In the present work, based on strain gradient theory, the free vibration analysis of tapered viscoelastic micro-rod resting on viscoPasternak foundation is investigated. The material properties of micro-rod are assumed the visco-elastic and modeled as the Kelvin-Voigt. Using Hamilton's principle and energy method, the governing equation of motion of viscoelastic micro-rods is derived, then this obtained equation using the differential quadrature method (DQM) for different boundary conditions is solved. In this study, the effects of various parameters such as the structural damping coefficient, Winkler and Pasternak foundation modulli, damping coefficient of the elastic medium and material length scale parameters on the non-dimensional natural frequencies of viscoelastic micro-rod are investigated. The results show that with an increase in the Winkler and Pasternak coefficients, the natural frequency increases as well as the obtained nondimensional natural frequencies by MCST and SGT decrease by increasing the material length to radius ratio. It can be seen that the nondimensional frequency for SGT is higher than that of the other theories. It is shown that the non-dimensional frequencies increase by increasing the damping coefficient for all theories. Moreover, at the specified value of damping coefficient of the elastic medium, the variation of non-dimensional natural frequency is approximately smooth.

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