# Harmonic balance method

An approximate method for the study of non-linear oscillating systems described by ordinary non-linear differential equations. The essence of the method is to replace the non-linear forces in the oscillating systems by specially-constructed linear functions, so that the theory of linear differential equations may be employed to find approximate solutions of the non-linear systems.

The linear functions are constructed by a special method, known as harmonic linearization. Let the given non-linear function (force) be

$$F ( x, \dot{x} ) \equiv \ \epsilon f ( x, \dot{x} ),\ \ \dot{x} = \frac{dx }{dt } ,$$

where $\epsilon$ is a small parameter. Harmonic linearization is the replacement of $F ( x, \dot{x} )$ by the linear function

$$F _ {l} ( x, \dot{x} ) = \ kx + \lambda \dot{x} ,$$

where the parameters $k$ and $\lambda$ are computed by the formulas

$$k ( a) = \ { \frac \epsilon {\pi a } } \int\limits _ { 0 } ^ { {2 } \pi } f ( a \cos \psi , - a \omega \sin \psi ) \cos \psi d \psi ,$$

$$\lambda ( a) = - { \frac \epsilon {\pi a \omega } } \int\limits _ { 0 } ^ { {2 } \pi } f ( a \ \cos \psi , - a \omega \sin \psi ) \sin \psi d \psi ,$$

$$\psi = \omega t + \theta .$$

If $x = a \cos ( \omega t + \theta )$, $a = \textrm{ const }$, $\omega = \textrm{ const }$, $\theta = \textrm{ const }$, the non-linear force $F( x, \dot{x} )$ is a periodic function of time, and its Fourier series expansion contains, generally speaking, an infinite number of harmonics, having the frequencies $n \omega$, $n = 1, 2 \dots$ i.e. it is of the form

$$\tag{1 } F ( x, \dot{x} ) = \ \sum _ {n = 0 } ^ \infty F _ {n} \cos ( n \omega t + \theta _ {n} ).$$

The term $F _ {1} \cos ( \omega t + \theta _ {1} )$ is called the fundamental harmonic of the expansion (1). The amplitude and the phase of the linear function $F _ {l}$ coincide with the respective characteristics of the fundamental harmonic of the non-linear force.

For the differential equation

$$\tag{2 } \dot{x} dot + \omega ^ {2} x + F ( x, \dot{x} ) = 0,$$

which is typical in the theory of quasi-linear oscillations, the harmonic balance method consists in replacing $F( x, \dot{x} )$ by the linear function $F _ {l}$; instead of equation (2), one then considers the equation

$$\tag{3 } \dot{x} dot + \lambda \dot{x} + k _ {1} x = 0,$$

where $k _ {1} = \omega ^ {2} + k$. It is usual to call $F _ {l}$ the equivalent linear force, $\lambda$ the equivalent damping coefficient and $k _ {1}$ the equivalent elasticity coefficient. It has been proved that if the non-linear equation (2) has a solution of the form

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

where

$$\dot{a} = O ( \epsilon ),\ \ \dot \omega = O ( \epsilon ),$$

then the order of the difference between the solutions of (2) and (3) is $\epsilon ^ {2}$. In the harmonic balance method the frequency of the oscillation depends on the amplitude $a$( through the quantities $k$ and $\lambda$).

The harmonic balance method is used to find periodic and quasi-periodic oscillations, periodic and quasi-periodic conditions in automatic control theory, as well as stationary conditions, and in the studies of their stability. It is extensively used in automatic control theory.

#### References

 [1] N.M. Krylov, N.N. Bogolyubov, "Introduction to non-linear mechanics" , Princeton Univ. Press (1947) (Translated from Russian) [2] N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi (1961) (Translated from Russian) [3] E.P. Popov, I.P. Pal'tov, "Approximate methods for studying non-linear automatic systems" , Translation Services , Ohio (1963) (Translated from Russian)
How to Cite This Entry:
Harmonic balance method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_balance_method&oldid=47177
This article was adapted from an original article by E.A. Grebenikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article