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### Research Papers

J. Comput. Nonlinear Dynam. 2018;13(12):121001-121001-10. doi:10.1115/1.4041577.

This paper reports theoretical and numerical results about the reinjection process in type V intermittency. The M function methodology is applied to a simple mathematical model to evaluate the reinjection process through the reinjection probability density function (RPD), the probability density of laminar lengths, and the characteristic relation. We have found that the RPD can be a discontinuous function and it is a sum of exponential functions. The RPD shows two reinjection behaviors. Also, the probability density of laminar lengths has two different behaviors following the RPD function. The dependence of the RPD function and the probability density of laminar lengths with the reinjection mechanisms and the lower boundary of return are considered. On the other hand, we have obtained, for the analyzed map, that the characteristic relation verifies $l¯≈ε−0.5$. Finally, we highlight that the M function methodology is a suitable tool to analyze type V intermittency and there is a very high accuracy between the new theoretical equations and the numerical data.

Topics: Density , Probability
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(12):121002-121002-7. doi:10.1115/1.4041417.

This paper aims to perform a complete Noether symmetry analysis of a generalized hyperbolic Lane–Emden system. Several constraints for which Noether symmetries exist are derived. In addition, we construct conservation laws associated with the admitted Noether symmetries. Thereafter, we briefly discuss the physical meaning of the derived conserved vectors.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(12):121003-121003-6. doi:10.1115/1.4041579.

The Maggi and Kane equations of motion are valid for systems with only nonholonomic constraints, but may fail when applied to systems with holonomic constraints. A tangent space ordinary differential equation (ODE) extension of the Maggi and Kane formulations that enforces holonomic constraints is presented and shown to be theoretically sound and computationally effective. Numerical examples are presented that demonstrate the extended formulation leads to solutions that satisfy position, velocity, and acceleration constraints for holonomic systems to near computer precision.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(12):121004-121004-8. doi:10.1115/1.4041621.

A typical phenomenon of the fractional order system is presented to describe the initial value problem from a brand-new perspective in this paper. Several simulation examples are given to introduce the named aberration phenomenon, which reflects the complexity and the importance of the initial value problem. Then, generalizations on the infinite dimensional property and the long memory property are proposed to reveal the nature of the phenomenon. As a result, the relationship between the pseudo state-space model and the infinite dimensional exact state-space model is demonstrated. It shows the inborn defects of the initial values of the fractional order system. Afterward, the pre-initial process and the initialization function are studied. Finally, specific methods to estimate exact state-space models and fit initialization functions are proposed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(12):121005-121005-8. doi:10.1115/1.4041681.

Elastic-driven slender filaments subjected to compressive follower forces provide a synthetic way to mimic the oscillatory beating of biological flagella and cilia. Here, we use a continuum model to study the dynamical, nonlinear buckling instabilities that arise due to the action of nonconservative follower forces on a prestressed slender rod clamped at both ends and allowed to move in a fluid. Stable oscillatory responses are observed as a result of the interplay between the structural elastic instability of the inextensible slender rod, geometric constraints that control the onset of instability, energy pumped into the system by the active follower forces, and motion-driven fluid dissipation. Initial buckling instabilities are initiated by the effect of the follower forces and inertia; fluid drag subsequently allows for the active energy pumped into the system to be dissipated away and results in self-limiting amplitudes. By integrating the equations of equilibrium and compatibility conditions with linear constitutive laws, we compute the critical follower forces for the onset of oscillations, emergent frequencies of these solutions, and the postcritical nonlinear rod shapes for two forms of the drag force, namely linear Stokes drag and quadratic Morrison drag. For a rod with fixed inertia and drag parameters, the minimum (critical) force required to initiate stable oscillations depends on the initial slack and weakly on the nature of the drag force. Emergent frequencies and the amplitudes postonset are determined by the extent of prestress as well as the nature of the fluid drag. Far from onset, for large follower forces, the frequency of the oscillations can be predicted by evoking a power balance between the energy input by the active forces and the dissipation due to fluid drag.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(12):121006-121006-12. doi:10.1115/1.4041472.

Component-centric reduced order models (ROMs) have recently been developed in the context of linear structural dynamics. They lead to an accurate prediction of the response of a part of structure (referred to as the β component) while not requiring a similar accuracy in the rest of the structure (referred to as the α component). The advantage of these ROMs over standard modal models is a significantly reduced number of generalized coordinates for structures with groups of close natural frequencies. This reduction is a very desirable feature for nonlinear geometric ROMs, and thus, the focus of the present investigation is on the formulation and validation of component-centric ROMs in the nonlinear geometric setting. The reduction in the number of generalized coordinates is achieved by rotating close frequency modes to achieve unobservable modes in the β component. In the linear case, these modes then completely disappear from the formulation owing to their orthogonality with the rest of the basis. In the nonlinear case, however, the generalized coordinates of these modes are still present in the nonlinear stiffness terms of the observable modes. A closure-type algorithm is then proposed to finally eliminate the unobserved generalized coordinates. This approach, its accuracy and computational savings, is demonstrated first on a simple beam model and then more completely on the 9-bay panel model considered in the linear investigation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(12):121007-121007-14. doi:10.1115/1.4041774.

Tooth friction is unavoidable and changes periodically in gear engagement. Friction excitation is an important excitation source of a gear transmission system. They are different than the friction coefficients of any two points on the same contact line of a helical/herringbone gear. In order to obtain the influence of the friction excitation on the dynamic response of a helical/herringbone planetary gear system, a method that uses piecewise solution and then summing them to analyze the friction force and frictional torque of tooth surfaces is proposed. Then, the friction coefficient is obtained based on the mixed elastohydrodynamic lubrication (EHL) theory. A dynamic model of a herringbone planetary gear system is established considering the friction, mesh stiffness, and meshing error excitation by the node finite element method. The influence of friction excitation on the dynamic response of the herringbone planetary gear is studied under different working conditions. The results show that friction excitation has a great influence on the vibration acceleration of the sun and planetary gear. However, the effect on the radial and tangential vibration acceleration of a planetary gear is the opposite. In addition, the friction excitation has a slight effect on the meshing force.

Commentary by Dr. Valentin Fuster