Rational Energy Balance Method to Nonlinear Oscillators With Cubic Term

Daeichin, Meysam; Ahmadpoor, Mohammad

doi:10.1142/s1793557113500198

Cited by 24 publications

(10 citation statements)

References 31 publications

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“…To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, such as, Perturbation Method [1], Homotopy Perturbation Method [2], Modified Homotopy Perturbation Method [3,4], Rational Homotopy Perturbation Method [5], He's Homotopy Perturbation Method [6], Modified He's homotopy Perturbation Method [7], Asymptotic Method [8][9][10][11], Optimal Iteration Perturbation Method [12], Generalization 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 [19], Parameter Expansion Method and Variational Iteration Method [20][21][22], Amplitude Frequency Formulation Method [23], Energy Balance Method [24,25], He's Energy Balance Method [26,27], Rational Energy Balance Method [28], Rational Harmonic Balance Method [29], Residue Harmonic Balance Method [30][31][32][33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems.…”

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

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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...

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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

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“…Their work also reports a nonlinear mathematical model for the developed fabricated energy harvesting device. The obtained nonlinear differential equation of their system has relativistic form, and its approximated form can be presented by the Duffing equation …”

Section: Techniques For Sensing and Energy Harvesting In Tirementioning

confidence: 99%

Tire Condition Monitoring and Intelligent Tires Using Nanogenerators Based on Piezoelectric, Electromagnetic, and Triboelectric Effects

Argani

Hashemi

Khajepour

et al. 2018

Adv Materials Technologies

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With the prospect of autonomous driving on roads in near future, it is paramount to make the vehicles safe on any driving and road condition. This is only possible by additional sensors to make up for the driver's cognitive and sensory system. Measuring road condition and tire forces especially in autonomous vehicles are vital in their safety, reliability, and public confidence in automated driving. Real time measurement of road condition and tire forces in buses and trucks can significantly improve the safety of road transportation system, and in miming/construction and off-road vehicles can improve performance, tire life, and reduce operational costs.There are three particular branches of research in the area of tire condition monitoring systems (TCMS) including estimation, sensing, and power harvesting. The main focus of this review article would be on progress in estimation techniques, energy harvesting, and also sensing approaches for TCMS based on piezoelectric, electromagnetic, and triboelectric nanogenerators (TENGs). Indeed, the fusion of these three areas in a singular package leads to designing future intelligent tires. The implementation of intelligent tires is highly advantageous for reduction of disastrous accidents,CO 2 , and noise emissions, and also fuel consumption. Figure 1 depicts an intelligent tire with its basic parts, its crucial applications, and advantages. In Figure 1, NGs, PEGs, and EMGs refer to nanogenerators, piezoelectric nanogenerators, and electromagnetic generators, respectively.Advances in applications of sensor technologies, sensor fusion, and cooperative estimation in intelligent transportation systems facilitate having reliable tire forces. Consequently, developing a flexible estimation structure to operate the available sensor data in production vehicles is a preeminent objective of the car manufacturers. In this direction, reliable estimation of tire forces at a reasonable cost is necessary for proper functioning of active safety systems in current vehicles. Longitudinal, lateral, and vertical tire forces make major contributions to velocity estimation, [3] traction and stability control, [4,5] and obstacle avoidance systems. [6,7] This article provides a comprehensive review on intelligent tire and TCMS. The first part of the article describes tire models, techniques for tire force, and tire-road friction The quest to utilize intelligent tires has prompted substantial multidisciplinary research including vehicle dynamics, control, estimation, energy harvesting, and even nanotechnology. This review article presents the progress in the area of tire condition monitoring systems (TCMS) and intelligent tires using devices fabricated based on piezoelectric, electromagnetic, and triboelectric effects. Three main branches of the research in this area are presented, including estimation techniques, sensing, and energy harvesting approaches. The authors delineate the importance of TCMS for vehicle active safety systems, its importance for transportation safety, and also its remarka...

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“…The corresponding angular frequencies of the gauging oscillations, corresponding to the above string, are 30,31 respectively, shown as follows:…”

Section: Gaugingmentioning

confidence: 99%

A Biologically-Inspired Smart Grid Replication

Stefanov¹,

Wang

2017

New Math. and Nat. Computation

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The characteristics of the replication is investigated in this paper. By invoking the Potts model versus holographic superconductors for \gauge" versus \string", the pair forms as a duality in natural manner. It can be shown that resulted characteristics of replication hence deserves to be called as an Autopoietic Smart Grid. Furthermore, we are able to trace the factors contributed to these characteristics; the autopoiesis is contributed via gauge self-energy; the surveillance is due to Maxwell's demon; the organizational adaptivity is due to the communication capacity of the string and the oscillations of the gauge. Finally, it can also be shown that such a Smart Grid replication exists as long as the string is stable and gauge is synchronized.

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Rational Energy Balance Method to Nonlinear Oscillators With Cubic Term

Cited by 24 publications

References 31 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

Tire Condition Monitoring and Intelligent Tires Using Nanogenerators Based on Piezoelectric, Electromagnetic, and Triboelectric Effects

A Biologically-Inspired Smart Grid Replication

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