Nonlinear modeling of compliant mechanisms incorporating circular flexure hinges with finite beam elements

Modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This work shows that nonlinear behavior plays an important role in the deformation and stress analysis, which consequently impacts the design of compliant mechanisms. In this study a nonlinear higher order finite beam element based modeling approach is presented strongly reducing the computation time of nonlinear models. Planar deformation and mechanical stress of a single circular flexure hinge under a wide range of loads is modeled and computed with the proposed approach. A comparison with a 3D-nonlinear finite element model shows very good agreement and validates the beam model. It is shown that the linear and nonlinear deformation behavior of a single flexure hinge deviate marginally so that linear modeling approaches are sufficient. Furthermore a planar displacement amplification mechanism incorporating circular flexure hinges is studied by means of the same method highlighting the distinct deviation of the behavior of the geometrically nonlinear model from its linear prediction. In conclusion the nonlinear behavior at the system level can not longer be neglected. Finally, a study shows that different designs of the displacement amplification mechanism are achieved when linear or nonlinear modeling approaches are applied.     

Accurate low DOF modeling of a planar compliant mechanism with flexure hinges: the equivalent beam methodology

A methodology for accurate and efficient finite elements method (FEM) simulations of planar compliant mechanisms with flexure hinges is presented. First, using symmetry/antisymmetry boundary conditions and 3D elements, one-eighth of a single hinge is simulated to determine its true stress/stiffness characteristics. A set of fictitious beams is derived, which have the identical characteristics. This set is used in conjunction with other beams that model relatively stiff links to generate an equivalent model of an entire mechanism consisting of the beam elements only. The model has a low number of degrees-of-freedom (DOF) and appears to be more accurate than any 2D FEM models, even those with very large number of DOF. The methodology has been developed specifically for the right circular flexure hinge; however, it can be applied to all types of revolute flexure hinges.     

On the modeling of flexure hinge mechanisms with finite beam elements of variable cross section

In this paper, a modeling technique for flexure hinge mechanisms is studied. Beam elements of variable cross sections are deployed within a finite element procedure to model a circular flexure hinge. The resulting finite element model has very few degrees of freedom and is accurate in both static and dynamic analysis. Furthermore the modeling approach is applied to an amplifier mechanism. Comparing the results of the proposed model with a 3D finite element reference model, high accuracy for a broad spectrum of hinge parameters is reported while reducing the number of degrees of freedom immensely.     

Dimensionless design graphs for flexure elements and a comparison between three flexure elements

This paper presents dimensionless design graphs for three types of flexure elements, based on finite element analysis. Using these graphs as a design tool, a designer can determine the optimal geometry, based on the stiffness and rotation demands of a flexure element. An example is given using the beam flexure hinge.Between the analyzed flexure hinges, a comparison is made on basis of equal hinge functionality: rotation. The result describes the maximum stiffness properties from different hinges in identical situations. A beam flexure element is preferred over a circular flexure hinge for stiffness demands in a single direction, while a cross flexure element enables medium stiffness in two perpendicular directions.     

椭圆导角混合柔性铰链的设计计算与性能分析

为满足柔顺机构的大柔度要求,设计了一类新型椭圆导角混合柔性铰链。首先,以卡氏第二定理为基础推导了柔度和回转精度的计算公式,在参数的极限条件下,椭圆导角结构演化出其他三种铰链形式：直圆导角、椭圆直圆和直圆柔性铰链,使得多种柔性铰链的柔度和回转精度的计算公式合并在一组方程中,通过有限元分析验证了计算公式的正确性。其次,讨论了结构参数对柔度、回转精度和柔度精度比的影响趋势,分析结果表明,柔度与回转精度随参数的变化趋势具有相反性,且减小最小厚度是提高柔度的最佳方式。再次,比较了所提四种柔性铰链的性能,椭圆导角混合柔性铰链具有最大的柔度但回转精度较低,而直圆柔性铰链具有较高的回转精度且综合性能也较优越,但柔度最小。最后,对椭圆导角和直圆柔性铰链进行了应用研究,研究结果表明,椭圆导角混合柔性铰链在回转能力和应力水平方面具有显著优势。     

Two general solutions of torsional compliance for variable rectangular cross-section hinges in compliant mechanisms

Several approaches exist for calculating the torsional compliance of rectangular cross-section beams, but most depend on the relative magnitude of the cross-section thickness and width, which might be changing during the design phase (especially for design optimization) or is variant for variable cross-section beams such as circular flexure hinges and tapered bars. After summarizing current equations and analyzing their computational accuracy, two new equations are proposed, which are thickness-to-width ratio independent, and suitable for variable cross-section beams and optimization design of torsional elements in compliant mechanisms. The closed-form equations for the torsional compliance of elliptical and circular flexure hinges are derived by using the new equations.     

Design and analysis of a multi-notched flexure hinge for compliant mechanisms

This article presents a new multi-notched flexure hinge, which consists of two right circular and two parabolic notches, for positioning stages based on compliant mechanisms. First, the configuration of the presented multi-notched hinge is obtained using topology optimization, and the final shape is proposed based on post-processing. Second, the dimensionless empirical equations for the stiffness, rotational precision and stress levels of the flexure hinges are developed using finite element analysis (FEA). Third, based on the established equations, the influences of the geometric parameters on the performance of the flexure hinge are investigated. Finally, to further understand the characteristics of this type of flexure hinge, comparisons with flexure hinges of various shapes are performed in terms of stiffness, rotational precision and stress levels.     

A novel analytical model for flexure-based proportion compliant mechanisms

This paper proposes a novel analytical model for flexure-based proportion compliant mechanisms. The displacement and stiffness calculations of such flexure-based compliant mechanisms are formulated based on the principle of virtual work and pseudo rigid body model (PRBM). According to the theory and method, a set of closed-form equations are deduced in this paper, which incorporate the stiffness characteristics of each flexure hinge, together with the other geometric and material properties of the compliant mechanism. The rotation center point for a corner-filleted flexure hinge is investigated based on the finite element analysis (FEA) and PRBM. An empirical equation for the rotational angle is fitted in this paper in order to calculate accurately the position of the end-point of the flexure hinge. The displacement proportion equation for such mechanisms is derived according to the new approach. Combining the new proposed design equation and the existed stiffness equation, a new proportion compliant mechanism with corner-filleted flexure hinges is designed by means of the least squares optimization. The designed models are verified by finite element analysis.     

A generalized analytical compliance model for transversely symmetric three-segment flexure hinges

This paper presents a generalized compliance model for a three-segment notch flexure hinge with transverse symmetry. This flexure hinge configuration is most frequently employed in planar-motion, small-displacement compliant mechanisms. The axial and bending compliances are derived for this flexure hinge based on the compliances of two flexure components. The derivation is generalized such that it can be applied to various segment geometries. Using this open-ended model, a three-segment right elliptical corner-filleted flexure hinge design was analyzed. This geometric configuration introduces additional geometric parameters, which can be used to optimize the compliance of the flexure hinge without modifying its gross dimensions. The results of the analysis were validated in part by modifying the geometric parameters of the center segment and elliptical corner fillets to form limiting cases corresponding to several previously investigated configurations, namely right elliptical, three-segment right circular corner-filleted, and right circular geometries. Finite element analysis simulation and experimental testing were used to further validate the three-segment right elliptical corner-filleted analytical model. Additional simulations based on the analytical model were performed to highlight the influence of geometric parameters on compliances and to investigate shear effects for short flexure hinges.     

直梁圆角形柔性铰链的柔度矩阵分析

对直梁圆角形柔性铰链的柔度矩阵进行了研究。首先,基于悬臂梁理论推导了直梁圆角形柔性铰链平面内变形的解析计算方法,建立了柔性铰链平面内柔度矩阵的闭环解析模型,并给出了rt(r为铰链圆角半径,t为铰链厚度)时柔度矩阵的简化计算公式。然后,建立了直梁圆角形柔性铰链的有限元模型,得到了柔性铰链结构参数r/t和l/t(l为铰链长度）变化时柔度矩阵解析值和有限元仿真值的相对误差,以及r/t变化时柔度矩阵简化解析值和仿真值的相对误差。结果表明:采用悬臂梁理论建立的柔性铰链柔度矩阵模型,当l/t≥4时,柔度矩阵各项参数的理论解析值与有限元仿真值相对误差在5.5%以内,当0.1≤r/t≤0.5时,两者的相对误差能够控制在9%以内,当0.2≤r/t≤0.3时,两者的相对误差能够控制在6.5%以内;当r/t≤0.3时,简化解析值与仿真值的相对误差控制在9%以内, 当r/t≤0.2时,简化解析值与仿真值的相对误差控制在7%以内,从而验证了柔度矩阵闭环解析模型的正确性。建立的直梁圆角形柔性铰链柔度矩阵闭环解析模型可为柔性铰链以及柔性体机构的设计和优化提供理论依据。     

基于虚功原理的3-RRPR柔性精密定位工作台动力学分析

为了实现柔性精密操作系统的高频控制,提出一种新型空间柔性并联精密定位工作台系统,该工作台采用多个压电陶瓷驱动方式,通过半圆形凹槽单自由度柔性铰链的弹性变形实现末端执行件3平动自由度的主动调整.根据机构伪刚体模型,采用矢量闭环方法建立其位置、速度以及加速度方程,并结合柔性铰链弹性应变能,采用虚功原理方法进行动力学分析,推...     

Three flexure hinges for compliant mechanism designs based on dimensionless graph analysis

This paper presents the dimensionless empirical equations and graph expressions of three flexure hinges for compliant mechanism designs. The in-plane and out-of-plane stiffnesses of the flexure hinges are developed. The rotational precision, denoted by the midpoint stiffness, is derived for the purpose of optimized geometric design. Based on the developed methodologies, the influences of the geometric parameters on the performance of the flexure hinges are investigated, and the performance comparisons among the flexure hinges are conducted to further understand the characteristics of these kinds of compliant mechanisms.     

Lattice flexures: Geometries for stiffness reduction of blade flexures

Reducing the motion-direction stiffness of compliant mechanisms reduces their actuation effort and simplifies associated static balancing mechanisms. This work introduces a flexure type called lattice flexures and evaluates some of their fundamental properties. Lattice flexures have a reduced bending stiffness when compared to traditional rectangular-section blade flexures of similar size. The motion-direction bending stiffness of two lattice flexure types, called X-type and V-type, are analytically derived, corroborated with finite element analysis, and validated with measurements of physical prototypes. The lattice flexure has the potential to reduce the bending stiffness of some compliant mechanisms by 60–80%, as demonstrated in devices manufactured using 3D printing technologies. It is shown that some lattice flexures exhibit a torsional/bending stiffness ratio as much as 1.7 times higher than an equal aspect-ratio blade flexure, and a transverse bending/motion-direction bending stiffness ratio up to 6.5 times higher than an equal aspect-ratio blade flexure.     

混合型柔性铰链构型设计与柔度建模

通过将对称型柔性铰链进行分段和命名,提出一种基于柔性铰链基段和镜像段串联组合的混合型柔性铰链构型设计方法。基于卡式第二定理,将基段柔度建模转化成与其槽口形状函数有关的定积分问题。推导得到镜像段与其基段间的柔度映射和多段基段的柔度矩阵,进而构建出形式统一的两段混合型柔性铰链通用柔度和精度模型。选取圆锥曲线、矩形梁和棱台型基段类型,基于组合设计共获得96种新型混合型柔性铰链。结合具体算例和借助有限元验证理论模型的正确性,并对所设计和现有的柔性铰链构型性能统一进行评价分析。研究工作可为柔性铰链构型优选和特性评价提供理论指导。     

A novel flexure beam module with low stiffness loss in compliant mechanisms

Flexure mechanisms provide guided motion via elastic deformation of thin beams. Due to the employment of compliant elements, these mechanisms cannot sufficiently maintain acceptable constraint stiffness level in the entire range of motion. The stiffness deterioration afflicts the performance of flexure mechanisms in terms of motion range, accuracy and constraint characteristics. This paper presents a novel flexure beam module with improved constraint behavior in beam-based flexure mechanisms. The proposed module alleviates the problem of stiffness loss in large displacements and provides a better motion performance. The mathematical model governing the static behavior of the module is developed using the principle of virtual work. The geometric nonlinearity associated with large midplane stretching is taken into account. Closed-form solutions are derived for load-displacement relationships, providing a powerful design tool for the novel flexure. Also a nonlinear expression is obtained for the strain energy of the flexure in terms of end displacements. The functionality of the presented module is exploited in a multi-beam parallelogram mechanism. The constraint behavior of the parallelogram is analytically quantified and considerable improvements in stiffnesses and error motions are observed. The analytical results provided in this paper are verified via finite element simulations. The proposed novel module can be used as the building block of more complex flexures to improve their stiffness characteristics, diminish their error motions and widen their stability region.     

Topological and Shape Optimization of Flexure Hinges for Designing Compliant Mechanisms Using the Level Set Method

A flexure hinge is a major component in designing compliant mechanisms that o ers unique possibilities in a wide range of application fields in which high positioning accuracy is required. Although various flexure hinges with di erent configurations have been successively proposed, they are often designed based on designers' experiences and inspirations. This study presents a systematic method for topological optimization of flexure hinges by using the level set method. Optimization formulations are developed by considering the functional requirements and geometrical constraints of flexure hinges. The functional requirements are first constructed by maximizing the compliance in the desired direction while minimizing the compliances in the other directions. The weighting sum method is used to construct an objective function in which a self-adjust method is used to set the weighting factors. A constraint on the symmetry of the obtained configuration is developed. Several numerical examples are presented to demonstrate the validity of the proposed method. The obtained results reveal that the design of a flexure hinge starting from the topology level can yield more choices for compliant mechanism design and obtain better designs that achieve higher performance.     

直圆椭圆复合型柔性铰链研究

提出一种新型的单轴柔性铰链结构型式——直圆椭圆复合柔性铰链,它由半个直圆柔性铰链和半个椭圆柔性铰链构成。给出了直圆椭圆复合柔性铰链的柔度设计计算公式,并通过有限元方法验证了此公式的正确性。分析计算了一组不同椭圆短半轴的复合铰链实例,结果表明直圆椭圆复合柔性铰链的性能高于直圆柔性铰链。设计的直圆椭圆复合柔性铰链非常适合于需要大位移、高精度传动的应用场合。     

传递矩阵法分析平面柔顺机构的振动特性

为了分析柔顺机构的动态性能和力学传递关系,提出一种新型的基于传递矩阵法的振动力学模型。柔顺机构由若干个柔性铰链与杆件顺序联接而成,采用传递矩阵法描述其力学状态量的传递关系。将柔性铰链视为拉伸和弯曲变形的弹性梁,应用材料力学理论建立反映其自振性能的传递矩阵。将杆件视为刚体,采用动量矩定理建立其动力学模型,得到描述振动刚体的传递矩阵。按照柔顺机构的联接形式,将每个元件的传递矩阵进行拼装,得到系统总传递矩阵。总传递方程以柔顺机构的边界点状态矢量为未知变量,矩阵中的元素为机构结构参数和频率的函数。应用边界条件,可得柔顺机构的特征方程,通过求解方程可得系统的固有频率和振型。将外力纳入传递矩阵,建立反映外力激励与系统的关系的扩展传递矩阵以求解系统的频率响应。通过2种常用平面柔顺机构的动态性能分析,结果表明所建立模型的正确性,能精确描述柔顺机构力学传递关系。     

Design and stiffness analysis of a compliant spherical chain with three degrees of freedom

This paper introduces and investigates a compliant spherical 3R open chain that is obtained by the in-series connection of three identical circularly-curved beam flexures with coincident centers of curvature and mutually orthogonal axes of maximum rotational compliance. The considered open chain is intended to be used directly as a spherical mechanism in pointing devices or as a complex spherical flexure for the development of spatial parallel manipulators. The compliance matrix of the proposed chain is first determined via an analytical procedure. After finite element validation, the obtained equations are used in a parametric study to assess the influence of circularly-curved beam flexure geometric parameters on the overall stiffness performances of the considered compliant spherical 3R open chain. In addition, comparison with an equivalent compliant spherical chain employing straight beam flexures is reported to highlight the added benefits of using circularly-curved beam flexures in terms of reduced parasitic motions.