Application of Udwadia-Kalaba method for rotorcraft analysis

Abstract

Modeling a rotorcraft as a system of interconnected bodies requires a systematic procedure to assemble the equations of motion of the individual components and enforce the constraints between them. Such systematic treatment often involves the use of a redundant set of coordinates to describe the motion of the system so that the modeling process could easily be automated. The kinematic constraints relating these redundant coordinates are then enforced through Lagrange multipliers, resulting in a system of differential algebraic equations (DAEs). However, higher the index of the DAEs, more arduous methods need to be employed for their numerical integration. Udwadia-Kalaba (U-K) approach allows for the reduction of the equations of motion of a constrained multibody system to a set of ordinary differential equations (ODEs) even when redundant coordinates are employed. Thus, U-K approach improves the ease of implementation of the multibody simulation engine as a computer program and allows for the use of explicit time integration schemes. In this work, the U-K approach is first used to model the benchmark problem of a double four bar mechanism. The benchmarking results indicate that the U-K method provides a better accuracy than the augmented Lagrangian formulation. Subsequently, the U-K method is used to model a system of interest to the rotorcraft community, a rigid rotating flapping blade. In this case, the simulated results from the U-K method are shown to be in good agreement with those from the minimal coordinates approach. This proves that the U-K method could be used for automatic development of the equations of motion in a comprehensive rotorcraft analyses package.