Design of nonlinear springs for prescribed load-displacement functions

A nonlinear spring has a defined nonlinear load-displacement function, which is also equivalent to its strain energy absorption rate. Various applications benefit from nonlinear springs, including prosthetics and microelectromechanical system devices. Since each nonlinear spring application requires a unique load-displacement function, spring configurations must be custom designed, and no generalized design methodology exists. In this paper, we present a generalized nonlinear spring synthesis methodology that (i) synthesizes a spring for any prescribed nonlinear load-displacement function and (ii) generates designs having distributed compliance. We introduce a design parametrization that is conducive to geometric nonlinearities, enabling individual beam segments to vary their effective stiffness as the spring deforms. Key features of our method include (i) a branching network of compliant beams used for topology synthesis rather than a ground structure or a continuum model based design parametrization, (ii) curved beams without sudden changes in cross section, offering a more even stress distribution, and (iii) boundary conditions that impose both axial and bending loads on the compliant members and enable large rotations while minimizing bending stresses. To generate nonlinear spring designs, the design parametrization is implemented into a genetic algorithm, and the objective function evaluates spring designs based on the prescribed load-displacement function. The designs are analyzed using nonlinear finite element analysis. Three nonlinear spring examples are presented. Each has a unique prescribed load-displacement function, including a (i) "J-shaped," (ii) "S-shaped," and (iii) constant-force function. A fourth example reveals the methodology's versatility by generating a large displacement linear spring. The results demonstrate the effectiveness of this generalized synthesis methodology for designing nonlinear springs for any given load-displacement function.