Nonlinear Equations Of Motion Research Articles

Nonlinear free vibration analysis of the rectangular conductive elastic plate in magnetic field based on homotopy perturbation method

AbstractThis article tries to investigate the nonlinear free vibrations of the rectangular conductive elastic plate in uniform magnetic fields under the classic plate theory considering nonlinear strain‐displacement. The formulation of the governing equations integrates the electromagnetic forces arising from the motion of the plate. The nonlinear motion equations are dimensionless, which takes the effect of in‐plane inertia into account. The equations are solved by the Galerkin method and homotopy perturbation method (HPM). The effectiveness of the solution is verified. According to the solutions by HPM, the effects of the initial conditions, length‐to‐thickness ratio, and magnetic induction intensity on the nonlinear free vibrations behavior of conductive elastic plates are discussed.

Dynamic response of a pipe conveying pulsating fluid under random vibration

This paper studies the dynamic response of a pipe conveying pulsating fluid under random vibration. Its nonlinear equation of motion is first developed using the method of Newton. The method of Galerkin is then used to discretize the equation of motion into a system of ordinary differential equations. Afterwards, the statistical moments and mean peak values of the displacement and velocity responses of the pipe are obtained using an efficient Monte Carlo simulation (MCS) method based on the explicit time-domain method (ETDM). Furthermore, the displacement-sensitive and velocity-sensitive locations of the pipe are obtained for different boundary conditions. The impacts of the fluid, random vibration, and structural parameters on the mean peak value of the dynamic response of the pipe are finally analyzed. The obtained results provide a reference for better understanding the dynamic characteristics of pulsating fluid transport pipes under random vibration and for accurately designing pipe structures.

Nonlinear dynamic analysis and vibration reduction of two sandwich beams connected by a joint with clearance

The dynamics and vibration reduction characteristics of the clamped–clamped two sandwich beams jointed with clearance is studied theoretically and experimentally. A transverse and torsional spring system with clearance is used to equivalent the joint model. The homogenization method is used to equivalent the core layer and Rayleigh-Ritz method is utilized to derive the mode function of the interconnected sandwich beam by using a sequence of orthogonal polynomials. The nonlinear motion equation of the two jointed sandwich beam structure with clearance is derived by the application of the Hamilton principle and then solved using an improved Newmark integration approach. In order to validate the accuracy of the natural frequency and vibration mode, the finite element model is established. This paper examines the impact of clearance on the amplitude frequency response and vibration transmission of the two jointed sandwich beam structure, it is found the jointed sandwich beams show obvious nonlinear characteristics and intermittent vibration transmission phenomenon due to the clearance. Moreover, the vibration transmission analysis reveals that the existed clearance demonstrates significant vibration reduction effect, for which an experiment is conducted to validate the results. In general, this work proposes a novel approach for modeling sandwich structures with clearance with improved vibration reduction performance.

Extraction of the Descriptive Function of a Folding Wing With Free-Play Using the Modal Test Results to Be Used in the Nonlinear Motion Equation of the Structure

Extraction of the Descriptive Function of a Folding Wing With Free-Play Using the Modal Test Results to Be Used in the Nonlinear Motion Equation of the Structure

Analytical Framework for Tension Characterization in Submerged Anchor Cables via Nonlinear In-Plane Free Vibrations

Submerged tensioned anchor cables (STACs) are pivotal components utilized extensively for anchoring and supporting offshore floating structures. Unlike tensioned cables in air, STACs exhibit distinctive nonlinear damping characteristics. Although existing studies on the free vibration response and tension identification of STACs often employ conventional Galerkin and average methods, the effect of the quadratic damping coefficient (QDC) on the vibration frequency remains unquantified. This paper re-examines the effect of bending stiffness on the static equilibrium configuration of STACs, and establishes the in-plane transverse free motion equations considering bending stiffness, sag, and hydrodynamic force. By introducing the bending stiffness influence coefficient and the Irvine parameter, the exact analytical solutions of symmetric and antisymmetric frequencies and modal shapes of STACs are derived. An improved Galerkin method is proposed to discretize the nonlinear free motion equations ensuring the accuracy and applicability of the analytical results. Additionally, this paper presents an analytical solution for the nonlinear free vibration response of the STACs using the improved averaging method, along with improved frequency formulas and tension identification methods considering the QDC. Through a case study, it is demonstrated that the improved methods introduced in this paper offer higher accuracy and wider applicability compared to the conventional approaches. These findings provide theoretical guidance and reference for the precise dynamic analysis, monitoring, and evaluation of marine anchor cable structures.

Nonlinear vibration of sandwich beams made of FGM faces and FGP core under multiple moving loads using a quasi-3D theory

This study aims to investigate nonlinear dynamic behavior of sandwich beams constituted by functionally graded material (FGM) faces and functionally graded porous (FGP) core under the action of multiple moving loads. The energy equations based on a quasi-3D beam theory considering several functions of shear deformation were established for producing the nonlinear equations of motion used for describing dynamic responses of beams with various boundary conditions. By using the hybrid processes of the Newton-Raphson iteration procedure, the time-integration approach of Newmark-β, and the Gram-Schmidt-Ritz method, the new results of nonlinear analysis were explored and proposed. Several prime parameters such as the porous coefficient, sandwich thickness ratio, power law index and others that significantly affect the dynamic response of the beams were taken into investigations. It can be concluded that beams under one moving load have larger dynamic deflections compared to that under many moving loads. This is because the distance between the moving loads can produce a bending moment against the beam deformation.

Dynamic and stability analysis of crescent geometry-possessing textured journal bearing using nanolubricant

PurposeThis study aims to find the dynamic performance parameters of the journal bearing with micro geometries patterning the arc (crescent) shape textures provided in three specific regions of the journal bearing: the full, the second half and the increasing pressure region. The dynamic behavior of textured journal bearings has been analyzed by computing dynamic parameters and linear and non-linear trajectories.Design/methodology/approachThe lubricant flows between the bearing and journal surface are governed by Reynold’s equation, which has been solved by finite the element method. The dynamic performance parameters such as stiffness, damping, threshold speed, critical mass and whirl frequency ratio are examined under various operating conditions by considering various ranges of eccentricity ratios and texture depths. Linear and non-linear equations of motion have been solved with Ranga–Kutta method to get journal motion trajectories. Also, the impact of adding aluminum oxide and copper oxide nanoparticles to the base lubricant in combination with arc-shaped textures is analyzed to further see any enhancement in the performance parameters.FindingsThe findings demonstrated that direct stiffness and damping parameters increased to their maximum level with six textures in the pressure-increasing region when compared with the untextured surface. Also, nanoparticle additives showed improvements above the highest value attained with no inclusion of additives in the same region or quantity of textures.Originality/valueEngineers may design bearings with improved stability and overall performance if they understand how texture form impacts dynamic properties.

Bidirectional pendulum-type tuned mass damper inerter for synchronously controlling the alongwind and crosswind responses of super-tall buildings

In the context of slender super-tall buildings, which have a high height-to-width ratio and low damping, significant bidirectional vibrations can occur under strong winds. Traditional TMDI device is mainly focused on controlling unidirectional wind-induced vibration responses. To address this issue, the present study introduces a bidirectional pendulum-type tuned mass damper inerter (BPTMDI) designed to synchronously mitigate alongwind and crosswind responses. The study derives nonlinear equations of motion for a multiple degrees of freedom (MDOF) structure integrated with the BPTMDI, using the Euler-Lagrange method. These equations capture the three-dimensional motion characteristics of the BPTMDI. Additionally, linear equations of motion are derived for a simplified planar model. The time-domain responses of the MDOF structure coupled with the BPTMDI are calculated by dividing the system into linear and nonlinear subsystems and applying the state space method and Runge-Kutta method, respectively. The effectiveness of the BPTMDI is evaluated through time history response analysis of a 400 m tall case-study building subjected to bidirectional wind loadings. Results show that the BPTMDI can significantly reduce both alongwind and crosswind responses, with reductions of at least 29.75 % in displacement and 48.64 % in acceleration, regardless of wind direction. Furthermore, the BPTMDI demonstrates improved control efficacy compared to BPTMD systems. Lastly, the simplified planar model of the BPTMDI shows basic similarity to the nonlinear planar-spherical model.

Nonlinear adaptive pose motion control of a servicer spacecraft in approximation with an accelerated tumbling target

Removing a limited number of large debris can significantly reduce space debris risks. These bodies are generally exposed to extreme environmental disturbance torques or consecutive accidents due to their large wet area, which causes them to experience accelerated high-rate tumbling motion. The existing literature has adequately explored the approximation operations with non-cooperative targets exhibiting 3-axis tumbling motion. However, the research gap lies in the lack of attention given to addressing this approximation for targets undergoing accelerated motion. Agile, accurate, and large-angle maneuvers are three common necessities for safely capturing such targets. Changes in the moment of inertia brought on by fuel slushing cannot be disregarded during such a maneuver. To deal with nonlinearities, adverse coupling effects, actuator saturation constraints, time-varying moment of inertia, and external disturbances that worsen during accelerated agile large-angle maneuvers, a novel adaptive control approach is developed in this paper. The controller's main advantage is its adjustable desired acceleration, which maintains its performance even when dealing with accelerated motion. The control law is directly synthesized from the nonlinear relative equations of motion, without any linearization or simplification of the system dynamics, making it robust to a variety of orbital elements and target behaviors. Adaptation laws are extracted from the Lyapunov stability theorem in a way that guarantees asymptotic stability. Moreover, control actuator roles (delay, saturation, and allocation) are accounted for in modeling and simulation. Finally, a comprehensive numerical simulation based on three different realistic and strict scenarios is carried out to demonstrate the effectiveness and performance of the proposed control approach. The controller's robustness against time-varying dynamic parameters (sharp and sudden change, smooth and slow change, and periodic change) is extensively demonstrated through simulation.

ALGORITHM FOR ESTIMATING THE PARAMETRIC ROLL MOTIONS

For a ship, both synchronous and parametric roll motions can be modelled by nonlinear second-order differential equations with the roll angle as dependent variable and the nonlinearities coming from the damping and restoring moments. In the absence of exact solution techniques, approximate solutions for such equations can be obtained, in principle, analytically or numerically. In this paper, we focused on a fast and accurate recursive scheme based on the segmentation of the time domain into a fine grid of intervals of equal lengths. Due to the smallness of each time subinterval, a reset of the terms of the initial nonlinear equation of motion allows replacing it with a linear one in which the constant coefficients represent approximate values of the former variable coefficients in the middle of each subinterval. The linear equation is solved via the Laplace transform and the resulting solution together with its first derivatives is used to generate the iterative algorithm. This technique was applied to the case of two typical roll equations. The first describes the synchronous roll of a vehicle ferry model equipped or not with bilge keels while the second estimates the parametric roll of a container ship model. In both cases, the analyzed recursive algorithm not only generated results in full agreement with those provided by the ode45 solver in Matlab but also lead to a gain in computation time and offered more flexibility in imposing some restrictive conditions associated, for example, with exceeding some oscillation amplitudes or to the ship capsizing. Key words: synchronous and parametric ship rolling, iterative scheme

Mechanical Neural Networks with Explicit and Robust Neurons.

Mechanical computing provides an information processing method to realize sensing-analyzing-actuation integrated mechanical intelligence and, when combined with neural networks, can be more efficient for data-rich cognitive tasks. The requirement of solving implicit and usually nonlinear equilibrium equations of motion in training mechanical neural networks makes computation challenging and costly. Here, an explicit mechanical neuron is developed of which the response can be directly determined without the need of solving equilibrium equations. A training method is proposed to ensure the robustness of the neuron, i.e., insensitivity to defects and perturbations. The explicitness and robustness of the neurons facilitate the assembly of various network structures. Two exemplified networks, a robust mechanical convolutional neural network and a mechanical recurrent neural network with long short-term memory capabilities for associative learning, are experimentally demonstrated. The introduction of the explicit and robust mechanical neuron streamlines the design of mechanical neural networks fulfilling robotic matter with a level of intelligence.

Postbuckling behavior of two-dimensional decagonal quasicrystal plates under biaxial compression

Two-dimensional (2D) decagonal quasicrystal (QC) plates are analyzed for their postbuckling behaviors when subjected to biaxial compressive loads in this research. The analysis using a combination of the first-order shear deformation theory (FSDT) along with von Kármán’s nonlinear equations of motion, while also accounting for the initial geometric imperfections. The governing equations for the QC plate are obtained through the application of the Hamiltonian variational principle, and the postbuckling equilibrium path for the four-sided plate with simply support is calculated using a two-step perturbation technique. The quasiperiodic structure and phonon-phason coupling in 2D decagonal QCs play a crucial role in determining their mechanical behavior. The numerical analysis explores the effects of geometric shape, phonon-phason coupling constants, initial geometric imperfections, and compressive forces on the postbuckling characteristics of the plates. Numerical simulations validate the theoretical model, demonstrating its accuracy. The application of FSDT, von Kármán theory, and the two-step perturbation method to QC plates, considering the complex interactions involving the phason field unique to quasicrystals, provides new insights into the intricate mechanical responses of QC plates under postbuckling conditions.

Nonlinear transient response of rotating graphene platelets reinforced metal foams blades with initial geometric imperfection

In this paper, a new mechanical model including initial geometric imperfection is developed to investigate the nonlinear transient response of graphene platelets reinforced metal foams (GPLRMF) blades subjected to blast load. Combined with higher-order shear deformation plate theory (HSDT) and von-Karman geometric nonlinearity, the nonlinear motion equations are established, in which the Coriolis force, centrifugal force as well as initial geometric imperfection are simulated. Considering cantilever boundary condition, the partial differential equations are transformed into a series of ordinary differential equations by Galerkin discretization. Using the four order Runge-Kutta algorithm, the nonlinear transient response is numerically researched, and the time histories and phase trajectories are obtained. In addition, the effectiveness of the current research is validated by comparing the results with the existing valuable literatures. The influences of rotating speed, presetting angle, initial geometric imperfection, material property, damping coefficient, and blast load parameter on the dynamic deflection and instantaneous speed are presented. Numerical results reveal that the nonlinear transient dynamics of the composite blades under blast load are significantly influenced by initial geometric imperfection.

Dynamics of bistable composite plates

Nonlinear asymmetric bi-stable composite laminates, characterized by multistable nature and rich dynamics, have seen increased application potential recently. This has necessitated a detailed dynamical analysis of bistable laminates. In this manuscript, the nonlinear behaviour of asymmetric [0n/90n] bistable laminates with free–free boundary conditions is investigated through simulations and experimental observations. A refined 17 degrees of freedom (dofs) analytical model is developed, fusing Raleigh–Ritz with Hamilton’s principle to obtain the governing nonlinear equations of motion. A nonlinear finite element (FE) model is also developed using ABAQUS®. Experiments are conducted by clamping the midpoint of the plate with a shaker and then exciting it. The composite laminate shows many intricate dynamics, such as sub-harmonic and super-harmonic oscillations, intra-well oscillation (periodic and chaotic), and inter-well snap-through (periodic and chaotic snap-through) prominently. The primary focus of this paper is to highlight the potential of bistable laminates by elucidating the existence of large-amplitude vibrations over a broad frequency range under different input and system parameters. Further, it has been observed that attached mass plays a significant role in modifying the response bandwidth for cross-well oscillation for a given excitation frequency and amplitude. Hence fine-tuning of masses can excite different nonlinear dynamic characteristics of the laminate, making it applicable in different engineering fields.

High-precision identification and compensation of nonlinear error in hemispherical resonator gyro

This paper presents a novel nonlinear error identification and compensation method for the Hemispherical Resonator Gyro (HRG) from the signal processing perspective. Firstly, the paper establishes an HRG nonlinear error model that reveals the quantitative correlation between angle measurement error and the amplitude-gap ratio. Subsequently, a dynamic analysis model for gyro drift is developed using nonlinear motion equations. Based on this, a scalar factor nonlinear identification model encompassing assembly and nonlinear errors is proposed. Nonlinear optimization is then used to identify error parameters accurately. Finally, a joint compensation method is devised to mitigate assembly and nonlinear error issues. Empirical findings provide concrete validation for the precision of the identification and compensation method. The experimental results demonstrate that reductions of 97%, 89%, and 40% after the joint compensation are achieved in the fourth harmonic component, scale factor nonlinearity, and bias instability of the gyro, respectively.

Seismic response of structures with friction pendulum inerter system (FPIS) under near-fault earthquakes

Near-fault ground motions with high acceleration peaks and long-period velocity pulses pose a serious threat to the reliability of engineering structures. To reduce the displacement of isolation layer and improve the seismic performance of superstructure under the near-fault ground motions, a composite isolation system, which is composed of friction pendulum system (FPS) and inerter system, namely FPIS, was used in this paper. Based on D’Alembert’s principle, the nonlinear motion equations of a base-isolated structure with FPIS were established. The damping effect of two different mechanical layouts of FPIS subsystems, i.e, series-parallel inerter system-I-FPS(SPIS-I-FPS) or series-parallel inerter system-II-FPS(SPIS-II-FPS), were investigated in this study. The strong nonlinearity of FPIS was considered, and the inerter system parameters were designed based on the principle of maximum damping enhancement. The fourth-order Runge-Kutta approach was used to solve the dynamic response of a multi-degree-of-freedom system under seismic excitations. The effectiveness of FPIS was verified by comparing the isolation layer displacement and the acceleration of superstructure calculated by MATLAB with the friction pendulum system and viscous damper (FPS-VD). The nonlinear time history analysis results indicate that within a certain additional damping range, the SPIS-I-FPS subsystem performs effectively than SPIS-II-FPS in reducing the superstructure acceleration and isolation layer displacement. To increase the energy dissipation efficiency of structures, it is recommended to increase the design parameters of the inerter system and control the friction coefficient of FPS within the range of 0.05∼0.10.

Experimental and numerical study of lateral vibration of a rotor–stator rubbing system

AbstractIn this paper, lateral vibration of a rotating system with rotor–stator rubbing is investigated experimentally and numerically. An experimental study is conducted using a simple test rig to model a horizontal rotating system with a fixed stator device to simulate rubbing. Furthermore, a lumped parameter model with two degrees of freedom is established with a viscoelastic contact model that considers the stiffness and damping properties of the stator. The fourth/fifth order Runge–Kutta technique is used to solve the nonlinear equations of motion of the rotor system. The effect of changing mass unbalance and radial clearance on the dynamic response is investigated experimentally and numerically. The observed results are presented in detail through bifurcation diagrams, frequency spectra, Poincaré’s points, orbit plots, and time waveforms. The experimental results show that changing the mass unbalance and radial clearance yield significant effects on the system response. The 2 × and 3 × harmonic components are found to serve as good indicators of increasing rub severity. The numerical results show the dynamic characteristics of rub responses such as periodic no-contact, periodic with contact, quasi-periodic, and chaotic responses. The response characteristics are very sensitive to the changes in system parameters and initial conditions. Both the experimental and numerical results show qualitatively that the lateral vibration response exhibits coexistence and alternation of periodic and chaotic responses. Also, quasiperiodic response shows up in some numerical case studies. The obtained results contribute toward a better diagnosis of rotor–stator rub fault in rotating machines.

Period-doubling cascade route to chaos in an initially curved microbeam resonator exposed to fringing-field electrostatic actuation

This paper explores the chaotic dynamics of a piezoelectrically laminated initially curved microbeam resonator subjected to fringing-field electrostatic actuation, for the first time. The resonator is fully clamped at both ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The nonlinear motion equation which is obtained by considering the nonlinear fringing-field electrostatic force, includes geometric nonlinearities due to the mid-plane stretching and initial curvature. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is obtained using the combination of shooting and continuation methods. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period-doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The bifurcation diagram is obtained for a wide range of excitation frequency and thus the exact range in which chaotic behavior occurs for the system is determined. The chaotic response of the system is regularized and controlled by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.

Geometrically nonlinear vibration of toroidal sandwich shells with auxetic honeycomb core under periodic/impulsive pressure

The main objective of this paper is to evaluate the dynamic behaviour of sandwich toroidal shells composed of a core with negative Poisson ratio (auxetic honeycomb material) and nanocomposite enriched coating layers under impulsive/periodic pressures considering the geometrical nonlinearity. The set of nonlinear equilibrium equations based on the first-order shear deformation theory are and the resultant nonlinear differential motion equations are approximately solved by the Galerkin method and 4th-order Runge–Kutta solution, respectively. The achieved responses show that the amplitude of outward deformations is intensively increased when the proposed sandwich shells are subjected to rectangular periodic and impulsive loadings which shows the destructive effects of this type dynamic loading. The performed parametric studies shows that these destructive effects can be controlled by changing the geometrical properties of the honeycomb cells and the volume fractions of the nanocomposites in the coating layers. It is notable that in the previous studies the nonlinear dynamic response of toroidal shells subjected to various periodic and impulsive pressures was not investigated. However, in the present work, the mentioned dynamic responses were achieved and investigated without spending time-consuming computational. In fact, the novelty of this work is to develop such a robust and efficient tool.

Studying the nonlinear response of incompressible hyperelastic thin circular cylindrical shells with geometric imperfections

This study presents a comprehensive analysis of hyperelastic thin cylindrical shells exhibiting initial geometrical imperfections. The nonlinear equations of motion are derived using an improved formulation of Donnell’s nonlinear shallow-shell theory and Lagrange’s equations, incorporating the small strain hypothesis. Mooney–Rivlin constitutive model is employed to capture the hyperelastic behavior of the material. The coupled nonlinear equations of motion are analytically solved using Multiple-Scale method, which effectively accounts for the inherent nonlinearity of the system. To ensure the model’s accuracy, the linear model is verified by comparing the results with those obtained through hybrid finite element method. Subsequently, the model with only geometrical nonlinearity is evaluated against other research works existing in the open literature to ensure its reliability and precision. Finally, the results of the model, considering both geometrical and physical nonlinearity, are verified against the results obtained from Abaqus software. The main objective of this research is to provide a detailed understanding of the response of hyperelastic thin cylindrical shells in the presence of initial geometric imperfections. In this order, the impact of three distinct geometric imperfections – axisymmetric, asymmetric, and a combination of driven and companion modes – on the natural frequency is examined. The behavior of each of these geometric imperfections is investigated by varying their respective coefficients. The numerical results indicate that geometric imperfections enhance the natural frequency, and employing different models for imperfections leads to a variation in this trend. In the amplitude response of hyperelastic cylindrical shells, two peaks coexist, reflecting the softening and hardening responses of the system. Distinct initial geometric imperfections influence these two peaks.