WO2024099923A1

WO2024099923A1 - Non-reciprocal micromechanical device

Info

Publication number: WO2024099923A1
Application number: PCT/EP2023/080759
Authority: WO
Inventors: Jacopo Marconi; Davide Enrico QUADRELLI; Francesco Braghin
Original assignee: Politecnico Di Milano
Priority date: 2022-11-07
Filing date: 2023-11-06
Publication date: 2024-05-16

Abstract

Unidirectional mechanical device comprising at least one array (10; 20) of three or more cells (100; 200). Each cell (100; 200) of the array comprises: a floating mass (101; 201); at least one first elastic element (102; 202) connecting the floating mass (101; 201) to a reference external structure for the unidirectional mechanical device, wherein the at least one first elastic element (102; 202) comprises at least one first folded beam; at least one second elastic element (103) connecting the floating mass (101) to a second floating mass of a preceding cell of the at least one array (10), wherein the at least one second elastic element (103) comprises at least one second folded beam; at least one third elastic element (104) connecting the floating mass (101) to a third floating mass of a succeeding cell of the at least one array (10), wherein the at least one third elastic element (104) comprises at least one third folded beam; an electric system configured to control a potential difference acting on the floating mass (101; 201), determining an overall electrostatic stiffness of the at least one array (10; 20).

Description

Title: Unidirectional mechanical device DESCRIPTION Technical field The present invention relates to a unidirectional mechanical device. In particular, the invention relates to a micrometer scale unidirectional mechanical device, which can be integrated in MEMS (MicroElectroMechanical Systems) systems. Such unidirectional mechanical device, in certain frequency bands, is able to make mechanical waves pass in just one direction, while blocking the mechanical waves traveling in the opposite direction. Prior art A reciprocity mechanical principle provides that a mechanical system stressed in a first point and observed in a second different point behaves in the same way as if stressed in the second point and observed in the first point. A way to constitute a non-reciprocal and therefore unidirectional mechanical system is to break the temporal invariance of the system. In the context of periodic structures, this unidirectional effect is expressed in the form of directional bandgaps and non-symmetric scatter diagrams. In recent years, scientific works have dealt with the theoretical and experimental description of unidirectional mechanical systems made through periodic structures of time-varying stiffness. Trainiti G and Ruzzene M Non-reciprocal elastic wave propagation in spatiotemporal periodic structures 2016 New Journal of Physics 18(8) 1- 22 ISSN 13672630. In this publication, the authors identify a non- reciprocal behaviour of periodic systems modulated in space and time. Such periodic systems provide for rod continuous elements. Vila J, Pal R K, Ruzzene M and Trainiti G A Bloch-based procedure for dispersion analysis of lattices with periodic time-varying properties 2017 Journal of Sound and Vibration 406 363-377. In this publication, the authors consider discrete systems modulated in space and time. Marconi J, Riva E, Di Ronco M, Cazzulani G, Braghin F and Ruzzene M Experimental Observation of Nonreciprocal Band Gaps in a Space-Time- Modulated Beam Using a Shunted Piezoelectric Array 2020 Physical Review Applied 13 031001. In this publication, the authors present an implementation solution to modulate the stiffness of a beam, using piezoelectric patches with appropriate circuits, aimed at modifying their equivalent stiffness properties. Chen Y, Li X, Nassar H, Norris A N, Daraio C and Huang G 2019 Physical Review Applied 11 064052. In this publication, the authors discuss about elastic beams with magnetic resonators driven by electric coils. Wang, Y., Yousefzadeh, B., Chen, H., Nassar, H., Huang, G., & Daraio, C. 2018 Physical Review Letters. In this publication, the authors discuss about an array of masses connected by driven magnets. Nassar H, Yousefzadeh B, Fleury R, Ruzzene M, Al`u A, Daraio C, Norris A N, Huang G and Haberman M R 2020 Nature Reviews Materials 5 667-685. In this publication, the authors offer a review of the recent literature in the non-reciprocity field. Discussed experimental evidence of nonreciprocal Bragg scattering in activated materials was observed in a structure of ring magnets sliding over a common axial rail and housed by grounded solenoids; each ring repels or attracts its host solenoid by a force proportional to the current input of the solenoid. Patent application US2022/166406A1 relates to a microelectromechanical MEMS resonator includes a spring-mass system having a first weight portion, a second weight portion, and a central spring portion in between the weight portions. The unidirectional systems of prior art are merely demonstrations aimed at observing the phenomenon without being effective technical solutions. Prior art solutions are complex, lying at the limits of stability, with many components that need to be manually adjusted from time to time. Moreover, the prior art solutions are of such size that it is impossible to imagine an effective practical use thereof. In general, the known unidirectional systems remain confined to the level of speculation and demonstrative prototypes created at the macro- scale, with complicated solutions that are difficult to implement. Summary of the invention An object of the present invention is to overcome drawbacks of the prior art. A further object of the present invention is to provide a unidirectional filter for devices that use elastic waves as the main means of transduction. A further object of the present invention is to provide a unidirectional sensor, in particular produced by MEMS technologies. A further object of the present invention is to provide a unidirectional device that is an effective technical implementation and is easily programmable. These and other objects are achieved by a unidirectional mechanical device according to the features of the attached claims, which are an integral part of the present description. A solution idea underlying the present invention is to provide a unidirectional mechanical device whose periodic structure has a stiffness that can be modulated over time and is adjustable through the appropriate variation of electrical voltages and a relative potential difference. Such device, in particular producible by MEMS technologies, uses an electrostatic principle for stiffness modulation. The device provides for the repetition of some elementary cells, made of springs (stiffnesses) and masses that house electrodes, in particular with parallel faces, for the electric modulation. The device uses a mechanical array of resonators, whose equivalent mechanical properties are modulated by controlling the electrostatic forces acting on masses at the interface of specific electrodes. This type of unidirectional mechanical device may be used both as a new sensor and as a filter, even inside more complex apparatuses. In particular, the invention provides a unidirectional mechanical device comprising at least one array of three or more cells. Each cell comprises a floating mass and at least one first elastic element that connects the floating mass to a reference external structure for the unidirectional mechanical device. The unidirectional mechanical device comprises an electric system configured to control a potential difference acting on the floating mass. The electric system determines an overall electrostatic stiffness of the at least one array by controlling the potential difference. In a first embodiment, as it will be further described, the potential difference acting on each floating mass is determined by a first electrode positioned on the floating mass itself and by a second electrode positioned on the reference external structure. The potential acting on the first electrode and on the second electrode determines the acting potential difference, controllable in a substantially independent manner for each floating mass. In a second embodiment, as it will be further described, the potential difference acting on each floating mass is determined by a first, and unique, electrode positioned on the floating mass itself. The potential acting on the first electrode determines the potential difference between the specific floating mass and further adjacent floating masses in the array, also placed at a respective potential that is controllable in a coordinated manner for the set of all of the floating masses in the array. Advantageously, the unidirectional mechanical device provides a solution able to interact and modify the response to mechanical waves passing therethrough. Advantageously, the unidirectional mechanical device provides a non- symmetrical response, according to a non-reciprocity of stress. Advantageously, the unidirectional mechanical device allows operating at lower frequencies compared to the electromagnetic waves used by the sensors which such device may be coupled to. Advantageously, the unidirectional mechanical device can be produced by using traditional MEMS technologies. Further features and advantages will become more apparent from the following detailed description, of preferred non-limiting embodiments of the present invention, and from the dependent claims which outline preferred and particularly advantageous embodiments of the invention. Brief description of the drawings The invention is illustrated with reference to the following figures, provided by way of non-limiting example, wherein: ^ Figure 1 exemplifies a first embodiment of a cell in an array of a unidirectional mechanical device. ^ Figure 2 exemplifies three cells according to the first embodiment in an array of a unidirectional mechanical device. ^ Figure 3 exemplifies three cells according to a second embodiment of an array of a unidirectional mechanical device. ^ Figure 4 illustrates an example of array corresponding to the first embodiment of cell. ^ Figure 5 illustrates a further example of array corresponding to the first embodiment of cell, which can be made by MEMS technologies. ^ Figure 6 illustrates an example of array corresponding to the second embodiment of cell. ^ Figure 7 illustrates a further example of array corresponding to the first embodiment of cell. ^ Figure 8 illustrates a further example of array corresponding to the first embodiment of cell. ^ Figure 9 illustrates a further example of array corresponding to the first embodiment of cell. ^ Figure 10 illustrates a further example of circular array corresponding to the first embodiment of cell. ^ Figure 11 illustrates a further example of circular array corresponding to the first embodiment of cell, which can be made by MEMS technologies. In the different figures, analogous elements will be identified by analogous reference numbers. Moreover, in the presence of a plurality of elements equivalent to each other in the same figure, only some of them will be indicated by the reference number, to improve the understanding of the figure. Detailed description The unidirectional mechanical device according to the present invention consists of a device that can be stressed in a non-reciprocal manner, that is usable as a wave guide or filter, is one-dimensional, that at certain frequencies allows mechanical waves traveling in a direction of the device to pass while blocking the mechanical waves traveling in the opposite direction. As it will be further described, the unidirectional mechanical device comprises an array of oscillators organized in cells. Figure 1 exemplifies a first embodiment of a cell 100 in an array of a unidirectional mechanical device. The cell 100 comprises a floating mass 101. The cell 100 comprises at least one first elastic element 102 connecting the floating mass 101 to a reference external structure for the unidirectional mechanical device; in these examples, the reference external structure will be schematized as a “ground” constraint. A part of an electric system (not represented) of the unidirectional mechanical device is associated with the cell 100. Such electric system is configured to control a potential difference acting on the floating mass 101. By controlling the potential difference, it is possible to determine an overall electrostatic stiffness of an array of cells, as it will be further described. The cell 100 further comprises at least one second elastic element 103 connecting the floating mass 101 to a second floating mass of a preceding cell (not represented in Figure 1, but visible in Figure 2). The cell 100 further comprises at least one third elastic element 104 connecting the floating mass 101 to a third floating mass of a succeeding cell (not represented in Figure 1, but visible in Figure 2). In general, the electric system comprises at least one first electrode (not represented) positioned on the floating mass 101. In this embodiment, the electric system further comprises at least one second electrode 105 positioned on the reference external structure (and thus, schematized as “ground” in the Figure). The electrode 105 is configured to electrically interact with the first electrode (not represented) positioned on the floating mass 101, thus interacting with the mass 101 itself. Still with reference to the nomenclature exposed in Figure 1, the cell 100 is the cell of coordinate xi and composed of a mass m that is free to move along the horizontal axis and corresponding to the floating mass 101. The floating mass 101 is connected to ground through a spring 102 having stiffness kg and to preceding xi-1 and successive xi+1 masses in the array of cells, through further springs 103 and 104 having stiffness k. In general, it is advantageous for the stiffness k or kg to be the same for all of the cells of the array, to simplify the design of the device; however, this is just a non-limiting example. A gap is arranged inside each mass 101 in order to house an electrode 105, preferably having faces parallel to the internal walls of the mass 101. In particular, the electrode 105 is fixed to the ground. Preferably, the i-th mass m, as well as the entire structure of the array, is placed at a constant voltage V0 level whereas a voltage V1, which is variable as a function of time, is imposed to the electrode 105, being thus The electrode 105 along the respective electrode on the mass 101 thus behave as a parallel-wall capacitance, with differential behavior. When the floating mass 101 is moved by an amount xi as indicated in the Figure, one face of the mass 101 approaches the electrode 105, whereas the opposite face of the mass 101 moves away. In this situation, electrostatic forces acting on the mass 101 arise, respectively: FL coming from the left of the mass and directed toward the right, as in the figure, and FR coming from the right of the mass and directed toward the left, as in the figure. Assuming that FL is greater than FR, the net effect is given by fe = FL - FR. Such net effect may be expressed as a function of the coordinate x of the mass 101, as:

where ΔV(t) = V₀-V₁(t), ε is the dielectric constant in vacuum, A the area of the electrode facing the walls, x₀ the initial space between mass and electrode under equilibrium conditions. Expanding this force to the third order with a Taylor series, we obtain the following expression, valid for small displacements around the equilibrium position

from which it is noted how the resultant of the electrostatic forces produces a linear stiffness and a cubic stiffness (at the third order), both modulated over time by ΔV(t). In particular, imposing ^₁(^) = ^₁cos (^_^ ^ + ^₁) we will obtain:

and consequently a linear stiffness _^^ ⁽ _^ ⁾ ₌ 2^^ ^2 4^^ ^ ^ ₁ + ^2 ^ ^ − ^ ^₀^₁^ cos⁽^_^ ^ + ^₁ ⁾ + ^ ^2^ cos⁽2^_^ ^ + 2^ ^2 2 ₀ 2 2 _{1 1} ⁾ ^^₀^^^^^^^^^ ^ ^ ^₀^^^^ ^ ^₀ ^^^ ^_^0 ^_^1 ^_^2 composed of a constant term, a harmonic having frequency ωm and one having frequency 2ωm. Under the hypothesis that V0 / V1 ≈ 10, it is reasonable to neglect the second harmonic, since kE1 = 40∙kE2. If we consider that the system can be approximated as linear, the motion equation for the i-th mass can be written as

Defining with k0 = kg – kE0 the total ground stiffness (constant) and with ^m = kE1 / kE0 the modulation amplitude, we can rewrite the previous equation as

From a design point of view, it is noted that it must be k0 > 0 and 0 ≤ ^m < 1 for the system to be mechanically stable. These considerations will become further clear by considering the description made herein below, in which a single cell 100 already described is associated with other cells, within an array provided in the unidirectional mechanical device according to the present invention. Figure 2 exemplifies three cells 100 in an array of a unidirectional mechanical device. The unidirectional mechanical device comprises at least one array 10 of three (or more) cells, each cell 100 of the array being made as above described. To observe a non-reciprocal effect of the array structure, in the unidirectional mechanical device according to the present invention, a number N equal to or greater than 3 cells is required, as it will be further described. In the example of Figure 2, the modulation of the voltage Vn(t) with n = 1, 2, 3 to consider all of the cells is appropriately phase-shifted by φn = 2πn/N with n ordinal of the single cell, thus modulating the overall electrostatic stiffness of the array over time. In general, although increasing the number N of cells brings the system closer to the ideal conditions, due to practical constraints it is often preferable to use the minimum number necessary to observe the desired effect. Therefore, hereinafter below we will dwell in the case of three cells, although all considerations will be valid and extendable to a greater number of cells. The voltages considered for each cell, imposed through the respective electrodes 105, are therefore:

It is therefore expected that the electric system of the unidirectional mechanical device is configured to vary the potential difference acting on the floating mass over time, modulating it periodically with a phase displacement between different cells. In particular, the phase displacement is dependent on an ordinal position n of the cell in the array and is further dependent on a total number N of cells in the array. The present invention thus provides a unidirectional mechanical device which, forced at one end, has different propagation features compared to the case in which it was forced from the other end. It is possible to evaluate the unidirectionality (or dispersion) features by applying a Plane-Wave Expansion Method (PWEM) to the lumped parameter model illustrated above, thus obtaining a non-symmetric dispersion for positive and negative wavenumbers. In other words, a unidirectional mechanical device structure controlled by a suitable electric system as described allows achieving at least partially unidirectional propagation, for certain frequencies, of mechanical elastic waves. Indeed, the electric system of the unidirectional mechanical device is able to control an electrostatic force acting between adjacent floating masses, to determine an overall electrostatic stiffness varying over time. Figure 3 exemplifies three cells according to a second embodiment of an array 20 of a unidirectional mechanical device. In this embodiment, the unidirectional mechanical device comprises an array of three (or more) cells 200. Each cell 200 comprises a floating mass 201 and at least one first elastic element 202 connecting the floating mass 201 to a reference external structure for the unidirectional mechanical device. Therefore it should be noted that this second embodiment is simplified compared to the first embodiment, since it does not provide for mechanical springs interposed between adjacent cells. The electric system of the unidirectional mechanical device is anyway configured to control a potential difference acting on the floating mass 201, thus determining an overall electrostatic stiffness of the array 20. As already described, such electric system is configured to vary the potential difference over time, modulating it periodically with a phase displacement between different cells. It should be noted that in this second embodiment it is provided to directly apply a voltage Vn(t) with n = 1, 2, 3 varying over time individually for each of the cells, thus defining three respective capacities C dependent on the mutual position (xn-1, xn) of the three cells, respectively considered two by two. In general, the electric system comprises at least one first electrode (not represented) positioned on the floating mass 201. In this configuration the ground springs 202 remain at a preferably constant stiffness kg whereas the other mechanical springs that join the masses and that are present in the first embodiment of Figure 2 are removed. Now an alternating voltage with a zero constant component is applied to each floating mass 201:

In this way, the square of the difference of the voltages applied between two adjacent masses becomes:

and the electromechanical stiffnesses between the masses can be obtained (likewise to what has already been done for the previous embodiment), thus obtaining the motion equation for the i-th mass: ^^̈_^ + ^^_^ − ^_^,^−1(^)(^_^ − ^_^−1) + ^_^+1,^(^_^+1 − ^_^) = ^(^) where k_ij(t) represents the equivalent electrostatic stiffness between the masses i and j and where F(t) represents an additional forcing term generated by the modulation of the voltage over time. If unwanted, the additional forcing term may be compensated by means of proper additional electrodes to be placed on the masses, according to a variant not shown. While requiring a more onerous control for the electric system, this second embodiment, compared to the first one, has the advantage to reduce the spaces and to increase the floating mass of the cells, for instance allowing the reduction in the frequencies of interest of the device, especially if used as a filter. In general, even in this second embodiment the phase displacement introduced by the electric system is dependent on an ordinal position of the cell in the array and is further dependent on a total number of cells in the array. Moreover, in general, the electric system is further configured to control an electrostatic force acting between adjacent floating masses to determine the electrostatic stiffness therebetween, forming a unidirectional system with an overall electrostatic stiffness, as described. Figure 4 illustrates an example of array 10 corresponding to the first embodiment of cell 100. The array 10 of three (or more) cells is repeated in periodic form. Each cell of the array 10 comprises a floating mass 101 and a first elastic element 102 connecting the floating mass to a reference external structure (herein exemplified as a ground bond). Preferably, the first elastic element 102 is shaped as a folded beam. In general, an alternative to the folded beam is the cantilever beam. Each cell of the array 10 comprises at least one second elastic element 103 connecting the floating mass 101 to a second floating mass 101 of a preceding cell in the array 10, and at least one third elastic element 104 connecting the floating mass 101 to a third floating mass 101 of a succeeding cell in the array 10. Preferably, even the at least one second elastic element 103 and the at least one third elastic element 104 comprise at least one second or third folded beam. In this embodiment, the second or third folded beam is outer to two succeeding floating masses 101 in the array 10. The electric system, as already described, comprises at least one first electrode 101b positioned on the floating mass 101 and further comprises at least one second electrode 105 positioned on the reference external structure (herein exemplified as ground bond) and configured to electrically interact with the at least one first electrode 101b. In this embodiment, the at least one second electrode 105 is at least partially inserted in a corresponding gap of the floating mass, in particular externally thereto. The exemplified structure of the array 10 is useful to constitute a unidirectional mechanical device. In particular, lumped parameter models can be used, such as the one exposed above, to design masses, springs, and electrodes in order to obtain the desired mechanical modulation features (in terms of filter or wave guide). The sizing of the elastic elements can be done by imposing the desired stiffness value and inverting the formula

where E is the elastic module, w, h and L the width, height and length of the rod, respectively, Nf the number of folds of the rod (equal to 1 for the cantilever rod). The electromechanical parameters, however, can be chosen by changing the size of the electrodes (x0, A) and the applied voltages (V0, V1). Figure 5 illustrates a further example of array 10 corresponding to the first embodiment of cell, which can comprise a MEMS structure made by MEMS technologies. The array 10 which makes up at least in part the unidirectional mechanical device comprises at least one initial cell 11 and one final cell 12, connected to an own floating mass, respectively. Such initial cell 11 and final cell 12 are in particular provided with respective actuation or measurement electrodes, as commonly provided in the MEMS systems. As exemplified, the array 10 provides for a symmetric arrangement for the three (or more) cells with respect to a development axis of the array. In other words, the elastic elements 102, 103 and 104 are replicated symmetrically on both sides of the floating mass 101, to improve the distribution of loads and inertia. Preferably, the array 10 of this example can comprise a MEMS structure produced by MEMS technologies. The unidirectional mechanical device which the array 10 is part of allows, depending on the propagation direction (from the initial cell 11 toward the final cell 12 or vice versa), defines different bandgaps. In the case of the array 10, the bandgaps are directional, namely different in propagation from the left to the right or from the right to the left (in the figure) by appropriate control by the electric system, as already described. Figure 6 illustrates an example of array 20 corresponding to the second embodiment of cell 200. The array 20 of three (or more) cells is repeated in periodic form. Each cell of the array 20 comprises a floating mass 201 and a first elastic element 202 connecting the floating mass to a reference external structure (herein exemplified as ground bond). Preferably, the first elastic element 202 is shaped as a folded beam. In general, an alternative to the folded beam is the cantilever beam. The electric system, as already described, comprises at least one first electrode (not represented) positioned on the floating mass 201. The electrodes of different cells are configured to interact with each other by electrostatic effects. Figure 7 illustrates a further example of array 10 corresponding to the first embodiment of cell 100. In this example, compared to what has been described in Figure 4, the second elastic element 103 and the third elastic element 104 are interposed between two succeeding floating masses 101 in the array 10, inside the profile of the cell. Figure 8 illustrates a further example of array 10 corresponding to the first embodiment of cell 100. In this example, compared to what has been described in Figure 4, the electric system comprises at least one first electrode (not represented) positioned on the floating mass 101 and further comprises at least one second electrode 105 positioned on the reference external structure (herein exemplified as ground bond) in a suitable central gap inside the floating mass 101. Figure 9 illustrates a further example of array corresponding to the first embodiment of cell 100. In this example, compared to what has been described in Figure 8, the second elastic element 103 and the third elastic element 104 are interposed between two succeeding floating masses 101 in the array 10, inside the profile of the cell. Figure 10 illustrates a further example of circular array 10 corresponding to the first embodiment of cell 100. In this case, the array 10 substantially corresponds to what has been already described, with a circular involute profile. Figure 11 illustrates a further example of circular array 10 corresponding to the first embodiment of cell 100, which can comprise a MEMS structure made by MEMS technologies. This example, as visible in the enlargement on the right, substantially corresponds to the example of Figure 10, except that it does not provide a symmetrical arrangement of the cells with respect to a circular development axis of the array, being the electrode 105 on the sole outer side. In the left global view, it can be noted that the array being circular comprises three respective cells that can be identified as initial cell 11 or final cell 12, respectively connected to a floating mass. Indeed, the circular array 10 allows intervening on waveforms that are expressed through clockwise or anti-clockwise rotations (circular movements), exhibiting unidirectionality in this direction of rotation through appropriate control of the electric system, similarly to what has already been described. In other words, the periodicity of the unidirectional mechanical device is not limited to linear shapes, but could also have other more complex shapes, circular as in the example or with different developments (squared, polygonal, etc.). This example of circular array 10 is particularly adapted to be produced by MEMS technologies. Industrial applicability The unidirectional mechanical device according to the present invention may be designed to be effective at various frequencies, preferably of the order of a few tens of kHz. The frequency response of the unidirectional mechanical device can be optimized by establishing appropriate stiffness values of the elastic elements, mass of the floating masses, as well as by carefully controlling the electric parameters of the electric system. Unidirectional mechanical devices according to the present invention can be used in modern telecommunications systems, in MEMS filters and in non-reciprocal mechanical filters. Unidirectional mechanical devices according to the present invention provide solutions that are small in size and can be effectively produced using currently available technologies. Considering the description herein reported, the person skilled in the art may devise further changes and variants, in order to meet contingent and specific needs. For instance, the specific geometry of each cell, of the arrays and of the resulting devices may be modified and varied based on the specific application. Moreover, a unidirectional mechanical device may comprise one or more arrays, based on the specific application. Therefore the embodiments herein described are to be considered as illustrative and non-limiting examples of the invention.

Claims

CLAIMS 1. Unidirectional mechanical device comprising at least one array (10; 20) of three or more cells (100; 200), each cell (100; 200) of said at least one array (10; 20) comprising: - a floating mass (101; 201); - at least one first elastic element (102; 202) connecting said floating mass (101; 201) to a reference external structure for said unidirectional mechanical device, wherein said at least one first elastic element (102; 202) comprises at least one first folded beam; - at least one second elastic element (103) connecting said floating mass (101) to a second floating mass of a preceding cell of said at least one array (10), wherein said at least one second elastic element (103) comprises at least one second folded beam; - at least one third elastic element (104) connecting said floating mass (101) to a third floating mass of a succeeding cell of said at least one array (10), wherein said at least one third elastic element (104) comprises at least one third folded beam; - an electric system configured for controlling a potential difference acting on said floating mass (101; 201), determining an overall electrostatic stiffness of said at least one array (10; 20).

2. Unidirectional mechanical device according to claim 1, wherein said electric system is further configured to vary over time said potential difference, said potential difference being periodically modulated with a phase displacement between different cells.

3. Unidirectional mechanical device according to claim 2, wherein said phase displacement is dependent on an ordinal position of said cell (100; 200) in said at least one array (10; 20) and is further dependent on a total number of cells in said at least one array (10; 20).

4. Unidirectional mechanical device according to any one of claims 1 to 3, wherein said at least one second folded beam (103) and/or said at least one third folded beam (104) are/is interposed between or outer to two successive floating masses (101) in said at least one array (10).

5. Unidirectional mechanical device according to any one of claims 1 to 4, wherein said electric system comprises at least one first electrode (101b) positioned on said floating mass (101).

6. Unidirectional mechanical device according to claim 5, wherein said electric system further comprises at least one second electrode (105) positioned on said reference external structure and configured to electrically interact with said at least one first electrode (101b).

7. Unidirectional mechanical device according to claim 6, wherein said at least one second electrode (105) is at least partially inserted in a corresponding gap of said floating mass (101).

8. Unidirectional mechanical device according to any one of claims 1 to 7, wherein said at least one array (10) comprises at least one initial cell (11) and one final cell (12), each connected to a respective said floating mass (101) and provided with respective actuation or measurement electrodes.

9. Unidirectional mechanical device according to any one of claims 1 to 8, wherein said electric system is further configured to control an electrostatic force acting between adjacent floating masses (101; 201) to determine said overall electrostatic stiffness.

10. Unidirectional mechanical device according to any one of claims 1 to 9, wherein said at least one array (10; 20) is provided with a substantially symmetrical arrangement for said three or more cells (100; 200) with respect to a development axis of said at least one array (10; 20).

11. Unidirectional mechanical device according to any one of claims 1 to 10, comprising a MEMS structure.