Compliance Equations Research Articles

Analytical Compliance Model for Right Circle Flexure Hinge Considering the Stress Concentration Effect

In this paper, an analytical compliance model for right circle flexure hinge (RCFH) is presented with the stress concentration in consideration. The stress concentration caused by changes in RCFH’s cross-section usually happens at the weakest point. It has been shown to seriously affect RCFH’s compliance calculation. Based on the virtual work theory, superposition relationship of the deformation, as well as Castigliano’s second theorem, RCFH’s analytical compliance model considering the stress concentration effect is established. The model is calculated as a series of closed-form equations which are related with geometric dimensions and employed material. Complicated definite integrals existing in these compliance equations are proved to be correctly calculated through comparisons with other literatures. Finally, in order to examine the validity of the established model, finite element analysis (FEA) is conducted. The relative errors between the theoretical values obtained by the established model and FEA results are found within 20% for a wide range of geometric dimensions.

Compliance Modeling of a Compliant Stage with Symmetric Configuration

This paper presents the compliance modeling of a compliant stage with symmetric configuration. Empirical compliance equations for the circular flexure hinge are first introduced. Using the matrix method, the output compliance of a compliant stage with symmetric configuration is then obtained. Finally, the compliances derived from the proposed theoretical model and finite element analysis (FEA) are compared. It indicates that the results calculated by the theoretical model are in good agreement with those derived from FEA, which demonstrates the accuracy of the theoretical model.

Compliance and Fatigue Life Analysis of U-shaped Flexure Hinge

This paper establishes four models of U-shaped flexure hinges with different notch shapes and structure parameters, and presents the close-form compliance equations for the four structure types of U-shaped flexure hinges. The compliance of the flexure hinges is developed based on the Castiglione’s second theorem and calculus theory. A relationship between compliances and structure parameters is deduced using the models. The influences of the notch structure parameters on the compliance of the flexure hinges are investigated. Moreover, fatigue life of U-shaped flexure hinges is studied by finite element analysis, the results show that the fatigue life of flexure hinge increases gradually with the increasing of flexure hinge center thickness t and hinge notch width m. With the increasing of the major axis of the ellipse a and semi minor axis of the ellipse b, the fatigue life of flexure hinge fluctuates locally, the general trend is a gradual decrease. The stress and fatigue life of U-shaped flexure hinges and arc flexure hinge are compared. The results show that the reliability of U-shaped flexure hinge is higher than that of circular arc flexure hinge.

Empirical compliance equations for conventional single-axis flexure hinges

In consideration of the stress concentration, unified compliance equations for conventional single-axis hinges are presented. The relationship between the stress concentration and the compliance of corner-filleted flexure hinges is first analyzed. Considering the stress concentration, coupled with a wide range of geometrical parameters, empirical compliance equations for conventional flexure hinges, are then obtained by using the exponential model. Subsequently, the proposed equations are unified. To verify the validity and accuracy of these equations, the characteristics of a bridge-type flexure-based mechanism are then analyzed by the proposed equations and finite element analysis, respectively. The results of compliances and displacement amplification ratios obtained by these two methods are in good agreement. It demonstrates that the empirical compliance equations could be obtained by exponential model, and these equations can be unified.

Verification of stress-intensity factor and compliance relations for a single edge notch specimen under tension-compression loading

Stress-intensity factor and compliance solutions developed for a single edge notch (SEN) specimen has been validated and verified for tension-compression loading. Fatigue crack growth tests were conducted on the SEN and a surface-crack specimen at stress ratios, R of −1.0 and −2.0 using IN718 forged nickel base superalloy material. (The surface-crack specimen is referred to as the KB-bar specimen in the literature.) Experimentally determined compliance values for the SEN specimen compared well with the analytically predicted values. In addition, crack lengths measured using the compliance equation compared well with crack lengths measured using potential drop and optical techniques. Finally, agreement was found between the SEN and KB specimens in the intermediate crack growth rate range.

An analytical solution for double cantilever beam based on elastic–plastic bilinear cohesive law: Analysis for mode I fracture of fibrous composites

The most difficult problem in mode I fracture test for fibrous composites using double cantilever beam (DCB) specimen is to identify crack tip without ambiguity. Consequently to evaluate fracture toughness through compliance method is a tricky matter. This study was conducted to solve this problem by developing an analytical solution for DCB fibrous composite specimen tested for mode I fracture. Cohesive law was employed to model separation constitution of cracking. Based on this analytical solution, the length of fracture process zone (FPZ) can be evaluated. Compliance equations of DCB specimen which take FPZ length into consideration were provided. Therefore, R-curve can be directly obtained by differentiating the compliance with respect to crack length. The model was validated by DCB tests of parallel strand bamboo (PSB), a bamboo-based fibrous composite. Good agreement between the predicted and test load–displacement were achieved. It was found that the R-curve behavior of PSB is similar to that of many other fibrous composites, which undergoes a nonlinear increase stage during FPZ formation and then reaches a constant after FPZ fully developed. Finally, general solutions for common used linear-type cohesive laws are provided.

A Generic Compliance Modeling Method for Two-Axis Elliptical-Arc-Filleted Flexure Hinges.

As a kind of important flexible joint, two-axis flexure hinges can realize in-plane and out-of-plane motions and can be used for constructing flexure-based spatial compliant mechanisms. The paper introduces a common two-axis elliptical-arc-filleted flexure hinge that is generated by two different elliptical-arc-filleted cutout profiles and that provides some new hinge types. The analytical compliance equations of both half-segments of the two-axis elliptical-arc flexure hinges are firstly formulated, and then, based on a generic compliance modeling method of a flexure serial chain, the closed-form compliance and precision matrices of two-axis elliptical-arc-filleted flexure hinges are established and validated by the finite element method. Some numerical simulations are conducted to compare the effect of different design geometric parameters on the performance of the two-axis flexure hinges.

A single edge notch specimen for fatigue, creep-fatigue and thermo-mechanical fatigue crack growth testing

The importance of capturing crack growth behavior under creep-fatigue and thermo-mechanical-fatigue is well known. Such service load environments are often simulated by performing tests under load and displacement control conditions where the specimen may be subjected to both tensile and compressive loads. A single edge notch specimen was developed to accommodate tension and compression loading in fatigue, creep-fatigue and thermo-mechanical fatigue crack growth experiments. The stress intensity factor and compliance solutions were developed for the single edge notch specimen using a 3D fracture code (FRANC3D) and 2D boundary element code (FADD2D). Both codes agreed well with each other. It is shown in this work that using simple approximations in a 2D code such as FADD2D, accurate stress-intensity and compliance solutions can be obtained for specimen geometry with varying cross sections. An importance of using correct boundary condition is emphasized by displaying the difference in stress intensity solutions for rectangular edge-cracked plate under uniform-stress, and uniform-displacement cases. Also, it is shown that stress intensity solution for a threaded edge-cracked specimen considered in this paper is different from the solution for rectangular edge-cracked plate under displacement control. Asa validation of the developed stress-intensity and compliance equations, fatigue crack growth tests were conducted using the single edge notch, compact type and a surface crack specimen. Tests were conducted under constant amplitude load control of 0.1 stress ratio, R for IN738 nickel base super alloy material. Agreement was found between the different specimen types in the intermediate crack growth rate range.

Effective Modulus for Crack Size Measurement With SE(T) Specimens using Unloading Compliance

Abstract Low-constraint toughness tests have been increasingly used in pipeline flaw assessment. A number of protocols have been published using a test employing a clamped single-edge tension (SE(T)) specimen with the distance between the clamps H equal to ten times the specimen width W, and specimen width W equal to thickness B. For this geometry, the constraint over the bulk of the specimen varies between plane stress and plane strain. This affects the elastic compliance, commonly used to estimate crack size. It was the intent of the present work to assess the effects of constraint on the compliance and to identify the best combination of compliance equation and modulus to estimate crack size from CMOD compliance where CMOD is the crack mouth opening displacement. To do this, values of compliance from finite element analyses were provided by the authors of this paper from three separate laboratories. 2D plane strain, 3D plane-sided, and 3D side-grooved specimens were analyzed. The data were used to assess available compliance equations and to propose a new one for the specific geometry of interest in this work.

Design and analysis of symmetric and compact 2R1T (in-plane 3-DOC) flexure parallel mechanisms

Abstract. Symmetry is very necessary in flexure mechanisms, which can eliminate parasitic motions, avoid buckling, and minimize thermal and manufacturing sensitivity. This paper proposes two symmetric and compact flexure designs, in-plane 3-DOC (degree of constraint) mechanisms, which are composed of 4 and 6 identical wire beams, respectively. Compared to traditional leaf-beam-based designs, the two present designs have lower stiffness in the primary motion directions, and have smaller stiffness reduction in the parasitic directions. Analytical modelling is conducted to derive the symbolic compliance equations, enabling quick analysis and comparisons of compliances of the two mechanisms. A prototype has been tested statically to compare with analytical models.

Design, analysis, and testing of kinematic mount for astronomical observation instrument used in space camera.

A statically determinate kinematic mount structure is designed for an astronomical observation instrument. The basic principle of the proposed kinematic mount is introduced in detail, including the design principle, its structure, and its degrees of freedom. The compliance equations for the single-axis right circle flexure hinge are deduced, and mathematical models of the compliances of the bipod in the X-axis and Z-axis directions are established. Based on the index requirements, the range of one design parameter (the hinge groove depth, R) for the kinematic mount is determined. Parametric design is performed, with the entire structure being the design object and the first three eigenfrequencies as the design objective; the final design parameter for the kinematic mount is 1.9 mm. The first three eigenfrequencies of the final structure are 36.49 Hz, 38.65 Hz, and 72.41 Hz, which meet the frequency requirements. The Z-direction deformation and the bipod compliances in the X-axis and Z-axis directions are analyzed through simulations and experiments. The results show that (1) the Z-direction deformation of the bipod meets the displacement requirement; (2) the deviations between the finite element results and the compliance equation Cx results, and between the finite element results and the compliance equation Cz results are 8.8% and 3.92%, respectively; (3) the deviation between the experimental results and the compliance equation Cz results is 10.3%. It is concluded that the bipod compliance equations in the X-axis and Z-axis directions are valid, and that the kinematic mount thus meets the design requirements.

Compliance modeling of planar flexure-based mechanisms and its application to micro-motion stages

This article presents the compliance modeling method of flexure-based mechanisms. The relationship between deformations and loads of the flexure members and the end effector is analyzed according to the zero virtual work principle. Using the matrix method, concise compliance equations for flexure serial and parallel structures are derived in conjunction with consideration of the compliance calculation errors of flexure hinges. Finally, relationships between the output compliances and geometrical parameters of a 3-revolute-revolute-revolute micro-motion stage are discussed. To validate the proposed compliance modeling method, the output compliances calculated by the empirical and theoretical compliance equations are subsequently compared with values derived from finite element analysis (FEA). The comparisons indicate that the results obtained from the empirical compliance equations are in good agreement with those derived from FEA, whereas the errors calculated by the theoretical equations are much larger, which demonstrate the accuracy of the empirical compliance equations and validate the proposed compliance modeling method.

Prediction of viscoelastic behavior of unidirectional polymer matrix composites

To develop a novel method predicting the viscoelastic behavior of polymer matrix composites according to the viscoelasticity of the matrix, we used the viscoelastic model of the matrix to build new models for unidirectional composites in both 0° and 90° directions. Viscoelastic parameters for both new models were derived, and the obtained equations shared the same form as the viscoelastic constitutive equation of matrix material. The viscoelastic behaviors of matrix material and unidirectional composites were also tested. Results showed that fitting parameters of creep compliance equation were close to the theoretical values of viscoelastic constitutive parameters of the unidirectional composites, proving the validity of the models. A new method was obtained to predict the viscoelastic property of the unidirectional composites based on the viscoelastic property of composite matrix and elastic property of the unidirectional composites. This method provides a theoretical basis for future studies on the viscoelasticity of composite laminates.

Design and analysis of a high-accuracy flexure hinge.

This paper designs and analyzes a new kind of flexure hinge obtained by using a topology optimization approach, namely, a quasi-V-shaped flexure hinge (QVFH). Flexure hinges are formed by three segments: the left and right segments with convex shapes and the middle segment with straight line. According to the results of topology optimization, the curve equations of profiles of the flexure hinges are developed by numerical fitting. The in-plane dimensionless compliance equations of the flexure hinges are derived based on Castigliano's second theorem. The accuracy of rotation, which is denoted by the compliance of the center of rotation that deviates from the midpoint, is derived. The equations for evaluating the maximum stresses are also provided. These dimensionless equations are verified by finite element analysis and experimentation. The analytical results are within 8% uncertainty compared to the finite element analysis results and within 9% uncertainty compared to the experimental measurement data. Compared with the filleted V-shaped flexure hinge, the QVFH has a higher accuracy of rotation and better ability of preserving the center of rotation position but smaller compliance.

Empirical Compliance Equations for Constant Rectangular Cross Section Flexure Hinges and Their Applications

This paper presents the derivation of empirical compliance equations of the constant rectangular cross section flexure hinge. The stress concentration caused by changes in cross section is analyzed based on finite element analysis results for the purpose of overcoming compliance calculation errors. It shows that the stress concentration has great influence on axial compliance calculation, while it has little influence on shear and bending compliance calculation. Then empirical compliance equations with a relative wide range ofh/Landt/Lare derived based on the exponential model in conjunction with consideration of all geometrical parameters of flexure hinges and the influence of the stress concentration on axial compliance calculation. Finally, in order to verify the validity of the empirical equations, the input/output compliance and displacement amplification ratios of bridge-type microdisplacement amplification mechanisms are analyzed. Meanwhile, an experimental platform of displacement amplification mechanisms is set up. The experimental results and finite element method (FEM) values are in good agreement with the theoretical arithmetic, which demonstrates the accuracy of the empirical compliance equations. It provides a reference point for further studies on the design and optimization of flexure hinges and compliant mechanisms.

Assessment of mode-II fracture tests for unidirectional fiber reinforced composite laminates

Three basic mode-II test methods (ENF, JIS, and ASTM D7905M-14) are assessed using the material system AS4/8552 carbon/epoxy unidirectional composite laminate to understand similarities and differences. The modified JIS method uses a PTFE film coated stainless steel rod instead of the PTFE strip that was proposed in JIS. The ASTM D7905M-14 test method determines FEP film crack front (NPC) and shear precrack front (PC) fracture toughnesses. Alternately, wedge precracked specimens were also tested to assess the shear versus opening mode precracking on mode-II fracture toughness. The analysis and test results revealed that the JIS method is a mixed-mode I-II test and result in lower value of mode-II fracture toughness. The GI loading is about 51 J/m2 for the material tested and GIIc measured by JIS is always less than pure mode-II fracture toughness. The GIIc measured from the ASTM D7905M-14 NPC and the ENF tests are almost identical, but the ASTM test offers a compliance equation that may be beneficial in fatigue crack growth studies. As suggested in ASTM standard, shear precracked specimen is appropriate to measure mode-II fracture toughness.

Derivation of empirical compliance equations for circular flexure hinge considering the effect of stress concentration

This paper presents the compliance modeling of circular flexure hinge. The calculation accuracies of several compliance equations are firstly compared with the finite element analysis (FEA) results. Then the influences of the stress concentration on the compliance calculations of the flexure hinge are analyzed systematically. It shows that the stress concentration has great influence on the axial compliance calculation. Utilizing the exponential model, empirical compliance equations of circular flexure hinge are derived, and the influences of the stress concentration on the axial compliance are considered. These empirical compliance equations are verified to be with high calculation accuracy for a wide range of t/R ratios. Finally, the empirical compliance equations are utilized to analyze the compliance and displacement amplification ratio of a bridge-type compliant amplifier. The results are compared with FEA software, and it proves the accuracy and effectiveness of these empirical equations once again. In addition, the modeling method presented in this paper is general and can also be applied to the compliance equations derivation of other types of flexure hinges.

Comparison of hand hygiene opportunities (HHOS) between a us study and in acute care facilities in three other countries

There is a lack of published data on HHOs/patient/day across countries. While HH compliance rates are often published based on observations by auditors, some recognised disadvantages are training required, valuable hours taken to collect representative samples and the Hawthorne effect. Emerging technologies have potential to improve data reliability, timeliness and density. A key challenge is to establish an accepted “denominator” (HHOs) in the compliance equation.

Impact of crystalline defects and size on X-ray line broadening: A phenomenological approach for tetragonal SnO2 nanocrystals

Nanocrystalline tin oxide (SnO2) powders with different grain size were prepared by chemical precipitation method. The reaction was carried out by varying the period of hydrolysis and the as-prepared samples were annealed at different temperatures. The samples were characterized using X-ray powder diffractometer and transmission electron microscopy. The microstrain and crystallite size were calculated for all the samples by using Williamson-Hall (W-H) models namely, isotropic strain model (ISM), anisotropic strain model (ASM) and uniform deformation energy density model (UDEDM). The morphology and particle size were determined using TEM micrographs. The directional dependant young’s modulus was modified as an equation relating elastic compliances (sij) and Miller indices of the lattice plane (hkl) for tetragonal crystal system and also the equation for elastic compliance in terms of stiffness constants was derived. The changes in crystallite size and microstrain due to lattice defects were observed while varying the hydrolysis time and the annealing temperature. The dependence of crystallite size on lattice strain was studied. The results were correlated with the available studies on electrical properties using impedance spectroscopy.

A Study on CMOD-Based Compliance Calibration Functions for Clamped Single-Edge Tension Specimens

Abstract The compliance calibration function for the clamped single-edge tension (SE(T)) specimen has not been reported in the literature. Based upon the inverse fitting of the three existing CMOD-based compliance equations, three CMOD-based compliance calibration equations are proposed for specimens with a crack length-to-width ratio 0.05 ≤ a/W ≤ 0.75. The fitting errors of the equations here employed are no greater than 1 %.