In the current study, the nonlinear free and forced vibration of functionally graded strain gradient Timoshenko microbeam are analytically investigated. The nonlinear equations governing the motion of the microbeam and the corresponded boundary conditions are derived using variational approach. A novel analytical approach is proposed to derive closed-form expressions for the mode shapes of the system. The analytical mode shapes are then closely verified via numerical simulations and the effect of different parameters on the fundamental natural frequency of the beam is studied. Multiple time scales (MTS) perturbation technique is employed to model nonlinear free vibration of the system based on a single mode approximation. The accuracy of the time history results are confirmed by numerical calculations. The nonlinear forced vibration of the Timoshenko beam is also simulated using MTS. More specifically, the primary resonance and the frequency responses of the system are derived analytically and the effect of different design parameters on their behavior is studied in detail. The results and approaches presented in this paper can be further employed for suggesting analytical predictions in more complex micro structures.

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