Free vibrations of a strain gradient beam by the method of initial values

We extend the application of the method of initial values (also known as the transfer matrix method) to find frequencies of free vibrations of a strain-gradient-dependent Euler–Bernoulli beam (EBB) under different boundary conditions at the two end faces of the beam. For the classical EBB, we find the exact matricant or the carry-over matrix but it is numerically evaluated for the strain-gradient-dependent EBB. For the numerically evaluated matricant, it is found that ten iterations give converged values of the first six frequencies for the classical and the strain-gradient-dependent EBB. For the strain-gradient EBB, the sixth-order ordinary differential equation for the lateral deflection and three boundary conditions at each end have been derived by using the Hamilton principle. The material characteristic length is found to noticeably affect frequencies of free vibrations. Thus, the difference between frequencies of the classical and the strain-gradient-dependent EBB can be used to determine the value of the material characteristic length for a nanobeam for which length scale effects are believed to be dominant.