An Efficient 3D Mathematical Model to Predict Structural Dynamics and Chatter in Cold Rolling Mills

The research described in this dissertation aims to provide a highly efficient predictive computational tool to improve the dimensional quality of cold rolled metal strip, particularly for high-value, thin specialty alloys. The corresponding objectives of this work are to (1) understand the transfer of high-fidelity roll grinding errors that may generate complex geometric defects on the strip, and to investigate a novel method for correction of such defects; and (2) develop a highly efficient 3D dynamic predictive model for the time-history of mill and strip transient behavior, including highly damaging chatter vibrations. First, a novel approach that can potentially correct for high-fidelity geometric defects in cold rolled strip is proposed. High- fidelity flatness defects in thin cold-rolled strip that arise from highly localized thickness strain variations present an ongoing challenge to the metals industry. A primary cause of such defects, based on rolling practice, but for which the effects have not been rigorously investigated, may be the transfer of localized diameter deviations from the work rolls that arise from roll grinding errors due to grinding performance inaccuracies. The proposed research addresses the effects of high-fidelity roll diameter deviation transfer to the strip, as well as their correction. Parametric case studies are first undertaken using a 4-high mill to investigate the influences that roll diameter, strip reduction, strip width, and material strength have on the 3D transfer of high- fidelity work roll diameter deviations to the rolled sheet. The studies are conducted with an efficient 3D mathematical roll-stack model that predicts the associated high-fidelity strip thickness profile deviations using the simplified-mixed finite element method (SM-FEM). Reduction deviations, which strongly correlate to strip flatness/shape defects, are first quantified and analyzed to understand the transfer characteristics of localized work-roll grinding deviations relative to benchmarked perfectly smooth work rolls. Results of the study reveal that the high- fidelity transfer depends not only on the specific roll grinding deviation amplitude and mill loading, but also on the location of the roll diameter deviations along the roll face length due to non-negligible 3D bulk roll-stack deformations, as well as the effective stiffness ratio between the work roll and the strip. The inability of conventional flatness control devices to correct for high-fidelity roll diameter deviations is also demonstrated. Based on this work, suggested is a novel corrective approach to identify customized work roll grinding profiles that are tailored to strip with specific pre-existing high-fidelity defect patterns generated in previous rolling passes. Using the described SM-FEM modeling technique, high-fidelity “corrective” roll diameter profiles could eventually be applied in-situ during rolling, whereby the profiles are “engineered” to account for the predicted 3D mill deflections, contact force distributions, and coupled micro/macro scale deformation mechanics. Following investigation of the SM-FEM formulation to high-fidelity static problems involving the transfer of roll grinding error to the strip, the SM- FEM method is adapted to create a general purpose 3D structural dynamics model capable of predicting the transient behavior of the mill components and strip thickness profile geometry. Over the last three decades, computational models have been developed and employed in effort to understand dynamic disturbances in the rolling operation. Such disturbances can potentially lead to self-excitation (chatter vibrations) and result in significant gauge variations in the exit strip, as well as strip rupture and/or damage to the mill in extreme cases. Numerous challenges exist, however, in adequately modeling the 3D dynamic behavior. For instance, highly coupled relationships exist between several rolling process parameters, including the rolling force/torque, strip entry/exit tensions, rolling speed, roll gap profile, friction, neutral point, etc. In addition, both “hard” and “soft” nonlinearities are present, including continuous changes in the roll/strip contact conditions, elastic-plastic deformation of the strip, and nonlinear elastic flattening the rolls. These factors make it very difficult to effectively and efficiently model the rolling operation even under a quasi-static (or steady-state) assumption. Moreover, the existing models that account for structural dynamics of the rolling process exploit many simplifications, such as modeling the mill structure as linear lumped parameter system, and symmetry in the motions of rolls, among other assumptions. Even with these assumptions, the current state-of-the-art models are generally not capable of accommodating conventional strip thickness profile/flatness control mechanisms, such as roll bending, roll shifting, or non-uniform machined roll profiles, which severely restricts the flexibility/applicability to accommodate complex mill structures. A 3D general purpose dynamic model that can incorporate profile/flatness control mechanisms in addition to complex mill configurations (like 12-high or 20-high cluster mills) can provide significant new insights into rolling dynamics and chatter investigations. Accordingly, the presented research combines the static SM-FEM mathematical formulation with a Newmark- Beta time integration technique to develop a highly-efficient, stable, high-fidelity global-stiffness based transient structural dynamics model. Case studies are carried out to demonstrate the features and capabilities of the presented model in addressing the aforementioned research gaps and challenges. Based on the results, the presented structural dynamics model is able to efficiently capture time histories due to discrete disturbances on both vertical and cluster-type mill configurations. For chatter investigation, however, an appropriate roll-bite model to capture the relationships among the coupled rolling process parameters that influence the roll-bite contact mechanics is required to be coupled to accommodate the real time interactions between the structural dynamics and roll-bite mechanics. In addition to the aforementioned assumptions and simplifications in mill structural dynamic modeling, presence of both hard, and soft non- linearities as well as material non-linearities in the roll bite contact mechanics has led to exclusive use of linearized relationship in the current state-of-the-art chatter models, while 3D chatter models are non-existent in the literature. Accordingly, following the development of general-purpose 3D dynamic model, the dynamic simplified mixed-finite element method (D- SM-FEM) is coupled with a roll-bite process model. A critical part in replicating the dynamic interaction between the rolling process and the mill structural dynamics in this work relates to real-time variations in the “working point” or “operating point” relationship between specific rolling force and the plastic strain of the rolled strip which changes according to perturbations in the roll gap, position of entry/exit plane, entry/exit velocity, and tensions. These variations in the working point are incorporated in the presented chatter model via the concept of a “dynamic strip modulus” based on the secant (or tangent) relationship between the specific rolling force and plastic strain at the working point, but where the strip modulus is updated at every time-step. Case studies are presented using 4-high mill with aim to demonstrate the ability of the presented 3D chatter model to address the lack of available chatter models in literature employing 3D bulk body deformation effects, and to address some of the limitations identified above. The capabilities of the model to predict the stability, or dynamic instability is also illustrated with accompanying 3D plots showing the true mode shapes. Case studies are also undertaken to demonstrate the effect of asymmetric (with varying lower housing stiffness) mill stand assumptions. The results reveal interesting phase relationships not elucidated in previous research, as well as detailed effects from the 3D modeling on the strip profile and shape/flatness.