Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

The present study theoretically investigates the free vibration problem of a discrete granular system. This can be considered as simple model to rigorously effects microstructure on dynamic behavior equivalent continuum structural model. consists uniform grains confined by elastic interactions, take into account lateral contributions. repetitive system referred Cosserat chain or lattice with shear interaction. First for simply supported beam resting Winkler foundations, due critical frequencies which concern nature results, natural are exactly calculated, starting from resolution linear difference eigenvalue problem. such analytically calculated whatever modes. It is shown that equations governed converge differential Bresse-Timoshenko foundation (also classified continuous model) an infinite number grains. A gradient constructed continualization equations. elasticity obtained polynomial rational expansion pseudo-differential operators, stemming process. Scale captured models compared those associated chain. results clarify dependency responses length ratio.