Deflection and potential energy of linear and nonlinear springs: approximate expressions in terms of generalized coordinates

The method for finding approximate expressions for the deflection and the potential energy of linear and certain nonlinear springs is presented. These expressions are given in terms of the constant static deflection, which can be zero or non-zero, and the deflection considered in two mutually orthogonal directions and the corresponding deflections, which need to be truncated to a certain order of the generalized coordinate. An example is given to demonstrate their use and benefits. These approximations can be useful for teaching courses on vibration. In addition, it is expected that they can be utilized by students and researchers to define the potential energy of springs as well as to derive the equations of motion more easily.