In this paper a nonlinear dynamic model for drill string in inclined well drilling is developed. Effects of drilling mud flow rate, weight on bit, angular velocity along with viscous damping on stability and vibration of the drill string are studied. Findings indicate the nonlinear effects are significant on the results. The effects of drilling mud flow rate and weight on bit on the natural frequencies and time responses are evaluated. Enhancement of drilling mud flow rate results in decreasing of natural frequencies and vibrational amplitude, while increasing the weight on bit, leads to decrease of the natural frequencies and increase the vibrational amplitude.

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A viscoelastic microcantilever beam is analytically analyzed based on the modified strain gradient theory. The Kelvin Voigt scheme is used to model the beam viscoelasticity. Applying Bernoulli-Euler inextensibility of the centerline condition via Hamilton's principle, the nonlinear equation of motion and related boundary conditions are derived based on shortening effect theory and discretized by Galerkin method. Inner damping, nonlinear curvature effect, and nonlinear inertia terms are applied. The generalized derived formulation in this article, allows modeling of any nonlinearity combinations such as nonlinear terms arises due to inertia, damping, and stiffness, as well as modeling the size effect via considering modified coupled stress or modified strain gradient theories. First mode nonlinear frequency and time response of the viscoelastic microcantilever beam are analytically evaluated utilizing multiple time scale method and validated by numerical findings. Results indicate that the nonlinear terms have an appreciable effect on natural frequency and time response of a viscoelastic microcantilever. Furthermore, the investigation reveals that due to the size effects, natural frequency enhances drastically, especially when the thickness of the beam and the length scale parameter are comparable. Outcomes clarify the importance of size effects in analyzing of the mechanical behavior of small scale structures.