http://en.wikipedia.org/wiki/Acoustics" \o "Acoustics" acoustical systems. Other analogous systems include electrical harmonic oscillators such as  HYPERLINK "http://en.wikipedia.org/wiki/RLC_circuit" \o "RLC circuit" RLC circuits (see  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Equivalent_systems#Equivalent_systems" \o "" Equivalent systems below). Contents  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Simple_harmonic_oscillator#Simple_harmonic_oscillator" 1 Simple harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Driven_harmonic_oscillator#Driven_harmonic_oscillator" 2 Driven harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Damped_harmonic_oscillator#Damped_harmonic_oscillator" 3 Damped harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Damped.2C_driven_harmonic_oscillator#Damped.2C_driven_harmonic_oscillator" 4 Damped, driven harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Full_mathematical_definition#Full_mathematical_definition" 5 Full mathematical definition  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Important_terms#Important_terms" 5.1 Important terms  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Simple_harmonic_oscillator_2#Simple_harmonic_oscillator_2" 6 Simple harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Universal_oscillator_equation#Universal_oscillator_equation" 7 Universal oscillator equation  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Transient_solution#Transient_solution" 7.1 Transient solution  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Steady-state_solution#Steady-state_solution" 7.2 Steady-state solution  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Amplitude_part#Amplitude_part" 7.2.1 Amplitude part  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Phase_part#Phase_part" 7.2.2 Phase part  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Full_solution#Full_solution" 7.3 Full solution  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Equivalent_systems#Equivalent_systems" 8 Equivalent systems  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Applications#Applications" 9 Applications  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Examples#Examples" 10 Examples  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Simple_pendulum#Simple_pendulum" 10.1 Simple pendulum  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Pendulum_swinging_over_turntable#Pendulum_swinging_over_turntable" 10.2 Pendulum swinging over turntable  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Spring-mass_system#Spring-mass_system" 10.3 Spring-mass system  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "References#References" 11 References  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "See_also#See_also" 12 See also  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "External_links#External_links" 13 External links Simple harmonic oscillator The simple harmonic oscillator has no driving force, and no  HYPERLINK "http://en.wikipedia.org/wiki/Friction" \o "Friction" friction ( HYPERLINK "http://en.wikipedia.org/wiki/Damping" \o "Damping" damping), so the net force is just: Using  HYPERLINK "http://en.wikipedia.org/wiki/Newton%27s_Second_Law" \o "Newton's Second Law" Newton's Second Law The acceleration, a is equal to the second derivative of x. If we define , then the equation can be written as follows, We observe that: and substituting integrating where K is the  HYPERLINK "http://en.wikipedia.org/wiki/Integration_constant" \o "Integration constant" integration constant, set K = (A 0)2 integrating, the results (including integration constant ) are and has the general solution where the  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude and the  HYPERLINK "http://en.wikipedia.org/wiki/Phase_%28waves%29" \o "Phase (waves)" phase are determined by the initial conditions. Alternatively, the general solution can be written as where the value of is shifted by relative to the previous form; or as where and are the constants which are determined by the initial conditions, instead of and in the previous forms. The  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency of the oscillations is given by The  HYPERLINK "http://en.wikipedia.org/wiki/Kinetic_energy" \o "Kinetic energy" kinetic energy is . and the  HYPERLINK "http://en.wikipedia.org/wiki/Potential_energy" \o "Potential energy" potential energy is so the  HYPERLINK "http://en.wikipedia.org/wiki/Total_energy" \o "Total energy" total energy of the system has the constant value Driven harmonic oscillator A driven harmonic oscillator satisfies the nonhomogeneous second order  HYPERLINK "http://en.wikipedia.org/wiki/Linear_differential_equation" \o "Linear differential equation" linear differential equation where A0 is the driving amplitude and  is the driving  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency for a sinusoidal driving mechanism. This type of system appears in  HYPERLINK "http://en.wikipedia.org/wiki/Alternating_current" \o "Alternating current" AC LC ( HYPERLINK "http://en.wikipedia.org/wiki/Inductor" \o "Inductor" inductor- HYPERLINK "http://en.wikipedia.org/wiki/Capacitor" \o "Capacitor" capacitor) circuits and idealized spring systems lacking internal mechanical resistance or external  HYPERLINK "http://en.wikipedia.org/wiki/Air_resistance" \o "Air resistance" air resistance. Damped harmonic oscillator Main article:  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \o "Damping" Damping A damped harmonic oscillator satisfies the second order differential equation where b is an experimentally determined damping constant satisfying the relationship F = " bv. An example of a system obeying this equation would be a weighted spring underwater if the damping force exerted by the water is assumed to be linearly proportional to v. The frequency of the damped harmonic oscillator is given by where Damped, driven harmonic oscillator This satisfies the equation The general solution is a sum of a  HYPERLINK "http://en.wikipedia.org/wiki/Transient" \o "Transient" transient (the solution for damped undriven harmonic oscillator,  HYPERLINK "http://en.wikipedia.org/wiki/Homogeneous_%28mathematics%29" \o "Homogeneous (mathematics)" homogeneous ODE) that depends on initial conditions, and a  HYPERLINK "http://en.wikipedia.org/wiki/Steady_state" \o "Steady state" steady state (particular solution of the nonhomogenous ODE) that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force, The steady-state solution is where is the absolute value of the  HYPERLINK "http://en.wikipedia.org/wiki/Mechanical_impedance" \o "Mechanical impedance" impedance or  HYPERLINK "http://en.wikipedia.org/wiki/Linear_response_function" \o "Linear response function" linear response function and is the  HYPERLINK "http://en.wikipedia.org/wiki/Phase_%28waves%29" \o "Phase (waves)" phase of the oscillation relative to the driving force. One might see that for a certain driving frequency, , the amplitude (relative to a given F0) is maximal. This occurs for the frequency and is called  HYPERLINK "http://en.wikipedia.org/wiki/Resonance" \o "Resonance" resonance of  HYPERLINK "http://en.wikipedia.org/wiki/Displacement_%28vector%29" \o "Displacement (vector)" displacement. In summary: at a steady state the frequency of the oscillation is the same as that of the driving force, but the oscillation is phase-offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred (resonant) frequency of the oscillating system. Example:  HYPERLINK "http://en.wikipedia.org/wiki/RLC_circuit" \o "RLC circuit" RLC circuit. Full mathematical definition Most harmonic oscillators, at least approximately, solve the differential equation: where t is time, b is the damping constant, o is the characteristic  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" angular frequency, and Aocos(t) represents something driving the system with amplitude Ao and angular frequency . x is the measurement that is oscillating; it can be position, current, or nearly anything else. The  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" angular frequency is related to the frequency, f, by Important terms  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" Amplitude: maximal displacement from the  HYPERLINK "http://en.wikipedia.org/wiki/Mechanical_equilibrium" \o "Mechanical equilibrium" equilibrium. Period: the time it takes the system to complete an oscillation cycle. Inverse of  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" frequency.  HYPERLINK "http://en.wikipedia.org/wiki/Frequency" \o "Frequency" Frequency: the number of cycles the system performs per unit time (usually measured in  HYPERLINK "http://en.wikipedia.org/wiki/Hertz" \o "Hertz" hertz = 1/s).  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" Angular frequency:  = 2f  HYPERLINK "http://en.wikipedia.org/wiki/Phase_%28waves%29" \o "Phase (waves)" Phase: how much of a cycle the system completed (system that begins is in phase zero, system which completed half a cycle is in phase ).  HYPERLINK "http://en.wikipedia.org/wiki/Initial_condition" \o "Initial condition" Initial conditions: the state of the system at t = 0, the beginning of oscillations. Simple harmonic oscillator A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is: Physically, the above never actually exists, since there will always be friction or some other resistance, but two approximate examples are a mass on a  HYPERLINK "http://en.wikipedia.org/wiki/Spring_%28device%29" \o "Spring (device)" spring and an  HYPERLINK "http://en.wikipedia.org/wiki/LC_circuit" \o "LC circuit" LC circuit. In the case of a mass attached to a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that: where k is the  HYPERLINK "http://en.wikipedia.org/wiki/Spring_constant" \o "Spring constant" spring constant m is the  HYPERLINK "http://en.wikipedia.org/wiki/Mass" \o "Mass" mass x is the position of the mass a is its  HYPERLINK "http://en.wikipedia.org/wiki/Acceleration" \o "Acceleration" acceleration. Because acceleration a is the second derivative of position x, we can rewrite the equation as follows: The most simple solution to the above  HYPERLINK "http://en.wikipedia.org/wiki/Differential_equation" \o "Differential equation" differential equation is and the second derivative of that is where A is the  HYPERLINK "http://en.wikipedia.org/wiki/Amplitude" \o "Amplitude" amplitude,  is the phase shift, and  is the  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" angular frequency. Plugging these back into the original differential equation, we have: Then, after dividing both sides by we get: or, as it is more commonly written: The above formula reveals that the  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" angular frequency  of the solution is only dependent upon the physical characteristics of the system, and not the initial conditions (those are represented by A and ). We will label this  as o from now on. This will become important later. Universal oscillator equation The equation is known as the universal oscillator equation since all second order linear oscillatory systems can be reduced to this form. This is done through  HYPERLINK "http://en.wikipedia.org/wiki/Nondimensionalization" \o "Nondimensionalization" nondimensionalization. If the forcing function is f(t) = cos(t) = cos(tc) = cos(), where  = tc, 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  HYPERLINK "http://en.wikipedia.org/wiki/Ordinary_differential_equation" \o "Ordinary differential equation" ordinary differential equation is for arbitrary constants c1 and c2 is The transient solution is independent of the forcing function. If the system is critically damped, the response is independent of the damping. Steady-state solution Apply the " HYPERLINK "http://en.wikipedia.org/wiki/Complex_analysis" \o "Complex analysis" complex variables method" by solving the  HYPERLINK "http://en.wikipedia.org/w/index.php?title=Auxiliary_equation&action=edit" \o "Auxiliary equation" auxiliary equation below and then finding the real part of its solution: Supposing the solution is of the form Its derivatives from zero to 2nd 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  HYPERLINK "http://en.wikipedia.org/wiki/Image:Harmonic_oscillator_gain.png" \o "Log-log plot of the frequency response of an ideal harmonic oscillator."   HYPERLINK "http://en.wikipedia.org/wiki/Image:Harmonic_oscillator_gain.png" \o "Enlarge"  Log-log plot of the frequency response of an ideal harmonic oscillator. Squaring both equations and adding them together gives By convention the positive root is taken since amplitude is usually considered a positive quantity. Therefore, Compare this result with the theory section on  HYPERLINK "http://en.wikipedia.org/wiki/Resonance" \o "Resonance" resonance, as well as the "magnitude part" of the  HYPERLINK "http://en.wikipedia.org/wiki/RLC_circuit" \o "RLC circuit" RLC circuit. This amplitude function is particularly important in the analysis and understanding of the  HYPERLINK "http://en.wikipedia.org/wiki/Frequency_response" \o "Frequency response" frequency response of second-order systems. Note that the variables in these equations ought to be identified before showing the equation. Phase part To solve for , divide both equations to get This phase function is particularly important in the analysis and understanding of the  HYPERLINK "http://en.wikipedia.org/wiki/Frequency_response" \o "Frequency response" 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  HYPERLINK "http://en.wikipedia.org/wiki/Superposition" \o "Superposition" superposition (sum) of the transient and steady-state solutions For a more complete description of how to solve the above equation, see  HYPERLINK "http://en.wikipedia.org/wiki/Ordinary_differential_equation" \l "Linear_ODEs_with_constant_coefficients" \o "Ordinary differential equation" linear ODEs with constant coefficients. Equivalent systems Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \l "Universal_oscillator_equation#Universal_oscillator_equation" \o "" 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 will be the same. Translational MechanicalTorsional MechanicalSeries RLC CircuitParallel RLC CircuitPosition Angle  HYPERLINK "http://en.wikipedia.org/wiki/Charge" \o "Charge" Charge  HYPERLINK "http://en.wikipedia.org/wiki/Voltage" \o "Voltage" Voltage  HYPERLINK "http://en.wikipedia.org/wiki/Velocity" \o "Velocity" Velocity  HYPERLINK "http://en.wikipedia.org/wiki/Angular_velocity" \o "Angular velocity" Angular velocity  HYPERLINK "http://en.wikipedia.org/wiki/Current" \o "Current" Current  HYPERLINK "http://en.wikipedia.org/wiki/Mass" \o "Mass" Mass  HYPERLINK "http://en.wikipedia.org/wiki/Moment_of_inertia" \o "Moment of inertia" Moment of inertia  HYPERLINK "http://en.wikipedia.org/wiki/Inductance" \o "Inductance" Inductance  HYPERLINK "http://en.wikipedia.org/wiki/Capacitance" \o "Capacitance" Capacitance  HYPERLINK "http://en.wikipedia.org/wiki/Hooke%27s_law" \o "Hooke's law" Spring constant  HYPERLINK "http://en.wikipedia.org/wiki/Torsion_spring" \l "Torsion_coefficient" \o "Torsion spring" Torsion constant  HYPERLINK "http://en.wikipedia.org/wiki/Capacitance" \l "Elastance" \o "Capacitance" Elastance  HYPERLINK "http://en.wikipedia.org/wiki/Susceptance" \o "Susceptance" Susceptance  HYPERLINK "http://en.wikipedia.org/wiki/Friction" \o "Friction" Friction  HYPERLINK "http://en.wikipedia.org/wiki/Torsion_spring" \l "Motion_of_torsion_balances_and_pendulums" \o "Torsion spring" Rotational friction  HYPERLINK "http://en.wikipedia.org/wiki/Electrical_resistance" \o "Electrical resistance" Resistance  HYPERLINK "http://en.wikipedia.org/wiki/Conductance" \o "Conductance" Conductance Drive  HYPERLINK "http://en.wikipedia.org/wiki/Force" \o "Force" force Drive  HYPERLINK "http://en.wikipedia.org/wiki/Torque" \o "Torque" torque Undamped  HYPERLINK "http://en.wikipedia.org/wiki/Resonance" \o "Resonance" resonant frequency :Differential equation:Applications The problem of the simple harmonic oscillator occurs frequently in physics because of the form of its potential energy function: Given an arbitrary potential energy function V(x), one can do a  HYPERLINK "http://en.wikipedia.org/wiki/Taylor_series" \o "Taylor series" Taylor expansion in terms of x around an energy minimum (x = x0) to model the behavior of small perturbations from equilibrium. Because V(x0) is a minimum, the first derivative evaluated at x0 must be zero, so the linear term drops out: The  HYPERLINK "http://en.wikipedia.org/wiki/Constant_term" \o "Constant term" 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 V(x) 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. Examples Simple pendulum A  HYPERLINK "http://en.wikipedia.org/wiki/Simple_pendulum" \o "Simple pendulum" simple pendulum exhibits simple harmonic motion under the conditions of no damping and small amplitude. Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is The solution to this equation is given by: where 0 is the largest angle attained by the pendulum. The  HYPERLINK "http://en.wikipedia.org/wiki/Sine" \o "Sine" period, the time for one complete oscillation , is given by 2 divided by whatever is multiplying the time in the argument of the cosine Pendulum swinging over turntable Simple harmonic motion can in some cases be considered to be the one-dimensional  HYPERLINK "http://en.wikipedia.org/wiki/Projection" \o "Projection" projection of two-dimensional  HYPERLINK "http://en.wikipedia.org/wiki/Circular_motion" \o "Circular motion" circular motion. Consider a long  HYPERLINK "http://en.wikipedia.org/wiki/Pendulum" \o "Pendulum" pendulum swinging over the turntable of a  HYPERLINK "http://en.wikipedia.org/wiki/Record_player" \o "Record player" record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line. It is possible to change the frequency of rotation of the turntable in order to have a perfect  HYPERLINK "http://en.wikipedia.org/wiki/Synchronization" \o "Synchronization" synchronization with the motion of the pendulum. The angular speed of the turntable is the pulsation of the pendulum. In general, the  HYPERLINK "http://en.wikipedia.org/wiki/Pulsation" \o "Pulsation" pulsation-also known as angular frequency, of a straight-line simple harmonic motion is the  HYPERLINK "http://en.wikipedia.org/wiki/Angular_speed" \o "Angular speed" angular speed of the corresponding circular motion. Therefore, a motion with period T and frequency f=1/T has pulsation In general, pulsation and angular speed are not synonymous. For instance the pulsation of a pendulum is not the angular speed of the pendulum itself, but it is the angular speed of the corresponding circular motion. Spring-mass system Spring-mass system in equilibrium (A), compressed (B) and stretched (C) states. When a spring is stretched or compressed by a mass, the spring develops a restoring force.  HYPERLINK "http://en.wikipedia.org/wiki/Hooke%27s_law" \o "Hooke's law" 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. This relationship shows that the distance of the spring is always opposite to the force of the spring. 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: If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by: Energy variation in the spring-damper system In terms of energy, all systems have two types of energy,  HYPERLINK "http://en.wikipedia.org/wiki/Potential_energy" \o "Potential energy" potential energy and  HYPERLINK "http://en.wikipedia.org/wiki/Kinetic_energy" \o "Kinetic energy" kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is 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 maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, the spring will try to reach back to equilibrium, and all its potential energy is converted into kinetic energy of the mass. References Serway, Raymond A.; Jewett, John W. (2003). Physics for Scientists and Engineers. Brooks/Cole.  HYPERLINK "http://en.wikipedia.org/w/index.php?title=Special:Booksources&isbn=0534408427" ISBN 0-534-40842-7. Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1, 4th ed., W. H. Freeman.  HYPERLINK "http://en.wikipedia.org/w/index.php?title=Special:Booksources&isbn=1572594926" ISBN 1-57259-492-6. Wylie, C. R. (1975). Advanced Engineering Mathematics, 4th ed., McGraw-Hill.  HYPERLINK "http://en.wikipedia.org/w/index.php?title=Special:Booksources&isbn=0070721807" ISBN 0-07-072180-7. See also  HYPERLINK "http://en.wikipedia.org/wiki/Q_factor" \o "Q factor" Q factor  HYPERLINK "http://en.wikipedia.org/wiki/Normal_mode" \o "Normal mode" Normal mode  HYPERLINK "http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" \o "Quantum harmonic oscillator" Quantum harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Anharmonic_oscillator" \o "Anharmonic oscillator" Anharmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Parametric_oscillator" \o "Parametric oscillator" Parametric oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Critical_speed" \o "Critical speed" Critical speed External links  HYPERLINK "http://hypertextbook.com/chaos/41.shtml" \o "http://hypertextbook.com/chaos/41.shtml" Harmonic Oscillator from The Chaos Hypertextbook  HYPERLINK "http://planetphysics.org/encyclopedia/SimpleHarmonicOscillator.html" \o "http://planetphysics.org/encyclopedia/SimpleHarmonicOscillator.html" Simple Harmonic oscillator on  HYPERLINK "http://en.wikipedia.org/w/index.php?title=PlanetPhysics&action=edit" \o "PlanetPhysics" PlanetPhysics  HYPERLINK "http://www.stationcrafts.net/physics" \o "http://www.stationcrafts.net/physics" A-level Physics experiment on the subject of Damped Harmonic Motion with solution curve graphs Retrieved from " HYPERLINK "http://en.wikipedia.org/wiki/Harmonic_oscillator" \l "Spring-mass_system" http://en.wikipedia.org/wiki/Harmonic_oscillator#Spring-mass_system" Damping From Wikipedia, the free encyclopedia Jump to:  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "column-one#column-one" navigation,  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "searchInput#searchInput" search  HYPERLINK "http://en.wikipedia.org/wiki/Image:Mergefrom.svg" \o "Mergefrom.svg" It has been suggested that  HYPERLINK "http://en.wikipedia.org/wiki/Damping_ratio" \o "Damping ratio" Damping ratio be  HYPERLINK "http://en.wikipedia.org/wiki/Wikipedia:Merging_and_moving_pages" \o "Wikipedia:Merging and moving pages" merged into this article or section. ( HYPERLINK "http://en.wikipedia.org/wiki/Talk:Damping_ratio" \l "Merger_proposal" \o "Talk:Damping ratio" Discuss) HYPERLINK "http://en.wikipedia.org/wiki/Image:Damped_spring.gif" \o "A damped spring-mass system."  INCLUDEPICTURE "http://upload.wikimedia.org/wikipedia/commons/2/2b/Damped_spring.gif" \* MERGEFORMATINET  A damped  HYPERLINK "http://en.wikipedia.org/wiki/Spring-mass_system" \o "Spring-mass system" spring-mass system. Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of  HYPERLINK "http://en.wikipedia.org/wiki/Oscillation" \o "Oscillation" oscillations of an oscillatory system. Contents [ HYPERLINK "javascript:toggleToc()" hide]  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Definition#Definition" 1 Definition  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Example:_mass-spring-damper#Example:_mass-spring-damper" 2 Example: mass-spring-damper  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Differential_equation#Differential_equation" 2.1 Differential equation  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "System_behavior#System_behavior" 2.2 System behavior  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Critical_damping#Critical_damping" 2.2.1 Critical damping  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Over-damping#Over-damping" 2.2.2 Over-damping  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Under-damping#Under-damping" 2.2.3 Under-damping  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "Alternative_models#Alternative_models" 3 Alternative models  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "See_also#See_also" 4 See also  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "References#References" 5 References  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "External_links#External_links" 6 External links Definition In  HYPERLINK "http://en.wikipedia.org/wiki/Physics" \o "Physics" physics and  HYPERLINK "http://en.wikipedia.org/wiki/Engineering" \o "Engineering" engineering, damping may be  HYPERLINK "http://en.wikipedia.org/wiki/Mathematical_model" \o "Mathematical model" mathematically modelled as a  HYPERLINK "http://en.wikipedia.org/wiki/Force" \o "Force" force synchronous with the  HYPERLINK "http://en.wikipedia.org/wiki/Velocity" \o "Velocity" velocity of the object but opposite in direction to it. Thus, for a simple mechanical damper, the force F may be related to the velocity v by where c is the viscous damping coefficient, given in units of newton-seconds per meter. This relationship is perfectly analogous to  HYPERLINK "http://en.wikipedia.org/wiki/Electrical_resistance" \o "Electrical resistance" electrical resistance. See  HYPERLINK "http://en.wikipedia.org/wiki/Ohm%27s_law" \o "Ohm's law" Ohm's law. This force is an (raw) approximation to the  HYPERLINK "http://en.wikipedia.org/wiki/Friction" \o "Friction" friction caused by  HYPERLINK "http://en.wikipedia.org/wiki/Drag_%28physics%29" \o "Drag (physics)" drag. In playing stringed instruments such as  HYPERLINK "http://en.wikipedia.org/wiki/Guitar" \o "Guitar" guitar or  HYPERLINK "http://en.wikipedia.org/wiki/Violin" \o "Violin" violin, damping is the quieting or abrupt silencing of the strings after they have been sounded, by pressing with the edge of the palm, or other parts of the hand such as the fingers on one or more strings near the bridge of the instrument. The strings themselves can be modelled as a continuum of infinitesimally small mass-spring-damper systems where the damping constant is much smaller than the resonance frequency, creating damped oscillations (see below). See also  HYPERLINK "http://en.wikipedia.org/wiki/Vibrating_string" \o "Vibrating string" Vibrating string. Example: mass-spring-damper  HYPERLINK "http://en.wikipedia.org/wiki/Image:Mass-Spring-Damper.png" \o "A mass attached to a spring and a damper. The damping coefficient, usually c, is represented by B in this case. The F in the diagram denotes an external force, which this example does not include."   HYPERLINK "http://en.wikipedia.org/wiki/Image:Mass-Spring-Damper.png" \o "Enlarge"  A mass attached to a spring and a damper. The damping coefficient, usually c, is represented by B in this case. The F in the diagram denotes an external force, which this example does not include. An ideal mass-spring-damper system with mass m (in  HYPERLINK "http://en.wikipedia.org/wiki/Kilograms" \o "Kilograms" kilograms), spring constant k (in  HYPERLINK "http://en.wikipedia.org/wiki/Newton" \o "Newton" newtons per  HYPERLINK "http://en.wikipedia.org/wiki/Meter" \o "Meter" meter) and viscous damper of damping coeficient c (in  HYPERLINK "http://en.wikipedia.org/wiki/Newton-second" \o "Newton-second" newton-seconds per meter) can be described with the following formula: Fs = " kx Treating the mass as a  HYPERLINK "http://en.wikipedia.org/wiki/Free_body" \o "Free body" free body and applying  HYPERLINK "http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion" \l "Newton.27s_second_law:_Law_of_Motion" \o "Newton's laws of motion" Newton's second law, we have: where a is the  HYPERLINK "http://en.wikipedia.org/wiki/Acceleration" \o "Acceleration" acceleration (in meters per second squared) of the mass and x is the  HYPERLINK "http://en.wikipedia.org/wiki/Displacement_%28vector%29" \o "Displacement (vector)" displacement (in meters) of the mass relative to a fixed point of reference. Differential equation The above equations combine to form the equation of motion, a second-order  HYPERLINK "http://en.wikipedia.org/wiki/Differential_equation" \o "Differential equation" differential equation for displacement x as a function of time t (in  HYPERLINK "http://en.wikipedia.org/wiki/Second" \o "Second" seconds): Rearranging, we have Next, to simplify the equation, we define the following parameters: and The first parameter, 0, is called the (undamped)  HYPERLINK "http://en.wikipedia.org/wiki/Resonance" \o "Resonance" natural frequency of the system . The second parameter, , is called the  HYPERLINK "http://en.wikipedia.org/wiki/Damping_ratio" \o "Damping ratio" damping ratio. The natural frequency represents an  HYPERLINK "http://en.wikipedia.org/wiki/Angular_frequency" \o "Angular frequency" angular frequency, expressed in  HYPERLINK "http://en.wikipedia.org/wiki/Radian" \o "Radian" radians per second. The damping ratio is a  HYPERLINK "http://en.wikipedia.org/wiki/Dimensionless_quantity" \o "Dimensionless quantity" dimensionless quantity. The differential equation now becomes Continuing, we can solve the equation by assuming a solution x such that: where the  HYPERLINK "http://en.wikipedia.org/wiki/Parameter" \o "Parameter" parameter is, in general, a  HYPERLINK "http://en.wikipedia.org/wiki/Complex_number" \o "Complex number" complex number. 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!"$%rs'(*+ŭƭJKXY[\*+ůƯȯЯMNXY[\hL 0JmH sH jhL UmH sH hL 5CJ\aJmH sH hL mH sH h_Jv>*B*mH ph"sH h_Jv0JmH sH h_Jv0JmH sH h_JvmH sH jh_JvUmH sH =*[ȯЯ WKkd2$$If02!634a$If $Ifed -D`M gdL -D`M gdL -D`M gdL -D`M gd_Jv & Fdd-D`1$M [$\$gd_Jv  $%pq~!"ijklmwxͳγ٪٪ˢ󑅑t jhL B*UmH phsH hL B*mH phsH jhL B*UmH phsH hL CJaJ hL 0JhL 0J6] hL 6]jhL 6U]hL OJQJ^JhL jhL UhL mH sH hL 0JmH sH jhL UmH sH -nɴҴ^p߶S&R & F dd$1$If[$\$gdL  & F dd$1$If[$\$gdL  & F dd$1$If[$\$gdL $IfgdL $IfgdL -D`M gdL -D`M gdL XYɴҴӴԴNOPQ[\^_εϵеѵSTWXmnpqȶɶ̶Ͷܶݶ߶9:?@PQSTýӵӵӵӵӵӵӵӵӵӵӵӵhL 0J>*B*phhL 0J>*B*phjhL U hL 0JjhL 0JU hL 0JhL hL 0JmH sH jhL UmH sH hL mH sH hL 5\mH sH >#$&'BCDEOPRSǹ˹̹ abmnպĺhL 0JmH sH jhL UmH sH hL mH sH hL 0JmH sH hL CJaJhL OJQJ^J hL 0JhL 0J>*B*phhL jhL UhL 0J>*B*ph=ȹwoL;Wly &u-D`M ^gdL -D`M gdL -D`M gdL -D`M gdL 8kd'$$If 634aպֺ0167MN%&.Ibclmݽ޽DEIJtu '(89;VWXiklmhL 0JmH sH hL 6]mH sH hL 5\mH sH hL mH sH hL 0JmH sH jhL UmH sH L$%89 $%bcjkpq01?@yz{:<,-56&'34uɾhL 0JmH sH hL 0JH*mH sH hL 0J6]mH sH hL 0JmH sH hL 6]mH sH hL mH sH jhL UmH sH Gu23HI[\styz,.df BC#$YZcdwxT0TNT͵UhL 0J6]mH sH jhL 6U]mH sH hL H*mH sH hL 6]mH sH hL 0JmH sH jhL UmH sH hL mH sH hL 0JmH sH Au )0TPThUVVYYr\l]h^^_` -D`M gdL -D`M gdL -D`M gdL -D`M gdL Solving for , we find: System behavior  HYPERLINK "http://en.wikipedia.org/wiki/Image:Damping.svg" \o "Dependence of the system behavior on the value of the damping ratio ."   HYPERLINK "http://en.wikipedia.org/wiki/Image:Damping.svg" \o "Enlarge"  Dependence of the system behavior on the value of the damping ratio . The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency 0 and the damping ratio . In particular, the qualitative behavior of the system depends crucially on whether the  HYPERLINK "http://en.wikipedia.org/wiki/Quadratic_equation" \o "Quadratic equation" quadratic equation for  has one real solution, two real solutions, or two complex conjugate solutions. Critical damping When  = 1, (defined above) is real, the system is said to be critically damped. An example of critical damping is the door-closer seen on many hinged doors in public buildings. An automobile suspension has a damping near critical damping (slightly higher for "hard" suspensions and slightly less for "soft" ones) In this case, the solution simplifies to HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "_note-0#_note-0" \o "" [1]: where A and B are determined by the initial conditions of the system (usually the initial position and velocity of the mass): Over-damping When  > 1, is still real, but now the system is said to be over-damped. An over-damped door-closer will take longer to close the door than a critically damped door closer. The solution to the motion equation is HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "_note-1#_note-1" \o "" [2]: where A and B are determined by the initial conditions of the system: Under-damping Finally, when 0 d"  < 1, is  HYPERLINK "http://en.wikipedia.org/wiki/Complex_number" \o "Complex number" complex, and the system is under-damped. In this situation, the system will oscillate at the natural damped frequency , which is a function of the natural frequency and the damping ratio. In this case, the solution can be generally written as HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "_note-2#_note-2" \o "" [3]: where represents the natural damped frequency of the system, and A and B are again determined by the initial conditions of the system: Alternative models Viscous damping models, although widely used, are not the only damping models. A wide range of models can be found in specialized literature, but one of them should be referred here: the so called "hysteretic damping model" or "structural damping model". When a metal beam is vibrating, the internal damping can be better described by a force proportional to the displacement but in phase with the velocity. In such case, the  HYPERLINK "http://en.wikipedia.org/wiki/Differential_equation" \o "Differential equation" differential equation that describes the free movement of a single-degree-of-freedom system becomes: where h is the hysteretic damping coefficient and i denotes the  HYPERLINK "http://en.wikipedia.org/wiki/Imaginary_unit" \o "Imaginary unit" imaginary unit; the presence of i is required to synchronize the damping force to the velocity ( xi being in phase with the velocity). This equation is more often written as: where  is the hysteretic damping ratio, that is, the fraction of energy lost in each cycle of the vibration. Although requiring  HYPERLINK "http://en.wikipedia.org/wiki/Complex_analysis" \o "Complex analysis" complex analysis to solve the equation, this model reproduces the real behaviour of many vibrating structures more closely than the viscous model. See also:  HYPERLINK "http://en.wikipedia.org/wiki/Friction" \o "Friction" Frictionand  HYPERLINK "http://en.wikipedia.org/wiki/Drag_%28physics%29" \o "Drag (physics)" Drag See also  HYPERLINK "http://en.wikipedia.org/wiki/Damping_ratio" \o "Damping ratio" Damping ratio  HYPERLINK "http://en.wikipedia.org/wiki/Damping_factor" \o "Damping factor" Damping factor  HYPERLINK "http://en.wikipedia.org/wiki/RLC_circuit" \o "RLC circuit" RLC circuit  HYPERLINK "http://en.wikipedia.org/wiki/Oscillator" \o "Oscillator" Oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Harmonic_oscillator" \o "Harmonic oscillator" Harmonic oscillator  HYPERLINK "http://en.wikipedia.org/wiki/Simple_harmonic_motion" \o "Simple harmonic motion" Simple harmonic motion  HYPERLINK "http://en.wikipedia.org/wiki/Resonance" \o "Resonance" Resonance  HYPERLINK "http://en.wikipedia.org/wiki/Impulse_excitation_technique" \o "Impulse excitation technique" Impulse excitation technique  HYPERLINK "http://en.wikipedia.org/wiki/Audio_system_measurements" \o "Audio system measurements" Audio system measurements  HYPERLINK "http://en.wikipedia.org/wiki/Tuned_mass_damper" \o "Tuned mass damper" Tuned mass damper  HYPERLINK "http://en.wikipedia.org/wiki/Vehicle_suspension" \o "Vehicle suspension" Vehicle suspension  HYPERLINK "http://en.wikipedia.org/wiki/Vibration" \o "Vibration" Vibration References  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "_ref-0#_ref-0" \o "" ^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Critical Damping." From MathWorld--A Wolfram Web Resource.  HYPERLINK "http://mathworld.wolfram.com/DampedSimpleHarmonicMotionCriticalDamping.html" \o "http://mathworld.wolfram.com/DampedSimpleHarmonicMotionCriticalDamping.html" [1]  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "_ref-1#_ref-1" \o "" ^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Overdamping." From MathWorld--A Wolfram Web Resource.  HYPERLINK "http://mathworld.wolfram.com/DampedSimpleHarmonicMotionOverdamping.html" \o "http://mathworld.wolfram.com/DampedSimpleHarmonicMotionOverdamping.html" [2]  HYPERLINK "http://en.wikipedia.org/wiki/Damping" \l "_ref-2#_ref-2" \o "" ^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Underdamping." From MathWorld--A Wolfram Web Resource.  HYPERLINK "http://mathworld.wolfram.com/DampedSimpleHarmonicMotionUnderdamping.html" \o "http://mathworld.wolfram.com/DampedSimpleHarmonicMotionUnderdamping.html" [3] External links  HYPERLINK "http://en.wikipedia.org/wiki/Image:Wiktionary-logo-en.png" \o "Wiktionary-logo-en.png"  Look up  HYPERLINK "http://en.wiktionary.org/wiki/Special:Search/damping" \o "wiktionary:Special:Search/damping" Damping in  HYPERLINK "http://en.wikipedia.org/wiki/Wiktionary" \o "Wiktionary" Wiktionary, the free dictionary.  HYPERLINK "http://www.sengpielaudio.com/calculator-bridging.htm" \o "http://www.sengpielaudio.com/calculator-bridging.htm" Calculation of the matching attenuation,the damping factor, and the damping of bridging  HYPERLINK "http://www.stationcrafts.net/physics" \o "http://www.stationcrafts.net/physics" A-level Physics experiment on the subject of Damped Harmonic Motion with solution curve graphs NTPTRTbUfUhUjUUVVVxWzW\X^XY Y.Y0Y:Y}xnx{B/h>}PNG  IHDRU >'PLTE/>8C?MF]ETFUKbWsWt_~-JT2NX1S^7T^@T[FTYS_ciiiesxwvvw}fkloopsuxy{|Ӄ9bKGDgIFgS7SgIFxNETSCAPE2.0$NP cmPPJCmp0712HsIDAThC[0pw@`)tC LQƦ;t_Pʙ4)}fR~y& Ik ;6lw6A˞NeO#Bt/dkvV)1>nhPXNTw:R$3[xTuvNTGpuqB2MRvO9y+UѫCL#yb hѣ0d)73߷bCGz\d)n j8 Y#T4МqR#V1,"0zoH64]t 0bSV@ cW3xq@--F/0ӵy'N Q 葝TL=1 ==j̊3ow8K7;^ /`LiW]?Tsg==s΁n A ă^p~W?{'>z޶W?HX;C@ofm fMs$GMHAVg{Y#hqtd\ YG\RMeGs^b/]1Kkqb080]'mh̴ч3wN:<[d'dpc~G`QGXD-&wcTf{1̦69Ճ @Ǐ]o-?t;nSchE &5ܖ `i7^τ4 )46`s~i5C՛*lN>kչ^E$}YȾ6O5<]lYtWrg]:{C%8:Z/:-!QY&i*ffR\Z<_ɴ< ssqmsOGMSOFFICE9.0GIF89aU/>8C?MF]ETFUKbWsWt_~-JT2NX1S^7T^@T[FTYS_ciiiesxwvvw}fkloopsuxy{|Ӄ!! 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