Accepted Manuscripts

Sunhua Huang and Bin Wang
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038443
This study is interested in the stability and stabilization of a class of fractional-order nonlinear systems with Caputo derivatives. Based on the properties of the Laplace transform, Mittag-Leffler function, Jordan Decomposition and Grönwall's inequality, sufficient conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with 1
TOPICS: Stability, Nonlinear systems, Laplace transforms
Mojtaba Hajipour, Amin Jajarmi and Dumitru Baleanu
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038444
In this paper we formulate a new non-standard finite difference scheme to study the dynamic treatments of a class of fractional chaotic systems. To design the new proposed scheme, an appropriate non-local framework is applied for the discretization of the nonlinear terms. This method is easy to implement and preserves some important physical properties of the considered model, e.g. fixed points and their stability. Additionally, this scheme is explicit and inexpensive to solve fractional differential equations. From a practical point of view, the stability analysis and chaotic behavior of three novel fractional systems are provided by the proposed approach. Numerical simulations and comparison results confirm that this scheme is also successful for the fractional chaotic systems with delay arguments.
TOPICS: Stability, Computer simulation, Design, Differential equations, Delays
Ranjan Kumar Mitra, A. K. Banik and Shyamal Chatterjee
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038445
Nonlinear dynamics, control and stability analysis of dry friction damped system under state feedback control with time delay are investigated. The dry friction damped system is harmonically excited and the nonlinearities in the equation of motion arise due to nonlinear damping and spring force. In this paper a frequency domain based method, viz. incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary responses which may be present in such system. The IHBC is then reformulated in a manner to treat the dry friction damped system under state feedback control with time delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses is obtained by Floquet's theory using Hsu' scheme and the stability of the controlled responses is obtained by applying a semi-discretization method for delay differential equation.
TOPICS: State feedback, Dry friction, Stability, Delays, Springs, Equations of motion, Damping, Delay differential equations, Nonlinear dynamics
Shiyu Wang, Penghui Zhang and Wenjia Sun
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038446
In-plane vibration of cyclically symmetric ring structures is examined with emphasis on the comparison of instabilities estimated by complete and simplified models. The aim of this paper is to understand under what conditions and to what degree the simplified models can approach the complete model. Previous studies develop time-variant models and employ perturbation method by assuming weak support. In this work, the rotating-load problem is cast into a non-rotating-load problem. A complete model with time-invariant coefficients is developed in rotating-support-fixed frame, where the bending and extensional deformations are incorporated. It is then reduced into two simplified ones based on different deformation restrictions. Due to the time-invariant effect observed in the rotating-support-fixed frame, the eigenvalues are formulated directly by using classical vibration theory and compared based on a sample structure. The comparisons verify that the two types of models are comparable only for weak support, though the simplified models cannot accurately predict all unstable regions in particular for strong support. Even in the comparable regions, the corresponding eigenvalues are still different. For verification purpose, the time-invariant models are transformed into time-variant ones in the inertial frame, based on which numerical calculations using Floquét theory are made to estimate instability. Consistence between the time-invariant and -variant models verifies the comparison between the complete and simplified models.
TOPICS: Vibration, Eigenvalues, Deformation, Stress
Chandan Bose, Vikas Reddy, Sayan Gupta and Sunetra Sarkar
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038447
This paper deals with the nonlinear fluid structure interaction (FSI) dynamics of a dipteran flight motor inspired flapping system in an inviscid fluid. In the present study, the FSI effects are incorporated to an existing forced Duffing oscillator model to gain a clear understanding of the nonlinear dynamical behaviour of the system in the presence of aerodynamic loads. The present FSI framework employs a potential flow solver to determine the aerodynamic loads and an explicit fourth order Runge-Kutta scheme to solve the structural governing equations. A bifurcation analysis has been carried out considering the amplitude of the wing actuation force as the control parameter to investigate different complex states of the system. Interesting dynamical behavior including period doubling, chaotic transients, periodic windows and finally an intermittent transition to stable chaotic attractor have been observed in the response with an increase in the bifurcation parameter. Similar dynamics is also reflected in the aerodynamic loads as well as in the trailing edge wake patterns.
TOPICS: Transients (Dynamics), Chaos, Flight, Fluid structure interaction, Stress, Bifurcation, Dynamics (Mechanics), Flow (Dynamics), Fluids, Engines, Motors, Wakes, Attractors, Wings
Touria Karite, Ali Boutoulout and Delfim F. M. Torres
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038450
We investigate exact enlarge controllability for time fractional diffusion systems of Riemann-Liouville type. The Hilbert uniqueness method is used to prove exact enlarge controllability for both cases of zone and pointwise actuators. A penalization method is given and the minimum energy control is characterized.
TOPICS: Diffusion (Physics), Actuators, Differential equations
Nicholas Candelino and Nader Jalili
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038331
There have been a variety of attempts to model the quasi-static and high energy impact dynamics of vertically-aligned carbon nanotube pads. However, very little work has focused on identifying the behavior at the mid-level frequencies that may occur in materials handling or vibration suppression applications. Moreover, the existing models are predominantly very complex, and yet provide only a very rough approximation of the bulk behavior. While several of the existing models make attempts at ascribing physical relevance, an adequate first principles approach has yet to be demonstrated. In this work, a close-fitting continuous model of these mid-frequency dynamics is developed utilizing a combination of phenomenological and identification based methodologies. First, a set of specially fabricated carbon nanotube pads are preconditioned and subjected to various position controlled compression experiments. The measured position and force responses are used to develop load-displacement curves, from which several characteristic features are identified. Based on these observations, a preliminary version of the proposed model is introduced. This simplified model is then systematically refined in order to demonstrate completely both the modeling approach and parameter identification scheme. The accuracy of the model is demonstrated through a comparison between the modeled and experimental responses including a normalized vector correlation of >0.998 across all sets of sinusoidal experimental data. A brief analysis utilizing a Lyapunov linearization approach follows, as well as a discussion of the advantages and limitations of the final model.
TOPICS: Modeling, Carbon nanotubes, Experimental analysis, Dynamics (Mechanics), Materials handling, Surface roughness, Stress, Compression, Displacement, Vibration suppression, Approximation, Fittings
Albert C.J. Luo and Siyu Guo
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038204
In this paper, the analytical solutions of periodic evolutions of the periodically diffused Brusselator are obtained through the general harmonic balanced method. Stable and unstable solutions of period-1 and period-2 evolutions in the Brusselator are discussed. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis, and the corresponding Hopf bifurcations are presented on the analytical bifurcation tree of periodic motion. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectrums show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.
TOPICS: Stability, Computer simulation, Bifurcation, Eigenvalues
Technical Brief  
Chandan Kumar and Somnath Sarangi
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037995
Planar dynamics of a rotor supported by long hydrodynamic journal bearing is investigated theoretically. An analytical model of the long journal bearing system is numerically integrated for analysis of fixed point and periodic oscillations. The nonlinearities in the system arise due to a nonlinear fluid film force acting on the journal. The influences of three dimensionless parameters viz. bearing parameter, unbalance, and rotor speed on the dynamic behavior of the rotor bearing system is studied and compared with the short journal bearing. For the same bearing parameter, short bearing has large operating speed compared to a long bearing. Results are presented in the form of a bifurcation diagram and Poincare´ map of the oscillations based on numerical computation. The considered unbalanced system shows periodic, multiperiodic and quasi-periodic motion in different speed range. Jumping phenomenon is also observed for a high value of bearing parameter with unbalance.
TOPICS: Fluid-dynamic forces, Bearings, Rotors, Dynamic response, Journal bearings, Oscillations, Dynamics (Mechanics), Fluid films, Bifurcation, Computation
Li Ma and Changpin Li
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037930
This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense, that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral which does not exist. Besides, our results also cover that some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.
Dumitru Baleanu, Mustafa Inc, Abdullahi Yusuf and Aliyu Isa Aliyu
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037765
In this work, Lie symmetry analysis for the time fractional third-order evolution (TOE) equation with Riemann-Liouville (RL) derivative is analyzed. We transform the time fractional TOE equation to nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. We obtain a kind of an explicit power series solution for the governing equation based on the power series theory. Using Ibragimov's nonlocal conservation method to time fractional partial differential equations, we compute conservation laws (Cls) for the TOE equation. 2-D, 3-D, and contour plots for the explicit power series solution are presented.
TOPICS: Differential equations, Partial differential equations
Jeremy Kolansky and Corina Sandu
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4031194
The Generalized Polynomial Chaos mathematical technique, when integrated with the Extended Kalman Filter method, provides a parameter estimation and state tracking method. The truncation of the series expansions degrades the link between parameter convergence and parameter uncertainty which the filter uses to perform the estimations. An empirically derived correction for this problem is implemented, that maintains the original parameter distributions. A comparison is performed to illustrate the improvements of the proposed approach. The method is demonstrated for parameter estimation on a regression system, where it is compared to the Recursive Least Squares method.
TOPICS: Kalman filters, Parameter estimation, Polynomials, Chaos, Filters, Uncertainty

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