In the field of biology and medicine, for example, in the use of microelectromechanical systems (MEMS) with microbeam structure in human implantation, the accuracy and stability of the equipment and instruments are greatly affected by the elastic foundation because the cellular environment in the body is similar to that of hydrogels. In order to analyze this problem, a beam dynamics model based on nonlinear beam is established in this paper, and the free vibration of a beam with a spring at any position and based on nonlinear spring is studied. By means of Laplace transform and linear superposition principle, the constrained Green’s function is obtained. Numerical calculation is carried out to verify the proposed solution, and the influence of various important physical parameters is studied. It is found that the position of the spring moves towards the midspan and the modal symmetry is broken. The spring stiffness increases as the modal order changes.