Notch Flexure Hinges Research Articles

Six-Bar Linkages With Compliant Mechanisms for Programmable Mechanical Structures

Abstract Programmable mechanical structures are formed by autonomous and adaptive cells and can reproduce meshes known from the finite element method. Furthermore, they can change their structure not only through morphing, but also by self-reconfiguration of the cells. A crucial component of the cells, which can preserve the underlying geometry of a triangular mesh, are six-bar linkages. The main part of the present contribution concerns the six-bar linkages as a fully 3D-printable compliant mechanism where each revolute joint of the six-bar linkage is replaced with a notch flexure hinge with the circular contour. The utilization of notch flexure hinges presents two significant drawbacks. First, notch flexure hinges do not maintain the center of rotation. Second, although compliance is an inherent and desirable characteristic of flexural hinges, it gives rise to secondary or parasitic motion. The compliance subsequently lead to alterations in the underlying geometry of a triangular mesh. For self-reconfiguration of the cells, an efficient model is needed to predict the positioning errors. Therefore, the flexure hinge is represented by three distinct models, namely a finite element model, a beam model, and a simplified linearized model based on translational and rotational spring elements. These models are compared and evaluated in succession first to identify the parameters of the simplified model and later on, the simplified model is used to show the deviations of a medium-scaled programmable structure with respect to the idealized behavior. The current work brings us closer to both the development of programmable mechanical structures and the prediction of positioning errors during self-reconfiguration.

Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges

Flexure-based compliant mechanisms can be used to achieve bio-imitability and adaptability in the applications of biomedical engineering. However, a nonlinear load-displacement profile increases the design complexity of this type of compliant mechanism, especially when the cross-section of the flexure hinge is not constant. This paper proposes two general analytical models by analyzing the compliance and stress characteristics of the semi-circular notch flexure hinge undergoing large deflections, which is a typical variable cross-section of a flexure hinge, based on the Castigliano’s second theorem and the finite elements analysis method. As a case study for verification, three compliant four-bar linkage mechanisms are designed based on the proposed design approach, the design method proposed by Howell, and the equations proposed by Lobontiu, respectively. The results show that the design accuracy is improved 36% in comparison with designs from Howell and Lobontiu. Finally, a flexure-based artificial finger is designed and manufactured based on the proposed optimization approach. The performance test of the prototype shows that the artificial finger has good bio-imitability and adaptability with respect to joint movements.

Compliance and precision modeling of general notch flexure hinges using a discrete-beam transfer matrix

Compliance and precision are two key aspects in designing flexure hinges for use in compliant mechanisms. A generalized method with the finite-discrete idea and beam transfer matrix is developed for quickly predicting the kinetostatic behaviors of single and multiple-axis notch flexure hinges. The transfer matrix of Timoshenko beams explicitly including the shear deformation effect is derived in a concise form of Taylor series expansion. The compliance and rotational precision matrices of general notch flexure hinges are then derived targeting for arbitrary cutout profiles with a step-by-step modeling procedure. It is easy to operate for comparing and synthesizing different flexure hinges within a unified modeling framework. All a designer has to do is to prepare the concerned profile function without being caught into the laborious and even unsolvable derivation of mechanics. Several groups of single and multiple-axis notch flexure hinges are adopted to verify the feasibility of the presented approach by comparing with the numerical and experimental results available in literature. The comparative results indicate the high prediction accuracy and easy operation of the proposed approach for a wide applicability of complex notch flexure hinges.

Bionic design of a curvature-adjustable flexure hinge inspired by red blood cells

Recent years have witnessed an ever-increasing research interest in experience-driven and topology-optimization design of flexure hinges for use in compliant mechanisms. In this paper, we report the bionic design of a new notch flexure hinge with adjustable curvatures inspired by the biconcave disk morphology of red blood cells. The biconcave shape maintains red blood cells excellent deformability and surface properties that motivates the bio-inspired design by learning from nature. By imitating the biconcave shape of red blood cells using the quartic Bézier curve, the contour of a cluster of bio-inspired flexure hinges in the presence of positive and negative curvatures can be parametrically adjusted. The closed-form equations of compliance, precision and the maximum stress of the bio-inspired flexure hinge are derived based on Castigliano's second theorem and Timoshenko beam theory. Comparative studies verify the well precision and low stress of such a new flexure hinge. At last, an amplified piezoelectric actuator is designed with the bio-inspired flexure hinge as an application case. Also interestingly, adjustable curvatures of the presented bio-inspired flexure hinge make the simultaneous shape and parameter optimization effective.

Compliance and Dynamic Modeling of General Notch Flexure Hinges Using a Beam Transfer Matrix

Compliance and Dynamic Modeling of General Notch Flexure Hinges Using a Beam Transfer Matrix

Design of an electrostatic balance mechanism to measure optical power of 100 kW.

A new instrument is required to accommodate the need for increased portability and accuracy in laser power measurement above 100 W. Reflection and absorption of laser light provide a measurable force from photon momentum exchange that is directly proportional to laser power, which can be measured with an electrostatic balance traceable to the SI. We aim for a relative uncertainty of 10-3 with coverage factor k = 2. For this purpose, we have designed a monolithic parallelogram 4-bar linkage incorporating elastic circular notch flexure hinges. The design is optimized to address the main factors driving force measurement uncertainty from the balance mechanism: corner loading errors, balance stiffness, stress in the flexure hinges, sensitivity to vibration, and sensitivity to thermal gradients. Parasitic rotations in the free end of the 4-bar linkage during arcuate motion are constrained by machining tolerances. An analytical model shows this affects the force measurement less than 0.01 percent. Incorporating an inverted pendulum reduces the stiffness of the system without unduly increasing tilt sensitivity. Finite element modeling of the flexures is used to determine the hinge orientation that minimizes stress which is therefore expected to minimize hysteresis. Thermal effects are mitigated using an external enclosure to minimize temperature gradients, although a quantitative analysis of this effect is not carried out. These analyses show the optimized mechanism is expected to contribute less than 1 × 10-3 relative uncertainty in the final laser power measurement.

Design and stiffness modeling of a four-degree-of-freedom nanopositioning stage based on six-branched-chain compliant parallel mechanisms.

Multi-degree-of-freedom (multi-DOF) nanopositioning stages (NPSs) have rapidly growing applications in the spatial micro-/nano-machining and manipulation. Compliant parallel mechanisms (CPMs) demonstrate advantages to achieve a large output stiffness and high payload. A four-DOF NPS based on six-branched-chain CPMs is proposed in this paper. First, a mechanism design approach is introduced. One primary vertical DOF is generated using three parallel-kinematic lever amplifiers. A three-revolute-revolute-revolute mechanism acts as the kinematic configuration to produce three secondary planar DOFs. Three types of single-axis and one type of double-axis notch flexure hinges (NFHs) are employed to realize the nanoscale displacement/movement guiding, transferring, and decoupling. Second, a stiffness modeling approach is derived. Combined with exact compliance matrices of 54 NFHs and 95 flexible beams, a four-DOF high-efficiency stiffness model of the six-branched-chain CPM is built. The calculation procedure of the whole input/output stiffnesses and coupling ratios takes 12.06 ms. Simulation and prototype test results validate the calculation accuracy. For example, the maximum calculation deviation of input stiffnesses is verified to be 4.52% and 8.18%, respectively. The two proposed approaches contribute to the statics parameter optimization of spatial multi-DOF NPSs.

Topology optimization of distributed flexure hinges with desired performance

ABSTRACTTraditional notch flexure hinges have been widely used in precision engineering, but the lumped notch configuration limits the range of motion. This article presents a synthesis method for designing flexure hinges with distributed configurations using topology optimization. In order to obtain a distributed configuration, an optimization model for the topology optimization of a distributed flexure hinge is developed first. The objective is to maximize the bending displacement while minimizing the axial displacement. The constraint for accomplishing the symmetry requirement is presented. Then flexure hinges having the desired performance are designed on the basis of a distributed configuration. In particular, this work offers optimization models for designing flexure hinges with prescribed compliances and with minimal parasitic motion. This method is implemented based on the popular Solid Isotropic Material with Penalization (SIMP) method, and a number of numerical examples are provided to demonstrate the validity of the proposed method. The results of the examples demonstrate that the flexure hinges obtained have distributed configurations and the desired performances (prescribed compliances or minimal parasitic motion).

DetasFLEX – A computational design tool for the analysis of various notch flexure hinges based on non-linear modeling

Abstract. Notch flexure hinges are commonly used in compliant mechanisms for precision engineering applications and yet important rotational properties of a hinge like the bending stiffness, maximum angular deflection and rotational precision are difficult to predict accurately and simultaneously. There exist some closed-form equations and a few design tool approaches for calculating flexure hinges with particular geometries, but apart from that no comprehensive calculation program for the contour-specific analysis is known to the authors. Developed in MATLAB, this paper presents a novel computational design tool using a non-linear analytical approach for large deflections of rod-like structures to calculate the elasto-kinematic flexure hinge properties by numerically solving a system of differential equations. Building on previous investigations, four certain hinge contours are implemented, the circular, the corner-filleted, the elliptical, and the power function-based contour with different exponents. In addition to the theoretical approach and the implementation it is exemplarily shown, that finite elements method (FEM) results correlate well with the analytical design tool results. For a given deflection angle of 10° and a corner-filleted contour as an example, the deviations of the bending stiffness are between 0.1 % and 9.4 % for typical parameter values. The presented design tool can be beneficial for the accelerated and systematic synthesis of compliant mechanisms with optimized flexure hinges.

Modeling and analysis of conical-shaped notch flexure hinges based on NURBS

An analytical compliance matrix model of a flexure hinge is established and the conical-shaped notch flexure hinge is innovatively parameterized using the non-uniform rational B-spline (NURBS) curve, which can be used to describe several different conical notch shapes, including the ellipse, parabola, and hyperbola. Parameterization is useful for performing integral operations on compliance models. Three translational directions of compliance are simulated using the finite element (FE) method and results show a maximum error of 5.39% between the theoretical value and simulation result, thus verifying the accuracy of our analytical compliance model. The influence of certain parameters on flexure hinge performance, including compliance, relative compliance, and axial shift, is analyzed.

Modeling and design of a two-axis elliptical notch flexure hinge.

As an important part of the joule balance system, the two-axis elliptical notch flexure hinge (TENFH) which typically consists of two single-axis elliptical notch flexure hinges was studied. First, a 6 degrees of freedom (6-DOF) compliance model was established based on the coordinate transformation method. In addition, the maximum stress of the TENFH was derived. The compliance and maximum stress model was verified using finite element analysis simulation. To decouple the attitude of the suspended coil system and reduce the offset between the centroid of the suspended coil mechanism and the mass comparator in the joule balance system, a new mechanical structure of TENFH was designed based on the compliance model and stress model proposed in this paper. The maximum rotation range is up to 10°, and the axial load is more than 5 kg, which meets the requirements of the system. The compliance model was also verified by deformation experimentation with the designed TENFH.

Analysis of Bridge-type Distributed-compliance Mechanism

This paper analyses a class of bridge-type distributed-compliance mechanism, which has better performances than traditional bridge-type mechanisms using notch flexure hinges. An analytical model for the displacement amplification ratio and input stiffness calculations of the bridge-type mechanism is established based on the stiffness matrix method. The finite element analysis results are then given to validate the correctness of the analytical model. The differences of the analytical results with respect to the finite element analysis results are less than 8%, which demonstrate the high accuracy of the analytical model. The influences of the geometric parameters on the amplification ratio and input stiffness of the mechanism are also investigated using the analytical model to provide theoretical guidelines for the practical design.

Analysis and design of symmetric notch flexure hinges

This article presents the analytic models of four types of flexure hinges (elliptic, circular, parabolic, and hyperbolic flexure hinges). The analytic models are developed based on the theory of el...

General design equations for the rotational stiffness, maximal angular deflection and rotational precision of various notch flexure hinges

Abstract. Notch flexure hinges are often used as revolute joints in high-precise compliant mechanisms, but their contour-dependent deformation and motion behaviour is currently difficult to predict. This paper presents general design equations for the calculation of the rotational stiffness, maximal angular elastic deflection and rotational precision of various notch flexure hinges in dependence of the geometric hinge parameters. The novel equations are obtained on the basis of a non-linear analytical model for a moment and a transverse force loaded beam with a variable contour height. Four flexure hinge contours are investigated, the semi-circular, the corner-filleted, the elliptical, and the recently introduced bi-quadratic polynomial contour. Depending on the contour, the error of the calculated results is in the range of less than 2 % to less than 16 % for the suggested parameter range compared with the analytical solution. Finite elements method (FEM) and experimental results correlate well with the predictions based on the comparatively simple and concise design equations.

Generalized Equations for Estimating Stress Concentration Factors of Various Notch Flexure Hinges

The flexure hinges are the most vulnerable parts in a flexure-based mechanism due to their smaller dimensions and stress concentration characteristics, therefore evaluating the maximum stresses generated in them is crucial for assessing the workspace and the fatigue life of the mechanism. Stress concentration factors characterize the stress concentrations in flexure hinges, providing an analytical and efficient way to evaluate the maximum stress. In this work, by using the ratio of the radius of curvature of the stress-concentrating feature to the minimum thickness as the only fitting variable, generalized equations for both the bending and tension stress concentration factors were obtained for two generalized models, the conic model and the elliptic-arc-fillet model, through fitting the finite element results. The equations are applicable to commonly used flexure hinges including circular, elliptic, parabolic, hyperbolic, and various corner-fillet flexure hinges, with acceptable errors. The empirical equations are tractable and easy to be employed in the design and optimization of flexure-based mechanisms. The case studies of the bridge-type displacement amplifiers demonstrated the effectiveness of the generalized equations for predicting the maximum stresses in flexure-based mechanisms.

Planar Compliances of Symmetric Notch Flexure Hinges: The Right Circularly Corner-Filleted Parabolic Design

This study proposes a general analytical compliance model for symmetric notch flexure hinges composed of segments with constant width and analytically defined variable thicknesses. Applying serial combination and longitudinal/transverse mirroring of base segments, the in-plane compliances of the full flexure are obtained as functions of one quarter-flexure compliances. The new right circularly corner-filleted parabolic flexure hinge is introduced as an illustration of the general analytical modeling algorithm. Experimental testing and finite element simulation confirm the analytical model predictions for an aluminum flexure prototype. The planar compliances sensitivity to the relevant geometric parameters is also studied.

Hybrid flexure hinges

This paper designs and analyzes the hybrid flexure hinge composed of half a hyperbolic flexure hinge and half a corner-filleted flexure hinge. As it is transversely asymmetric, it has different performance when the fixed and free ends switch. Considering the diversion of rotation center from midpoint, closed-form equations are formulated to characterize both the active rotation and all other in-plane parasitic motion by the Castigliano's second theorem. The maximum stress is evaluated as well. These equations are verified by the finite element analysis and experimentation. The compliance precision ratios are proposed to indicate flexure hinges' ability of preserving the rotation center when they have the same displacement at the free end. The hybrid flexure hinges are compared with five kinds of common notch flexure hinges (circular, corner-filleted, elliptical, hyperbolic, and parabolic flexure hinges) quantitatively based on compliance, precision, compliance precision ratios, and the maximum stress. Conclusions are drawn regarding the performance of these six kinds of flexure hinges.

Planar Compliances of Thin Circular-Axis Notch Flexure Hinges with Midpoint Radial Symmetry

The new class of flexure hinges with circular longitudinal axis and midpoint radial symmetry is introduced. Using rotation and mirroring, the symmetric flexure hinge is obtained from one half flexure. The six planar-bending compliances of the full hinge are determined analytically for small deformations by combining only three compliances of the half flexure. To illustrate the general flexure hinge category, the novel circular-axis, right circularly corner-filleted design is introduced. Experimental and finite element results correlate well with the analytical model predictions. The new flexure hinge design is compared to the circular-axis, constant-thickness flexure and the straight-axis, right circularly corner-filleted hinge.

In-plane elastic response of two-segment circular-axis symmetric notch flexure hinges: The right circular design

Abstract This work presents a general analytical model of the planar compliances for two-segment circular-axis symmetric notch flexure hinges. Compared to a similar straight-axis flexure design, the circular-axis configuration enhances the design parameter domain by adding the median circle radius. Six compliances are formulated with respect to a central reference frame and to an end reference frame in terms of a smaller number of half-flexure compliances. The model covers both relatively large radius-to-thickness ratio designs (thin flexures) and small radius-to-thickness ratio configurations (thick flexures). The new circular-axis right circular flexure is introduced to illustrate the general model. Predictions of the analytical compliance model for this flexure were confirmed by finite element analysis and experimental testing. The analytical model is further utilized to investigate the influence of normal and shear forces in the thick-member model and to assess the accuracy of the simpler thin-beam model, which considers only bending. A comparison is also performed between the corresponding compliances of the new circular-axis flexure design and the existing straight-axis right circular flexure hinge in terms of defining geometric parameters.

Symmetry-Based Compliance Model of Multisegment Notch Flexure Hinges

An analytical procedure is proposed to calculate the compliances of planar-motion notch flexure hinges with transverse or/and axial symmetry by employing base segments of known compliances. Mirroring, translation, and serial combination of individual segments are used to obtain the flexure hinge compliances that are necessary to evaluate its capacity of deformation and precision of rotation. Two case studies are illustrated: one derives the compliances of the known right circular flexure hinge while the other introduces a new flexure hinge with both axial and transverse symmetry. For the latter flexure design, the numeric values of compliances are confirmed by finite element simulation.