Eigen oscillation

free oscillation

An oscillation occurring in a dynamical system in the absence of an external action by perturbing it at the initial moment by an "external action" from a state of equilibrium. The nature of eigen oscillations is determined mainly by the internal forces determined by the physical structure of the system. The energy necessary for the movement enters the system from the "external action" at the initial moment of motion.

An example of eigen oscillations are the small oscillations of a conservative system with $ n $ degrees of freedom around a state of stable equilibrium. The equations of motion have the form

$$ \tag{1 } \sum _ { i= 1} ^ { n } ( a _ {si} \ddot{q} _ {i} + c _ {si} q _ {i} ) = 0 ,\ \ s = 1, \dots, n , $$

where the $ q _ {i} $ are generalized coordinates and the $ a _ {si} $, $ c _ {si} $ are constant coefficients. The general solution of (1) consists of the sum of $ n $ harmonic oscillations:

$$ q _ {i} = \sum _ { j= 1} ^ { n } A _ {j} \Delta _ {i} ( k _ {j} ^ {2} ) \ \sin ( k _ {j} t + \beta _ {j} ) ,\ \ i = 1, \dots, n , $$

where $ A _ {j} $, $ \beta _ {j} $ are constants of integration, $ k _ {j} $ are eigen frequencies, i.e. roots of the frequency equation

$$ \tag{2 } \mathop{\rm det} \left ( \begin{array}{ccc} c _ {11} - a _ {11} k ^ {2} &\dots &c _ {1n} - a _ {1n} k ^ {2} \\ \vdots &\ddots &\vdots \\ c _ {n1} - a _ {n1} k ^ {2} &\dots &c _ {nn} - a _ {nn} k ^ {2} \\ \end{array} \right ) = 0 $$

(where it is assumed that there are no zero or multiple frequencies), and $ \Delta _ {i} ( k _ {j} ^ {2} ) $ is the minor corresponding to the $ i $-th column and last row of the determinant (2). The variables $ A _ {j} \Delta _ {i} ( k _ {j} ^ {2} ) $, $ k _ {j} t + \beta _ {j} $ and $ \beta _ {j} $ are the amplitude, phase and initial phase of the $ j $-th harmonic, respectively. It follows from this example that harmonic oscillations of the same frequency for all coordinates arise in phase or contra-phase, and the distribution of amplitudes of oscillations of a given eigen frequency in the coordinates is determined by the physical structure of the system.

References

[1]	I.M. Babakov, "Oscillation theory" , Moscow (1965) (In Russian)
[2]	N.V. Butenin, "The theory of oscillations" , Moscow (1963) (In Russian)
[3]	S.P. Strelkov, "Introduction to oscillation theory" , Moscow (1964) (In Russian)
[4]	A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Pergamon (1966) (Translated from Russian)