Analytical derivatives of flexible multibody dynamics with the floating frame of reference formulation

In this paper, the analytical derivatives of flexible multibody dynamics with the floating frame of reference formulation are derived in a new way using the invariants and their sensitivities. This enables the decoupling of the sensitivity analysis of flexible multibody dynamics from the finite-element solver and guarantees high accuracy and efficiency of the sensitivity computations. The invariants are shown with both consistent and lumped mass approaches. The latter allows generality towards the formulation of a finite-element type, including beams, shells, and solids. The expressions are fully derived with lumped masses, showing for the first time the compensation term of inertia due to the non-consideration of the mass distribution with this approach. It is then shown that the expressions of the system parameters in the lumped case with the newly introduced inertia compensation term correspond to the general case, and, therefore, the derived approach and equations are of general nature. Crucial for the decoupling of the sensitivity analysis are the analytical derivatives of the system parameters that contain the derivatives of the invariants and whose analytical expressions are derived and provided here for the first time. The partial derivatives arise in the sensitivity analysis with both the direct differentiation method and the adjoint variable method, and the former is shown here. In addition, the partial derivatives arise in the Jacobian matrix of the nonlinear solver for the transient solution of flexible multibody systems.