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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.
 
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
 
The linear functions are constructed by a special method, known as harmonic linearization. Let the given non-linear function (force) be
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464301.png" /></td> </tr></table>
+
$$
 +
F ( x, \dot{x} )  \equiv \
 +
\epsilon f ( x, \dot{x} ),\ \
 +
\dot{x} =  
 +
\frac{dx }{dt }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464302.png" /> is a small parameter. Harmonic linearization is the replacement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464303.png" /> by the linear function
+
where $  \epsilon $
 +
is a small parameter. Harmonic linearization is the replacement of $  F ( x, \dot{x} ) $
 +
by the linear function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464304.png" /></td> </tr></table>
+
$$
 +
F _ {l} ( x, \dot{x} )  = \
 +
kx + \lambda \dot{x} ,
 +
$$
  
where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464306.png" /> are computed by the formulas
+
where the parameters $  k $
 +
and $  \lambda $
 +
are computed by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464307.png" /></td> </tr></table>
+
$$
 +
k ( a)  = \
 +
{
 +
\frac \epsilon {\pi a }
 +
}
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
f ( a  \cos  \psi , - a \omega  \sin  \psi )  \cos  \psi  d \psi ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464308.png" /></td> </tr></table>
+
$$
 +
\lambda ( a)  = - {
 +
\frac \epsilon {\pi a \omega }
 +
} \int\limits _ { 0 } ^ { {2 }  \pi } f ( a \
 +
\cos  \psi , - a \omega  \sin  \psi )  \sin  \psi  d \psi ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h0464309.png" /></td> </tr></table>
+
$$
 +
\psi  = \omega t + \theta .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643013.png" />, the non-linear force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643014.png" /> is a periodic function of time, and its Fourier series expansion contains, generally speaking, an infinite number of [[Harmonics|harmonics]], having the frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643016.png" /> i.e. it is of the form
+
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|harmonics]], having the frequencies $  n \omega $,
 +
$  n = 1, 2 \dots $
 +
i.e. it is of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F ( x, \dot{x} )  = \
 +
\sum _ {n = 0 } ^  \infty 
 +
F _ {n}  \cos  ( n \omega t + \theta _ {n} ).
 +
$$
  
The term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643018.png" /> is called the fundamental harmonic of the expansion (1). The amplitude and the phase of the linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643019.png" /> coincide with the respective characteristics of the fundamental harmonic of the non-linear force.
+
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
 
For the differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x} dot + \omega  ^ {2} x + F ( x, \dot{x} )  = 0,
 +
$$
  
which is typical in the theory of quasi-linear oscillations, the harmonic balance method consists in replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643021.png" /> by the linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643022.png" />; instead of equation (2), one then considers the equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\dot{x} dot + \lambda \dot{x} + k _ {1} x  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643024.png" />. It is usual to call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643025.png" /> the equivalent linear force, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643026.png" /> the equivalent damping coefficient and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643027.png" /> the equivalent elasticity coefficient. It has been proved that if the non-linear equation (2) has a solution of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643028.png" /></td> </tr></table>
+
$$
 +
= a  \cos  ( \omega t + \theta ),
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643029.png" /></td> </tr></table>
+
$$
 +
\dot{a}  = O ( \epsilon ),\ \
 +
\dot \omega  = O ( \epsilon ),
 +
$$
  
then the order of the difference between the solutions of (2) and (3) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643030.png" />. In the harmonic balance method the frequency of the oscillation depends on the amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643031.png" /> (through the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046430/h04643033.png" />).
+
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|automatic control theory]].
 
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|automatic control theory]].

Latest revision as of 19:43, 5 June 2020


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=14745
This article was adapted from an original article by E.A. Grebenikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article