Advanced materials are widely used in machine mechanisms, along with the requirement that the mechanical structure of the mechanism must be compact and slim to reduce inertia, save energy and increase working speed. However, this leads to elastic deformation affecting movement, accuracy and causing delays in sequential operations. In this paper, a six-bar linkage planar mechanism is combined from two four-bar linkage mechanisms. In which the connecting rods are considered as general elastic elements that are both compressive and bending. By representing the elastic rods by the modal synthesis technology, combined with the use of multi-body dynamics methods, which are the multiplier Lagrange equations, the paper has established a nonlinear differential-algebraic system of motion. From this system of equations, the study has numerically investigated some cases of forward dynamics to make assessments and comments on the influence of elastic deformation on the motion, the accuracy of the working position of the mechanism, and the significance of numerical investigation in the design of mechanical structures. This dynamic model can be used for other problems such as periodic oscillation of elastic mechanisms, linearization and investigation of motion stability, or control of motion and elastic oscillation.