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Journal articles on the topic 'Harmonic balance methods'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Handy, C. R. "Combining the methods of harmonic balance and Kryloff-Bogoliuboff." Journal of Sound and Vibration 102, no. 2 (1985): 243–46. http://dx.doi.org/10.1016/s0022-460x(85)80056-0.

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2

Hall, Kenneth C., Kivanc Ekici, Jeffrey P. Thomas, and Earl H. Dowell. "Harmonic balance methods applied to computational fluid dynamics problems." International Journal of Computational Fluid Dynamics 27, no. 2 (2013): 52–67. http://dx.doi.org/10.1080/10618562.2012.742512.

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3

ROBBIO, FEDERICO I., DIEGO M. ALONSO, and JORGE L. MOIOLA. "DETECTION OF LIMIT CYCLE BIFURCATIONS USING HARMONIC BALANCE METHODS." International Journal of Bifurcation and Chaos 14, no. 10 (2004): 3647–54. http://dx.doi.org/10.1142/s0218127404011491.

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In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can

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4

Pierre, C., A. A. Ferri, and E. H. Dowell. "Multi-Harmonic Analysis of Dry Friction Damped Systems Using an Incremental Harmonic Balance Method." Journal of Applied Mechanics 52, no. 4 (1985): 958–64. http://dx.doi.org/10.1115/1.3169175.

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A multi-harmonic, frequency domain analysis of dry friction damped systems is presented which uses an incremental harmonic balance method. When compared with time domain solution methods, it is found that the incremental harmonic balance method can yield very accurate results with some advantages over the time domain methods. Both one and two degree-of-freedom systems are studied.

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5

Handy, C. R. "Harmonic balance methods and the theory of generalized Padé approximants." Journal of Sound and Vibration 102, no. 2 (1985): 247–57. http://dx.doi.org/10.1016/s0022-460x(85)80057-2.

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6

Wei Dong and Peng Li. "Hierarchical Harmonic-Balance Methods for Frequency-Domain Analog-Circuit Analysis." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 26, no. 12 (2007): 2089–101. http://dx.doi.org/10.1109/tcad.2007.907018.

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7

Hiamang, S., and R. E. Mickens. "Harmonic Balance: Comparison of Equation of Motion and Energy Methods." Journal of Sound and Vibration 164, no. 1 (1993): 179–81. http://dx.doi.org/10.1006/jsvi.1993.1204.

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8

YTTERDAL, TROND, TOR A. FJELDLY, and MICHAEL S. SHUR. "BEYOND SPICE, A REVIEW OF MODERN ANALOG CIRCUIT SIMULATION TECHNIQUES." International Journal of High Speed Electronics and Systems 09, no. 03 (1998): 783–805. http://dx.doi.org/10.1142/s0129156498000324.

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We present a review of modern analog simulation techniques based on time- and frequency-domain algorithms. For time-domain techniques, important topics such as circuit decomposition, relaxation methods, latency, multirate integration, continuation methods, parallel algorithms, and finite difference time-domain methods are discussed. Frequency-domain simulation techniques included are harmonic balance, harmonic relaxation, harmonic-Newton, spectral balance, methods for quasiperiodic circuits, and device modeling for frequency domain simulators. Also included are examples of modern simulators.

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9

Gorgadze, Svetlana F., and Anton A. Maximov. "HARMONIC BALANCE THEORY FOR SCHEME TECHNICAL DESIGN." T-Comm 14, no. 11 (2020): 21–32. http://dx.doi.org/10.36724/2072-8735-2020-14-11-21-32.

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The analysis and generalization of the main publications on the methods of synthesis and analysis of non-linear active microwave circuits based on the use of the harmonic balance method are presented. As a result of some classification of mathematical approaches and techniques used in the context of this method, a selection and review of basic algorithms was made, the sequential application of which makes it possible to obtain the final result for a scheme of any complexity. The principles of drawing up the initial system of differential equations for electronic circuits and reducing it to a s

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10

Attar, Peter J. "Using Pseudotime Solution Framework in Harmonic Balance Methods for Aperiodic Problems." AIAA Journal 51, no. 12 (2013): 2982–87. http://dx.doi.org/10.2514/1.j052635.

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11

Dufour, Guillaume, Frédéric Sicot, Guillaume Puigt, Cédric Liauzun, and Alain Dugeai. "Contrasting the Harmonic Balance and Linearized Methods for Oscillating-Flap Simulations." AIAA Journal 48, no. 4 (2010): 788–97. http://dx.doi.org/10.2514/1.43401.

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12

Brazil, T., S. El-rabaie, E. Choo, V. Fusco, and C. Stewart. "Large-signal FET simulation using time domain and harmonic balance methods." IEE Proceedings H Microwaves, Antennas and Propagation 133, no. 5 (1986): 363. http://dx.doi.org/10.1049/ip-h-2.1986.0065.

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13

Dinh, Nguyen Van. "A note on the method of harmonic balance." Vietnam Journal of Mechanics 23, no. 1 (2001): 34–38. http://dx.doi.org/10.15625/0866-7136/9938.

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Free oscillation period of the Duffing oscillator with cubic non-linearity was examined. A comparison between the exact period and those obtained by the asymptotic and the harmonic balance methods was done. It was shown that the remits given by the harmonic balance are acceptable even for large oscillations whereas the asymptotic method can only be applied for small oscillations.

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14

Ko, Sangho, and Robert R. Bitmead. "IDENTIFIABILITY ANALYSIS FOR NONLINEAR FEEDBACK LIMIT-CYCLE SYSTEMS USING HARMONIC BALANCE METHODS." IFAC Proceedings Volumes 39, no. 1 (2006): 1341–46. http://dx.doi.org/10.3182/20060329-3-au-2901.00217.

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15

Genesio, R., and A. Tesi. "Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems." Automatica 28, no. 3 (1992): 531–48. http://dx.doi.org/10.1016/0005-1098(92)90177-h.

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16

Ekici, Kivanc, Kenneth C. Hall, and Earl H. Dowell. "Computationally fast harmonic balance methods for unsteady aerodynamic predictions of helicopter rotors." Journal of Computational Physics 227, no. 12 (2008): 6206–25. http://dx.doi.org/10.1016/j.jcp.2008.02.028.

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17

Rao, B. Nageswara. "Comments on "Harmonic Balance: Comparison of Equation of Motion and Energy Methods"." Journal of Sound and Vibration 172, no. 5 (1994): 697–99. http://dx.doi.org/10.1006/jsvi.1994.1207.

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18

Yao, Hongliang, Qian Zhao, Qi Xu, and Bangchun Wen. "Comparison of Common Methods in Dynamic Response Predictions of Rotor Systems with Malfunctions." International Journal of Rotating Machinery 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/126843.

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The efficiency and accuracy of common time and frequency domain methods that are used to simulate the response of a rotor system with malfunctions are compared and analyzed. The Newmark method and the incremental harmonic balance method are selected as typical representatives of time and frequency domain methods, respectively. To improve the simulation efficiency, the fixed interface component mode synthesis approach is combined with the Newmark method and the receptance approach is combined with the incremental harmonic balance method. Numerical simulations are performed for rotor systems wit

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19

Rahmani, Masoud, Amin Moslemi Petrudi, and Ionut Cristian Scurtu. "Analytical Study of Nonlinear Vibrations of the Tympanic Membrane by Homotopy Perturbation Method." Technium: Romanian Journal of Applied Sciences and Technology 2, no. 5 (2020): 187–92. http://dx.doi.org/10.47577/technium.v2i5.1967.

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Nonlinear vibration problems are generally of great importance in physics, mechanical structures, and other engineering research. This type of equation is very difficult and time-consuming. Researchers, therefore, focus on analytical and numerical methods; In this paper, using the new Newton-Harmonic Balance analytical method, which includes a combination of Newton methods and the harmonic balance method, the common Homotopy Perturbation method is used to solve the nonlinear equation of human tympanic membrane vibration and are compared and validated.

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20

Narayana Murty, S. V. S., and B. Nageswara Rao. "Further comments on “harmonic balance comparison of equation of motion and energy methods”." Journal of Sound and Vibration 183, no. 3 (1995): 563–65. http://dx.doi.org/10.1006/jsvi.1995.0271.

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21

Xue, Xin, Pei Jun Ju, and Dan Sun. "A Global Residue Harmonic Balance Method for a Class of Nonlinear Jerk Equation." Applied Mechanics and Materials 353-356 (August 2013): 3324–27. http://dx.doi.org/10.4028/www.scientific.net/amm.353-356.3324.

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A new approach, namely the global residue harmonic balance, was developed to determine the accurately approximate periodic solution of a class of nonlinear Jerk equation containing velocity times acceleration-squared and velocity. Unlike other improved harmonic balance methods, all the forward harmonic residuals were considered in the present approximation to improve the accuracy. Comparison of the results obtained using this approach with the exact one and the existing results reveals that the high accuracy, simplicity and efficiency of the presented solution procedure. The method can be easi

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22

Peng, Z. K., and Z. Q. Lang. "Relationship between harmonic balance method and non-linear output frequency response function approach." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 221, no. 12 (2007): 1533–43. http://dx.doi.org/10.1243/09544062jmes785.

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The current paper is concerned with the investigation of the relationship between the harmonic balance method (HBM) and the non-linear output frequency response function (NOFRF) approach in the analysis of non-linear systems. Both are applied to the Duffing's oscillator to demonstrate their relationships. The results reveal that, if the Volterra series representation of a non-linear system is convergent, the harmonic components calculated by the NOFRFs are a solution of the HBM. Moreover, the simulation studies show that, in the convergent cases, the NOFRF method can give more accurate results

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23

Magagnato, Franco, and Jinfeng Zhang. "Simulation of a Centrifugal Pump by Using the Harmonic Balance Method." International Journal of Rotating Machinery 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/729140.

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The harmonic balance method was used for the flow simulation in a centrifugal pump. Independence studies have been done to choose proper number of harmonic modes and inlet eddy viscosity ratio value. The results from harmonic balance method show good agreements with PIV experiments and unsteady calculation results (which is based on the dual time stepping method) for the predicted head and the phase-averaged velocity. A detailed analysis of the flow fields at different flow rates shows that the flow rate has an evident influence on the flow fields. At 0.6Qd, some vortices begin to appear in th

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24

GENESIO, R., and A. TESI. "A HARMONIC BALANCE APPROACH FOR CHAOS PREDICTION: CHUA’S CIRCUIT." International Journal of Bifurcation and Chaos 02, no. 01 (1992): 61–79. http://dx.doi.org/10.1142/s0218127492000070.

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The paper proposes a practical engineering approach for predicting chaotic dynamics in an important class of nonlinear systems. The aim of this approach is to provide a heuristic method of analysis which can give reasonably accurate answers but is far simpler to apply than other more rigorous methods based on nonlinear dynamics. Our approach is founded on the harmonic balance principle and uses standard describing function techniques well known to design engineers. Our method consists of a synergism of two independent techniques, each one constituting a possible mechanism for chaos. These two

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25

Molla, Md Helal Uddin, Md Abdur Razzak, and M. S. Alam. "An analytical technique for solving quadratic nonlinear oscillator." Multidiscipline Modeling in Materials and Structures 13, no. 3 (2017): 424–33. http://dx.doi.org/10.1108/mmms-02-2017-0008.

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Purpose The purpose of this paper is to present an analytical technique, based on the He’s energy balance method (an improved version recently presented by Khan et al.), to obtain the approximate solution of quadratic nonlinear oscillator (QNO). Design/methodology/approach This oscillator (QNO) is used as a mathematical model of the human eardrum oscillation. Findings It has been shown that the results by the present technique are very close to the numerical solution. Originality/value The results obtained in this paper are compared with those obtained by Hu (harmonic balance method) and Khan

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26

Jacques, N., E. M. Daya, and M. Potier-Ferry. "Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods." Journal of Sound and Vibration 329, no. 20 (2010): 4251–65. http://dx.doi.org/10.1016/j.jsv.2010.04.021.

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27

Portilla, J., J. Jugo, A. Anakabe, F. J. Casas, J. P. Pascual, and J. M. Collantes. "Analysis of nonlinear RF and microwave circuits using harmonic balance and system identification methods." International Journal of RF and Microwave Computer-Aided Engineering 12, no. 5 (2002): 448–59. http://dx.doi.org/10.1002/mmce.10043.

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28

Siew, C.-C., M. Hill, R. Holmes, and M. Brennan. "A theoretical and experimental investigation into the vibration response of a flexible rotor and squeeze-film damper assembly." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 215, no. 10 (2001): 1251–69. http://dx.doi.org/10.1177/095440620121500109.

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This paper presents two efficient methods to calculate the unbalance vibration response of a flexible rotor provided with a squeeze-film damper (SFD) with retainer springs. Both methods are iterative and combine the harmonic balance and receptance approaches. The first method, called the modified iteration method (MIM), is suitable for predicting the three-dimensional mode shapes of a concentric SFD-rotor system. The second method, called the modified harmonic balance method (MHBM), is developed to calculate the non-linear vibration response of a flexible shaft provided with either a concentri

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29

Ma, Q. L., A. Kahraman, J. Perret-Liaudet, and E. Rigaud. "An Investigation of Steady-State Dynamic Response of a Sphere-Plane Contact Interface With Contact Loss." Journal of Applied Mechanics 74, no. 2 (2006): 249–55. http://dx.doi.org/10.1115/1.2190230.

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In this study, the dynamic behavior of an elastic sphere-plane contact interface is studied analytically and experimentally. The analytical model includes both a continuous nonlinearity associated with the Hertzian contact and a clearance-type nonlinearity due to contact loss. The dimensionless governing equation is solved analytically by using multi-term harmonic balance method in conjunction with discrete Fourier transforms. The accuracy of the dynamic model and solution methods is demonstrated through comparisons with experimental data and numerical solutions for both harmonic amplitudes of

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30

Sharaiha, A. "Harmonic and intermodulation distortion analysis by perturbation and harmonic balance methods for in-line photodetection in a semiconductor optical amplifier." IEEE Photonics Technology Letters 10, no. 3 (1998): 421–23. http://dx.doi.org/10.1109/68.661430.

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31

Tannir, Dani A. "Computation of intermodulation distortion in RF MEMS variable capacitor circuits using moments." International Journal of Microwave and Wireless Technologies 8, no. 2 (2015): 193–204. http://dx.doi.org/10.1017/s1759078714001548.

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This paper introduces a new technique for the efficient computation of intermodulation distortion in radio frequency circuits that contain microelectromechanical system (MEMS) variable capacitors using moments analysis. This method is applied to an extended harmonic balance formulation, which contains the nonlinear equations that describe the dynamic mechanical behavior of MEMS variable capacitors, in addition to the nonlinear electric circuit equations. As a result, the moments method becomes a general multi-domain simulation method for quantifying nonlinear intermodulation distortion, while

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32

Heners, Jan Philipp, Stephan Stotz, Annette Krosse, Detlef Korte, Maximilian Beck, and Damian Vogt. "Prediction of Transient Pressure Fluctuations within a Low-Pressure Turbine Cascade Using a Lanczos-Filtered Harmonic Balance Method." International Journal of Turbomachinery, Propulsion and Power 6, no. 3 (2021): 25. http://dx.doi.org/10.3390/ijtpp6030025.

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Unsteady pressure fluctuations measured by fast-response pressure transducers mounted in a low-pressure turbine cascade are compared to unsteady simulation results. Three differing simulation approaches are considered, one time-integration method and two harmonic balance methods either resolving or averaging the time-dependent components within the turbulence model. The observations are used to evaluate the capability of the harmonic balance solver to predict the transient pressure fluctuations acting on the investigated stator surface. Wakes of an upstream rotor are generated by moving cylind

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33

EL-Sayed, Tamer S., and S. EL-Rabaie. "Comparative Study of Time Domain, Harmonic Balance and Volterra Series Methods for Nonlinear Circuit Simulation." Menoufia Journal of Electronic Engineering Research 21, no. 2 (2011): 167–86. http://dx.doi.org/10.21608/mjeer.2011.66820.

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34

Peyton Jones, J. C., and K. S. A. Yaser. "Recent advances and comparisons between harmonic balance and Volterra-based nonlinear frequency response analysis methods." Nonlinear Dynamics 91, no. 1 (2017): 131–45. http://dx.doi.org/10.1007/s11071-017-3860-z.

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35

Ni, Jiayao, Ling Xu, Feng Ding, Ya Gu, Ahmed Alsaedi, and Tasawar Hayat. "Parameter estimation for time‐delay systems based on the frequency responses and harmonic balance methods." International Journal of Adaptive Control and Signal Processing 34, no. 12 (2020): 1779–98. http://dx.doi.org/10.1002/acs.3180.

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36

Daeichin, M., M. A. Ahmadpoor, H. Askari, and A. Yildirim. "RATIONAL ENERGY BALANCE METHOD TO NONLINEAR OSCILLATORS WITH CUBIC TERM." Asian-European Journal of Mathematics 06, no. 02 (2013): 1350019. http://dx.doi.org/10.1142/s1793557113500198.

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In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity in the

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37

Lee, Y. Y. "Analysis of the Nonlinear Structural-Acoustic Resonant Frequencies of a Rectangular Tube with a Flexible End Using Harmonic Balance and Homotopy Perturbation Methods." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/391584.

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The structural acoustic problem considered in this study is the nonlinear resonant frequencies of a rectangular tube with one open end, one flexible end, and four rigid side walls A multiacoustic single structural modal formulation is derived from two coupled partial differential equations which represent the large amplitude structural vibration of the flexible end and acoustic pressure induced within the tube. The results obtained from the harmonic balance and homotopy perturbation approaches verified each other. The effects of vibration amplitude, aspect ratio, the numbers of acoustic modes

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38

Yang, Xiao-Dong, Ming Liu, Wei Zhang, Ying-Jing Qian, and Roderick V. N. Melnik. "On the Perturbation Methods for Vibration Analysis of Linear Time-Varying Systems." International Journal of Applied Mechanics 08, no. 03 (2016): 1650035. http://dx.doi.org/10.1142/s1758825116500356.

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Some perturbation methods in the studying vibrations of the linear time-varying (LTV) system are discussed. Three classical perturbation methods, namely, averaging method, harmonic balance method, and multiple scales method with linear scales, have been used from a new perspective based on analytical approximations to the corresponding LTV ordinary differential equations. The deploying beam model has been taken as an example to validate the explicit approximate solutions obtained by these perturbation methods. It is demonstrated that such approximate solutions have good agreement with numerica

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39

Obara, P., and W. Gilewski. "Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods." Bulletin of the Polish Academy of Sciences Technical Sciences 64, no. 4 (2016): 739–50. http://dx.doi.org/10.1515/bpasts-2016-0083.

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Abstract The objective of this paper is to determine dynamic instability areas of moderately thick beams and frames. The effect of moderate thickness on resonance frequencies is considered, with transverse shear deformation and rotatory inertia taken into account. These relationships are investigated using the Timoshenko beam theory. Two methods, the harmonic balance method (HBM) and the perturbation method (PM) are used for analysis. This study also examines the influence of linear dumping on induced parametric vibration. Symbolic calculations are performed in the Mathematica programme enviro

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40

Brito, Leonardo da Cunha, and Paulo Henrique Portela de Carvalho. "Simulation of RF Analog Circuits: A Systematic Exposition of the Harmonic Balance and Complex Envelope Methods." Journal of Communication and Information Systems 19, no. 1 (2004): 41–52. http://dx.doi.org/10.14209/jcis.2004.21.

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41

Fontanela, F., A. Grolet, L. Salles, and N. Hoffmann. "Computation of quasi-periodic localised vibrations in nonlinear cyclic and symmetric structures using harmonic balance methods." Journal of Sound and Vibration 438 (January 2019): 54–65. http://dx.doi.org/10.1016/j.jsv.2018.09.002.

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42

Yamgoue, Serge Bruno, Olivier Tiokeng Lekeufack, and Timoleon Crepin Kofane. "HARMONIC BALANCE FOR NON-PERIODIC HYPERBOLIC SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 22, no. 2 (2017): 140–56. http://dx.doi.org/10.3846/13926292.2017.1276983.

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In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to const

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43

GUGLIELMI, NICOLA. "INEXACT NEWTON METHODS FOR THE STEADY STATE ANALYSIS OF NONLINEAR CIRCUITS." Mathematical Models and Methods in Applied Sciences 06, no. 01 (1996): 43–57. http://dx.doi.org/10.1142/s0218202596000043.

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In this paper numerical problems arising from steady state analysis of nonlinear circuits with quasiperiodic excitation are discussed. The approach we consider is based on the piecewise harmonic balance techniques8,9 (HB), a methodology which has its theoretical foundations in Galerkin's procedure (see the paper by Urabe12). The original problem, which can be expressed in the form of a system of integro-differential equations in the time domain, is changed into a nonlinear algebraic system through a natural projection technique. Thus, one of the main issues we have investigated consists in the

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44

Hui, Han, Li Na Hao, Zhang Qi, and Gao Xiang. "The Trapezoid Counterweight Method in Dynamic Balance of Symmetric Flexible Rotor." Applied Mechanics and Materials 868 (July 2017): 201–6. http://dx.doi.org/10.4028/www.scientific.net/amm.868.201.

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Steam turbine generator unit, water pump and other high speed revolution symmetric flexible rotor were regarded as research objects in this paper. According to variation characteristic of rotor shaft in rigid and flexible working mode, nine-reel high pressure water pump rotor was analyzed. The former four-order intrinsic frequency of flexible rotor was obtained by modal analysis and harmonic response analysis. The methods of reaction force response and unbalance response were been studied after first order and second order resonance region eliminating in different modes of simple harmonic exci

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45

Chen, Huaxiong, and Mingkang Ni. "Iterative Homotopy Harmonic Balance Approach for Determining the Periodic Solution of a Strongly Nonlinear Oscillator." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/675176.

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A novel approach about iterative homotopy harmonic balancing is presented to determine the periodic solution for a strongly nonlinear oscillator. This approach does not depend upon the small/large parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. Importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. With this procedure, the higher-order approximate frequency and corresponding periodic solution ca

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46

Rahmani, Masoud, Ionut Cristian Scurtu, and Amin Moslemi Petrudi. "Analytical and Dynamic study of Pulled Mass Nonlinear Vibration by Two Cables using Newton's Harmonic Balance Method." Technium: Romanian Journal of Applied Sciences and Technology 2, no. 2 (2020): 79–86. http://dx.doi.org/10.47577/technium.v2i2.343.

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In recent years, much research has been done on nonlinear vibrations, and analytical and numerical methods have been used to solve complex nonlinear equations. The behavior of nonlinear oscillating equations is discussed until the second order is approximated. Harmonic balance method, which itself has limitations in application. This method continues to be able to study a wider range of nonlinear differential equations. In general, nonlinear vibration problems are of great importance in physics, mechanical structures, and other engineering research. First, the equation of nonlinear vibrations

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47

Jones, J. C. Peyton, and I. Qankaya. "A Note on the Computation of Maximum Roll Amplitudes in Regular Beam Seas." Journal of Ship Research 41, no. 03 (1997): 224–29. http://dx.doi.org/10.5957/jsr.1997.41.3.224.

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Generalized methods for deriving the harmonic balance equations are illustrated using a ship roll model with angle dependent cubic damping, and cubic stiffness terms. The balance equations are then solved subject to a phase constraint which identifies the resonant solution only. The amplitude of the response at resonance, and the frequency at which resonance occurs, can therefore be obtained without needing to compute the response over a wide frequency range. This provides an efficient tool for investigating the dependence of the resonant response on the level of input excitation, with results

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48

Al-Qaisia, A. A., M. N. Hamdan, and B. O. Al-Bedoor. "On the Steady State Response of a Cantilever Beam Partially Immersed in a Fluid and Carrying an Intermediate Mass." Shock and Vibration 7, no. 4 (2000): 179–94. http://dx.doi.org/10.1155/2000/937247.

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This paper presents a study on the nonlinear steady state response of a slender beam partially immersed in a fluid and carrying an intermediate mass. The model is developed based on the large deformation theory with the constraint of inextensible beam, which is valid for most engineering structures. The Lagrangian dynamics in conjunction with the assumed mode method is utilized in deriving the non-linear unimodal temporal equation of motion. The distributed and concentrated sinusoidal loads are accounted for in a consistent manner using the assumed mode method. The non-linear equation of motio

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Kim, Pilkee, Wan Sun, and Jong Won Seok. "High-Dimensional Harmonic Balance Analysis for a Turning Process with State-Dependent Delay." Advanced Materials Research 655-657 (January 2013): 515–20. http://dx.doi.org/10.4028/www.scientific.net/amr.655-657.515.

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Abstract:

The present study considers the state-dependent delay differential equations (SD-DDEs) for the turning process. In general, series expansion of the SD-DDEs turning system is essential in the nonlinear analysis such as the conventional methods of multiple scales and harmonic balance. Unfortunately, the mathematical theory of SD-DDEs, especially for those with an implicit function of delay, was just recently developed and any rigorous mathematical theory has not yet been proven. As one approach for the nonlinear analysis of the SD-DDEs, physically reasonable results could be obtained by extendin

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Hu, Shuiqing, Stephen Howell, Arvind Raman, Ron Reifenberger, and Matthew Franchek. "Frequency Domain Identification of Tip-sample van der Waals Interactions in Resonant Atomic Force Microcantilevers." Journal of Vibration and Acoustics 126, no. 3 (2004): 343–51. http://dx.doi.org/10.1115/1.1760560.

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Abstract:

Hamaker constants are characteristic material properties that determine the magnitude of the nonlinear van der Waals force between atoms, molecules and nanoscale aggregates of atoms. This paper explores the novel possibility of using Harmonic Balance based nonlinear system identification methods to extract from the nonlinear vibration spectrum of resonant atomic force silicon microcantilevers, the Hamaker constants between a few atoms at the tip of the microcantilever and graphite, gold and silicon carbide samples. First, the nonlinear dynamics of a diving board microcantilever coupled to the

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