Modeling Flexible Multi-Body Systems Within the Udwadia–Kalaba Framework, a Lumped Parameter Approach

It is common practice within multi-body dynamic (MBD) modeling to assume that individual bodies are rigid however this can be an oversimplification especially when slender bodies are present resulting in inaccurate estimations of the system's natural frequencies and overall behavior. To address this shortfall, we extend the Udwadia–Kalaba (U–K) MBD formulation in this paper to model flexible multibody systems for purposes of exploring system natural frequencies. To model the flexibility, a lumped parameter approach is proposed, which in this work idealizes a flexible beam as a series of discrete rigid elements connected by torsional springs. In the U–K formulation, a mechanical system can also be discretized into rigid elements and adapted. This is viewed as a benefit for incorporating a lumped parameter approach within the U–K formulation to model flexible multibody systems. A flexible crank-slider mechanism is introduced and modeled within the Udwadia–Kalaba formulation to capture the dynamics of flexibility through linkage compliance. The model is validated against an alternatively formulated MBD model and system natural frequencies and mode shapes numerically predicted. Results of the study show the effectiveness and potential of extending the application of the Udwadia–Kalaba formulation by using a lumped parameter approach to dynamically model flexible multibody systems.