Second-order Approximate Solution Research Articles

High-Order Models for Gas Transport in Multiscale Coal Rock.

The coal reservoirs exhibit great heterogeneity and strong anisotropy in multiscale pore/fracture structures. Developing highly accurate multiscale models for real-time prediction of microgaseous flow in complicated porous media with pronounced contrast in transport coefficients is crucial but not yet available, which is time-consuming, expensive, and even computationally impossible. In this study, a multiscale approximate solution of the gas flow and pressure field is derived in this paper for predicting coalbed methane (CBM) transport in macro-microscopic two-scale porous media of typical coal rocks in Guizhou Province, China, and the detailed finite element algorithm is established. The multiscale approximate solutions are constructed from the first- and second-order auxiliary cell functions and low- or high-order derivatives of the homogenized pressure field, which can accurately approximate the solution of the original problem to a certain extent. The numerical accuracy and validity of the multiscale approximate solutions under various working conditions are analyzed in detail by comparing and verifying the results with the direct numerical simulation results of fine-mesh high-precision finite elements. The results show that the homogenized approximate solution can only roughly capture the characteristics of the macroscopic pressure evolution, the first-order approximate solution can partially reproduce the macro-microscopic pressure oscillation phenomenon, and the second-order approximate solution can accurately predict the pressure profile and its evolution law with time in porous media with macro-micro configuration under various complex conditions. The second- and high-order two-scale approximate solutions constructed in this paper exhibit high numerical accuracy and excellent computational efficiency. We also develop multiscale approximate solutions for predicting microgaseous flow at a low reservoir pressure where the slippage effects are very pronounced. Potential applications of our high-order models to the coal reservoirs with a hierarchical matrix and fractures are discussed. The developed multiscale models exhibit excellent applicability to investigating the crossflow effects and coupling mechanisms between the matrix and cleats or fractures with multiple configurations in the CBM and shale gas reservoirs, which is crucial for the effective implementation of hydraulic fracturing technology. This study provides a fundamental understanding of gas transport in multiscale matrix-fracture networks, which helps to improve the optimization design of hydraulic fracturing in tight reservoirs.

Resonance and attraction domain analysis of asymmetric duffing systems with fractional damping in two degrees of freedom

This article establishes a two degree of freedom asymmetric Duffing system model with fractional damping through the study of the Jeffcott rotor system with horizontal support, and analyzes the dynamic behavior of its resonance and attraction domain structures. The second-order approximate solution of the system equation is obtained using a multiscale method, and the slow flow modulation equations of both the amplitudes and phases are extracted. The comparison between approximate and numerical solutions has shown their high consistency and the Routh Hurwitz criterion is utilized to study the stability of the system. Finally, we explore the effect of fractional damping on the vibration behavior of the system using amplitude frequency curves, attractive watersheds, and time trajectory plots, including both negative and positive values of the nonlinear stiffness coefficient. The analysis shows that if the coefficient and order of the fractional damping term are reasonably selected, the horizontal and vertical displacements of the system vibration can be effectively controlled. The importance of this work lies in its application in rotor vibration, which has significant implications for the study of the system under investigation.

Moment Lyapunov exponent and stochastic stability of a vibro-impact system driven by non-Gaussian colored noise

Vibro-impact phenomena are prevalent in practical engineering, making research on their stochastic dynamic characteristic of great practical significance. However, the research on stochastic stability and bifurcation of vibro-impact systems is still limited, especially the moment stability. In this paper, based on the pth moment Lyapunov exponent, the stochastic stability of a vibro-impact system driven by non-Gaussian colored noise is investigated. Firstly, the smooth stochastic dynamic system is obtained making use of a non-smooth transformation and the non-Gaussian colored noise is simplified to an Ornstein-Uhlenbeck process by utilizing the path-integral method. Thereafter, through applying the L.Arnold perturbation method, the second-order approximate solution of the pth moment Lyapunov exponent is calculated, which agree well with the simulation results given by the Monte Carlo method. Finally, the effects of the noise parameters, natural frequency, coefficient of restitution, and damping coefficient on the stochastic stability of the vibro-impact are studied. Due to the existence of impact factor, the natural frequency has a direct and significant effect on the stochastic stability of the system.

Method of directly defining the inverse mapping for nonlinear fuzzy heat-like partial differential equations

Natural phenomena or physical systems can be described using Partial Differential Equations (PDEs), such as wave equations, heat equations, Poisson’s equation, and so on. Consequently, investigations of PDEs have become one of the key areas of modern mathematical analyses, attracting a lot of attention. Many authors have recently expressed an interest in researching the theoretical framework of fuzzy Initial Value Problems (IVPs). The Method of Directly Defining the inverse Mapping (MDDiM) was effectively employed in this research to obtain the second-order approximate fuzzy solution of heatlike equations in one and two dimensions, and the results were compared with exact solutions. In each illustrated example, all the results achieved using Maple 16 were graphically depicted. This is the first time MDDiM was utilized to solve nonlinear Fuzzy Partial Differential Equations (FPDEs).

Investigation on the free vibration characteristics of vertical quasi-zero stiffness isolation system considering the disc spring's loading position

This study investigates a quasi-zero stiffness (QZS) isolation system designed for vertical vibration control in isolation structures. The QZS characteristics are achieved by connecting a negative stiffness disc spring in parallel with the linear positive stiffness spiral spring. Initially, the force-displacement relationship of the disc spring is established, taking into account the loading position, and validated through static experiments. Subsequently, the free vibration equations of the QZS system are derived. The Jacobi elliptic functions and the Newton-harmonic balance (NHB) method are then employed to study its free vibration characteristics without considering damping. The results demonstrate that the natural vibration period of the QZS system is related to the nondimensional amplitude and the height of the loaded inner cone. Furthermore, the second-order analytical approximate solution obtained via the NHB method exhibits remarkable accuracy. Finally, the critical damping ratio of the damped QZS isolation system is investigated using a numerical approach. The findings indicate that the critical damping ratio is only correlated with the nondimensional amplitude. This study addresses the gaps in existing research and provide recommendations for the parametric design of vertical QZS isolation system.

Second-Order Approximate Reflection Coefficient of Thin Interbeds with Vertical Fractures

The horizontal fractures in the strata will close in the compaction effect of overlying strata, while the vertical cracks are widely developed, which can be equivalent to HTI (transverse isotropy with a horizontal axis of symmetry) medium. When an S-wave propagates into HTI media, the shear wave will divide into two types of waves: a fast S-wave and slow S-wave. When the strata of HTI are thin and overlapping, called the thin interbeds model, the wave field exhibits complex primary reflections, converted waves, and multiples. We introduce a new second-order approximation of the total reflection coefficient, with the incidence angle lower than the critical angle in thin-interbed HTI media using a recursive algorithm. We verify the effectiveness of the second-order approximation by analyzing the energy of multiples. Comparing the second-order approximate solution that degenerates the HTI medium into isotropic and Kennett’s exact solution, we find that our solution has an accuracy of over 99.9% in any azimuth, with the incidence angle lower than the critical angle under P-wave incidence. However, our solution of the SP wave field is suitable for incidence azimuth angles between 0–75° and 120–180°, with the lowest accuracy occurring at an incidence angle of 25° and a relative error of 6.4%. The approximate solution in the SS wave field has the same applicable range as the SP wave, with the maximum error of 6.3% occurring at the incident angle of 1°. This new second-order approximate formula for the total reflection coefficient of thin interbeds composed of HTI helps us to understand the reflection characteristics of complex thin interbeds. It also lays a theoretical foundation for the development of AVO (Amplitude Versus Offset) analysis and inversion techniques for lithological and stratigraphic oil and gas reservoirs.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Analytic Osculating Frozen Orbits Under Perturbation

This work provides a set of closed-form analytical expressions to define osculating frozen orbits under the perturbation effects of the oblateness of the main celestial body. To this end, an analytical perturbation method based on osculating elements is proposed to characterize, define, and study the three existing families of frozen orbits in closed form: the two families of frozen orbits close to the critical inclination and the family of frozen orbits appearing at low eccentricity values. As such, this work aims to complement other analytical approaches based on mean elements by providing an alternative methodology based on the more natural osculating elements that is able to generate closed-form expressions for all known frozen conditions in the main satellite problem. Additionally, this work includes the first- and second-order approximate solutions of the proposed perturbation method, including their applications to the analytical definition of frozen orbits, repeating ground-track orbits, and sun-synchronous orbits under this perturbation. Examples of applications are also provided to show the expected error performance of the proposed approach.

Singular Perturbation Method Applied to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Initial Value Problem (IVP) and Boundary Value Problem (BVP). It is applied to two state switching power converters to provide rigorous justification of the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for IVP and BVP. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

Analytical study of the vibrating double-sided quintic nonlinear nano-torsional actuator using higher-order Hamiltonian approach

In this work, we investigate and apply higher-order Hamiltonian approach (HA) as one of the novelty techniques to find out the approximate analytical solution for vibrating double-sided quintic nonlinear nano-torsional actuator. Periodic solutions are analytically verified, and consequently, the relationship between the initial amplitude and the natural frequency are obtained in a novel analytical way. The HA is then extended to the second-order to find more accurate results. To show the accuracy and applicability of the technique, the approximated results are compared with the homotopy perturbation method and numerical solution. According to the numerical results, it is highly remarkable that the second-order approximate solutions produce better than previously existing results and almost similar in comparing with the numerical solutions.

Accurate analytical approximation to post-buckling of column with Ramberg−Osgood constitutive law

The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

Act of nonlinear proportional derivative controller for MFC laminated shell

This manuscript presents connect a Proportional-Derivative (PD) controller with a macro-fiber composite (MFC) laminated shell system. Nonlinear equations of the MFC laminated shell have been established. The simultaneous resonance case considered as worst resonance. The main motivation of this research is applying a suitable vibration control to reduce or diminish the high vibration caused by the model. So, we used here PD controller in controlling the behavior MFC laminated shell structure. The goal of using the PD controller is to increase the system stability by improving control since it has an ability to predict the future error of the framework response. This control method makes stability increased, maximum peak overshoot decreased, settling time decreases. This controller is applied to recover the framework transient response. A second-order approximate solution has gained via applying multiple time scale perturbation (MTSP) technique. Every expected resonance case at this approximation order is extracted. After adding the PD-controller of the current framework, it eliminates the vibration amplitudes of the considered structure. Frequency (angular speed) response curves examined at special uncontrolled and controlled coefficients of the system. Finally, the conclusion shows a high efficiency of the measured control algorithm for eliminating vibrations of the modified system. A comparison between other approaches control algorithms had been prepared. The results are compared with the numerical simulations, and this shows a good verification for the approximate solution and for the control algorithm used in this manuscript. A comparison with earlier available papers is involved at the end.

Analysis of Vibration Control of Nonlinear Beam Using a Time-Delayed PPF Controller

This paper presents a study on the performance of a positive position feedback (PPF) controller to suppress the vibration of a horizontal beam under vertical excitation. Time delays in the control loop are taken into consideration to study their effects on the controller performance and the stable region. The integral iterative method is conducted to obtain a second-order approximate solution and the corresponding amplitude equations for the considered system. The stability of the steady-state solutions is ascertained using a combination of Floquet theory and Hill’s determinant. The maximum limits of time delays at which the system remains stable have been determined for different values of control parameters. And the effects of various control parameters on the existence of multiple-solution region are investigated. The analysis illustrates that the appearance of time delay and the elimination of controller damping coefficient are the two main factors to enhance the nonlinear characteristics of the controlled system. The points at which the steady-state amplitude of the main system reaches its minimum are studied analytically. The analyses show that the analytical results are in excellent agreement with the numerical simulations.

A multiscale parallel algorithm for dual-phase-lagging heat conduction equation in composite materials

In this paper, a new numerical scheme which combines the multiscale asymptotic method and the Laplace transformation, is presented for solving the 3-D dual-phase-lagging equation in composite materials. The convergence results of the truncated first-order and second-order multiscale approximate solutions are given rigorously. The numerical experiments are carried out to validate the theoretical results of this paper. It is pointed out that the proposed method allows us to choose a relative coarse grid and solve problems in parallel, and therefore it can greatly save computer memory storage and CPU time.

Singular Perturbation Method for Boundary Value and Optimal Problems to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Boundary Value Problem (BVP) and Optimal Problem. It is applied to two state switching power converters to provide rigorous justification of\ the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for BVP and optimal control problems. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

Crashworthiness design of graded cellular materials: An asymptotic solution considering loading rate sensitivity

Introducing graded cellular materials into energy absorber may improve the crashworthiness of protective structures. In order to meet the crashworthiness requirement that the impact force does not exceed the value that the protected object can afford, a backward strategy guiding the design of relative density distribution of cellular material is improved by considering the loading rate sensitivity in a shock model. Asymptotic solutions of the relative density distribution are obtained for the cases with or without considering the loading rate sensitivity, when a constant impact force is required. It is found that the second-order approximate solutions of the relative density distribution are good enough to approximate the exact solutions calculated by the fourth-order Runge-Kutta scheme. The validity of the design method is further verified by the cell-based finite element method, and the results indicate that the deformation localization at the proximal end is caused by both the density gradient and inertial effect. It is shown that the average impact force from the design without considering loading rate sensitivity exceeds the affordable value of the protected object, but that considering loading rate sensitivity can well meet the crashworthiness requirement. Therefore, the asymptotic solution of the relative density distribution from the design strategy considering loading rate sensitivity is suitable and convenient to guide the crashworthiness design in practical engineering.

Effect of soil mass motion on nonlinear forced vibrations of beams on elastic foundation

Based on the nonlinear equation of motion of the beam on the Winkler foundation with the consideration of finite-depth soil mass motion, the nonlinear forced vibrations of the beam were investigated. Applying the Galerkin method and the method of multiple scales, the frequency-response equation and the second-order approximate solution of the primary resonance of the beam were obtained. Then, by means of the numerical calculation and parameter analysis, the effects of parameters about the soil mass motion on the frequency-response curves of the beam on the Winkler foundation were explored, such as foundation depth, soil mass, and Winkler foundation stiffness. The results show that the effect of the soil mass motion on the nonlinear forced vibration of the beam on the Winkler foundation is significant. When the effect of soil mass motion is introduced into the equation of motion of the beam, the range and the softening behavior of primary harmonic resonance of the system are reduced.

Simultaneous Resonances of Suspended Cables Subjected to Primary and Super-Harmonic Excitations in Thermal Environments

This paper concerns with a suspended cable in thermal environments under bi-frequency harmonic excitations, with a focus placed on the effect of temperature changes on one type of simultaneous resonance. First, the nonlinear equation of motion in thermal environments is obtained for the in-plane displacement of the cable. Then, the Galerkin method is employed to reduce the partial differential equation to an ordinary one. Second, based on the discretized form of the governing equation, the method of multiple scales is employed to obtain the second-order approximate solutions, with the stability characteristics determined. Third, numerical results are presented by using the perturbation method, together with numerical integration by the following means: frequency-response curves, time-displacement curves, phase-plane diagrams, and Poincare sections. The direct integration method is utilized to verify the results obtained by the perturbation method, while revealing more nonlinear dynamic behaviors induced by temperature changes. Both the softening and/or hardening behaviors, and the switching between them are observed for the cable in thermal environments. The response amplitude of the cable is very sensitive to temperature changes, but the number of circles in the phase diagrams and the number of cluster points in Poincaré sections is independent of the thermal effects in most cases. Finally, the vibration characteristics of the cable for different thermal expansion coefficients and temperature-dependent Young’s moduli are also investigated.

Thermo-mechanical analysis of nonlinear heterogeneous materials by second-order reduced asymptotic expansion approach

An effective second-order reduced asymptotic expansion (SRAE) approach is proposed to analyze the thermo-mechanical coupling problems of nonlinear periodic heterogeneous materials. The first-order and second-order nonlinear unit cell solutions at a microscale obtained by calculating the different multiscale cell functions are given at first. Then, the homogenization coefficients are evaluated, and the nonlinear homogenized equations defined on whole structure are solved, successively. Also, the temperature and displacement fields are constructed as second-order multiscale approximate solutions by assembling the distinct local cell solutions and homogenized solutions. The main features of the proposed approach are: (i) an effective model reduction scheme for computing high-order nonlinear unit cell problems and (ii) an asymptotic high-order homogenization solution that does not need high-order continuity of the macroscale solutions. Further, the corresponding finite element-difference algorithms based on the SRAE approach are given in detail. Finally, by some typical numerical examples, the availability and correctness of the proposed algorithms are confirmed. The computational results clearly demonstrate that the SRAE approach reported in this work are efficient and valid to predict the macroscopic thermo-mechanical properties, and can catch the microscale information of the heterogenous materials accurately.

Nonlinear normal modes and primary resonance for permanent magnet synchronous motors with a nonlinear restoring force and an unbalanced magnetic pull

The unbalanced magnetic pull (UMP) in permanent magnet synchronous motors (PMSMs) leads to the generation of vibrations and noise because of mechanical and magnetic coupling effects. This paper investigates the nonlinear oscillations of a PMSM based on a Jeffcott rotor-bearing system, and the effects of the UMP, nonlinear restoring forces due to the Hertz contact force and bearing clearance, rotor weight and rotor mass eccentricity are considered. A second-order approximate solution in the primary resonance case is obtained by applying the multiple-scale perturbation method. The Routh–Hurwitz criterion is utilized to investigate the stability of the obtained solution. The linear natural frequencies of the horizontal and vertical modes show little difference based on the constructed model. The localized oscillations and nonlocalized oscillations of the rotor-bearing system are analyzed through its frequency response curves by introducing different values of system parameters. The response curves exhibit hard spring characteristics with unstable regions and jump phenomena. Detailed numerical research including Poincare map and bifurcation diagram reveals the effect of excitation amplitude on the system. Additionally, this work provides some theoretical and practical guidance for the design of PMSMs with strong dynamic behaviors.