Special Issue "Application of Non-linear Dynamics"

A special issue of Applied Sciences (ISSN 2076-3417). This special issue belongs to the section "Mechanical Engineering".

Deadline for manuscript submissions: closed (20 May 2022) | Viewed by 13912

Special Issue Editors

Dr. Roman Starosta
E-Mail Website
Guest Editor
Institute of Applied Mechanics, Poznan University of Technology, Poznan, Poland
Interests: analytical mechanics; non-linear dynamics; asymptotic methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear dynamics was originally developed as a new area of research in applied physics and mathematics. Now, it is known that most physical processes and phenomena are non-linear. Although many issues are roughly taken as linear and that approach gives satisfactory results, there are some practical problems where non-linear effects need to be considered. Some observable phenomena like chaos, synchronization, multistability, jump or hysteresis in resonance, etc. can only be captured in the non-linear models. Sometimes, nonlinear behavior of the system is desired and could give many advantages: increase efficiency, stability, or control.

This Special Issue is focused on contemporary applications, innovative theories, and challenges related to nonlinear dynamics in various branches of science and technology.

We invite scientists and engineers who contribute to the development of non-linear dynamics to send their work to the Special Issue of the MDPI journal "applied sciences".

Dr. Roman Starosta
Prof. Dr. Jan Awrejcewicz
Guest Editors

Manuscript Submission Information

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Keywords

  • nonlinear dynamic in structural, mechanical, aeronautical, ocean and electrical engineering
  • nonlinear effects in physical systems (chaos, synchronization, bifurcation, quasi-periodicity, etc.)
  • analytical, computational, and experimental techniques applied to the nonlinear dynamics
  • nonlinear models related to control systems, identification, energy harvesting, friction and damping
  • nonlinear vibration of vibro-impact, self-induced, parametric or non-ideal systems

Published Papers (23 papers)

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Research

Article
Non-Linear Interactions of Jeffcott-Rotor System Controlled by a Radial PD-Control Algorithm and Eight-Pole Magnetic Bearings Actuator
Appl. Sci. 2022, 12(13), 6688; https://doi.org/10.3390/app12136688 - 01 Jul 2022
Viewed by 184
Abstract
Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole [...] Read more.
Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole of the magnetic actuator generates an attractive magnetic force proportional to the radial displacement and radial velocity of the rotating shaft in the direction of that pole. According to the proposed control mechanism, the mathematical model that governs the non-linear interactions between the Jeffcott system and the magnetic actuator has been established. Then, an analytical solution for the obtained non-linear dynamic model has been derived using perturbation analysis. Based on the extracted analytical solution, the motion bifurcation of the Jeffcott system has been investigated before and after control via plotting the different response curves. The obtained results illustrate that the uncontrolled Jeffcott-rotor behaves like a hard-spring duffing oscillator and responds with bi-stable periodic oscillation when the rotor angular speed is higher than the system’s natural frequency. It is alsomfound that the system, before control, can exhibit stable symmetric motion with high vibration amplitudes in both the horizontal and vertical directions, regardless of the eccentricity magnitude. In addition, the acquired results demonstrate that the introduced control technique can eliminate catastrophic bifurcation behaviors and undesired vibration of the system when the control parameters are designed properly. However, it is reported that the improper design of the controller gains may destabilize the Jeffcott system and force it to perform either chaotic or quasi-periodic motions depending on the magnitudes of both the shaft eccentricity and the control parameters. Finally, to validate the accuracy of the obtained results, numerical simulations for all response curves have been introduced which have been in excellent agreement with the analytical investigations. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Nonlinear Dynamics of an Elastic Stop System and Its Application in a Rotor System
Appl. Sci. 2022, 12(10), 5103; https://doi.org/10.3390/app12105103 - 19 May 2022
Viewed by 273
Abstract
Impact dampers or vibration systems with gaps are common in engineering applications, and the impact effects introduced by the gaps make such systems strongly nonlinear. In this paper, a model with an elastic stop is established, considering the stiffness and damping characteristics of [...] Read more.
Impact dampers or vibration systems with gaps are common in engineering applications, and the impact effects introduced by the gaps make such systems strongly nonlinear. In this paper, a model with an elastic stop is established, considering the stiffness and damping characteristics of the stop, which is a novel kind of impact damper and can be applied in a rotor system. The amplitude–frequency and phase–frequency response of the system at different gaps are obtained by the harmonic balance method with the alternating frequency–time scheme (HBM-AFT). The stability of the periodic solution is analyzed by the Floquet theory, and the time history and frequency spectra of the unstable point are analyzed by the numerical integration method. In the results, there can be more than one steady-state response at unstable points for a given excitation frequency, and the jump phenomenon occurs. The elastic stop is effective in the vibration amplitude suppression if its stiffness has been designed properly. This study provides an insight into the dynamic responses and its applications of the system with gaps, which is guidance for the analysis of pedestal looseness faults and vibration suppress methods. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Particle Filter Design for Robust Nonlinear Control System of Uncertain Heat Exchange Process with Sensor Noise and Communication Time Delay
Appl. Sci. 2022, 12(5), 2495; https://doi.org/10.3390/app12052495 - 27 Feb 2022
Viewed by 498
Abstract
In this paper, a particle filter design scheme for a robust nonlinear control system of uncertain heat exchange process against noise and communication time delay is presented. The particle filter employs a cluster of particles and associated weights to approximate the posterior distribution [...] Read more.
In this paper, a particle filter design scheme for a robust nonlinear control system of uncertain heat exchange process against noise and communication time delay is presented. The particle filter employs a cluster of particles and associated weights to approximate the posterior distribution of states and is capable of handling nonlinear and non-Gaussian issues. However, when the realistic given noise is much larger than that of the one modeled by the particle filter, the estimated posterior distribution is no longer reliable. Considering that, the exponential weights take the place of the original absolute particle weights in this paper, which act as an adjustment to the particle filter weights for a better state estimation. This adjustment for the weight of the particle filter takes into account the practical significance and can ensure the stability, tracking performance, and continuous operation of the control process incorporated with the particle filter. The simulation verifies the feasibility and usefulness of the method. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction
Appl. Sci. 2022, 12(5), 2333; https://doi.org/10.3390/app12052333 - 23 Feb 2022
Viewed by 313
Abstract
In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to [...] Read more.
In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to reformulate the original nonlinear dynamics into a different, more simplified format. Additionally, we study the problem of complexity/model reduction of large-scale lifted models of nonlinear systems from data. The method under consideration is the Loewner framework, an established data-driven approach that requires samples of input–output mappings. The latter are known as generalized transfer functions, which are appropriately defined for both bilinear and quadratic-bilinear systems. We show connections between these mappings as well as between the matrices of reduced-order models. Finally, we illustrate the theoretical discussion with two numerical examples. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network
Appl. Sci. 2022, 12(4), 1942; https://doi.org/10.3390/app12041942 - 12 Feb 2022
Viewed by 409
Abstract
This research explores the effect of synaptic pruning on a ring-shaped neural network of non-locally coupled FitzHugh–Nagumo (FHN) oscillators. The neurons in the pruned region synchronize with each other, and they repel the coherent domain of the chimera states. Furthermore, the width of [...] Read more.
This research explores the effect of synaptic pruning on a ring-shaped neural network of non-locally coupled FitzHugh–Nagumo (FHN) oscillators. The neurons in the pruned region synchronize with each other, and they repel the coherent domain of the chimera states. Furthermore, the width of the pruned region decides the precision and efficiency of the control effect on the position of coherent domains. This phenomenon gives a systematic comprehension of the relation between pruning and synchronization in neural networks from a new aspect that has never been addressed. An explanation of this mechanism is also given. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control
Appl. Sci. 2022, 12(3), 1113; https://doi.org/10.3390/app12031113 - 21 Jan 2022
Viewed by 428
Abstract
In this paper, we propose a model-based method for the reconstruction of not directly measured epidemiological data. To solve this task, we developed a generic optimization-based approach to compute unknown time-dependent quantities (such as states, inputs, and parameters) of discrete-time stochastic nonlinear models [...] Read more.
In this paper, we propose a model-based method for the reconstruction of not directly measured epidemiological data. To solve this task, we developed a generic optimization-based approach to compute unknown time-dependent quantities (such as states, inputs, and parameters) of discrete-time stochastic nonlinear models using a sequence of output measurements. The problem was reformulated as a stochastic nonlinear model predictive control computation, where the unknown inputs and parameters were searched as functions of the uncertain states, such that the model output followed the observations. The unknown data were approximated by Gaussian distributions. The predictive control problem was solved over a relatively long time window in three steps. First, we approximated the expected trajectories of the unknown quantities through a nonlinear deterministic problem. In the next step, we fixed the expected trajectories and computed the corresponding variances using closed-form expressions. Finally, the obtained mean and variance values were used as an initial guess to solve the stochastic problem. To reduce the estimated uncertainty of the computed states, a closed-loop input policy was considered during the optimization, where the state-dependent gain values were determined heuristically. The applicability of the approach is illustrated through the estimation of the epidemiological data of the COVID-19 pandemic in Hungary. To describe the epidemic spread, we used a slightly modified version of a previously published and validated compartmental model, in which the vaccination process was taken into account. The mean and the variance of the unknown data (e.g., the number of susceptible, infected, or recovered people) were estimated using only the daily number of hospitalized patients. The problem was reformulated as a finite-horizon predictive control problem, where the unknown time-dependent parameter, the daily transmission rate of the disease, was computed such that the expected value of the computed number of hospitalized patients fit the truly observed data as much as possible. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Modeling and Stability Analysis for the Vibrating Motion of Three Degrees-of-Freedom Dynamical System Near Resonance
Appl. Sci. 2021, 11(24), 11943; https://doi.org/10.3390/app112411943 - 15 Dec 2021
Cited by 6 | Viewed by 336
Abstract
The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using [...] Read more.
The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using Lagrange’s equations, which is then solved analytically up to the third approximation employing the approach of multiple scales (AMS). The emerging cases of resonance are categorized according to the solvability requirements wherein the modulation equations (ME) have been found. The stability areas and the instability ones are examined utilizing the Routh–Hurwitz criteria (RHC) and analyzed in line with the solutions at the steady state. The obtained results, resonance responses, and stability regions are addressed and graphically depicted to explore the positive influence of the various inputs of the physical parameters on the rheological behavior of the inspected system. The significance of the present work stems from its numerous applications in theoretical physics and engineering. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance
Appl. Sci. 2021, 11(20), 9520; https://doi.org/10.3390/app11209520 - 13 Oct 2021
Cited by 7 | Viewed by 446
Abstract
This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations [...] Read more.
This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
Appl. Sci. 2021, 11(18), 8658; https://doi.org/10.3390/app11188658 - 17 Sep 2021
Cited by 7 | Viewed by 609
Abstract
Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system [...] Read more.
Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system that correlates with two devices, namely a piezoelectric device and an electromagnetic one, to produce two novel models. These devices are connected to a nonlinear damping spring pendulum with two degrees of freedom. The damping spring pendulum is supported by a point moving in a circular orbit. Lagrange’s equations of the second kind were utilized to obtain the equations of motion. The asymptotic solutions of these equations were acquired up to the third approximation using the approach of multiple scales. The comparison between the approximate and the numerical solutions reveals high consistency between them. The steady-state solutions were investigated, and their stabilities were checked. The influences of excitation amplitudes, damping coefficients, and the different frequencies on energy-harvesting device outputs are examined and discussed. Finally, the nonlinear stability analysis of the modulation equations is discussed through the stability and instability ranges of the frequency response curves. The work is significant due to its real-life applications, such as a power supply of sensors, charging electronic devices, and medical applications. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Comparison of Nonlinear and Linear Controllers for Magnetic Levitation System
Appl. Sci. 2021, 11(17), 7795; https://doi.org/10.3390/app11177795 - 24 Aug 2021
Cited by 1 | Viewed by 543
Abstract
Nonlinear system control belongs to advanced control problems important for real plants control design. Various techniques have been developed in this field. In this paper we compare two different approaches to a nonlinear unstable Magnetic levitation system control. The first control design approach [...] Read more.
Nonlinear system control belongs to advanced control problems important for real plants control design. Various techniques have been developed in this field. In this paper we compare two different approaches to a nonlinear unstable Magnetic levitation system control. The first control design approach further develops our recent results on robust discrete-time pole-placement, based on convex DR-regions. The second studied approach is based on feedback linearization and the simplified development of the corresponding nonlinear control law is provided. Both approaches are compared and evaluated. The efficiency of robust discrete-time pole-placement controller is shown as well as its competitiveness in comparison with nonlinear control for Magnetic levitation system. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Analysis of the Dynamic Stiffness, Hysteresis Resonances and Complex Responses for Nonlinear Spring Systems in Power-Form Order
Appl. Sci. 2021, 11(16), 7722; https://doi.org/10.3390/app11167722 - 22 Aug 2021
Viewed by 491
Abstract
Power-form nonlinear contact force models are widely adopted in relatively moving parts of macro (e.g., rolling bearings considering Hertzian contact restoring force between rolling elements and bearing raceways) or micro (e.g., the micro cantilever probe system of atomic force microscopy) scale mechanical systems, [...] Read more.
Power-form nonlinear contact force models are widely adopted in relatively moving parts of macro (e.g., rolling bearings considering Hertzian contact restoring force between rolling elements and bearing raceways) or micro (e.g., the micro cantilever probe system of atomic force microscopy) scale mechanical systems, and contact resonance could cause serious problems of wear, contact fatigue, vibration, and noise, which has attracted widespread attention. In the present paper, the softening/hardening stiffness characteristics of continuous and one-sided contact power-form nonlinear spring models are addressed, respectively, by the analysis of the monotone features of resonant frequency-response skeleton lines. Herein, the period-n solution branch and its stability characteristics are obtained by the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory. Compared with previous studies, this paper will furtherly clarify the influences of externally normal load, the power form exponent term, and excitation amplitude on the softening/hardening stiffness characteristics of general power-form spring systems. In addition, for a power-form system with a one-sided contact, the phenomena of primary and super/sub-harmonic hysteretic resonances inducing period-doubling, folding bifurcation, the coexistence of multiple solutions are demonstrated. Besides, it gives the evolution mechanism of two types of intermittency chaos in a one-sided contact system. The overall results may have certain basic theoretical significance and engineering values for the control of vibration and noise in contact mechanical systems. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems
Appl. Sci. 2021, 11(15), 6955; https://doi.org/10.3390/app11156955 - 28 Jul 2021
Cited by 4 | Viewed by 494
Abstract
This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the [...] Read more.
This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study
Appl. Sci. 2021, 11(9), 3742; https://doi.org/10.3390/app11093742 - 21 Apr 2021
Viewed by 529
Abstract
Applications of a novel time-integration technique to the non-linear and linear dynamics of mechanical structures are presented, using an extended Picard-type iteration. Explicit discrete-mechanics approximations are taken as starting guess for the iteration. Iteration and necessary symbolic operations need to be performed only [...] Read more.
Applications of a novel time-integration technique to the non-linear and linear dynamics of mechanical structures are presented, using an extended Picard-type iteration. Explicit discrete-mechanics approximations are taken as starting guess for the iteration. Iteration and necessary symbolic operations need to be performed only before time-stepping procedure starts. In a previous investigation, we demonstrated computational advantages for free vibrations of a hanging pendulum. In the present paper, we first study forced non-linear vibrations of a tower-like mechanical structure, modeled by a standing pendulum with a non-linear restoring moment, due to harmonic excitation in primary parametric vertical resonance, and due to excitation recordings from a real earthquake. Our technique is realized in the symbolic computer languages Mathematica and Maple, and outcomes are successfully compared against the numerical time-integration tool NDSolve of Mathematica. For out method, substantially smaller computation times, smaller also than the real observation time, are found on a standard computer. We finally present the application to free vibrations of a hanging double pendulum. Excellent accuracy with respect to the exact solution is found for comparatively large observation periods. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
Appl. Sci. 2021, 11(7), 3213; https://doi.org/10.3390/app11073213 - 03 Apr 2021
Cited by 1 | Viewed by 602
Abstract
A base isolation system, aimed to passively control the nonlinear dynamics of an internally resonant tower, exposed to turbulent wind flow, is studied. A continuous visco-elastic beam, constrained at the bottom end by a nonlinear visco-elastic device and free at the top end, [...] Read more.
A base isolation system, aimed to passively control the nonlinear dynamics of an internally resonant tower, exposed to turbulent wind flow, is studied. A continuous visco-elastic beam, constrained at the bottom end by a nonlinear visco-elastic device and free at the top end, is considered. All the nonlinearities, structural, inertial and aeroelastic, these latter computed via the quasi-static theory, are accounted in the model. The interaction between self- and parametric excitations, triggered by the mean wind velocity and the turbulent component, respectively, are analyzed. The Multiple Scale Method is applied to the partial differential equations of motion, to investigate critical and post-critical behaviors, when two modes in internal 1:3 resonance are involved in the response. The first mode is found to lead the phenomenon, while the second mode is marginally involved. The effectiveness of the visco-elastic nonlinear isolation system is assessed, both in increasing the mean wind bifurcation value and in reducing the limit-cycle amplitude. The contribution of structural nonlinearities is found to weakly affect the response. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
High-Fidelity Fin–Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect
Appl. Sci. 2021, 11(7), 3057; https://doi.org/10.3390/app11073057 - 29 Mar 2021
Cited by 3 | Viewed by 511
Abstract
Both the dynamic characteristics and structural nonlinearities of an actuator will affect the flutter boundary of a fin–actuator system. The actuator models used in past research are not universal, the accuracy is difficult to guarantee, and the consideration of nonlinearity is not adequate. [...] Read more.
Both the dynamic characteristics and structural nonlinearities of an actuator will affect the flutter boundary of a fin–actuator system. The actuator models used in past research are not universal, the accuracy is difficult to guarantee, and the consideration of nonlinearity is not adequate. Based on modularization, a high-fidelity modeling method for an actuator is proposed in this paper. This model considers both freeplay and friction, which is easy to expand. It can be directly used to analyze actuator characteristics and perform aeroelastic analysis of fin–actuator systems. Friction can improve the aeroelastic stability, but the mechanism of its influence on the aeroelastic characteristics of the system has not been reported. In this paper, the LuGre model, which can better reflect the friction characteristics, was integrated into the actuator. The influence of the initial condition, freeplay, and friction on the aeroelastic characteristics of the system was analyzed. The comparison of the results with the previous research shows that oversimplified friction models are not accurate enough to reflect the mechanism of friction’s influence. By changing the loads, material, and geometry of contact surfaces, flutter can be effectively suppressed, and the power loss caused by friction can be minimized. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Fast-Scale Instability and Stabilization by Adaptive Slope Compensation of a PV-Fed Differential Boost Inverter
Appl. Sci. 2021, 11(5), 2106; https://doi.org/10.3390/app11052106 - 27 Feb 2021
Cited by 1 | Viewed by 598
Abstract
Numerical simulations reveal that a single-stage differential boost AC module supplied from a PV module under an Maximum Power Point Tracking (MPPT) control at the input DC port and with current synchronization at the AC grid port might exhibit bifurcation phenomena under some [...] Read more.
Numerical simulations reveal that a single-stage differential boost AC module supplied from a PV module under an Maximum Power Point Tracking (MPPT) control at the input DC port and with current synchronization at the AC grid port might exhibit bifurcation phenomena under some weather conditions leading to subharmonic oscillation at the fast-switching scale. This paper will use discrete-time approach to characterize such behavior and to identify the onset of fast-scale instability. Slope compensation is used in the inner current loop to improve the stability of the system. The compensation slope values needed to guarantee stability for the full range of operating duty cycle and leading to an optimal deadbeat response are determined. The validity of the followed procedures is finally validated by a numerical simulations performed on a detailed circuit-level switched model of the AC module. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Non-Linear Qualitative Dynamic Analysis of Supercritical Water-Heated Channels under External Vertical Accelerations
Appl. Sci. 2021, 11(4), 1695; https://doi.org/10.3390/app11041695 - 14 Feb 2021
Cited by 2 | Viewed by 580
Abstract
Employing the external force method to regard seismic impact and the three-region methodology to analyze the supercritical heated channel, a non-linear dynamic model was developed to investigate the transient characteristics of single channel or parallel channels under the impacts of vertical sinusoidal and [...] Read more.
Employing the external force method to regard seismic impact and the three-region methodology to analyze the supercritical heated channel, a non-linear dynamic model was developed to investigate the transient characteristics of single channel or parallel channels under the impacts of vertical sinusoidal and seismic accelerations. The present model was validated against the experimental data, which could suitably estimate the additional pressure drop caused by the vertical vibrations. The influences of parameters on the seismic-induced oscillation conducted in a supercritical heated channel indicated that a longer heated length, uprating operation power and a larger outlet loss coefficient all exhibit unstable effects, while the increase of inlet loss coefficient, a larger tube diameter and a lower inlet fluid temperature would tend to stabilize the system. Moreover, the supercritical fluid would present a high natural frequency in the very small NP-SUB region. The parametric effects on the parallel channel system are related to the inherent stability nature of initial state and the interactions among channels. The more uneven heat flux distribution among channels would cause a larger vibration-induced oscillation. In particular, when it is combined with the resonance effect, the system may exhibit much larger oscillations than in the case of non-resonance. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
A Method for the Design and Optimization of Nonlinear Tuned Damping Concepts to Mitigate Self-Excited Drill String Vibrations Using Multiple Scales Lindstedt-Poincaré
Appl. Sci. 2021, 11(4), 1559; https://doi.org/10.3390/app11041559 - 09 Feb 2021
Cited by 8 | Viewed by 947
Abstract
In downhole drilling systems, self-excited torsional vibrations caused by the bit-rock interactions can affect the drilling process and lead to the premature failure of components. Especially self-excited oscillations of higher-order modes lead to critical dynamic loads. The slim drill string design and the [...] Read more.
In downhole drilling systems, self-excited torsional vibrations caused by the bit-rock interactions can affect the drilling process and lead to the premature failure of components. Especially self-excited oscillations of higher-order modes lead to critical dynamic loads. The slim drill string design and the naturally limited drilled borehole diameter limit the installation space, power supply and lead to numerous potentially critical self-excited torsional modes. Consequently, small and robust passive damping concepts are required. The variety of possible downhole boundary conditions and potential damper designs necessitates analytical solutions for effective damper design and optimization. In this paper, two nonlinear passive damper concepts are investigated regarding design and effectiveness to reduce self-excited high-frequency torsional oscillations in drill string dynamics. Based on a finite element model of a drill string, a suitable minimal model based on the identified critical mode is generated and solved analytically using the Multiple Scales Lindstedt-Poincaré (MSLP) method. The advantages of MSLP compared to conventional MS methods are shown for this example. On the basis of the analytical solution, parameter influences are determined, and design equations are derived. The analytical results are transferred to self-excited drill string vibrations and discussed using time domain simulations of the drill string model. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter
Appl. Sci. 2021, 11(4), 1395; https://doi.org/10.3390/app11041395 - 04 Feb 2021
Viewed by 563
Abstract
This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. [...] Read more.
This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
Appl. Sci. 2021, 11(2), 741; https://doi.org/10.3390/app11020741 - 14 Jan 2021
Viewed by 551
Abstract
Time integration is commonly used to obtain accurate system responses, such as the limit cycle oscillations (LCOs) for an aeroelastic system with freeplay. However, the integrations that start with various initial conditions (I.C.s) are usually studied case by case, so only a few [...] Read more.
Time integration is commonly used to obtain accurate system responses, such as the limit cycle oscillations (LCOs) for an aeroelastic system with freeplay. However, the integrations that start with various initial conditions (I.C.s) are usually studied case by case, so only a few system states can possibly be focused on. This paper proposes a state space iterating (SSI) scheme to find LCO solutions using time integration by using another method. First, a large number of arbitrary I.C. cases are used for time integrations, but only a very short integration time is required for each I.C. case. Second, system behaviors are depicted visually through a method that combines a modified Poincaré map and Lorenz map, in which the LCO solutions are found as fixed points via visual inspections. To verify the SSI scheme’s ability to find LCOs, a typical plunge–pitch wing section is established numerically. Time integrations with both the classic scheme and the proposed SSI scheme are carried out. The LCO results of the SSI scheme are well-aligned with those from the classic scheme. The SSI scheme visualizes the patterns of system responses using arbitrary I.C. cases and analyzes the LCO stability, which provides more mathematical insights into an aeroelastic system with freeplay. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Adaptive Tracking PID and FOPID Speed Control of an Elastically Attached Load Driven by a DC Motor at Almost Step Disturbance of Loading Torque and Parametric Excitation
Appl. Sci. 2021, 11(2), 679; https://doi.org/10.3390/app11020679 - 12 Jan 2021
Cited by 3 | Viewed by 839
Abstract
Adaptive tracking control of the speed of a very elastically attached circular load driven by a direct current motor accompanied with an adaptive conventional and a fractional-order Proportional Integral Derivative (PID) controller is studied. It refers to improving the closed-loop control system response [...] Read more.
Adaptive tracking control of the speed of a very elastically attached circular load driven by a direct current motor accompanied with an adaptive conventional and a fractional-order Proportional Integral Derivative (PID) controller is studied. It refers to improving the closed-loop control system response of elastically coupled components of drivelines. The motor and the load mechatronic models and the block diagrams are constructed. Parameters of the PID controller in the model reference control are both constant, as well as vary in time. The adaptive control method is improved by the application of a new closed-loop control structure canceling error dynamics. A few competing control strategies are tested based on the application of two types low and high frequency stepwise increasing variations of loading torque and damping coefficient of motion. Moreover, the performance of the control strategies is verified by Integral Time-Weighted Absolute Error (ITAE) index, since their robustness is evaluated by applying a sine modulated triangle waves of selected electric parameters. Therefore, a dynamic forcing and parameter uncertainty is applied. Simulation results are compared for checking the proposed methods. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Mechanism and Characteristics of Global Varying Compliance Parametric Resonances in a Ball Bearing
Appl. Sci. 2020, 10(21), 7849; https://doi.org/10.3390/app10217849 - 05 Nov 2020
Cited by 3 | Viewed by 798
Abstract
Varying compliance (VC) is an unavoidable form of parametric excitation in rolling bearings and can affect the stability and safety of the bearing and its supporting rotor system. To date, we have investigated VC primary resonance in ball bearings, and in this paper [...] Read more.
Varying compliance (VC) is an unavoidable form of parametric excitation in rolling bearings and can affect the stability and safety of the bearing and its supporting rotor system. To date, we have investigated VC primary resonance in ball bearings, and in this paper other parametric VC resonance types are addressed. For a classical ball bearing model with Hertzian contact and clearance nonlinearities between the rolling elements and raceway, the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory are adopted to analyze the VC parametric resonances and their stabilities. It is found that the 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various VC combination resonances, such as the 1-order and 2-order summed types, can be excited, thus resulting in period-1, period-2, period-4, period-8, period-35, quasi-period, and even chaotic VC motions in the system. Furthermore, the bifurcation and hysteresis characteristics of complex VC resonant responses are discussed, in which cyclic fold, period doubling, and the second Hopf bifurcation can occur. Finally, the global involution of VC resonances around bearing clearance-free operations (i.e., adjusting the bearing clearance to zero or one with low interference) are provided. The overall results extend the investigation of VC parametric resonance cases in rolling bearings. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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Article
Harmonic-Modal Hybrid Reduced Order Model for the Efficient Integration of Non-Linear Soil Dynamics
Appl. Sci. 2020, 10(19), 6778; https://doi.org/10.3390/app10196778 - 27 Sep 2020
Cited by 3 | Viewed by 738
Abstract
Nonlinear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis, and more concretely laboratory data, indicate that the stress-strain relationship of soils is nonlinear and exhibits hysteresis. An equivalent linearization method, in which [...] Read more.
Nonlinear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis, and more concretely laboratory data, indicate that the stress-strain relationship of soils is nonlinear and exhibits hysteresis. An equivalent linearization method, in which non-linear characteristics of shear modulus and damping factor of soils are modeled as equivalent linear relations of the shear strain is usually applied, but this assumption, however, may lead to a conservative approach of the seismic design. In this paper, we propose an alternative analysis formulation, able to address forced response simulation of soils exhibiting their characteristic nonlinear behavior. The proposed approach combines ingredients of modal and harmonic analyses enabling efficient time-integration of nonlinear soil behaviors based on the offline construction of a dynamic response parametric solution by using Proper Generalized Decomposition (PGD)-based model order reduction technique. Full article
(This article belongs to the Special Issue Application of Non-linear Dynamics)
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