A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

Wu, Biao; Liu, Weijia; Zhong, Huanhuan; Lim, C.W.

doi:10.1007/s42417-019-00176-3

Cited by 8 publications

(4 citation statements)

References 15 publications

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“…Dehrouyeh-Semnani et al [52] applied the HB method to analyze the nonlinear vibrations of functionally graded pipes conveying hot fluid subject to harmonic external loading. Wu et al [53] proposed a modified NHB method for analyzing nonlinear vibrations in oscillators with odd nonlinearities. They applied two simplified Newton steps to reduce the amount of linearization in the restring force function.…”

Section: Introductionmentioning

confidence: 99%

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Mohammadian,

Ismail

2024

Phys. Scr.

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The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

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Section: Introductionmentioning

confidence: 99%

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Mohammadian,

Ismail

2024

Phys. Scr.

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“…A wide variety of modifications to HBM have been developed over the last 40 years, including the incremental harmonic balance (IHB) [41][42][43][44][45], the alternating f requency/time domain method using harmonic balance (AFTHB) [46], the adaptive harmonic balance (AHB) [47,48], and various further improvements and modifications of these methods [49][50][51][52][53][54][55]. Comprehensive reviews of these methods can be found in Refs.…”

Section: Introductionmentioning

confidence: 99%

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

Gonzalez

2024

Nonlinear Dyn

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Recently, the sinosoidal output response in power series (SORPS) formalism was presented for system identification and simulation. Based on the concept of characteristic curves (CCs), it establishes a mathematical connection between power series and Fourier series for a first-order nonlinear system [F. J. Gonzalez, Sci. Rep. 13, 1955]. However, the system identification procedure discussed there, based on f ast Fourier transform (FFT), presents the limitations of requiring a sinusoidal single tone for the dynamical variable and equally spaced time steps for the input-output dataset (DS). These limitations are here addressed by introducing a different approach: we use a power series-based model (referred to as model 1) for system modeling instead of FFT, where two hyperparameters Â0 and Â1 are optimally defined depending on the DS. Subsequently, two additional models are obtained from parameters obtained in model 1: another power series-based model (model 2) and a Fourier analysis-based model (model 3). These models are useful for comparing parameters obtained from different DSs. Through an illustrative example, we show that while the predicted values from the models are the same due to a mathematical equivalence, the parameters obtained for each model vary to a greater or lesser extent depending on the DS used for system estimation. Hence, the parameters of the Fourier analysis-based model exhibit notably less variation compared to those of the power series-based model, highlighting the reliability of using the Fourier analysis-based model for comparing model parameters obtained from different DSs. Overall, this work expands the applicability of the SORPS formalism to system identification from arbitrary inputoutput data and represents a groundbreaking contribution relying on the concept of CCs, which can be straightforwardly applied to higher-order nonlinear systems. The method of CCs can be considered as complementary to the commonly used approach (such as NARMAX-models and sparse regression techniques) that emphasizes the estimation of the individual parameter values of the model. Instead, the CCs-based methods emphasize the computation of the CCs as a whole. CCs-based models present the advantages that the system identification is uniquely defined, and that it can be applied for any system without additional algebraic operations. Thus, the parsimonious principle defined by the NARMAX-philosophy is extended from the concept of a model with as few parameters as possible to the concept of finding the lowest model order that correctly describe the input-output data. This opens up a wide variety of potential applications, covering areas such as vibration analysis, structural dynamics, viscoelastic materials, design and modeling of nonlinear electric circuits, voltammetry techniques in electrochemistry, structural health monitoring, and fault diagnosis.

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“…Even though the restoring resistances provided by the springs are linear, their resulting actions on the mass can be irrationally nonlinear due to the geometrical configurations [6]. In the presence of damping and external excitation, these oscillatory systems can exhibit rich dynamical behaviors [7].…”

Section: Introductionmentioning

confidence: 99%

Global Dynamics and Bifurcations of an Oscillator with Symmetric Irrational Nonlinearities

Liu,

Shang

2023

Fractal Fract

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This study’s objective is an irrationally nonlinear oscillating system, whose bifurcations and consequent multi-stability under the circumstances of single potential well and double potential wells are investigated in detail to further reveal the mechanism of the transition of resonance and its utilization. First, static bifurcations of its nondimensional system are discussed. It is found that variations of two structural parameters can induce different numbers and natures of potential wells. Next, the cases of mono-potential wells and double wells are explored. The forms and stabilities of the resonant responses within each potential well and the inter-well resonant responses are discussed via different theoretical methods. The results show that the natural frequencies and trends of frequency responses in the cases of mono- and double-potential wells are totally different; as a result of the saddle-node bifurcations of resonant solutions, raising the excitation level or frequency can lead to the coexistence of bistable responses within each well and cause an inter-well periodic response. Moreover, in addition to verifying the accuracy of the theoretical prediction, numerical results considering the disturbance of initial conditions are presented to detect complicated dynamical behaviors such as jump between coexisting resonant responses, intra-well period-two responses and chaos. The results herein provide a theoretical foundation for designing and utilizing the multi-stable behaviors of irrationally nonlinear oscillators.

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A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

Cited by 8 publications

References 15 publications

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

Global Dynamics and Bifurcations of an Oscillator with Symmetric Irrational Nonlinearities

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