A Graph Based Design Methodology for Compliant Mechanisms (Non Linear Springs) to More Fully Explore and Exploit the Design Domain
Authors: Rehman Tanzeel ur , Qaiser Zeeshan , Ou Haihua , Yi Haiping , Johnson Shane 
Published: January 2022

Abstract
Nonlinear springs are compliant mechanisms that may provide desired force versus displacement relations that give rise to improved energy storage, and improved accuracy in position control beneficial for applications in aerospace, automobile, MEMS, robotics, etc. Compliant mechanisms may be designed by 1) topology optimization, 2) shape optimization and 3) path optimization techniques. Some of these techniques may inherently result in exhaustive computations, stress intensities or design space inefficiencies. Path optimization techniques help in avoiding stress intensities. In this paper a new design approach that uses 1) paths from a graph based method to avoid stress intensities 2) explores these paths to reduce the computational cost and 3) exploitation with optimization of a down-selected explored path to fully utilize the design space is proposed. After proposing the design method a case study of a constant force spring (CFS) is presented in the paper where all the paths of a 9 node grid are explored using modified depth-first search M-DFS algorithm; these paths are then analyzed to show path dependencies to narrow down the paths to parent paths. A parent path represents various paths that are either 1) symmetric in geometry and loading or 2) has nodal positions that can be obtained by perturbing the nodal positions of the parent paths. Amongst the parent paths, one down selected parent path is exploited through optimization where the section properties and the nodal location are perturbed within the limits of the design space in the grid to fully utilize the design space. The optimization utilizes FEA of a 2D beam structure to analyze the structural configuration described by perturbed parent paths. Strength-based failure and interference analysis of the structure is analyzed using FEA. For the case study, the objective was to maximize the percentage of the force versus displacement relationship with constant force of the CFS. The optimized path is then validated experimentally and compared with a similar mechanism in the literature. It is observed that the proposed method results in a CFS with higher initial stiffness (2.18 times stiffer as compared to literature), a higher percentage of constant force displacement (77.6% as compared to 75%), and similar maximum displacement (83% of maximum displacement in literature) within the design space. The proposed method is also found to reduce the number of design variables required when compared with methods in the literature for path optimization resulting in a reduced computational cost design approach. The M-DFS algorithm in the proposed method defines all the paths in the grid structure of the domain, these paths are reduced during the exploration of the domain by 50 percent paths using symmetry, and in this case study; the perturbation of nodes reduced the paths further by more than 95 percent paths. This results in the exploitation of 2 percent remaining paths, such a method helps in reducing the computational cost of design that utilizes the complete domain. The proposed design method can be used to design various nonlinear springs with computation and design space efficiency while avoiding stress concentrations.