Flexural-flexural internal resonances 3:1 in initially straight, extensible Timoshenko beams with an axial spring

This work investigates the dynamics of a hinged–simply supported beam with an axial spring in the neighbourhood of a 3:1 internal resonance between two successive transversal vibration modes. Multiple time scale method is applied to the analytical model of the planar extensible-shearable beam with associated boundary conditions, and a second order approximate solution is obtained. The phenomena that we investigate occur on the resonant branch of the frequency response curve around the main natural frequency, and consists of a localized extra (bent) peak on that branch. Also some detached branches have been theoretically detected in the neighbourhood of this extra peak. The analytical frequency response curves are analysed extensively and have been compared with finite element numerical simulations, finding an overall good agreement. Stability of the analytical solutions is checked by computing the eigenvalues of the Jacobian matrix and looking at the sign of their real part. Strain energies due to flexural, axial, shear deformability and axial spring are determined and compared between each other to ascertain their relative importance.