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

Notch flexure hinges experience a growing application in industry and research. Due to their advantages of high reproducibility, high resolution, and clearance-free or frictionless motion they are widely used in compliant mechanisms (Howell et al., 2013; Zentner, 2014) for precision engineering, micromechanics or measurement technology tasks. For these applications flexure hinges with specific notch geometries are the most common form of compliant segments used to realize a rotation (Lobontiu, 2003). According to bending of the materially coherent joint, the angular deflection is limited because of the resulting maximum strain. Further, a small shift of the axis of rotation results which affects the motion behavior as well as guidance accuracy of a compliant mechanism over its rigid-body counterpart (Venanzi et al., 2005).

Nevertheless, predicting the motion of an individual single-axis flexure hinge is a non-trivial task due to geometric non-linearities caused by large deflections, leading to a system of differential equations that need numeric solving approaches for accurately determining the hinge performance properties. When looking for an optimal flexure hinge design for a specific mechanism motion task predominantly time-consuming simulations and iterative procedures become a necessity and only a few detailed guidelines are known to the authors. In literature closed-form equations exist for the prediction of the bending stiffness and deflection in most cases (e.g. Paros and Weisbord, 1965; Smith et al., 1987; Tseytlin, 2002; Wu and Zhou, 2002; Lobontiu, 2003; Schotborgh et al., 2005). Rarely, simple and contour-independent design equations for calculating various elasto-kinematic hinge properties are suggested (e.g. Linß et al., 2017a, b). Otherwise, when optimal hinge contours are looked for, design graphs may be used to figure out an appropriate order for polynomial flexure hinges (Linß, 2015).

To date specific tools for the analysis of flexure hinges are sparsely found. One author for example provides scripts that can be downloaded and used within MATHCAD to calculate flexure hinges and compliant mechanisms using simplified design-equations in a limited parameter range (Janssen, 2018). Other software-based applications for the analysis of circular notch flexure hinges are coming with a simple graphical user interface (Vink, 2018; van Beek, 2018). Additionally, inaccessible applications which are developed for published research are reported briefly (Ivanov, 2016). Most of these applications are based on the implementation of empirical design equations derived from analytical considerations or FEM analysis. When analyzing flexure hinges using these tools, the analyzing options are mostly very restricted. Most of these tools apply compromised design equations only within a small parameter range. Apart from that, more complex tools for the analysis (Megaro et al., 2017) and synthesis (Turkkan and Su, 2016; Culpepper and Kim, 2004) of compliant mechanisms with distributed compliance are state of the art. On the contrary they do neither offer the possibility to regard several notch flexure hinges nor calculate the rotational precision or axis shift. Moreover, it has to be mentioned that some of these software applications require licenses for commercial software like MATHCAD or MATLAB, but others are available for free. With this paper the authors try to fill the gap regarding a comprehensive software application for the accurate and non-linear analysis of various notch flexure hinges with the most important contours and broad evaluation criteria which may be advantageous especially for precision engineering to enable an intuitive and quick design process.

According to a plane rotation caused by bending due to a moment or transverse force load, this paper presents a novel design tool for the contour-specific quasi-static analysis of notch flexure hinges. The tool offers the calculation of parameters like bending stiffness, rotational axis shift, maximum elastic strain and outer fiber strain distribution, maximum angular deflection, and deformation of the neutral axis for extensive geometric specifications – computable for a specified load or rotational angle. The calculations will be possible by numerically solving a system of differential equations for large deflections of thin rod-like structures within a few seconds. Further, the design tool provides a plot of the hinge for the given geometry.

The following sections are organized as follows. In Sect. 2 the investigated types and regarded contours of flexure hinges are presented and characterized. In Sect. 3 the analytical approach for the characterization of flexure hinges with the theory of large deflections of rod-like structures based on non-linear modeling is described. Moreover, specific parameters to calculate the motion and strain are investigated, too. In Sect. 4 the development and implementation of the design tool are outlined. Also, the algorithm for the analyzing procedure is illustrated. In Sect. 5 an FEM-based characterization is done for four different hinge contours and results are compared to the design tool-based solution. Finally, conclusions are drawn in Sect. 6.

In the past numerous designs of flexure hinges with no limit to the geometric shape have been developed. It can mainly be differentiated between cut-out geometries and more complex compositions of the compliant segment. An example for a complex geometry would be the proposed butterfly hinge (e.g. Henein et al., 2003; Pei and Yu, 2011) or a topology optimization-based contour (Zhu et al., 2014). However, in this paper the focus is on notch flexure hinges with distinct contour shapes. For this purpose, many notch geometries are being outlined in literature. By far the most commonly used and easy to manufacture circular flexure hinge contour (e.g. Paros and Weisbord, 1965; Wu and Zhou, 2002) is well known for its large bending stiffness and high precision but also for high maximum strain values. On the contrary, a well distributed strain can be achieved with the corner-filleted contour (e.g. Lobontiu, 2003; Meng et al., 2013). As a compromise between high precision and low strain the elliptical contour has been applied (e.g. Smith et al., 1997; Chen et al., 2008). Moreover, parabolic or hyperbolic (e.g. Lobontiu, 2003; Chen et al., 2009) and cycloidal contours (Tian et al., 2010) have been utilized as well as combinations of the mentioned basic geometries (e.g. Zelenika et al., 2009; Lobontiu et al., 2011; Chen et al., 2011).

Beyond that, more complex mathematical functions like the spline contour (Christen and Pfefferkorn, 1998; De Bona and Munteanu, 2005), the power function-based 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) are rarely taken into consideration. Lately higher order polynomial functions were proposed to regulate the bending stiffness, precision and elastic strain distribution to an optimum (Linß et al., 2011a, 2015; Gräser, et al., 2018). For combining the advantages of different notch contours, asymmetric flexure hinges were previously considered (Chen et al., 2005). Also the kinematic behavior in terms of an ideal axis of rotation can be realized with asymmetric flexure hinges (Linß et al., 2011b; Lin et al., 2013). However, in this paper only transversal and axial symmetric notch flexure hinges are investigated because they allow a holistic and intuitive design with regard to the mechanism synthesis. The investigated design and modeling of notch flexure hinges are described in the following subsections.

Subject of this research is a notch flexure hinge fixed at one end and loaded
at the free end as it is shown in Fig. 1 for the initial and the deflected
state. With respect to the application in a mechanism and according to
literature, the flexure hinge with its notch contour design domain is
generally modeled with adjacent link segments (Yong et al., 2008). The
deflection can be modeled in two ways by either specifying the load (a moment
load

Notch flexure hinge with a variable hinge height within the contour design domain, the geometric parameters and the deflected state as a result of a moment and/or a transverse force load.

In this paper only directionally constant forces are accounted and the
equations are based on them. The system of differential equations will vary
for follower forces and may be adjusted for those applications. The vectorial
direction for a given moment is

Regarding Fig. 1, the design domain for the variable notch contour ranges
from point

Geometric flexure hinge parameters related to the total hinge height and their parameter ranges.

The four implemented flexure hinges with their contour-specific
parameters:

In the design tool three typical flexure hinge contours are considered, the circular (Fig. 2a), corner-filleted (Fig. 2b) and elliptical contour (Fig. 2c). Besides that, the advantages of the polynomial contour (Linß et al., 2011a) are implemented and extended to a power function (Fig. 2d) to offer a wider range of possible contours. The power function-based hinge contour offers a great spectrum for the flexure hinge design (Linß, 2015). Due to the variable height function, the contour design may be adjusted from circular-type to corner-filleted-type contours to achieve a desired angular deflection or axis shift.

Three different cases for designing a circular flexure hinge in
dependence of the input parameter values (all shown for

Three different cases for designing an elliptical flexure hinge in
dependence of the input parameter values (all shown for

Height functions of the four implemented flexure hinge contours.

In Table 2 the according height functions of the four implemented flexure
hinge contours are listed. Contour-specific parameters for the exact
definition of each flexure hinge are proposed with

In contrast to form and force-closed joints the materially coherent connection of flexure hinges leads to a restoring force when bent (bending stiffness) – which can be advantageous in technical systems, too. Apart from that, the angular deflection of a flexure hinge is limited by reaching admissible material stress or strain values (maximum angular deflection). Moreover, 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). When applying flexure hinges in a compliant mechanism, this can lead to path deviations compared to the rigid-body mechanism, which are not negligible especially in precision engineering (Venanzi et al., 2005; Linß, 2015).

In this section, the approach of the non-linear analytical characterization of a notch flexure hinge and the equations for its bending stiffness, strain distribution/maximum angular elastic deflection, and rotational precision are presented.

As long as the dimensions of a cross section are small compared to the rod length, the non-linear theory for large deflections of rod-like structures is sufficient to describe the motion behavior of compliant systems (Zentner, 2014). If a flexure hinge is modeled together with adjacent deformable link segments as a bent beam with a variable height, this theory is assumed to be suitable for the calculations in this paper, too. Therefore, the assumption is made that Bernoulli's hypothesis, Saint-Venant's principle, and Hooke's law apply. Shear deformation according to Timoshenko as investigated in (Dirksen and Lammering, 2011) and the effect of anticlastic bending (Campanile et al., 2011) are neglected, because first investigations with the applied theory show a good correlation between the theoretical results and FEM simulations for different flexure hinges (Linß et al., 2017b).

A stationary coordinate system

Parameters for the theoretical characterization of a flexure hinge (depiction of the initial and deflected position) with the model for the determination of the rotational axis shift based on guiding the center with a constant distance, the fixed center approach.

On the contrary to the compliance of a flexure hinge, the bending stiffness
around an axis of a system of coordinates is a measure for the resistance of
an object against deformation under external loads. Due to the fact that
flexure hinges in this paper are mainly exposed to bending around the

The bending stress

For a moment load, the maximum strain occurs, related to the

Due to the fact that flexure hinges do not have a stationary rotation axis,
the term rotational precision is introduced. Notably, in precision
engineering, the rotational precision of a flexure hinge is a very important
performance criterion. Because of the serial connection of several flexure
hinges in the kinematic chain of a compliant mechanism, the rotational axis
shift

Graphical user interface of the PC program detasFLEX (design tool for the analysis of flexure hinges), shown using the example of a corner-filleted flexure hinge.

The absolute value of the rotational axis shift

Furthermore, to minimize the axis shift, the user should know, that
independent from the hinge contour a transverse force leads to a significant
larger axis shift than a moment load for the same angle

In this section the development of a computer program for the analysis of notch flexure hinges with different hinge contours will be described. For this purpose, the previously described theory for large deflections of rod-like structures will be implemented in a graphical user interface (GUI) developed with MATLAB. Therefore, a solution for the non-linear analytical characterization will be received and evaluated. In the following subsections it is shown how the different inputs and selections are realized throughout the GUI (front end) and how they are internally processed by the software (back end) to produce the results which are then displayed in fields and diagrams. Further it is explained how the software will be distributed for license-free public usage. For the development of the GUI the MATLAB version R2017b was used.

The developed GUI for the analysis of notch flexure hinges is shown in Fig. 6. For public usage it was generated as a standalone application called “detasFLEX” using the MATLAB deploy tool for WINDOWS 64 bit computers. Due to this process, no MATLAB license is needed to run the program. Although, the file comes with a Runtime – a database including all the important MATLAB functions – that needs to be downloaded and installed to the computer.

The development of the GUI has been made with the use of the GUI development environment, called GUIDE. It basically is a layout editor where one can graphically design the appearance of an application using input and output text fields, push buttons, sliders, axes and more. GUIDE automatically generates the MATLAB code for modifying the program behavior. This way it is possible to process input data, solve the system of differential equations and generate all result data and diagrams.

For a better overall understanding for a user to analyze a specific flexure hinge using the given design tool, the modular composition of the program is shown in Fig. 7 which can be derived from the GUI in Fig. 6. Mainly the interface is divided into four groups. To start with, there are text fields for the input of data and information about the hinge notation and valid geometries for the chosen contour. In this area the material, the hinge contour, the dimensional and the contour-specific geometric parameters of the flexure hinge may be set. Next the analysis settings can be made. Thereby the program user can choose between the selection of a given deflection angle or a given load and specify its value, set the load case and finally press the “calculate”-button or reset everything to default values. Afterwards, when all results are obtained by the solver, the result values are presented in a separate section of the interface. Also, the geometry, bending line, bending stiffness and precision as well as the strain distribution are charted in diagrams. In addition, features to export data and diagrams into files, print the geometry and zoom into the notation are provided.

Modular structure of the PC program detasFLEX (Analysis).

Detailed flowchart of the PC program detasFLEX.

Internally an algorithm based on the user input and analysis settings is
executed to compute a solution for the boundary value problem and supply all
the important results and diagrams. The underlying procedure is showcased in
the form of a flowchart in Fig. 8 and will be briefly described. When first
executing the program “detasFLEX.exe” a user may specify input values for
material and basic geometry, choose a contour and manipulate contour specific
parameters. These values and selections get read and converted into SI units
for further processing. To make sure all input data comply with the geometric
bounds and are numeric real positive data greater zero, a query is
programmed. In the case of invalid inputs, a warning will be displayed and
parameters need to be adjusted by the user. Following, a constant step size
along

In case a deflection angle is specified, firstly the angle value is divided
into five load steps. It has been shown that the motion behavior for very
large angular deflections may accurately be calculated with the used theory
(Zentner et al., 2017) which is why the angle was limited to

In a following step, when a solution for the system of differential
equations was found, the deflected state of the neutral axis is plotted into
the diagram. Also the bending stiffness and axis shift are evaluated with
the given equations described in Sect. 3 and outlined in their respective
chart. Next, the outer fiber strain distribution and maximum value are being
calculated with the step size

As it is already shown in Fig. 2, the flexure hinges are split up into several sections for the numerical solution. Beam sections with a constant hinge height are solved by referring to a constant geometrical moment of inertia whilst sections containing a variable contour function are referred to the solution of the corresponding height function found in Table 2. The system of differential equations therefore varies for each section. The circular, the elliptical and the power function-based contour are split up into three sections (cf. Fig. 2a, c and d) and the corner-filleted contour is split up into five separate sections when being calculated (cf. Fig. 2b). The numerical solution at the end of each section serves as a set of initial values for the next section. This method guarantees a continuous progression of the neutral axis. Furthermore, using this approach, transition points arise at the exact location in between sections along the neutral axis which enable the evaluation of results precisely at these points. This is especially important when it comes to examining the rotational precision of the regarded flexure hinge.

To demonstrate some of the functionalities the design tool offers, examples are given in this subsection. The program enables a wide variety of different geometry, material, contour and analysis settings selections so that numerous notch flexure hinges for diverse tasks in industry and research may be analyzed within a few seconds with respect to the geometric lower and upper bounds.

Example results of the deformed neutral axis and the outer fiber
strain distribution for a force load computed for an angle input of

Example results of the deformed neutral axis and the outer fiber
strain distribution for different force load inputs to equal

The following examples are done for all four different contours with the same
typical geometric ratios (

FEM-based verification and comparison of the contour-specific and
load-dependent results for a discrete angular deflection of

In another case study, which is presented in Fig. 10, the same four flexure
hinges are regarded but with different analysis settings. For this
application example, the maximum angular deflection that is derived from the
previous example is taken as an input for

FEM-based characterization of a flexure hinge:

To confirm the analytical design tool-based results and therefore the
usability of the provided PC program, finite elements method is considered to
compare and verify the implementation of the non-linear deformation theory.
For the FEM-based 3-D structural simulation ANSYS Workbench 18.2 was used.
The CAD model and FEM model are shown in Fig. 11. For the determination of
the rotational precision in FEM, the same approach as it is described in
Sect. 3.4 is considered by adding an additional part onto the CAD model
according to the chosen fixed center approach (cf. Fig. 5) to measure the
distance between

According to literature, flexure hinges with very variable dimensions are
generally modeled as a 3-D solid structure (Zettl et al., 2005) and if
possible with adjacent link segments (Yong et al., 2008) in all FEM
simulations in this paper. The latter accounts for the considerations for the
analytical characterization in Sect. 3, too. Moreover, large deflections are
also considered in the FEM analysis settings for an accurate comparison with
the analytical calculations due to the non-linear beam theory. Other
assumptions are a linear material behavior and a comparable and fine
discretization of the hinge for all the different contours. The mesh of the
FEM model is chosen to be finely divided in areas of the notch and especially
in areas of the minimum hinge height

In an example FEM analysis for typical geometric parameter values (

Parameter study of a corner-filleted flexure hinge with

Generally, the results are in good correlation with one another. Regarding
the bending stiffness, the maximum deviation of the FEM results compared to
the analytical results is 6.2 % which is indicated to be very precise.
Regarding the three criteria load (

In particular, greater differences may be found in terms of the rotational
precision, especially when the hinge is deflected with a force load. With
reference to the rotational precision it becomes clear that smaller absolute
values for the axis shift in the micrometer range generally lead to higher
deviations between FEM and analytical approaches. For example very small
values of approximately

Another reason for these differences may be the fact that in the analytical
approach no elongation of the neutral axis by tensile forces is considered,
whereas they are possible in the FEM analysis. Because these strains are of
the same magnitude as the axis shift, the discrepancy may be explained. In
addition to that, the results in Table 3 provide insight in a correlation
between the necessary load for achieving a deflection angle of

Another investigation is exemplarily done for corner-filleted hinges with
regard to the bending stiffness for a force load and different total hinge
length ratios

Overall, the stiffness errors are in a range of 0.1 % and 9.4 %.
Thus, the non-linear analytical approach used within the design tool may be
accounted appropriate for the implemented typical parameter value ranges. It
may be expected to receive larger deviations between FEM and the analytical
method for deflection angles larger than

In this paper a non-linear analytical approach for modeling various notch
flexure hinges realizing an in-plane rotation is considered and implemented
in the form of a design tool using the theory for large deflections of
rod-like structures. The design tool was developed with MATLAB as a
stand-alone software application which only requires the license-free Runtime
environment. Four different certain hinge contours are chosen and implemented
in the design tool, the circular, corner-filleted, elliptical and power
function-based contour. Various geometric and material parameters may be
realized to allow for a broad usability in different cases. The analysis is
possible for a moment and a transverse force load case as well as both loads
combined for different lengths of both hinge sides. All three cases may be
computed with a given load or deflection angle up to 45

The presented design tool contributes to an accelerated contour-specific quasi-static analysis of the elasto-kinematic properties of notch flexure hinges with no need for iterative and time-consuming simulations. It therefore may be used for the systematic angle-dependent synthesis of compliant mechanisms with differently optimized flexure hinges (Linß et al., 2015; Gräser et al., 2018). Further research may earmark a synthesis part in the design tool so that required hinge properties may be predefined and an appropriate hinge contour with its suitable geometric parameter values will be proposed by the design tool. Other considerations like shear deformation, which make the program results more accurate, especially for specific cases of non-typical geometries, may be implemented, too. The influence of geometric scaling which is important for precision applications may also be considered (Linß et al., 2018). Additional features like the import of existing arbitrary geometries are possible.

The design tool can
be requested for free on the following website:

SH and SL developed the software detasFLEX and wrote the paper. SL did the FEM-based verification, reviewed the results and the paper. LZ supported the implementation of the theory for large deflections of rod-like structures.

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 714/10-2. Edited by: Marek Wojtyra Reviewed by: two anonymous referees