Free harmonic oscillation

A sinusoidal oscillation. If a mechanical or physical quantity $ x ( t) $, where $ t $ denotes time, varies in accordance with the law

$$ \tag{1 } x ( t) = \ A \cos ( \omega t + \phi ), $$

then it is said that $ x ( t) $ performs a free harmonic oscillation. Here $ A > 0 $, $ \omega > 0 $ and $ \phi $ are real constants, called, respectively, the amplitude, frequency and phase of the free harmonic oscillation. The period is $ T = 2 \pi / \omega $. The following terminology is often used in physics and engineering. A free harmonic oscillation is called a harmonic oscillation, or a simple harmonic oscillation; functions of the form (1) are called harmonics, the variable $ \omega t + \phi $ is called the instantaneous phase, and $ \phi $ is called the initial phase. The quantity $ \omega $ is also called the circular or cyclic frequency, and $ f = \omega /2 \pi $ is called the frequency. A free harmonic oscillation (1) can be written as

$$ x ( t) = \ a \cos \omega t + b \sin \omega t, $$

where $ a, b $ and $ A, \phi $ are connected by the relations

$$ A = \ \sqrt {a ^ {2} + b ^ {2} } ,\ \ \cos \phi = \ { \frac{a}{\sqrt {a ^ {2} + b ^ {2} } } } ,\ \ \sin \phi = \ { \frac{b}{\sqrt {a ^ {2} + b ^ {2} } } } , $$

or by

$$ x ( t) = \ \mathop{\rm Re} ( Ae ^ {i ( \omega t + \phi ) } ). $$

Often the phase is taken to be $ - \phi $ and not $ \phi $.

Small oscillations of mechanical or physical systems with one degree of freedom near a stable non-degenerate equilibrium position are free harmonic oscillations, to a large degree of exactness. For example, small oscillations of a pendulum, oscillations of a load suspended by a string, oscillations of a tuning fork, the variation of the direction and strength of the current in an oscillating electrical circuit, the rolling of a ship, etc. A system performing free harmonic oscillation is called a linear harmonic oscillator, and its oscillation is described by the equation

$$ \dot{x} dot + \omega ^ {2} x = 0. $$

For a mathematical pendulum of length $ l $ and mass $ m $, $ \omega ^ {2} = g/l $; for a load of mass $ m $ on a string with elasticity coefficient $ k $, $ \omega ^ {2} = k/m $; for an oscillating electrical circuit of capacity $ C $ and inductance $ L $, $ \omega ^ {2} = 1/CL $. The equilibrium position $ x = 0 $, $ \dot{x} = 0 $ in the phase plane $ ( x, \dot{x} ) $ for a free harmonic oscillator is the centre, and the phase trajectories are circles.

The sum $ x _ {1} ( t) + x _ {2} ( t) $ of two free harmonic oscillations, where

$$ x _ {j} ( t) = \ A _ {j} \cos \ ( \omega _ {j} t + \phi _ {j} ),\ \ j = 1, 2, $$

with commensurable frequencies $ \omega _ {1} $ and $ \omega _ {2} $ is a free harmonic oscillation. If $ \omega _ {1} $ and $ \omega _ {2} $ are incommensurable, then $ x _ {1} ( t) + x _ {2} ( t) $ is an almost-periodic function, and

$$ \sup _ {t \in \mathbf R } \ ( x _ {1} ( t) + x _ {2} ( t)) = \ A _ {1} + A _ {2} = \ - \inf _ {t \in \mathbf R } \ ( x _ {1} ( t) + x _ {2} ( t)). $$

The sum of $ n $ free harmonic oscillations with rationally-independent frequencies $ \omega _ {1} \dots \omega _ {n} $ is also almost-periodic. For the sum of two free harmonic oscillations, $ \Omega = | \omega _ {1} - \omega _ {2} | $ is called the derangement. If $ \Omega $ is small, $ \Omega / \omega _ {1} \ll 1 $, and if $ \omega _ {1} $ and $ \omega _ {2} $ have the same order of magnitude, then

$$ x _ {1} ( t) + x _ {2} ( t) = \ A ( t) \cos ( \omega _ {1} t + \phi ( t)), $$

$$ A ^ {2} ( t) = A _ {1} ^ {2} + A _ {2} ^ {2} + 2A _ {1} A _ {2} \cos ( \psi ( t) - \phi _ {1} ), $$

$$ \psi ( t) = \Omega t + \phi _ {2} . $$

The "amplitude" $ A ( t) $ is a slowly-varying function of period $ 2 \pi / \Omega $, and $ A ^ {2} ( t) $ varies from $ ( A _ {1} - A _ {2} ) ^ {2} $ to $ ( A _ {1} + A _ {2} ) ^ {2} $. The oscillation $ x _ {1} ( t) + x _ {2} ( t) $ is called a beat, and the "amplitude" $ A ( t) $ alternatingly increases and decreases. This case is important in the analysis of receiving devices.

Suppose one has a system of $ n $ equations

$$ M \dot{x} dot + Kx = 0,\ \ x \in \mathbf R ^ {n} , $$

where $ M $ and $ K $ are real symmetric positive-definite matrices with constant elements. By using an invertible transformation $ x = Ty $ this system is transformed into the decomposed system

$$ \dot{y} dot _ {j} + \omega _ {j} ^ {2} y _ {j} = 0,\ \ j = 1 \dots n. $$

The coordinates $ y _ {1} \dots y _ {n} $ are called normal. In normal coordinates $ x ( t) $ is the vector sum of free harmonic oscillations along the coordinate axes.

References

[1]	A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)
[2]	G.S. Gorelik, "Oscillations and waves" , Moscow-Leningrad (1950) (In Russian)
[3]	L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1965) (Translated from Russian)