A new formalism leading to closed-form formulation of equations for controlled elastic multibody systems is presented. The method is derived from the virtual work principle and includes the effects of a moving base and rigid body dynamics. The elastic members are treated as Euler-Bernoulli beams and the assumed-mode method is adopted. The equations of motion are expanded in a closed form with a minimum set of variables using the generalized coordinate partitioning and a Jacobian matrix expansion. The inertia matrix, nonlinear coupling vector, generalized force vector and other terms containing the velocity and acceleration effects of a moving base are formulated separately. The formalism facilitates matrix computations and is very suitable for symbolic processing. The method is very systematic and general and can be applied to a multibody system subject to control and constraint conditions. Derivation of the formalism is presented in part I of the article, and symbolic implementation and application of the formalism to various elastic mechanical systems are presented in part II.