MS12.1: Nonlinear Dynamics for Engineering Design

This study concerns the effect of uncertain tooth profile modifications on the primary resonance of a spur gear pair induced by the backlash nonlinearity. It should be noted that these corrections are micro geometric within tolerances. This primary resonance can exhibit dynamic responses with vibro-impacts between teeth. In this context, we have developed an original approach based on tracking the saddle-node bifurcations of the primary resonance. The equation of motion is solved using the harmonic balance method combined with an arc-length continuation method and a bordering technique. The control parameters are the meshing frequency associated with the rotation speed, and the tip relief associated with the profile correction. The evolution of the bifurcation points with respect to the parameters is calculated deterministically. To characterize the severity of the resonance, we introduce two criteria, namely the amplitude of the response at the resonance peak and the width of the resonance hysteresis. Uncertainty is then introduced by means of probability densities of the tip relief within the tolerance interval. Monte Carlo simulations are then used to conclude on the optimum profile correction, which differs significantly from that recommended in the purely deterministic case.

Since rolling bearings frequently operate in variable speed conditions, the elastic deformation of the cage occurs intensely due to the enhanced excitation, which may lead to dynamic instability or even failure of the bearing. To this end, a flexible cage model based on the semi-flexible body element is developed in this paper. The force excitation from the ball and the guiding ring is considered to be applied to the cage. By coupling with the previous bearing model, a bearing dynamics model with flexible cage is presented. The results predicted by this model are closer to the experimental values than those of the rigid model. The differences between the two models in predicting the bearing dynamic performance under typical variable speed conditions are compared. The effect of preload on its dynamic behavior was further investigated with the flexible model. The results show that a slightly smaller radius of the cage orbit is predicted by the flexible model than that of the rigid model. Compared with the rigid model, more stability of cage motion is predicted by the flexible model. A similar phenomenon can be observed in the force excitation on the cage. There is not significant difference between these two models in predicting the velocity of the bearing assembly. The work in this paper provides a new idea for modeling flexible cages.

This work proposes a methodology for simulating the elastoplastic strain behaviors of a steel sheet under the impact of one hammer drop. The intention was to aid in the optimization of the hammer design and material performance for an enhanced steelpan building process. It navigates the complexities of the nonlinear strain behavior for the two modes: 1) the duration when the hammer deforms the sheet to create a dimple, and 2) the time after the hammer loses contact, leaving the entire system to freely vibrate. A finite element analysis using COMSOL Multiphysics software was employed to simulate the dynamic system. The model determined the strain behavior of the material during the two modes previously described. An understanding of the mechanics of deformation is crucial to the choice of an effective universal standard procedure and the appropriate tools and equipment for the sinking of a steelpan. The steelpan industry’s present choice of sinking tools ranges between separate sizes of hammers and/or shotput balls. In most cases, a sledgehammer is modified by grinding the face edge, resembling the proximity of a smooth hemisphere. A previous investigation surveyed qualitative and quantitative data on the procedure to sink the steelpan, which led to this investigation. The active interplay between experimentation and simulation revealed strains at the highest magnitudes nearer the center of the sheet. As hammer diameter increased, the strains were lower and diminished in the direction toward the end of the blank. The simulation also exposed that increasing hammer diameter produced higher compressive strains during deformation at the center.

Optical systems that feature some sort of coupling between various subsystems offer a flexible platform to observe a diverse range of rich nonlinear dynamics. Often, the vector fields that describe the dynamics of such coupled systems have within them a natural symmetry that can lead to either delocalization or symmetry breaking. We consider a microresonator that supports two interacting electric fields, generated by two laser beams of equal properties. Mathematically, this system is modeled by a four-dimensional $\mathbb{Z}_2$-equivariant vector field with strength and detuning of the input light as parameters. We identify a symmetric pair of heteroclinic cycles as an organizing center in the parameter plane. Different types of dynamics nearby are determined by means of the identification and continuation of global bifurcations in combination with the computation of kneading invariants and Lyapunov exponents. In this way, we provide a numerical unfolding of this codimension-two global bifurcation and show how it involves infinitely many further global bifurcations.