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Modeling and Analysis in Dynamical Systems and Bistability

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (15 September 2023) | Viewed by 12348

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Special Issue Editors

Prof. Dr. Wei Zhang

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Guest Editor

College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China
Interests: chaotic theory and bifurcation theory; vibration of composite plate and shell structures; bi-stable laminate structures

Prof. Dr. Yuxin Hao

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Guest Editor

College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing 100192, China
Interests: nonlinear dynamics; structural dynamics and stability; vibration of composite plate and shell structures; bi-stable laminate structures; bifurcation analysis

Prof. Dr. Youhua Qian

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Guest Editor

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Interests: homotopy analysis method; chaotic theory and bifurcation theory; bursting oscillation
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Special Issue Information

Dear Colleagues,

Dynamical systems are pervasive in various areas of science whose research has been the focus of recent studies, leveraging a valuable integration of theoretical and applied investigations.

Bi-stable dynamic systems exhibit two stable states, they include various thin laminate structures, post-buckled beams, bi-stable mechanical systems, etc. Additionally, bi-stable structures can be utilized in many engineering fields such as commercial and fighter morphing aircrafts, energy harvesters, and shock absorbers. It is very important for researchers and engineers to get knowledge of the dynamic properties and dynamic responses of bi-stable systems. Establishing mathematical models of bi-stable systems and structures to find and analyze the solutions is crucial for their use in contemporary studies.

The primary purpose of this Special Issue is to present original research articles and review articles, which will help with the understanding of how to establish mathematical models and methods for analyzing dynamical systems with a particular interest in systems with bi-stable structures, furthermore, contributing to the understanding of their unique dynamic behaviors and reveal the mechanism behind them, as well as the utilization of the results in advanced science applications.

We are particularly interested in articles from experts and active researchers to present their recent achievements and progress in this area of scientific research, including methods in modeling and analysis in dynamical systems as well as bi-stable structures, potential applications for dynamical systems, especially bi-stable structures in engineering and mechanics.

The topics of interest include but are not limited to the following:

New researches on modeling and numerical simulations of dynamical systems;
Dynamics and control of complex systems;
Advanced modeling and analysis for bi-stable systems;
Natural vibration of bi-stability laminate structures;
Recent research on nonlinear dynamics of bi-stability structures;
Recent advances about bifurcation behaviors of dynamical systems and bi-stable systems;
Recent advances about analysis methods of bi-stable dynamic systems;
Studies on the applications of bi-stability structures in energy harvesters;
Studies on the applications of bi-stability structures in shock absorbers.

Prof. Dr. Wei Zhang
Prof. Dr. Yuxin Hao
Prof. Dr. Youhua Qian
Guest Editors

Manuscript Submission Information

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Keywords

dynamical systems
bi-stability
structural dynamics
natural vibration
vibration analysis
damping
acoustic waves
bifurcation
snap-through
bi-stable dynamical systems
mathematical modeling
numerical methods

Published Papers (10 papers)

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Research

18 pages, 9122 KiB

Open AccessFeature PaperArticle

An Analytical Expression for the Fundamental Frequency of a Long Free-Spanning Submarine Pipeline

by Ty Phuor, Pavel A. Trapper and Avshalom Ganz

Mathematics 2023, 11(21), 4481; https://doi.org/10.3390/math11214481 - 30 Oct 2023

Viewed by 994

Abstract

The DNVGL-RP-F105 guidelines provide essential techniques for the preliminary design of undersea pipelines. However, its approximations for static displacement and the natural frequency of the pipe are restricted to cases where the ratio of the pipe’s diameter to its length (L/D) is less than 140. This limitation poses challenges for longer spans, which, although rare, can sometimes be unavoidable. This study introduces a novel analytical method, rooted in the energy method and cable theory, for computing the static deformation and natural frequency of long free-span underwater pipelines. We conducted a comprehensive verification of our proposed method by comparing its outcomes with those of 212 finite element analysis simulations. The results reveal excellent predictions for long spans. However, for shorter spans, traditional methods remain more accurate. Additionally, we explored the influence of pipeline’s diameter, thickness, and boundary conditions on both static displacement and frequency, providing valuable insights for design considerations. We found that the boundary conditions’ impact on the fundamental frequency becomes negligible for long spans, with up to a 10% difference between pinned–pinned and fixed–fixed conditions. In essence, this research offers a vital enhancement to the existing DNV guidelines, becoming particularly beneficial during the preliminary design phases of pipelines with L/D ratios exceeding 140. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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30 pages, 14308 KiB

Open AccessArticle

Nonlinear Oscillations of a Composite Stepped Piezoelectric Cantilever Plate with Aerodynamic Force and External Excitation

by Yan Liu and Wensai Ma

Mathematics 2023, 11(13), 3034; https://doi.org/10.3390/math11133034 - 07 Jul 2023

Cited by 1 | Viewed by 847

Abstract

Axially moving wing aircraft can better adapt to the flight environment, improve flight performance, reduce flight resistance, and improve flight distance. This paper simplifies the fully unfolded axially moving wing into a stepped cantilever plate model, analyzes the structural nonlinearity of the system, and studies the influence of aerodynamic nonlinearity on system vibration. The model is affected by aerodynamic forces, piezoelectric excitation, and in-plane excitation. Due to Hamilton’s principle of least action, the mathematical model is established based on Reddy’s higher-order shear deformation theory, and using Galerkin’s method, the governing dimensionless partial differential equations of the system are simplified to two nonlinear ordinary differential equations, and then a study of the influence of the various engineering parameters on the nonlinear oscillations and frequency responses of this model is conducted by the method of multiple scales. It was found that the different values of

a_{5}

a_{6}

b_{6}

and

b_{8}

can change the shape of the amplitude–frequency response curve and size of the plate, while different symbols

a_{7}

and

b_{7}

can change the rigidity of the model. The excitations greatly impact the nonlinear dynamic responses of the plate. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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17 pages, 5920 KiB

Open AccessArticle

Multi-Frequency Homotopy Analysis Method for Coupled Van der Pol-Duffing System with Time Delay

by Youhua Qian, Shuli Wang and Shuping Chen

Mathematics 2023, 11(2), 407; https://doi.org/10.3390/math11020407 - 12 Jan 2023

Cited by 1 | Viewed by 763

Abstract

This paper mainly studied the analytical solutions of three types of Van der Pol-Duffing equations. For a system with parametric excitation frequency, we knew that the ordinary homotopy analysis method would be unable to find the analytical solution. Thus, we primarily used the multi-frequency homotopy analysis method (MFHAM). First, the MFHAM was introduced, and the solution of the system was expressed by constructing auxiliary linear operators. Then, the method was applied to three specific systems. We compared the numerical solution obtained using the Runge–Kutta method with the analytical solution to verify the correctness of the latter. Periodic solutions, with and without time delay, were also compared under the same parameters. The results demonstrated that it was both effective and correct to use the MFHAM to find analytical solutions to Van der Pol-Duffing systems, which were classical systems. By comparison, the MFHAM proved to be effective for time delay systems. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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21 pages, 8596 KiB

Open AccessArticle

Slow–Fast Dynamics Behaviors under the Comprehensive Effect of Rest Spike Bistability and Timescale Difference in a Filippov Slow–Fast Modified Chua’s Circuit Model

by Shaolong Li, Weipeng Lv, Zhenyang Chen, Miao Xue and Qinsheng Bi

Mathematics 2022, 10(23), 4606; https://doi.org/10.3390/math10234606 - 05 Dec 2022

Cited by 1 | Viewed by 1239

Abstract

Since the famous slow–fast dynamical system referred to as the Hodgkin–Huxley model was proposed to describe the threshold behaviors of neuronal axons, the study of various slow–fast dynamical behaviors and their generation mechanisms has remained a popular topic in modern nonlinear science. The primary purpose of this paper is to introduce a novel transition route induced by the comprehensive effect of special rest spike bistability and timescale difference rather than a common bifurcation via a modified Chua’s circuit model with an external low-frequency excitation. In this paper, we attempt to explain the dynamical mechanism behind this novel transition route through quantitative calculations and qualitative analyses of the nonsmooth dynamics on the discontinuity boundary. Our work shows that the whole system responses may tend to be various and complicated when this transition route is triggered, exhibiting rich slow–fast dynamics behaviors even with a very slight change in excitation frequency, which is described well by using Poincaré maps in numerical simulations. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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25 pages, 6019 KiB

Open AccessArticle

Chaos of the Six-Dimensional Non-Autonomous System for the Circular Mesh Antenna

by Ying Sun, Jingyi Liu, Junhua Zhang, Aiwen Wang and Fei Mao

Mathematics 2022, 10(23), 4454; https://doi.org/10.3390/math10234454 - 25 Nov 2022

Viewed by 841

Abstract

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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16 pages, 7235 KiB

Open AccessArticle

Non-Smooth Dynamic Behaviors as well as the Generation Mechanisms in a Modified Filippov-Type Chua’s Circuit with a Low-Frequency External Excitation

by Hongfang Han, Shaolong Li and Qinsheng Bi

Mathematics 2022, 10(19), 3613; https://doi.org/10.3390/math10193613 - 02 Oct 2022

Cited by 1 | Viewed by 1189

Abstract

The main purpose of this paper is to study point-cycle type bistability as well as induced periodic bursting oscillations by taking a modified Filippov-type Chua’s circuit system with a low-frequency external excitation as an example. Two different kinds of bistable structures in the fast subsystem are obtained via conventional bifurcation analyses; meanwhile, nonconventional bifurcations are also employed to explain the nonsmooth structures in the bistability. In the following numerical investigations, dynamic evolutions of the full system are presented by regarding the excitation amplitude and frequency as analysis parameters. As a consequence, we can find that the classification method for periodic bursting oscillations in smooth systems is not completely applicable when nonconventional bifurcations such as the sliding bifurcations and persistence bifurcation are involved; in addition, it should be pointed out that the emergence of the bursting oscillation does not completely depend on bifurcations under the point-cycle bistable structure in this paper. It is predicted that there may be other unrevealed slow–fast transition mechanisms worthy of further study. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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19 pages, 16231 KiB

Open AccessArticle

Nonlinear Bending of Sandwich Plates with Graphene Nanoplatelets Reinforced Porous Composite Core under Various Loads and Boundary Conditions

by Xudong Fan, Aiwen Wang, Pengcheng Jiang, Sijin Wu and Ying Sun

Mathematics 2022, 10(18), 3396; https://doi.org/10.3390/math10183396 - 19 Sep 2022

Cited by 4 | Viewed by 1348

Abstract

The nonlinear bending of the sandwich plates with graphene nanoplatelets (GPLs) reinforced porous composite (GNRPC) core and two metal skins subjected to different boundary conditions and various loads, such as the concentrated load at the center, linear loads with different slopes passing through the center, linear eccentric loads, uniform loads, and trapezoidal loads, has been presented. The popular four-unknown refined theory accounting for the thickness stretching effects has been employed to model the mechanics of the sandwich plates. The governing equations have been derived from the nonlinear Von Karman strain–displacement relationship and principle of virtual work with subsequent solution by employing the classical finite element method in combination with the Newton downhill method. The convergence of the numerical results has been checked. The accuracy and efficiency of the theory have been confirmed by comparing the obtained results with those available in the literature. Furthermore, a parametric study has been carried out to analyze the effects of load type, boundary conditions, porosity coefficient, GPLs weight fraction, GPLs geometry, and concentrated load radius on the nonlinear central bending deflections of the sandwich plates. In addition, the numerical results reveal that the adopted higher order theory can significantly improve the simulation of the transverse deflection in the thickness direction. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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18 pages, 2680 KiB

Open AccessArticle

Research on the Dynamic Characteristics of the Double Slings System with Elastic Connection Considering Boundary Conditions

by Xia Liao, Danhui Dan, Fei Han and Rui Zhao

Mathematics 2022, 10(17), 3129; https://doi.org/10.3390/math10173129 - 01 Sep 2022

Cited by 1 | Viewed by 1121

Abstract

As the length of single sling increases, double slings with transverse connections are gradually becoming one of the effective measures to control the undesirable vibration of single slings. In the analysis of the dynamic characteristics of the double sling system, this paper firstly deduces the dynamic stiffness matrix of the elastically connected double sling system by the dynamic stiffness method (DSM), solves the frequency equation evolved from the dynamic stiffness matrix by using the Wittrick-Williams (W-W) algorithm, and obtains the systematic analysis and calculation of the dynamic characteristics of the double sling system under arbitrary boundary conditions. Secondly, a complete and accurate analysis method of the dynamic characteristics of the double sling system is obtained by comprehensively considering the bending stiffness and boundary conditions of the sling, and the accuracy of the calculation can be verified by the actual measurement data. Finally, the best installation position and quantity of transverse sling clamps in the double sling system are obtained by the parametric analysis of transverse sling clamps. The analysis of this paper will provide a theoretical basis for the design and optimization of slings, and further promote the wide application of the double sling system. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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15 pages, 3706 KiB

Open AccessEditor’s ChoiceArticle

Nonlinear Transient Dynamics of Graphene Nanoplatelets Reinforced Pipes Conveying Fluid under Blast Loads and Thermal Environment

by Siyu Liu, Aiwen Wang, Wei Li, Hongyan Chen, Yufen Xie and Dongmei Wang

Mathematics 2022, 10(13), 2349; https://doi.org/10.3390/math10132349 - 05 Jul 2022

Cited by 4 | Viewed by 1529

Abstract

This work aims at investigating the nonlinear transient response of fluid-conveying pipes made of graphene nanoplatelet (GPL)-reinforced composite (GPLRC) under blast loads and in a thermal environment. A modified Halpin–Tsai model is used to approximate the effective Young’s modulus of the GPLRC pipes conveying fluid; the mass density and Poisson’s ratio are determined by using the Voigt model. A slender Euler–Bernoulli beam is considered for modeling the pipes conveying fluid. The vibration control equation of the GPLRC pipes conveying fluid under blast loads is obtained by using Hamilton’s principle. A set of second-order ordinary differential equations are obtained by using the second-order Galerkin discrete method and are solved by using the adaptive Runge–Kutta method. Numerical experiments show that GPL distribution and temperature; GPL weight fraction; pipe length-to-thickness ratio; flow velocity; and blast load parameters have important effects on the nonlinear transient response of the GPLRC pipes conveying fluid. The numerical results also show that due to the fluid–structure interaction, the vibration amplitudes of the GPLRC pipes conveying fluid decay after the impact of blast loads. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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25 pages, 5582 KiB

Open AccessArticle

Static and Dynamic Stability of Carbon Fiber Reinforced Polymer Cylindrical Shell Subject to Non-Normal Boundary Condition with One Generatrix Clamped

by Shaowu Yang, Yuxin Hao, Wei Zhang, Lingtao Liu and Wensai Ma

Mathematics 2022, 10(9), 1531; https://doi.org/10.3390/math10091531 - 03 May 2022

Cited by 6 | Viewed by 1169

Abstract

In this paper, static and dynamic stability analyses taking axial excitation into account are presented for a laminated carbon fiber reinforced polymer (CFRP) cylindrical shell under a non-normal boundary condition. The non-normal boundary condition is put forward to signify that both ends of the cylindrical shell are free and one generatrix of the shell is clamped. The partial differential motion governing the equations of the laminated CFRP cylindrical shell with a non-normal boundary condition is derived using the Hamilton principle, nonlinear von-Karman relationships and first-order deformation shell theory. Then, nonlinear, two-freedom, ordinary differential equations on the radial displacement of the cylindrical shell are obtained utilizing Galerkin method. The Newton-Raphson method is applied to numerically solve the equilibrium point. The stability of the equilibrium point is determined by analyzing the eigenvalue of the Jacobian matrix. The solution of the Mathieu equation describes the dynamic unstable behavior of the CFRP laminated cylindrical shells. The unstable regions are determined using the Bolotin method. The influences of the radial line load, the ratio of radius to thickness, the ratio of length to thickness, the number of layers and the temperature field of the laminated CFRP cylindrical shell on static and dynamic stability are investigated. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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