Employing the nonlocal theory can be helpful to model contact shear forces besides long-range interactions in the set of interconnected dampers. Nonlocal damping is also beneficial to optimize the damping capacity in large-scale structures. Numerous researchers have considered viscous damping to simplify studying the dissipation behavior of the systems. However, non-viscous damping models depend on more parameters and show better agreement with experimental results and the responses of the real structures. In the recent study, the external damping at any point of AFG Euler Bernoulli beam with one end fixed is influenced by the past history of the velocity and long-range forces. Employing the Galerkin method and after applying Laplace transformation, the integro-partial differential equation of the beam turns into an ordinary differential equation. Now, the distributed system is equivalent to a discrete system with finite degrees of freedom. Undamped mode shapes of the homogenous beam are selected as suitable admissible functions to expand the corresponding trial solution. These admissible functions satisfy essential boundary conditions as a concern. Subsequently, mass, stiffness, and damping matrices are determined with respect to the generalized coordinates by minimizing the weighted residual. Afterward, the determinant of the dynamic stiffness matrix is set equal to zero. Consequently, the elastic and non-viscous eigenvalues are obtained. The physical and mechanical characteristics of the material are assumed to vary continuously in a gradual manner along the longitudinal axis of the beam according to the power-law gradient assumption. Recently, AFG materials are used widely in the aerospace and automotive industry due to the promising mechanical and thermal advantages over traditional homogenous materials and layered composite systems. The numerical results of the recent study are compared to the special cases to confirm the needed accuracy of the method.