# 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.

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