Research Papers

J. Comput. Nonlinear Dynam. 2019;14(8):081001-081001-12. doi:10.1115/1.4043565.

Fractional Bloch equation is a generalized form of the integer order Bloch equation. It governs the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance (NMR). Scale-3 (S-3) Haar wavelet operational matrix along with quasi-linearization is applied first time to detect the spin flow of fractional Bloch equations. A comparative analysis of performance of classical scale-2 (S-2) and novel scale-3 Haar wavelets (S-3 HW) has been carried out. The analysis shows that scale-3 Haar wavelets give better solutions on coarser grid point in less computation time. Error analysis shows that as we increase the level of the S-3 Haar wavelets, error goes to zero. Numerical experiments have been conducted on five test problems to illustrate the merits of the proposed novel scheme. Maximum absolute errors, comparison of exact solutions, and S-2 Haar wavelet and S-3 Haar wavelet solutions, are reported. The physical behaviors of computed solutions are also depicted graphically.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081002-081002-11. doi:10.1115/1.4043453.

This paper provides a comprehensive numerical analysis of a simple 2D model of running, the spring-loaded inverted pendulum (SLIP). The model consists of a point-mass attached to a massless spring leg; the leg angle at touch-down is fixed during the motion. We employ numerical continuation methods combined with extensive simulations to find all periodic motions of this model, determine their stability, and compute the basins of attraction of the stable solutions. The result is a detailed and complete analysis of all possible SLIP model behavior, which expands upon and unifies a range of prior studies. In particular, we demonstrate and explain the following effects: (i) saddle-node bifurcations, which lead to two distinct solution families for a range of energies and touch-down angles; (ii) period-doubling (PD) bifurcations which lead to chaotic behavior of the model; and (iii) fractal structures within the basins of attraction. In contrast to prior work, these effects are found in a single model with a single set of parameters while taking into account the full nonlinear dynamics of the SLIP model.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081003-081003-11. doi:10.1115/1.4043452.

Motivated by modeling directional drilling dynamics where planar curved beams undergo small displacements, withstand high compression forces, and are in contact with an external wall, this paper presents an finite element method (FEM) modeling framework to describe planar curved beam dynamics under loading. The shape functions of the planar curved beam are obtained using the assumed strain field method. Based on the shape functions, the stiffness and mass matrices of a planar curved beam element are derived using the Euler–Lagrange equations, and the nonlinearities of the beam strain are modeled through a geometric stiffness matrix. The contact effects between curved beams and the external wall are also modeled, and corresponding numerical methods are discussed. Simulations are carried out using the developed element to analyze the dynamics and statics of planar curved structures under small displacements. The numerical simulation converges to the analytical solution as the number of elements increases. Modeling using curved beam elements achieves higher accuracy in both static and dynamic analyses compared to the approximation made by using straight beam elements. To show the utility of the developed FEM framework, the post-buckling condition of a directional drill string is analyzed. The drill pipe undergoes spiral buckling under high compression forces, which agrees with experiments and field observations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081004-081004-10. doi:10.1115/1.4043617.

In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081005-081005-5. doi:10.1115/1.4043525.

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081006-081006-12. doi:10.1115/1.4043526.

Thin curved rings used mostly as seals, including in internal combustion engines undergo complex elastodynamic behavior when subjected to a combination of normal radial loading and tangential shear with friction. In turn, their complex modal behavior often results in loss of sealing, increased friction, and power loss. This paper presents a new finite difference approach to determine the response of thin incomplete circular rings. Two interchangeable approaches are presented; one embedding mass and stiffness components in a unified frequency-dependent matrix, and the other making use of equivalent mass and stiffness matrices for the ring structure. The versatility of the developed finite difference formulation can also allow for efficient modification to account for multiple dynamically changing ring support locations around its structure. Very good agreement is observed between the numerical predictions and experimental measurements, particularly with new precision noncontact measurements using laser Doppler vibrometry. The influence of geometric parameters on the frequency response of a high performance motorsport engine's piston compression ring demonstrates the degree of importance of various geometrical parameters on ring dynamic response.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081007-081007-11. doi:10.1115/1.4043564.

Torsional stick–slip vibrations easily occur when the drill bit encounters a hard or a hard-soft staggered formation during drilling process. Moreover, serious stick–slip vibrations of the drill string is the main factor leading to low drilling efficiency or even causing the downhole tools failure. Therefore, establishing the stick–slip theoretical model, which is more consistent with the actual field conditions, is the key point for new drilling technology. Based on this, a new torsional vibration tool is proposed in this paper, then the multidegree-of-freedom torsional vibrations model and nonlinear dynamic model of the drill string are established. Combined with the actual working conditions in the drilling process, the stick–slip reduction mechanism of the drill string is studied. The research results show that the higher rotational speed of the top drive, smaller viscous damping of the drill bit, and smaller WOB (weight on bit) will prevent the stick–slip vibration to happen. Moreover, the new torsional vibration tool has excellent stick–slip reduction effect. The research results and the model established in this paper can provide important references for reducing the stick–slip vibrations of the drill string and improving the rock-breaking efficiency.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081008-081008-11. doi:10.1115/1.4043450.

Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.

Topics: Manifolds
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081009-081009-14. doi:10.1115/1.4043669.

Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation (FPE) to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the FPE numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(8):081010-081010-11. doi:10.1115/1.4043670.

Although a large number of hidden chaotic attractors have been studied in recent years, most studies only refer to integer-order chaotic systems and neglect the relationships among chaotic attractors. In this paper, we first extend LE1 of sprott from integer-order chaotic systems to fractional-order chaotic systems, and we add two constant controllers which could produce a novel fractional-order chaotic system with hidden chaotic attractors. Second, we discuss its complicated dynamic characteristics with the help of projection pictures and bifurcation diagrams. The new fractional-order chaotic system can exhibit self-excited attractor and three different types of hidden attractors. Moreover, based on fractional-order finite time stability theory, we design finite time synchronization scheme of this new system. And combination synchronization of three fractional-order chaotic systems with hidden chaotic attractors is also derived. Finally, numerical simulations demonstrate the effectiveness of the proposed synchronization methods.

Commentary by Dr. Valentin Fuster

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