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In mechanics and physics, simple harmonic motion (sometimes abbreviated as SHM) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position.
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 ...
Figure 2: A simple harmonic oscillator with small periodic damping term given by ¨ + ˙ + =, =, ˙ =; =.The numerical simulation of the original equation (blue solid line) is compared with averaging system (orange dashed line) and the crude averaged system (green dash-dotted line). The left plot displays the solution evolved in time and ...
In physics, a system with a set of conservative forces and an equilibrium point can be approximated as a harmonic oscillator near equilibrium. An example of this is the Lennard-Jones potential, where the potential is given by: = [() ()]
Natural frequency, measured in terms of eigenfrequency, is the rate at which an oscillatory system tends to oscillate in the absence of disturbance. A foundational example pertains to simple harmonic oscillators, such as an idealized spring with no energy loss wherein the system exhibits constant-amplitude oscillations with a constant frequency.
The Kepler problem and the simple harmonic oscillator problem are the two most fundamental problems in classical mechanics. They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (Bertrand's theorem). [1]: 92
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
The mass–spring–damper model is an example of a simple harmonic oscillator. The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the RLC circuit. Note: This article does not include the step-by-step mathematical derivations, but focuses on major vibration analysis equations and concepts.