Topic

# Harmonic balance

About: Harmonic balance is a research topic. Over the lifetime, 4978 publications have been published within this topic receiving 79148 citations.

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TL;DR: The basic conceptual framework of variational iteration technique with application to nonlinear problems is outlined and a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested.

Abstract: Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.

722 citations

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TL;DR: In this article, a 64-element dual-circularly-polarized spiral rectenna array is designed and characterized over a frequency range of 2-18 GHz with single-tone and multitone incident waves.

Abstract: This paper presents a study of reception and rectification of broad-band statistically time-varying low-power-density microwave radiation. The applications are in wireless powering of industrial sensors and recycling of ambient RF energy. A 64-element dual-circularly-polarized spiral rectenna array is designed and characterized over a frequency range of 2-18 GHz with single-tone and multitone incident waves. The integrated design of the antenna and rectifier, using a combination of full-wave electromagnetic field analysis and harmonic balance nonlinear circuit analysis, eliminates matching and filtering circuits, allowing for a compact element design. The rectified dc power and efficiency is characterized as a function of dc load and dc circuit topology, RF frequency, polarization, and incidence angle for power densities between 10/sup -5/-10/sup -1/ mW/cm/sup 2/. In addition, the increase in rectenna efficiency for multitone input waves is presented.

687 citations

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Duke University

^{1}TL;DR: In this paper, a harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented, which exploits the fact that many unstaidy e ow variables are periodic in time.

Abstract: A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.

673 citations

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Bell Labs

^{1}TL;DR: In this paper, two independent equations relating the average powers at the different frequencies in nonlinear inductors and capacitors are derived, which are independent of the particular shape of this characteristic, of the power levels at the various frequencies, and of the external circuit in which the nonlinear reactor is connected.

Abstract: Two independent equations relating the average powers at the different frequencies in nonlinear inductors and capacitors are derived. These equations are a consequence only of the assumption of a single-valued characteristic for the nonlinear element. They are independent of the particular shape of this characteristic, of the power levels at the various frequencies, and of the external circuit in which the nonlinear reactor is connected. These general energy relations give information regarding the gain and stability of nonlinear reactor modulators and demodulators, and consequently of magnetic and dielectric amplifiers, without requirng detailed information about these devices. The utility of these equations is illustrated by discussing the gain and stability of the simplest types of nonlinear reactor modulators and demodulators and harmonic generators. This analysis for nonlinear inductors and capacitors is extended to include the effects of hysteresis in the nonlinear characteristic in the special case where the operating hysteresis loop is no more than double-valued. A similar analysis applied to nonlinear resistors yields two equations relating the reactive powers at the different frequencies, rather than the real powers as above. The interpretation of reactive power under these conditions is not clear.

654 citations

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04 Apr 2011

TL;DR: In this article, the authors present a survey of the literature on nonlinear dynamics of pendulum and nonlinear oscillators, including a brief biography of Georg Duffing, and some of the most relevant works.

Abstract: List of Contributors. Preface. 1 Background: On Georg Duffing and the Duffing Equation (Ivana Kovacic and Michael J. Brennan). 1.1 Introduction. 1.2 Historical perspective. 1.3 A brief biography of Georg Duffing. 1.4 The work of Georg Duffing. 1.5 Contents of Duffing's book. 1.6 Research inspired by Duffing s work. 1.7 Some other books on nonlinear dynamics. 1.8 Overview of this book. References. 2 Examples of Physical Systems Described by the Duffing Equation (Michael J. Brennan and Ivana Kovacic). 2.1 Introduction. 2.2 Nonlinear stiffness. 2.3 The pendulum. 2.4 Example of geometrical nonlinearity. 2.5 A system consisting of the pendulum and nonlinear stiffness. 2.6 Snap-through mechanism. 2.7 Nonlinear isolator. 2.8 Large deflection of a beam with nonlinear stiffness. 2.9 Beam with nonlinear stiffness due to inplane tension. 2.10 Nonlinear cable vibrations. 2.11 Nonlinear electrical circuit. 2.12 Summary. References. 3 Free Vibration of a Duffing Oscillator with Viscous Damping (Hiroshi Yabuno). 3.1 Introduction. 3.2 Fixed points and their stability. 3.3 Local bifurcation analysis. 3.4 Global analysis for softening nonlinear stiffness ( < 0). 3.5 Global analysis for hardening nonlinear stiffness ( < 0). 3.6 Summary. Acknowledgments. References. 4 Analysis Techniques for the Various Forms of the Duffing Equation (Livija Cveticanin). 4.1 Introduction. 4.2 Exact solution for free oscillations of the Duffing equation with cubic nonlinearity. 4.3 The elliptic harmonic balance method. 4.4 The elliptic Galerkin method. 4.5 The straightforward expansion method. 4.6 The elliptic Lindstedt Poincare method. 4.7 Averaging methods. 4.8 Elliptic homotopy methods. 4.9 Summary. References. Appendix AI: Jacob elliptic function and elliptic integrals. Appendix 4AII: The best L2 norm approximation. 5 Forced Harmonic Vibration of a Duffing Oscillator with Linear Viscous Damping (Tamas Kalmar-Nagy and Balakumar Balachandran). 5.1 Introduction. 5.2 Free and forced responses of the linear oscillator. 5.3 Amplitude and phase responses of the Duffing oscillator. 5.4 Periodic solutions, Poincare sections, and bifurcations. 5.5 Global dynamics. 5.6 Summary. References. 6 Forced Harmonic Vibration of a Duffing Oscillator with Different Damping Mechanisms (Asok Kumar Mallik). 6.1 Introduction. 6.2 Classification of nonlinear characteristics. 6.3 Harmonically excited Duffing oscillator with generalised damping. 6.4 Viscous damping. 6.5 Nonlinear damping in a hardening system. 6.6 Nonlinear damping in a softening system. 6.7 Nonlinear damping in a double-well potential oscillator. 6.8 Summary. Acknowledgments. References. 7 Forced Harmonic Vibration in a Duffing Oscillator with Negative Linear Stiffness and Linear Viscous Damping (Stefano Lenci and Giuseppe Rega). 7.1 Introduction. 7.2 Literature survey. 7.3 Dynamics of conservative and nonconservative systems. 7.4 Nonlinear periodic oscillations. 7.5 Transition to complex response. 7.6 Nonclassical analyses. 7.7 Summary. References. 8 Forced Harmonic Vibration of an Asymmetric Duffing Oscillator (Ivana Kovacic and Michael J. Brennan). 8.1 Introduction. 8.2 Models of the systems under consideration. 8.3 Regular response of the pure cubic oscillator. 8.4 Regular response of the single-well Helmholtz Duffing oscillator. 8.5 Chaotic response of the pure cubic oscillator. 8.6 Chaotic response of the single-well Helmholtz Duffing oscillator. 8.7 Summary. References. Appendix Translation of Sections from Duffing's Original Book (Keith Worden and Heather Worden). Glossary. Index.

650 citations