Nonlinear Dynamic Problems Research Articles

The role of a pre-wetting layer in resolving the contact line paradox of thin films in infinite elastic domains

Hydraulic fracturing for oil and gas production from shale formations, as well as natural geological phenomena, involve the propagation of thin viscous films within elastic media. For viscous fluids, stress diverges as the thickness of the film tends to zero, arresting the propagation of the film, and thus implying the contact line paradox. For free-surface films, this paradox is resolved by considering a precursor film, leading to Tanner's law. This approach was extended recently for viscous films between a thin elastic plate and a rigid solid, allowing calculation of the film propagation rate. In this work, we examine the effect of a pre-wetting layer on the rate of propagation of a viscous flow within an infinitely deep and long domain. We analyse the linear and nonlinear dynamic problems, and perform a self-similarity analysis. We find that peeling front propagation scales as time to the power of $1/9$ and $1/3$ for thin and thick pre-wetting layer limits, respectively. Our results contribute to the understanding of the contact line paradox in elastic media and the crucial role of the pre-wetting layer in resolving it.

Crash topology optimization for front-end safety parts of battery electric vehicle using an improved equivalent static loads method

Crash topology optimization is a typical nonlinear dynamic response structural topology optimization problem, which is one of the most difficult problem in the structural design field. The equivalent static load method (ESLM) provides a well-defined pattern to solve such difficult problems, which can convert a nonlinear dynamic response optimization into multi-load steps optimization problem with the equivalent static loads (ESLs). However, due to the large deformation in the crash condition and nodal characteristics of the ESLs, it is hard to solve the crash topology optimization directly using the standard ESLM. To expand the application scope of the ESLM, an improved ESLs calculation method is proposed by using the model order reduction method and energy principle, which only acting on some nodes and can be scaled adaptively. Correspondingly, to enrich the connotation of topology optimization, a crash topology optimization method is proposed by using the improved ESLs, which can solve the numerical problems in the design domain by guaranteeing the topology optimization with the improved ESLs perform in the linear rang. First, the principle of the standard ESLM is introduced, and corresponding problems and deficiencies in solving the crash topology optimization are summarized. Then, to solve the above problems, the improved ESLs calculation method is proposed. Meanwhile, the corresponding crash topology strategy is proposed based on the improved ESLs. Finally, to verify the effectiveness and engineering application value of the proposed crash topology optimization method, a test-verified frontal crash simulation model of the BEV front-end is established, and their safety parts are redesigned by using the proposed method. The results show that, the proposed method can effectively solve the crash topology optimization of thin-walled structures under large deformation crash condition. This method also provides a new idea and practical method for the crashworthiness and lightweight design of automobile structures.

Higher-Order Accurate Explicit Time Schemes with Improved Dissipation Properties

In this paper, a novel family of fourth-order accurate explicit time integration schemes is developed by combining the new time approximations and the explicit fourth-order Runge–Kutta (RK4) method. Inaccurate predictions due to the presence of excessive numerical dissipation are often observed in practical analyses of structural dynamics when the RK4 is employed for both displacement and velocity approximations. To remedy this, novel time approximations with adjustable algorithmic parameters are employed for the displacement vectors while the velocity vectors are approximated by using the RK4. For the complete elimination of numerical dissipation, algorithmic parameters are unconventionally determined by taking the determinant of the amplification matrix as unity. A set of algorithmic parameters obtained from this process makes the new schemes completely non-dissipative while keeping the computational cost the same as the RK4. Due to this improvement, the new schemes have enhanced total energy-conserving capabilities for nonlinear systems and give noticeably more accurate predictions in practical analyses. Until now, controllable numerical dissipation and fourth-order accuracy are not attained simultaneously in a unified set of time schemes. In the new schemes, however, a systematic way to adjust the level of numerical dissipation is also presented while attaining fourth-order accuracy. The numerical results of various test problems show that the new time schemes can provide more accurate predictions for nonlinear conservative dynamic problems than the existing time schemes.

Analytical approach to solving the problems of nonlinear dynamics of systems with time-varying parameters under the conditions of the external environment reaction

Purpose. The development and application of analytical (including approximate) methods of analyzing nonlinear mathematical models of system dynamics, which have complex behavior due to the presence of time-dependent characteristics, is relevant for solving various classes of engineering problems. Therefore, the purpose of this work is mathematical modeling of problems of nonlinear dynamics of such systems, which allows determining the trajectory of the system's movement over time and other dynamic characteristics in accordance with their setting. Method. Based on the modern achievements of analytical, in particular asymptotic and numerical research methods based on existing software complexes, the possibility of non-local research and the formation of a sufficiently complete representation of the peculiarities of the behavior of nonlinear systems is considered. To achieve the goal, a mathematical model of the nonlinear dynamics of a system with time-varying properties is considered, provided that the reaction of the environment depends on the function of the speed of movement of the system to the n degree. The results. The solution of some engineering problems of the nonlinear dynamics of systems with time-varying characteristics of dependence for n=2 and systems whose response to the external environment can be a function of both whole and fractional degrees allow to determine the trajectory of the system's movement over time and other dynamic characteristics in accordance with before their production. Scientific novelty. The analytical dependence of the speed function of the dynamic process of the system with time-varying parameters was obtained for the nonlinearity parameter of the studied system most used in practice, n=2, for the reaction function of the external environment. Practical significance. The obtained analytical dependencies can be applied in a sufficiently wide range of research. Approximate analytical methods based on asymptotic approaches based on hybrid methods (perturbation, phase integrals in combination with Galerkin's principle) are applied. The use of asymptotic and numerical methods of research based on existing software complexes opens up the possibility of non-local research and the formation of a sufficiently complete representation of the peculiarities of the behavior of nonlinear systems with variable characteristics of materials, in particular, composite and functionally gradient ones.

Dynamic Analysis of the High Speed Train–Track Spatial Nonlinear Coupling System Under Track Irregularity Excitation

Track irregularity is the main factor that affects the safety and comfort of high-speed railway. It is important to study the influence of the track irregularity on the safety service of high speed train–track coupling system. In this paper, a vehicle–track spatial nonlinear coupled system dynamics model is developed with FEM. The model includes vehicle subsystem and track subsystem, which are coupled by wheel–rail interface. The vehicle subsystem is a 31-DOF spatial vehicle model, and the track subsystem is a three-dimensional slab track model composed of the rail, rail pad and track slab. Considering wheel tread profile and rail head profile, the wheel–rail spatial contact geometry relationship is established, and the “Trace method” and the “Minimum distance method” are used to search for wheel–rail contact points. Considering the nonlinear contact between the wheel and the rail, a numerical approach is presented, where the “Trace method” is embedded into the algorithm for solving the vehicle–track nonlinear coupling dynamics equation by cross iteration, realizing searching the wheel–rail contact points and solving the vehicle–track nonlinear coupling equation simultaneously, which greatly improves the efficiency of numerical analysis. As application examples, the dynamic responses of a high speed train–track spatial nonlinear coupled system excited by the track V-shaped local profile irregularity and the track torsional irregularity are analyzed. The results show that the algorithm is effective in solving the vehicle–track spatial nonlinear coupled system dynamics problem, and the track irregularity has significant influence on the dynamic response of vehicle and track structure.

Nonlinear dynamic thermal-mechanical behaviors of framed structures under temperature variations

An abnormal change in dynamic characteristics is considered an indicator of structural damage and can help the optimal design of structures suffering from dynamic loads, but these characteristics are also affected by ambient temperature. Therefore, a novel advanced computational method based on the fiber-based approach is proposed for the first time to predict the nonlinear dynamic behaviors of frames under environmental temperature changes with linear and nonlinear profiles by using Fortran programming language. The cross-section will be divided into a matrix of fibers, and by assigning the different temperatures to each fiber, all linear and nonlinear temperature gradients in both vertical and horizontal directions will be operated. The material nonlinearity is traced by the distributed plasticity model using the same mesh as the fiber-based approach whereas the effects of geometric nonlinearity due to member curvature and member chord rotation are considered by using the stability functions and a geometric matrix, respectively. A sequential nonlinear thermal incremental-iterative solution scheme based on the Newton-Raphson algorithm and Newmark-β method is also developed to address the nonlinear dynamic problems due to temperature variations. The accuracy and computational performance of the proposed method are validated by comparing the obtained results with those from experiments and the Abaqus. Results obtained show that the proposed method is exact for the nonlinear inelastic dynamic thermo-mechanical analysis of framed structures. It would offer a tool for thermal-structural design practice instead of using time and cost-consuming commercial software.

Exponential Tracking and Disturbance Rejection for Nonlinear Reaction Convection Diffusion Equations via Boundary Control

This paper studies the problem of exponential tracking and disturbance rejection for nonlinear reaction convection diffusion equations via Dirichlet or Neumann boundary control. Disturbances are present both in the equation and on the boundary. The tracking problem is simply a nonlinear dynamic regulator problem. If an initial condition is not compatible with a reference, the nonlinear dynamic regulator problem does not have a solution. Thus, we split the tracking problem into two problems to relax the compatibility condition. We choose one problem to be exponentially stable and leave everything else in a dynamic regulator problem. Then a controller is just a solution component of the dynamic regulator problem, while the tracking error is given by the solution of the exponentially stable problem. The controller can be obtained by integrating the dynamic regulator equation and using the reference condition. The dynamic regulator problem with the obtained controller becomes an initial boundary value problem. We then transfer the initial boundary value problem into an equivalent integral equation and prove that the integral equation has a unique global solution if the nonlinearity is globally Lipschitz. In addition, a numerical example is given to illustrate the theoretical results.

Composite Observer-Based Optimal Attitude-Tracking Control With Reinforcement Learning for Hypersonic Vehicles.

This article proposes an observer-based reinforcement learning (RL) control approach to address the optimal attitude-tracking problem and application for hypersonic vehicles in the reentry phase. Due to the unknown uncertainty and nonlinearity caused by parameter perturbation and external disturbance, accurate model information of hypersonic vehicles in the reentry phase is generally unavailable. For this reason, a novel synchronous estimation is proposed to construct a composite observer for hypersonic vehicles, which consists of a neural-network (NN)-based Luenberger-type observer and a synchronous disturbance observer. This solves the identification problem of nonlinear dynamics in the reference control and realizes the estimation of the system state when unknown nonlinear dynamics and unknown disturbance exist at the same time. By synthesizing the information from the composite observer, an RL tracking controller is developed to solve the optimal attitude-tracking control problem. To improve the convergence performance of critic network weights, concurrent learning is employed to replace the traditional persistent excitation condition with a historical experience replay manner. In addition, this article proves that the weight estimation error is bounded when the learning rate satisfies the given sufficient condition. Finally, the numerical simulation demonstrates the effectiveness and superiority of the proposed approaches to attitude-tracking control systems for hypersonic vehicles.

α-Sumudu Transformation Homotopy Perturbation Technique on Fractional Gas Dynamical Equation

Transformation and many other substitution methods have been used to solve non-linear differential fractional equations. In this present work, the homotopy perturbation method to solve the non-linear differential fractional equation with the help of He’s Polynomials is provided as the transformation plays an essential role in solving differential linear and non-linear equations. Here is the α-Sumudu technique to find the relevant results of the gas dynamics equation in fractional order. To calculate the non-linear fractional gas dynamical problem, a consumer method created on the new homotopy perturbation a-Sumudu transformation method (HP TM) is suggested. In the Caputo type, the derivative is evaluated. a-Sumudu homotopy perturbation technique and He’s polynomials are all incorporated in the HPSaTM. The availability of He’s polynomials could be used to conveniently manage the non-linearity. The suggested approach shows that the strategy is simple to implement and provides results that can be compared to the results gained from any other transformation technique.

Detecting unstable periodic points of chaotic maps by stability transformation of reservoir

Detecting unstable periodic orbits in chaotic systems based on the time series is a fundamental problem in nonlinear dynamics, but it often becomes extremely challenging one. In this study, we propose a new approach for detecting unstable periodic points using reservoir computing and stability transformation method. We connects reservoir computing, which is a well-known machine learning technique, and stability transformation method, which can detect unstable periodic points in chaotic dynamical systems, to perform unstable periodic points detection in a data-driven and model-free process. In this paper, we use an example of the Hénon map to demonstrate detecting unstable fixed point and unstable 2-periodic points.

Взаимодействие подсистем в задачах нелинейной динамики. Различные фазы хаоса

Взаимодействие подсистем в задачах нелинейной динамики. Различные фазы хаоса

On opto-thermally excited parametric oscillations of microbeam resonators. I

The present article is the first part of the work devoted to investigation of the nonlinear dynamics of parametrically excited flexural vibrations of a clamped-clamped microbeam - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. An analytical solution of the heat transfer problem is found for the steady harmonic distribution of temperature in the volume of the resonator. The static and dynamic components of temperature-induced axial force and bending moment are determined. Using the Galerkin method, the discretization of nonlinear coupled partial differential equations describing the longitudinal-flexural oscillations of the resonator is performed. Using the asymptotic method of multiple time-scales, an approximate analytical solution is obtained for the nonlinear dynamics problem under the conditions of primary parametric resonance.

Method of numerical modeling in the problems of predicting the effect from air gap in blasthole charge design

Introduction. The article presents the results of numerical modeling of the effect made the air gap in the borehole charge design on the seismic impact of blasting, base work and the stemming ejection speed. Intensification of mining drives the need for the design of blasting operations with a gentle impact on the rock mass while maintaining the required quality of the explosion. The assessment of the major blast impact on a rock mass may include both the results of collecting statistical observational data in specific mining and technological conditions, and the results of numerical modeling of crush zone formation, seismic impact, and the speed of stemming ejection during a major blast. Numerical modeling allows not only to make a preliminary forecast, but also to conduct a series of computational experiments to select the appropriate parameters of drilling and blasting operations in order to achieve the expected effect. The position and size of the air gap in the borehole charge are considered as such a parameter Research methods. The work uses an integrated method combining analysis, scientific generalization of theoretical and experimental studies relating to air gap in borehole charges, and the use of numerical modeling tools and digital technologies in mining. To simulate the action of an air gap during an explosion with expandable and destructible rock mass walls, a software package was used, consisting of the ANSYS AUTODYN software package, which has proven itself in solving non-stationary problems of nonlinear dynamics, and the software package developed by the members of the Institute of Communications and Information Technologies of the Kyrgyz-Russian Slavic University, to assess the stress-strain state of rocks using parallel computing technology. Results. Gas-dynamic processes relating to air gap were analyzed by the method of numerical simulation; the crush zone was estimated. The simulation tool was adjusted and calibrated based on numerical simulation results comparison with the results of laboratory studies of the air gap effect in wells during an explosion in organic glass. Satisfactory convergence of the results made it possible to use this tool for modeling explosive loading in conditions close to industrial scales. Discussion. The modeling performed allows us to state that the bottom work, the reduction in the stemming ejection speed and in the seismic impact are achieved through the use of an air gap and a reduced proportion of explosive. Thus, the use of numerical simulation techniques in mining problems to solve a wide range of production problems to optimize the charge design is justified. Scope of results. The results are relevant for the problems of optimizing drilling and blasting operations in the mining industry in order to ensure a gentle effect of blast on the environment

Parametric dynamic mode decomposition for reduced order modeling

Dynamic Mode Decomposition (DMD) is a model-order reduction approach, whereby spatial modes of fixed temporal frequencies are extracted from numerical or experimental data sets. The DMD low-rank or reduced operator is typically obtained by singular value decomposition of the temporal data sets. For parameter-dependent models, as found in many multi-query applications such as uncertainty quantification or design optimization, the only parametric DMD technique developed was a stacked approach, with data sets at multiple parameter values were aggregated together, increasing the computational work needed to devise low-rank dynamical reduced-order models. In this paper, we present two novel approach to carry out parametric DMD: one based on the interpolation of the reduced-order DMD eigen-pair and the other based on the interpolation of the reduced DMD (Koopman) operator. Numerical results are presented for diffusion-dominated nonlinear dynamical problems, including a multiphysics radiative transfer example. All three parametric DMD approaches are compared.

Retraction Note: An Implicit Unconditionally Stable Integration Method for Nonlinear Structural Dynamics and Earthquake Engineering Problems

Retraction Note: An Implicit Unconditionally Stable Integration Method for Nonlinear Structural Dynamics and Earthquake Engineering Problems

A novel sub-step explicit time integration method based on cubic B-spline interpolation for linear and nonlinear dynamics

A novel sub-step explicit time integration method is proposed. In the procedure, uniform cubic B-spline function interpolation is employed to achieve high-order accuracy. The proposed method has multiple free parameters which can be used to control algorithm properties. The proposed method shows at least second-order accuracy for linear systems with and without damping, and especially, when proper algorithmic parameters are selected, it can achieve third-order accuracy for no damping case. Numerical example analysis confirms higher accuracy than other classical explicit methods. For both linear and nonlinear dynamic problems, the proposed method shows desirable accuracy and efficiency.

On the problem of nonlinear dynamics of shell systems influenced by the internal pressure local impulse

Purpose. Critical infrastructure facilities, including pipelines and tanks, are an integral part of ensuring the functioning of industrial and civil life support facilities. External factors that may affect their functioning can be both the cause of force majeure and targeted actions by third parties. Therefore, the purpose of this work is to establish a mathematical description of the dynamic behavior of cylindrical shells under the action of a local impulse caused by internal pressure. Methodology. To achieve the goal of the work, the provisions of the nonlinear dynamics of the system of forces are applied, taking into account approximate analytical and asymptotic methods. Results. The results obtained make it possible to describe nonlinear processes caused by local pulses of internal pressure in shell systems, which can be interpolated for use in numerous methods for determining the parameters of the strength and bearing capacity of the corresponding structure, taking into account the physical and mechanical parameters of the materials used and the nature of the force action. Interpolation of the results to similar technical systems will allow scaling the mathematical approach in solving engineering problems. Scientific novelty. The existing dynamic models are mostly represented by simplified calculation models without taking into account the totality of real loads and characteristic design and technological factors. In this paper, it is proposed to take into account the nonlinearity of dynamic processes due to the shape of the impulse action, the power exponent of the nonlinearity of displacements, and the Dirac delta function. This approach is new of scientific and practical significance. Practical significance. The resulting singular inhomogeneous model of interaction in the form of a differential equation with variable coefficients makes it possible to provide numerical simulation of interaction processes with asymptotically approximate results and to establish the parameters of dynamic behavior to impulse action. As a result, it is possible to obtain new technical solutions for shell structures resistant to dynamic impact, which will have improved technical and operational characteristics.

Design and investigation of a quad-stable piezoelectric vibration energy harvester by using geometric nonlinearity of springs

In this paper, a quad-stable piezoelectric vibration energy harvester induced by the geometric nonlinearity of springs is designed and investigated. The novelty is that by replacing the permanent magnets with a combination of springs, the system can be effectively protected from electromagnetic interference in the working environment. The innovative design of three hinged springs attached to the tip of the cantilever beam makes the stiffness of the structure inherently nonlinear, which can induce the multi-stability of the system. The Euler-Lagrange equations are applied to establish the governing equation of the energy harvester. The static bifurcation analysis reveals the evolution process of multi-stability. The complex dynamic frequency (CDF) method, an efficient approach to solving strongly nonlinear dynamic problems with multi-well systems, is then applied to deduce the system's amplitude-frequency response. Combinations of theoretical and numerical results show that among a wide frequency bandwidth, the system can exhibit such as single-, double- and quadruple-well periodic motions. Finally, experiments are set up to verify the advantage of inter-well motion on harvesting power. Result shows that geometric nonlinearity can effectively make the energy harvester exhibit multi-stability, generate different movement patterns, and obtain high harvesting power.

Experimental investigation on nonlinear dynamic response of towing cable under vessel motion

Marine towing cable is a crucial equipment of the underwater towing system which can acquire the marine resources distribution in the sea. The marine towing cable will experience nonlinear dynamic problems due to the vortex shedding and vessel motion during the towing movement. In order to investigate the dynamic response characteristics of the towing cable during the working process, a set of experimental devices for towing cable is designed and a PPR (Polypropylene-Random) pipe with the aspect ratio of 150 is tested with coupled uniform towing and vessel motion. The characteristics of response amplitude and frequency under uniform towing are compared and analyzed comprehensively with different towing speeds and towing depths. The key characteristics of nonlinear dynamic response under vessel motion are then discussed. Some phenomena involving multi-modal participation, frequency transition and amplitude modulation are captured. The vessel motion will excite ωCT - ωT, ωCT + ωT, ωT frequency components in the CF (cross-flow) direction which may be caused by the parametric excitation, and wave frequency ωT dominates the IL (in-line) direction response. Some extreme motion conditions may enlarge the response displacement in the IL and CF direction. It is suggested that the vessel motion should be taken into account in the evaluation of the nonlinear dynamic analysis of marine towing cable.

Multi-scale computational method for nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties

For most of materials, the properties of the materials are usually dependent on the temperature, which causes the nonlinear physical and mechanical material behaviors. In this paper, an innovative multi-scale computational method is developed for efficiently simulating nonlinear dynamic thermo-mechanical problems of composite materials. The nonlinearities of these multi-scale problems were caused by the temperature-dependent properties of each component material in the composites and the heterogeneities of composite materials are taken into account by periodic distributions of representative unit cells on the micro-scale. Firstly, a second-order two-scale (SOTS) computational model for accurately analyzing nonlinear dynamic thermo-mechanical behaviors of composite materials is established based on multi-scale asymptotic analysis and Taylor series method. The proposed SOTS computational model includes the first-order and second-order auxiliary cell problems on micro-scale, homogenized problem on macro-scale and macro–micro coupled SOTS asymptotic solutions. Then we theoretically explain the crucial importance of developing the SOTS solutions by the error analysis in the point-wise sense. Next, a rigorous error analysis with an explicit rate of the SOTS approximate solutions is given under some simplifications and assumptions in the integral sense. In addition, a multi-scale numerical algorithm is proposed in detail to effectively simulate these nonlinear multi-scale problems. Finally, several typical numerical examples are carried out to confirm the effectiveness and correctness of the proposed multi-scale numerical algorithm, especially for large-scale engineering structures. This study offers an efficient and high-accuracy multi-scale computational scheme that enables the effective simulation and analysis of nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties.