Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy. The term vibration is precisely used to describe a mechanical oscillation.


Oscillation, especially rapid oscillation, may be an undesirable phenomenon in process control and control theory (e.g. in sliding mode control), where the aim is convergence to stable state. In these cases it is called chattering or flapping, as in valve chatter, and route flapping.

Small oscillation approximation[edit]

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: $${\displaystyle U(r)=U_{0}\left[\left({\frac {r_{0}}{r}}\right)^{12}-\left({\frac {r_{0}}{r}}\right)^{6}\right]}$$


The equilibrium points of the function are then found: $${\displaystyle {\begin{aligned}{\frac {dU}{dr}}&=0=U_{0}\left[-12r_{0}^{12}r^{-13}+6r_{0}^{6}r^{-7}\right]\\\Rightarrow r&\approx r_{0}\end{aligned}}}$$


The second derivative is then found, and used to be the effective potential constant: $${\displaystyle {\begin{aligned}\gamma _{\text{eff}}&=\left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}}=U_{0}\left[12(13)r_{0}^{12}r^{-14}-6(7)r_{0}^{6}r^{-8}\right]\\[1ex]&={\frac {114U_{0}}{r^{2}}}\end{aligned}}}$$


The system will undergo oscillations near the equilibrium point. The force that creates these oscillations is derived from the effective potential constant above: $${\displaystyle F=-\gamma _{\text{eff}}(r-r_{0})=m_{\text{eff}}{\ddot {r}}}$$


This differential equation can be re-written in the form of a simple harmonic oscillator: $${\displaystyle {\ddot {r}}+{\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}(r-r_{0})=0}$$


Thus, the frequency of small oscillations is: $${\displaystyle \omega _{0}={\sqrt {\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}}={\sqrt {\frac {114U_{0}}{r^{2}m_{\text{eff}}}}}}$$


Or, in general form^[3] $${\displaystyle \omega _{0}={\sqrt {\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}}}$$


This approximation can be better understood by looking at the potential curve of the system. By thinking of the potential curve as a hill, in which, if one placed a ball anywhere on the curve, the ball would roll down with the slope of the potential curve. This is true due to the relationship between potential energy and force. $${\displaystyle {\frac {dU}{dt}}=-F(r)}$$


By thinking of the potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between ${\displaystyle r_{\text{min}}}$ and ${\displaystyle r_{\text{max}}}$. This approximation is also useful for thinking of Kepler orbits.

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