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In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion an object experiences due to a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely (if uninhibited by friction or any other dissipation of energy).

Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle approximation). Simple harmonic motion can also be used to model molecular vibration.

Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of Fourier analysis.

Introduction

The motion of a particle moving along a straight line with an acceleration whose direction is always towards a fixed point on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.^[1]

In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law.

Mathematically, the restoring force F is given by

where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m⁻¹), and x is the displacement from the equilibrium position (m).

For any simple mechanical harmonic oscillator:

• When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the mass has momentum because of the acceleration that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has no energy loss, the mass continues to oscillate. Thus simple harmonic motion is a type of periodic motion. If energy is lost in the system, then the mass exhibits damped oscillation.

Note if the real space and phase space plot are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum.

Dynamics

In Newtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's 2nd law and Hooke's law for a mass on a spring.

where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is a constant (the spring constant for a mass on a spring).

Therefore,

Solving the differential equation above produces a solution that is a sinusoidal function: ${\displaystyle x(t)=c_{1}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right),}$ where ${\textstyle \omega ={\sqrt {{k}/{m}}}.}$ The meaning of the constants ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ can be easily found: setting ${\displaystyle t=0}$ on the equation above we see that ${\displaystyle x(0)=c_{1}}$, so that ${\displaystyle c_{1}}$ is the initial position of the particle, ${\displaystyle c_{1}=x_{0}}$; taking the derivative of that equation and evaluating at zero we get that ${\displaystyle {\dot {x}}(0)=\omega c_{2}}$, so that ${\displaystyle c_{2}}$ is the initial speed of the particle divided by the angular frequency, ${\displaystyle c_{2}={\frac {v_{0}}{\omega }}}$. Thus we can write:

This equation can also be written in the form:

where

• ${\displaystyle A={\sqrt {{c_{1}}^{2}+{c_{2}}^{2}}}}$
• ${\displaystyle \tan \varphi ={\frac {c_{2}}{c_{1}}},}$
• ${\displaystyle \sin \varphi ={\frac {c_{2}}{A}},\;\cos \varphi ={\frac {c_{1}}{A}}}$

or equivalently

• ${\displaystyle A=|c_1}$Zdroj:https://en.wikipedia.org?pojem=Simple_harmonic_oscillator
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