A dynamic-stiffness approach for damped locally-resonant Timoshenko beams

A dynamic-stiffness matrix approach is presented for locally-resonant Timoshenko beams with a periodic array of viscously-damped multi-degree-of-freedom resonators. First, an exact dynamic condensation of the resonator degrees of freedom is pursued, expressing the resonator reaction forces in terms of the deflections of the attachment points via pertinent frequency-dependent stiffness terms. On this basis, a direct integration deriving from the theory of generalized functions provides the exact dynamic-stiffness matrix of the beam, whose size is 4 × 4 for any the number of resonators and degrees of freedom within the resonators. The dynamic-stiffness matrix is used to calculate the complex eigenvalues of the beam by a recently introduced contour-integral algorithm, without missing anyone. Further, it provides the transmittance for an insight into the elastic wave attenuation properties of the beam.