Vibration analysis of nonlocal beams using higher-order theory and comparison with classical models.

New higher-order models are developed for plane rods and beams based on the linear theory of nonlocal elasticity. The one-dimensional higher-order theory is based on two-dimensional equations of the nonlocal theory of elasticity and expansion of the equations of the nonlocal theory of elasticity into a Fourier series of Legendre polynomials in a thickness coordinate. The higher-order models developed are then used in the analysis of the tension-compression and transverse bending modes of nonlocal rod and beam vibration. The equation of motion for each mode is analyzed separately. Free and forced vibration are analyzed using the eigenfunction expansion and Navier's analytic closed-form solution. An analysis and comparison with well-known theories is performed using computer algebra system MATHEMATICA. The proposed models consider nonlocal effects and can be used for vibration analysis of rods and beams at macroscales, microscales, and nanoscales.