Accepted Manuscripts

Technical Brief  
Sandeep Reddy and Ashitava Ghosal
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4040022
This paper deals with the issue of robustness in control of robots using the proportional plus derivative (PD) controller and the augmented PD controller. In literature, a variety of PD and model based controllers for multi-link serial manipulator have been claimed to be asymptotically stable for trajectory tracking, in the sense of Lyapunov, as long as the controller gains are positive. In this paper, we first establish that for a simple PD controllers, the criteria of positive controller gains is insufficient to establish asymptotic stability, and secondly that for the augmented PD controller the criteria of positive controller gains is valid only when there is no uncertainty in the model parameters. We show both these results for a simple planar two-degree-of-freedom robot with two rotary (R) joints, following a desired periodic trajectory, using the Floquet theory. We provide numerical simulation results which conclusively demonstrate the same.
TOPICS: Control equipment, Robots, Trajectories (Physics), Robustness, Uncertainty, Stability, Manipulators, Computer simulation
Shobhit Jain and Paolo Tiso
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4040021
We present an efficient method to significantly reduce the offline cost associated to the construction of training sets for hyper-reduction of geometrically nonlinear, finite element discretized, structural dynamics problems. The reduced order model is obtained by projecting the governing equation onto a basis formed by vibration modes and corresponding modal derivatives, thus avoiding cumbersome manual selection of high-frequency modes to represent nonlinear coupling effects. Cost-effective hyper-reduction is then achieved by lifting inexpensive linear modal transient analysis to a quadratic manifold constructed with dominant modes and related modal derivatives. The training forces are then computed from the so obtained representative displacement sets. In this manner, the need of full simulations required by traditional, proper orthogonal decomposition based projection and training is completely avoided. In addition to significantly reducing the offline cost, this technique selects a smaller hyper-reduced mesh as compared to proper orthogonal decomposition based training and therefore leads to larger online speedups as well. The proposed method constitutes a solid alternative to direct methods for the construction of the reduced order model, which suffer from either high intrusiveness into the finite element code or expensive offline nonlinear evaluations for the determination of the nonlinear coefficients.
TOPICS: Structural dynamics, Simulation, Manifolds, Principal component analysis, Construction, Finite element analysis, Vibration, Displacement, Transient analysis
Sen Zhang, Yi Cheng Zeng and Zhi Jun Li
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039980
y using a simple state feedback control technique and introducing two new nonlinear functions into a modified Sprott B system, a novel 4D no-equilibrium hyper-chaotic system with grid multi-wing hyper-chaotic hidden attractors is proposed in this paper. One remarkable feature of the new presented system is that it has no equilibrium points and therefore, Shil'nikov theorem is not suitable to demonstrate the existence of chaos for lacking of hetero-clinic or homo-clinic trajectory. But grid multi-wing hyper-chaotic hidden attractors can be obtained from this new system. The complex hidden dynamic behaviors of this system are analyzed by phase portraits, the time domain waveform, Lyapunov exponent spectra and the Kaplan-York dimension. In particular, the Lyapunov exponent spectra are investigated in detail. Interestingly, when changing the newly introduced nonlinear functions of the new hyper-chaotic system, the number of wings increases. And with the number of wings increasing, the region of the hyper-chaos is getting larger, which proves that this novel proposed hyper-chaotic system has very rich and complicated hidden dynamic properties. Furthermore, A corresponding improved module-based electronic circuit is designed and simulated via Multisim software. Finally, the obtained experimental results are presented, which are in agreement with the numerical simulations of the same system on the Matlab platform.
TOPICS: Equilibrium (Physics), Wings, Attractors, Spectra (Spectroscopy), Chaos, Circuits, Computer software, Matlab, State feedback, Computer simulation, Dimensions, Trajectories (Physics), Theorems (Mathematics)
Csaba Budai, Laszlo Kovacs and Jozsef Kovecses
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039962
Dissipation mechanisms and dissipative forces play a pivotal role in the operations and performance of human-machine interfaces and particularly in haptic systems. Dissipation is a very difficult phenomenon to model. Coulomb friction in general can be the most influential element in systems involving multiple direct contact connections such as joints with transmissions or mechanically guided components. Coulomb friction includes non-smooth discontinuity and can induce complex dynamic behaviours. The effect of Coulomb friction is often neglected in haptics. The part of the literature which deals with friction mainly focusing on friction compensation and/or simulation of friction for haptic rendering. In this paper, the nature of the dynamic behaviour caused by Coulomb friction in haptic sampled-data systems is illustrated by experiment, analysis and simulation. It is also demonstrated that a simple model can represent this behaviour, and show the effects of the haptic system parameters on this dynamics.
TOPICS: Coulombs, System dynamics, Friction, Haptics, Simulation, Energy dissipation, Experimental analysis, Rendering, Machinery, Dynamics (Mechanics)
Takashi Ikeda, Yuji Harata and Yukio Ishida
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039899
Nonlinear vibration characteristics of three-blade wind turbines are theoretically investigated. The wind turbine is modeled as a coupled system, consisting of a flexible tower with two degrees of freedom (2DOF), and three blades, each with a single degree of freedom (SDOF). The blades are subjected to steady winds. The wind velocity increases proportionally with height due to vertical wind shear. The natural frequency diagram is calculated with respect to the rotational speed of the wind turbine. The corresponding linear system with parametric excitation terms is analyzed to determine the rotational speeds where unstable vibrations appear and to predict at what rotational speeds the blades may vibrate at high amplitudes in a real wind turbine. The frequency response curves are then obtained by applying the swept-sine test to the equations of motion for the nonlinear system. They exhibit softening behavior due to the nonlinear restoring moments acting on the blades. Stationary time histories and their fast Fourier transform (FFT) results are also calculated. In the numerical simulations, localization phenomena are observed, where the three blades vibrate at different amplitudes. Basins of attraction (BOAs) are also calculated to examine the influence of a disturbance on the appearance of localization phenomena.
TOPICS: Vibration, Blades, Wind turbines, Degrees of freedom, Nonlinear systems, Nonlinear vibration, Excitation, Fast Fourier transforms, Frequency response, Linear systems, Wind shear, Computer simulation, Wind velocity, Equations of motion
Sohrab Effati, Seyed Ali Rakhshan and Samane Saqi
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039900
This article presents the approximation for solving the fractional optimal control problems with delays in state and control variables. The fractional derivative is considerd in the Grunwald-Letnikov sense. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the multi-delay fractional optimal control problem. For numerical computation, the multi-delay fractional dynamic system are approximated using the Grunwald-Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. We illustrate the effectiveness of the procedure with four examples.
TOPICS: Optimal control, Delays, Variational techniques, Algebra, Approximation, Computation, Dynamic systems
Laura Menini, Corrado Possieri and Antonio Tornambè
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039876
The main goal of this paper is to design a state feedback control that makes a point mass track a non-Zeno reference trajectory in a planar billiard. This objective is achieved by firstly determining a continuous-time dynamical model, whose trajectories approximate the solutions of the hybrid system. Hence, a state feedback that makes the hybrid system track a reference trajectory of the continuous-time one is proposed. Finally, these two techniques are combined in order to find a state feedback that achieves tracking of the trajectories of the unforced system. Examples are reported all throughout the paper to illustrate the theoretical results.
TOPICS: Trajectories (Physics), Design, Approximation, State feedback
Elena G. Tolkacheva, Brian Feeny and Xiaopeng Zhao
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039840
Dynamic modeling and analysis have broad relevance to many biological processes and biomedical applications, such as heart dynamics, DNA/RNA, cell mobility, surgical robotics, and so on. Various analytical and numerical techniques have been developed to qualitatively and quantitatively study dynamics associated with design, diagnosis, and control in these problems. The main focus of this special issue is to promote the exchange of new ideas, methods (primarily analytical and numerical) and their use in studies of nonlinear phenomena, computational modeling, dynamic analysis, and control for clinical diagnosis, patient health monitoring, drug administration, and bio-signal assisted rehabilitation. Recent years have seen a significant increase in research activities in these areas within diverse specialties including mechanical, electrical, biomedical, and computational engineering. However, there is a lack of integrated approaches to biomedical research from a dynamical systems point of view. This special issue calls for latest innovative and integrated approaches to develop a deeper understanding of the dynamics of biological and biomedical systems using techniques from nonlinear dynamics and computational fields.
TOPICS: Dynamics (Mechanics), Biomedicine, DNA, RNA, Mechanical admittance, Computer simulation, Design, Dynamic analysis, Dynamic systems, Robotics, Surgery, Drugs, Signals, Dynamic modeling, Nonlinear dynamics
Samuel Jung, Tae-Yun Kim and Wan-Suk Yoo
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039838
Dynamic relaxation (DR) is the most widely used approach for static equilibrium analyses. Specifically, DR compels dynamic systems to converge to a static equilibrium through the addition of fictitious damping. DR methods are classified by the method in which fictitious damping is applied. Conventional DR methods use a fictitious mass matrix to increase the fictitious damping while maintaining numerical stability. There are many calculation methods for the fictitious mass matrix; however, it is difficult to select the appropriate method. In addition, these methods require a stiffness matrix of a model, which makes it difficult to apply nonlinear models. To resolve these problems, a new DR method that uses continuous kinetic damping is proposed in this study. The proposed method does not require the fictitious mass matrix and any tuning coefficients, and it possesses a second-order convergence rate. The aforementioned advantages are unique and significant when compared to those of conventional methods. The stability and convergence rate were analyzed by using an eigenvalue analysis and demonstrated by simulating nonlinear models of a pendulum and cable. Simple but representative models were used to clearly demonstrate the features of the proposed DR method and to enable the reproducibility of the verification results.
TOPICS: Algorithms, Damping, Relaxation (Physics), Equilibrium (Physics), Stability, Cables, Dynamic systems, Eigenvalues, Numerical stability, Pendulums, Stiffness, Computational methods
Manashita Borah
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039841
This paper proposes new fractional-order models of seven non-equilibrium or stable equilibrium systems and investigates the existence of chaos and hyperchaos in them. It thereby challenges the conventional generation of chaos that involves starting the orbits from the vicinity of unstable manifold. This is followed by the discovery of coexisting hidden attractors in fractional dynamics. All the seven newly proposed fractional-order chaotic systems (FOCSs) ranging from minimum fractional dimension (n_f) of 2.76 to 4.95, exhibit multiple hidden attractors, such as periodic orbits, stable foci, and strange attractors, often coexisting together. To the best of the author's knowledge, this phenomenon of prevalence of fractional-order (FO) coexisting hidden attractors in multidimensional FOCSs is reported for the first time. These findings have significant practical relevance because the attractors are discovered in real-life physical systems such as the FO homopolar disc dynamo, FO memristive system, FO model of the modulation instability in a dissipative medium, etc., as analysed in this work. Numerical simulation results confirm the theoretical analyses and comply with the fact that multistability of hidden attractors does exist in the proposed FO models.
TOPICS: Attractors, Chaos, Equilibrium (Physics), Disks, Dynamics (Mechanics), Computer simulation, Dimensions, Manifolds, Theoretical analysis
Pitcha Khamsuwan, Teerawat Sangpet and Suwat Kuntanapreeda
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039681
This paper deals with the problem of master-slave synchronization of fractional-order chaotic systems with input saturation. Sufficient stability conditions for achieving the synchronization are derived from the basis of a fractional-order extension of the Lyapunov direct method, a new lemma of the Caputo fractional derivative, and a local sector condition. The stability conditions are formulated in Linear Matrix Inequality (LMI) forms and therefore are readily solved. The fractional-order chaotic Lorenz and hyperchaotic Lü systems with input saturation are utilized as illustrative examples. The feasibility of the proposed synchronization scheme is demonstrated through numerical simulations.
TOPICS: Stability, Computer simulation, Chaos synchronization, Linear matrix inequalities, Synchronization
Han Kyul Joo, Mustafa Mohamad and Themistoklis Sapsis
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4039309
We characterize the complex, heavy-tailed probability distribution functions (pdf) describing the response and its local extrema for structural systems subjected to random forcing that includes extreme events. Our approach is based on the recent probabilistic decomposition-synthesis technique, where we decouple rare events regimes from the background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems) our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom structural system undergoing rare events due to intermittent forcing. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean, through comparisons with expensive Monte-Carlo simulations.
TOPICS: Simulation, Fluctuations (Physics), Engineering simulation, Approximation, Displacement, Statistical distributions, Excitation
Technical Brief  
Edmon Perkins and Timothy Fitzgerald
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038895
For stochastic systems, the Fokker-Planck equation (FPE) is used to describe the system dynamics. The FPE is a partial differential equation, which is a function of all the variable in state space and of time. To solve the FPE, several methods are used, including finite elements, moment neglect methods, and cumulant neglect methods. This paper will study the cumulant neglect equations, which are derived from the FPE. It will be shown that the cumulant neglect method, while being a useful and popular tool for studying the system response, introduces several nonphysical artifacts. This paper extends the continuation method technique, typically employed on nonlinear deterministic systems, to a stochastic system. Employing the continuation method to stochastic systems in this way could further develop a bifurcation theory of stochastic systems.
TOPICS: System dynamics, Finite element analysis, Bifurcation, Fokker-Planck equation, Partial differential equations, Stochastic systems
Carla Pinto and Ana Carvalho
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4038643
We introduce a fractional order model for HIV dynamics where time-varying drug-exposure and drug-resistance are assumed. We derive conditions for the local and global asymptotic stability of the disease-free equilibrium. We find periodic stable endemic states for certain parameter values, for sinusoidal drug efficacies and when considering a density dependent decay rate for the $T$ cells. Other classes of periodic drug-efficacies are considered and the effect of the phases of these functions on the dynamics of the model are also studied. The order of the fractional derivative plays an important role in the severity of the epidemics.
TOPICS: Dynamics (Mechanics), Drugs, Density, Stability, Equilibrium (Physics), Diseases
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.

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