Abstract: Based on the potential energy principle, a frictionless elastic contact problem quadratic program-ming model is constructed by treating the nodal displacements as the variables and the contact conditions as the constraint equation. On the basis of the model, the particular solution and the base solutions of the linear constraint equation of the force balance are used to construct the singular modal matrix, and then the freedom reduction method based on a singular coordinate transformation is proposed, which greatly reduces the dimension of the quadratic programming, and leaves the quadratic programming with no dominance equality constraint. According to the characteristics of the elastic contact mechanical system, and the contact freedom displacement mode generally assumed，a simple and eflcient singular modal matrix calculation method is pro-posed. The solution of the elastic contact problem is verified through comparison and analysis of two numerical cases of the contact problems of two cylinders and the axle hole clearance fit. The reduction of the quadratic programming method overcomes the problems of large amount of work, the sensitivity to the parameter settings and the convergence diflculties inherent in the traditional method.