Model complexity of distributed parameter systems: An energy-based approach

Abstract

Continuous components, like the cantilever beam, are widely used in numerous engineering systems with their geometric and material properties varying depending on the application. Calculating the dynamic behavior of a cantilever beam is a challenging task since the essential physical phenomena and interactions vary significantly based on the geometry of the beam. There exist a number of theories that can be used to predict the transverse motion of a cantilever beam of which the two most commonly used are the Timoshenko and Euler-Bernoulli theories. The Euler-Bernoulli theory is simpler and thus preferred, however depending on the beam’s parameters and operating conditions can lead to erroneous results and thus the more complex Timoshenko theory has to be used. Currently, selecting the theory to use depends on some heuristics or rules that are based on experience and the accuracy requirements of the predictions. It is the purpose of this chapter to address the model complexity of a cantilever beam through a systematic modeling methodology. A new approach is presented for selecting the appropriate theory to use for modeling a cantilever beam. The beam is discretized through a finite segment approach and modeled using the bond graph formulation. Then the previously developed energy-based modeling metric, activity, and the Model Order Reduction Algorithm are used to determine when and where the shear and rotary inertial effects, of the Timoshenko theory, need to be included in the model in order to have accurate predictions of the dynamic behavior. An illustrative example is provided to demonstrate the new methodology. © Springer International Publishing Switzerland 2017.