Elasto-kinematic comparison of flexure hinges undergoing large displacement

Mechanism and Machine Theory 110 (2017) 50–60

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Elasto-kinematic comparison of flexure hinges undergoing large displacement

MARK

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Pier Paolo Valentini , Ettore Pennestrì Department of Enterprise Engineering, University of Rome Tor Vergata, Via del Politecnico, 1, 00133 Rome, Italy

A R T I C L E I N F O

ABSTRACT

Keywords: Compliant mechanism Flexure hinge Dynamic-spline Elasto-kinematics Motion invariant

The design of precision compliant mechanisms requires the assessment of the elasto-kinematic characteristics of the flexure hinges connecting bulky parts. Due to geometrical nonlinearities, a comprehensive analysis requires the functional properties to be evaluated within the feasible range of relative displacement. This investigation proposes the use of kinematic invariants to characterize the main features of a compliant mechanism. In particular, this paper offers the comparison of five common flexure hinges in terms of the relative motion kinematic invariants. By using the dynamic spline formulation, for each hinge typology, a flexible multibody model is developed to obtain the fixed and moving centrodes and the diameter of the inflection circle of the relative motion. In order to fully characterize the elasto-kinematic behaviour, the simulative models are also used to compute the equivalent stiffness as a function of the rotation angle. The results show that the differences among the various types of hinges is relevant in terms of both kinematic and compliance characteristics. The findings and the methodology herein outlined are new tools for the optimal design and synthesis of flexure joints.

1. Introduction The mobility of a compliant mechanism is due to the deformation of its parts rather than to the inclusion of kinematic pairs, as in traditional mechanism [1]. A compliant mechanism is often manufactured as a single piece of material and it does not possess any discrete degree of freedom in a strict sense. These mechanisms are widely used in different fields of application, from robotics, to micromechanics, to aerospace and biomechanics. In order to simplify the design of such devices, engineers often localize deformation zones, as to mimic the presence of standard kinematic pairs (i.e. virtual hinges). Mechanisms with lumped compliance emulate the traditional rigid-link ones, where kinematic joints are replaced with flexure hinges. Consequently, methods conceived to design traditional mechanisms can be adapted accordingly. In fact, the well-known pseudo-rigid body approach is based on the outlined criterion [2,3]. Generally speaking, flexure hinges are able to undergo large deformation and connect stiff links (almost undeformable). When compared to classical joints, they have several advantages, including the absence of assembly procedure, negligible friction loss, clearance and wear (there are no sliding parts). Moreover, they have an intrinsic elasto-kinematic behaviour that has both positive and negative consequences. As with a standard spring elements, flexure hinges generate a reaction against the deformation. This property can be useful because, in the same compliant joint, there are the capabilities of movement and reaction using a single structure. On the other side, the kinematic behaviour is affected by the loads acting on the hinge. In precision mechanics and control systems this is often undesirable. These considerations motivated many scientific contributions on the flexure hinge performances,

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Corresponding author. E-mail address: [email protected] (P.P. Valentini).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.12.006 Received 3 October 2016; Received in revised form 28 November 2016; Accepted 20 December 2016 0094-114X/ © 2016 Elsevier Ltd. All rights reserved.

Mechanism and Machine Theory 110 (2017) 50–60

P.P. Valentini, E. Pennestrì

especially when the displacements become large [4,5], and specific design methodologies are available [6–9]. In most of the cases, numerical approaches based on finite elements are used [10,11], but scientific literature also reports analytical studies for simple cases [12,13]. The above mentioned studies are mainly focused on structural and elastic performance of the flexure hinges. On the other hand, in precise applications, it is also important to assess the elasto-kinematic behaviour in the whole range of deformation. Most of the contributions in literature investigate the relative motion of two bodies connected by a flexure hinge in terms of rotation pole (finite or instantaneous) [14]. In particular, on this topic, Venanzi et al. [15] pointed out the importance of the position analysis of compliant mechanism with different types of flexure hinges. Few years later, Linb et al. [16] assessed the influence of asymmetric flexure hinge on the location of the axis of relative motion. More recently, Guo et al. [17] proposed an equivalent pin model in order to address the simulation of leaf hinge taking into account the moving rotation centre. Verotti et al. investigated the possibility of designing a flexure hinge based on the theory of conjugate surfaces [18]. On year later, he also suggested an analytical expression to locate the centre of rotation in a curved flexure hinge [19]. The main original contribution of the present paper is the comparison, based on instantaneous kinematic invariants [20–23], of the elasto-kinematic behaviour for five types of common flexure hinges connecting two bulky rigid bodies. According to the authors’ best knowledge, the analysis of kinematics invariants in flexure hinge design and a direct comparison among different solutions have never been addressed in previous studies. This investigation is motivated by the observation that the invariants, such as the fixed and moving centrodes and the inflection circle, characterize the relative motion, independently from the actual structural embodiment [21]. Therefore, the relative motion between the rigid bodies coupled with the hinge, under predefined boundary conditions, is analysed in terms of instantaneous kinematics invariants. The equivalent overall elastic characteristics (the spring-like reaction of the hinges), within the whole range of deformation, is also assessed. Due to the necessity of including geometrical nonlinearities of large displacements, the investigation makes use of numerical models developed with the dynamic spline flexible multibody formulation [24]. This results into a compact and efficient simulations of slender flexible structures undergoing large displacements, as in other compliant mechanisms analyses [25,26]. The paper is organized as follows. In the first part, the flexure hinge types under investigation are introduced. In the second part, the details of the numerical simulative models are presented. In the third part, the results of both kinematics and compliance investigations are presented and discussed. 2. Flexure hinge description and modelling strategy The five compliant hinges considered in the study, depicted in Fig. 1, are the circular profile flexure hinge, the elliptical profile flexure hinge, the leaf flexure hinge, the solid flexure cross hinge and the two-flexure cross hinge. These joints are widely used in compliant mechanisms. The circular and elliptical profile flexure hinges are characterized by a localized necking in the middle. They differ due to the

Fig. 1. The five flexure hinges under investigation.

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variation of the thickness of the material between the two connecting elements (body 1 and body 2 in Fig. 1). The leaf flexure hinge is manufactured with a constant thickness beam which connects the two bulky elements. In some embodiments, rounded edges are included in the attachment regions. The cross hinges are characterized by the presence of two constant-thickness beams arranged in a cross. In the solid-flexure variant, the beams are joined in the middle (more precisely the cross is a continuum body), producing a single structure; in the two-flexure variant, the beams are independent and overlapping. Due to the necessity of evaluating the performance for large displacements, in which the beams of the hinges undergo large deflection, the investigation is based on the use of flexible multibody dynamics approach. Considering the planar nature of the relative motion controlled by the hinges under investigation, a two-dimensional simulation environment is chosen. The two bulky bodies connected by the hinges are included as two rigid bodies with three degrees-of-freedom each. One of them is considered fixed (body 1 in Fig. 1) and the other one (body 2) is constrained by a prescribed rotation about an axis perpendicular to the plane of motion (two degrees-of-freedom are left free). This ensures that on the moving bulky body only an external torque is acting, namely the reaction due to the enforced rotation. This gives a specific loading condition to achieve a comparable elasto-kinematic behaviour. This boundary condition is also representative of the common hypotheses adopted for the experimental testing and simulation of the hinges where the bending and stretching loads are usually neglected to assess elastokinematic properties [27]. The enforced rotation is set as a linear ramp from 0° to 90° in 10 s and it is imposed through a rheonomic constraint at position level. To limit the effects of inertia on the bodies and achieve a quasi-static simulation a low velocity is chosen. The hinges are composed of beams modelled by means of dynamic spline formulation. This tool is motivated by its capability to describe large displacements with a limited set of degrees of freedom [24,28]. In particular, the deformed shape of the neutral beam fibre can be described with a geometrical spline. A spline, a very common entity in computer-aided geometric design, is a piecewise polynomial p(u ) of a curve expressed in parametric form and defined by a set of m control points { P0 ... Pm −1} as: m −1

∑ bi (u) Pi

p (u ) =

(1)

i =0

where bi (u ) are the blending functions (polynomial functions of the scalar parameter u ) fitting the control points. For simulation purposes, the control points of the spline can be considered as the degrees-of-freedom of the beam. The overall deformed shape can be reproduced by adjusting the position vector of these points. One should observe that the control points are not necessarily physical points of the beam (as it happens in finite element formulation) but they are mere geometrical descriptors. For a 2D curve with m control points, we have 2m degrees-of-freedom (the position of each point is located with the coordinates q2i , q2i −1). The equations of motion of a dynamic spline are expressed in terms of these coordinates combining Lagrange differential equations and position constraints:

⎧ ⎪ d ∂T − ⎨ dt ∂qi̇ ⎪ ⎩

∂T ∂qi

⎧ ∂ψ ⎫T − ⎨ ∂q ⎬ λ = Qi ⎩ i⎭ ψ=0

i = 1..2m (2)

where:

qi is the i -th generalized spline coordinate; dq q̇i = dti is the time derivative of the i -th generalized spline coordinate; T is the kinetic energy of the spline system; ψ is the vector containing the constraint equations, written in terms of generalized coordinates (constraint between two splines or between a spline and a rigid body); λ is the vector containing the Lagrange's multipliers associated to each constraint; Qi is the sum of all generalized forces acting on the i -th spline coordinate qi . Since in a flexure hinge the beam does not have any applied external force, the generalized force vector includes only the contribution of the elastic forces (stretching and bending) generated by the deformation of the beam (torsion is negligible for planar simulations). The i -th elastic force component can be computed as the derivative of the elastic energy with respect to the i -th coordinate qi as:

Qi = Qi,bending + Qi,stretching = −

1 2

2

2

∫ spline

EI

∂(εb − εb0 ) 1 ds − ∂qi 2

∫ spline

EA

∂(εs − εs0 ) ds ∂qi

(3)

εb0

εs0

and are the bending and extension where εb and εs are the bending and extension strains along axial direction, respectively, strains at the reference configuration (initial state), respectively, E is the Young's modulus of the material; I is the moment of inertia of the beam cross section with respect to the axis of bending; A is the area of the cross section. The kinetic energy is expressed as:

T=

1 2

∫

ṗ (u )T Mṗ (u ) ds (4)

spline

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Fig. 2. Nomenclature and modelling details of compliant hinges using dynamic splines. Circular profile, elliptical profile and leaf hinges are modelled using a single spline (drawing at the top left), solid flexure cross hinge is modelled using two connected splines (drawing at the top right) and two-flexure cross hinge is modelled using two splines (drawing at the bottom). Control points, control polygon and main geometrical parameters are highlighted. d p(u )

where M is the inertia matrix of the cross section of the beam, and ṗ (u ) = dt . Full details about how to compute the kinetic energy and the partial derivatives of the elastic energy are reported in Ref. [24]. Since the equations of motion of a dynamic spline is written in terms of Lagrange's equations according to Eq. (2), they can be directly included in standard rigid-body equations of a multibody system based on the same Lagrange's formulation. Therefore, the equations can be solved for the location of control points of each spline, which represent the unknowns of the direct dynamic problem. 3. Modelling details of each hinge Due to the difficulty in achieving analytical closed-form solution of flexure hinge properties, the investigation is accomplished using numerical models within a reference scenario. Although the investigation is carried out for specific cases, this should not affect the generality of results. The modelling strategy is the following. For each hinge, the main elastic elements are recognized. For each of them a dynamic spline is drawn choosing n=6 equally spaced control points. The splines are drawn at the initial condition (undeformed beams). Appropriate kinematic constraints (details are provided later in the section) are introduced in order to preserve the attachment conditions to the bulky bodies (body 1 and body 2). An additional driving constraint is enforced to the body 2 in order to produce a constant-velocity rotation. All the modelled hinges are assumed to be made of elastic material (Young's modulus E = 20000 MPa , Poisson's ratio ν = 0.45). The thickness along the direction orthogonal to the plane of motion is assumed to be 1 mm (constant for all the cases). The initial horizontal distance between the two rigid bodies edges is 100 mm and the center of the hinges (midpoint between the two rigid bodies) is located at {100 0 }T of the global reference system. Fig. 2 depicts modelling details and nomenclature for all the simulated hinges (note that circular profile, elliptical profile and leaf hinges are modelled using the same scheme but using different crosssection parameters). 3.1. Circular rounded hinge The hinge deformed shape is modelled as a single spline p(u ) with six control points { P0 ... P5 } (see Fig. 2). The variable thickness t (s ) along the spline arc length s is expressed according to: 53

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t (s ) = (d + tmin ) − 2 sd − s 2

(5)

where d is the initial distance between the connected bodies edges (100 mm) and tmin is the minimum thickness of the notch (2 mm). The arc length is [24]:

s=

∫0

u

d p(u ) du du

(6)

}T

Since the hinge is clamped to each end A = { 50 0 and B = {150 0 four constraint equations (6 scalar equations) can be established:

}T

of the corresponding bulky bodies 1 and 2 (see Fig. 1),

P0 − A = 0 P5 − B = 0 d p(u ) du

u =0

d p(u ) du

u =1

× mA = 0 × mB = 0

(7)

The first two constraint equations follow from the incidence of the first and last control points to the corresponding attachment points A and B on the bulky bodies. The last two express the orthogonality between the initial and final spline tangent vectors and two unit vectors mA = { 0 1}T and mB = { 0 1}T parallel to the edge of the attachment point on the corresponding rigid body. 3.2. Elliptical rounded hinge The hinge is modelled in the same way as the circular rounded hinge. The only difference regards the law of variation of thickness in the plane of motion:

t (s ) = (d + tmin ) − 2 ρsd − ρ2 s 2

(8)

ρ2

where the parameters are the same of those in Eqs. (5) and (6), and is the ratio between the two radii of the ellipse (assumed to be 2). The same constraint scheme (see Eq. (7)) of the circular rounded hinge is applied. 3.3. Leaf flexure hinge The hinge is modelled in the same way as circular and elliptical rounded hinges. The only difference is that in this case the thickness in the plane of motion is set constant to 2 mm. The same constraint scheme of circular and elliptical rounded hinges is applied (see Eq. (7)). Rounded edges at the boundaries are not considered. 3.4. Solid-flexure cross hinge The hinge is modelled using two splines p(u ) and q(v ), with six control points each { P0 ... P5 } and { Q0 ... Q5 } (see Fig. 2). Both splines have a constant thickness of 1 mm in the plane of motion. Each beam is rigidly fixed to the rigid bodies with the same constraint scheme of the previous hinges, considering as attachment ⎧ ⎫T 2 ⎬ , points A1 = { 50 50 }T , A2 = { 50 − 50 }T , B1 = {150 50 }T and B2 = {150 − 50 }T . In this case, the edge vectors m A1 = ⎨ 2 2 ⎭ ⎩ 2 ⎫T ⎧ ⎧ ⎫T ⎧ ⎫T 2 ⎬ , m B1 = ⎨ 2 − 2 ⎬ , m B2 = ⎨− 2 − 2 ⎬ are inclined of 45° in order to fulfill the initial cross configuration: m A2 = ⎨− 2 2 2 2 2 2 2 ⎭ ⎩ ⎩ ⎭ ⎩ ⎭ P0 − A2 = 0 Q0 − A1 = 0 P5 − B1 = 0 Q 5 − B2 = 0 d p(u ) du d p(u ) du d q(u ) du d q(u ) du

⋅m A2 = 0 u =0

⋅m B1 = 0 u =1

⋅m A1 = 0 v =0

⋅m B2 = 0

(9)

v =1

The system of Eq. (9) includes 12 scalar equations. Three other constraint equations (3 scalar equations) are added in order to 54

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take into account the solidity in the middle of the beam, described as the perpendicularity of the tangent vectors of the two spline at the common point:

p (u ) u =0.5 − q (v ) d p (u ) du

v =0.5

d q (u ) ⋅ dv

v =0.5

=0

=0

(10)

v =0.5

3.5. Two-flexure cross hinge The hinge is modelled as the solid-flexure hinge using six control points with each spline as depicted in Fig. 2. The difference with respect to the previous hinge is the absence of the three scalar constraint equations (Eq. (10)) imposing the central connections between the two beams. They are therefore independent. Both splines have a constant thickness in the plane of motion of 1 mm. 4. Evaluation of instantaneous kinematic invariants: results and discussion The kinematic performance under evaluation is the capability of the hinge to reproduce the behavior of an ideal revolute joint (virtual hinge). In a revolute joint, the joint center lies where the relative axis of rotation and the plane of motion intersect. For observers on the bodies connected by the joint, its location is independent from the relative displacement and on the applied load. Conversely, in a compliant hinge, the relative axis changes its position depending on the applied loads, also for observers on the bodies connected by the hinge. As previously mentioned, in all the simulated cases the applied load is a torque. There are not vertical or horizontal forces acting on the bodies. To represent the kinematic proprieties of the relative motion, the comparison among different types of hinges is performed on the basis of instantaneous kinematic invariants [20–23,29], namely the fixed and moving centrodes and the inflection circle diameter. The centrodes are the paths traced by the instantaneous center of rotation of a rigid body moving in a plane. The fixed centrode refers to the trace in the fixed reference frame, the moving one refers to a reference system attached to the moving body. For a rigid body moving in a plane, the inflection circle (often called first Bresse circle) is the locus of the points whose acceleration is parallel to velocity (their normal acceleration vanishes). Therefore, the trajectory of the inflection circle points has zero curvature. The rolling without slipping of the moving centrode with respect to the fixed centrode reproduces the relative motion between two bodies. Hence, for different hinges, the properties of the relative motion can be objectively compared on the basis of centrodes shapes and geometry in a manner that is not influenced by the topological arrangement. Given a planar motion of a body (body 2) with respect to a fixed body (body 1), the equation of fixed centrode c1 can be expressed in Cartesian parametric form [20,22,23]:

⎧ ⎧ c1, x ⎫ ⎪ x2 − c1 = ⎨ c ⎬ = ⎨ ⎩ 1, y ⎭ ⎪ y + ⎩ 2

dy2 ⎫ d ϑ2 ⎪ ⎬ dx 2 ⎪ d ϑ2 ⎭

⎧ ⎪ x2 − =⎨ ⎪ y2 + ⎩

dy2 dt ⎫ dt d ϑ 2 ⎪ ⎬ dx 2 dy2 ⎪ dt d ϑ 2 ⎭

⎧ ⎪ x2 − =⎨ ⎪ y2 + ⎩

dy2 1 ⎫ dt ω2 ⎪ ⎬ dx 2 1 ⎪ dt ω2 ⎭

(11)

ϑ2 and ω 2 are the coordinates of the origin, the rotation angle and the angular velocity of a reference frame where O = { x2 y2 attached to body 2, respectively. Similarly, for the moving centrode c2 we have: }T ,

dy2 dy2 1 ⎧ dx 2 ⎫ ⎧ dx 2 1 ⎫ ⎧ c2, x ⎫ ⎪ d ϑ sin ϑ2 − d ϑ2 cos ϑ2 ⎪ ⎪ dt ω2 sin ϑ2 − dt ω2 cos ϑ2 ⎪ ⎬=⎨ ⎬ c2 = ⎨ c ⎬ = ⎨ 2 ⎩ 2, y ⎭ ⎪ dx 2 cos ϑ2 + dy2 sin ϑ2 ⎪ ⎪ dx 2 1 cos ϑ2 + dy2 1 sin ϑ2 ⎪ ⎭ ⎭ ⎩ dt ω2 ⎩ d ϑ2 d ϑ2 dt ω2

(12)

Another important invariant for describing the properties of the relative motion is the diameter of the inflection circle δ , whose length depends on the curvature radii of the two centrodes during the rolling [20]:

1 1 1 = − δ rc2 rc1

(13)

where rc1 and rc2 are the radii of curvature of the fixed and moving centrodes, respectively. Assuming a canonical reference frame [20], the diameter of the inflection circle can be computed as follows:

δ= E= F=

E2 + F2 d 2x

2

d ϑ 22 d 2y2 d ϑ 22

+

dy2 d ϑ2

=

d 2x 2 1 dt 2 ω22

+

dy2 1 dt ω2

−

dx 2 d ϑ2

=

d 2y2 1 dt 2 ω22

−

dx 2 1 dt ω2

(14)

The comparison of the centrodes among the five hinges is reported in Fig. 3 where both fixed and moving curves are plotted using the Eqs. (11) and (12), assuming a rotation from 0° to 90°. The reference frame on the moving body is chosen with the origin O initially located at { 200 0 }T and the x axis aligned to the fixed frame X axis. Due to their invariance, the centrodes shape is not 55

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Fig. 3. Fixed and moving centrodes of the investigated hinges for a relative rotation between the connected bodies in a range 0°−90°, overall view including connecting bodies (above) and detailed view (below) – (colour figure in the online version).

affected by the choice of the moving reference frame. Eqs. (11), (12) and (14) can be evaluated by means of the results of the integration of the Eq. (2) in terms of the control points of the spline(s). The coordinates of the origin O = { x2 y2 }T are computed as follows: 1. For the single element hinges (leaf, circular and elliptical)

⎧ B − O cos ϑ2 ⎫ ⎬ O=B+⎨ ⎩ B − O sin ϑ2 ⎭

(15)

2. For the two-element hinges (two-flexure and solid-flexure)

⎧ M − O cos ϑ2 ⎫ ⎬ O=M+⎨ ⎩ M − O sin ϑ2 ⎭ M=

B1 + B2 2

(16)

The computation of the derivatives of (15) and (16), necessary to evaluate the Eqs. (11), (12) and (14), is straightforward. For all the hinges, the location of the axis of rotation changes during the motion. Except for the solid-cross flexure hinge, the instantaneous centre of rotation moves up following the driven body (body 2) during the imposed motion. The fixed and moving 56

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Fig. 4. Distance from the midsize location {100 0 }T of the instantaneous centre of rotation of the moving body versus the relative rotation angle – (colour figure in the online version).

centrodes of the circular and elliptical profile flexure hinges are very short compared to those of the other hinges. The centrodes of the circular profile, elliptical profile and leaf hinge are close to be a portion of a circumference and they are almost symmetrical with respect to the physical mid-plane of the arrangement. This qualitative assessment is then confirmed by the evaluation of the inflection circle diameter (see Fig. 5 and subsequent discussion). In the first part of the motion, the centre of rotation of the solidcross hinge displaces almost vertically and then horizontally. It sketches like an “L” in the fixed body reference frame and a “U” in the moving body reference frame. Unlike the other hinges, the instantaneous axis of rotations migrates in the opposite direction of the driven body displacement. The two-flexure cross hinge shows the largest centrodes among the investigated hinges with variable concavity. In order to evaluate the behaviour of the hinges during the rotation, Fig. 4 reports the plot of the distance Δ = (c1, x − 100)2 + c1,2 y of the instantaneous centre of rotation c1 (the point of contact and tangency between the centrodes) with respect to the location of an ideal revolute joint in the middle of the two bodies at {100 0 }T . For the circular profile, elliptical profile and the solid-cross hinge, the instantaneous axis of rotation migrates of less than 5% of the distance between the connected bodies. On the other side, the leaf hinge and the two flexure cross hinge have greater displacement. At the end of imposed rotation, the instantaneous axis of rotation for the leaf hinge is displaced more than 20% of the distance between the connected bodies and that of the two-flexure cross hinge more than 40%. For these reasons, the leaf hinge and the two flexure cross hinge are far from satisfying the condition of virtual revolute joint. Fig. 5 reports the length of the diameter of the inflection circle for the five hinges in the entire range of relative rotation. It can be observed that the circular and elliptical profile hinge and the leaf hinge show an almost constant inflection circle diameter. This occurrence means that the difference of the geometric curvatures of the fixed and moving centrodes does not change as well. On the other side, the two cross hinges show a more relevant variation during the enforced motion.

Fig. 5. Length of the inflection circle diameter of the relative motion versus the relative rotation angle (spikes at the beginning are to numerical processing) – (colour figure in the online version).

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Fig. 6. Inflection circles of the five hinges at the beginning of the imposed rotation (ϑ 2 = 0°) – (colour figure in the online version).

Fig. 6 depicts the inflection circles of the five hinges at the beginning of the imposed rotation (ϑ2 = 0°). For all except the solid cross hinge, the inflection circle lays on the semi-plane towards the moving body (body 2).

5. Evaluation of elastic compliance: results and discussion Another interesting assessment is about the equivalent elastic compliance of the hinges. This property is deduced from the reaction torque exerted by the kinematic driving constraint that imposes the rotation of the body 2. With reference to the general equations of the multibody system (Eq. (2)), the reaction torque Tdriving can be computed as:

Tdriving = λdriving

(17)

where λdriving is the Lagrange's multiplier associated to the rheonomic constraint. Fig. 7 reports the plots of the reaction torques which are normalized to each maximum value (the value at the maximum displacement at the end of the imposed rotation) in order to make comparison on a common basis. It can be observed that the circular profile, elliptical profile and the solid cross hinge have a very similar trend of the reaction torque with negligible differences. The leaf hinge shows comparable reaction torque but with a lower slope. The two-flexure hinge has a completely different behaviour. Its reaction torque is always lower (up to 18%) than those of the others and its shape is concave. In order to directly evaluate the elastic compliance of the hinges, it is also useful to compute the equivalent stiffness khinge as the derivative of the torque Tdriving with respect to the angle of rotation ϑ :

Fig. 7. Dimensionless reaction torques of the driving constraint that imposes the rotation of body 2, as functions of the relative rotation angle.

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Fig. 8. Equivalent stiffness of the investigated hinges, as a function of the relative rotation angle.

khinge =

∂Tdriving (18)

∂ϑ

The results of the computation are summarized in Fig. 8. They refer to the finite differences numerical differentiation of the normalized torques plotted in Fig. 7. None of the hinges has a constant equivalent stiffness. Except from the two-flexure cross hinge, the equivalent stiffness decreases when the angle of rotation increases and the trend is almost linear. The elliptical profile hinge has the smallest variation of stiffness (bounded to 13%). The variation is bounded to 25% for the circular profile and for the leaf hinge. Circular profile hinge, elliptical profile hinge, leaf hinge and solid cross hinge have comparable equivalent stiffness. The two-flexure hinge has the greatest variation of the equivalent stiffness among all the investigated hinges. Moreover, this variation has a parabolic trend which evidences relevant geometrical nonlinearities. 6. Conclusion Five types of flexure hinge commonly used in compliant mechanisms are investigated in terms of elasto-kinematic behaviour when subjected to large displacements. Specific flexible multibody models for five types of flexure hinges are developed. They are based on the use of dynamic spline formulation that has confirmed to be suitable for addressing compliant mechanisms simulations. All the investigated hinges show complex behaviours far from the simplification of virtual revolute joint which is often used in the synthesis and analysis of compliant mechanisms. On the one hand, the assessment of the centrodes offers better insights about the motion of the centre of relative rotation between the bodies connected by the hinge. The circular profile hinge and the elliptical profile hinge show the closer behaviour to a virtual revolute joint. The leaf hinge and the two-flexure solid hinge show the largest displacement of the instantaneous axis of rotation. The circular profile hinge, the elliptical profile hinge and the leaf hinge show an almost constant diameter of the inflection circle and therefore the connected bodies evidenced an epicycloidal type of relative motion. To the best of the authors’ knowledge, this type of behaviour has not been proposed or observed in previous investigations. This should open new possibilities in the search of kinematic equivalence between traditional mechanisms and compliant ones. Considering the elastic behaviour, variable stiffness characteristic was observed for all the hinges. With the exception of the twoflexure cross hinge, the equivalent stiffness decreases as the angle of rotation increases. The circular profile hinge, the elliptical profile hinge and the solid cross hinge show similar trend and amplitude of the equivalent stiffness. The elliptical profile hinge shows the smallest variation and the solid cross hinge the largest among the three. The two-flexure cross hinge has a completely different behaviour with an increasing nonlinear trend of the equivalent stiffness and a variation of more than 150% during the articulation. It is hoped that the observations herein reported offer guidelines to designers for the choice of flexure hinges in compliant mechanisms. The results of the study encourage a deeper investigation on the use of motion invariants for the study, comparison and simulation of compliant mechanisms. The opinion of the authors is that they can open a new field for addressing the design of compliant mechanisms. References [1] L.L. Howell, Compliant Mechanisms, John Wiley and Sons, Inc, USA, 2001. [2] L.L. Howell, A. Midha, T.W. Norton, Evaluation of equivalent spring stiffness for use in a pseudo-rigid-body model of large-deflection compliant mechanisms, J. Mech. Des. Trans. ASME 118 (1) (1996) 126–131. [3] Y.-Q. Yu, L.L. Howell, C. Lusk, Y. Yue, M.-G. He, Dynamic modeling of compliant mechanisms based on the pseudo-rigid-body model, J. Mech. Des. Trans. ASME 127 (2005) 760–765. [4] H. Ahuett-Garza, O. Chaides, P.N. Garcia, P. Urbina, Studies about the use of semicircular beams as hinges in large deflection planar compliant mechanisms,

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