This paper concerns about modeling of tethered satellites systems (TSS). In this paper, tethers of TSS are modeled as chains of elastic rods connected by rigid joints. A group of spatial vectors are used to describe displacement, orientation and deformation of the rods, which can avoid the coupling of trigonometric functions of Euler angles. The coordinates of spatial vectors are chosen as the generalized coordinates of system. Equations of motion are derived and written in explicit form. A computationally efficient algorithm is developed to solve equations of motion. Computational complexity of the developed algorithm is reduced to O( n) ( n denotes the number of rods). A numerical experiment is performed. Results about energy show energy conservation principle is obeyed. The same numerical experiment is also performed utilizing multibody software ADAMS. Validity of the presented algorithm is demonstrated by comparing numerical results of the two methods.