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Journal articles on the topic 'Nonlinear Dynamic Equations'

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Author: Grafiati

Published: 6 September 2023

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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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2

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

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3

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 (February 1, 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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4

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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5

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 (October 25, 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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6

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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7

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 (December 7, 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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8

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 (June 2006): 1042–48. http://dx.doi.org/10.1088/0253-6102/45/6/018.

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9

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

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10

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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11

Jafari, Hossein, and Hassan Kamil Jassim. "Approximate Solution for Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operator." Journal of Zankoy Sulaimani - Part A 18, no. 1 (August 30, 2015): 127–32. http://dx.doi.org/10.17656/jzs.10456.

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12

Chang, Tai Ping. "Stochastic Nonlinear Vibration of Fluid-Loaded Double-Walled Carbon Nanotubes." Applied Mechanics and Materials 284-287 (January 2013): 362–66. http://dx.doi.org/10.4028/www.scientific.net/amm.284-287.362.

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This paper investigates the stochastic dynamic behaviors of nonlinear vibration of the fluid-loaded double-walled carbon nanotubes (DWCNTs) by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. The nonlinear governing equations of the fluid-conveying DWCNTs are formulated based on the Hamilton’s principle. The Young’s modulus of elasticity of the DWCNTs is assumed as stochastic with respect to the position to actually describe the random material properties of the DWCNTs. By utilizing the perturbation technique, the nonlinear governing equations of the fluid-conveying can be decomposed into two sets of nonlinear differential equations involving the mean value of the displacement and the first variation of the displacement separately. Then we adopt the harmonic balance method in conjunction with Galerkin’s method to solve the nonlinear differential equations successively. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the amplitude of the displacement are computed. It is concluded that the mean value and standard deviation of the amplitude of the displacement increase nonlinearly with the increase of the frequencies.

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13

Zhang, Junhua, Xiaodong Yang, and Wei Zhang. "Free Vibrations and Nonlinear Responses for a Cantilever Honeycomb Sandwich Plate." Advances in Materials Science and Engineering 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/8162873.

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Dynamics of a cantilever honeycomb sandwich plate are studied in this paper. The governing equations of the composite plate subjected to both in-plane and transverse excitations are derived by using Hamilton’s principle and Reddy’s third-order shear deformation theory. Based on the Rayleigh–Ritz method, some modes of natural frequencies for the cantilever honeycomb sandwich plate are obtained. The relations between the natural frequencies and the parameters of the plate are investigated. Further, the Galerkin method is used to transform the nonlinear partial differential equations into a set of nonlinear ordinary differential equations. Nonlinear dynamic responses of the cantilever honeycomb sandwich plate to such external and parametric excitations are discussed by using the numerical method. The results show that in-plane and transverse excitations have an important influence on nonlinear dynamic characteristics. Rich dynamics, such as periodic, multiperiodic, quasiperiodic, and chaotic motions, are located and studied by the bifurcation diagram for some specific parameters.

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14

Miková, Lubica. "LINEARIZATION OF A NONLINEAR VEHICLE MODEL." TECHNICAL SCIENCES AND TECHNOLOGIES, no. 2(24) (2021): 33–37. http://dx.doi.org/10.25140/2411-5363-2021-2(24)-33-37.

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The purpose of this article is to create a mathematical model of a vehicle using dynamic equations of motion and simulation of perturbations acting on a vehicle. It is assumed that the tire in the car model behaves linearly. Because the vehicle model is nonlinear, the model will need to be linearized in order to find the transfer function between the angle of rotation of the front wheel and the lateral position of the vehicle. For this purpose, simple dynamic models of the car were created, which reflect its lateral and longitudinal dynamics. These types of models are usually used with a linearized form of mechanical and mathematical equations that are required when designing controllers, active suspension and other driver assistance systems.

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15

Nie, J. F., M. L. Zheng, G. B. Yu, J. M. Wen, and B. Dai. "The Method of Multiple Scales in Solving Nonlinear Dynamic Differential Equations of Gear Systems." Applied Mechanics and Materials 274 (January 2013): 324–27. http://dx.doi.org/10.4028/www.scientific.net/amm.274.324.

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To obtain exact analytical solutions of differential equations of gear system dynamics due to the difficulty of solving complicated differential equations. Only the approximate analytical solutions can be determined. The method of multiple scales is one of the most powerful, popular perturbation methods. The dynamic model which describes the torsional vibration behaviors of gear system has been introduced accurately in this paper. The differential equation of gear system nonlinear dynamics exhibiting combined nonlinearity influence such as time-varying stiffness, tooth backlash and dynamic transmission error (DTE) has been proposed. The theory of multiple scales method has been presented in solving nonlinear differential equations of gear systems and the frequency response equation has been obtained. The fact that the approximate analytical solution by using the method of multiple scales is in good agreement with the exact solutions by numerically integrating differential equations has proved that the method of multiple scales is one of the most frequently used methods in solving differential equations, especially for large and complicated differential equations.

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16

Chiou, B. C., and M. Shahinpoor. "Dynamic Stability Analysis of a Two-Link Force-Controlled Flexible Manipulator." Journal of Dynamic Systems, Measurement, and Control 112, no. 4 (December 1, 1990): 661–66. http://dx.doi.org/10.1115/1.2896192.

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This study investigates the effect of link flexibility on the dynamic stability of a two-link force-controlled robot manipulator. The nonlinear open-loop equations for the compliant motion are derived first. By employing the hybrid force/position control law, the closed-loop dynamic equations are then explicitly derived. The nonlinear closed-loop equations are linearized about some equilibrium configurations. Stability analyses are carried out by computing the eigenvalues of the linearized system equations. Results are verified by the numerical simulations using the complete nonlinear dynamic equations. The effect of the wrist force sensor stiffness on the dynamic stability is also investigated. Results show that the link flexibility is indeed an important source of dynamic instability in the motion of force-controlled manipulators. Moreover, the system stability is dominated by the effect of the distributed flexibility of the first link.

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17

XU, KUN, and ZHAOLI GUO. "GENERALIZED GAS DYNAMIC EQUATIONS WITH MULTIPLE TRANSLATIONAL TEMPERATURES." Modern Physics Letters B 23, no. 03 (January 30, 2009): 237–40. http://dx.doi.org/10.1142/s0217984909018096.

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Based on a multiple stage BGK-type collision model and the Chapman–Enskog expansion, the corresponding macroscopic gas dynamics equations in three-dimensional space will be derived. The new gas dynamic equations have the same structure as the Navier–Stokes equations, but the stress strain relationship in the Navier–Stokes equations is replaced by an algebraic equation with temperature differences. In the continuum flow regime, the new gas dynamic equations automatically recover the standard Navier–Stokes equations. The current gas dynamic equations are natural extension of the Navier–Stokes equations to the near continuum flow regime and can be used for near continuum flow study.

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18

Cui, Ming, Yanxin Su, and Dong Liang. "High-Order Finite Volume Methods for Aerosol Dynamic Equations." Advances in Applied Mathematics and Mechanics 8, no. 2 (January 27, 2016): 213–35. http://dx.doi.org/10.4208/aamm.2013.m362.

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AbstractAerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aerosol dynamic equation where the condensation and removal processes are considered. Numerical examples are provided to show the efficiency of these numerical methods.

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19

Li, Chunqiu, Desheng Li, and Zhijun Zhang. "Dynamic Bifurcation from Infinity of Nonlinear Evolution Equations." SIAM Journal on Applied Dynamical Systems 16, no. 4 (January 2017): 1831–68. http://dx.doi.org/10.1137/16m1107358.

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20

Jiang, Yao-Lin. "Periodic waveform relaxation solutions of nonlinear dynamic equations." Applied Mathematics and Computation 135, no. 2-3 (March 2003): 219–26. http://dx.doi.org/10.1016/s0096-3003(01)00324-1.

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21

Saker, S. H. "Oscillation of nonlinear dynamic equations on time scales." Applied Mathematics and Computation 148, no. 1 (January 2004): 81–91. http://dx.doi.org/10.1016/s0096-3003(02)00829-9.

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22

Bohner, Martin, and TongXing Li. "Kamenev-type criteria for nonlinear damped dynamic equations." Science China Mathematics 58, no. 7 (February 6, 2015): 1445–52. http://dx.doi.org/10.1007/s11425-015-4974-8.

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23

Akın-Bohner, Elvan, Martin Bohner, Smaïl Djebali, and Toufik Moussaoui. "On the Asymptotic Integration of Nonlinear Dynamic Equations." Advances in Difference Equations 2008 (2008): 1–18. http://dx.doi.org/10.1155/2008/739602.

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24

Zhang, Chenghui, Tongxing Li, Ravi P. Agarwal, and Martin Bohner. "Oscillation results for fourth-order nonlinear dynamic equations." Applied Mathematics Letters 25, no. 12 (December 2012): 2058–65. http://dx.doi.org/10.1016/j.aml.2012.04.018.

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25

Grace, Said R., Ravi P. Agarwal, Billûr Kaymakçalan, and Wichuta Sae-jie. "Oscillation theorems for second order nonlinear dynamic equations." Journal of Applied Mathematics and Computing 32, no. 1 (February 13, 2009): 205–18. http://dx.doi.org/10.1007/s12190-009-0244-7.

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26

Grace, Said R., and Taher S. Hassan. "Oscillation criteria for higher order nonlinear dynamic equations." Mathematische Nachrichten 287, no. 14-15 (May 3, 2014): 1659–73. http://dx.doi.org/10.1002/mana.201300157.

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27

Li, Xue-Qin, Guang-Chen Bai, Lu-Kai Song, and Wei Zhang. "Nonlinear Vibration Analysis for Stiffened Cylindrical Shells Subjected to Electromagnetic Environment." Shock and Vibration 2021 (July 19, 2021): 1–26. http://dx.doi.org/10.1155/2021/9983459.

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The nonlinear vibration behaviors of stiffened cylindrical shells under electromagnetic excitations, transverse excitations, and in-plane excitations are studied for the first time in this paper. Given the first-order shear deformation theory and Hamilton principle, the nonlinear partial differential governing equations of motion are derived with considering the von Karman geometric nonlinearity. By employing the Galerkin discretization procedure, the partial differential equations are diverted to a set of coupled nonlinear ordinary differential equations of motion. Based on the case of 1 : 2 internal resonance and principal resonance-1/2 subharmonic parametric resonance, the multiscale method of perturbation analysis is employed to precisely acquire the four-dimensional nonlinear averaged equations. From the resonant response analysis and nonlinear dynamic simulation, we discovered that the unstable regions of stiffened cylindrical shells can be narrowed by decreasing the external excitation or increasing the magnetic intensity, and their working frequency range can be expanded by reducing the in-plane excitation. Moreover, the different nonlinear dynamic responses of the stiffened cylindrical shell are acquired by controlling stiffener number, stiffener size, and aspect ratio. The presented approach in this paper can provide an eﬃcient analytical framework for nonlinear dynamics theories of stiffened cylindrical shells and will shed light on complex structure design in vibration test engineering.

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28

Chen, Zi Li, Xiao Liang You, Chang Ping Chen, and Ao Ling Ma. "Dynamic Characteristics Research of Structure with Nonlinear TMD System." Advanced Materials Research 639-640 (January 2013): 812–17. http://dx.doi.org/10.4028/www.scientific.net/amr.639-640.812.

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The dynamic control equations are derived for the mechanical model with two degrees of freedom that the soften spring is considered in the dynamic vibration absorber. In order to facilitate the computation to the equations, through the integral of the dynamic control equations which became a four order ordinary differential equation. The amplitude frequency response curves of the primary structure excited by a harmonic force were drawn with harmonic balance method, the influence of the dynamic characteristics of primary structure about this kind of nonlinear dynamic vibration absorber was discussed.

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29

Hwang, Yunn Lin, Shen Jenn Hwang, Zi Gui Huang, Ming Tzong Lin, Yen Chien Mao, and Pei Yu Wang. "Computational Analysis of Multibody Dynamic Systems Using Nonlinear Recursive Formulation." Key Engineering Materials 419-420 (October 2009): 289–92. http://dx.doi.org/10.4028/www.scientific.net/kem.419-420.289.

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. In this paper the computer implementation of the nonlinear recursive formulation in multibody dynamics systems is described. The organization of the computer algorithm which is used to automatically construct and numerically solve the system of loosely coupled dynamic equations expressed in terms of the absolute and joint coordinates is discussed. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. By using the inertia matrix structure of deformable mechanical systems and the fact that the joint reaction forces associated with the elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as the joint reaction forces. The multibody flexible four-bar system is used as an example to demonstrate the use of the procedure discussed in this paper.

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30

Mitchell, L. D., and J. W. David. "Proposed Solution Methodology for the Dynamically Coupled Nonlinear Geared Rotor Mechanics Equations." Journal of Vibration and Acoustics 107, no. 1 (January 1, 1985): 112–16. http://dx.doi.org/10.1115/1.3274700.

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The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.

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31

Li, Chong, Jichun Xing, Jiwen Fang, and Zhong Zhao. "Numerical Analysis on Chaotic Vibration of Drive System for a Movable Tooth Piezoelectric Motor." Shock and Vibration 2017 (2017): 1–17. http://dx.doi.org/10.1155/2017/3216010.

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The nonlinear dynamic equations of the drive system for movable tooth piezoelectric motor are established. Using these equations, the chaotic vibrations of the system are investigated. The results show that chaotic vibrations occur in the movable tooth drive system under some parameters. The average mesh stiffness, theoretical radius, and wave generator offset significantly influence the nonlinear chaotic vibrations of the drive system of the movable tooth piezoelectric motor. The ranges for the system parameters that lead to a motor with bad dynamics are shown. The results can be used to predict the dynamic load and optimize power density of the proposed piezoelectric motor.

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32

Cui, Li, and Jian Rong Zheng. "Study on Bifurcation of Rigid Rotor Roller Bearings System Considering Nonlinear Dynamic Behavior." Applied Mechanics and Materials 34-35 (October 2010): 467–71. http://dx.doi.org/10.4028/www.scientific.net/amm.34-35.467.

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Rigid rotor roller bearing system displays complicated nonlinear dynamic behavior due to nonlinear Hertzian force of bearing. Nonlinear bearing forces of roller bearing and dynamic equations of rotor bearing system are established. The bifurcation and stability of the periodic motion of the system in radial clearance-rotating speed and ellipticity-rotating speed parametric domains are studied by use of continuation-shooting algorithm for periodic solutions of nonlinear non-autonomous dynamics system. Results show that the parameters of rotor bearing system should be designed carefully in order to obtain period-1 motion.

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33

Wu, Li Juan, Jin Yuan Tang, and Si Yu Chen. "Three Kinds of Gear Transmission Nonlinear Dynamics Models." Advanced Materials Research 787 (September 2013): 765–70. http://dx.doi.org/10.4028/www.scientific.net/amr.787.765.

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Based on the Lagranges equations, a new nonlinear dynamic gear model is established by introducing two variables of relative rotation angleand mean rotation angle. The motion equations derived with Lagranges equation exhibit nonlinear terms which are absent in the equations derived on Newtons equations. Combining with the numerical simulation, the dynamic responses in time domain and frequency domain are deduced, and it can be concluded that the responses at low speed of three different models are different. However, they are similar at the designed speed without the consideration of dissipation energy. On the contrary, the dynamic responses are similar at low speed and the simplified Newtons equation differs at the designed speed including dissipation energy.

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34

DasGupta, A., and H. Hatwal. "Dynamics and Nonlinear Coordination Control of Multifingered Mechanical Hands." Journal of Dynamic Systems, Measurement, and Control 120, no. 2 (June 1, 1998): 275–81. http://dx.doi.org/10.1115/1.2802419.

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This paper presents a study of the dynamics and nonlinear coordination control of multifingered mechanical hands. Considering the dynamics of the object and the fingers, the equations of motion are derived in the finger joint space resulting in a set of differential equations with some algebraic constraints. Using dynamic extension, the equations are converted to the state space form, which results in a nonlinear system affine in control. For the purpose of control, certain output equations are defined. Using the tools of differential geometric control theory, some important properties of the system are shown. Using these properties, a nonlinear input-output linearizing controller is synthesized which yields a decoupled linear system. The poles of the resulting linearized system are then placed appropriately to render desirable features to the system. The theory is validated with an example of a three-fingered spatial hand manipulating a cuboidal object.

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35

Hao, Yu Xin, Wei Zhang, L. Yang, and J. H. Wang. "Dynamic Response of Cantilever FGM Cylindrical Shell." Applied Mechanics and Materials 130-134 (October 2011): 3986–93. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.3986.

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An analysis on the nonlinear dynamics of a cantilever functionally graded materials (FGM) cylindrical shell subjected to the transversal excitation is presented in thermal environment.Material properties are assumed to be temperature-dependent. Based on the Reddy’s first-order shell theory,the nonlinear governing equations of motion for the FGM cylindrical shell are derived using the Hamilton’s principle. The Galerkin’s method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. It is our desirable to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the cantilever FGM cylindrical shell. Numerical method is used to find that in the case of non-internal resonance the transverse amplitude are decreased by increasing the volume fraction index N.

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36

Jin, Liang, Peter N. Nikiforuk, and Madan M. Gupta. "Dynamic Recurrent Neural Networks for Control of Unknown Nonlinear Systems." Journal of Dynamic Systems, Measurement, and Control 116, no. 4 (December 1, 1994): 567–76. http://dx.doi.org/10.1115/1.2899254.

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A scheme of dynamic recurrent neural networks (DRNNs) is discussed in this paper, which provides the potential for the learning and control of a general class of unknown discrete-time nonlinear systems which are treated as “black boxes” with multi-inputs and multi-outputs (MIMO). A model of the DRNNs is described by a set of nonlinear difference equations, and a suitable analysis for the input-output dynamics of the model is performed to obtain the inverse dynamics. The ability of a DRNN structure to model arbitrary dynamic nonlinear systems is incorporated to approximate the unknown nonlinear input-output relationship using a dynamic back propagation (DBP) learning algorithm. An equivalent control concept is introduced to develop a model based learning control architecture with simultaneous on-line identification and control for unknown nonlinear plants. The potentials of the proposed methods are demonstrated by simulation results.

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37

Wang, Yu. "Prediction of Periodic Response of Rotor Dynamic Systems With Nonlinear Supports." Journal of Vibration and Acoustics 119, no. 3 (July 1, 1997): 346–53. http://dx.doi.org/10.1115/1.2889730.

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A numerical-analytical method for estimating steady-state periodic behavior of nonlinear rotordynamic systems is presented. Based on a finite element formulation in the time domain, this method transforms the nonlinear differential equations governing the motion of large rotor dynamic systems with nonlinear supports into a set of nonlinear algebraic equations with unknown temporal nodal displacements. A procedure is proposed to reduce the resulting problem to solving nonlinear algebraic equations in terms of the coordinates associated with the nonlinear supports only. The result is a simple and efficient approach for predicting all possible fundamental and sub-harmonic responses. Stability of the periodic response is readily determined by a direct use of Floquet’s theory. The feasibility and advantages of the proposed method are illustrated with two examples of rotor-bearing systems of deadband supports and squeeze film dampers, respectively.

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38

Sarkisov, S. V., S. Z. El-Salim, A. V. Bondarev, A. N. Korpusov, and P. A. Putilin. "Signal processing of nonlinear dynamic systems." Journal of Physics: Conference Series 2094, no. 2 (November 1, 2021): 022057. http://dx.doi.org/10.1088/1742-6596/2094/2/022057.

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Abstract The paper considers Hermite polynomials that act as a self-similar basis for the decomposition of functions in phase space. It is shown that the equations of behavior of nonlinear dynamical systems are simplified. It is also noted that the wavelet decomposition over Hermite polynomials reduces the number of approximation coefficients and improves the quality of approximation.

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39

MAZHUKIN, A. V. "Dynamic Adaptation In Convection — Diffusion Equations." Computational Methods in Applied Mathematics 8, no. 2 (2008): 171–86. http://dx.doi.org/10.2478/cmam-2008-0012.

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Abstract With the example of solving some known modeling problems the fea-tures of constructing grids adapted to the solution of parabolic equations are consid-ered. Convection-diffusion problems are described by nonlinear Burgers and Buckley — Leverette equations. A detailed analysis of the differential approximations and numerical results shows that the idea of using an arbitrary time-dependent system of coordinates for adapted grid generation in combination with the principle of quasi-stationarity makes the dy-namic adaptation method universal, effective, and algorithmically simple. Universality is achieved due to the use of an arbitrary time-dependent system of co-ordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for time-dependent problems of mathematical physics with various mathematical features. Among these features are large gradients, and propagation of weak and strong discontinuities in known nonlinear transfer problems. The efficiency is determined by automatically fitting the velocity of the moving nodes to the dynamics of the solution. The close relationship between the adaptation mechanism and the structure of parabolic equations allows one to automatically control nodes motion of nodes so that their trajectories do not intersect. The simplicity of the algorithm is achieved due to the general approach to the adaptive grid generation that is independent of the form and type of the differential equations.

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Niu, Yan, Yuxin Hao, Minghui Yao, Wei Zhang, and Shaowu Yang. "Nonlinear Dynamics of Imperfect FGM Conical Panel." Shock and Vibration 2018 (2018): 1–20. http://dx.doi.org/10.1155/2018/4187386.

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Structures composed of functionally graded materials (FGM) can satisfy many rigorous requisitions in engineering application. In this paper, the nonlinear dynamics of a simply supported FGM conical panel with different forms of initial imperfections are investigated. The conical panel is subjected to the simple harmonic excitation along the radial direction and the parametric excitation in the meridian direction. The small initial geometric imperfection of the conical panel is expressed by the form of the Cosine functions. According to a power-law distribution, the effective material properties are assumed to be graded along the thickness direction. Based on the first-order shear deformation theory and von Karman type nonlinear geometric relationship, the nonlinear equations of motion are established by using the Hamilton principle. The nonlinear partial differential governing equations are truncated by Galerkin method to obtain the ordinary differential equations along the radial displacement. The effects of imperfection types, half-wave numbers of the imperfection, amplitudes of the imperfection, and damping on the dynamic behaviors are studied by numerical simulation. Maximum Lyapunov exponents, bifurcation diagrams, time histories, phase portraits, and Poincare maps are obtained to show the dynamic responses of the system.

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Didonna, Marco, Merten Stender, Antonio Papangelo, Filipe Fontanela, Michele Ciavarella, and Norbert Hoffmann. "Reconstruction of Governing Equations from Vibration Measurements for Geometrically Nonlinear Systems." Lubricants 7, no. 8 (August 5, 2019): 64. http://dx.doi.org/10.3390/lubricants7080064.

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Data-driven system identification procedures have recently enabled the reconstruction of governing differential equations from vibration signal recordings. In this contribution, the sparse identification of nonlinear dynamics is applied to structural dynamics of a geometrically nonlinear system. First, the methodology is validated against the forced Duffing oscillator to evaluate its robustness against noise and limited data. Then, differential equations governing the dynamics of two weakly coupled cantilever beams with base excitation are reconstructed from experimental data. Results indicate the appealing abilities of data-driven system identification: underlying equations are successfully reconstructed and (non-)linear dynamic terms are identified for two experimental setups which are comprised of a quasi-linear system and a system with impacts to replicate a piecewise hardening behavior, as commonly observed in contacts.

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Rega, Giuseppe, and Narakorn Srinil. "Nonlinear Hybrid-Mode Resonant Forced Oscillations of Sagged Inclined Cables at Avoidances." Journal of Computational and Nonlinear Dynamics 2, no. 4 (March 28, 2007): 324–36. http://dx.doi.org/10.1115/1.2756064.

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We investigate nonlinear forced oscillations of sagged inclined cables under planar 1:1 internal resonance at avoidance. To account for frequency avoidance phenomena and associated hybrid modes, actually distinguishing inclined cables from horizontal cables, asymmetric inclined static configurations are considered. Emphasis is placed on highlighting nearly tuned 1:1 resonant interactions involving coupled hybrid modes. The inclined cable is subjected to a uniformly distributed vertical harmonic excitation at primary resonance of a high-frequency mode. Approximate nonlinear partial-differential equations of motion, capturing overall displacement coupling and dynamic extensibility effect, are analytically solved based on a multimode discretization and a second-order multiple scale approach. Bifurcation analyses of both equilibrium and dynamic solutions are carried out via a continuation technique, highlighting the influence of system parameters on internally resonant forced dynamics of avoidance cables. Direct numerical integrations of modulation equations are also performed to validate the continuation prediction and characterize nonlinear coupled dynamics in post-bifurcation states. Depending on the elasto-geometric (cable sag and inclination) and control parameters, and on assigned initial conditions, the hybrid modal interactions undergo several kinds of bifurcations and nonlinear phenomena, along with meaningful transition from periodic to quasiperiodic and chaotic responses. Moreover, corresponding spatio-temporal distributions of cable nonlinear dynamic displacement and tension are manifested.

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Wang, Yu-Chi, Donglong Sheu, and Chin-E. Lin. "A Unified Approach to Nonlinear Dynamic Inversion Control with Parameter Determination by Eigenvalue Assignment." Mathematical Problems in Engineering 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/548050.

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This paper presents a unified approach to nonlinear dynamic inversion control algorithm with the parameters for desired dynamics determined by using an eigenvalue assignment method, which may be applied in a very straightforward and convenient way. By using this method, it is not necessary to transform the nonlinear equations into linear equations by feedback linearization before beginning control designs. The applications of this method are not limited to affine nonlinear control systems or limited to minimum phase problems if the eigenvalues of error dynamics are carefully assigned so that the desired dynamics is stable. The control design by using this method is shown to be robust to modeling uncertainties. To validate the theory, the design of a UAV control system is presented as an example. Numerical simulations show the performance of the design to be quite remarkable.

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Avramov, K. V., I. V. Biblik, I. V. Hrebennik, and I. A. Urniaieva. "Reducing the dimension of a nonlinear dynamic system to simulate a multi-walled nanotube." Technical mechanics 2023, no. 2 (June 15, 2023): 105–20. http://dx.doi.org/10.15407/itm2023.02.105.

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A system of nonlinear partial differential equations is derived to describe the vibrations of a multi-walled nanotube. The system reduces to a nonlinear dynamic system with а large number of degrees of freedom (DOFs). To reduce its dimension, the nonlinear modal analysis method is used to give 2-DOF dynamic system, which is studied by the asymptotic multiple scale method. This gives a system of modulation equations, whose fixed points describe the free vibrations of the nanotube. The fixed points are described by nonlinear algebraic equations, whose solutions are given on a backbone curve. Use is made of the Sanders–Koiter shell model to describe the nonlinear deformation of the nanotube and Hook’s nonlocal anisotropic law to simulate its vibrations. Notice that the elastic constants of the nanotube walls differ. The nanotube model is a system of nonlinear ordinary differential equations, which is obtained by applying the weighed residuals method to the nonlinear partial equations. Three types of nonlinearities are accounted for in the nanotube model. First, the Van der Waals forces are nonlinear functions of the radial displacements. Second, the displacements of the nanotube walls are assumed to be moderate, which is described by a geometrically nonlinear model. Third, since the resultant forces are nonlinear functions of the displacements, the use of natural boundary conditions in the weighted residuals method results in additional nonlinear terms. A finite-DOF nonlinear dynamical system is derived. The free nonlinear vibrations of the nanotube are analyzed. The calculated results are shown on a backbone curve.

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Putra, Supriadi, M. Imran, Ayunda Putri, and Rike Marjulisa. "Dynamic Comparison of Variations of Newton’s Methods with Different Types of Means for Solving Nonlinear Equations." International Journal of Mathematics And Computer Research 10, no. 11 (November 30, 2022): 2969–74. http://dx.doi.org/10.47191/ijmcr/v10i11.04.

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This article discusses dynamic comparison of variations of Newton’s methods with different types of means for solving nonlinear equations. There are two factors that are considered to affect the shape of the basin of attraction of a method namely the size of the determined convergence area and number of partitions. The computation results of some functions show that harmonic mean Newton’s method (HMN) has small divergence area. On the other hand, contra harmonic mean Newton’s method (CMN) has the largest divergence area and is considered to be the least effective method

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DIAZ-GUILERA, ALBERT. "NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS AND SELF-ORGANIZED CRITICALITY." Fractals 01, no. 04 (December 1993): 963–67. http://dx.doi.org/10.1142/s0218348x93001039.

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Several nonlinear stochastic differential equations have been proposed in connection with self-organized critical phenomena. Due to the threshold condition involved in its dynamic evolution, an infinite number of nonlinearities arise in a hydrodynamic description. We study two models with different noise correlations which make all nonlinear contributions to be equally relevant below the upper critical dimension. The asymptotic values of the critical exponents are estimated from a systematic expansion in the number of coupling constants by means of the dynamic renormalization group.

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Su, Hua, Lishan Liu, and Xinjun Wang. "Higher-Order Dynamic Delay Differential Equations on Time Scales." Journal of Applied Mathematics 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/939162.

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We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-orderp-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem withp-Laplacian operator are obtained.

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Apartsyn, A. S., S. V. Solodusha, and V. A. Spiryaev. "Modeling of Nonlinear Dynamic Systems with Volterra Polynomials." International Journal of Energy Optimization and Engineering 2, no. 4 (October 2013): 16–43. http://dx.doi.org/10.4018/ijeoe.2013100102.

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The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multi-parameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.

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Wei, Jin, Tao Yu, Dongping Jin, Mei Liu, Dengqing Cao, and Jinjie Wang. "Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass." Materials 14, no. 23 (November 28, 2021): 7279. http://dx.doi.org/10.3390/ma14237279.

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A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.

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Fu, Yi Ming, and Xian Qiao Wang. "Nonlinear Dynamic Response of Piezoelectric Plates Considering Damage Effects." Key Engineering Materials 324-325 (November 2006): 299–302. http://dx.doi.org/10.4028/www.scientific.net/kem.324-325.299.

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Based on the Talreja’s tensor valued internal state variables damage model and the Helmhotlz free energy of piezoelectric material, the constitutive relations of the piezoelectric plates with damage are derived. Then, the nonlinear dynamic equations of the piezoelectric plates considering damage are established. By using the finite difference method and the Newmark scheme, these equations are solved and the effects of damage and electric loads on the nonlinear dynamic response of piezoelectric plates are discussed.

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