Joshua LeGrande, A. Malla, M. Bukhari, Oumar Barry
Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasiperiodic metamaterial with periodic local resonator attachments. We model a 1-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasiperiodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasiperiodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.
{"title":"Introduction of Local Resonators to a Nonlinear Metamaterial with Topological Features","authors":"Joshua LeGrande, A. Malla, M. Bukhari, Oumar Barry","doi":"10.1115/1.4064726","DOIUrl":"https://doi.org/10.1115/1.4064726","url":null,"abstract":"\u0000 Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasiperiodic metamaterial with periodic local resonator attachments. We model a 1-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasiperiodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasiperiodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"61 26","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139844300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The COVID-19 virus emerged suddenly in early 2020 and spread rapidly, causing a significant impact on national health. To achieve our goal, we introduce a time-delay factor based on the traditional SIR/SIRD model and perform a sliding average on the data collected from the official website during the data preprocessing stage. The results of this study are in very good agreement with the actual evolution of COVID-19, and the prediction accuracy can all be controlled within 3%. From our model parameter perspective, under strict isolation policies, the transmission rate of COVID-19 in China is relatively low and still significantly reduced, indicating that government intervention has had a positive effect on epidemic prevention in the country. Besides, our model is also successfully applied to predict the outbreaks caused by the SARS virus in 2003 and the COVID-19 outbreak induced by the Omicron virus in 2022, demonstrating its wide application and effectiveness. This work facilitates timely measures and adjustment of medical resources in various regions, ultimately helping to reduce economic and social losses.
{"title":"Long-Term Prediction of Large-Scale and Sporadic COVID-19 Epidemics Induced by the Original Strain in China Based On the Improved Non-Autonomous Delayed SIRD and SIR Models","authors":"Xin Xie, Lijun Pei","doi":"10.1115/1.4064720","DOIUrl":"https://doi.org/10.1115/1.4064720","url":null,"abstract":"\u0000 The COVID-19 virus emerged suddenly in early 2020 and spread rapidly, causing a significant impact on national health. To achieve our goal, we introduce a time-delay factor based on the traditional SIR/SIRD model and perform a sliding average on the data collected from the official website during the data preprocessing stage. The results of this study are in very good agreement with the actual evolution of COVID-19, and the prediction accuracy can all be controlled within 3%. From our model parameter perspective, under strict isolation policies, the transmission rate of COVID-19 in China is relatively low and still significantly reduced, indicating that government intervention has had a positive effect on epidemic prevention in the country. Besides, our model is also successfully applied to predict the outbreaks caused by the SARS virus in 2003 and the COVID-19 outbreak induced by the Omicron virus in 2022, demonstrating its wide application and effectiveness. This work facilitates timely measures and adjustment of medical resources in various regions, ultimately helping to reduce economic and social losses.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"29 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139784850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Baochen Meng, Chencheng Lian, Ji Wang, Huimin Jing, Rongxing Wu, Ji Lin, Isaac Elishakoff
The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross-section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.
{"title":"The Approximate Analysis of Higher-Order Frequencies of Nonlinear Vibrations of a Cantilever Beam with the Extended Galerkin Method","authors":"Baochen Meng, Chencheng Lian, Ji Wang, Huimin Jing, Rongxing Wu, Ji Lin, Isaac Elishakoff","doi":"10.1115/1.4064724","DOIUrl":"https://doi.org/10.1115/1.4064724","url":null,"abstract":"\u0000 The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross-section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"55 18","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139844514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Roy Kimura, Tsuyoshi Inoue, Hiroo Taura, Akira Heya
The rotordynamic (RD) fluid force generated in fluid elements such as seals in turbomachinery affects the stability of turbomachinery and causes shaft vibrations. Various studies have been conducted to clarify the effects of seals on the stability of rotor systems. Many studies have investigated the rotor dynamics of horizontal shaft systems, considering the RD fluid force generated in the seals, and in these studies, the stability of horizontal shaft systems has been assessed via eigenvalue analysis using RD coefficients. However, few studies have analyzed vertical shaft systems. The dynamic behavior of vertical shafts differs significantly from that of horizontal shafts because the weight of the rotor does not act on the seal in a vertical shaft system. Vertical shaft systems are generally prone to instability because of the fluid film whirl, and the amplitude of the shaft whirl tends to be large. When the amplitude is large, the RD fluid force cannot be linearized around the equilibrium point using RD coefficients. Therefore, destabilization and stabilization phenomena that appear in vertical shaft systems cannot be predicted using eigenvalue analysis. Fluid-structure interaction (FSI) analysis that considers the interaction between the shaft vibration and the RD fluid force generated in seals is required to predict such phenomena. This study used FSI analysis to investigate the effects of unbalance and differential pressure on the stability of a vertical shaft system subjected to RD fluid force generated in the seal.
{"title":"Influence of Unbalance and Differential Pressure On the Stability of Vertical Rotor-seal System","authors":"S. Roy Kimura, Tsuyoshi Inoue, Hiroo Taura, Akira Heya","doi":"10.1115/1.4064725","DOIUrl":"https://doi.org/10.1115/1.4064725","url":null,"abstract":"\u0000 The rotordynamic (RD) fluid force generated in fluid elements such as seals in turbomachinery affects the stability of turbomachinery and causes shaft vibrations. Various studies have been conducted to clarify the effects of seals on the stability of rotor systems. Many studies have investigated the rotor dynamics of horizontal shaft systems, considering the RD fluid force generated in the seals, and in these studies, the stability of horizontal shaft systems has been assessed via eigenvalue analysis using RD coefficients. However, few studies have analyzed vertical shaft systems. The dynamic behavior of vertical shafts differs significantly from that of horizontal shafts because the weight of the rotor does not act on the seal in a vertical shaft system. Vertical shaft systems are generally prone to instability because of the fluid film whirl, and the amplitude of the shaft whirl tends to be large. When the amplitude is large, the RD fluid force cannot be linearized around the equilibrium point using RD coefficients. Therefore, destabilization and stabilization phenomena that appear in vertical shaft systems cannot be predicted using eigenvalue analysis. Fluid-structure interaction (FSI) analysis that considers the interaction between the shaft vibration and the RD fluid force generated in seals is required to predict such phenomena. This study used FSI analysis to investigate the effects of unbalance and differential pressure on the stability of a vertical shaft system subjected to RD fluid force generated in the seal.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"60 47","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139844771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Duffing oscillator with delayed feedbacks is widely used in engineering. Chaos in such system plays an important role in the dynamic response of the system, which may lead to the collapse of the system. Therefore, it is necessary and significant to study the chaotic dynamical behaviors of such systems. Chaotic dynamics of the Duffing oscillator subjected to periodic external and nonlinear parameter excitations with delayed feedbacks are investigated both analytically and numerically in this paper. With the Melnikov method, the critical value of chaos arising from heteroclinic intersection is derived analytically. The feature of the critical curves separating chaotic and non-chaotic regions on the excitation frequency and the time delay is investigated analytically in detail. Under the corresponding system parameters, the monotonicity of the critical value to the excitation frequency, displacement time delay and velocity time delay is obtained rigorously. The chaos threshold obtained by the analytical method is verified by numerical simulations.
{"title":"Chaotic Dynamics of a Duffing Oscillator Subjected to External and Nonlinear Parametric Excitations with Delayed Feedbacks","authors":"Aijia Ding, Sengen Hu, Liangqiang Zhou","doi":"10.1115/1.4064723","DOIUrl":"https://doi.org/10.1115/1.4064723","url":null,"abstract":"\u0000 Duffing oscillator with delayed feedbacks is widely used in engineering. Chaos in such system plays an important role in the dynamic response of the system, which may lead to the collapse of the system. Therefore, it is necessary and significant to study the chaotic dynamical behaviors of such systems. Chaotic dynamics of the Duffing oscillator subjected to periodic external and nonlinear parameter excitations with delayed feedbacks are investigated both analytically and numerically in this paper. With the Melnikov method, the critical value of chaos arising from heteroclinic intersection is derived analytically. The feature of the critical curves separating chaotic and non-chaotic regions on the excitation frequency and the time delay is investigated analytically in detail. Under the corresponding system parameters, the monotonicity of the critical value to the excitation frequency, displacement time delay and velocity time delay is obtained rigorously. The chaos threshold obtained by the analytical method is verified by numerical simulations.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"43 15","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139783969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sondos M. Syam, Z. Siri, Sami Altoum, M. Aigo, R. Md Kasmani
In this paper, we present a novel numerical approach for solving nonlinear problems with a singular kernel. We prove the existence and uniqueness of the solution for these models as well as the uniform convergence of the function sequence produced by our novel approach to the unique solution. Additionally, we offer a closed form and prove these results for a specific class of these problems where the free term is a fractional polynomial, an exponential, or a trigonometric function. These findings are new to the best of our knowledge. To demonstrate the effectiveness of our numerical method and how to apply our theoretical findings, we solved a number of physical problems. Comparisons with various researchers are reported. Findings demonstrate that our approach is more effective and accurate. In addition, compared to methods that address this type of problems, our approach is simple to implement and has lower computing costs.
{"title":"A New Method for Solving Physical Problems with Nonlinear Phoneme within Fractional Derivatives with Singular Kernel","authors":"Sondos M. Syam, Z. Siri, Sami Altoum, M. Aigo, R. Md Kasmani","doi":"10.1115/1.4064719","DOIUrl":"https://doi.org/10.1115/1.4064719","url":null,"abstract":"\u0000 In this paper, we present a novel numerical approach for solving nonlinear problems with a singular kernel. We prove the existence and uniqueness of the solution for these models as well as the uniform convergence of the function sequence produced by our novel approach to the unique solution. Additionally, we offer a closed form and prove these results for a specific class of these problems where the free term is a fractional polynomial, an exponential, or a trigonometric function. These findings are new to the best of our knowledge. To demonstrate the effectiveness of our numerical method and how to apply our theoretical findings, we solved a number of physical problems. Comparisons with various researchers are reported. Findings demonstrate that our approach is more effective and accurate. In addition, compared to methods that address this type of problems, our approach is simple to implement and has lower computing costs.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"27 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139783494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The COVID-19 virus emerged suddenly in early 2020 and spread rapidly, causing a significant impact on national health. To achieve our goal, we introduce a time-delay factor based on the traditional SIR/SIRD model and perform a sliding average on the data collected from the official website during the data preprocessing stage. The results of this study are in very good agreement with the actual evolution of COVID-19, and the prediction accuracy can all be controlled within 3%. From our model parameter perspective, under strict isolation policies, the transmission rate of COVID-19 in China is relatively low and still significantly reduced, indicating that government intervention has had a positive effect on epidemic prevention in the country. Besides, our model is also successfully applied to predict the outbreaks caused by the SARS virus in 2003 and the COVID-19 outbreak induced by the Omicron virus in 2022, demonstrating its wide application and effectiveness. This work facilitates timely measures and adjustment of medical resources in various regions, ultimately helping to reduce economic and social losses.
{"title":"Long-Term Prediction of Large-Scale and Sporadic COVID-19 Epidemics Induced by the Original Strain in China Based On the Improved Non-Autonomous Delayed SIRD and SIR Models","authors":"Xin Xie, Lijun Pei","doi":"10.1115/1.4064720","DOIUrl":"https://doi.org/10.1115/1.4064720","url":null,"abstract":"\u0000 The COVID-19 virus emerged suddenly in early 2020 and spread rapidly, causing a significant impact on national health. To achieve our goal, we introduce a time-delay factor based on the traditional SIR/SIRD model and perform a sliding average on the data collected from the official website during the data preprocessing stage. The results of this study are in very good agreement with the actual evolution of COVID-19, and the prediction accuracy can all be controlled within 3%. From our model parameter perspective, under strict isolation policies, the transmission rate of COVID-19 in China is relatively low and still significantly reduced, indicating that government intervention has had a positive effect on epidemic prevention in the country. Besides, our model is also successfully applied to predict the outbreaks caused by the SARS virus in 2003 and the COVID-19 outbreak induced by the Omicron virus in 2022, demonstrating its wide application and effectiveness. This work facilitates timely measures and adjustment of medical resources in various regions, ultimately helping to reduce economic and social losses.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"56 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139844726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees of freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses from periodic to quasiperiodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees of freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude.
这项研究探讨了粘弹性如何影响双稳态 von Mises 桁架的整块参数模型的动态行为。该系统由一个线性一阶方程和一个二阶非线性达芬方程控制，其中的二次非线性控制着机械行为。二阶方程控制机械振荡，而线性一阶方程控制粘弹力的演变。这两个方程结合在一起，就形成了一个控制系统动力学的三阶抽搐方程。粘弹性为材料行为增加了时间尺度和自由度，使其有别于仅有粘度的系统。由于谐波激励，系统表现出从周期到准周期再到混沌的各种动态响应。为此，我们探索了带有粘性阻尼的谐波强迫 von Mises 桁架的动力学。我们展示了该系统因驱动振幅变化而产生的丰富动态行为。这有助于解释粘弹性系统的行为。粘弹性单元取代了粘性阻尼器，我们证明，虽然粘性阻尼仅仅改变了轨迹衰减到吸引子的速度，但粘弹性同时改变了能量景观和衰减速度。在传统的线性固体模型中，三个粘弹性参数控制着系统的行为，而不是纯粘滞阻尼中的一个参数。这增加了影响系统动力学的自由度。我们介绍了混沌行为的参数空间以及从规则运动到不规则运动的转变。最后，梅尔尼科夫标准确定了规则-混沌阈值。系统的粘性和弹性成分会影响混沌阈值振幅。
{"title":"The Presence of Chaos in a Viscoelastic Harmonically Forced Von Mises Truss","authors":"Pritam Ghoshal, James Gibert, Anil K. Bajaj","doi":"10.1115/1.4064554","DOIUrl":"https://doi.org/10.1115/1.4064554","url":null,"abstract":"\u0000 This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees of freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses from periodic to quasiperiodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees of freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"56 31","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139598736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present Picard's iterative method for solving time-space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The present method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.
{"title":"Solutions of Time-Space Fractional PDEs Using Picard's Iterative Method","authors":"Manoj Kumar, Aman Jhinga, J. T. Majithia","doi":"10.1115/1.4064553","DOIUrl":"https://doi.org/10.1115/1.4064553","url":null,"abstract":"\u0000 In this paper, we present Picard's iterative method for solving time-space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The present method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"14 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139597504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dheeraj Varma Manthena, C. P. Vyasarayani, Anindya Chatterjee
In this paper we study a particle sliding on a horizontally vibrating frictional table. This dynamical system has applications in parts manipulation within robotics and manufacturing. We first show numerically that, upon vibrating the table in a specific open-loop way, the particle moves to a target location under Coulomb friction alone. We then turn to analytical treatment of the system. The governing equations have strong nonlinearities due to the friction. We apply the method of multiple scales (MMS) near a 1:2 resonance in two frequencies relevant to the forcing. Application of the MMS requires some difficult integrals, for which we develop asymptotic approximations. The MMS slow flow has logarithmic nonlinearities, is valid near the target location on the table, and is easy to integrate numerically since it retains parametric excitation only in slow time. The slow flow matches very well with full numerical solutions. This problem has a practical motivation, novel elements in the application of the MMS, a satisfactory slow flow, and potential for further analytical simplification.
{"title":"Open-Loop Centering of a Point Mass on a Horizontally Vibrating Frictional Table","authors":"Dheeraj Varma Manthena, C. P. Vyasarayani, Anindya Chatterjee","doi":"10.1115/1.4064552","DOIUrl":"https://doi.org/10.1115/1.4064552","url":null,"abstract":"\u0000 In this paper we study a particle sliding on a horizontally vibrating frictional table. This dynamical system has applications in parts manipulation within robotics and manufacturing. We first show numerically that, upon vibrating the table in a specific open-loop way, the particle moves to a target location under Coulomb friction alone. We then turn to analytical treatment of the system. The governing equations have strong nonlinearities due to the friction. We apply the method of multiple scales (MMS) near a 1:2 resonance in two frequencies relevant to the forcing. Application of the MMS requires some difficult integrals, for which we develop asymptotic approximations. The MMS slow flow has logarithmic nonlinearities, is valid near the target location on the table, and is easy to integrate numerically since it retains parametric excitation only in slow time. The slow flow matches very well with full numerical solutions. This problem has a practical motivation, novel elements in the application of the MMS, a satisfactory slow flow, and potential for further analytical simplification.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"59 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139597050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：