This paper presents a general method to obtain the mathematical model for the elasto-dynamic behavior of the multi-body branched systems developed in the welcoming frame of the Computational Mechanics. This particular class of multi-body systems generally named branched systems refers to the mechanical systems with a number of branches activated and controlled by the driving links of a common driver, taking into consideration the flexibility of the elements (the "rigid body system" assumption does not hold). The method allows us to consider the essentially nonlinear position functions of the driver mechanisms (one in each branch) in their real form, without any linearization. So, the achieved mathematical model is a system of second order nonlinear differential equations with non-constant coefficients. As the mathematical model is achieved in a symbolic form, it serves all the targets as an analytic one, but it is obtained for any number of branches (and, by consequence, for any number of degrees of freedom).
