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Compliant mechanisms have been well designed to reach an ultra-high accuracy in positioning systems. However, the displacement of compliant mechanisms is still a major problem that restricts practical applications. Hence, a new flexure hinge array (FHA) is proposed to improve its displacement in this article. This paper is aimed to design and optimize the FHA. The structure of FHA is constructed by series-parallel array. Analytical calculations of the FHA are derived so as to analyze the stiffness and deformation. The displacement of the FHA is optimized by moth-flame optimization algorithm. The results determined that optimal parameters are found at _{t1} of 20.58 mm,

Compliant mechanism (CM) is a monolithic structure; its motion is a cause of elastic deformations of flexure hinge [

LEMs also have advantages similar to CMs but it has a motion moving from fabrication plane [

In order to analyze the stiffness and displacement of LETs, several analytical methods are proposed, including integral method, pseudorigid-body model [

The goal of this article is to design and optimize a new FHA. The structure of FHA is constructed by using a leaf hinge array in series-parallel connection. Analytical models are formulated to analyze the stiffness and displacement of the FHA, and then, the displacement of the FHA is optimized by using moth-flame optimization algorithm [

The rest of this paper is arranged as follows: Section

A conceptual design of the FHA is built by arranging three LET hinges in a series to bear a single load in a direction or combined loads in multiple directions. A LET hinge can recognize bending elements and torsional elements. In order to create the FHA, it initializes from a basic LET hinge which consists of four types of torsional elements 1, four types of torsional elements 2, and two types of bending elements 1, as illustrated in Figure

A basic LET: (a) 3D structure; (b) 2D dimensions.

Dimensions of the FHA (unit: mm).

Symbol | _{t1} | _{t2} | _{b1} | _{b2} | _{b3} | ||||
---|---|---|---|---|---|---|---|---|---|

Value | Variable | 23 | 13 | 13 | 10 | 2 | 2 | 8 | Variable |

Symbol | _{1} | _{2} | _{3} | ||||||

Value | 407 | 97 | 1 | 65 | 20 | 40 | 30 | 137 | Variable |

In order to generate a large displacement, three basic LETs are connected in series to make an array of flexure hinges, so-called FHA. Three TLET flexure hinges are combined through a bending element 3, as illustrated in Figure _{t1},

Model of the proposed flexure hinge array.

Analytical equations of the FHA are established to demonstrate the stiffness and displacement. From scheme of basic LETs (see in Figure

Stiffness plot of basic LET.

Figure

Stiffness plot of basic LET.

The stiffness of the torsional segment is determined by

Two springs of bending element 1 are in a parallel system. The equivalent stiffness of these two springs is computed as_{eqb1} is the equivalent stiffness of the parallel system, and _{b1} is the stiffness of bending element 1.

The equivalent stiffness of bending element 2 is as follows:

where _{eqb2} and _{b2} are the equivalent stiffness and stiffness of bending element 2, respectively.

The bending stiffness of a basic LET (Figure _{b1}), (_{b2}), and (_{b3}) are the width and length of bending element 1, bending element 2, and bending element 3, respectively.

The stiffness of torsion element 1 and torsion element 2 is as follows [_{t1}) and (_{t2}) are the width and thickness of segment, respectively.

As the aforementioned design (see in Figure _{b3} (see in Figure _{I}, one of the second system II is _{II}, and one of the third one is _{III} (see in Figure

Stiffness diagram of the flexure hinge array.

The stiffness of springs system (I) is similar to the equivalent stiffness of torsional elements and yielded as_{I} represents the equivalent stiffness of the springs systems (I).

Overall stiffness of the flexure hinge array is determined as_{eq-FHA} is the stiffness of the flexure hinge array.

Displacement (Δ) versus load (

The purpose of deformation analysis is to consider the maximum range of displacement and load in which the FHA can withstand to ensure reliability in operation.

Stainless steel, ABS, PE, and AL are potential materials of the FHA (see in Table

Results of displacement of the flexure hinge array.

Material | Yield strength, _{l} (MPa) | Young’s modulus (MPa) | Max force (N) | Max stress, _{m} (MPa) | Max displacement (mm) | Comparison |
---|---|---|---|---|---|---|

Stainless steel | 207 | 193000 | 10 | 193.29 | 1.84 | _{l} > _{m} |

ABS | 43.6 | 2300 | 2 | 38.5 | 30.85 | _{l} > _{m} |

PE | 25 | 1100 | 1 | 19.33 | 32.23 | _{l} > _{m} |

Al | 503 | 71700 | 25 | 481.81 | 12.39 | _{l} > _{m} |

Meshed model of flexure hinge array.

By using the FHA made of stainless steel, it can supper a maximum force of 10 N in the

The FHA with almost materials is in safety area of elastic limitation. However, the FHA with aluminum permits the best displacement with a max force of 25 N. When increasing the force over 30 N, the FHA is over elastic area of material due to the max stress (_{m} of 578, 18 MPa) over a limited stress (_{l} of 503 MPa).

It is noted that if the _{t1},

Sensitivity plot of design variables to the displacement.

Recently, a metaheuristic algorithm is often adopted to enhance the quality or specification of a product. For instance, particle swarm optimization (PSO) and genetic algorithm (GA) were combined with artificial neural network to predict the pile bearing capacity [

The MFO. (a) Flowchart; (b) pseudocode.

The optimization process is programmed in MATLAB R2019b; the optimal parameters are found at _{t1} of 20.58 mm,

A comparison of the MFO with common algorithms (GA, PSO, and DE) is conducted by a nonparameter statistical method, Kruskal–Wallis test [

Kruskal–Wallis test: MFO versus other algorithms.

Algorithms | Number of runs | Median displacement | Mean rank | |
---|---|---|---|---|

DE | 35 | 26.89 | 53.0 | −2.95 |

GA | 35 | 26.85 | 18.0 | −8.84 |

MFO | 35 | 27.02 | 123.0 | 8.84 |

PSO | 35 | 26.92 | 88.0 | 2.95 |

Overall | 140 | 70.5 | ||

130.32 | ||||

<0.001 |

Through aforementioned simulations and analyses, Al material is suitable for the proposed FHA. A prototype of basic LET and a prototype of FHA are made by Al, as given in Figure

Prototype: (a) LET; (b) flexure hinge array.

A force from a force gauge is acted to free end of the FHA; the FHA moves in the horizontal direction. At the same time, the FHA emerges the

Deformation of flexure hinge array in the

Deformation of flexure hinge array in the

In this section, validity for theoretical models is verified by simulations and experiment. A prototype of the FHA is made of Al material. Boundary conditions include a fixed end and a free end is acted by a load of 25 N. The errors between the computation and simulation (_{c}) and errors among the theoretical computation and the experiment (_{e}) are calculated.

As given in Table _{c} is about 5.66% while _{e} is around 7.11%. It verifies that the analytical models are relatively good.

Comparison results.

Force (N) | 5 | 10 | 15 | 20 | 25 |

Δ-theory (mm) | 2.61 | 5.23 | 7.85 | 10.47 | 13.10 |

Δ-simulation (mm) | 2.47 | 4.95 | 7.43 | 9.91 | 12.39 |

Δ-experiment (mm) | 2.81 | 5.64 | 8.47 | 11.29 | 14.12 |

_{c} (%) | 5.66 | ||||

_{e} (%) | 7.11 |

In conforming the behaviors of the FHA, another LET_{C} (common LET hinge) with its shape in Figure

A LET_{C}: (a) 2D CAD; (b) prototype.

The experimental results are then treated as the normalized values for comparison among different types of hinges. Normalization is determined by_{i}, _{min}, and _{max} are the

Among the three types, the displacement of the FHA is the highest (see in Figure

Comparison of displacement.

Figure

Diagram of

A relationship of driving force in

Relationship of driving force in

Buckling is a popularly unstable failure of a rigid structure. The structures are designed to avoid this phenomenon. In this study, the structure FHA is desired to reach a large displacement in the

On the contrary, simplified equation is as follows:

A range of loads from 5 N to 25 N is applied to the FHA in the

Comparison of buckling behavior.

This article presented a new design of flexure hinge array. In this paper, a few TLET hinges were connected in series to create a new FHA. The proposed FHA was capable of providing a large displacement in the desired motion directions. The theoretical equations for understanding the stiffness and displacement of the proposed FHA were deduced. Furthermore, the static modeling of the FHA was given. In order to enhance the displacement of the FHA, the MFO was applied to search the best parameters for designing the FHA.

The results found that the optimal displacement is about 27.02 mm along the _{t1} of 20.58 mm,

In selecting a suitable material, the Al material was chosen for fabricating the FHA in comparison with stainless steel, ABS, and PE. Furthermore, the result determined that the FHA can reach a large displacement of 12 mm with a max force of 25 N. Additionally, the displacement and buckling value of FHA were better than those of other hinges. It also found that the FHA can achieve the

Data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 107.01-2019.14.