Simple Discrete Lumped Parameter Models: Insight Into Large Displacement Continuous Beam Models

Abstract

Modeling considerations for large lateral displacement beam dynamics are investigated by means of discrete lumped parameter models. This investigation is limited to only geometric non-linear behavior; thus material non-linearity is not considered. The beam is cantilevered to a rigid rotating table at one end and free at the other end. The work demonstrates the ability of lumped parameter models to accomplish two major functions. First, the models can act as a validation tool for more complex models. Second, the models can be used as a means of determining the most appropriate non-linear expansion that is required to enhance the predictive capabilities of a linear continuous beam formulation. Equations of motion are developed, and the solution to a dynamic beam problem is presented. These tasks are first undertaken for a four degree-offreedom lumped parameter model which employs axial and rotational degrees of freedom. The resulting linear and non-linear models are compared to an existing continuous shape function beam model. This comparison demonstrates the failure of the linear continuous model to describe the non-linear large lateral displacement behavior. A second lumped parameter model using axial and lateral degrees of freedom, which is more closely analogous to the continuous model, is then developed. This model is employed to determine the contributions of non-linear elastic (static) terms and non-linear dynamic terms to the description of the large displacement behavior. The results of this portion of the investigation are that static elastic terms of up to third degree non-linearity are required to describe the large lateral displacement behavior. The validity and accuracy of the two and four degree-of-freedom lumped parameter models are investigated by comparison to a previously published closed-form solution for a static, large lateral displacement beam problem with a continuous beam model. Finally, it is shown that the insight gained in this study leads directly to improvements in a continuous beam model that enabled it to accurately describe non-linear large lateral displacement behavior.