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.

In precision engineering applications, micromechanical systems or special applications often compliant mechanisms (Howell, 2013; Zentner, 2014) are used instead of rigid-body mechanisms. Due to their monolithic design, compliant mechanisms are suitable to realize high reproducible motion without clearance, external friction and wear. In compliant mechanisms with lumped compliance mostly flexure hinges (Lobontiu, 2003) are used as material coherent revolute joints. Among a variety of flexure hinge types the notch flexure hinge is state of the art. Therefore, lots of cut-out geometries are described in literatures, while the circular or corner-filleted flexure hinge contour are mainly used in technical systems.

Based on the rigid-body model (Howell and Midha, 1994) the rigid-body replacement method is widely used for the synthesis of compliant mechanisms, especially in precision engineering. In comparison to the optimal synthesis with a continuum model, a better guiding accuracy of a coupler point is possible with this method (Pavlović et al., 2010). In contrast to the synthesis of rigid-body mechanisms the stress and deformation behaviour as well as the motion behaviour must be considered as multi-objective design criteria in the compliant mechanisms synthesis. Starting from the rigid-body mechanism, this leads to a complex and iterative model-based design process for compliant mechanisms. Therefore, usually numerical methods and simulations are needed. Regarding the required mechanism properties, the step of the geometric design of the notch flexure hinges is a key aspect in the synthesis of a compliant mechanism. Therefore, many different approaches exist, but nevertheless no detailed guidelines or concise design equations for the contour-dependent and multi-criterial calculation of the flexure hinge characteristics are known to the author. Existing design equations are not concise, have complex structural form, and are limited to commonly used hinge contours (cf. Sect. 4). In addition, the rotational precision depends on the approach to model the position of the rotation axis during motion. Hence, a suitable approach for the model-based investigations must be used. In conclusion, simple and concise design equations for flexure hinges would be of great benefit to the accelerated and goal-oriented synthesis of compliant mechanisms without computer-aided simulations or the test of manufactured prototypes.

Regarding a plane rotational motion due to a bending moment or a transverse force load, this paper addresses the development of new general design equations for the simplified calculation of the rotational stiffness, maximal angular elastic deflection and rotational precision of various notch flexure hinges with a semi-circular, corner-filleted, elliptical, and polynomial contour for an appropriate range of the basic hinge parameters. The design equations are derived according to results of the analytical solution based on the non-linear theory for modelling of large deflections of rods.

The remaining sections are organized as follows. In Sect. 2, the state of the art of flexure hinges and their geometrical notch design is presented together with the investigated flexure hinge contours in this paper. In Sect. 3, the analytical characterization of the flexure hinges is described regarding the three mentioned performance criteria in dependence of the hinge contour and the geometric parameters. In Sect. 4, the method and results for the derived design equations are presented. In Sect. 5, the equation-based results are discussed and compared with the exact analytical solution and the results of a FEM-based and experimental characterization of the regarded flexure hinges. Finally, conclusions are drawn in Sect. 6.

In contrast to form- and force-closed joints a flexure hinge enables a restoring force which can be advantageous in technical systems (this performance criterion is named rotational stiffness). According to the material coherent connection, the angular deflection of a flexure hinge is limited by reaching admissible material stress respective elastic strain values (maximal angular deflection). Thus, the motion range of a compliant mechanism is limited too? by the hinge in the kinematic chain with largest rotation angle. In addition, no exact relative rotation is possible with a flexure hinge because always a shift of its axis of rotation occurs in dependence of geometric and load parameters (rotational precision). In turn, this can lead to path deviations of the compliant mechanism compared to the rigid-body mechanism, which are not negligible especially in precision engineering (Venanzi et al., 2005; Linß et al., 2014).

As a flexure hinge in this paper a monolithic, small-length and elastic deformable segment of a compliant mechanism is meant, which realizes the function of a relative rotation of two adjacent links mainly due to bending. The demand for a larger angular deflection and a low shift of the rotational axis during the rotation results in a variety of sometimes very complex flexure hinge types, like the butterfly hinge (e.g. Henein et al., 2003; Pei and Xu, 2011) for example. However, the investigations in this paper are focused on notch flexure hinges. Due to their low complexity they are easy to manufacture and therefore mainly used in plane compliant mechanisms, especially for kinematic chains with a higher link number. Furthermore, notch flexure hinges enable optimization potential regarding the rotational precision and possible deflection as equivalent objectives, which is not used yet. Hence, generalized design equations would be of a great benefit.

In the past, notch flexure hinges have been designed very frequently so that various cut-out geometries are proposed to describe the hinge contour, see Fig. 1. Mostly, there are predefined basic geometry elements, which lead to three main notch flexure hinge types, each with a typical characteristic: The precise hinge with a semi-circular contour (e.g. Paros and Weisbord, 1965; Wu and Zhou, 2002), the large-deflective hinge with a corner-filleted contour (e.g. Lobontiu, 2003; Meng et al., 2013) or the elliptical hinge (e.g. Smith et al., 1997; Chen et al., 2008) as a compromise. Furthermore, flexure hinges are designed with other elementary geometries to realize a special characteristic, like the parabolic or hyperbolic contour (e.g. Lobontiu, 2003; Chen et al., 2009), and cycloidal contour (Tian et al., 2010). Increasingly flexure hinges are designed with a combination of the mentioned basic geometries (e.g. Zelenika et al., 2009; Lobontiu et al., 2011; Chen et al., 2011). Rarely special mathematical functions are used that allow more precise shape variations of the partial or whole hinge contour due to a higher number of geometric parameters, like the spline contour (Christen and Pfefferkorn, 1998; De Bona and Munteanu, 2005), the power-function contour (Li et al., 2013), the exponent-sine contour (Wang et al., 2013), the Lamé contour (Desrochers, 2008), and the Bézier contour (Vallance et al., 2008). The design with undefined freeform geometries based on topology optimization (Zhu et al., 2014) is a very complex, non-intuitive and not a general design process.

Approaches for the geometric design of a flexure hinge contour
with notches on both sides:

Nevertheless, special higher order polynomial functions, which was suggested by author (Linß et al., 2011b), are not state of the art. Among the variety of cut-out geometries especially polynomial contours offer high potential for optimization while a comparatively simple contour modelling is possible. Depending on the polynomial order and the coefficients arbitrary complex curves can be realized. Furthermore, nearly any elementary geometry could be approximated.

While usually completely symmetric flexure hinges are used, there are a several studies on transversal and axial symmetric hinges. Especially axial symmetric flexure hinges are realised mostly as so-called hybrid hinges because they allow combining the advantages of right circular and corner-filleted flexure hinges (Chen et al., 2005). Further it is known, that a better kinematic behaviour up to an ideal rotation axis can be realized due to a smaller radius at the loaded hinge side (Linß et al., 2011a; Lin et al., 2013). However, in this paper only transversal and axial symmetric notch flexure hinges are investigated at first because they allow a holistic and intuitive design with regard to the mechanism synthesis.

Subject of the investigations in this paper is a separate notch flexure
hinge, which is fixed at one end, see Fig. 2a. A given moment

Investigated flexure hinge:

The variable hinge contour height

The determination of typical functional parameters of the hinge contour,
e.g. radii, can be done by selection, design of experiments, or
optimization, where this design step is not in the focus of this paper. For
the investigations, a flexure hinge with the following characteristics of
the hinge contour

There exist different suggestions for the suitable design of the geometric
parameters of corner-filleted flexure hinges aiming for low stress values by
using a special ratio of the fillet radius and the minimal hinge height,
e.g.

Therefore corner-filleted contours with a stress optimal fillet radius of

Quarter model of a flexure hinge with a corner-filleted contour
with the stress optimal fillet radius

Quarter model of a flexure hinge with a semi-circular contour with

To ensure principle similarity of circular contours for the varying hinge
dimensions

According to this, the notch length

For modelling similar elliptical contours the same approach is used as for
the semi-circular contours. Next, the elliptical flexure hinges are always
considered with the two radii

Quarter model of a flexure hinge with an elliptical contour with

According to this, the notch length

For modelling flexure hinges special polynomial contours are suitable too
(Linß et al., 2011b). In this case, the contour function

Quarter model of a flexure hinge with polynomial contours of a
different order

The used 4th-order polynomial contour function

Parameters of the characterized flexure hinge (drawing of the initial and deflected position) with the model for the determination of the rotational axis shift based on guiding the centre with a constant distance (fixed centre approach).

Influence of the hinge contour – analytical results for the
rotational stiffness of a flexure hinge with various contours
(

Influence of the hinge dimensions – analytical results for the
influence of

Influence of the hinge dimensions – analytical results for the
influence of

In this section, the approach of the non-linear analytical characterization
of a notch flexure hinge and the results for its rotational stiffness,
strain distribution, maximal angular elastic deflection, and rotational
precision are presented in dependence of the hinge contour and the geometric
parameters

The analytical characterization is based on the non-linear theory for
modelling large deflections of curved rods, e.g. Zentner (2014), for which
the dimensions of a cross-section are small compared to the rod length.
Equilibrium equations are used to describe a rod element on the basis of the
assumptions of a static problem for a slender structure with an axial
inextensible line and the val idity of Bernoulli hypothesis, Saint-Venant's
principle, and Hooke's law. Thus, for describing a flexure hinge as a beam
four non-linear differential equations result:

Required load for a discrete angular deflection – analytical
results for a semi-circular flexure hinge at

The following analytical characterization for the rotational stiffness, the
strain, and the rotational precision is exemplified for the hard aluminium
alloy EN AW 7075 with a Young's modulus of

Analytical results for the strain distribution of a flexure hinge
with various contours along

Analytical results for the maximal angular deflection

As rotational stiffness of a flexure hinge in this paper the

The principal load-angle-behaviour is almost linear, which leads to a
constant stiffness for the regarded small angular deflections up to
5

The influence of the basic hinge dimensions on the rotational stiffness is
investigated for all hinge contours. The influence of

The rotational stiffness increases with a decreasing hinge length ratio

The rotational stiffness increases with an increasing hinge height ratio

The rotational stiffness increases with an increasing hinge width ratio

The characteristic is qualitative and quantitative similar for both load cases.

Because flexure hinges are used in compliant mechanisms, the load
characteristic for a discrete angular deflection is interesting too. The
resulting loads

In this section, additionally the bending stress is analysed after linear
beam theory to characterize the maximum stress of the entire flexure hinge
for a given deflection as a result of the moment or force load:

Influence of the hinge contour – analytical results for the
rotational precision of a flexure hinge with various contours
(

Among the four regarded contours, the semi-circular contour always leads to the highest strain values. In this case, the maximum admissible elastic strain would be exceeded for the aluminium material AW 7075. According to the following order – of using the elliptical, the polynomial, and the corner-filleted contour – the maximum strain value can be reduced.

For a moment load it is obvious, that the maximum strain occurs in the hinge
centre in general. In contrast to this, the critical coordinate

Furthermore, regarding a concrete application in a compliant mechanism, the
admissible elastic strain

Influence of the hinge dimensions – analytical results for the
influence of

Influence of the hinge dimensions – analytical results for the
influence of

Rotational axis shift for a discrete angular deflection –
analytical results for a semi-circular flexure hinge at

Because the maximum strain value limits the deflection, the maximum rotation
angle of a flexure hinge always is possible with a corner-filleted contour,
while a semi-circular contour leads to the lowest possible angles. The
rotation angle of corner-filleted contour is more than five respectively
four times higher than of a semi-circular contour. Especially for flexure
hinges made from plastic, with admissible strain values higher than 1 %,
a large angular deflection can be realized in dependence of the chosen
minimal height

In particular in precision engineering, the rotational precision of a
flexure hinge is a very important performance criterion for the kinematic
behaviour of a compliant mechanism. Because of the serial connection of
several flexure hinges in the kinematic chain, the rotational axis shift

Since there is no stationary rotation axis, the rotational axis shift

In this case, the absolute value of the rotational axis shift

The influence of the hinge contour on the rotational precision is shown in
Fig. 14. The qualitative axis shift-angle-behaviour is non-linear for a
moment load and almost linear for a force load. Furthermore, the load case
has an influence on the absolute value: Independent from the hinge contour,
a transverse force leads to a significant larger axis shift than a moment
load for an equal angle

The hinge contour has a strong influence on the axis shift, which can be in
the range of several micrometres up to the millimetre range in dependence of
the basic dimensions

The influence of the basic hinge dimensions

The rotational precision increases (the axis shift decreases) with a
decreasing hinge length ratio

The rotational precision increases with an increasing hinge height ratio

The rotational precision and the kinematic behaviour are independent of the
hinge width ratio

The characteristic is qualitative and quantitative different for both load cases.

The influence of the hinge contour decreases with an increasing hinge height
ratio

Finally, the rotational axis shift for a discrete angular deflection of

In this section, novel general design equations for the rotational stiffness, the maximal angular elastic deflection, and the rotational precision of a notch flexure hinge in dependence of the load case are derived based on the analytical results. The approach and the method are described briefly and the developed design equations are presented together with the contour-specific coefficients.

In literature, design equations exist in particular for the rotational stiffness of a flexure hinge with a semi-circular, corner-filleted or elliptical contour (e.g. Paros and Weisbord, 1965; Tseytlin, 2002; Wu and Zhou, 2002; Lobontiu, 2003; Chen et al., 2008, 2011; Meng et al., 2013). Design equations for calculating the contour-dependent maximal angular elastic deflection are not state of the art, because the few existing design equations consider solely the stress (e.g. Chen et al., 2014) or the correlation between the load and the strain (Tres, 1995; Kunz, 2007; Dirksen, 2013), but not explicit the rotation angle. Especially regarding the rotational precision, the presented design equations are limited due to the use of standard hinge contours and a different approach to model the axis of rotation. In literature, closed-form equations are mainly presented for the approach of the offset of the hinge centre point (e.g. Lobontiu, 2003; Chen et al., 2009; Hu et al., 2012; Li et al., 2013). Rarely, empirical design equations based on FEM simulation results are suggested for semi-circular (Yong et al., 2008) and corner-filleted (Meng et al., 2013) flexure hinges.

FEM-based characterization of a flexure hinge:

However, all suggested design equations are characterized by a long expression and complex structural form, and they are only valid for a special group of flexure hinge contours. Simple and concise analytically derived design equations, whose principal structural form is independent from the notch geometry, are not known to the authors.

To develop the novel design equations, first the analytical results for the
rotational stiffness and rotational precision of a flexure hinge are
calculated, as in Sect. 3 described, for the following parameters and ranges
of variation:

four flexure hinge contours (semi-circular, corner-filleted, elliptical and bi-quadratic polynomial contour);

three basic hinge dimensions of

the angular deflection of

the load case of a moment load or a transverse force load close to the hinge
centre at

Second, with the help of MATLAB the contour-specific coefficients of a power
function are determined based on a fitting procedure in order to realize the
smallest maximum error over all calculated result points. The approach is
explained in Sect. 4.1 exemplarily for the rotational stiffness and the
moment load case. The remaining design equations for the contour-dependent
maximal angular deflection are obtained by conversion and the additional
introduction of a correction factor for considering the location of the
critical strain. Finally, the rounded coefficients (two digits) are
determined again for a reduced parameter range for

Coefficients for design Eqs. (21)–(26) in dependence of the
flexure hinge contour, valid for appropriate hinge dimensions
(

Based on the numerically calculated analytical results, a power function is
used to express the functional load-angle-correlation

Experimental characterization of a force loaded flexure hinge:

Following the dimensional analysis theory, we consider the geometric hinge
parameters according to the defined dimensionless ratios

Comparison of results for the rotational stiffness of a flexure
hinge with various contours (

Comparison of results with existing design equations for the
rotational stiffness of a moment loaded semi-circular flexure hinge
(

Comparison of results for the maximal angular deflection of a
flexure hinge with various contours in dependence of the admissible material
strain (

Comparison of results for the rotational precision of a flexure
hinge with various contours (

To obtain the optimal coefficients of the non-linear power function a curve
fit was realized based on the Nelder-Mead simplex algorithm for each hinge
contour. This optimization has been implemented in MATLAB with the function
fminsearch in order to minimize the maximum error of all calculated results of the
design equation compared with each analytical result over the parameter
range. Since a special correlation between the both coefficients

Hence, for a moment load we finally obtain the contour-independent design
equation for the rotational stiffness of a flexure hinge as

To obtain the design equations for the

For a moment load, the maximum strain occurs contour-independent in the
hinge centre. Therefore, the critical hinge height

The design equations for the

Hence, for a moment load we obtain the contour-independent design equation
for the rotational precision of a flexure hinge as:

In this section, the design equation results are compared with the analytical results, and with further results based on FEM simulations and experimental tests of selected flexure hinge contours. The FEM model and the experimental setup are described briefly. Then, the results are discussed for appropriate hinge dimensions.

For the FEM-based simulation of the flexure hinges ANSYS Workbench 16.2 is
used. The CAD model and the FEM model are shown in Fig. 18. The CAD model
includes an additional part to realize the direct determination of the
rotational axis

According to the literature, the flexure hinge is modelled as a solid structure (Zhang and Hu, 2009) with adjacent link segments (Zettl et al., 2005; Yong et al., 2008), like they are considered for the analytical characterization too. In the FEM simulation large deflection is considered due to non-linear beam theory. Further assumptions are linear material behaviour for the used aluminium AW 7075 material (with the same parameters like in Sect. 3) and a comparable and fine discretisation of the hinge for all the different contours.

For the experimental investigation of the deflected state of the flexure
hinge a coordinate-reading microscope (Carl Zeiss ZKM 01-250C) was used. The
realized experimental setup with the clamped flexure hinge is shown in Fig. 19. The frame of the experimental setup is fixed to a

Because the load case of an ideal moment cannot be realized due to the
inherent rotational axis shift with the required accuracy, a transverse
force is regarded only. In order to measure the rotational precision and the
rotational stiffness, the force load can be implemented displacement driven
(micrometre screw drive, Owis MS 6-12) or force driven (dead weight drive).
The force load is transmitted from the input by a pushing rod and a
cylindrical pin to a second orthogonal pin, which is fixed to the flexure
hinge and which is coincident with the load acting point

For the comparison of the design equation results for various notch flexure
hinges with the results of the analytical, FEM-based and experimental
characterization, typical parameter values for the hinge dimensions (

Generally, regarding all three performance criteria, the equation-based results are in good correlation with the analytical solution as well as with the FEM simulation and the measured results (qualitative as well as quantitative). Thus, the principle conclusions of Sect. 3 concerning the influence of the hinge contour and the geometric parameters on the rotational precision, the maximal angular deflection and the rotational precision have been confirmed by different methods. The little differences, especially to the FEM results, can be explained in particular with a more accurate modelling of a flexure hinge by means of FEM simulation. Regarding the experimental investigation, especially the measurement of the rotational axis shift in the micrometre range is challenging, but the principle absolute value could be confirmed for a flexure hinge with a semi-circular and a corner-filleted contour.

The comparison of results for the rotational stiffness of a moment loaded semi-circular flexure hinge with existing design equations from literature (cf. Fig. 21) and the FEM result shows, that Eq. (21) is accurate and very close to the simplified expression of Paros and Weisbord (1965). The used equations from the literature are mentioned in Eqs. (A1), (A2) and (A3).

In conclusion, the suggested design equations are suitable to predict the
deformation and motion behaviour of a flexure hinge in dependence of the
hinge contour within the investigated parameter range. The presented design
equations are advantageous, because they are concise, and with only two
coefficients their structural form is simple, contour-independent and, with
the exception of the parameter

Regarding the compliant mechanism application, the contour comparison confirms the potential of elliptical and bi-quadratic polynomial contours to realize a large motion range with high precision. A strong influence of the contour on the flexure hinge performance exists in particular for thin hinges, which are suitable for application due to low strain values and thus larger angular deflections. Furthermore, the use of different flexure hinges in the same mechanism is very promising especially in terms of variable higher order polynomial contours, cf. Eq. (7). In this case, the polynomial order can be adjusted easily in dependence of the relative rotation angle in the mechanism by means of using design graphs (Linß et al., 2015).

In this paper, 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 are suggested and evaluated. The power function based equations are derived for a moment and a transverse force loaded beam by fitting the analytical results which are obtained due to non-linear modelling with the theory of large deflections of rods. For the accurate model-based investigation of the rotational precision the fixed centre approach is used to define the axis of rotation. Among the variety of existing notch geometries, four flexure hinge contours are selected and investigated: Three usual contours, the semi-circular, corner-filleted and elliptical contour, and the recently introduced bi-quadratic polynomial contour of 4th order, which simultaneously provides a large angular deflection and a high rotational precision. Depending on the contour, the maximum root mean squared error of the calculated results is in the range of less than 2 % to less than 16 % for an appropriate parameter range compared with the analytical solution. Furthermore, the FEM simulations and experimental results correlate well with the predictions based on the design equations. The presented equations are advantageous because with only two coefficients their structural form is simple, concise, contour-independent and dimensionless. Thus, the novel design equations contribute to the accelerated and goal-oriented synthesis of compliant mechanisms with the most commonly used hinge contours or the promising polynomial flexure hinges. More hinge contours, like higher order polynomial contours of a different order, can also be considered by determining the values of their coefficients in further research.

Paros and Weisbord (1965), simplified expression:

The authors declare that they have no conflict of interest.

The authors would like to gratefully acknowledge the support of the German Research Foundation (DFG) under Grant no. ZE 714/10-1. Edited by: G. Hao Reviewed by: two anonymous referees