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Academic literature on the topic 'Nonlinear Dynamic Equations'

Author: Grafiati

Published: 6 September 2023

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Journal articles on the topic "Nonlinear Dynamic Equations"

1

Feireisl, Eduard. "Dynamic von Kármán equations involving nonlinear damping: Time-periodic solutions." Applications of Mathematics 34, no. 1 (1989): 46–56. http://dx.doi.org/10.21136/am.1989.104333.

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MA, TIAN, and SHOUHONG WANG. "DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS." Chinese Annals of Mathematics 26, no. 02 (2005): 185–206. http://dx.doi.org/10.1142/s0252959905000166.

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Yang, Min, Weiming Xiao, Erjing Han, Junjuan Zhao, Wenjiang Wang, and Yunan Liu. "Dynamic analysis of negative stiffness noise absorber with magnet." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 265, no. 7 (2023): 183–88. http://dx.doi.org/10.3397/in_2022_0031.

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In the paper, the negative stiffness membrane absorber with magnet has been taken as a nonlinear noise absorber. The dynamic characteristics of the nonlinear noise absorber have been studied by nonlinear dynamics theory and numerical simulation. The dynamic equations of the system were established under harmonic excitation. The slow flow equations of the system are derived by using complexification averaging method, and the nonlinear equations which describe the steady-state response are obtained. Bifurcation diagram, amplitude frequency diagram and phase diagram are used to study the nonlinear response of structures under different excitation conditions. The effects of excitation amplitude, excitation frequency, nonlinear term and structural parameters on the nonlinear dynamic characteristics and sound absorption characteristics of the structure are studied. The resulting equations are verified by comparing the results which respectively obtained from complexification-averaging method and Runge-Kutta method. It is helpful to optimize the structural parameters and further improve the sound absorption performance to study the variation of the sound absorption performance of magnet negative stiffness membrane absorber system with its structural parameters.

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Tie, Yu Jia, Wei Yang, and Hao Yu Tan. "Spacecraft Attitude and Orbit Coupled Nonlinear Adaptive Synchronization Control." Advanced Materials Research 327 (September 2011): 6–11. http://dx.doi.org/10.4028/www.scientific.net/amr.327.6.

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Precise dynamic model of spacecraft is essential for the space missions, to be completed successfully. Nevertheless, the independent orbit or attitude dynamic models can not meet high precision tasks. This paper developed a 6-DOF relative coupling dynamic model based upon the nonlinear relative motion dynamics equations and attitude kinematics equations described by MRP. Nonlinear synchronization control law was designed for the coupled nonlinear dynamic model, whose close-loop system was proved to be global asymptotic stable by Lyapunov direct method. Finallly, the simulation results illustrate that the nonlinear adaptive synchronization control algorithm can robustly drive the orbit and attitude errors to converge to zero.

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Gebrel, Ibrahim F., and Samuel F. Asokanthan. "Influence of System and Actuator Nonlinearities on the Dynamics of Ring-Type MEMS Gyroscopes." Vibration 4, no. 4 (2021): 805–21. http://dx.doi.org/10.3390/vibration4040045.

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This study investigates the nonlinear dynamic response behavior of a rotating ring that forms an essential element of MEMS (Micro Electro Mechanical Systems) ring-based vibratory gyroscopes that utilize oscillatory nonlinear electrostatic forces. For this purpose, the dynamic behavior due to nonlinear system characteristics and nonlinear external forces was studied in detail. The partial differential equations that represent the ring dynamics are reduced to coupled nonlinear ordinary differential equations by suitable addition of nonlinear mode functions and application of Galerkin’s procedure. Understanding the effects of nonlinear actuator dynamics is essential for characterizing the dynamic behavior of such devices. For this purpose, a suitable theoretical model to generate a nonlinear electrostatic force acting on the MEMS ring structure is formulated. Nonlinear dynamic responses in the driving and sensing directions are examined via time response, phase diagram, and Poincare’s map when the input angular motion and nonlinear electrostatic force are considered simultaneously. The analysis is envisaged to aid ongoing research associated with the fabrication of this type of device and provide design improvements in MEMS ring-based gyroscopes.

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Shan, Li Jun, Xue Fang, and Wei Dong He. "Nonlinear Dynamic Model and Equations of RV Transmission System." Advanced Materials Research 510 (April 2012): 536–40. http://dx.doi.org/10.4028/www.scientific.net/amr.510.536.

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The nonlinear dynamics model of gearing system is developed based on RV transmission system. The influence of the nonlinear factors as time-varying meshing stiffness, backlash of the gear pairs and errors is considered. By means of the Lagrange equation the multi-degree-of-freedom differential equations of motion are derived. The differential equations are very hard to solve for which are characterized by positive semi-definition, time-variation and backlash-type nonlinearity. And linear and nonlinear restoring force are coexist in the equations. In order to solve easily, the differential equations are transformed to identical dimensionless nonlinear differential equations in matrix form. The establishment of the nonlinear differential equations laid a foundation for The Solution of differential equations and the analysis of the nonlinearity characteristics.

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Piprek, Patrick, Michael M. Marb, Pranav Bhardwaj, and Florian Holzapfel. "Trajectory/Path-Following Controller Based on Nonlinear Jerk-Level Error Dynamics." Applied Sciences 10, no. 23 (2020): 8760. http://dx.doi.org/10.3390/app10238760.

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This study proposes a novel, nonlinear trajectory/path-following controller based on jerk-level error dynamics. Therefore, at first the nonlinear acceleration-based kinematic equations of motion of a dynamic system are differentiated with respect to time to obtain a representation connecting the translation jerk with the (specific) force derivative. Furthermore, the path deviation, i.e., the difference between the planned and the actual path, is formulated as nonlinear error dynamics based on the jerk error. Combining the derived equations of motion with the nonlinear error dynamics as well as employing nonlinear dynamic inversion, a control law can be derived that provides force derivative commands, which may be commanded to an inner loop for trajectory control. This command ensures an increased smoothness and faster reaction time compared to traditional approaches based on a force directly. Furthermore, the nonlinear parts in the error dynamic are feedforward components that improve the general performance due to their physical connection with the real dynamics. The validity and performance of the proposed trajectory/path-following controller are shown in an aircraft-related application example.

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Xia, Xie, Huang Hong-Bin, Qian Feng, Zhang Ya-Jun, Yang Peng, and Qi Guan-Xiao. "Dynamic Equations and Nonlinear Dynamics of Cascade Two-Photon Laser." Communications in Theoretical Physics 45, no. 6 (2006): 1042–48. http://dx.doi.org/10.1088/0253-6102/45/6/018.

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Bohner, M., and S. H. Saker. "Oscillation criteria for perturbed nonlinear dynamic equations." Mathematical and Computer Modelling 40, no. 3-4 (2004): 249–60. http://dx.doi.org/10.1016/j.mcm.2004.03.002.

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Ma, Tian, and Shouhong Wang. "Bifurcation of Nonlinear Equations: II. Dynamic Bifurcation." Methods and Applications of Analysis 11, no. 2 (2004): 179–210. http://dx.doi.org/10.4310/maa.2004.v11.n2.a2.

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Dissertations / Theses on the topic "Nonlinear Dynamic Equations"

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Peters, James Edward II. "Group analysis of the nonlinear dynamic equations of elastic strings." Diss., Georgia Institute of Technology, 1988. http://hdl.handle.net/1853/29348.

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Sotoudeh, Zahra. "Nonlinear static and dynamic analysis of beam structures using fully intrinsic equations." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41179.

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Beams are structural members with one dimension much larger than the other two. Examples of beams include propeller blades, helicopter rotor blades, and high aspect-ratio aircraft wings in aerospace engineering; shafts and wind turbine blades in mechanical engineering; towers, highways and bridges in civil engineering; and DNA modeling in biomedical engineering. Beam analysis includes two sets of equations: a generally linear two-dimensional problem over the cross-sectional plane and a nonlinear, global one-dimensional analysis. This research work deals with a relatively new set of equations for one-dimensional beam analysis, namely the so-called fully intrinsic equations. Fully intrinsic equations comprise a set of geometrically exact, nonlinear, first-order partial differential equations that is suitable for analyzing initially curved and twisted anisotropic beams. A fully intrinsic formulation is devoid of displacement and rotation variables, making it especially attractive because of the absence of singularities, infinite-degree nonlinearities, and other undesirable features associated with finite rotation variables. In spite of the advantages of these equations, using them with certain boundary conditions presents significant challenges. This research work will take a broad look at these challenges of modeling various boundary conditions when using the fully intrinsic equations. Hopefully it will clear the path for wider and easier use of the fully intrinsic equations in future research. This work also includes application of fully intrinsic equations in structural analysis of joined-wing aircraft, different rotor blade configuration and LCO analysis of HALE aircraft.

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See, Chong Wee Simon. "Numerical methods for the simulation of dynamic discontinuous systems." Thesis, University of Salford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358276.

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Zigic, Jovan. "Optimization Methods for Dynamic Mode Decomposition of Nonlinear Partial Differential Equations." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103862.

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Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.<br>Master of Science<br>The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

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Brown, Andrew M. "Design, construction and analysis of a chaotic vibratory system." Thesis, Georgia Institute of Technology, 1985. http://hdl.handle.net/1853/18172.

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SOAVE, NICOLA. "Variational and geometric methods for nonlinear differential equations." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2014. http://hdl.handle.net/10281/49889.

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This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.

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Qu, Zheng. "Nonlinear Perron-Frobenius theory and max-plus numerical methods for Hamilton-Jacobi equations." Palaiseau, Ecole polytechnique, 2013. http://pastel.archives-ouvertes.fr/docs/00/92/71/22/PDF/thesis.pdf.

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Une approche fondamentale pour la résolution de problémes de contrôle optimal est basée sur le principe de programmation dynamique. Ce principe conduit aux équations d'Hamilton-Jacobi, qui peuvent être résolues numériquement par des méthodes classiques comme la méthode des différences finies, les méthodes semi-lagrangiennes, ou les schémas antidiffusifs. À cause de la discrétisation de l'espace d'état, la dimension des problèmes de contrôle pouvant être abordés par ces méthodes classiques est souvent limitée à 3 ou 4. Ce phénomène est appellé malédiction de la dimension. Cette thèse porte sur les méthodes numériques max-plus en contôle optimal deterministe et ses analyses de convergence. Nous étudions et developpons des méthodes numériques destinées à attenuer la malédiction de la dimension, pour lesquelles nous obtenons des estimations théoriques de complexité. Les preuves reposent sur des résultats de théorie de Perron-Frobenius non linéaire. En particulier, nous étudions les propriétés de contraction des opérateurs monotones et non expansifs, pour différentes métriques de Finsler sur un cône (métrique de Thompson, métrique projective d'Hilbert). Nous donnons par ailleurs une généralisation du "coefficient d'ergodicité de Dobrushin" à des opérateurs de Markov sur un cône général. Nous appliquons ces résultats aux systèmes de consensus ainsi qu'aux équations de Riccati généralisées apparaissant en contrôle stochastique<br>Dynamic programming is one of the main approaches to solve optimal control problems. It reduces the latter problems to Hamilton-Jacobi partial differential equations (PDE). Several techniques have been proposed in the literature to solve these PDE. We mention, for example, finite difference schemes, the so-called discrete dynamic programming method or semi-Lagrangian method, or the antidiffusive schemes. All these methods are grid-based, i. E. , they require a discretization of the state space, and thus suffer from the so-called curse of dimensionality. The present thesis focuses on max-plus numerical solutions and convergence analysis for medium to high dimensional deterministic optimal control problems. We develop here max-plus based numerical algorithms for which we establish theoretical complexity estimates. The proof of these estimates is based on results of nonlinear Perron-Frobenius theory. In particular, we study the contraction properties of monotone or non-expansive nonlinear operators, with respect to several classical metrics on cones (Thompson's metric, Hilbert's projective metric), and obtain nonlinear or non-commutative generalizations of the "ergodicity coefficients" arising in the theory of Markov chains. These results have applications in consensus theory and also to the generalized Riccati equations arising in stochastic optimal control

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Ferrara, Joseph. "A Study of Nonlinear Dynamics in Mathematical Biology." UNF Digital Commons, 2013. http://digitalcommons.unf.edu/etd/448.

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We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.

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Larson, David F. H. "Modeling nonlinear stochastic ocean loads as diffusive stochastic differential equations to derive the dynamic responses of offshore wind turbines." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/105690.

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Thesis: S.B., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2016.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (page 54).<br>A procedure is developed for modeling stochastic ocean wave and wind loads as diffusive stochastic differential equations (SDE) in a state space form to derive the response statistics of offshore structures, specifically wind turbines. Often, severe wind and wave systems are highly nonlinear and thus treatment as linear systems is not applicable, leading to computationally expensive Monte Carlo simulations. Using Stratonovich-form diffusive stochastic differential equations, both linear and nonlinear components of the wind thrust can be modeled as 2 state SDE. These processes can be superposed with both the linear and nonlinear (inertial and viscous) wave forces, also modeled as a multi-dimensional state space SDE. Furthermore, upon implementing the ESPRIT algorithm to fit the autocorrelation function of any real sea state spectrum, a simple 2-state space model can be derived to completely describe the wave forces. The resulting compound state-space SDE model forms the input to a multi-dimension state-space Fokker-Planck equation, governing the dynamical response of the wind turbine structure. Its solution yields response, fatigue and failure statistics-information critical to the design of any offshore structure. The resulting Fokker-Planck equation can be solved using existing numerical schemes.<br>by David F.H. Larson.<br>S.B.

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Challa, Subhash. "Nonlinear state estimation and filtering with applications to target tracking problems." Thesis, Queensland University of Technology, 1998.

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Books on the topic "Nonlinear Dynamic Equations"

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Grusa, K. U. Mathematical analysis of nonlinear dynamic processes. Longman Scientific & Technical, 1988.

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Oscillations in planar dynamic systems. World Scientific, 1996.

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Papageorgiou, Evangelos C. Development of a dynamic model for a UAV. Naval Postgraduate School, 1997.

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Murthy, V. R. Linear and nonlinear dynamic analysis of redundant load path bearingless rotor systems. National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1994.

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Grusa, Karl-Ulrich. Mathematical analysis of nonlinear dynamic processes: An introduction to processes governed by partial differential equations. Longman Scientific & Technical, 1988.

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UNESCO. Working Group on Systems Analysis. Meeting. Lotka-Volterra-approach to cooperation and competition in dynamic systems: Proceedings of the 5th Meeting of UNESCO's Working Group on System Theory held on the Wartburg, Eisenach (GDR), March 5-9, 1984. Akademie-Verlag, 1985.

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Maurice, Holt, Packard Andrew, and Institute for Computer Applications in Science and Engineering., eds. Simulation of a controlled airfoil with jets. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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1941-, Brunner H., Zhao Xiao-Qiang, and Zou Xingfu 1958-, eds. Nonlinear dynamics and evolution equations. American Mathematical Society, 2006.

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Verhulst, F. Nonlinear differential equations and dynamical systems. Springer-Verlag, 1990.

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Verhulst, Ferdinand. Nonlinear Differential Equations and Dynamical Systems. Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-97149-5.

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Book chapters on the topic "Nonlinear Dynamic Equations"

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Simonovits, András. "Nonlinear Difference Equations." In Mathematical Methods in Dynamic Economics. Palgrave Macmillan UK, 2000. http://dx.doi.org/10.1057/9780230513532_4.

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Georgiev, Svetlin G. "Nonlinear Dynamic Equations and Optimal Control Problems." In Fuzzy Dynamic Equations, Dynamic Inclusions, and Optimal Control Problems on Time Scales. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76132-5_15.

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Wang, Chao, Ravi P. Agarwal, Donal O’Regan, and Rathinasamy Sakthivel. "Nonlinear Dynamic Equations on Translation Time Scales." In Theory of Translation Closedness for Time Scales. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-38644-3_6.

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Georgiev, Svetlin G. "Oscillations of Second-Order Nonlinear Functional Dynamic Equations." In Functional Dynamic Equations on Time Scales. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-15420-2_9.

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Georgiev, Svetlin G. "Nonlinear Integro-Dynamic Equations and Optimal Control Problems." In Fuzzy Dynamic Equations, Dynamic Inclusions, and Optimal Control Problems on Time Scales. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76132-5_16.

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Socha, Leslaw. "Moment Equations for Nonlinear Stochastic Dynamic Systems (NSDS)." In Linearization Methods for Stochastic Dynamic Systems. Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-72997-6_4.

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Święch, Andrzej. "HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control." In Nonlinear Partial Differential Equations for Future Applications. Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-33-4822-6_5.

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Mas-Gallic, S. "A Particle in Cell Method for the Isentropic Gas Dynamic System." In Navier—Stokes Equations and Related Nonlinear Problems. Springer US, 1995. http://dx.doi.org/10.1007/978-1-4899-1415-6_29.

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Georgiev, Svetlin G., and Khaled Zennir. "Boundary Value Problems for Nonlinear First Order Dynamic Equations." In Boundary Value Problems on Time Scales, Volume I. Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003173557-1.

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Georgiev, Svetlin G., and Khaled Zennir. "Boundary Value Problems for Nonlinear Second Order Dynamic Equations." In Boundary Value Problems on Time Scales, Volume I. Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003173557-5.

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Conference papers on the topic "Nonlinear Dynamic Equations"

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Siranosian, Antranik A., Miroslav Krstic, Andrey Smyshlyaev, and Matt Bement. "Gain Scheduling-Inspired Control for Nonlinear Partial Differential Equations." In ASME 2009 Dynamic Systems and Control Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/dscc2009-2532.

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We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with a destabilizing in-domain nonlinearity is considered first. For this system a nonlinear feedback law based on gain scheduling is derived explicitly, and a statement of stability is presented for the closed-loop system. Control designs are then presented for a string and shear beam PDE, both with Kelvin-Voigt damping and potentially destabilizing free-end nonlinearities. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization-based design.

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Caruntu, Dumitru I. "On Internal Resonance of Nonlinear Nonuniform Beams." In ASME 2009 Dynamic Systems and Control Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/dscc2009-2647.

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This paper reports the case of internal resonance three-to-one with frequency of excitation near natural frequency in the case of bending vibrations of nonuniform cantilever with small damping. The case of nonlinear curvature, moderately large amplitudes, is considered. The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions. The phase-amplitude equations are analytically determined. Steady-state response is reported.

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Bava, G. P., P. Debernadri, L. A. Lugiato, and F. Castelli. "Dynamic Model for Optical Bistability in Multiple Quantum-Well Structures." In Nonlinear Dynamics in Optical Systems. Optica Publishing Group, 1990. http://dx.doi.org/10.1364/nldos.1990.tdsls66.

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We formulate a set of dynamical equations, which govern the dynamical evolution of optically bistable systems based on Multiple Quantum Well Structures, in conditions of quasi-resonance with an excitonic line. The steady state diagrams indicate the possibility of bistability in this system.

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Xianmin, Zhang, and Guo Xuemei. "Nonlinear Dynamic Performance Analysis of Elastic Linkage Mechanisms." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4206.

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Abstract In this paper, the generalized nonlinear equations of motion for elastic linkage mechanism systems are presented, in which the gross motion and elastic deformation coupling terms and the geometric nonlinearity effects are taken into account. The equations of motion are period and time-varying nonlinear equations. According to the characteristics, solution method for this kind nonlinear equations is investigated, and an efficient closed-form iterative procedure is presented. The effects of geometric nonlinearity on linkage mechanisms are studied. The results of this study are important for dynamic design of linkage mechanism systems.

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Bélanger, Nicolas, and Pierre-André Bélanger. "Cascadable rms characteristics and average dynamic of pulses in dispersive nonlinear lossy fibers." In Nonlinear Guided Waves and Their Applications. Optica Publishing Group, 1998. http://dx.doi.org/10.1364/nlgw.1998.nsnps.p8.

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More accurate equations describing the propagation law of the three rms parameters in average- soliton regime are presented. These equations are cascadable from a piece of fiber to another making them useful for designing dispersion maps. Finally, the average soliton dynamic is generalized to any pulse shape even if the hyperbolic secant seems to be the optimal case.

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Ru, P., P. K. Jakobsen, and J. V. Moloney. "Nonlocal Adiabatic Elimination in the Maxwell-Bloch Equation." In Nonlinear Dynamics in Optical Systems. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nldos.1992.mc6.

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Adiabatic elimination is a standard procedure applied to the Maxwell-Bloch laser equations when one variable or more is slaved to the remaining variables. An important case in point is a laser with an extremely large gain bandwidth satisfying the condition γ⊥ ≫ γ||, k where γ⊥ is the polarization dephasing rate, γ|| the de-energization rate and k the cavity damping constant. For example, color center gain media satisfy this criterion and support hundreds of thousands of longitudinal modes in synchronous pumped mode-locking operation. For simple single mode plane wave models the crude adiabatic elimination step of setting the derivative of the polarization variable to zero can be avoided by using center manifold techniques [1]. In this general class of singular perturbation problem, the idea is to coordinatize the problem using linear stability analysis about some known solution and then to construct an approximation to the center manifold on which the (possibly dynamic) solution remains for all time. This procedure has been successfully applied to the Maxwell-Bloch equations describing a single mode homogenously broadened ring laser [2]. Extension of the procedure to nonlinear partial differential equations is very difficult in general as the resulting center manifold may be an infinite dimensional object. When transverse (or additional longitudinal) degrees of freedom are introduced in the Maxwell-Bloch equations in order to investigate spatial pattern formation (or mode-locking dynamics) we find that a crude adiabatic elimination (henceforth referred to as standard adiabatic elimination SAE) leads to nonphysical high transverse (or longitudinal) spatial wavenumber instabilities [3]. Recent attempts to apply the center manifold technique to the transverse problem have met with mixed success [4]. In fact the high transverse wavenumber instability shows an even stronger divergence than the SAE case for positive sign of the laser-atom detuning. Moreover, the analysis becomes unwieldy even in situations when the center manifold approach appears to work.

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Song, X., and Q. X. Zhang. "Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales." In 2015 International Conference on Electrical, Automation and Mechanical Engineering. Atlantis Press, 2015. http://dx.doi.org/10.2991/eame-15.2015.221.

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Müller, R. "Dynamic Behavior of Directly Modulated Single-Quantum-Well Semiconductor Lasers." In Nonlinear Dynamics in Optical Systems. Optica Publishing Group, 1990. http://dx.doi.org/10.1364/nldos.1990.tdsls86.

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This paper deals theoretically with light emission from GRIN-SCH single-quantum-well diode lasers where the optical transitions between the first subbands (n=1) as well as between the second ones (n=2) are taken into account, see Fig.1. The mathematical model consists of three rate equations describing the time evolution of the carrier concentration N and the photon densities P1, P2 at the frequencies γ1 = E1/h and γ2 = E2/h, Moreover, nonlinear gain suppression depending on P1 and P2 and noise terms are considered too. In addition, algebraic equations are used giving the gain at the two frequencies of interest in dependence on the time-varying electron and heavy-hole concentration via the Fermi-Dirac distribution functions. Since the nonlinear gain flattening effect /1/ is larger for γ1 than for γ2 the gain at γ2 will succeed that at γ1 for sufficiently large values of N. Hence it depends on the threshold value of N (or on the laser losses respectively) if stationary laser action is reached at frequency γ1 or γ2. Fig. 2 shows the static maximum modal gain g1mod (first subband transition, solid curves) and g2mod (second subband transition, broken curves) versus N for various well widths Lz (10, 15, and 20 nm). Some loss levels are indicated by broken horizontal lines (a,…,d) crossing both the gain curves for each well width. For a constant injection current these crossing points relate to stationary stable or unstable solutions. For example, the points marked C1 and D2 belong to stable stationary solutions for P1 (loss level c) and P2 (level d) at Lz = 20 nm, respectively, while C2 and D1 relate to unstable solutions for P2 and P1. Relaxation oscillations to stable stationary states and the behavior of the light output in case of sinusoidal modulation of the applied current have been investigated by numerical calculations.

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Crouch, David D., Diana M. Lininger, and Dana Z. Anderson. "Theory of Bistability and Self-Pulsing in an Optical Ring Circuit Having Saturable Photorefractive Gain, Loss, and Photorefractive Feedback." In Nonlinear Dynamics in Optical Systems. Optica Publishing Group, 1990. http://dx.doi.org/10.1364/nldos.1990.dmmpcps483.

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We present a theory for an optical ring circuit in which gain, loss, and feedback are provided by means of refractive-index gratings in photorefractive crystals. Maxwell’s equations and the Kukhtarev charge transport model describe the evolution of the optical fields and the gratings, respectively. Steady-state solutions of the equations exhibit bistability. Dynamic solutions, obtained numerically, exhibit either history-dependent bistability due to the dependence of the feedback coupling element on its past, or self pulsing, depending on the relative speeds of the photorefractive gain and loss.

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Gudetti, Jacinth Philemon, Seyed Jamaleddin Mostafavi Yazdi, Javad Baqersad, Diane Peters, and Mohammad Ghamari. "Data-Driven Modeling of Linear and Nonlinear Dynamic Systems for Noise and Vibration Applications." In Noise and Vibration Conference & Exhibition. SAE International, 2023. http://dx.doi.org/10.4271/2023-01-1078.

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<div class="section abstract"><div class="htmlview paragraph">Data-driven modeling can help improve understanding of the governing equations for systems that are challenging to model. In the current work, the Sparse Identification of Nonlinear Dynamical systems (SINDy) is used to predict the dynamic behavior of dynamic problems for NVH applications. To show the merit of the approach, the paper demonstrates how the equations of motions for linear and nonlinear multi-degree of freedom systems can be obtained. First, the SINDy method is utilized to capture the dynamic behavior of linear systems. Second, the accuracy of the SINDy algorithm is investigated with nonlinear dynamic systems. SINDy can output differential equations that correspond to the data. This method can be used to find equations for dynamical systems that have not yet been discovered or to study current systems to compare with our current understanding of the dynamical system. With this amount of flexibility, SINDy can be used for NVH applications to help analyze vibration-related datasets as the study shows that SINDy results are consistent with ODE solutions. This study demonstrates how SINDy can accurately replicate mature known dynamical system models to highlight its potential to extract equations for more complex systems whose dynamic equations are challenging or impossible to obtain.</div></div>

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Reports on the topic "Nonlinear Dynamic Equations"

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Michalopoulos, C. D. PR-175-420-R01 Submarine Pipeline Analysis - Theoretical Manual. Pipeline Research Council International, Inc. (PRCI), 1985. http://dx.doi.org/10.55274/r0012171.

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Describes the computer program SPAN which computes the nonlinear transient response of a submarine pipeline, in contact with the ocean floor, to wave and current excitation. The dynamic response of a pipeline to impact loads, such as loads from trawl gear of fishing vessels, may also be computed. In addition, thermal expansion problems for submarine pipelines may be solved using SPAN. Beam finite element theory is used for spatial discretization of the partial differential equations governing the motion of a submarine pipeline. Large-deflection, small-strain theory is employed. The formulation involves a consistent basis and added mass matrix. Quadratic drag is computed using a nonconventional approach that involves the beam shape functions. Soil-resistance loads are computed using unique pipeline-soil interaction models which take into account coupling of axial and lateral soil forces. The nonlinear governing equations are solved numerically using the Newmark Method. This manual presents the discretized equations of motion, the methods used in determining hydrodynamic and soil-resistance forces, and the solution method.

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Hale, Jack, Constantine M. Dafermos, John Mallet-Paret, Panagiotis E. Souganidis, and Walter Strauss. Dynamical Systems and Nonlinear Partial Differential Equations. Defense Technical Information Center, 1989. http://dx.doi.org/10.21236/ada255356.

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Dafermos, Constantine M., John Mallet-Paret, Panagiotis E. Souganidis, and Walter Strauss. Dynamical Systems and Nonlinear Partial Differential Equations. Defense Technical Information Center, 1993. http://dx.doi.org/10.21236/ada271514.

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Archambault, M. R., and C. F. Edwards. Computation of Spray Dynamics by Direct Solution of Moment Transport Equations Inclusion of Nonlinear Momentum Exchange. Defense Technical Information Center, 2000. http://dx.doi.org/10.21236/ada381371.

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Wu, Yingjie, Selim Gunay, and Khalid Mosalam. Hybrid Simulations for the Seismic Evaluation of Resilient Highway Bridge Systems. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, 2020. http://dx.doi.org/10.55461/ytgv8834.

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Bridges often serve as key links in local and national transportation networks. Bridge closures can result in severe costs, not only in the form of repair or replacement, but also in the form of economic losses related to medium- and long-term interruption of businesses and disruption to surrounding communities. In addition, continuous functionality of bridges is very important after any seismic event for emergency response and recovery purposes. Considering the importance of these structures, the associated structural design philosophy is shifting from collapse prevention to maintaining functionality in the aftermath of moderate to strong earthquakes, referred to as “resiliency” in earthquake engineering research. Moreover, the associated construction philosophy is being modernized with the utilization of accelerated bridge construction (ABC) techniques, which strive to reduce the impact of construction on traffic, society, economy and on-site safety. This report presents two bridge systems that target the aforementioned issues. A study that combined numerical and experimental research was undertaken to characterize the seismic performance of these bridge systems. The first part of the study focuses on the structural system-level response of highway bridges that incorporate a class of innovative connecting devices called the “V-connector,”, which can be used to connect two components in a structural system, e.g., the column and the bridge deck, or the column and its foundation. This device, designed by ACII, Inc., results in an isolation surface at the connection plane via a connector rod placed in a V-shaped tube that is embedded into the concrete. Energy dissipation is provided by friction between a special washer located around the V-shaped tube and a top plate. Because of the period elongation due to the isolation layer and the limited amount of force transferred by the relatively flexible connector rod, bridge columns are protected from experiencing damage, thus leading to improved seismic behavior. The V-connector system also facilitates the ABC by allowing on-site assembly of prefabricated structural parts including those of the V-connector. A single-column, two-span highway bridge located in Northern California was used for the proof-of-concept of the proposed V-connector protective system. The V-connector was designed to result in an elastic bridge response based on nonlinear dynamic analyses of the bridge model with the V-connector. Accordingly, a one-third scale V-connector was fabricated based on a set of selected design parameters. A quasi-static cyclic test was first conducted to characterize the force-displacement relationship of the V-connector, followed by a hybrid simulation (HS) test in the longitudinal direction of the bridge to verify the intended linear elastic response of the bridge system. In the HS test, all bridge components were analytically modeled except for the V-connector, which was simulated as the experimental substructure in a specially designed and constructed test setup. Linear elastic bridge response was confirmed according to the HS results. The response of the bridge with the V-connector was compared against that of the as-built bridge without the V-connector, which experienced significant column damage. These results justified the effectiveness of this innovative device. The second part of the study presents the HS test conducted on a one-third scale two-column bridge bent with self-centering columns (broadly defined as “resilient columns” in this study) to reduce (or ultimately eliminate) any residual drifts. The comparison of the HS test with a previously conducted shaking table test on an identical bridge bent is one of the highlights of this study. The concept of resiliency was incorporated in the design of the bridge bent columns characterized by a well-balanced combination of self-centering, rocking, and energy-dissipating mechanisms. This combination is expected to lead to minimum damage and low levels of residual drifts. The ABC is achieved by utilizing precast columns and end members (cap beam and foundation) through an innovative socket connection. In order to conduct the HS test, a new hybrid simulation system (HSS) was developed, utilizing commonly available software and hardware components in most structural laboratories including: a computational platform using Matlab/Simulink [MathWorks 2015], an interface hardware/software platform dSPACE [2017], and MTS controllers and data acquisition (DAQ) system for the utilized actuators and sensors. Proper operation of the HSS was verified using a trial run without the test specimen before the actual HS test. In the conducted HS test, the two-column bridge bent was simulated as the experimental substructure while modeling the horizontal and vertical inertia masses and corresponding mass proportional damping in the computer. The same ground motions from the shaking table test, consisting of one horizontal component and the vertical component, were applied as input excitations to the equations of motion in the HS. Good matching was obtained between the shaking table and the HS test results, demonstrating the appropriateness of the defined governing equations of motion and the employed damping model, in addition to the reliability of the developed HSS with minimum simulation errors. The small residual drifts and the minimum level of structural damage at large peak drift levels demonstrated the superior seismic response of the innovative design of the bridge bent with self-centering columns. The reliability of the developed HS approach motivated performing a follow-up HS study focusing on the transverse direction of the bridge, where the entire two-span bridge deck and its abutments represented the computational substructure, while the two-column bridge bent was the physical substructure. This investigation was effective in shedding light on the system-level performance of the entire bridge system that incorporated innovative bridge bent design beyond what can be achieved via shaking table tests, which are usually limited by large-scale bridge system testing capacities.

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