We propose a method for finding approximate analytic solutions to autonomous single degree-of-freedom nonlinear oscillator equations. It consists of the harmonic balance with linearization in which Jacobian elliptic functions are used instead of circular trigonometric functions. We show that a simple change of independent variable followed by a careful choice of the form of anharmonic solution enable to obtain highly accurate approximate solutions. In particular our examples show that the proposed method is as easy to use as existing harmonic balance based methods and yet provides substantially greater accuracy.
Ordinary differential equations (ODEs in short) are ubiquitous in fundamental science as well as in engineering. Indeed they commonly arise as direct results of the application of some fundamental laws in the various fields of science or engineering. A classical example here is Newton’s second law of motion. They also often arise indirectly, for example, in the intermediate steps of solving other types of problems such as partial differential equations (PDEs). Solving ODEs, especially analytically, thus appears to be of great importance for gaining insights into real-world or engineering problems. This is yet a very challenging task since the ODEs of interest, being usually nonlinear, are rarely susceptible to exact analytical solutions. In fact, the lack of a general and systematic methodology for solving these nonlinear equations is probably the most important difficulty in the determination of their analytical solutions. So, for the important class of ODEs associated with oscillatory behaviors, numerous techniques which enable to obtain some approximations to the desired solutions have been developed. These techniques may be classified in two groups: perturbation and non-perturbation.
The most famous of perturbation approaches include the Lindstedt-Poincaré (LP) method, the method of multiple scales (MMS) and the Krylov-Bogoliubov-Mitropolsky (KBM) method. These now classical methods are known to have some serious limitations. For instance they are useless for equations which describe essentially nonlinear oscillators, such as (1) below with
In contrast, non-perturbation techniques of which the method of harmonic balance (HB) is a well-known example, do not suffer these limitations. But the straightforward application of this last method leads to systems of nonlinear algebraic equations for the coefficients of the truncated Fourier series assumed for the solutions; which are still very difficult to solve. However the acuteness of this difficulty can now be reduced considerably thanks to the idea proposed initially by Wu and Li [
Another important class of non-perturbative techniques involves the approximation of the nonlinear restoring force in a given oscillator ODE by some simple forms for which the exact solution can be readily obtained. The most common technique in this class is the method of equivalent linearization. In this approach the approximate ODE has the simple form of a linear harmonic oscillator equation [
In this paper, we investigate a further generalization of the HBwL by using Jacobian elliptic functions instead of circular trigonometric functions [
The paper is organized as follows. The main idea sustaining the elliptic harmonic balance is introduced in the next section. The illustrations are presented next in Section 3, where examples of oscillators with polynomial and rational restoring force functions are considered. Section 4 contains the conclusion of the paper.
We consider a nonlinear second-order differential equation of the general form
which governs the time (t) evolution of the state of the state (x) of a conservative single degree-of-freedom system. Here, an overdot denotes differentiation with respect to time t. Thus,
Here, T and w are the unknown time-period and angular frequency of the sought periodic solution. Under this scaling, (1) becomes
where we have set
In this paper we use a different time-scaling than (2) which is simply linear in τ. Following Yuste Bravo [
where am is the Jacobian elliptic amplitude function of parameter m while K is the complete elliptic integral of the first kind. In this paper we adopt the notation of [
one can easily show that the nonlinear time-scaling (4) transforms (1) into
Notice that (6) reduces to (3) for m = 0. The former differs from the latter only by two terms which are parametrically linear in the derivatives. Thus if the HB can be applied to the latter, which requires that one be able to compute the Fourier series of
Then, by making the substitutions
and
are applicable respectively for m < 0 and m > 1. Thus the results that would be obtained using the proposed approach are valid for all real values of such that
in which
Two examples are now presented for illustration.
Our first example involve the cubic-quintic Duffing oscillator described by (1) with f defined such that
Since
Substituting this in (6) with the above form of f and equating the coefficients of
The solution of this set of equations is found to be (recall that
The explicit solution associated with (12) is as in (18a) below. Some limiting cases can be verified readily: For
The results obtained at this level are exactly the same that would be obtained by cubication [
where
The method of cubication therefore approximates (1) with (11) by
with
The solution of (17a) can be written explicitly as [
with
One can easily verify that for μ and λ defined by (17b) the expressions of and in (18b) and (14) are the same. This shows that the method of cubication and the lowest order harmonic balance in our proposed approach are equivalent. But unlike the method of cubication whose results would be limited to the above, the transformation proposed herein allows for further improvments. For instance, the approximate solution to (6) can be sought directly in the form
The required Fourier series expansion is quite easily calculated for polynomial restoring force functions. By equating the coefficients of
we can express w and m as follows
where d solves the polynomial equation
The coefficients of the above polynomial equations are as follows:
Since
Using (7a) for
It appears obviously from (23) that
It deserves to point out here that one could have considered the improved solution in the form
as was suggested by Yuste Bravo [
As can be seen, this expression is independent of the coefficient of the linear term α in contrast to (23). The consequence of this is that (25) is less versatile than (19) to accurately approximate the solutions of the Duffing oscillator for all combinations of the coefficients of its terms. In effect, we have noticed that (25) is very poor when
To further appreciate the accuracy of the various approximations obtained above, we use the relative error
to compare the approximate period
to the exact period
Consider first the case
Calculations using the approximate expressions derived above show that the relative errors of the first-order and second-order analytical approximations as compared to the exact solution are respectively less than 0.37% and 0.072%. In comparison it would be necessary to proceed up to the third-order approximation of the standard HBwL to obtain a result just better than our least accurate first-order result; with a relative error of 0.23% [
In
A further comparison is presented in
As a second example, we consider a conservative nonlinear oscillatory system in which the restoring force has a rational form. Specifically we choose
A peculiarity of this function which is worth noting is that it is not dominated by odd-power monomials in both of the limiting cases
For this form of the restoring force function the expressions of μ and λ in the cubic Duffing equation (17a) approximating (1) with (31) are given by
It follows that the expressions of the parameters m and w of the approximate solution of the problem using the method of cubication are
Again these expressions can also be obtained through the lowest order harmonic balance method for (6) with (31). They solve exactly the coefficients of the fundamental and third order harmonics in the residual of this equation when
and
with
To appreciate the accuracy of the two approximate results in (33) and (34) we have plotted in
as a function of A. It appears clearly from this figure that the result obtained from the method of cubication so diverges from the exact result for large oscillation amplitudes that it can simply be considered invalid. However a good match is observed between the result of our proposed approach and the exact result.
In this paper we have investigated the approximation of periodic solutions to autonomous single degree-of- freedom oscillators equations using the Jacobian elliptic function with the objective of improving the method of cubication. To this end we have shown that the properties of these functions can be exploited to put such ODEs
in a form for which the standard HB and like methods can be applied, while the result is however expressed in terms of Jacobian elliptic functions. Our investigation reveals that in the HBwL the harmonics to include in the correction to a given stage should be higher than or equal to the least harmonic of the residual terms of that stage. With the change of variable sustaining our approach, the analysis can be carried out to higher order, just as in the standard HBwL. For our examples, comparison to the standard harmonic balance indicates that our approach is generally the best, except for periodic orbits too close to a heteroclinic orbit. However, even in this case, the standard HBwL has to be carried to higher order than our approach to gain this advantage. Such a higher order analysis can be quite intricate for non polynomial restoring force. A further interesting property of the solution resulting from our approach is that it also approximates all the harmonics of the exact solution. While the method of rational harmonic balance and the method of cubication also offer solutions with this property, they do not include a mean to carrying the approximations to higher orders as in the present work. In fact our method encompasses the method of cubication which is equivalent to ist first order application.
Serge Bruno Yamgoué,Bonaventure Nana,Olivier Tiokeng Lekeufack, (2015) Improvement of Harmonic Balance Using Jacobian Elliptic Functions. Journal of Applied Mathematics and Physics,03,680-690. doi: 10.4236/jamp.2015.36081