\r\naccounts for nonlocal stress field, strain gradient field and higher

\r\norder inertia force field, is derived based on the nonlocal strain

\r\ngradient theory considering velocity gradient effect. The governing

\r\nequations and boundary conditions are derived both in dimensional

\r\nand dimensionless form by employed the Hamilton principle. The

\r\nanalytical solutions based on different continuum theories are

\r\ncompared. The effect of higher order inertia terms is extremely

\r\nsignificant in high frequency range. It is found that there exists

\r\nan asymptotic frequency for the proposed beam model, while for

\r\nthe nonlocal strain gradient theory the solutions diverge. The effect

\r\nof strain gradient field in thickness direction is significant in low

\r\nfrequencies domain and it cannot be neglected when the material

\r\nstrain length scale parameter is considerable with beam thickness.

\r\nThe influence of each of three size effect parameters on the natural

\r\nfrequencies are investigated. The natural frequencies increase with

\r\nthe increasing material strain gradient length scale parameter or

\r\ndecreasing velocity gradient length scale parameter and nonlocal

\r\nparameter.", "references": null, "publisher": "World Academy of Science, Engineering and Technology", "index": "International Science Index 123, 2017" }