Period-3 Motions in a Parametrically Exited Inverted Pendulum

In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

Modal Reduction Procedures for Flexible Multibody System Applications

The comprehensive simulation of flexible multibody systems calls for the ability to model various types of structural components such as rigid bodies, beams, plates, and kinematic joints. Modal components test offer additional modeling versatility by enabling the treatment of complex, three-dimensional structures via modal reduction procedures based on the small deformation assumption. In this paper, the problem is formulated within the framework of the motion formalism. The kinematic description involves simple, straightforward frame transformations and leads to deformation measures that are both objective and tensorial. Derivatives are expressed in the material frame, which results in the remarkable property that the tangent matrices are independent of the configuration of the modal component with respect to an inertial frame. This implies a reduced level of geometric nonlinearity as compared to standard description. In particular, geometrically nonlinear problems can be solved with the constant tangent matrices of the reference configuration, without re-evaluation and re-factorization.

Finite Element Models for Flexible Cosserat Solids

The application of the finite element method to the modeling of Cosserat solids is investigated in detail. In two- and three-dimensional elasticity problems, the nodal unknowns are the components of the displacement vector, which form a linear field. In contrast, when dealing with Cosserat solids, the nodal unknowns form the special Euclidean group SE(3), a nonlinear manifold. This observation has numerous implications on the implementation of the finite element method and raises numerous questions: (1) What is the most suitable representation of this nonlinear manifold? (2) How is it interpolated over one element? (3) How is the associated strain field interpolated? (4) What is the most efficient way to obtain the discrete equations of motion? All these questions are, of course intertwined. This paper shows that reliable schemes are available for the interpolation of the motion and curvature fields. The interpolated fields depend on relative nodal motions only, and hence, are both objective and tensorial. Because these schemes depend on relative nodal motions only, only local parameterization is required, thereby avoiding the occurrence of singularities. For Cosserat solids, it is preferable to perform the discretization operation first, followed by the variation operation. This approach leads to considerable computation efficiency and simplicity.

Numerical Procedure for Non-Hertzian Wheel-Rail Contact Model Integrated in Quasi-Steady Railway Vehicle Motion Solver

Although the Hertzian contact theory is widely utilized in railway vehicle simulations with new wheel and rail profiles, the Hertzian contact assumptions would lead to inaccurate contact prediction for severely worn wheel and rail profiles due to their geometric conformity, causing non-elliptical contact shapes as well as pressure distribution. For this reason, various non-Hertzian contact models have been studied for use in vehicle dynamics simulations. Among others, a method proposed by Piotrowski and Kik has gained acceptance in predicting non-elliptical wheel-rail contact for vehicle dynamics simulations. Despite the elegant formulation and its accuracy, detailed online geometric calculation for non-elliptical contact shape is required for all the contact patches at every iteration, along with iterative evaluation of the force-deflection relationship. It leads to computation burdens for use in long-distance vehicle simulations. Therefore, in this study, an off-line based numerical procedure for non-Hertzian contact model is developed and integrated in the quasi-steady railway vehicle motion solver.

On Periodic Motions in a Periodically Driven van der Pol-Duffing Oscillator

In this paper, the symmetric and asymmetric period-1 motions on the bifurcation tree are obtained for a periodically driven van der Pol-Duffing hardening oscillator through a semi-analytical method. Such a semi-analytical method develops an implicit mapping with prescribed accuracy. Based on the implicit mapping, the mapping structures are used to determine periodic motions in the van der Pol-Duffing oscillator. The symmetry breaks of period-1 motion are determined through saddle-node bifurcations, and the corresponding asymmetric period-1 motions are generated. The bifurcation and stability of period-1 motions are determined through eigenvalue analysis. To verify the semi-analytical solutions, numerical simulations are also carried out.

Consistent and Inertia-Shape-Integral-Free Invariants of the Floating Frame of Reference Formulation

The conventional continuum-mechanics-based floating frame of reference formulation involves unhandy so-called inertia-shape-integrals in the equations of motion, which is why, commercial multibody software codes resort to a lumped mass approximation to avoid the evaluation of these integrals in their computer implementations. This paper recaps the conventional continuum mechanics floating frame of reference formulation and addresses its drawbacks by summarizing recent developments of the so-called nodal-based floating frame of reference formulation, which avoids inertia shape integrals ab initio, does not rely on a lumped mass approximation, and exhibits a way to calculate the so-called invariants, which are constant “ingredients” required to set up the equations of motion, in a consistent way.

Data Driven Model Identification for a Chaotic Pendulum With Variable Interaction Potential

Data driven model identification methods have grown increasingly popular due to enhancements in measuring devices and data mining. They provide a useful approach for comparing the performance of a device to the simplified model that was used in the design phase. One of the modern, popular methods for model identification is Sparse Identification of Nonlinear Dynamics (SINDy). Although this approach has been widely investigated in the literature using mostly numerical models, its applicability and performance with physical systems is still a topic of current research. In this paper we extend SINDy to identify the mathematical model of a complicated physical experiment of a chaotic pendulum with a varying potential interaction. We also test the approach using a simulated model of a nonlinear, simple pendulum. The input to the approach is a time series, and estimates of its derivatives. While the standard approach in SINDy is to use the Total Variation Regularization (TVR) for derivative estimates, we show some caveats for using this route, and we benchmark the performance of TVR against other methods for derivative estimation. Our results show that the estimated model coefficients and their resulting fit are sensitive to the selection of the TVR parameters, and that most of the available derivative estimation methods are easier to tune than TVR. We also highlight other guidelines for utilizing SINDy to avoid overfitting, and we point out that the fitted model may not yield accurate results over long time scales. We test the performance of each method for noisy data sets and provide both experimental and simulation results. We also post the files needed to build and reproduce our experiment in a public repository.

A Non-Prismatic Beam Element for the Optimization of Flexure Mechanisms

Flexure joints are rapidly gaining ground in precision engineering because of their predictable behavior. However the range of motion of flexure joints is limited due to loss of support stiffness in deformed configurations. Most of the common flexure joints consist of prismatic leaf springs. This paper presents a simple non-prismatic beam formulation that can be used for the efficient modelling of non-prismatic leaf springs. The resulting stiffness and stress computed by the non-prismatic beam element are compared to the results of a finite element analysis. The paper shows that the support stiffness of two typical flexure joints can be increased up to a factor of 1.9 by using non-prismatic instead of prismatic leaf springs.

A Jupyter Notebook Environment for Multibody Dynamics

DARTS is a rigid/flexible multibody dynamics toolkit for the modeling and simulation of aerospace and robotic vehicles for engineering applications. In this paper we describe an on-line, browser-based environment using Jupyter notebooks to support training needs for the DARTS software. The suite of curated tutorial notebooks is organized into different topic areas, and into multiple themes within each topic area. The notebooks within a theme use a progression of examples for users to expand their understanding of the software. The topic areas include one on the DARTS multibody dynamics software and another one on the theory underlying the multibody dynamics formulation. We also describe a number of Jupyter extensions that were used — and some developed in house — to enhance the notebook interface for use with the dynamics simulation software. One significant extension we implemented allows the embedding of live 3D visualizations within simulation notebooks.

Mechanical System Modelling Using Physics-Based and Data-Based Subsystems

Mechanical systems have been traditionally represented using parametric physics-based models. In this work, we introduce a novel concept, in this part of the mechanical system is represented using data-based subsystem models, and the overall mechanical system model is composed of these data-based and other, physics-based subsystems. A core element is the interfacing of the subsystems, which gives rise to interaction forces. The interfacing problem is formulated in a way that makes it possible to give a general representation to the interaction forces. We demonstrate that from the point of view of the physics-based subsystems the important element is that the data-based models can represent the interaction force systems properly. The data-based subsystems are developed using deep recurrent neural networks, and the training data is generated based on simulations using the fully parametric physics-based model of the system. Such training data could also be obtained through physical experimentation.