A theoretical basis for the Harmonic Balance Method

García-Saldaña, Johanna D.; Gasull, Armengol

doi:10.1016/j.jde.2012.09.011

Cited by 21 publications

(18 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Due to his contribution he is known as father of HBM. Afterwards, Belendez et al [40] and others researchers [41][42][43] has significantly improved the HBM. The HBM provides a general technique for calculating approximations to the periodic solutions of linear and NDEs.…”

Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

View full text Add to dashboard Cite

a b s t r a c tIn the present paper, a complicated strongly nonlinear oscillator with cubic and harmonic restoring force, has been analysed and solved completely by harmonic balance method (HBM). Investigating analytically such kinds of oscillator is very difficult task and cumbersome. In this study, the offered technique gives desired results and to avoid numerical complexity. An excellent agreement was found between approximate and numerical solutions, which prove that HBM is very efficient and produces high accuracy results. It is remarkably important that, second-order approximate results are almost same with exact solutions. The advantage of this method is its simple procedure and applicable for many other oscillatory problems arising in science and engineering. IntroductionNonlinear oscillations are important fact in physical science, mechanical structures and other engineering problems. Naturally, all differential equations involving engineering and physical phenomena are nonlinear. The methods of solutions of linear differential equations are comparatively easy and well established. On the contrary, the techniques of solutions of nonlinear differential equations (NDEs) are less available and have no exact solution and, in general, linear approximations are frequently used. Nowadays, NDEs have been the subject of all-embracing studies in various branches of nonlinear science and engineering. A special class of analytical solutions named strongly nonlinear oscillator with cubic and harmonic restoring force has a lot of importance, because, most of the phenomena that arise in mathematical physics and engineering fields can be described by NDEs. Therefore, investigating strongly nonlinear oscillator with cubic and harmonic restoring force solutions is becoming increasingly attractive in nonlinear sciences. Moreover, obtaining exact solutions for nonlinear oscillatory problems has many difficulties. It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [30-33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems. Borges et al. [35] and Bobylev et al. [36] first provided overviews of HBM. Mickens [37-39] was first applied HBM in truly nonlinear oscillators. Due to his contribution he is known as father of HBM. Afterwards, Belendez et al. [40] and others researchers [41-43] has significantly improved the HBM.The HBM provides a general technique for calculating approximations to the periodic solutio...

show abstract

Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

View full text Add to dashboard Cite

show abstract

“…by the periods of the trigonometric polynomials obtained from applying the N -th order HBM to (4). Our results are given in next theorem, where [a] denotes the integer part of a.…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

“…We propose it following the definition of accuracy of an approximated solution used in [4] and inspired in the classical works [18,20].…”

mentioning

confidence: 99%

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

García-Saldaña

Gasull

2017

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable A Beléndez, E Gimeno, E Fernández et al. Abstract. We prove that the differential equation xẍ + 1 = 0 has continuous weak periodic solutions and compute their periods. Then, we use the Harmonic Balance Method until order six to approximate these periods and to illustrate how the accuracy of the method increases with the order. Our computations rely on the Gröbner basis approach.

show abstract

“…It is mainly applied with practical purposes, although in many cases there is no theoretical justification. In most of the applications this method is used to approach isolated periodic solutions, see for instance [16,22,[27][28][29][30]. Since the HBM also provides an approximation of the angular frequency of the searched periodic solution, it can be used to get its period as well.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The period function and the Harmonic Balance Method

García-Saldaña

Gasull

2015

Bulletin des Sciences Mathématiques

Self Cite

View full text Add to dashboard Cite

In this paper we consider several families of potential non-isochronous systems and study their associated period functions. Firstly, we prove some properties of these functions, like their local behavior near the critical point or infinity, or their global monotonicity. Secondly, we show that these properties are also present when we approach to the same questions using the Harmonic Balance Method.

show abstract

A theoretical basis for the Harmonic Balance Method

Cited by 21 publications

References 7 publications

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

The period function and the Harmonic Balance Method

Contact Info

Product

Resources

About