The main purpose of this work is to study the motion of 2-DOF of an
auto-parametric dynamical system attached with a damped system. The governing
equations of motion are gained utilizing Lagrange’s equations in terms of the
generalized coordinates. The method of multiple scales (MS) is used to obtain the
solutions of the governing equations up to the third order of approximation. The
primary external resonance simultaneously with the internal one are investigated
to establish the solvability conditions and the modulation equations. The graphical
representations of the time histories together with the amplitude and phases of the
dynamical system are represented in some plots to describe the motion of the system
at any instance. The stability of the solution has been made with use of Mathematica.
