Nonlinear Equations Of Motion Research Articles

Nonlinear vibrations of graphene nanoplates with arbitrarily orientated crack located in magnetic field using nonlocal elasticity theory

PurposeThe study explores frequency curves and natural frequencies as functions of crack length, crack angle, magnetic field strength and small size effects under the three boundary conditions.Design/methodology/approachThis study investigates the nonlinear dynamics of a single-layered graphene nanoplate with an arbitrarily oriented crack under the influence of a magnetic field. The research focuses on three boundary conditions: simply supported, clamped and clamped-simply supported. The crack effect is modeled by incorporating membrane forces and additional flexural moments created by the crack into the equation of motion.FindingsResults reveal that increasing the crack length, small size effects and magnetic field intensity reduces the flexural stiffness of the nanoplate, increases the compressive load and lowers its natural frequency. Additionally, excessive magnetic field intensity may lead to static buckling. The critical dimensionless magnetic fields are found to be 33.6, 95.1 and 72.3 for All edges of the nanoplate are simply supported (SSSS), fully clamped edges (CCCC) and two opposite edges are clamped and the other are simply supported (CSCS) nanoplates, respectively. Furthermore, for SSSS and CCCC boundary conditions, an increase in the crack angle results in a softening behavior of the hard spring. In contrast, the SCSC boundary condition exhibits the opposite behavior. These findings emphasize the importance of considering the effects of angled cracks and electromagnetic loads in the analysis and design of graphene-based nanostructures.Originality/valueNovel equations are derived to account for the applied loads induced by the magnetic field. The nonlinear equation of motion is discretized using the Galerkin technique, and its analytical response is obtained via the multiple time-scales perturbation technique.

Nonlinearity Harmonic Error Compensation Method Based on Intelligent Identification for Rate Integrating Resonator Gyroscope

This paper presents an improved intelligent optimizing algorithm based on parameter identification and drift compensation for the rate-integrating resonator gyroscope (RIRG). Besides damping and frequency imperfections, the RIRG measurement accuracy still suffers from limitations due to nonlinear error. Therefore, the dynamic nonlinear error model for RIRG has been established to reveal the relationship between the pattern angle and harmonic drifts. Based on this, a dynamic analysis of the gyroscope operating state is carried out using nonlinear motion equations. The optimal harmonic error parameters are then identified by a particle swarm optimization (PSO) algorithm for the drift error compensation. To further improve the measurement accuracy, chaotic technology is integrated with PSO, leading to more precise identification. Subsequently, the harmonic parameters of the bias drifts are efficiently compensated. Experimental results demonstrate that the bias drift is reduced by over 90% after harmonic error compensation, demonstrating the validity of the proposed method in enhancing the measurement accuracy of RIRGs.

Pendulum between two springs using Ms- DTM

Abstract This study provides a comprehensive analysis of a pendulum system suspended between two horizontal springs, focusing on its dynamic behavior and the application of advanced mathematical methods. The research begins by formulating the Lagrangian of the system, which serves as the foundation for deriving the equations of motion, offering a rigorous mathematical framework for understanding the system’s dynamics. The nonlinear equations of motion are then solved using the multi-step differential transformation method (Ms-DTM), whose accuracy and effectiveness are evaluated through comparison with the well-established Runge–Kutta method. This comparative analysis highlights the strengths and limitations of each numerical approach in managing the complexity of the system. The study also includes a series of simulated plots that depict the system’s behavior under various initial conditions and parameter settings. These visualizations provide valuable insights into the system’s dynamic responses, revealing phenomena such as chaotic motion and periodic oscillations. The results underscore the utility of sophisticated mathematical methods in addressing complex physical problems and enhancing our understanding of nonlinear dynamic systems.

On the Efficacy of Sparse Representation Approaches for Determining Nonlinear Structural System Equations of Motion

Abstract A sparsity-based optimization approach is presented for determining the equations of motion of stochastically excited nonlinear structural systems. This is done by utilizing measured excitation-response realizations in the formulation of the related optimization problem, and by considering a library of candidate functions for representing the system governing dynamics. Note that a novel aspect of the approach relates to treating, also, systems endowed with fractional derivative elements. Clearly, this is of significant importance to a multitude of diverse applications in engineering mechanics taking into account the enhanced modeling capabilities of fractional calculus. Further, the fundamental theoretical and computational aspects of various representative, state-of-the-art, numerical schemes for solving the derived sparsity-based optimization problem are reviewed and discussed. A Bayesian compressive sampling approach that exhibits the additional advantage of quantifying the uncertainty of the estimates is considered as well. Furthermore, comparisons and a critical assessment of the employed numerical schemes are provided with respect to their efficacy in determining the nonlinear structural system equations of motion. In this regard, two illustrative numerical examples are considered pertaining to a nonlinear tuned mass-damper–inerter vibration control system and to a nonlinear electromechanical energy harvester, both endowed with fractional derivative elements.

Analysis of electron spectra dynamics in a moving periodical ponderomotive potential

Abstract The interaction between freely propagating electrons and light waves is typically described using an approximation in which we assume that the electron’s velocity nearly does not change during the interaction. In this article we analytically describe the dynamics of electrons in an interaction potential generated by an optical beat wave beyond this regime and find a structure of sharp electron distribution peaks that periodically alternate in the energy/momentum spectrum. In classical description we analytically solve the nonlinear equation of motion, which is an analogy to mathematical pendulum. While addressing the problem using quantum mechanics we first use parabolic approximation of the interaction potential and then we also study the evolution of electron wavepacket in an infinite periodical potential. Using numerical simulations we show the classical and quantum evolution of the electron spectra during the interaction for different conditions and experimental settings.

Non-linear equation of motion for higher curvature semiclassical gravity

Non-linear equation of motion for higher curvature semiclassical gravity

Nonlinear dynamic thermal buckling behavior of FG-GPLRC spherical shells and circular plates with porous core

A semi-analytical approach for dynamic buckling of sandwich functionally graded graphene platelet-reinforced composite (FG-GPLRC) spherical shells and circular plates under dynamic thermal loads with porous core is reported in the present research. Based on the higher-order shear deformation theory (HSDT), the formulations are established, and the large deflection nonlinearity of von Karman with the visco-elastic model of the nonlinear foundation is applied. The structure's nonlinear equations of motion can be obtained utilizing the Euler-Lagrange equations combined with Rayleigh dispersion functions. The dynamic thermal load is assumed to be a linear function of time. Numerical studies are investigated employing the Runge-Kutta method to obtain the dynamic thermal behavior, and the dynamic criterion of Budiansky-Roth can be used to determine the critical buckling temperature. Significant remarks on the dynamic thermal behavior of structures are presented through the investigated examples.

Model of sensitivity to the initial conditions of the equations of motion in the electric fieldn

A numerical method for calculating ste¬ady periodic processes of nonlinear differential equations of motion in an electric field is proposed based on their integration under initial conditions that exclude transi¬ent reaction. The calculation of such initial conditions is based on Newton's iterative formula. At each iteration, together with the basic equations of motion, the corresponding equations of the first variation are integrated over the periodicity time interval, which represent a model of sensitivity to the initial conditions, In addi¬tion, the number of unknown variational equations has been reduced, which makes it possible to lower their or¬der, and, at the same time, the order of the monodromy matrix. All this contributes to the simplification of the relevant computational algorithms and computer programs. The method takes into account the finite rate of electric field propagation.

Size-dependent nonlinear dynamics of two-directional functionally graded microbeams in thermal environment under a moving mass

The size-dependent nonlinear dynamics of two-directional functionally graded (2D-FG) microbeams in a thermal environment under a moving mass is studied for the first time in the framework of the refined higher-order shear deformation theory (HSDT) and modified couple stress theory (MCST). The variation of the temperature-dependent material properties in the axial and thickness directions is estimated by the Mori–Tanaka homogenization scheme. Considering the rotary inertia and shear deformation, the nonlinear differential equations of motion, obtained from Hamilton’s principle, are discretized using an enriched beam element. Numerical results determined by the Newmark method in combination with Newton–Raphson iterative procedure confirm the efficiency of the derived beam formulation in predicting the nonlinear dynamics. The enriched beam element, which is derived herein by using hierarchical functions to enrich the conventional shape functions, is capable of furnishing accurate nonlinear dynamic characteristics with fewer elements compared to the conventional one. It is shown that the difference between the dynamic response predicted by the linear analysis and the nonlinear one increases by increasing the temperature, but it decreases by increasing the size scale parameter. The effects of the material gradation, temperature rise and micro-scale parameter on the nonlinear dynamic behavior of the 2D-FG microbeams are studied in detail and highlighted.

Asymptotic Numerical Method for dynamic buckling of shell structures with follower pressure

Asymptotic Numerical Method (ANM) is applied to non-linear dynamics of thin-shells subjected to conservative and non-conservative loads such as follower pressure. ANM is decomposed into several stages: the finite element discretization of the non-linear equations of motion of the shell dynamics, a homotopy transformation of the semi-discrete non-linear equations, a perturbation technique to expand the quantities into Taylor series according to the homotopy parameter and the time integration scheme to solve the series of linear problems resulting from the perturbation technique. ANM is applied here with the 7-parameter shell elements thanks to the Enhanced Assumed Strain (EAS) concept and implicit Newmark integration. In the case of non-conservative force, follower pressure also requires to be decomposed in either Taylor series or rational Padé approximants. The academic case of the cylindrical roof with dynamic snap-through phenomenon is investigated for the purpose of comparing ANM strategies and the classical Newton–Raphson (NR) method. Two engineering cases including an I-shaped thin-walled beam and a closed thin-shell cylinder under dynamic external follower pressure are also investigated. ANM turns out to be accurate, robust and efficient in terms of computation time, providing an alternative method to the well-established Newton–Raphson method.

Vibration suppression of suspended cables with three-to-one internal resonances via time-delay feedback

Based on the time-delay feedback control, the vibration suppression of suspended cables with three-to-one internal resonances are investigated. Initially, the nonlinear differential equation of motion for a suspended cable under time-delay feedback control is considered, and a discrete model is derived using the Galerkin method. Subsequently, the method of multiple scales is employed to perturbatively solve the discrete time-delay differential equation, determining the modulation equations around the first primary resonance. Steady-state and periodic solutions of the modulation equations are detected numerically. Numerical results indicate that the internal resonance enhances the nonlinear dynamical complexity of the controlled suspended cable. It is observed that the time delay and control gain affect the controlled system: in particular, an increase in control gain leads to a reduction in response amplitude. By adjusting the time delay and control gain, the critical excitation can be altered, an aspect that could be very useful from a practical point of view. This research sheds light on the intricate dynamics of suspended cable and provides a theoretical foundation for designing more effective control strategies in engineering applications.

Chaotic vibration control of an axially moving string of multidimensional nonlinear dynamic system with an improved FSMC

A new control approach based on fuzzy sliding mode control (FSMC) is proposed to regulate the chaotic vibration of an axial string. Hamilton’s principle is used to formulate the nonlinear equation of motion of the axial translation string, and the von Kármán equations are used to analyse the geometric nonlinearity. The governing equations are nondimensionalized as partial differential equations and transformed into a nonlinear 3-dimensional system via the third-order Galerkin approach. An active control technique based on the FSMC approach is suggested for the derived dynamic system. By using a recurrent neural network model, we can accurately predict and effectively apply a control strategy to suppress chaotic movements. The necessity of the suggested active control method in the regulation of the nonlinear axial translation string system is proven using different chaotic vibrations. The results show that the study of the chaotic vibrations of axially translating strings requires nonlinear multidimensional dynamic systems of axially moving strings; the validity of the proposed control strategy in controlling the chaotic vibration of axially moving strings in a multidimensional form is demonstrated.

A Higher-Order Theory for Nonlinear Dynamic of an FG Porous Piezoelectric Microtube Exposed to a Periodic Load

This paper investigates the nonlinear dynamic deflection, natural frequency, and wave propagation in functionally graded (FG) porous piezoelectric microscale tubes under periodic load, hygrothermal conditions, and an external electric field. The piezoelectric material used to make the smart microtubes has pores that may be smoothly changed or uniformly distributed over the tube wall. Here, three types of porosity distribution are taken into consideration. The nonlinear motion equations are constructed using a novel shear deformation beam theory and the modified couple stress theory (MCST). The nonlinear motion equations are solved using the fourth-order Runge–Kutta technique and the Galerkin approach. The effects of various geometric parameters, porosity distribution type, porosity factor, periodic load amplitude and frequency, material length scale parameter, moisture, and temperature on the nonlinear dynamic deflection, natural frequency, and wave frequency of FG porous piezoelectric microtubes are explored through a number of parametric investigations.

Dynamic analysis of a high-speed micro compressor supported by spiral-grooved hydrodynamic air bearing

Spiral-grooved hydrodynamic air bearings are used in high-speed machines due to their advantages of oil-free, low friction, and compact structure. This paper aims to investigate the dynamic characteristics of a high-speed rotor supported by spiral-grooved hydrodynamic air bearings and the effects of spiral groove parameters. A mathematical model of the rotor-bearing system is developed by coupling the rotor nonlinear motion equations and the bearing dynamic Reynolds equation. The finite element method combined with the Runge-Kutta method is employed simultaneously to obtain the nonlinear trajectory of the rotor-bearing system. The theoretical model is verified by experiments on the built micro compressor prototype. The numerical predicted onset speed of the sub-synchronous motion of the rotor-bearing system agrees well with the experimental observations. Furthermore, the effects of radius clearance, groove depth, groove angle, groove axial and circumferential width on the dynamic performance of the rotor-bearing system are analyzed. The results show that radius clearance has the most significant influence, followed by groove depth, groove axial, and then circumferential width. The groove angle has the least impact.

A novel approach to solving Euler’s nonlinear equations for a 3DOF dynamical motion of a rigid body under gyrostatic and constant torques

This research focuses on approaching a new technique to solve the rotary movement of a three degrees-of-freedom (DOF) semi-symmetric rigid body (RB) affected by a gyrostatic torque (GT) and under the influence of the RB’s constant-fixed torques (RBCFTs). The governing nonlinear differential equations (NDEs) for the RB are derived from the famous Euler’s equations, along with another system to calculate Euler’s angles, which gives an accurate solution for the RB’s position during its motion. In the case of a semi-symmetric body about its minor axis, novel analytical solutions for the RB’s angular velocities are presented in addition to the solutions of Euler’s angles. Our procedure to obtain these solutions arises through decoupling the governing NDEs into two coupled DEs, and the averaged time parameter is used. General solutions of the later DEs are presented in view of series expansions. The impact of the internal and external torques on the body’s motion is graphically simulated, which gives an overview of the periodicity of the derived solution and the effect of various values of these torques on the solution. Moreover, these simulations present other prospects for the body’s stability by analyzing them using the famous Lyapunov function. The importance of this study arises from its wide applications in our lives in many fields, such as mathematics and physics. It also holds immense potential for enhancing mechanical systems, elucidating celestial motion, and enhancing spacecraft efficiency.

Fractional mimetic dark matter: A fractional action-like variational approach

In this paper, we propose a nonlocal extension of the mimetic dark matter model based on the FALVA implementation of fractional calculus. Our primary objective to explore how certain properties of dark matter can be modeled within the FALVA framework rather than formulate a phenomenological model. We begin by constructing the action functional of the cosmological extension of the mimetic dark model and deriving the nonlinear equations of motion in the general case. Next, we focus on a fractional-power homogeneous mimetic dark field with an exponential expansion factor. We derive the equation of motion for the lapse field and obtain its general solution. Analytical solutions are then obtained for small time intervals and arbitrary values of the fractionality parameter. These solutions enable us to establish the physical line element of spacetime, incorporating the mimetic dark matter field. From this line element, we derive the Ricci tensor, Ricci scalar, and geodesic equations.

Equivalent spring-like system for two nonlinear springs in series: application in metastructure units design

The paper deals with the problem of design of unit in auxetic metastructure. The unit is modeled as a two-part spring-like system where each part is with individual stiffness. To overcome the problem of analyzing of each of parts separately, the equivalent spring is suggested to be introduced. In the paper, a method for obtaining the equivalent elastic force of the unit is developed. The method is the generalization of the procedure suggested for substitution of a hard and a soft spring in series with an equivalent one. The nonlinearity of original springs is of quadratic order. As a results, it is obtained that the equivalent elastic force for two equal springs remains of the same type as of the original springs (soft or hard). For two opposite type springs in series with equal coefficients, the equivalent force is soft. The method is applicable for any hard and soft nonlinear springs or spring-like systems. Thus the hexagonal auxetic unit which contains a soft and a hard part in series is analyzed. In the paper, a new analytic method for determination of the frequency of vibration for the unit under action of a constant compression force acting along the unit axis is introduced. The method is applied for units which contain two parts: hard–hard, soft–soft, hard–linear, soft–linear and opposite. The obtained approximate vibration results are compared with numerically obtained ones and show good agreement. The advantage of the method is its simplicity as it does not require the nonlinear equation of motion to be solved.

A semi-analytical methodology for predicting the vibroacoustic response of functionally graded nanoplates under thermal loads

In this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of nanoplates made of functionally graded materials considering the nonlocal strain gradient theory is presented. It is presumed that the properties of the functionally graded nanoplate are in the form of the simple power law scheme and continuous along the thickness, under nonlinear temperature rise and incident oblique acoustic wave as well as the first-order shear deformation theory. For this purpose, applying Hamilton’s principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the von Kármán nonlinear strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to the Duffing equation. Then, applying the homotopy analysis method, the equation of the transverse movement of the nanoplate is solved semi-analytically to obtain the nonlinear frequencies. Lastly, the nonlinear vibration and acoustic response of functionally graded nanoplate are investigated by considering the variation of the significant factors such as material gradient indices, nonlocal parameters, length scale parameters, aspect ratio, dimensionless amplitude, nonlinear temperature changes, and sound characteristics of the functionally graded nanoplate.

Periodic free vibrations of composite laminates with curvilinear fibres and CNTs

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young’s modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.