Design and analyze of flexure hinges based on triply periodic minimal surface lattice

doi:10.1016/j.precisioneng.2020.12.019

Precision Engineering

Volume 68, March 2021, Pages 338-350

https://doi.org/10.1016/j.precisioneng.2020.12.019 Get rights and content

Highlights

•
The compliance characteristics of P-lattice, D-lattice, G-lattice and I-WP lattice are obtained.
•
Simplified model of single-lattice and one-dimensional parallel structure composed of several P-lattices are proposed.
•
The compliance and compliance ratio are greatly improved compared with traditional leaf flexure hinge.

Abstract

The triply periodic minimal surface lattice structure is innovatively introduced into the design of flexure hinges in this paper. Four types of triply periodic minimal surface lattices are generated by approximate mathematical expressions. The compliance characteristics of these four lattices are simulated by finite element analysis (FEA), and it is found that the primitive lattice (P-lattice) is the most suitable lattice for flexure hinges. Simplified model of single P-lattice and one-dimensional parallel structure composed of several P-lattices are proposed. Finally, the P-lattice is integrated into the beam portion of flexure hinges by Boolean operation, and this new type of flexure hinge is additively manufactured. The FEA and experimental results show that the compliance and compliance ratio of this new type of leaf flexure hinges are greatly improved.

Introduction

Flexure hinge is the mechanical connection structure that transmit force, energy and motion by elastic deformation of materials [1]. It has the characteristics of high displacement resolution, simple structure, small design space and easy for overall processing. It has been widely used in precision positioning [2,3], micromanipulators [4], microscopes [5], robots [6], vibration detection needle [7] and other high-precision engineering fields.

Mechanical property of flexure hinge is determined by its structure. Notched flexure hinge is the most widely used type of flexure hinges. Besides the conic curve notched flexure hinge [1,8], there are many other notch curves such as exponential-sine [9], corner-filleted [10], power-function-shape [11], NURBS [12] and piecewise curves [13]. The notched flexure hinge is difficult to achieve high motion accuracy and large movement stroke at the same time. Therefore, many flexure hinges of complex structures such as triple-cross-spring flexure hinges [14], two-axis elliptical notched flexure hinges [15], cartwheel flexure hinges [16], butterfly-shaped flexure hinges [17] are proposed. The more extensive application scenarios of flexure hinges are, the higher performance of flexure hinge (stroke, damping and so on) is required. Compliant mechanism with high performance can be obtained by the combination of flexure hinges [[18], [19], [20], [21]]. Although complex engineering requirements can be met, the big design space is need. The Complex structures can be introduced into flexure hinges, which can meet the same requirements in a small space. Topology optimization (TO) is an important way to solve this problem [44]. The structure obtained by TO can be manufactured easily by additive manufacturing (AM). For example, E.G. Merriam et al. [22] proposed a flexure mechanism based on truss-like lattices, which significantly improved the compliance. Z. Chen et al. [23] researched the comb-shaped flexure hinge, which can increase the motion damping and suppress the first-order bending mode. J. Pinskier [24] optimized beam structure resembled trusses by TO, which improved the compliance ratio and reduced the mass.

Lattice structure is a typical structure in AM. It is a topologically ordered and three-dimensional open cell structure composed of one or more repeating cells [25,26]. The physical response of lattice structure can be significantly changed by adjusting the parameters of the lattice structure, and the characteristics of lattice structure cannot be achieved by its original materials for the same structure [[27], [28], [29], [30]]. The advantages included high strength [31], rapid heat dissipation [32], and high damping ratio [33]. Lattice structures are usually created from truss structure or minimal surface. Truss lattice is mostly generated manually. Lattice meshes are manually generated by using of beam structures, but complex post-processing is required to connect each unit seamlessly [34]. Triply periodic minimal surface (TPMS) lattices do not require post-processing because they can be generated by mathematical formulas and are periodic in three-dimensional space [35]. Compared with truss lattices, TPMS lattices have smoother surface [34], more parametric and more controllable structure. The TPMS has become a promising lattice structure used in medical stents [36], microreactor [37] and mechanical isolation [38] and other applications.

The key to designing the lattice structure is to select appropriate lattice variables. Materials, cell types, and relative densities play crucial roles in determining structural stiffness and strength. TPMS lattice is a new lattice structure, and there are few research results of the performance analysis of TPMS. Therefore, analytical methods are needed to establish a reliable relationship between TPMS lattice geometry and performance. I. Maskery et al. [39] have used a method which combined mechanical tests and finite element analysis (FEA) to research the behavior of three TPMS lattices (Primitive, Diamond, and Gyroid) under compressive loads. L. Zhang et al. [40] researched failure behavior, fracture strength, and energy absorption capacity of the Diamond lattice by a quasi-static compression test. R. Ambu et al. [41] have used FEA method to numerically simulate a biological simulation scaffold composed of Primitive lattices (P-lattices), and analyzed its compressive load. However, the analysis of mechanical properties of the TPMS lattice is major about compressive deformation and vibration frequency. And there are few studies on the deformation caused by bending force of the TPMS lattice. Its compliance characteristic is unknown.

This paper researched the compliance of four TPMS lattices, and proved that they can be used in flexure hinges. In Section 2, TPMS lattices are generated by mathematical formula. In Section 3, the compliance of four types of lattices is obtained by FEA, and the influence of lattice parameters on the compliance of P-lattice is researched. In Section 4, a simplified compliance model of single P-lattice and one-dimensional parallel P-lattices structure are established. In Section 5, the leaf flexures containing P-lattices are designed and their compliance is obtained by FEA. In Section 6, the compliance of this new type of flexure hinge is measured. This paper is concluded in Section 7.

Section snippets

Generation of TPMS lattice

“Minimal surface” is a surface where the average curvature of every point on the surface is zero. The average curvature is average of the principal curvatures. Principal curvatures are the maximum curvature and the minimum curvature of a point on the surface. “Triply periodic” means that its shape is periodic in three independent directions.

There are two main methods to represent minimal surfaces. One method is to calculate the coordinates of points on the surface accurately by Weierstrass

FEA of single lattice compliance

After the TPMS lattice solid structure was generated, compliance of the lattice was analyzed by using ABAQUS/Standard FEA solver. The bottom of the lattice was fixed, and the top was subjected to shear stress and normal stress, as shown in Fig. 5. The defined load on the lattice is represented by Equation (5), the corresponding deformation is represented by Equation (6), and the compliance of the linear displacement is represented by Equation (7). $F = {[F_{x}, F_{y}, F_{z}]}^{'}$ $P = {[Δ_{x}, Δ_{y}, Δ_{z}]}^{'}$ ${\begin{cases} C_{x} = Δ_{x} / F_{x} \\ C_{y} = Δ_{y} / F_{y} \\ C_{z} = Δ_{z} / \end{cases}$

Simplified model for compliance of single lattice

Due to good compliance characteristic of the P-lattice, it is necessary to propose a simplified model to analysis relationship between structure parameters and compliance. The P-lattice can be approximated by a truss lattice composed of three orthogonal cylinders, as shown in Fig. 13.

The size of the bottom surface of cylinder is determined by the following formula $π {r_{1}}^{2} = a S_{0}$ where $r_{1}$ is radius of the bottom of the cylinder, and $S_{0}$ is the area of the end surface of P-lattice. Because the area of the

Design and FEA of flexure hinges

The dimensions of the leaf flexures are shown in Fig. 18. The x axis is direction of the length of the beam, the y axis is direction of the thickness of the beam, and the z axis is direction of the width of the beam. Due to the structure of the leaf flexures is simple, the lattice structure is integrated into the flexure hinge using the method of Boolean operation. The process is shown in Fig. 19 and completed by Solidworks.

Four flexure hinges with P-lattices were designed. Lattice parameters

Compliance measurement experiment of new flexure hinges

The device of the compliance measurement experiment is shown in Fig. 21. One end of the flexure hinge is fixed on the connector with a screw, and the other end was applied with a force by hanging weights, and the deformation is measured with the optoNCDT2300-50 laser sensor.

During the experiment, the position of the robot arm with the point laser sensor is adjusted firstly to ensure the laser is perpendicular to the experimental platform. The light spot was adjusted to shine at a stable

Conclusions

The FEA is used to obtain the compliance of four lattices, and the compliance of different lattices differs greatly. The G-lattice and D-lattices deforms nonuniformly when withstanding shear force and stress. The I-WP lattices have less compliance and compliance ratio than P-lattices. Therefore, P-lattice is selected to design flexure hinges. Lattice parameters include relative density and spatial frequency. They control geometry of the lattice, and then affect compliance of lattices. The

Data availability statements

The data that supports the findings of this study are available within the article.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China under Grant 51675136, National Research Council of Science and Technology Major Project under Grant 2017ZX02101006-005 and the Heilongjiang National Science Foundation under Grant E2017032.

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View full text

Design and analyze of flexure hinges based on triply periodic minimal surface lattice

Highlights

Abstract

Introduction

Section snippets

Generation of TPMS lattice

FEA of single lattice compliance

Simplified model for compliance of single lattice

Design and FEA of flexure hinges

Compliance measurement experiment of new flexure hinges

Conclusions

Data availability statements

Declaration of competing interest

Acknowledgments

Precis Eng

Mech Mach Theor

Precis Eng

Int J Mech Sci

Mech Mach Theor

Precis Eng

Precis Eng

Precis Eng

Acta Biomater

Curr Opin Solid State Mater Sci

Scripta Mater

J Mech Phys Solid

Comput Methods Appl Mech Eng

Comput Methods Appl Math

Int J Hydrogen Energy

Polymer

J Eur Ceram Soc

Med Image Anal

Chem Phys Lett

Mech Mach Theor

How to design flexure hinge

Mach Des

A compliant parallel xy micromotion stage with complete kinematic decoupling

IEEE Trans Autom Sci Eng