Purpose: The motion of three degrees-of-freedom (DOF) of an automatic parametric pendulum attached with a damped system has been investigated. The kinematics equations of this system have been derived employing Lagrange’s equations in accordance to it’s the generalized coordinates. Methods: The method of multiple scales (MMS) has been used to obtain the solutions of the controlling equations up to the third-order of approximation. The solvability criteria and modulation equations for primary external resonance have been explored simultaneously. Results: The non-linear stability approach has been used to analyze the stability of the considered system according to its different parameters. Time histories of the amplitudes and the phases of this system have been graphed to characterize the motion of the system at any given occurrence. Conclusions: The different zones of stability and instability of this study have been checked and examined, in which the system's behavior has been revealed to be stable for various values of its variables.

In this article, we solve the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) using an accelerated form of the Adomian decomposition method (ADM). Based on $(\alpha ,\delta ,\beta )$-cut, we convert the NVPFIE to the nonlinear Volterra integral equations in a crisp form. An accelerated version of the ADM is used to solve this transformed system, which is based on a new formula for the Adomian polynomial. The sufficient condition that guarantees a unique solution is obtained using this new Adomian polynomial, error estimates are given, and the convergence of the series solution is proven. Numerical cases are discussed to illustrate the effectiveness of this approach.