Fully analytical sensitivity analysis of flexible multibody systems in natural coordinates

Abstract In this work, a fully analytical formulation for the sensitivity analysis of flexible multibody systems in natural coordinates is derived, extending the analytical sensitivity formulations for rigid bodies, already available in the bibliography. For the representation of the flexibility, the floating frame of reference (FFR) method is used. The flexible approach intends to be general, within the framework of FFR, and therefore no assumptions on the floating frame or on the deformation modes are adopted and a continuum mechanics approach based on invariant shape integrals is adopted for the inertia terms coming from the flexible bodies. The implementation of the sensitivity equations was coded in MBSLIM, an in house library for the dynamics and sensitivity analysis of multibody systems. The general implementation aimed, exposed the limitations of the natural coordinates formulations published in the bibliography, which lacked the generality required here, motivating the thorough derivation accomplished here, not only for sensitivities but also for the dynamics equations of motion in natural coordinates. For simplicity, this work focuses only on the natural coordinates ALI3-P equations of motion with direct sensitivities, but the library features also adjoint sensitivities and many other formulations in natural coordinates and recursive formulations in joint coordinates. For this reason, the multidimensional matrix algebra (which might seem unnecessary complex at first) is in the core of the equations and it is crucial for simplifying expressions and for keeping a minimal number of terms and sharing code between very different formulations, which make use of the same exact pieces but arranged differently in direct sensitivities vs. adjoint and/or between different sets of equations of motion.