Constrained port-Hamiltonian modeling and structure-preserving discretization of the Rayleigh beam

This paper addresses the port-Hamiltonian modeling of the Rayleigh beam, which bridges the gap between the Euler-Bernoulli and Timoshenko beam theories. This balance makes the Rayleigh model particularly suitable for scenarios where Euler-Bernoulli assumptions are insufficient, but Timoshenko’s complexity is unnecessary, such as in cases of moderate oscillations. The originality of the approach lies in deriving the Rayleigh beam model from the displacement field of the Timoshenko beam and incorporating an algebraic constraint consistent with Rayleigh beam theory. The resulting model is formulated as an infinite-dimensional port-Hamiltonian differential-algebraic equation (PH-DAE). A structure-preserving spatial discretization strategy is developed using the mixed finite element method, ensuring the preservation of the PH-DAE structure in the finite-dimensional setting. Numerical simulations demonstrate the accuracy and effectiveness of the proposed model and discretization approach.