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IN THIS ISSUE

### Research Papers

J. Comput. Nonlinear Dynam. 2016;12(3):031001-031001-7. doi:10.1115/1.4034834.

Deoxyribonucleic acid (DNA) is a long flexible polyelectrolyte that is housed in the aqueous environment within the cell of an organism. When a length of torsionally relaxed (untwisted) DNA is held in tension, such as is the case in many single molecule experiments, the thermal fluctuations arising from the constant bombardment of the DNA by the surrounding fluid molecules induce bending in it, while the applied tension tends to keep it extended. The combined effect of these influences is that DNA is never at its full extension but eventually attains an equilibrium value of end-to-end extension under these influences. An analytical model was developed to estimate the tension-dependent value of this extension. This model, however, does not provide any insight into the dynamics of the extensional response of DNA to applied tension nor the kinetics of DNA at equilibrium under said tension. This paper reports the results of Brownian dynamics simulations using a discrete wormlike-chain model of DNA that provide some insight into these dynamics and kinetics.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031002-031002-9. doi:10.1115/1.4034565.

As a especial type of synchronous method, compound synchronization is designed by multiple drive systems and response systems. In this paper, a new type of compound synchronization of three drive systems and two response systems is investigated. According to synchronous control of five memristive cellular neural networks (CNNs), the theoretical analysis and demonstration are given out by using Lyapunov stability theory. The corresponding numerical simulations and synchronous performance analysis are supplied to verify the feasibility and scalability of compound synchronization design.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031003-031003-15. doi:10.1115/1.4034733.

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031004-031004-4. doi:10.1115/1.4035193.

A numerical method for finding spherically symmetric pseudobreathers of a nonlinear wave equation is presented. The algorithm, based on pseudospectral methods, is applied to find quasi-periodic solutions with force terms being continuous approximations of the signum function. The obtained pseudobreathers slowly radiate energy and decay after some (usually long) time depending on the period that characterizes (unambiguously) the initial configuration.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031005-031005-13. doi:10.1115/1.4034044.

A structural decomposition method based on symbol operation for solving differential algebraic equations (DAEs) is developed. Constrained dynamical systems are represented in terms of DAEs. State-space methods are universal for solving DAEs in general forms, but for complex systems with multiple degrees-of-freedom, these methods will become difficult and time consuming because they involve detecting Jacobian singularities and reselecting the state variables. Therefore, we adopted a strategy of dividing and conquering. A large-scale system with multiple degrees-of-freedom can be divided into several subsystems based on the topology. Next, the problem of selecting all of the state variables from the whole system can be transformed into selecting one or several from each subsystem successively. At the same time, Jacobian singularities can also be easily detected in each subsystem. To decompose the original dynamical system completely, as the algebraic constraint equations are underdetermined, we proposed a principle of minimum variable reference degree to achieve the bipartite matching. Subsequently, the subsystems are determined by aggregating the strongly connected components in the algebraic constraint equations. After that determination, the free variables remain; therefore, a merging algorithm is proposed to allocate these variables into each subsystem optimally. Several examples are given to show that the proposed method is not only easy to implement but also efficient.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031006-031006-8. doi:10.1115/1.4035194.

In this paper, we investigate the traveling wave solutions of a two-component Dullin–Gottwald–Holm (DGH) system. By qualitative analysis methods of planar systems, we investigate completely the topological behavior of the solutions of the traveling wave system, which is derived from the two-component Dullin–Gottwald–Holm system, and show the corresponding phase portraits. We prove the topological types of degenerate equilibria by the technique of desingularization. According to the dynamical behaviors of the solutions, we give all the bounded exact traveling wave solutions of the system, including solitary wave solutions, periodic wave solutions, cusp solitary wave solutions, periodic cusp wave solutions, compactonlike wave solutions, and kinklike and antikinklike wave solutions. Furthermore, to verify the correctness of our results, we simulate these bounded wave solutions using the software maple version 18.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031007-031007-9. doi:10.1115/1.4034736.

The nonlinear dynamic response of nanocomposite microcantilevers is investigated. The microbeams are made of a polymeric hosting matrix (e.g., epoxy, polyether ether ketone (PEEK), and polycarbonate) reinforced by longitudinally aligned carbon nanotubes (CNTs). The 3D transversely isotropic elastic constitutive equations for the nanocomposite material are based on the equivalent inclusion theory of Eshelby and the Mori–Tanaka homogenization approach. The beam-generalized stress resultants, obtained in accordance with the Saint-Venant principle, are expressed in terms of the generalized strains making use of the equivalent constitutive laws. These equations depend on both the hosting matrix and CNTs elastic properties as well as on the CNTs volume fraction, geometry, and orientation. The description of the geometry of deformation and the balance equations for the microbeams are based on the geometrically exact Euler–Bernoulli beam theory specialized to incorporate the additional inextensibility constraint due to the relevant boundary conditions of microcantilevers. The obtained equations of motion are discretized via the Galerkin method retaining an arbitrary number of eigenfunctions. A path following algorithm is then employed to obtain the nonlinear frequency response for different excitation levels and for increasing volume fractions of carbon nanotubes. The fold lines delimiting the multistability regions of the frequency responses are also discussed. The volume fraction is shown to play a key role in shifting the linear frequencies of the beam flexural modes to higher values. The CNT volume fraction further shifts the fold lines to higher excitation amplitudes, while it does not affect the backbones of the modes (i.e., oscillation frequency–amplitude curves).

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031008-031008-13. doi:10.1115/1.4034492.

In this paper, the consistent rotation-based formulation (CRBF) is used to develop a new fully parametrized plate finite element (FE) based on the kinematic description of the absolute nodal coordinate formulation (ANCF). The ANCF/CRBF plate element has a general geometric description which is consistent with the basic principles of continuum mechanics, defines a unique rotation field, ensures the continuity of the rotation and strains at the element nodes, can describe an arbitrarily large displacement, and is consistent with computational geometry methods allowing for correctly describing complex shapes as demonstrated in this paper. The proposed ANCF/CRBF finite element does not suffer from the serious and fundamental problems encountered when using other large rotation vector formulations (LRVF) including the coordinate redundancy and violation of the principle of non-commutativity of the finite rotations which cannot be treated as vectors. The proposed bi-cubic ANCF/CRBF plate element employs, as nodal coordinates, three position coordinates and three finite rotation parameters. This element is obtained from a fully parameterized ANCF plate element by writing the position vector gradients of the ANCF plate element in terms of three finite rotation parameters using a nonlinear velocity transformation that systematically reduces the number of the element coordinates. The resulting element captures stretch, bending, and twist deformation modes and it allows for systematically describing complex curved geometry. Because of the lower dimensionality of the resulting ANCF/CRBF plate element, it does not capture complex deformation modes that can be captured using the more general ANCF finite elements. Furthermore, the ANCF/CRBF element mass matrix is not constant, leading to nonlinear Coriolis and centrifugal inertia forces. The new element is validated by examining its performance using several examples that include pendulum plate, free falling plate, and tire models. The results obtained using this new element are compared with the results obtained using the bi-cubic fully parameterized ANCF plate element, the bi-linear shell element, and the conventional solid element implemented in the commercial software ANSYS.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031009-031009-9. doi:10.1115/1.4034998.

In this paper, a new method of state space reconstruction is proposed for the nonstationary time-series. The nonstationary time-series is first converted into its analytical form via the Hilbert transform, which retains both the nonstationarity and the nonlinear dynamics of the original time-series. The instantaneous phase angle θ is then extracted from the time-series. The first- and second-order derivatives $θ˙$, $θ¨$ of phase angle θ are calculated. It is mathematically proved that the vector field $[θ,θ˙,θ¨]$ is the state space of the original time-series. The proposed method does not rely on the stationarity of the time-series, and it is available for both the stationary and nonstationary time-series. The simulation tests have been conducted on the stationary and nonstationary chaotic time-series, and a powerful tool, i.e., the scale-dependent Lyapunov exponent (SDLE), is introduced for the identification of nonstationarity and chaotic motion embedded in the time-series. The effectiveness of the proposed method is validated.

Topics: Signals , Time series
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031010-031010-7. doi:10.1115/1.4035195.

In this work, we investigate both the mathematical and numerical studies of the fractional reaction–diffusion system consisting of spatial interactions of three components’ species. Our main result is based on the analysis of the model for linear stability. Mathematical analysis of the main equation shows that the dynamical system is both locally and globally asymptotically stable. We further propose a theorem which guarantees the existence and permanence of the three species. We formulate a viable numerical methods in space and time. By adopting the Fourier spectral approach to discretize in space, the issue of stiffness associated with the fractional-order spatial derivatives in such system is removed. The resulting system of ordinary differential equations (ODEs) is advanced with the exponential time-differencing method of ADAMS-type. The complexity of the dynamics in the system which we discussed theoretically are numerically presented through some numerical simulations in 1D, 2D, and 3D to address the points and queries that may naturally arise.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031011-031011-5. doi:10.1115/1.4034868.

Model-based control methods such as inverse dynamics control and computed torque control encounter difficulties if actuator saturation occurs. However, saturation is a common phenomenon in robotics leading to significant nonlinearity in system behavior. In this study, the saturation of the actuator torques is considered as a temporary reduction of the number of independent control inputs. The reduction of the number of actuators leads to an underactuated control problem which typically involves the handling of differential algebraic equation systems. The saturated system may become especially complex when intricate combinations of the actuator saturations appear. A servoconstraint-based inverse dynamics control method for underactuated multibody systems is applied for the treatment of actuator torque saturation. In case of human-friendly robots, the problem of saturation cannot be avoided on the level of trajectory planning because unexpected human perturbations may take place, which result in such abrupt changes in the desired trajectory that lead to saturation at some actuators. A case study for the service robot Acroboter shows the applicability of the proposed approach.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031012-031012-11. doi:10.1115/1.4034869.

In many cases of rotating systems, such as jet engines, two or more coaxial shafts are used for power transmission between a high/low-pressure turbine and a compressor. The major purpose of this study is to predict the nonlinear dynamic behavior of a coaxial rotor system supported by two active magnetic bearings (AMBs) and contact with two auxiliary bearings. The model of the system is formulated by ten degrees-of-freedom in two different planes. This model includes gyroscopic moments of disks and geometric coupling of the magnetic actuators. The nonlinear equations of motion are developed by the Lagrange's equations and solved using the Runge–Kutta method. The effects of speed parameter, speed ratio of shafts, and gravity parameter on the dynamic behavior of the coaxial rotor–AMB system are investigated by the dynamic trajectories, power spectra analysis, Poincaré maps, bifurcation diagrams, and the maximum Lyapunov exponent. Also, the contact forces between the inner shaft and auxiliary bearings are studied. The results indicate that the speed parameter, speed ratio of shafts, and gravity parameter have significant effects on the dynamic responses and can be used as effective control parameters for the coaxial rotor–AMB system. Also, the results of analysis reveal a variety of nonlinear dynamical behaviors such as periodic, quasi-periodic, period-4, and chaotic vibrations, as well as jump phenomena. The obtained results of this research can give some insight to engineers and researchers in designing and studying the coaxial rotor–AMB systems or some turbomachinery in the future.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031013-031013-9. doi:10.1115/1.4034679.

The present paper studies the nonlinear free flexural vibration of stiffened plates. The analysis is performed using a superparametric element. This element consists of an ACM plate-bending element along with in-plane displacements to represent the displacement field, and cubic serendipity shape function is used to define the geometry. The element can accommodate any arbitrary geometry, and the stiffeners either straight or curvilinear are modeled such that these can be placed anywhere on the plate. A number of numerical examples are presented to show its efficacy.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031014-031014-6. doi:10.1115/1.4035196.

This paper addresses the design of a robust fractional-order dynamic output feedback sliding mode controller (FDOF-SMC) for a general class of uncertain fractional systems subject to saturation element. The control law is composed of two components, one linear and one nonlinear. The linear component corresponds to the fractional-order dynamics of the FDOF-SMC, while the nonlinear component is associated with the switching control algorithm. The closed-loop system exhibits asymptotical stability and the system states approach the sliding surface in a finite time. In order to design the controller, a linear matrix inequality (LMI)-based procedure is also derived. Simulation results demonstrate the feasibility of the FDOF-SMC strategy.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031015-031015-9. doi:10.1115/1.4034735.

The exit problem and global stability of a nonlinear oscillator excited by an ergodic real noise and harmonic excitations are examined. The real noise is assumed to be a scalar stochastic function of an n-dimensional Ornstein–Uhlenbeck vector process which is the output of a linear filter system. Due to the existence of t-dependent excitation, two two-dimensional Fokker–Planck–Kolmogorov (FPK) equations governing the van der Pol variables process and the amplitude-phase process, respectively, are obtained and discussed through a perturbation method and the spectrum representations of the FPK operator and its adjoint operator of the linear filter system, while the detailed balance condition and the strong mixing condition are removed. Based on these FPK equations, the global properties of one-dimensional nonlinear oscillators with external or (and) internal periodic excitations under external or (and) internal real noises can be examined. Finally, a Duffing oscillator excited by a parametric real noise and parametric harmonic excitations is presented as an example, and the mean first-passage time (MFPT) about the oscillator's exit behavior between limit cycles is obtained under both wide-band noise and narrow-band noise excitations. The analytical result is verified by digital simulation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):031016-031016-10. doi:10.1115/1.4035197.

The adjoint method is a very efficient way to compute the gradient of a cost functional associated to a dynamical system depending on a set of input signals. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. An alternative and maybe more natural approach is the discrete adjoint method (DAM), which constructs a finite difference scheme for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the exact gradient of the discretized cost functional subjected to the discretized equations of motion. For the application of the discrete adjoint method to the forward solver, several matrices are necessary. In this contribution, the matrices are derived for the simple Euler explicit method and for the classical implicit Hilber–Hughes–Taylor (HHT) solver.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(3):031017-031017-11. doi:10.1115/1.4035060.

A nonlocal Bernoulli–Euler p-version finite-element (p-FE) is developed to investigate nonlinear modes of vibration and to analyze internal resonances of beams with dimensions of a few nanometers. The time domain equations of motion are transformed to the frequency domain via the harmonic balance method (HBM), and then, the equations of motion are solved by an arc-length continuation method. After comparisons with published data on beams with rectangular cross section and on carbon nanotubes (CNTs), the study focuses on the nonlinear modes of vibration of CNTs. It is verified that the p-FE proposed, which keeps the advantageous flexibility of the FEM, leads to accurate discretizations with a small number of degrees-of-freedom. The first three nonlinear modes of vibration are studied and it is found that higher order modes are more influenced by nonlocal effects than the first mode. Several harmonics are considered in the harmonic balance procedure, allowing us to discover modal interactions due to internal resonances. It is shown that the nonlocal effects alter the characteristics of the internal resonances. Furthermore, it is demonstrated that, due to the internal resonances, the nonlocal effects are still noticeable at lengths that are longer than what has been previously found.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(3):031018-031018-7. doi:10.1115/1.4035267.

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(3):031019-031019-6. doi:10.1115/1.4035191.

This work investigates the modification of the Nóse–Hoover thermostat, a well-known tool for controlling system temperature in nanoscale dynamical simulations. Nóse–Hoover response is characterized by a mean temperature converging to a target temperature. However, oscillations in the actual system temperature consistently appear over time. To reduce these oscillations, the Nóse–Hoover control law is modified to resemble a proportional–derivative controller. The modified thermostat is compared to the standard and shown to significantly reduce deviations. Gains are varied and compared to show effects on response and simulation time. Work–energy calculations show the modified dynamics drive the system to a low-energy state significantly faster than the standard. The behavior of the modified thermostat is illustrated using a simulation of a molten salt solution.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(3):031020-031020-11. doi:10.1115/1.4034732.

In this paper, we extend a methodology developed recently to study type-III intermittency considering different values of the noise intensity and the lower boundary of reinjection (LBR). We obtain accurate analytic expressions for the reinjection probability density (RPD). The proposed RPD has a piecewise definition depending on the nonlinear behavior, the LBR value, and the noise intensity. The new RPD is a sum of exponential functions with exponent α + 2, where α is the exponent of the noiseless RPD. The theoretical results are verified with the numerical simulations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407.

The Chaplygin sleigh is a canonical problem of mechanical systems with nonholonomic constraints. Such constraints often arise due to the role of a no-slip requirement imposed by friction. In the case of the Chaplygin sleigh, it is well known that its asymptotic motion is that of pure translation along a straight line. Any perturbations in angular velocity decay and result in an increase in asymptotic speed of the sleigh. Such motion of the sleigh is under the assumption that the magnitude of friction is as high as necessary to prevent slipping. We relax this assumption by setting a maximum value to the friction. The Chaplygin sleigh is then under a piecewise-smooth nonholonomic constraint and transitions between “slip” and “stick” modes. We investigate these transitions and the resulting nonsmooth dynamics of the system. We show that the reduced state space of the system can be partitioned into sets of distinct dynamics and that the stick–slip transitions can be explained in terms of transitions of the state of the system between these sets.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(3):031022-031022-6. doi:10.1115/1.4032258.

This paper deals with a newly born fractional derivative and integral on time scales. A chain rule is derived, and the given indefinite integral is being discussed. Also, an application to the traffic flow problem with a fractional Burger's equation is presented.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Comput. Nonlinear Dynam. 2016;12(3):034501-034501-3. doi:10.1115/1.4035058.

The limitations of the Taylor series approximations of the delayed variables have been documented by means of examples in several works. This note unifies these previous comments through a higher-level analysis of the issue. It is shown that the use of the truncated Taylor series expansion of the (inverse) advance operator does not feature the same drawbacks.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2016;12(3):034502-034502-12. doi:10.1115/1.4033896.

Dynamic analysis of a geared infinitely variable transmission (IVT) that can generate a continuous output-to-input speed ratio from zero to a certain value is studied for vehicle and wind turbine applications. With the IVT considered as a multirigid-body system, the Lagrangian approach is used to analyze its speeds and accelerations, and the Newtonian approach is used to conduct force analysis of each part of the IVT. Instantaneous input and output speeds and accelerations of the IVT have variations in one rotation of its input shaft. This work shows that the instantaneous input speed has less variation than the instantaneous output speed when the inertia on the input side is larger than that on the output side and vice versa. The maximum torque on the output shaft that is a critical part of the IVT increases with the input speed.

Commentary by Dr. Valentin Fuster