Nonlinear Equations Of Motion Research Articles

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.

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.

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.

Time‐optimal trajectory generation and observer‐based hierarchical sliding mode control for ballbots with system constraints

AbstractThis paper introduces a comprehensive motion planning–tracking–safety constraint scheme for a 3D ballbot system. A nonlinear control for the 3D ballbot system is designed based on three separate planes by utilizing extended state observer (ESO) to estimate coupling mechanisms. Three virtual control signals are generated from these distinct planes and can be used for formulating actual control signals. To overcome the complexity of nonlinear motion equations, flatness theory is used to construct the time‐optimal trajectory through an optimization problem, facilitating smooth movement of the ballbot, and obstacle avoidance based on RRT* waypoints. Furthermore, our work manipulates the hierarchical sliding mode controller (HSMC) as the nominal controller to ensure that the ballbot tracks to the optimal trajectory, unifying with the exponential control barrier function (ECBF) to address safety constraints in the body's deflection angle. Through extensive simulations and comparative analysis, the system demonstrates its effectiveness and safe operation in various working conditions.

Seismic response analysis of single‐degree‐of‐freedom structure coupled with tuned self‐centering wall

AbstractThis paper investigates the possibility of using self‐centering walls (SCWs) as tuned mass dampers (TMDs) to control the seismic response of structures. The basic characteristics of this system are investigated using a two‐degree‐of‐freedom (2‐DOF) model, representing the main structure as a single‐degree‐of‐freedom (SDOF) oscillator interconnected with an SCW through viscous and elastic devices. The nonlinear equations of motion for the proposed coupled system are derived and then utilized to determine the displacement amplification response. The coupling parameters are optimized to minimize the dynamic response of the oscillator using a numerical method. A simulation study is conducted to investigate the response of the proposed system by considering six recorded earthquakes. Several parameters are studied, including the mass ratio, the influence of prestressing, and the slenderness angle of the wall. The displacement spectra are generated using the optimized parameters and compared to those of the uncoupled system and the traditional coupled system, which features a rigid connection between the structure and the wall. The results reveal that the proposed system effectively suppresses both maximum and root mean square responses of structures, outperforming the traditional coupled system in most cases. Improved performance for the proposed system can be achieved by increasing the wall's mass ratio and decreasing the slenderness angle. Moreover, prestressing has an adverse impact on the system's displacement response. Finally, the influence of wall flexibility is examined using a finite element model, revealing a minimal effect on the system's response.

An improved Mickens’ solution for nonlinear vibrations

The Duffing equation and other nonlinear equations of motion of nonlinear vibrations are widely applied in the fields of engineering and science, especially mechanical and electrical engineering. The modified Mickens extended iteration method has been utilized in this article to investigate for further accurate solutions to the Duffing equation and an equation of motion of nonlinear vibrations. The third approximate frequency shows good agreement with the exact result for both oscillators. For simplicity in algebraic calculations, the increasing harmonic terms have been taken to the fifth order. In order to establish the reliability and effectiveness of the proposed approach, the results obtained through this method are compared with those available in the literature. The comparison of the obtained solutions with numerical solutions shows a remarkable accuracy. Also, it becomes evident that the modified Mickens extended iteration method is significantly simpler, more accurate, efficient, and straightforward than other existing methods. The highest percentage error in the third approximate frequency of an equation of motion of nonlinear vibrations is less than 0.0488, and the maximal percentage error in the third approximate frequency of the Duffing oscillator is less than 0.0065. The proposed technique is principally exhibited in nonlinear models where the nonlinear terms are strong, but it can also be broadly applicable to other problems arising in engineering. Mathematics Subject Codes34A34, 34C25, 37N15 (AMS Mathematics Subject Classification 2010)

Entropy Generation For Non-Newtonian Fluid With Constant Viscosity

In this research, entropy generation for non-Newtonian fluid flow with constant viscosity is considered. The governing nonlinear equations of motion are solved analytically with regular perturbation techniques. Third grade fluid is employed to account for the non-Newtonian influence. The influence of some physical parameters involved in the analysis is studied. Results show that the parameter has the tendency of increasing the velocity of the fluid flow as well as the temperature of the cylindrical pipe. It is observed that as the Brinkman parameter increases, the cylindrical wall temperature is enhanced. Entropy generation number for both heat transfer and fluid friction for various values of and is examined. Results indicate that increase in these parameters increases the temperature of the cylinder, thereby increasing the entropy generation number.

Joint Resonance Analysis in Multiple Modes of Soft Ferromagnetic Rectangular Thin Plate

In this article, the nonlinear principal and internal resonance properties of a soft ferromagnetic rectangular thin plate are investigated in a magnetic field environment. The nonlinear partial differential equation of motion of a soft ferromagnetic rectangular thin plate is derived under the effect of homogeneous simple harmonic excitation. The system of nonlinear differential equations with multiple degrees of freedom is established by the assumed one-sided fixed trilateral simply support condition using the Galerkin’s method. The system of nonlinear differential equations is solved by the multiscale method to obtain the response of two modes under the simple harmonic force at the principal and internal resonance. The numerical results of the system response show that when the frequency of the simple harmonic force is close to one of the modes (first-order or second-order mode) causing it to resonate, the other mode will also resonate internally. The magnetic field can have an inhibiting effect on the resonant response of the system and also affect the kinematic state of the system. The internal resonance provides a mechanism for transferring energy from a high mode to a lower mode.

Nonlinear dynamic buckling analysis of the FG-GPLRC cylindrical and sinusoid panels with porous core under time-dependent axial compression

The nonlinear dynamic buckling responses of functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and sinusoid panels with porous core are presented in this paper. The governing formulations are established by applying the nonlinear higher-order shear deformation theory (HSDT). an approximation technique is used to determine the stress function for complexly curved panels. Euler-Lagrange equations can be used to achieve the nonlinear motion equation. Numerical investigations are considered using the Runge-Kutta method for dynamic responses, and using the Budiansky-Roth criterion for critical dynamic buckling loads. Some discussions on the dynamic buckling responses of panels with porous core can be achieved from the numerical examples.

Vibration Characteristics of a Functionally Graded Viscoelastic Fluid-Conveying Pipe with Initial Geometric Defects under Thermal–Magnetic Coupling Fields

In this study, the Kevin–Voigt viscoelastic constitutive relationship is used to investigate the vibration characteristics and stability of a functionally graded viscoelastic(FGV) fluid-conveying pipe with initial geometric defects under thermal–magnetic coupling fields. First, the nonlinear dimensionless differential equations of motion are derived by applying Timoshenko beam theory. Second, by solving the equilibrium position of the system, the nonlinear term in the differential equations of motion is approximated as the sum of the longitudinal displacement at the current time and longitudinal displacement relative to the position, and the equations are linearized. Third, these equations are discretized using the Galerkin method and are numerically solved under simply supported conditions. Finally, the effects of dimensionless temperature field parameters, dimensionless magnetic field parameters, thermal–magnetic coupling, initial geometric defect types, and the power-law exponent on the complex frequency of the pipe are examined. Results show that increasing the magnetic field intensity enhances the critical velocity of first-order mode instability, whereas a heightened temperature variation reduces the critical velocity of first-order diverge instability. Under thermal–magnetic fields, when the magnetic field intensity and temperature difference are simultaneously increased, their effects on the complex frequency can partially offset each other. Increasing the initial geometric defect amplitude increases the imaginary parts of the complex frequencies; however, for different types of initial geometric defect tubes, it exhibits the most distinct influence only on a certain order.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Formulation and free vibration analysis of spinning Timoshenko composite beams using nonlinear geometrically exact theory

AbstractIn this study, a set of generalized nonlinear equations of motion for Timoshenko composite shafts is derived using the geometrically exact approach. These partial differential equations of motion are discretized by Galerkin method and the free vibration of an orthotropic spinning shaft with hinged‐hinged boundary conditions is investigated using the multiple‐scale method. We find that in the case of two mode discretization, composite couplings can have significant linear and nonlinear effects on the behavior of the shaft and that these effects were not predictable in the classical model. Also, in the investigating of composite layering we find that the effects of the fibers angle as well as the composite couplings can be noticeable, and these effects are greater in the case of the asymmetrical layering.

Transient Response of Sandwich Plates with Corrugated Core Under Mechanical-Thermal Loads

The nonlinear dynamic responses of the corrugated sandwich plate under mechanical-thermal loads are studied and analyzed. Based on hyperbolic parabola shear deformation plate theory (HPSDT), the partial differential equation of the sandwich plate with the corrugated core is established. Using the Galerkin truncation method, the nonlinear motion equation is derived. By the solution, the response of the corrugation plate with simply supported four edges at different temperatures is obtained and validated. Then the transient responses of the corrugated sandwich plate under different impact loads are analyzed, and the effects of the base materials, the corrugation types and the structural parameters of the corrugated plate on the transient responses are discussed in detail. The research provides a reference for improving the impact resistance of the sandwich plate in practical application.