DESIGN METHOD

DESCRIPTION

Technical field of the invention

The present invention relates to a nonlinear spring mechanism for actuation systems and its design method.

Background

The background of the invention are mechanisms for actuation systems, in particular those for robots and wearable robotic devices envisioned for disaster response scenarios, as well as personal assistance, in every-day work and household environments. In disaster response scenarios, robots and robotic devices shall provide technical support and protect/save human and animal lives, as well as protect, recover and safeguard objects, buildings and sites of great value, reducing risks that can be attributed to human interventions.

Inside household and every-day work environments, robots are designed to improve or restore quality and independence of life after accidents or in old age. Springs in the transmission system of the actuators used as prime drivers represent a key component of this robotic technology.

A first type of actuator is known by the name of Series Elastic Actuators (SEA). The measured deflection of the spring, together with the known stiffness of the spring itself, allows the robot or robotic device to "perceive" forces and torques exchanged thereby with the environment. In addition to active control of these forces and torques, SEAs offer general perspectives of intrinsic impact resilience as well as energy storage and release during recurrent movement tasks.

Until now, SEAs have been almost exclusively designed using linear passive springs, i.e. , with the feature linking the spring’s deflection to the force or torque of action characterized by a single proportional constant. This facilitates actuator calibration and torque control, which can already become challenging in the presence of modelling uncertainties and disturbances.

Nevertheless, the solution space being only a fixed linear spring stiffness fixed as a design parameter is limited. The designer must compensate for torque resolution, impedance rendering, impact absorption and decoupling, as well as energy storage capacity in relation to the achievable drive bandwidth and in relation to the peak torque. This inevitably leads to performance limitations, which prevent robots and robotic devices from achieving soft but also powerful physical interaction performance comparable to biological idols. In fact, biological muscles have been long known to show progressive stiffening behaviour when stretched passively or through electrical stimulation.

A second type of actuator, named Variable Stiffness Actuators (VSA), exploits the progressive and nonlinear characteristics of the torque deflection, offering additional design freedom, e.g. to achieve high torque resolution at lower load levels. This would enable robots to be soft and gentle during physical interaction in delicate environments, but also fast and strong during powerful manipulation and movement tasks.

The so-called VSAs actively modify the passive stiffness properties of the actuation mechanism using a secondary actuator.

Disadvantages of VSAs include an increase in volume, mass and complexity of the mechanism due to the increased number of actuators and the need for additional mechanisms for regulating rigidity.

In particular, the mass of a VSA is drastically high, which, in turn, requires the use of more powerful, therefore more dangerous and more energy-consuming, actuators, within a robot.

Several ideas or concepts of a third type of actuators named fixed passive nonlinear SEA have been reported in literature. The authors of these concepts use nonlinear rubber springs with hysteresis properties, which cause hurdles during control, may lead to instabilities at higher operating frequencies and involve energy losses. The concept is based on a linear spring tensioned via a cable running on a cam profile. Similarly, another concept employs a linear spring deformed through a hypocycloid mechanism (FlypoSEA).

While this approach offers comfortable design freedom to realize the nonlinear progressive torque-deflection characteristic, the mechanism requires space and involves many moving parts increasing its complexity of construction. Often, the used linear spring needs to be pre-tensioned to avoid play when the load is zero.

Summary of the invention

The technical problem posed and solved by the present invention is therefore to provide a nonlinear spring mechanism for actuation systems and its design method which allow to overcome the drawbacks above mentioned with reference to the prior art.

A purpose of the present invention is to devise a nonlinear spring mechanism for actuation systems and its design method capable of implementing a spring with nonlinear progressive stiffening characteristics in a monolithic component forming a right prism.

This problem is solved by a nonlinear spring mechanism for actuation systems and its design method according to claim 1 and claim 14.

For the sake of simplicity, the mechanism devised in this discussion is also indicated by the wording "Rolling Deflecture Mechanism" (ROFL).

Preferred features of the present invention are object of the dependent claims.

The present invention provides several significant advantages with respect to several drawbacks findable in the prior art.

■ Design limitations with linear passive springs: the invention uses shelves having a fitted end and a free end as deflectable beams. The stiffness k of such deflectable beams increases in inverse proportion to the third power of the length L of the shelf. Small length variations imply large nonlinear variations of stiffness, which is promising for the design of compact nonlinear actuators. In this invention, the variation in the active length of the deflectable beam is passively integrated into the mechanism itself in the form of curved bending supporting profiles whereon a deflectable beam subjected to a load rolls and curves. The shape of the supporting profile can be designed using the devised method, in order to obtain the advantage of realizing a wide range of progressive torque bending characteristics and to reduce the design compromises and restrictions associated with passive linear elastic actuators.

^■ Hysteresis in nonlinear passive springs: hysteresis appears when the bending mechanisms are subjected to internal or external friction damping. The invention alters the local stiffness of the deflectable beams by continually varying their free length for loads between zero and the maximum stress capacity of the material. The contact between the deflectable beam and the supporting profile are only rolling contacts which inherently have very small friction effects. By choosing standard materials with low internal damping (such as metals and many plastics) using the method described in the invention, the hysteresis in the rolling bending mechanism is reduced.

^■ Mechanical play in passive nonlinear progressive springs: when the deflection is zero, the invented "rolling bending mechanism" already shows the constant linear rigidity of the dimensioned deflectable beam. The supporting profile of the invention can be designed in such a way that only an infinitesimal load force is required for the deflectable beam to come into contact with the supporting profile, which generates the progressive increase of the "torque- displacement" curve over the entire loading capacity. The non-zero rigidity, when the deflection is zero, always creates a restoring force towards the equilibrium position, which in principle prevents any mechanical play.

^■ Complex and cumbersome design and assembly: the invention realizes the nonlinear progressive spring in a single two-dimensional (2D) monolithic part. The part can be manufactured with arbitrary materials using conventional production methods such as laser or water cutting, electric discharge machining, even stamping or pressing. The monolithic structure does not involve additional moving parts, which would be subjected to friction and wear, or would require the housing of additional bearings or the assortment of relative assembly tolerances to ensure functional integrity. Similarly, no further active components, e.g. motors, are needed to modify the effective length of the deflectable beam. This allows a light and compact integration of the invented device also in existing implementation concepts, not only robotic ones. Other advantages, characteristics and methods of use of the present invention will become apparent from the following detailed description of several embodiments, presented by way of non-limiting example.

Brief description of the drawings

Reference will be made to the figures of the attached drawings, in which:

^■ Figure 1 shows a plan view of a first preferred embodiment of a mechanism according to the present invention;

■ Figure 2 schematically shows the analytical principle of the mechanism of Figure 1 ;

^■ Figure 3 shows characteristic curves relating to the distribution of the deformation y(x), the bending moment T(X) and the bending force f(x) in the sections of a mechanism according to the invention having a circular supporting profile;

^■ Figure 4 shows a plan view of a second preferred embodiment of a mechanism according to the present invention;

^■ Figure 5 shows characteristic curves relating to the distribution of the deformation y(x), the bending moment T(X) and the bending force f(x) in the sections of a mechanism according to the invention having a clothoidal supporting profile.

^■ Figure 6 shows an axonometric exploded view of an actuation system comprising a mechanism according to the invention;

^■ Figure 7 illustrates a detail of a different embodiment of the actuation system of Figure 6;

^■ Figure 8 illustrates a detail of a further different embodiment of the actuation system of Figure 6.

Thicknesses and curves illustrated in the figures above introduced are to be intended as purely exemplary and not necessarily shown in proportion.

Detailed description of preferred embodiments

Various embodiments and variants of the invention will be described below, and with reference to the figures introduced above.

Similar components are denoted in the different figures with the same numerical reference.

In the detailed following description, embodiments and further variants with respect to embodiments and variants already treated within the description itself will be illustrated limitedly with respect to differences with what set forth above. Furthermore, the different embodiments and variants described below are liable to be used in combination, where compatible.

With initial reference to Figure 1 , according to an embodiment of the invention, a nonlinear spring mechanism for actuation systems is overall denoted by 1.

^■ Analytical operating principle: the analytical operating principle which the spring mechanism 1 is based on is described below.

A cantilever beam without mass, with a fitted end and a free end, of length L with constant rectangular cross section of width b and height h, is considered. Bernoulli's beam theory is assumed to be applied.

A position along the beam is expressed by the distance x measured from the end blocked along the beam, at the undeformed neutral fiber.

The deflection of the beam yb(x) is measured perpendicular thereto. An fi_ force loads the free end.

The beam reacts by an internal bending force fb(x) which results in the distribution of the bending moment Tb(x):

For small deviations, the bending moment Tb causes a curvature of the beam Kb(x) * y _b ”(x):

where E is Young's modulus of the beam material and I is the moment of inertia of the beam section area.

By integrating twice with respect to x, the deflection yb(x) is obtained: The integration constants Ci and C2 vanish for the clamped boundary condition relating to the joint (yb (0) = 0). From the above written equation, it appears that the deflection at the free end yi_ = yb (L) in response to the load force fi_ is characterized by the equivalent constant stiffness k.

The stiffness k increases in inverse proportion to the third power of the cantilever length L.

Small length variations affect large variations of nonlinear stiffness.

This observation motivates the concept of deflectable rolling beam illustrated in Fig. 2, that is, a beam which, by deflecting, rolls on a supporting profile.

The clamped beam end connects to a rotary actuator shaft and the free end a load flange.

A sliding hinge ensures that only transverse load forces fi_ are transmitted by the load flange to the free end.

As the beam bends, it rolls onto a supporting profile surrounding itself.

At the transition point x _s , the beam length splits into a supported segment and an unsupported segment. The supporting profile partially unloads the beam across the length of the supported segment.

The unsupported segment resembles a "new" cantilever beam, of shorter length L and therefore higher rigidity.

The desired progressive stiffening can be designed through the shape of the supporting profile.

The mathematical bases to describe this mechanism are illustrated below and the parameterization of the supporting profile is illustrated using two examples. For the sake of simplicity, the derivations are carried out with reference to the neutral fiber of the beam.

The supporting profile is always chosen to be tangential to the beam at x = 0. The design tasks consist of finding the supporting profile with curvature K _S (X) leading to the desired progressive stiffening behaviour. With increasing load force, the beam naturally deflects as described by the equations above until it comes into contact with the supporting profile. This occurs as soon as the curvature Kb of the beam matches the curvature K _S (X) of the profile. At this moment, the beam establishes contact with the supporting profile at point Xs obtained by providing (3) with K _S (X):

(l ap) h EIH X.) ^ 0

Depending on the nature of K _S (X), the above equation can only admit a numerical solution for x _s. The contact is initially established when the load force fi_ reaches the level f _s , such that x _s = 0 and:

for x which tends to 0.

For smooth supporting profiles the transition point x _s continuously moves along x and the beam rolls on the supporting profile if fi_> fs and Kb (x _s ) ³ K _S (XS).

For this reason, the implementations of the inventive concept proposed in this document are called "rolling flexure mechanisms" or "ROFL mechanisms".

For the length of the supported beam, i.e. the section resting on the supporting profile, therefore considering an xi such that 0 < x-i < x _s , the deflection and the slope of the beam are imposed by the supporting profile itself:

The deflection of the unsupported end is defined with the X2 coordinate subjected to 0 < X2 £ L - Xs, such that

The integration constants are defined by the continuous transition between the two segments of the beam

C¾ s p _s |¾|

The nonlinear and progressive force-deflection relation is obtained by calculating yb for X2 = L-Xs and solving for the load force fi_. The differential and progressive stiffness k for the supported beam is calculated as a function of the deflections k w

L

^■ Nonlinear spring mechanism description: the nonlinear spring mechanism

1 is described below in its preferred, but not exclusive, embodiments.

The nonlinear spring mechanism 1 for actuation systems has a monolithic main body 2.

In the present discussion, the term "monolithic" means a complex structure in which several parts can be identified as joining together, not being detachable, forming a single body.

In this regard, the main body 2 can be made with arbitrary materials, for example metals, polymers, composites and other similar materials, and its realization can be performed by conventional production methods such as laser or water cutting, electrical discharge, molding or pressing, aimed at obtaining a single body having the characteristics below indicated.

Advantageously, the main body 2 is substantially flat, having a thickness substantially lower with respect to two transverse dimensions.

The main body 2, therefore, defines a deflection plane according to the two transverse dimensions above indicated.

Usefully, the main body 2 can comprise at least one connection seat 3 which can be used to connect the mechanism 1 to an actuation system or to other mechanisms.

According to the invention, the main body 2 comprises at least one cantilever beam 4 having a free longitudinal end 5 and a constrained end 6.

The cantilever beam 4 is configured to deflect upon application of a tangential load.

In particular, the cantilever beam 4 is configured to deflect in the above mentioned deflection plane.

More particularly, the free longitudinal end 5, hereinafter referred to as the free end 5, is configured to receive a load. In particular, the free end 5 is configured to receive a point load acting tangentially with respect to the free end itself, in order to deflect the cantilever beam 5 in the deflection plane.

Preferably, as illustrated in figure 1 , the free end 5 may have a substantially tubular shape with a development substantially orthogonal with respect to the deflection plane.

Thereby the free end 5 is able to receive connection bars for insertion, for example those indicated below by the reference number 14, or other similar elements in the free end 5 and mechanically configured to receive and transmit a load.

The tangential load is applied to the free end 5, indeed reproducing a cantilever beam scheme having a clamped end and the free end subjected to a concentrated load.

The behaviour of the cantilever beam 4 subjected to a load, therefore, is completely analogous to the behaviour of the cantilever beam previously described in the above mentioned analytical operating principle.

Preferably, the cantilever beam 4 is shaped as a right prism.

As illustrated in the figures, the cantilever beam 4 can be usefully shaped as a right prism having a rectangular section.

Alternative solutions are not excluded wherein the cantilever beam 4 has a conical or pyramidal shape, or wherein it is shaped as a prism whose dimensions vary along its longitudinal development, for example whose cross section narrows as longitudinally proceeding, giving to the cantilever beam 4 a triangular or trapezoidal shape in the deflection plane defined by the main body 2.

Further according to the invention, the main body 2 comprises at least one supporting profile 7 fixed at the constrained end 6.

In particular, the supporting profile 7 is arranged tangentially with respect to the cantilever beam 4 at the constrained end 6.

Preferably, the supporting profile 7 and the cantilever beam 4 are collected around a core 8 of the main body 2.

Advantageously, the core 8 can be configured in such a way as to be associated with an actuator device, for example with a rotary actuator shaft.

Usefully, the core 8 can have an annular shape, as illustrated in figure 1 , thereby being wearable, for example, by a rotary shaft.

Different configurations for the core 8 are not excluded.

For example, the core 8 may be defined by a full body portion on which connection holes that can be associated with an actuator device can be obtained, or it may be provided with quick coupling means.

The supporting profile 7 is configured to allow a selective and progressive supporting rolling contact with the cantilever beam 4 upon deflection of the latter after the application of a load.

In particular, the configuration is such that, following the contact between the cantilever beam 4 and the supporting profile 7, the cantilever beam 4 deflected has a supported portion 9 from the supporting profile 7 and an unsupported portion 10 which defines an effective deflection length of the cantilever beam 4. The effective length has stiffness greater than the full length of the cantilever beam 4 unsupported, with stiffness varying according to a nonlinear relation with the variation in length of the unsupported portion 10.

The supported portion 9 is unloaded from the supporting profile 7, such that the effective stiffness of the cantilever beam 4 deflected is given only by the unsupported portion 10.

An increase in the applied load implies a greater deflection of the cantilever beam 4 which translates into a progressive rolling of the latter on the supporting profile

7, with a resultant increase in the length of the supported portion 9 and a mutual decrease in the effective length of the unsupported portion 10.

As a result, the stiffness of the cantilever beam 4 increases, as the effective length of the unsupported portion 10 decreases.

The stiffness of the cantilever beam, in fact, varies according to the formula:

where L is the value of said effective length, E is the elastic module of the cantilever beam 4, I the moment of inertia relative to the cross section of the cantilever beam 4, fi_ is the load acting on the free end 5, yi_ is the displacement of the free end 5 with respect to an initial equilibrium configuration in which fi_ is zero.

As illustrated in figure 1 , the supporting profile 7 can be a circular profile. Referring to the analytical principle above described, the surface of a circular supporting profile 7 with radius r _c can be expressed in the coordinates of the cantilever element 4 (or of the cantilever beam) with the following:

The supporting profile 7 is tangential to the neutral fiber of the cantilever beam 4 not deformed in x = 0 and curves away therefrom with the constant curvature K _C = 1/r _c.

Considering this relation, the closed solution is obtained for the point of transition Xs and of the level of transition force f _s

The rolling deflection with circular support shows constant linear stiffness up to the load force f _s and a progressive increase in stiffness for greater loads.

As an example, consider a cantilever beam 4 having fixed geometry and properties, for example length L = 35 mm, with constant rectangular cross section of width b = 8 mm and height h = 2 mm.

Figure 3 illustrates the bending curve and the internal loads of the cantilever beam for the ROFL mechanism with a circular supporting profile 7 of radius r _c = 1 /KC = 55 mm and load force fi_ = 10 N. Area A and lines in Fig. 3 correspond to supporting profile 7. Black lines represent the equivalent deflection of the cantilever beam without interference by a supporting profile. Area B indicates the length of the supported portion 9, while area C indicates the unsupported portion 10.

The supporting profile 7 partially unloads the cantilever beam 4 from the internal bending force.

Consequently, the maximum internal bending moment and the stress are directly limited by the constant curvature of the support.

This is a relevant property, since it allows the designer to intuitively dimension the cantilever beam 4 to take full advantage of the load capacities of the material, homogeneously loading the entire length of the cantilever beam 4 up to a desired stress and up to a corresponding deformation energy.

In the scenario illustrated in Fig. 3, the cantilever beam 4 itself, without the support of the profile, would have already exceeded its maximum yield stress limit.

Independently of the shape of the supporting profile 7, in the embodiment illustrated in figure 1 , the main body 2 comprises four cantilever beams 4 and at least four respective supporting profiles 7 symmetrically arranged in a Cartesian plane system.

In particular, the supporting profiles 7 are 8 in number and arranged in such a way that each cantilever beam 4 is placed between two of them.

This configuration allows to obtain a progressive stiffening for perpendicular loads in both directions, deflecting the cantilever beam 4 both clockwise and counter clockwise.

Preferably, as illustrated in figure 1 , two supporting profiles 7 are arranged between two cantilever beams 4.

Such supporting profiles 7 define a support body 1 1 .

Usefully, the support body 1 1 comprises one or more connection seats 3.

Advantageously, the support body 1 1 can comprise lightening cavities 12.

Thereby the mechanism 1 is lighter, while maintaining the solidity required for its use.

Alternative solutions wherein the support body 1 1 is full, without lightening cavities, are not excluded.

Figure 4 shows a second embodiment of the nonlinear spring mechanism 1 .

This second embodiment is entirely analogous to the first described embodiment and differs in that the supporting profile 7 is a clothoidal profile.

The performance of the supporting profile 7, therefore, is shaped as a clothoid spiral, that is, the curvature thereof varies linearly along its length.

Referring to the principle above described and considering the condition for which the curvature k varies linearly with the length L, the following contact point is obtained In comparison to the circular case, the clothoidal supporting profile does not show a linear deflection regime of the forces, although shows an immediate progressive stiffening. Since the initial curvature of the clothoid is K _C (0) = 0, only an infinitesimal load force is required for the cantilever beam 4 to come into contact with the supporting profile 7.

Fig. 5 shows the bending curve and the internal loads for a ROFL mechanism with clothoidal support.

The cantilever beam 4, the loading conditions and the representation of the supporting profile 7, i.e. supported length and free length are identical to those referred to in Fig. 2.

In comparison to the circular support, a residual load force fi_ remains in the length of the supported portion 9.

This leads to a bending moment in this portion, which increases linearly in accordance with the k _c curvature of the clothoid, ensuring that the cantilever beam 4 remains in contact with the supporting profile 7.

In case of same cantilever beam 4 and same load conditions, the clothoidal profile shows an anticipated progression of stiffness which translates into a lower deflection of the cantilever beam 4 compared to the circular profile.

^■ Actuation system description: figure 6 illustrates an actuation system 13 comprising at least one nonlinear spring mechanism previously described and indicated by reference number 1.

The mechanism 1 illustrated in Figure 6 is entirely analogous to that illustrated in Figure 1 and previously described.

Usefully, the free ends 5 of each cantilever beam 4 are provided with connection bars 14 apt to configure the free ends 5 to receive a tangential load.

Furthermore, the actuation system 13 comprises an actuator device 15 connected to the mechanism 1 at the constrained end 6 of the cantilever beam 4. The actuator device 15 can be of the type of an actuation flange that can be moved by a robotic actuator to perform various types of movements, preferably rotary movements.

Different solutions wherein the actuator device 15 is different, for example a rotary shaft moved by a motor or a movement group associated with a robotic actuator, are not excluded.

Usefully, the actuator device 15 comprises connection elements 16 inserted in the connection seats 3 in order to form an integral coupling between the mechanism 1 and the actuator device itself.

The connecting elements 16 illustrated are of the screws type, but technically equivalent solutions, known in the state of the art and different from the illustrated screws, are not excluded.

Preferably, the actuation system 13 comprises a load flange 17 coupled, or configured to be coupled, with the free end 5 of the cantilever beam 4.

In the embodiment of figure 6, the load flange 17 comprises a first portion 17a and a second portion 17b which are mirrored to each other.

The two portions 17a, 17b are coupled into a sandwich, with the mechanism 1 being interposed therebetween.

The first portion 17a faces outwards, while the second portion 17b is coupled to the actuator device 15.

Usefully, the load flange 17 comprises a radius 18 for each cantilever beam 4.

In particular, each portion 17a, 17b comprises a radius 18 for each cantilever beam 4.

Load transmission means 19 is placed at the ends of each radius 18, adapted to receive rotary movements and convert the latter into linear movements.

The load transmission means 19 is associated with the connection bars 14 in such a way as to interact with the cantilever beams 4 in order to create the conditions of a point load able to deflect the same in the deflection plane.

The load transmission means 19 has the function of coming into contact with various external stresses and transforming the same into tangential loads to be transmitted to the free ends 5 of the cantilever beams 4.

The load transmission means 19 illustrated in figure 6 is Robert-type mechanisms configured in such a way as to convert approximately a rotary motion into a linear motion.

By using this principle, the loads acting on the transmission means 19 are transmitted tangentially with respect to the free ends 5, independently of the acting load.

Solutions wherein different mechanisms are used in place of Robert mechanisms are not excluded, although still configured in such a way as to approximately transform a rotary motion into a linear motion, for example Chebyshev-type mechanisms. Figure 7 illustrates in a simplified way a detail of a different embodiment of the actuation system 13.

The actuation system of figure 7 is entirely analogous to that previously described and differs in that it comprises a plurality of nonlinear spring mechanisms 1 connected in parallel.

The configuration is such that mechanisms 1 are jointed and free ends 5 are integrally coupled to each other.

Thereby, stiffnesses of the individual mechanisms are combined.

This solution is useful in the case of, for production needs, an increase in stiffness is to be combined while maintaining the height of the mechanisms unvaried.

Figure 8 illustrates in a simplified way a detail of a further different embodiment of the actuation system 13.

The actuation system of figure 8 is entirely analogous to that previously described and differs in that it comprises a plurality of nonlinear spring mechanisms 1 connected in series.

The configuration is such that mechanisms 1 can be rotated in opposite directions and free ends 5 are integrally coupled to each other.

Thereby, the possibility of deflecting for cantilever beams 4 can be increased. ^■ Description of a robotic manipulator: a robotic manipulator, for the sake of simplicity not illustrated in the figures, can comprise at least one nonlinear spring mechanism 1 such as those described in the embodiments previously described. Thereby, the robotic manipulator will have a higher sensitivity in response to different stresses, in such a way as to perform delicate or powerful actions depending on the type of sensed load.

Similarly, a robotic manipulator can comprise the actuation system 13 previously described in the various embodiments.

^■ Description of a method for designing mechanism 1 : a method, computer implemented, for designing a nonlinear spring mechanism 1 is the following.

The method comprises a step of dimensioning a cantilever beam 4 having a free longitudinal end 5 configured to receive a load and a constrained end 6, with the cantilever beam 4 being configured to deflect upon application of a tangential load.

The dimensioning of the cantilever beam 4 comprises at least the determination of parameters characterizing the mechanical behaviour of the cantilever beam itself.

In particular, the parameters comprise at least the following: stiffness module E of the material constituting the cantilever beam 4, geometric shape of the cross section of the cantilever beam 4, total length of the cantilever beam 4.

The method, implemented by means of an electronic computer, further comprises a step of dimensioning a supporting profile 7 fixed to the constrained end 6.

In particular, the dimensioning of the supporting profile 7 comprises a geometric modelling of the supporting profile itself.

The supporting profile 7 is configured to allow a selective and progressive supporting rolling contact with the cantilever beam 4 upon deflection of the latter after the application of a load.

After the above contact, the cantilever beam 4 deflected has a supported portion 9 from the supporting profile 7 and an unsupported portion 10 which defines an effective deflection length of the cantilever beam 4 having higher stiffness than a full length of the cantilever beam 4 completely unsupported, with stiffness varying according to a nonlinear relation with the effective length variation of the unsupported portion 10.

Dimensioning a cantilever beam 4 and dimensioning a supporting profile 7 satisfy the following relation: , .... fh . „ El

.

Vb LT

where k is the effective stiffness of the cantilever beam 4, L is the value of said effective length, E is the elastic module of the cantilever beam 4, I is the moment of inertia relative to the cross section of the cantilever beam 4, fi_ is the force acting on said free end, yi_ is the displacement of the free end 5 of the cantilever beam 4 with respect to an initial equilibrium configuration in which fi_ is zero.

The present invention has been hitherto described with reference to preferred embodiments. It is to be understood that other embodiments belonging to the same inventive core, as defined by the scope of protection of the claims set forth below, may exist.