Optimal lumped mass matrices by minimization of modal errors for beam elements

This paper clarifies the reason why error occurs in the mass lumping procedure and presents a new approach to construct lumped mass matrices for Euler-Bernoulli beam elements, which contain both translational and rotational degrees of freedom. Lumped mass matrices provide the proper translational inertia but change the rotational inertia compared with the continuous mass representation. Therefore, the optimal lumped mass matrices are expressed through the adoption of a variable rotational inertia parameter to counterbalance the decreased or increased rotational inertia. The goal of this study is to propose lumped mass matrices to minimize the modal error for beam elements. The accuracy of the new mass matrices is validated by a number of numerical tests.