User:Calliso2022/Oscillation

Oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. The term vibration is precisely used to describe a mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.

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.

Simple harmonic
The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension, and without friction. Such a system may be approximated on an air table or ice surface. The system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired momentum which keeps it moving beyond that position, establishing a new restoring force in the opposite sense. If a constant force such as gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory period.

The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion. In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely the existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium.

In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is:

$$F=-kx$$

By using Newton's second law, the differential equation can be derived.

$$\ddot{x} = -\frac km x = -\omega^2x$$,

where $$\omega = \sqrt \frac km$$

The solution to this differential equation produces a sinusoidal position function.

$$x(t) = A \cos (\omega t - \delta)$$

'''where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and -1 infinitely, our spring-mass system would oscillate between the positive and negative amplitude forever without friction.'''

Two-dimensional oscillators
'''In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions.'''

$$F = -k\vec{r}$$

This produces a similar solution, but now there is a different equation for every direction.

$$x(t) = A_x \cos(\omega t - \delta _x)$$, $$y(t) = A_y \cos(\omega t - \delta_y)$$, [...]

Anisotropic oscillators
'''With anisotropic oscillators, different directions have different constants of restoring forces. The solution is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is quasiperiodic. This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat.'''

Damped oscillations
All real-world oscillator systems are thermodynamically irreversible. This means there are dissipative processes such as friction or electrical resistance which continually convert some of the energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator.

'''Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant b. This example assumes a linear dependence on velocity.'''

$$m\ddot{x} + b\dot{x} + kx = 0$$

This equation can be rewritten as before.

$$\ddot{x} + 2 \beta \dot{x} + \omega_0^2x = 0$$, where $$2 \beta = \frac b m$$

This produces the general solution:

$$x(t) = e^{- \beta t} (C_1e^{\omega _1 t} + C_2 e^{- \omega_1t})$$,

where $$\omega_1 = \sqrt{\beta^2 - \omega_0^2}$$

'''The exponential term outside of the parenthesis is the decay function and β is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where β < ω0; over-damped, where β > ω0; and critically damped, where β = ω0.'''

Driven oscillations
In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source. In this case the oscillation is said to be driven. The simplest example of this is s spring-mass system with a sinusoidal driving force.

$$\ddot{x} + 2 \beta\dot{x} + \omega_0^2x = f(t)$$, where $$f(t) = f_0 \cos(\omega t + \delta)$$

This gives the solution:

$$x(t) = A \cos(\omega t - \delta) + A_{tr} \cos(\omega_1 t - \delta_{tr})$$,

where $$A = \sqrt{\frac {f_0^2} {(\omega_0^2 - \omega ^2) + 2 \beta \omega}}$$ and  $$\delta = \tan^{-1}(\frac {2 \beta \omega} {\omega_0^2 - \omega^2})$$

'''The second term of x(t) is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system.'''

Resonance
'''Resonance occurs in a damped driven oscillator when ω = ω0, that is, when the driving frequency is equal to the natural frequency of the system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations.'''

Coupled oscillations
The harmonic oscillator and the systems it models have a single degree of freedom. More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of the others. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronize. This phenomenon was first observed by Christiaan Huygens in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into normal modes.

'''The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses.'''

$$m_1 \ddot{x}_1 = -(k_1 + k_2)x_1 + k_2x_2$$,     $$m_2\ddot{x}_2 = k_2x_1 - (k_2+k_3)x_2$$,

The equations are then generalized into matrix form.

$$F = M\ddot{x} = kx$$,

where $$M=\begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}$$,    $$x = \begin{bmatrix} x_1  \\ x_2  \end{bmatrix}$$,   and  $$k = \begin{bmatrix} k_1+k_2 & -k_2 \\ -k_2 & k_2+k_3 \end{bmatrix}$$

The values of k and m can be substituted into the matrices.

$$m_1=m_2=m $$,    $$k_1=k_2=k_3=k$$,

$$M = \begin{bmatrix} m & 0 \\ 0 & m \end{bmatrix}$$, $$k=\begin{bmatrix} 2k & -k \\ -k & 2k \end{bmatrix}$$

These matrices can now be plugged into the general solution.

$$(k-M \omega^2)a = 0$$

$$\begin{bmatrix} 2k-m \omega^2 & -k \\ -k & 2k - m \omega^2 \end{bmatrix} = 0$$

The determinant of this matrix yields a quadratic equation.

$$(3k-m \omega^2)(k-m \omega^2)= 0$$

$$\omega_1 = \sqrt{\frac km}$$,   $$\omega_2 = \sqrt{\frac {3k} m}$$

'''Depending on the starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses are started in opposite directions, the second, faster frequency is the frequency of the system.'''

More special cases are the coupled oscillators where energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum, where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring.

Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to the occurrence of a single, entrained oscillation state, where both oscillate with a compromise frequency. Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as Arnold Tongues, can lead to highly complex phenomena as for instance known as chaotic dynamics.

Small oscillation approximation
'''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:'''

$$U(r)=U_0 \left[ \left(\frac{r_0} r \right)^{12} - \left(\frac{r_0} r \right)^6 \right]$$

The equilibrium points of the function are then found.

$$\frac{dU}{dr} = 0 =U_0[-12 r_0^{12}r^{-13} + 6r_0^6r^{-7}]$$

$$\Rightarrow r_0 \approx r$$

The second derivative is then found, and used to be the effective potential constant.

$$\gamma_{eff} = \frac{d^2U}{dr^2} \vert_{r=r_0}=U_0 [12(13)r_0^{12}r^{-14}-6(7)r_0^6r^{-8}]$$

$$\gamma_{eff} = \frac{114 U_0}{r^2}$$

'''The system will undergo oscillations near the equilibrium point. The force that creates these oscillations is derived from the effective potential constant above.'''

$$F= - \gamma_{eff}(r-r_0) = m_{eff} \ddot r$$

This differential equation can be re-written in the form of a simple harmonic oscillator.

$$\ddot r + \frac {\gamma_{eff}} {m_{eff}} (r-r_0) = 0$$

Thus, the frequency of small oscillations is:

$$\omega_0 = \sqrt { \frac {\gamma_{eff}} {m_{eff}}} = \sqrt {\frac {114 U_0} {r^2m_{eff}}}$$

Or, in general form

$$\omega_0 = \sqrt{\frac {d^2U} {dU^2} \vert_{r=r_0}}$$