José A. Ignácio da Silva, Flávio D. Marques, L. Sanches, G. Michon
Galloping is a self-excited vibration problem that structures immersed in fluid flow can experience. Due to its essential nonlinear phenomena, the structure exhibits limit cycle oscillations (LCOs), which, at high levels, can lead to failure of the systems. This work proposes an investigation of electromagnetic-enhanced Nonlinear Energy Sinks (NES-EH) for the hybrid control of aeroelastic LCOs and energy harvesting. The study focuses on a prismatic bluff body with a linear suspension immersed in the airflow, using classical steady nonlinear modeling for aerodynamic loads. The conventional NES approach is adopted, employing cubic stiffness and linear damping. Additionally, a linear electromagnetic transducer is included in the assembly for the energy harvesting process. By combining the method of multiple scales with the Harmonic Balance Method, analytical solutions are derived to characterize the system's dynamics under the influence of the device. The different response domains and their respective boundaries induced by the NES-EH are characterized based on the bifurcation diagrams. Furthermore, a Slow Invariant Manifold characterization is presented for each induced response domain, and its significant features are discussed. Parametric studies are carried out based on bifurcation analyses to assess the effect of NES-EH parameters on the galloping system dynamics, which allows for designing the absorber parameters. The electrical resistance is optimized to maximize the harvested power. The optimal design of NES-EH is then compared with classical energy harvesting solutions for the galloping problem. Additionally, a thorough analysis of the Target Energy Transfer phenomenon is performed.
{"title":"An Enhanced Nonlinear Energy Sink for Hybrid Bifurcation Control and Energy Harvesting From Aeroelastic Galloping Phenomena","authors":"José A. Ignácio da Silva, Flávio D. Marques, L. Sanches, G. Michon","doi":"10.1115/1.4064721","DOIUrl":"https://doi.org/10.1115/1.4064721","url":null,"abstract":"\u0000 Galloping is a self-excited vibration problem that structures immersed in fluid flow can experience. Due to its essential nonlinear phenomena, the structure exhibits limit cycle oscillations (LCOs), which, at high levels, can lead to failure of the systems. This work proposes an investigation of electromagnetic-enhanced Nonlinear Energy Sinks (NES-EH) for the hybrid control of aeroelastic LCOs and energy harvesting. The study focuses on a prismatic bluff body with a linear suspension immersed in the airflow, using classical steady nonlinear modeling for aerodynamic loads. The conventional NES approach is adopted, employing cubic stiffness and linear damping. Additionally, a linear electromagnetic transducer is included in the assembly for the energy harvesting process. By combining the method of multiple scales with the Harmonic Balance Method, analytical solutions are derived to characterize the system's dynamics under the influence of the device. The different response domains and their respective boundaries induced by the NES-EH are characterized based on the bifurcation diagrams. Furthermore, a Slow Invariant Manifold characterization is presented for each induced response domain, and its significant features are discussed. Parametric studies are carried out based on bifurcation analyses to assess the effect of NES-EH parameters on the galloping system dynamics, which allows for designing the absorber parameters. The electrical resistance is optimized to maximize the harvested power. The optimal design of NES-EH is then compared with classical energy harvesting solutions for the galloping problem. Additionally, a thorough analysis of the Target Energy Transfer phenomenon is performed.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"11 47","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140442903","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}
A model describing propagation of waves in a pre-stressed granular media is considered. The model, having the form of evolutionary PDE, is obtained from the system of ODEs describing dynamics of a chain of pre-stressed granules by means of formal asymptotic expansion. It is shown in our previous papers, that in the lowest asymptotic approximation, in which both nonlinear effects and the presence of media structure are taken into account, the model equation possesses traveling wave (TW) solutions with compact support (compactons) manifesting soliton properties. In this paper, we study a higher-order evolutionary PDE obtained by taking into account previously discarded terms of the asymptotic expansion, as well as another PDE (called analogue), differing from the original one in the values of parameters, and having compacton solutions expressed in analytical form. Numerical and analytical studies of both the higher-order model and its analogue allow to conclude that both models have compacton solutions exhibiting some properties of “true” solitons. This, in turn, testifies the stability of the previously used model with respect to the inclusion of the discarded terms of the asymptotic expansion.
{"title":"Compactons in Higher-order Nesterenko's-Type Equations","authors":"Vsevolod A. Vladimirov, S. Skurativskyi","doi":"10.1115/1.4064796","DOIUrl":"https://doi.org/10.1115/1.4064796","url":null,"abstract":"\u0000 A model describing propagation of waves in a pre-stressed granular media is considered. The model, having the form of evolutionary PDE, is obtained from the system of ODEs describing dynamics of a chain of pre-stressed granules by means of formal asymptotic expansion. It is shown in our previous papers, that in the lowest asymptotic approximation, in which both nonlinear effects and the presence of media structure are taken into account, the model equation possesses traveling wave (TW) solutions with compact support (compactons) manifesting soliton properties.\u0000 In this paper, we study a higher-order evolutionary PDE obtained by taking into account previously discarded terms of the asymptotic expansion, as well as another PDE (called analogue), differing from the original one in the values of parameters, and having compacton solutions expressed in analytical form. Numerical and analytical studies of both the higher-order model and its analogue allow to conclude that both models have compacton solutions exhibiting some properties of “true” solitons. This, in turn, testifies the stability of the previously used model with respect to the inclusion of the discarded terms of the asymptotic expansion.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"238 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140447019","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}
L. Pyrhönen, Thijs Willems, A. Mikkola, Frank Naets
This study investigates the use of linear-regression-based identification in rigid multibody system applications. A multibody system model, originally described with differential-algebraic equations, is transformed into a set of ordinary differential equations using coordinate partitioning. This allows the identification framework (where the system is described with ordinary differential equations) to be applied to rigid multibody systems described with non-minimal coordinates. The methodology is demonstrated via numerical and experimental validation on a slider-crank mechanism. The results show that the presented methodology is capable of accurately identifying the system's inertial parameters even with a short motion trajectory used for training. The presented linear-regression-based identification approach opens new opportunities to develop more accurate multibody models. The resulting updated multibody models can be considered especially useful for state-estimation and the control of multibody systems.
{"title":"Inertial Parameter Identification for Closed-Loop Mechanisms: Adaptation of Linear Regression for Coordinate Partitioning","authors":"L. Pyrhönen, Thijs Willems, A. Mikkola, Frank Naets","doi":"10.1115/1.4064794","DOIUrl":"https://doi.org/10.1115/1.4064794","url":null,"abstract":"\u0000 This study investigates the use of linear-regression-based identification in rigid multibody system applications. A multibody system model, originally described with differential-algebraic equations, is transformed into a set of ordinary differential equations using coordinate partitioning. This allows the identification framework (where the system is described with ordinary differential equations) to be applied to rigid multibody systems described with non-minimal coordinates. The methodology is demonstrated via numerical and experimental validation on a slider-crank mechanism. The results show that the presented methodology is capable of accurately identifying the system's inertial parameters even with a short motion trajectory used for training. The presented linear-regression-based identification approach opens new opportunities to develop more accurate multibody models. The resulting updated multibody models can be considered especially useful for state-estimation and the control of multibody systems.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"704 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140446447","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}
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":"86 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139784421","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":"10 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139843217","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}
Lagrange-D'Alembert principle is based on the concept of the non-actual and non-measurable virtual displacement and the assumption that the system-constraint forces are workless. Because time is not considered in defining the virtual displacements, virtual changes in prescribed displacements that characterize rheonomic constraints, referred to as driving constraints, are zero. Consequently, Lagrange-D'Alembert principle does not account systematically, from the outset, for rheonomic constraints, which are not workless and have power associated with them. In multibody system (MBS) implementations, rheonomic-constraint forces are considered as constraint forces and not as applied forces. Consequently, the statement of the virtual-work principle that virtual work of the system-inertia forces is equal to the virtual work of the system-applied forces because the virtual work of system-constraint forces is zero omits inclusion of rheonomic constraints forces. This paper discusses using alternate forms to Lagrange-D'Alembert's principle to account for rheonomic constraints from the outset by using actual and measurable variables to replace the virtual displacements. The analysis presented in this paper, which is applicable to both holonomic and non-holonomic systems, shows that the power of the system-inertia forces is equal to the power of the system-applied forces plus the power of rheonomic-constraint forces. It is shown that when redundant coordinates are used, the effect of the rheonomic constraints appears explicitly in the constraint equations; while this effect appears as generalized inertia forces when using the independent coordinates.
{"title":"Alternate Forms to Lagrange-D'Alembert Principle for Treatment of Rheonomic Constraints","authors":"A. Shabana","doi":"10.1115/1.4064722","DOIUrl":"https://doi.org/10.1115/1.4064722","url":null,"abstract":"\u0000 Lagrange-D'Alembert principle is based on the concept of the non-actual and non-measurable virtual displacement and the assumption that the system-constraint forces are workless. Because time is not considered in defining the virtual displacements, virtual changes in prescribed displacements that characterize rheonomic constraints, referred to as driving constraints, are zero. Consequently, Lagrange-D'Alembert principle does not account systematically, from the outset, for rheonomic constraints, which are not workless and have power associated with them. In multibody system (MBS) implementations, rheonomic-constraint forces are considered as constraint forces and not as applied forces. Consequently, the statement of the virtual-work principle that virtual work of the system-inertia forces is equal to the virtual work of the system-applied forces because the virtual work of system-constraint forces is zero omits inclusion of rheonomic constraints forces. This paper discusses using alternate forms to Lagrange-D'Alembert's principle to account for rheonomic constraints from the outset by using actual and measurable variables to replace the virtual displacements. The analysis presented in this paper, which is applicable to both holonomic and non-holonomic systems, shows that the power of the system-inertia forces is equal to the power of the system-applied forces plus the power of rheonomic-constraint forces. It is shown that when redundant coordinates are used, the effect of the rheonomic constraints appears explicitly in the constraint equations; while this effect appears as generalized inertia forces when using the independent coordinates.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"61 28","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139844298","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":"67 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139784810","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}
Lagrange-D'Alembert principle is based on the concept of the non-actual and non-measurable virtual displacement and the assumption that the system-constraint forces are workless. Because time is not considered in defining the virtual displacements, virtual changes in prescribed displacements that characterize rheonomic constraints, referred to as driving constraints, are zero. Consequently, Lagrange-D'Alembert principle does not account systematically, from the outset, for rheonomic constraints, which are not workless and have power associated with them. In multibody system (MBS) implementations, rheonomic-constraint forces are considered as constraint forces and not as applied forces. Consequently, the statement of the virtual-work principle that virtual work of the system-inertia forces is equal to the virtual work of the system-applied forces because the virtual work of system-constraint forces is zero omits inclusion of rheonomic constraints forces. This paper discusses using alternate forms to Lagrange-D'Alembert's principle to account for rheonomic constraints from the outset by using actual and measurable variables to replace the virtual displacements. The analysis presented in this paper, which is applicable to both holonomic and non-holonomic systems, shows that the power of the system-inertia forces is equal to the power of the system-applied forces plus the power of rheonomic-constraint forces. It is shown that when redundant coordinates are used, the effect of the rheonomic constraints appears explicitly in the constraint equations; while this effect appears as generalized inertia forces when using the independent coordinates.
{"title":"Alternate Forms to Lagrange-D'Alembert Principle for Treatment of Rheonomic Constraints","authors":"A. Shabana","doi":"10.1115/1.4064722","DOIUrl":"https://doi.org/10.1115/1.4064722","url":null,"abstract":"\u0000 Lagrange-D'Alembert principle is based on the concept of the non-actual and non-measurable virtual displacement and the assumption that the system-constraint forces are workless. Because time is not considered in defining the virtual displacements, virtual changes in prescribed displacements that characterize rheonomic constraints, referred to as driving constraints, are zero. Consequently, Lagrange-D'Alembert principle does not account systematically, from the outset, for rheonomic constraints, which are not workless and have power associated with them. In multibody system (MBS) implementations, rheonomic-constraint forces are considered as constraint forces and not as applied forces. Consequently, the statement of the virtual-work principle that virtual work of the system-inertia forces is equal to the virtual work of the system-applied forces because the virtual work of system-constraint forces is zero omits inclusion of rheonomic constraints forces. This paper discusses using alternate forms to Lagrange-D'Alembert's principle to account for rheonomic constraints from the outset by using actual and measurable variables to replace the virtual displacements. The analysis presented in this paper, which is applicable to both holonomic and non-holonomic systems, shows that the power of the system-inertia forces is equal to the power of the system-applied forces plus the power of rheonomic-constraint forces. It is shown that when redundant coordinates are used, the effect of the rheonomic constraints appears explicitly in the constraint equations; while this effect appears as generalized inertia forces when using the independent coordinates.","PeriodicalId":506262,"journal":{"name":"Journal of Computational and Nonlinear Dynamics","volume":"86 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139784419","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":"74 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139843956","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":"22 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139784970","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}
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