Research on Theoretical Basis of Design Optimization of Flexible Hinges and Flexible Body Mechanisms1

Abstract: This research focuses on studying the flexibility matrix of straight beam rounded flexure hinges. The analytical calculation method for in-plane deformation of the hinge is derived based on the cantilever beam theory. The closed-loop analytical model for the flexibility matrix is established, and a simplified calculation formula for the flexibility matrix is provided when considering the corner radius and hinge thickness. Additionally, a finite element model of the hinge is developed to verify the accuracy of the analytical model. The relative error between the analytical and simulation values of the flexibility matrix parameters is analyzed for different hinge structure parameters. The results demonstrate that the analytical model is accurate, and the relative errors can be controlled within acceptable limits.

Flexible hinges are widely used in precision devices due to their advantages of high motion resolution, no friction, and simple manufacturing process. These hinges rely on their own elastic deformation to transmit or convert motion, force, or energy, eliminating the need for rigid components. The key parameters of a flexible hinge directly affect its dynamic characteristics and end positioning accuracy. Previous research has focused on different types of flexible hinges, but limited studies have been conducted on straight beam rounded flexure hinges. This paper aims to fill this research gap by studying the flexibility matrix of such hinges.

1. Flexibility matrix of straight beam rounded flexible hinges:

The straight beam rounded flexible hinge is a sheet structure with rounded corners to avoid stress concentration. The hinge's geometric parameters include height, length, thickness, and fillet radius. A closed-loop analytical model for the flexibility matrix of the hinge is established based on the derived analytical calculation method for in-plane deformation. The flexibility matrix parameters are analyzed for different hinge structure parameters, and the relative error between the analytical and simulation values is calculated.

2. Finite element verification of the flexibility matrix:

To validate the accuracy of the analytical model, a finite element model of the hinge is created using UGNX NASTRAN software. The simulation results of the hinge loaded with unit force/moment are compared with the analytical values. The relative error between the analytical and simulation values of the flexibility matrix parameters is analyzed for different ratios of hinge length to thickness (l/t) and corner radius to thickness (r/t).

2.1 Effect of l/t on flexibility matrix parameters:

The relative error between the analytical and simulation values of the flexibility matrix parameters is found to be within 5.5% when the ratio l/t is greater than or equal to 4. For ratios less than 4, the relative error increases significantly due to the limitations of the slender beam assumption. Therefore, the closed-loop analytical model is suitable for hinges with larger l/t ratios.

2.2 Effect of r/t on flexibility matrix parameters:

The relative error between the analytical and simulation values of the flexibility matrix parameters increases with the increase in the ratio r/t. For ratios between 0.1 and 0.5, the relative error can be controlled within 9%. For ratios between 0.2 and 0.3, the relative error can be controlled within 6.5%.

2.3 Effect of r/t on simplified flexibility matrix parameters:

Simplified analytical formulas for the flexibility matrix parameters are provided considering the ratio r/t. The relative error between the simplified analytical values and the simulation values increases with an increase in the ratio r/t. For ratios between 0.3 and 0.2, the relative error can be controlled within 9% and 7%, respectively.

The developed closed-loop analytical model of the flexibility matrix for straight beam rounded flexure hinges provides a theoretical basis for the design and optimization of flexible hinges and mechanisms. The accuracy of the model is validated through finite element simulations, and the relative errors are within acceptable limits for different hinge structure parameters. This research contributes to the understanding and application of straight beam rounded flexure hinges in various precision devices.