---

Nonlinear vibrations analysis of beams has its own specific importance in industrial, building and civil engineering applications. To achieve a propr design, the understanding of transverse vibrations of double end jointed beams and their nature frequencies, are very useful. In this paper the transverse vibrations analysis of beams with influence of constant axial load are investigated.the effects of mid-plane stretching and nonlinear damping terms in two case of primary and secondary resonance are also addressed. Multiple time scales method is used to make this contribution. In order to verify the accuracy of this method, the results are compared with the results of four order Runge-Kutta numerical method, which has a good accuracy. Comparison of this method and the homotopy method shoes that the convergence of this method is faster than the homotopy method.

---

In this paper, an atomic force microscope is modeled based on non-classical nonlocal theory and nonlinear vibration of the system is analyzed and controlled. In this modeling, the Hamilton principle is used to derive the governing equation of Euler-Bernoulli nanocantilever based on the Eringen nonlocal elasticity theory considering Von-Karman geometric non-linearity. In the next step, using the Galerkin method, the governing dynamics differential equation of the atomic force microscope is obtained in the presence of attractive and repulsive van der Waals forces. The governing nonlinear equation is solved by employing multiple time scales method, and primary and secondary resonance of the atomic force microscope is studied. In this regard, the frequency response and excitation amplitude curves of primary, superharmonic and subharmonic resonances are plotted for different values ​​of the nonlocal parameter. Accordingly, it is shown that primary, superharmonic and subharmonic resonances of atomic force microscope are significantly affected by the nonlocal parameter. The results show that the use of nonlocal theory is a fundamental necessity for analyzing nonlinear vibrations of the atomic force microscope. Then, in addition to dynamic analysis, the chaotic vibrations are completely controlled and removed in the nonlocal model of the atomic force microscope by designing and implementing the robust adaptive fuzzy controller. For this task, the robust adaptive fuzzy controller which is considered as a powerful method of chaos controlling is used in the nonlocal model of atomic force microscope. The obtained results are used in the design and control process of the atomic force microscope.