Research Papers

J. Comput. Nonlinear Dynam. 2018;13(4):041001-041001-10. doi:10.1115/1.4038991.

The application of Hopf bifurcation is essential to rail vehicle dynamics because it corresponds to the linear critical speed. In engineering, researchers always wonder which vehicle parameters are sensitive to it. With the nonlinear singularity theory's development, it has been widely applied in many other engineering areas. This paper mainly studies the singularity theory applied in nonlinear rail vehicle dynamics. First, the bifurcation norm forms of wheelset and bogie system are, respectively, deduced. Then the universal unfolding is obtained and the influences of perturbation on bifurcation are investigated. By the analysis of a simple bar-spring system, the relationship between the unfolding and original perturbation parameters can be found. But this may be difficult to calculate for the case in vehicle system because of higher degrees-of-freedom (DOFs) and indicate that can explain the influence of all possible parameters perturbations on vehicle bifurcation.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(4):041002-041002-10. doi:10.1115/1.4039191.

This paper presents the coupled axial-transverse-rotational nonlinear forced vibrations of Timoshenko tapered beams made of an axially functionally graded (AFG) material subjected to an external harmonic excitation. Two sources of nonlinearities are considered in modeling and numerical simulations: (i) the geometric nonlinearities arising from induced nonlinear tension due to the clamped–clamped boundary conditions and large deformations, and (ii) nonlinear expressions to address the nonuniform geometry and mechanical properties of the beam along the length. More specifically, a nonlinear model is developed based on the Timoshenko beam theory accounting for shear deformation and rotational inertia. Exponential distributions are presumed for the cross-sectional area, moduli of elasticity, mass density, and Poisson's ratio of the AFG tapered Timoshenko beam. The kinetic and potential energies, the virtual work of the external harmonic distributed load, and the one done by damping are implemented into Hamilton's energy principle. The resultant nonuniform nonlinearly coupled partial differential equations are discretized into a set of nonlinear ordinary differential equations utilizing Galerkin's technique. In the discretization scheme, a large number of modes, both symmetric and asymmetric, are employed due to the asymmetric characteristic of the nonuniform beam with respect to its length. The effect of different parameters, including the gradient index and different taper ratios, on the force-vibration-amplitude and frequency-vibration-amplitude diagrams is examined; the effect of these parameters on the natural frequencies, internal resonances, and asymmetric characteristics of the AFG system is investigated as well.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(4):041003-041003-12. doi:10.1115/1.4039241.

The nonlinear characteristics of slender wings have been studied for many years, and the influences of the geometric structural nonlinearity on the postflutter responses of the wing have been received significant attention. In this paper, the effects of the external store on the nonlinear responses of the slender wing will be discussed. Based on the Hodges–Dowell beam model, the dynamical equations of the wing which include the geometric structural nonlinearity and store effects are constructed. The unsteady aerodynamic loading of the wing will be calculated by employing Wagner function and strip theory. The slender body theory is adopted to get the aerodynamic forces of the store. The Galerkin method is used to obtain the state equations of the system and the appropriate mode combination is obtained for the cases studied in this paper. Numerical simulations are given to show that the store spanwise position and the distance between the store mass center and the elastic center of the wing are two important factors which will affect the nonlinear characteristics of the wing. These two parameters will induce the occurrence of quasi-periodic motion and branch structure in bifurcation diagrams to the system. The peak of postflutter response is also related to these parameters and the lower response peak can be obtained when the store mass center is in front of the elastic center. The models and results are helpful to the design procedure of the slender wing with store in the preliminary stage.

Topics: Wings
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(4):041004-041004-9. doi:10.1115/1.4039310.

This paper introduces a new parallel co-simulation method to study vehicle-track dynamic interactions. The new method uses the transmission control protocol/internet protocol (TCP/IP) to enable co-simulation between a detailed in-house track dynamics simulation package and a commercial vehicle system dynamics simulation package. The exchanged information are wheel-rail contact forces and rail kinematics. Then, the message passing interface (MPI) technique is used to enable the model to process track dynamics simulations and vehicle dynamics simulations in parallel. The parallel co-simulation technique has multiple advantages: (1) access to the advantages of both in-house and commercial simulation packages; (2) new model parts can be easily added in as new parallel processes; and (3) saving of computing time. The original track model used in this paper was significantly improved in terms of computing speed. The improved model is now more than ten times faster than the original model. Two simulations were conducted to model a locomotive negotiating a section of track with and without unsupported sleepers. The results show that the vertical rail deflections, wheel-rail contact forces and vehicle suspension forces are evidently larger when unsupported sleepers are present. The simulations have demonstrated the effectiveness of the proposed parallel co-simulation method for vehicle-track dynamic interaction studies.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(4):041005-041005-12. doi:10.1115/1.4039192.

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(4):041006-041006-12. doi:10.1115/1.4039129.

Rotation-free shell formulations were proved to be an effective approach to speed up solving large-scaled problems. It reduces systems' degrees-of-freedom (DOF) and avoids shortages of using rotational DOF, such as singular problem and rotational interpolation. The rotation-free element can be extended for solving geometrically nonlinear problems using a corotational (CR) frame. However, its accuracy may be lost if the approach is used directly. Therefore, a new nonlinear rotation-free shell element is formulated to improve the accuracy of the local bending strain energy using a CR frame. The linear strain for bending is obtained by combining two re-derived elements, while the nonlinear part is deduced with the side rotation concept. Furthermore, a local frame is presented to correct the conventional local CR frame. An explicit tangential stiffness matrix is derived based on plane polar decomposition local frame. Simple elemental rotation tests show that the stiffness matrix and the proposed local frame are both correct. Several numerical examples and the application of drape simulations are given to verify the accuracy of nonlinear behavior of the presented element, and some of the results show that the presented method only requires few elements to obtain an accurate solution to the problem studied.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(4):041007-041007-12. doi:10.1115/1.4039059.

The floating frame of reference (FFR) formulation is widely used in multibody system (MBS) simulations for the deformation analysis. Nonetheless, the use of elastic degrees-of-freedom (DOF) in the deformation analysis can increase significantly the problem dimension. For this reason, modal reduction techniques have been proposed in order to define a proper set of assumed body deformation modes. Crucial to the proper definition of these modes when the finite element (FE) FFR formulation is used is the concept of the reference conditions, which define the nature of the deformable body coordinate system. Substructuring techniques, such as the Craig–Bampton (CB) method, on the other hand, have been proposed for developing efficient models using an assembly of their lower order substructure models. In this study, the appropriateness and generality of using the CB method in MBS algorithms are discussed. It is shown that, when a set of reference conditions are not applied, the CB transformation leads to the free–free deformation modes. Because a square CB transformation is equivalent to a similarity transformation that does not alter the problem to be solved, the motivation of using the CB method in MBS codes to improve the solution is examined. This paper demonstrates that free–free deformation modes cannot be used in all applications, shedding light on the importance of the concept of the FE/FFR reference conditions. It is demonstrated numerically that a unique model resonance frequency is achieved using different modes associated with different reference conditions if the shapes are similar.

Commentary by Dr. Valentin Fuster

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