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2024 | Book

Perspectives in Dynamical Systems II — Numerical and Analytical Approaches

DSTA, Łódź, Poland December 6–9, 2021

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About this book

This proceedings volume gathers selected, peer-reviewed papers presented at the Dynamical Systems Theory and Applications International Conference - DSTA 2021, held virtually on December 6-9, 2021, organized by the Department of Automation, Biomechanics, and Mechatronics at Lodz University of Technology, Poland. This volume focuses on numerical and analytical approaches, while Volume I concentrates on studies on applications.
Being a truly international conference, this 16th iteration of DSTA received submissions from authors representing 52 countries. The program covered both theoretical and experimental approaches to widely understood dynamical systems, including topics devoted to bifurcations and chaos, control in dynamical systems, asymptotic methods in nonlinear dynamics, stability of dynamical systems, lumped mass and continuous systems vibrations, original numerical methods of vibration analysis, non-smooth systems, dynamics in life sciences and bioengineering, as well as engineering systems and differential equations.

DSTA conferences aim to provide a common platform for exchanging new ideas and results of recent research in scientific and technological advances in modern dynamical systems. Works contained in this volume can appeal to researchers in the field, whether in mathematics or applied sciences, and practitioners in myriad industries.

Table of Contents

Frontmatter
Modeling and Analyzing a Spring Pendulum Motion in the Presence of Energy Harvesting Devices

This paper investigates a dynamical system associated with two devices: a piezoelectric device and an electromagnetic one. These devices are connected with a nonlinear damping spring pendulum with 2DOF, in which its supported point moves in a circular path. The governing system of motion is derived using Lagrange’s equations of the second kind. The analytic solutions of this system are obtained up to the second approximation using the approach of multiple scales. The comparison between these solutions and the numerical ones reveals high consistency between them. The steady-state solutions are obtained, and their stabilities ranges are tested. The influences of the excitation amplitudes, the damping coefficients, and the different frequencies on energy harvesting devices outputs are examined and discussed. The developed methodology and obtained results can be useful in various applications like power supply of sensors and charging electronic devices.

M. K. Abohamer, J. Awrejcewicz, R. Starosta, T. S. Amer, M. A. Bek
Asymptotic Solutions of the Boundary Value Problem of Convective Diffusion Around Drops with Volumetric Nonlinear Chemical Reaction

We consider a stationary problem of convective diffusion around a droplet, which is streamlined by a liquid flow at low Reynolds numbers, taking into account a nonlinear volumetric chemical reaction. The characteristic feature of the problem is the presence of two dimensionless parameters: a constant of rate of the volumetric chemical reaction kv, and Peclet number Pe which determine the concentration distribution in the flow. The quantity constant of rate of the volumetric chemical reaction kv and Peclet number Pe assumed to have a constant value. It is a boundary value problem for a quasilinear partial elliptical equation with a small parameter multiplying in higher derivatives. Small parameter corresponds to large Peclet numbers. The limiting equation, when the small parameter is equal to zero, has singular points of the saddle type. Several boundary layers appear outside the drop. The matching conditions for solutions are formulated at the boundaries between neighboring areas. The principal terms of the asymptotics of the solution are constructed around the drop. MSC 76M45.

R. G. Akhmetov
Mathematical Model of Double Row Self-Aligning Ball Bearing

Ball bearings are widely used in variety of rotary mechanisms expecting their maintenance-free operation. One of the main factors influencing bearing life is radial internal clearance (RIC), which can be simply described as relative movement of ball and raceway perpendicular to the bearing axis. Mentioned parameter has a significant impact on dynamic response of applied ball bearing in the system and brings strong nonlinear effects. In the paper nonlinear 2 degrees of freedom (2-DOF) model of double-row self-aligning ball bearing is established. Mentioned type of bearing is very important from operation point of view, because it is used in applications, where strong misalignments are expected coming from assembly of shaft deflection. Model input data are based on characteristics of bearing 2309SK. Results of bearing dynamic response are studied for different values of radial internal clearance. For quantitative and qualitative analysis of nonlinear time series a few recurrence quantificators, providing information on bearing’s behavior regarding RIC’s value. Results can be used in the validation process with acceleration response coming from the real bearing node.

Bartłomiej Ambrożkiewicz, Grzegorz Litak, Anthimos Georgiadis, Arkadiusz Syta, Nicolas Meier, Alexander Gassner
Dynamics of Railway Wheelsets with a Nonsmooth Contact Force Model

The dynamics of a railway wheelset is investigated, focusing on the effect of the contact force models. For large values of creep velocities, the Coulomb model can be used as an asymptotic approximation of wheel-rail contact forces. Then, we get a nonsmooth dynamical system with codimension-2 discontinuities. At the discontinuity of the phase space, the so-called limit directions can be found, which correspond to the possible transitions between slipping and rolling. By this analysis, the nonsmooth model can complement the usual linear creep force model from the opposite direction, and we can explore more details about the qualitative behaviour of the wheelset.

Mate Antali
Studies of the Interaction Dynamics in Albumin–Chondroitin Sulfate Systems by Recurrence Method

The physicochemical basis of lubrication of articular cartilage is still not fully understood. However, the synergy between components of the synovial fluid can be a crucial factor that could explain this phenomenon. This work presents a nonlinear data analysis technique named the recurrence method, applied to the system involved two of such synovial fluid’s components: albumin and chondroitin sulfate (CS) immersed in a water environment. This analysis s performed in order to obtain the “statistical fingerprint” of the dynamics of the interaction between the molecules, and to answer whether the variables are more deterministic or more random. The system simulation has been prepared by molecular docking method followed by molecular dynamics simulations. The already mentioned recurrence method has been applied to time series of the energy of binding, and time series of number of intermolecular hydrogen bonds, as these features describe well binding between the two molecules. In detail, the time delay approach and embedded dimension approach have been applied to extract meaningful records from time series. Then, by means of recurrence plots and entropy approach, we discuss the similarities and differences between the molecular systems consisting of CS-4 and CS-6 molecules. Our main finding is higher affinity of chondroitin sulfate IV to albumin as compared with chondroitin sulfate VI.

Piotr Bełdowski, Piotr Weber, Adam Gadomski, Piotr Sionkowski, Natalia Kruszewska, Krzysztof Domino
Dynamic Integrity of Hyperelastic Spherical Membranes

Thin-walled elastomeric membranes are found in many engineering fields, being usually subjected to large deformation. Both geometrical and material nonlinearities play an important role in their static and dynamic behavior. Spherical membrane under internal pressure is a rather common problem and may present various coexisting stable solutions due to their high nonlinearity. Their response depends on the adopted constitutive law and the related material parameters. The aim of this work is to study the static and dynamic nonlinear behavior of a spherical membrane under internal pressure and investigate how the competing stable solutions may influence their dynamic integrity. Here, the Ogden model is adopted given its generality, and the nonlinear equation of motion of a preloaded membrane is obtained and solved by numerical integration and continuation techniques. To study the dynamic integrity of competing solutions, the global (GIM) and local (LIM) integrity measures and the integrity factor (IF) are adopted. For this, a numerical strategy based on the Monte Carlo method is proposed. The numerical results demonstrate the influence of competing solutions on the global dynamic behavior and safety of the structure. Furthermore, they validate the Monte Carlo technique as an efficient numerical tool for integrity quantification.

Kaio C. B. Benedetti, Frederico M. A. da Silva, Renata M. Soares, Paulo B. Gonçalves
Nonstationary Stochastic Analysis of Fractional Viscoelastic Euler-Bernoulli Beams

In this work a dynamic analysis of a viscoelastic beam subjected to nonstationary stochastic load is led; the latter is modeled as the product of a white noise process and a deterministic modulating function. The considered beam is made up of a viscoelastic material, whose constitutive law involves linear fractional operators. The partial fractional differential equation governing the beam deflection, written according to the Euler-Bernoulli hypothesis, is solved by adopting a Galerkin approach; this involves the linear modes of the corresponding elastic beam and some generalized displacements. Accordingly, a set of uncoupled fractional differential equations for the generalized displacements is obtained. These equations are solved employing a proposed numerical approach that relies on the Grunwald-Letnikov discretization scheme; in this way, the statistics of the beam response are easily computed. Finally, the proposed numerical method is validated for the stationary case exploiting an analytical solution derived from a frequency domain approach.

Andrea Burlon, Vincenzo Sucato, Giuseppe Failla, Mario Di Paola
Zermelo Navigation Problem with State Constraints

The article uses the example of the Zermelo navigation problem to illustrate a simple way to address state constraints of a certain type. The problem of the horizontal distance maximization of an autonomous aircraft operating in a steady, homogeneous flow field is considered. Process time is fixed beforehand. The simple particle model of the aircraft is considered. The particle moves in a horizontal plane with a constant modulus velocity relative to the flow of the medium. The angular velocity of rotation of the particle velocity vector is considered as the control variable. The angle between the velocity vector and the horizontal axis is subjected to a phase constraint. The structure of the dynamic system allows to reduce the optimal problem to the problem of a smaller dimension. In reduced problem the state constraints transform to the constraints on the control variables. For the reduced problem, the optimal synthesis is designed. Next, for the original problem, the sequence and the number of the arcs with motion along state constraints are determined. The control law in the initial problem is established.

Oleg Cherkasov, Egor Malykh, Nina Smirnova
On the Attacker-Defender-Target Optimal Problem

The Target-Attacker-Defender problem is considered. Assumed that all participants move in a horizontal plane with velocities of constant modulus. The Attacker uses the pure chase method to pursue the Target. The Defender launched from the Target’s wingman and the role of the Target is to minimize the distance between the Defender and the Attacker when the Attacker approaches the Target at a given distance. The Defender’s strategy is also a method of pure pursuit. The angular velocity of rotation of the Target velocity vector is considered as a control variable. The structure of the dynamic system allows to reduce it to a system of less dimension. In the reduced system, the angle between velocity vector and line-of-sight Target-Attacker is considered as a new control variable. The Pontryagin maximum principal procedure allows to reduce the optimal control problem to a boundary-value problem (BVP) for a system of nonlinear differential equations of the fourth order. The system of the BVP consists of the initial variables and doesn’t include conjugate variables. For solving the BVP, the shooting method is applied. The results of solving the BVP for various values of parameters are demonstrated.

Oleg Cherkasov, Egor Malykh, Elina Makieva
The Symbolic Description of Feedbacks in Nonlinear Control Problems with a Parameter Using Approximation Theory Methods

In this paper the algorithms for constructing parametric families of solutions for several classes of nonlinear dynamic problems on a finite time interval with a parameter on the basis of the state-dependent differential Riccati equation (SDDRE) approach and the Padé approximation (PA) are developed. Two-point Padé approximations of the differential Riccati equation solution are constructed using the pairs of local asymptotic approximations: for small and large values of the parameter and asymptotic approximations in the neighborhood of some fixed points. Padé approximations are applicable in a wider interval of parameter variation, then local asymptotic approximations. The proposed algorithms also use the approximation theory techniques, such as extrapolations and spline approximations and optimization. The possibility for increasing the accuracy of approximations and the improvement of the interpolation and extrapolation properties of two-point Spline Padé approximations (SPA) in comparison with local asymptotic approximations, SDDRE regulator is demonstrated on numerical experiments.

Yulia Danik, Mikhail Dmitriev
Reference Models of 4WS Vehicle Lateral Dynamics for the Synthesis of Steering Algorithms

The presented paper describes selected reference models describing the kinematics and dynamics of four-wheel steering (4WS) vehicle motion in a plane of the road. These models are based on well known “bicycle model” (two second order linear differential equations) supplemented by non-linear equations of trigonometric transformation of variables from local to global coordinate system. After linearization and Laplace transformation, these models acquire the forms of transfer functions which, as shown in computational examples presented in the paper, greatly facilitates the analysis of vehicle motion, as well as the synthesis of control algorithms. Especially, transmittance form of the reference models facilitate sensitivity analysis of vehicle lateral dynamics and steering system algorithms, because their parameters are analytical functions of “mechanical” parameters. For synthesis of steering system algorithms which should be adequately effective for on-line computations, the transfer functions can be reduced in several ways. One of them is discussed in the paper.

Andrzej Dębowski, Jakub Jan Faryński, Dariusz Piotr Żardecki
Deficient RC Slabs Strengthened with Combined FRP Layer and High-Performance Fiber-Reinforced Cementitious Composite

Nowadays, the strengthening of concrete structures to withstand excessive loads and increase the structure’s ductility, etc., using high-performance fiber-reinforced cementitious composite (HPFRCC) and fiber-reinforced polymer composite (FRP) is very common. In this study, a reinforced concrete (RC) slab under vertical load with different strengthening methods is investigated using the finite element method (FEM). If the concrete strength is 60% lower than the standard design status, the flexural stiffness of the slab is reduced by 75%, and the need for reinforcement is felt. By changing the width of the FRP layer with a strip arrangement from 50 to 100 cm and changing the thickness from 2 to 7 mm per slab with a width of 4 m, the maximum stiffness and bearing capacity are experienced with an increase of 23% and 10%. Also, by changing the width of the FRP layers in the checkered arrangement from 50 to 100 cm and changing the thickness from 2 to 7 mm, the maximum hardness and bearing capacity are experienced with an increase of 22% and 25%. It can be concluded that the use of the checkered arrangement is more effective in increasing the bearing capacity.

Mehdi Ebadi-Jamkhaneh, Masoud Ahmadi, Denise-Penelope N. Kontoni
On Alphabetical Shaped Soliton for Intrinsic Fractional Coupled Nonlinear Electrical Transmission Lattice Using Sine-Cosine Method

This present work deals with the effect of fractional order (FO) on alphabetical shaped solitons solutions for intrinsic fractional (2+1)-D coupled nonlinear electrical transmission lattice using sine-cosine method in addition to the higher order of dispersion. We attained some exotical, alphabetical and new types of soliton solutions by performing the fractional nonlinear partial differential circuit equation including the fourth-order spatial dispersion -due to the semi-discrete approximation- with the simplest sine-cosine method. We derive the fractional nonlinear differential circuit equation by applying the Kirchhoff’s laws and aiding by the fractional complex transform in the modified Riemann-Liouville derivatives sense we establish a nonlinear ordinary differential equation. Thus, we carry out some novel FO’s effects. We even show that, the FO including the fourth-order spatial dispersion reveal the existence of M-shaped solitons, bi soliton-like in addition to the known effect like the amplitude’s increasing. We reach for the studied circuit that, the derived solutions by means of the sine-cosine method are functions of all the capacitor’s nonlinearities (quadratic and cubic) if and only if, we use the fourth-order spatial dispersion (FOSD) during the continuous media approximation. In contrast, in the absence of the FOSD term, the solutions only exist if, either the quadratic or the cubic nonlinearity is considered separately. Moreover, the fractional order also displays on the soliton’s width.

Emmanuel Fendzi-Donfack, Nathan Nkouessi Tchepemen, Eric Tala-Tebue, Aurélien Kenfack-Jiotsa
The Effect of Damping on the Energy Transfer in the Spherical Pendulum with Fractional Damping in a Pivot Point

Nonlinear vibrations of a system with three degrees of freedom with a spherical pendulum are investigated. The system contains an oscillator and a spherical pendulum suspended from the oscillator. The damping at the pendulum pivot point is assumed to be modelled by a fractional derivative. The viscoelastic damping properties are described using the fractional Caputo derivative of order 0 < α ≤ 1 $$0 <\alpha \le 1$$ . Vibrations in the vicinity of the internal and external resonance are considered. The effect of the order of the fractional derivative on the vibrations of the autoparametric system is studied. Responses of the system, the internal and external resonance, bifurcation diagrams, Poincaré maps and the Lyapunov exponents have been calculated for various orders of fractional derivatives. Chaotic motion has been found for some system parameters.

Jan Freundlich, Danuta Sado
Lyapunov Functions by Interpolating Numerical Quadrature: Proof of Convergence

Lyapunov functions for nonlinear systems, whose dynamics are defined by ordinary differential equations, are computed by solving linear programming feasibility problems in the CPA method. Further, the CPA method is constructive and can generate a Lyapunov function on any compact subset of the basin of attraction of an asymptotically stable equilibrium. Instead of solving the linear programming feasibility problem, one can use converse theorems to determine a candidate solution and then verify the constraints of the feasibility problem. This procedure has the advantage of being usually much faster. Further, a partial solution to the feasibility problem that violates the constraints in some areas can be analyzed, whereas a solver either generates a feasible solution or assures that a feasible solution does not exist. In this paper we prove that the numerical quadrature of numerically integrated solutions will deliver a feasible solution to the linear programming problem, given that the time horizon is large enough and the time steps are small enough in the numerical integration and quadrature. Further, the relevant theorems are general enough to allow for considerable flexibility in the particular implementation as they cover a wider range of numerical methods both for integration and quadrature.

Peter Giesl, Sigurdur Hafstein
Proposal of a Control Hardware Architecture for Implementation of Fractional-Order Controllers

The objective of this paper is to present a proposal for a new control hardware architecture that supports the practical implementation of integer- and fractional-order controllers. This control hardware architecture exhibits various significant features: From the hardware point of view, the proposed architecture offers the possibility of using different conventional control technologies, namely, computer control, real-time microcontroller-based control, and FPGA-based control. From the software point of view, LabVIEW is used as the unique programming language, regardless of the selected control technology. Some examples are presented in which fractional-order controllers are implemented using different control technologies in order to demonstrate the applicability and effectiveness of the proposed control hardware architecture for this purpose. Some hardware implementation issues are also discussed in this context. Finally, some final remarks and conclusions are provided.

Juan J. Gude, Pablo García Bringas
Role of Time-Varying Physical Forcing in Plankton Ecosystems Dynamics

In the past decades, the coupling of ocean physics and of ecosystem dynamics has been investigated at length, in particular in terms of dynamical systems, but the impact of physics on the evolution of the ecosystem has been less often studied mathematically. The present contribution addresses the influence of a time-varying physical forcing of the nutrients, on the evolution of a simple ecosystem with three trophic levels. The intrinsic dynamics of the ecosystem is computed analytically first and the influence of physics is then studied numerically. A harmonic forcing introduced in the presence of a limit cycle of the ecosystem leads to a resonance in its evolution. A sub-harmonic forcing introduces a resonance only for a strong perturbation. The phase shift of the forcing to the oscillation has an impact on the amplitude of the oscillations. Thus, close to marginality, the system is sensitive to the physics driving the input of nutrients.

Anna Jaillet, Pascal Rivière, Xavier Carton
On Nonlinear Lateral Vibrations of Rotating Rods Under the Effect of Initial Stress

In this paper, we study nonlinear lateral vibrations of a simply supported rotating rod under the effect of initial stress and external loadings. The foundations of Novozhilov’s nonlinear theory of elasticity and Biot’s general theory for initially stressed solids are used when deriving a generalized elastic potential, based on which the nonlinear mathematical model is developed. The Bubnov-Galerkin approach and Wolfram Language built-in numerical method are utilized to find a solution of the model. The effect of the initial stress field on the rod spatial vibrations is analyzed and graphs for various values of the initial stress are plotted. Accounting for the tensile initial stresses allowed reducing the amplitude of the rod vibrations, whereas the compressive stresses resulted in their increase with retaining stability of the oscillatory process at optimal system parameters. The assessment of the influence of the initial stress field on the oscillatory process showed the need to take them into account when studying rotating rods under the influence of external loadings.

Lelya Khajiyeva, Askar Kudaibergenov, Askat Kudaibergenov, Aliya Umbetkulova
Optimisation Potentials of Laminated Composites Using Semi-analytical Vibro-Acoustic Models

Light and stiff composites such as fibre-reinforced plastics are sensitive to propagate structure borne sound but simultaneously can be manipulated to adjust the material behaviour. Furthermore, stiffness and damping of such composites have contradictory material properties in relation to the fibre orientation. Composite designs are normally analysed based on FEA that require special modelling efforts. However, a multi-dimensional optimisation approach of laminates with numerous layers consisting of different materials and different orientations requires very fast numerical solutions for numerous repetitions. Here, the FEA is extended by a strain energy-based modal with a damping approach for the layerwise accumulation of the anisotropic damping. The radiated sound power is determined by a velocity-based approach directly on steady state structural simulations avoiding a complex multi-physical modelling. Moreover, the frequency dependent radiation is consolidated to a single scalar optimisation objective using a fast and efficient semi-analytic approach. This efficient simulation methodology is applied to design a vibro-acoustic composite oil pan. The achieved results show the optimisation potentials of thermoplastic composites with various fibre and matrix materials in comparison to a steel as reference case.

Matthias Klaerner, Salma Binsilm, Steffen Marburg, Lothar Kroll
Validation of Numerical Models Describing the Stress-Strain Characteristics in the Strength Tests of Composite Materials on a Metal Matrix Using the Elastooptic Method

Testing the strength properties of materials intended as impact energy absorbers requires appropriate identification. The scope of tested becomes laborious when composites are the material used to build impact energy dissipation absorbers. Various methods are used for this. One of them is the method based on the analytical model or Eshelby model or the method based on the finite element method. The analysis was based on AC-44200 alloy reinforced with 20% vol. and 30% vol. of Al2O3 particles. Numerical analyses were carried out in the ABAQUS/Explicit environment based on composite material samples subjected to loads on a testing machine based on the fracture mechanics. The obtained results of the maximum stress distribution were compared with the results obtained using the elastooptic method. The high agreement of the results proves the correctly developed numerical models and the adopted boundary conditions. The developed conclusions from the research were used for further analysis in the field of modelling the impact load of a new group of materials characterized by appropriate ballistic parameters.

Adam Kurzawa, Dariusz Pyka, Mirosław Bocian, Ludomir Jankowski, Marcin Bajkowski, Kayode Olaleye, Krzysztof Jamroziak
Forced Vibrations in a Dynamic System That Is Damped By a Mechanism Which Trans-Pass Through Its Singular Position

The paper focuses on forced vibrations of a mechanical system. The system is composed of two structurally different parts: multibody modelled and finite elements modelled. To improve its numerical behaviour, author-proposed technique of tuning of modal properties is proposed. To combine the two sub-models, constraint equations are introduced and dynamics equations are extended with appropriate Lagrange multipliers. A slightly modified author-proposed technique of elimination of the multipliers is also presented. Assuming vibrations as undesired, effective method of energy dissipation is investigated. The present method is based on the modal disparity process. Due to specificity of configuration of the multibody part (its poses are close to its singular position), significant mathematical non-linearities are present. Physical properties of the model, and reactions to its harmonic excitation, are investigated. Calculations are performed for various sets of parameters expressing both: the model and the harmonic excitation. Efficiency of the damping method at the investigated region of the first resonance is confirmed.

Krzysztof Lipinski
Spectral Analysis of Chimney Vibrations

In the paper, the response of chimney in turbulent wind flow with use of the spectral method is investigated. The analysis is performed for wind flow model that reflects the real conditions. Numerical analysis investigates the vibrations of the chimney due to different parameters of turbulence. Spectra of longitudinal wind velocity for the numerical case, as well as the spectra of Karman, FSU are analyzed. The structure response for selected steel strengths and chimney cross-section reduction is analyzed. The frequency response functions for are performed.

Marcela R. Machado, Maciej Dutkiewicz
Non-smooth Dynamics in Ramp-Controlled and Sine-Controlled Buck Converters

The DC-DC Buck converter is a power converter system that transforms high DC voltages to low DC voltages and is widely used in several applications. However, a plethora of non-smooth dynamics have been found especially in the ramp-controlled buck converter configuration that lead the system to undesired behaviors, mainly due to the non-smooth nature of the ramp signal. As a consequence, we propose to change the ramp signal by a sine waveform as a smooth alternative to the non-smoothness of the ramp waveform, and thus try to avoid the non-smooth behaviors of the buck converter system without resorting to advanced control strategies. Nevertheless, we found that non-smooth dynamics remain present in the system even by using a smooth sine waveform instead of a non-smooth ramp waveform. In fact, a new variety of non-smooth bifurcations were found, cascade of border collision bifurcations and different non-smooth transitions from nT-periodic orbits to another nT-periodic orbits and chaotic bands, behaviors only observed in power inverter systems controlled by ramp signals. Finally, the simulation results are validated by computing the Lyapunov exponents.

Jose D. Morcillo, Juan-Guillermo Muñoz, Gerard Olivar Tost
Discovery and Interactive Representation of the Dimensionless Parameter-Space of the Spring-Loaded Inverted Pendulum Model of Legged Locomotion Using Surface Interpolation

The spring-loaded inverted pendulum is a widely used model of legged locomotion. However, a complete map of the dimensionless parameter regions that correspond to the stable periodic solutions cannot be found in the literature. In this work, the three-dimensional space of two dimensionless physical parameters and the dimensionless total mechanical energy of the conservative system was discovered by means of numerical continuation. The fundament of the stability analysis of the piecewise-smooth system was provided by the numerical calculation of the fundamental solution matrices and the monodromy matrix. An effective iteration procedure based on the Nelder-Mead method is presented which tunes the model parameters in order to imitate the motion characteristics of specific animals and locomotion types such as running, trotting and galloping. The results are available online in the form of an interactive platform.

Ábel Mihály Nagy, Dóra Patkó, Ambrus Zelei
Chaos and Fractal-Based Information Hiding Techniques as Applied to Steganography

Exploiting chaos and fractals is a field of great interest and has attracted a lot of attention in the last decades as far as steganography and cryptography are concerned. Their dynamic and sensitive properties lead to numerous applications; thus, research is gradually increasing through the years. The exploration of the field and the proposed applications found in bibliography, through an effort to classify directions and techniques, is the main purpose of this work. The wide spread of networking and vast amount of data circulating everyday through internet reveal both opportunities and dangers as far as security is concerned. The opportunity arising for steganography is obvious allowing to select different channels to transmit information hidden in various multimedia files. The most frequently used files are images, as transmission is daily thriving, while efficiently provide the properties required for hiding information. The main goals in steganography are explored, that is robustness, tamper resistance, hiding capacity and perceptual transparency, presenting the role fractals and chaos theory play.

Nikolaos Ntaoulas, Vasileios Drakopoulos
Analysis of Practical Application Aspects for an Active Control Strategy to Civil Engineering Structures

Last years have brought the use of information technology in all domains. For Civil Engineering, the Smart City concept, involving the new technologies, is pointing to (among other aspects) the community safety. In this line, a structural active control strategy based on full states LQR optimal control has been relatively recent improved, numerically tested and verified. In order to apply this newer strategy, practical aspects are in the views of the present work as reduced order controller in two versions and their consequences on the effectiveness of the control strategy. The numerical simulations are performed on a multi-story building loaded by strong earthquakes’ recorded accelerations. Using in-house software under Matlab, the seismic response of the structures is simulated and evaluated after introduction of Active Tuned Mass dampers. Time and frequency domain responses are obtained. The viability of the improved strategy involving application of practical considerations is determined and discussed.

Fideliu Paulet-Crainiceanu, Vitalie Florea, Septimiu George Luca, Cristian Pastia, Octavian Victor Rosca
Nonlinear Dynamic Model of the Oculo-Motor System Human Based on the Volterra Series

Eye tracking is a powerful tool that decodes eye movements and translates them into insights. Information such as pupil position, the gaze vector, and gaze point form physical characteristics that can be used in biometric systems. The data of the human oculo-motor system (OMS) responses to the test visual stimuli was obtained by use of the Tobii Pro TX300 screen-based eye tracker. The purpose of this experiment was to test the effectiveness of a new method of biometric identification based upon the definition of the integral Volterra model of the OMS in accordance with “input-output” research. Pursuant to the data received, determined first, second and third orders transient functions for two people. The informativeness of the proposed heuristic features was investigated to build a personality classifier. Resistant to computational errors pairs of features were discovered—the probability of correct recognition (PCR) value is in the range 0.92–0.97.

Vitaliy Pavlenko, Tetiana Shamanina, Vladyslav Chori
On Piezoelectric Energy Harvesting Using a Nonlinear Vibration Absorber

The mechanical system is considered which consists of two coupled oscillators (nonlinear absorber connected with primary mass) and piezoelectric element attached. Two goals are pursued: the mitigation of the responses of the main mass and maximizing the amount of energy extracted from vibrations. The influence of nonlinear stiffness component is discussed. It is shown that the piezoelectric element allows the effective energy harvesting and at the same has very limited influence on reducing the amplitude of oscillations of the main mass.

Volodymyr Puzyrov, Jan Awrejcewicz, Nataliya Losyeva
Estimation of Regions of Attraction of Dynamical Systems via Polynomial Lyapunov Function

While studying the behavior of a dynamical system, the problem of the stability of a stationary regime plays an important role. This task includes finding the stationary points or limit cycles, determining their stability or instability, and identifying regions of attraction (RoAs) of attractors. There are several classical methods for obtaining RoAs estimates, which can be divided into Lyapunov and non-Lyapunov methods; at the same time, due to the limitations of existing methods, the identification of a complex RoA boundary is practically impossible, and also leads to high computational costs. Existing methods are quite effective for systems of the second and third orders, however, an increase in the dimension of the system or the presence of uncertain mechanical parameters leads to an exponential increase in the required calculations. In this regard, it is important to develop algorithms that are fairly simple in terms of the number of necessary operations and at the same time give acceptable estimates of RoAs from a practical point of view. In this paper, we consider the problem of obtaining estimates for the domains of attraction and stability of a nonlinear dynamical system with a polynomial right hand side. The procedure is illustrated with examples previously studied by various authors.

Volodymyr Puzyrov, Jan Awrejcewicz, Nataliya Losyeva, Nina Savchenko, Oksana Nikolaieva
Super-Twisting Sliding Mode Control for a Formation of Fully-Actuated Multirotor Aerial Vehicles

The present paper is concerned with the robust attitude-position tracking control for a formation of heterogeneous fully actuated multirotor aerial vehicles equipped with fixed rotors and subject to matched model uncertainties and Lipschitz disturbances. Based on a geometrically consistent description of the control error in SE(3), a joint geometric attitude-position control law is designed using a super-twisting sliding mode approach. Trajectory commands for the formation are generated using a second-order polynomial S-curve model, which are designed in such a way that allow setting different time duration for the formation acquisition, position, and attitude commanded motions. The method is evaluated via numerical simulations using a formation of non-planar fully actuated hexacopters equipped with fixed rotors, showing to be effective and simple to implement and tune.

Jorge Antonio Ricardo Jr, Davi Antonio Santos
Parameter Identification for Two-Axis Gimbal System and Its Kinematic Calibration

This paper presents a parameter identification procedure for two-axis gimbal system and its kinematic calibration. The quality of gimbal system used for tracking, aiming or positioning highly depends on accuracy of its line-of-sight (LOS). A model of non-orthogonal two-axis gimbal system is introduced in the text. A kinematic calibration for identifying the model’s parameters is shown as a step-by-step algorithm. Forward and inverse kinematic problems considered herein are used to express potential non-orthogonality conditions between the axes among other errors captured in the transformation matrices. Results of numerical experiments, which take into account randomized errors, show the potential of the proposed calibration algorithm to improve system accuracy.

Lukasz Rówienicz, Pawel Malczyk
A Multi-Agent Computer Program for Automatic Investigation the Behavior of a Nonlinear Dynamic System in Real-Time

In this paper the general conception of the intellectual investigation of the regular and chaotic behavior of the dynamical systems with Hamilton structure are investigated. The computer system of the analysis of behavior of nonlinear dynamic system in real-time are developed. The computer system of multi-agents are implementing functions of an interface with user, an processing of data, a scheduling and a control of calculation, a control of time and special functions of recognition. The specification of the multi-agent system, its agents and database, the structure, functions and operation of the system are presented and are discussed.

Alexander Ruchkin, Constantin Ruchkin
Asymptotic Approach to Motion of Physical Pendulum with an Extended Model of Damping

In the paper, the plane motion of a physical pendulum involving the interactions with the surrounding air is considered. These interactions are described employing the model consisting of three components. The linear and quadratic terms are proportional to the magnitude of the velocity and its square, respectively. The last component is proportional to the tangential component of the acceleration. According to the semi-empirical Morison equation, the quadratic term and acceleration dependent component depict the total force exerted on the body i.e. the drag force and inertia force including the concept of mass added. The multiple scales method (MSM) is used to obtain the approximate asymptotic solution to the problem. A slight change in the natural frequency is caused by the inertial component of the total damping force. In turn, the occurrence of the absolute value of velocity in the damping model complicates the solving procedure. The derived asymptotic solution for the experimentally verified model of the air resistance force is a good basis for further qualitative analysis of the dynamic behaviour of the system. The model of interactions and the presented methodology of the asymptotic approach can be used to study the dynamics of other mechanical systems, including multibody systems. Two methods of assessing asymptotic solutions have been proposed. Both show that the applied MSM solves the governing equation to a high degree of accuracy.

Robert Salamon, Grażyna Sypniewska-Kamińska, Henryk Kamiński
Vibration Characterization of a Tubular Chemical Reactor

An essential element of the chemical industry operating system is the chemical reactor, which employs potentially dangerous materials and processes. The chemical sector employs potentially dangerous materials and processes. Thus, negligence or misfortune can easily result in devastating consequences like human health, the environment, the economy, and the industry’s reputation. Therefore, the vibration characterization of this system is essential and directly associated with its physical properties such as mass, damping, and stiffness. The numerical model is based on the Spectral Element Method (SEM), and numerical investigations are conducted regarding the effects of internal fluid on the reactor. This paper concerns the vibration characteristics of a tubular chemical reactor, and its vibration signatures represented by the acceptance response are used to characterize the reactor dynamic.

Juliana C. Santos, Marcela R. Machado, Lamiae Vernieres-Hassimi, Leila Khalij
On Dynamics of an Aerodynamic Pendulum with Multiple Links

Oscillations of different aeroelastic systems are of interest from the perspective of both practice and theory. One of examples of such systems is an aerodynamic pendulum with several elastically connected links. The last link carries a symmetrical wing which interacts with the incoming flow. This system has a “natural” equilibrium when all links are stretched along the flow. The influence of different parameters of the pendulum upon the stability of this equilibrium is studied. It is shown that the increase in the radius of inertia of the wing (the last link) contributes to destabilization of the equilibrium, while the increase in the distance between the last joint of the pendulum and the center of pressure the wind results in stabilization. Evolution of oscillations arising in the system with the increase of the wind speed is studied for pendulums with two and three links. It is shown that there exist two families of attracting solutions in both cases.

Yury Selyutskiy, Andrei Holub, Ching-Huei Lin
A Method to Improve the Accuracy of Bridge Cranes Overload Protection Using the Signal Graph

Often during the bridge cranes operation, there occur the crane major components’ breakdowns, due to overload. Therefore, to prevent these breakdowns, overhead cranes are equipped with safety devices to protect the mechanism from the overload.The system of bridge cranes protection against overload should expediently provide the crane securing against peak overloads as well as systematic overloads, another necessary requirement being to ensure such protection high accuracy, assessed using the accuracy coefficient.To determine the overhead crane overload protection accuracy coefficient, the “crane–limiter–load” system movement is presented in the form of a signal graph.Transfer functions’ dependencies are found by determining the dynamic loads applied to on the hoisting ropes.A method has been developed to improve the accuracy of bridge cranes protection from systematic and peak overloads by the means of a quasi-zero stiffness load limiter designed.It is proposed to use in this load limiter design a roller transmission mechanism that allows to achieve the load limiter’s quasi-zero stiffness.

Volodymyr Semenyuk, Vasil Martsenyuk, Valeriy Lingur, Nadiia Kazakova, Nataliia Punchenko, Pawel Falat, Kornel Warwas
Shock Torsion Wave in an Elastic Rod with Decreasing Function of Viscoplastic External Friction

The wave problem of propagation and deceleration of shock torsion perturbation in semi-infinite round elastic rod interacting with the medium is investigated using the model of viscoplastic friction with decreasing relation between shear stress and jump of velocity on the lateral surface. After linearization, an exact solution of the initial-boundary problem describing the effect of “negative viscosity” is obtained using the Laplace transforms. A wave pattern of perturbation including the prefront zone of rest, the area of motion and the domain of stationary residual stresses has been built. The three-dimensional diagrams for nonstationary fields of velocity and stresses have been constructed too.

Ivan Shatskyi, Vasyl Perepichka
Effect of Porosity on Free Vibration of FG Shallow Shells with Complex Plan Form

This paper presents application of the R-functions method for investigation of free vibrations of shallow shells with an arbitrary plan form. It is assumed that shell is fabricated of functionally graded materials with porosities. Two types of porosity are considered: evenly and unevenly distributed porosities. The volume fractions of metal and ceramic are described by the power law. The first order shear deformation theory is applied to describe mathematical formulation of the problem. Solution of the problem is carried out by the variational Ritz method and the R-functions theory. A very good agreement with available results is shown for FG shell with rectangular plan form. Detailed numerical study for shallow shell with complex plan form is fulfilled to show effectiveness of the proposed approach. In particular effects of porosity coefficient, the power law index, boundary conditions on fundamental frequencies are examined.

Tetyana Shmatko
Brachistochrone Problem with Thrust and State Constraints of Certain Type

The problem of maximizing the horizontal coordinate of a point mass moving in a vertical plane under the action of gravity forces, viscous friction, support reaction of the curve and the thrust is considered. The penalty for the control expenditures is included in the goal function. Assumed that inequality-type constraints are imposed on the slope angle of the trajectory. The system of equation belongs to a certain type that allows reduce the optimal control problem with state constraints to the optimal problem with control constraints. As a result, the sequence and the number of the arcs with motion along the phase constraints are determined and the synthesis of the optimal control is designed. It is shown that optimal trajectory of the Brachistochrone problem with viscous friction contains no more than one section of motion along the lower constraint and no more than two sections of motion along the upper one.

Nina Smirnova, Oleg Cherkasov
Evaluation of Forces in Dynamically Loaded Journal Bearings Using Feedforward Neural Networks

This paper explores the usage of artificial neural networks to evaluate forces acting in dynamically loaded finite-length journal bearings. Unlike standard numerical approaches, which require solving a hydrodynamic pressure field, the network predicts the forces directly from relative displacements and velocities of a rotating journal to a stationary bearing shell. This practice can significantly accelerate transient simulations of systems supported on such bearings without compromising their nonlinear properties. The proposed method utilises feedforward neural networks, which use a precomputed database of nondimensional forces for training. This database is generated using a finite difference method and supplemented with the corresponding relative displacements and velocities. The performance of the trained networks is also analysed.

Luboš Smolík, Jan Rendl, Radek Bulín
Time-Variable Normal Contact Force Influence on Dry-Friction Damping of Self-excited Vibration of Bladed Turbine Wheel

This conference paper deals with dry friction contacts in tie-boss couplings on reduced modal synthesis model of turbine bladed wheel with 66 blades. Due to deformation of the cascade caused by forced excitation from the stator flow and self-excitation instability modelled by Van der Pol model, the motion in friction contacts causes slipping of the surfaces and also changes in normal contact forces. Both friction and variable normal forces are considered in this contribution and effect of contact parameters, such as normal force and angle of contact surfaces, are studied.

Pavel Šnábl, Luděk Pešek, Chandra Shekhar Prasad
On the General Decay Stability of Coupled System of Stochastic Neural Networks with Impulses, Markovian Switching and Node and Interconnection Delays

In this paper we study pth moment ( p ≥ 2 $$p\geq 2$$ ) stability of a model of coupled system of stochastic neural networks. The model includes impulses, Markovian switching and time varying delays present both in the nodes and interconnection functions. This model generalizes many models in the literature and to the best of our knowledge has not been analyzed before. The presented results are based on M-matrix theory and known inequality techniques. Additionally, the stability is discussed with respect to a general decay function which includes exponential, but also more general lower rate decay functions as the polynomial and the logarithmic ones. With this we can study general decay stability, even in the cases when the exponential stability cannot be discussed.

Biljana Tojtovska, Panche Ribarski
Bifurcations in Inertial Focusing of a Particle Suspended in Flow Through Curved Rectangular Ducts

Particles suspended in a fluid flow through a curved duct can focus to specific locations within the duct cross-section. This particle focusing is a result of a balance between two dominant forces acting on the particle: (i) the inertial lift force arising from small but non-negligible inertia of the fluid, and (ii) the secondary drag force due to the cross-sectional vortices induced by the curvature of the duct. By adopting a simplified particle dynamics model developed by Ha et al. (Dynamics of small particle inertial migration in curved square ducts, SIAM J. Appl. Dyn. Syst., vol 21, 2022, pp 714–734), we investigate both analytically and numerically, the particle equilibria and their bifurcations when a small particle is suspended in low-flow-rate fluid flow through a curved duct having a 2 × 1 $$2\times 1$$ and a 1 × 2 $$1\times 2$$ rectangular cross-section. In certain parameter regimes of the model, we analytically obtain the particle equilibria and deduce their stability, while for other parameter regimes, we numerically calculate the particle equilibria and stability. Moreover, we observe a number of different bifurcations in particle equilibria such as saddle-node, pitchfork and Hopf, as the model parameters are varied. These results may aid in the design of inertial microfluidic devices aimed at particle separation by size.

Rahil N. Valani, Brendan Harding, Yvonne M. Stokes
Strategies for Amplitude Control in a Ring of Self-excited Oscillators

We study different strategies for amplitude death in a closed loop of mutually coupled limit cycle oscillators in this paper. An improvised generalised shooting and averaged model equations are used to implement the proposed strategies with ease in symmetrical and unsymmetrical models. The consequences of having a dissimilar oscillator and an external controller in an unsymmetrical network are also discussed in more detail. Numerical results show the predominant inphase attractors above a critical nonlinearity and a faster amplitude mitigation at an optimum coupling strength.

Vinod V., Bipin Balaram
Statistical Method for Analysis of Interactions Between Chosen Protein and Chondroitin Sulfate in an Aqueous Environment

We present the statistical method to study the interaction between a chosen protein and another molecule (e.g., both being components of lubricin found in synovial fluid) in a water environment. The research is performed on the example of univariate time series of chosen features of the dynamics of mucin, which interact with chondroitin sulfate (4 and 6) in four different saline solutions. Our statistical approach is based on recurrence methods to analyze chosen features of molecular dynamics. Such recurrence methods are usually applied to reconstruct the evolution of a molecular system in its reduced phase space, where the most important variables in the process are taken into account. In detail, the analyzed time-series are spitted onto sub-series of records that are expected to carry meaningful information about the system of molecules. Elements of sub-series are splinted by the constant delay-time lag (that is the parameter determined by statistical testing in our case), and the length of sub-series is the embedded dimension parameter (using the Cao method). We use the recurrent plots approach combined with the Shannon entropy approach to analyze the robustness of the sub-series determination. We hypothesize that the robustness of the sub-series determines some specifics of the dynamics of the system of molecules. We analyze rather highly noised features to demonstrate that such features lead to recurrence plots that graphically look similar. From the recurrence plots, the Shannon entropy has been computed. We have, however, demonstrated that the Shannon entropy value is highly dependent on the delay time value for analyzed features. Hence, elaboration of a more precise method of the recurrence plot analysis is required. For this reason, we suggest the random walk method that can be applied to analyze the recurrence plots automatically.

Piotr Weber, Piotr Bełdowski, Adam Gadomski, Krzysztof Domino, Piotr Sionkowski, Damian Ledziński
Development of a Cardiovascular Mathematical Model Considering the Thermal Environment

Recently, traffic accidents caused by the drowsy driving are frequently featured by medias. The system capable of catching the sign of drowsiness and alert the driver to prevent him from drowsy driving is highly demanded. The purpose of this research is to reproduce the impact on driver drowsiness by the cabin thermal environment using cardiovascular mathematical model.In this research, the cardiovascular model considering cabin thermal environment was built based on the cardiovascular model developed by Kotani et al. (Phys Rev E 65:051923, 2002) and the thermoregulation model developed by Gagge et al. (ASHRAE Trans 77:247–262, 1971). In this model, the effect of the cabin temperature on the circulatory system is mainly reflected by the change of peripheral resistance (TPR) which is influenced mainly by sympathetic nerves activity (Okada, Med Sci Int 1:54–55, 2013). Finally, the model was evaluated by comparing with the experiment value.

Zhouzheng Xia, Yusuke Ishikawa, Shigehiko Kaneko, Jin Kusaka
On Electrodynamic Attitude Stabilization of a Spacecraft in the Natural Magneto-Velocity Coordinate System

A spacecraft with electrodynamic attitude control system in a low Earth orbit is under consideration. The problem of the spacecraft attitude stabilization is studied in the natural magneto-velocity coordinate system associated with the geomagnetic induction and Lorentz force. The nonlinear stability analysis based on the Lyapunov direct method is applied in the problem. A rigorous mathematical justification is proposed, which makes it possible to obtain specific estimates for the control parameters that provide a solution to the problem of spacecraft attitude stabilization without any limitations on the Earth’s magnetic field model. With the aid of the proposed Lyapunov function, sufficient conditions for the asymptotic stability of the spacecraft equilibrium position are derived. These conditions are formulated in terms of explicit and simple inequalities on the control parameters. Thus, a constructive approach to the design of stabilizing electrodynamic attitude control in the natural magneto-velocity coordinate system is formulated.

Alexander Yu. Aleksandrov, Alexey A. Tikhonov
Influence of Fractional Order Parameter on the Dynamics of Different Vibrating Systems

The aim of this work is to investigate the fractional differential equations associated to different vibration phenomena. More specifically, we will discuss Bagley-Torvik equation, composite fractional relaxation oscillator, the motion of a linear oscillator and generalized form of Scott-Blair oscillator and Kelvin-Voigt oscillator using fractional derivative operator in the sense of Atangana-Baleanu. To maintain the consistency of the physical systems the value of the fractional order parameters (that specify the existence of non-integer order structures of the system) lies within unit interval. The solutions of the non-integer order differential equation are obtained by using integral transforms and Gacer-Stefest’s algorithm. Moreover, the classical cases could be recovered by using the interpolation property of non-integer order derivative operator. Furthermore, we will compare and analyze the control of the fractional order parameters on the dynamics of the models under consideration. Finally, useful conclusions and recommendations are recorded.

Azhar Ali Zafar, Jan Awrejcewicz
Method of Inversion of Laplace Transform in Some Problems of Dynamic Elasticity

The novelty of the proposed paper is in the workingout of new analytical approach for the inversion of Laplace transform. The necessity of it is coerced by widely application of Laplace transform in many problems of mechanics, elasticity, thermoelasticity, poroelasticity etc. The theorems are formulated and proved in the general form with the help of complex analysis. The transforms are expanded in Taylor series and Laplace transform inversion is applied term-by-term. The corollaries with the proves for the series converges are done using the generalized functions theory and real analysis. The verification of proposed approach is provided by the comparison with the formulas previously known from literature.

Zinaida Zhuravlova
Metadata
Title
Perspectives in Dynamical Systems II — Numerical and Analytical Approaches
Editor
Jan Awrejcewicz
Copyright Year
2024
Electronic ISBN
978-3-031-56496-3
Print ISBN
978-3-031-56495-6
DOI
https://doi.org/10.1007/978-3-031-56496-3

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