Higher-order Approximate Solutions Research Articles

Complex Multi-nonlinearity for piezoelectric energy harvesting systems based on an accurate higher-order perturbation method

A theoretical model of a complex nonlinear coupling energy harvester under hybrid excitations is established with analytical and numerical solutions. Different orders of the method of multiple scales are derived to approximate and analyze the equation of motion for the electromechanical coupling system, thus obtaining higher-order approximate analytical solutions for its operating status and output performance indicators. The significant effects of external excitation, linear and nonlinear damping coefficients, as well as impedance on the transverse displacement, voltage, and power of higher-order and lower-order methods are analyzed. The results show that higher-order solution methodology are more sensitive to parameter excitation. For a large damping, the higher-order analysis has more excellent performance in capturing energy. The higher order method shows more pronounced nonlinear softening phenomenon and it is more suitable for low frequency environments. The higher order methods improves solution accuracy of piezoelectric energy harvesting systems. Consequently, it is easier to derive the nonlinear performance of the harvesting systems, and thus improves the average output power.

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

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.

Higher-order averaging procedure for performance analysis of a mono-stable Duffing piezoelectric energy harvesting system under white noise excitation

This work is concerned with the energy harvesting of mono-stable Duffing oscillators with piezoelectric coupling under white noise excitation. This is the first time that the higher-order averaging procedure is extended to find the first-, second- and third-order approximate stationary joint probability density functions of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. Then the mean-square electric voltage and mean output power are derived through the approximate relation between the electric voltage and the mechanical states. The analytical results are verified by Monte Carlo simulation. The effects of system parameters, such as excitation intensity, mechanical damping term, nonlinear stiffness coefficients, and the resistance ratio of the resistive–capacitive circuit on the mean-square displacement and mean output power are discussed in detail. It has been shown that the averaging procedure of the averaging method of coefficients in FPK equation can be considered as a particular case of the proposed higher-order averaging procedure and the higher-order approximate solutions are much more accurate than the one obtained by the averaging method of coefficients in FPK equation.© 2022 The Authors. Published by Elsevier Ltd.

Nonlinear Vibration of Electrostatically Actuated Microbeam

In this paper, an analytical technique based on the global residue harmonic balance method (GRHBM) is applied in order to obtain higher-order approximate analytical solutions of an electrostatically actuated micro-beam. To illustrate the applicability and accuracy of the method, a high level of accuracy was established for the analytical solutions by comparing the results of the solutions with the numerical solution as well as the already published literature, such as the variational approach (VA), Hamiltonian approach (HA), energy balance method (EBM), and homotopy analysis method (HAM). It is shown that the GRHB method can be easily applied to nonlinear problems and provides solutions with a higher precision than existing methods. The obtained analytical expressions are employed to study the effects of axial force, initial gape, and electrostatic load on nonlinear frequency.

A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.

Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations

The nonlinear couple wave equations, which are extensively applied in scientific fields, such as, solid state physics, quantum mechanics, nonlinear optics, are a kind of important evolution equations. This paper is concerned with their numerical solutions via the combinations of compact difference method, Predictor–Corrector (P–C) iterative methods and Richardson extrapolation methods (REMs). Firstly, fourth-order compact difference methods are used to discrete temporal and spatial derivatives, thus forming a nonlinear fully discrete compact difference scheme. By utilizing the discrete energy analysis method and fixed point theorem, we can prove that under the condition of accepted stable criterion this scheme is conditionally convergent with an order of O(τ4+hx4+hy4) in H1-norm, and solvable. For avoiding solving the system of nonlinear algebraic equations, a P–C iterative method is introduced to save time cost and make the implementation simple. Besides, REMs are further applied to attain higher-order accurate approximate solutions. Finally, numerical findings confirm the exactness of theoretical results and efficiency of the algorithms.

Analysis of Dynamic Characteristics of a Fractional-Order Spur Gear Pair With Internal and External Excitations

Abstract The nonlinear frequency response characteristics of a spur gear pair with fractional-order derivative under combined internal and external excitations are investigated based on the incremental harmonic balance (IHB) method. First, a pure torsional vibration model is proposed that contains various complex factors, such as the time-varying mesh stiffness, transmission error, the fluctuation of input torque, backlash. Then, the IHB method is developed to calculate the higher-order approximate solution of the system, and the correctness of the results is verified by comparing with numerical simulation results obtained by the power series expansion (PSE) method. Furthermore, the types of various impact situations and their judgment conditions are discussed, and the different impact behaviors are analyzed in detail when ω∈[0, 1.5] by using phase diagrams and amplitude–frequency response curves. The influence of important parameters on the dynamic characteristics of gear pair is analyzed at last. The results indicate that the analytical solution derived by IHB method is sufficiently precise. Significantly, the dynamic characteristics of the system could be effectively controlled by adjusting time-varying mesh stiffness coefficient, the order and coefficient of fractional-order term, and the amplitude of internal excitation or external excitation. As a part of the theory of fractional-order mechanical system, the impact performance of fractional-order gear pair is approached for the first time by analytical method.

Global Residue Harmonic Balance Method for a System of Strongly Nonlinear Oscillator

In this paper, the global residue harmonic balance method is applied to obtain the approximate periodic solution and frequency for a well-known system of strongly nonlinear oscillator in engineering. This method can improve accuracy by considering all the residual errors in deriving each order approximation. With this procedure, the expressions of the higher-order approximate solution and corresponding frequency for the considered system can be determined easily. The comparison of the obtained results with previously existing and corresponding exact solutions shows the high accuracy and efficiency of the method.

High-Order Approximation to Two-Level Systems with Quasiresonant Control

In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the effectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.

A Modified Incremental Harmonic Balance Method for 2-DOF Airfoil Aeroelastic Systems with Nonsmooth Structural Nonlinearities

A modified incremental harmonic balance method is presented to analyze the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity. The current method, which combines the traditional incremental harmonic balance method and a fast Fourier transform, can be used to obtain the higher-order approximate solution for the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity using significantly fewer linearized algebraic equations than the traditional method, and the dominant frequency components of the response can be obtained by a fast Fourier transform of the numerical solution. Thus, periodic solutions can be obtained, and the calculation process can be simplified. Furthermore, the nonsmooth nonlinearity was expanded into a Fourier series. The procedures of the modified incremental harmonic balance method were demonstrated using systems with hysteresis and free play nonlinearities. The modified incremental harmonic balance method was validated by comparing with the numerical solutions. The effect of the number of harmonics on the solution precision as well as the effect of the free-play and stiffness ratio on the response amplitude is discussed.

OPTIMAL ITERATIVE PERTURBATION TECHNIQUE FOR SOLVING JEFFERY–HAMEL FLOW WITH HIGH MAGNETIC FIELD AND NANOPARTICLE

In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeffery–Hamel flow is conducted via the newly developed technique. We use the optimal iterative perturbation method with multiple parameters to see the effects of the magnetic field and nanoparticle on the Jeffery–Hamel flow. Comparing our new approximate solutions with some earlier works proved the excellent accuracy of the newly proposed technique. Convergence analysis of the proposed method is also discussed and error estimation is given to anticipate the accuracy of higher-order approximate solutions.

Implicit Order 3/2 Strong Method for Numerical Solutions of Stratonovich Stochastic Differential Equations

Due to that the explicit methods in solving stochastic differential equations give instability and inaccurate results, the aim of this paper is to derive an effective implicit method gives higher-order approximate solutions for a stiff stochastic differential equations by using Runge-Kutta method. It relies on the Stratonovich-Taylor expansion and uses the notion of perturbation and coupling to carry out the method. The validity of this new approximation method is shown by implementing in MATLAB and, showing the convergence of the method graphically.

Higher-order approximate periodic solution for the oscillator with strong nonlinearity of polynomial type

In this paper the harmonic balance method (HBM) is adopted for solving a special group of oscillators with strong nonlinear damping and elastic forces. The nonlinearity is of polynomial type. The motion is described with a strong nonlinear differential equation, whose approximate solution is assumed as a suitable sum of trigonometric functions. To find the most convenient combination of trigonometric functions as the probe function is the most important part of this investigation. Introducing the procedure of equating the terms with the same order of the trigonometric functions to zero, the problem is transformed into solving a system of nonlinear algebraic equations. Solving these equations the parameters of the solution up to high-order approximation are obtained. In the paper, the suggested solving procedure is applied for two nonlinear oscillator problems: free vibrations of a restrained uniform beam carrying an intermediate lumped mass and of a particle on a rotating parabola. The obtained approximate analytic solutions are compared with the already published results and with the numerically obtained solution. The solution up to third-order approximation is calculated. It is proved that the HBM with the suggested function gives more accurate results than the previous applied ones. Besides, the difference between the third-order approximate analytical solution and the numerical one is negligible. The method works well for different, even high, values of initial amplitudes.

Modified harmonic balance method for solving strongly nonlinear oscillators

In this paper, a modified harmonic balance method is applied to obtain higher-order approximate periodic solutions of strongly nonlinear oscillatory systems having a rational force. Approximate frequency and periodic solution are obtained. Frequency and solutions are analytically analyzed. Obtained results are compared with several authors and corresponding numerical results. Error analysis is carried out and found satisfactory results for the proposed method. The new technique also avoids the necessity of solving sets of equations with very complex nonlinearities numerically as in the classical harmonic balance method. Effectiveness of the method found from comparison with other articles. The method is illustrated by examples.

Higher-order approximate solutions of fractional stochastic point kinetics equations in nuclear reactor dynamics

Stochastic point kinetics equations (SPKEs) are a system of Ito stochastic differential equations whose solution has been obtained by higher-order approximation. In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.

An Iterative Approach to Obtaining the Nonlinear Frequency of a Conservative Oscillator with Strong Nonlinearities

An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but, unlike them, the errors obtained from the previous step are considered at the current step to increase the accuracy of the solution. A comparison of results with those obtained by exact solution and other approximate analytical techniques confirms the accuracy of the method. It is shown that the achieved approximate solutions are valid for both small and large amplitudes of oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and quintic terms on the frequency is discussed.

Discontinuous Galerkin method in numerical simulation of two-dimensional thermoelasticity problem with single stabilization parameter

The aim of the paper is the development of discontinuous Galerkin with finite difference rules (DGFD) to a two-dimensional stationary and non-stationary thermoelasticity problem. Displacement and temperature fields are approximated on the same mesh frame but with various approximation orders, which are set independently for each of the fields. Because the DGFD method does not use nodes, special attention needs to be paid to applying boundary conditions. Various types of thermal and mechanical boundary conditions are considered. In the presented approach only one stabilization parameter for the coupled problem needs to be evaluated in the DGFD method. The same parameter used in thermal and in mechanical part. The considered domain is discretized by a polygonal mesh in which the polygonal elements may have arbitrary shapes, such as e.g. a fish shape, as well as typical rectangular shapes. The orthogonality of Chebyshev basis functions may be utilized for rectangular elements. Very high-order approximate solution can be obtained in such case. In the coupled problem, the same element may be high-order for displacement field while low-order to approximate temperature. The argument contained in the paper is illustrated with few examples.

Resonance Analysis of Fractional-Order Mathieu Oscillator

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.

Effects of temperature variations on nonlinear planar free and forced oscillations at primary resonances of suspended cables

This paper is concerned with the analysis of the free and forced nonlinear vibrations of the suspended cable with thermal effects. By introducing the new thermal stressed configuration, the effect of temperature variation on the nonlinear vibration equations of motion is reflected by a non-dimensional cable tension variation factor. The partial differential equations of the planar motion are discretized to the ordinary equations via the Galerkin method, and the single-mode discretization is investigated. The Lindstedt–Poincare method and multiple scales method are applied to obtain the higher-order approximate solutions of the nonlinear free vibrations and primary resonances, respectively. Parametric investigations of temperature effects on the linear and nonlinear vibration characteristics of the suspended cable with different sag-to-span ratios and excitation amplitudes are performed. The results of the perturbation analysis show that the nonlinear free and forced vibration characteristics would be changed by the temperature variations qualitatively and quantitatively, depending on the sag-to-span ratio and the excitation amplitude. The asymmetric phenomena between the effects of warming and cooling conditions on the vibration characteristics can be observed. The crossover points between next two mode frequencies are shifted under the temperature variation, and these would significantly influence the internal resonances of the suspended cable. Finally, temperature effects on the time-history diagram of the cable axial total tension force are investigated.

Application of discontinuous Galerkin method to mechanical 2D problem with arbitrary polygonal and very high-order finite elements

Discontinuous Galerkin with finite difference rules (DGFD) is applied to mechanical plane stress state problem. The considered domain is discretized by polygonal mesh. The polygonal elements can be for example a hexagon, pentagon or just quadrangle or triangle. They do not have to be convex and a fish mesh, where the elements have fish shapes, is used. When the elements are rectangular then the orthogonality of Chebyshev basis functions can be utilized. In such a case very high-order approximate solution can be obtained. In this work the approximation order exceeds 10 and reaches 60, which in the latter case means 3600 numbers of degrees of freedom in a single element. The paper is illustrated by a benchmark example in which the exact solution is recovered by DGFD method for various meshes. In the other example the stress concentration is easily recovered by very high-order version of DGFD method.