Patent Publication Number: US-2011074247-A1

Title: Systems and Methods for Resonance Frequency Tuning of Micromachined Structures

Description:
FIELD 
     The present application discloses methods for tuning a resonance frequency of resonant structures, such as for example energy harvesters or sensors. Furthermore it relates to devices, e.g. micromachined devices, with tunable resonance frequency. 
     BACKGROUND 
     Future wireless sensor networks will comprise sensor nodes which occupy a volume of typically a few cm 3 . The scaling down of batteries for powering these sensor nodes faces technological restrictions as well as a loss in storage density. For this reason a worldwide effort is ongoing to replace batteries with more efficient, miniaturized power sources. Energy harvesters based on the recuperation of wasted ambient energy are a possible alternative to batteries. Several harvesting concepts have been proposed, based on the conversion of thermal energy, pressure energy or kinetic energy. 
     Kinetic energy harvesters or vibration energy harvesters convert energy in the form of mechanical movement (e.g. in the form of vibrations or random displacements) into electrical energy. For conversion of kinetic energy into electrical energy, different conversion mechanisms may be employed, for example based on piezoelectric, electrostatic or electromagnetic mechanisms. Piezoelectric harvesters employ active materials that generate a charge when mechanically stressed. Electrostatic harvesters utilize the relative movement between electrically isolated charged capacitor plates to generate electrical power. Electromagnetic harvesters are based on Faraday&#39;s law of electromagnetic induction and generate electrical power from the relative motion between a magnetic flux gradient and a conductor. 
     A vibration energy harvester has its maximum power output when input vibrations closely match its resonance frequency, which is determined by the material properties and the dimensions of the harvester&#39;s parts. However, in practical applications, input vibrations may occur in a wide frequency spectrum. The input vibration frequencies may vary over time or may initially be unknown. In such situations the vibration energy harvester structure may stay out of resonance, and generate very low power or no power at all. It would be clearly advantageous to have a single device that operates effectively over a range of input vibration frequencies. This may be obtained by developing resonators with wider bandwidths or by providing actuators to tune the harvester&#39;s resonance frequency. 
     There are a limited number of studies related to tuning the resonance frequency of vibration energy harvesters. For example, in “Resonance tuning of piezoelectric vibration energy scavenging generators using compressive axial preload”, Smart Materials and Structures 15, 2006, 1413-1420, Leland et al describe a method wherein axial forces are applied on the suspension of a macroscopic double-clamped beam for changing the effective stiffness of the oscillating structure. In this approach, a simply supported piezoelectric bimorph is used as an active element, with a proof mass mounted at the bimorph&#39;s center. A compressive axial preload is applied to the bimorph, reducing its stiffness and thus the resonance frequency of the device. The stiffness decrease is a function of the load applied in a given system. However, the change in the apparent stiffness is inversely proportional to the initial stiffness. This means that the largest tuning effect can be achieved by starting with a low stiffness. Besides the relatively low stiffness requirement, the maximum axial load applied to the structure is determined by the buckling load, which is lower in a low stiffness system. In addition, it may be difficult to fabricate the system proposed by means of micromachining methods. The system obtained is operated ‘passively’, which means that it could be adjusted periodically to tune its resonance frequency but would require no continuous actuation. 
     In V. R. Challa et al, “A vibration energy harvesting device with bidirectional resonance frequency tunability”, Smart Materials and Structures 17, p. 015035, 2008 a magnetic force is used to alter the overall stiffness and thus the resonance frequency of an energy harvesting device. Resonance frequency tuning is based on adjusting the position of permanent magnets. The position of the magnets is controlled externally. Resonance frequency tuning of a MEMS resonator using this approach may be unfeasible. 
     Resonance frequency tuning of oscillating structures is used in various MEMS resonators. More in particular, for resonators based on a comb structure (comprising interdigitated fingers) methods for post-fabrication resonance frequency tuning have been proposed. For example, in “A closed-form approach for frequency tunable comb resonators with curved finger contour”, Sensors and Actuators A 141, p 523-529, 2008, K. B. Lee et al describe frequency tunable resonators with curved finger contours. However, fabricating the ideal design is not easy and it may require a lot of space. 
     In “Vertically-shaped tunable MEMS resonators”, Journal of Microelectromechanical Systems, Vol. 17, No 1, p. 85, 2008, B. Morgan et al describe a tunable MEMS resonator with vertically-shaped comb fingers. However, for fabricating such structures gray scale lithography is needed. 
     In “A frequency selective silicon vibration sensor with direct electrostatic stiffness modulation”, Analog Integrated Circuits and Signal Processing, 37, pp 35-43, 2003, D. Scheibner et al describe a vibration sensor based on a MEMS comb resonator wherein electrostatic resonance tuning is implemented by means of a tuning actuator comprising a comb system with a linearly varying finger length. The tuning actuator and the resonator are co-fabricated. The electrostatic tuning mechanism proposed by Scheibner can only be used for energy harvesters with in-plane motion of the seismic mass, i.e. wherein the seismic mass moves in a plane parallel to the plane of the combs. 
     SUMMARY 
     The present application discloses methods for electrically tuning a resonance frequency of a resonant structure, e.g. MEMS or NEMS resonant structure. The disclosed methods can for example be used for tuning a resonance frequency of a vibration energy harvester or a sensor. It is an advantage of the disclosed tuning methods that it can be used for tuning a resonance frequency of different types of resonant structures, e.g. resonant structures with in-plane motion of a seismic mass as well as resonant structures with out-of-plane motion of a seismic mass. 
     In a first aspect, the present application also discloses devices, e.g. micromachined devices, with electrically tuneable resonance frequency. It is an advantage of disclosed devices that they can be compact and that they can easily be fabricated with standard micromachining techniques. Furthermore, the power consumption for tuning can be low. It is another advantage that the resonant structure and the tuning actuator can be fabricated separately. This allows optimizing the fabrication process and the properties of both the resonator structure and the tuning actuator independently. 
     A first embodiment includes a microstructure comprising a substrate, a resonant structure having a resonance frequency, and comprising a seismic mass and suspension elements for suspending the seismic mass at two opposite sides onto the substrate. The substrate is adapted for functioning as a tuning actuator, preferably a linear tuning actuator, adapted for applying stress onto the suspension elements, thus changing the stiffness of the suspension elements. It is an advantage of at least some of the disclosed microstructures that they provide a compact micromachined device for electrostatic conversion of kinetic energy into electrical energy, such as e.g. a micromachined energy harvester, which can easily be brought into an adapted resonance frequency, chosen depending on the input vibration frequencies which may vary over time or may initially be unknown. By adapting the resonance frequency of the resonant structure, a suitable power may be generated, even in circumstances of variable input frequencies. It is furthermore an advantage that the resonance frequency can be tuned continuously, e.g. also during actual use of the microstructure as energy harvester or sensor, and not only during design, manufacturing or calibration phases. 
     According to some embodiments, the substrate may be made from piezoelectric material forming the tuning actuator. Piezoelectric actuation can cause a precise frequency shift of the resonant structure. 
     In particular embodiments, the substrate may comprise piezoelectric bulk material with in-plane extension. Hereto, the substrate may have a top and a bottom major surface, and the microstructure may furthermore comprise a first electrode at the top major surface and a second electrode at the bottom major surface of the substrate for actuating the piezoelectric material. 
     In alternative embodiments, the substrate may comprise two piezoelectric shear actuators. In such embodiments, the microstructure may furthermore comprise interdigitated electrodes for actuating the piezoelectric material. Such interdigitated electrodes may be provided at a same major surface of the substrate, thus simplifying the manufacturing process. 
     In a microstructure according to some embodiments, the suspension elements may comprise a straight beam and an anchor point, the straight beam being attached between the seismic mass and the anchor point. It is an advantage of embodiments of the present invention that the suspension elements are straight beams, because if they would be bent (e.g. L- or U-shaped) the axial stress applied for tuning the resonant structure could relax by twisting the joints. 
     According to some embodiments, the anchor point is mechanically attached to the substrate. The anchor point transfers the movement of the substrate into a stress onto the beam which is part of the suspension element. 
     According to some embodiments, the resonant structure may be any of a MEMS (Micro-Electro-Mechanical Systems) or a NEMS (Nano-Electro-Mechanical Systems) structure. It may be manufactured for example by semiconductor processing. 
     In a microstructure according to some embodiments, the resonant structure may be a double clamped structure, i.e. a resonant structure clamped at two opposite sides. 
     In a second aspect, the present invention provides the use of a microstructure according to embodiments of the first aspect of the present invention in any of a vibration energy harvester or a wireless sensor network, where it replaces batteries. 
     In a third aspect, the present application discloses a method for tuning a resonance frequency of a resonant structure anchored to a substrate at two opposite sides by means of suspension elements. The method comprises applying a stress on the suspension elements by means of the substrate, thus changing the stiffness of the suspension elements. By changing the stiffness of the suspension elements, the resonance frequency of the resonant structure is changed and hence its possibilities to convert kinetic energy of input vibrations of different frequencies into electrical energy. 
     In a method according to some embodiments, applying a stress by means of the substrate may include actuating a piezoelectric substrate. Actuating a piezoelectric substrate may comprise shear actuation of the piezoelectric substrate, or alternatively it may comprise applying an electric field over a thickness of the piezoelectric substrate. 
     In yet another aspect, the present application discloses the use of a method according to embodiments of the third aspect, for tuning a resonance frequency of a vibration energy harvester or a sensor. 
     For purposes of summarizing the disclosed embodiments and the advantages achieved over the prior art, certain objects and advantages of the embodiments have been described herein above. Of course, it is to be understood that not necessarily all such objects or advantages may be achieved in accordance with any particular embodiment. Thus, for example, those skilled in the art will recognize that the some embodiments may be embodied or carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other objects or advantages as may be taught or suggested herein. 
     The disclosed embodiments, both as to organization and method of operation, together with features and advantages thereof, may best be understood by reference to the following detailed description when read with reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1(   a ) and  FIG. 1(   b ) illustrate an embodiment wherein a resonant structure or resonator is attached to a piezoelectric actuator fabricated from a piezoelectric bulk material with in-plane extension.  FIG. 1(   a ) shows a top view and  FIG. 1(   b ) shows a cross section along line A-A′. 
         FIG. 2  illustrates a top view of an embodiment wherein a resonant structure or resonator is attached to two piezoelectric shear actuators featuring interdigitated electrodes. 
         FIG. 3  illustrates an embodiment where the resonant structure is provided with only two suspension elements in the longitudinal direction of the resonant structure. 
         FIG. 4  shows a stray field electrode configuration for shear actuation of piezoelectric shear actuators, e.g. PZT-5A actuators. 
         FIG. 5(   a ) and  FIG. 5(   b ) show cross-sections of the device of  FIG. 2  along line V-V′, in non-actuated and actuated state, respectively. 
     
    
    
     The drawings described are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. The dimensions and the relative dimensions do not correspond to actual reductions to practice of the embodiments. 
     Any reference signs in the claims shall not be construed as limiting the scope. 
     In the different drawings, the same reference signs refer to the same or analogous elements. 
     DETAILED DESCRIPTION 
     The present invention will be described with respect to particular embodiments and with reference to certain drawings but the invention is not limited to the disclosed embodiment, but only by the claims. 
     The terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequence, either temporally, spatially, in ranking or in any other manner. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments described herein are capable of operation in other sequences than described or illustrated herein. 
     Furthermore, the terms top, bottom, over, under and the like in the description and the claims are used for descriptive purposes and not necessarily for describing relative positions. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments described herein are capable of operation in other orientations than described or illustrated herein. 
     It is to be noticed that the term “comprising”, used in the claims, should not be interpreted as being restricted to the means listed thereafter; it does not exclude other elements or steps. It is thus to be interpreted as specifying the presence of the stated features, integers, steps or components as referred to, but does not preclude the presence or addition of one or more other features, integers, steps or components, or groups thereof. Thus, the scope of the expression “a device comprising means A and B” should not be limited to devices consisting only of components A and B. It means that with respect to the present invention, the only relevant components of the device are A and B. 
     Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment, but may. Furthermore, the particular features, structures or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments. 
     Similarly it should be appreciated that in the description of exemplary embodiments, various features are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure and aiding in the understanding of one or more of the various inventive aspects. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed invention requires more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects may lie in less than all features of a single foregoing disclosed embodiment. Thus, the claims following the detailed description are hereby expressly incorporated into this detailed description, with each claim standing on its own as a separate embodiment of this invention. 
     Furthermore, while some embodiments described herein include some but not other features included in other embodiments, combinations of features of different embodiments are meant to be within the scope of the invention, and form different embodiments, as would be understood by those in the art. For example, in the following claims, any of the claimed embodiments can be used in any combination. 
     In the description provided herein, numerous specific details are set forth. However, it is understood that some embodiments may be practiced without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description. 
     In the context of the present invention, the term ‘tuning’ is used to indicate an electrically controlled non-permanent (reversible) modification of a device property after fabrication of the device. In particular, in embodiments of the present invention, the resonance frequency of a resonant structure is tuned. 
     Systems and methods will now be described by a detailed description of several embodiments. It is clear that other embodiments can be configured according to the knowledge of persons skilled in the art without departing from the true spirit or technical teaching of the disclosure, the invention being limited only by the terms of the appended claims. 
     In one aspect, the application discloses a method for electrically tuning a resonance frequency of a resonant structure, e.g. MEMS (Micro-Electro-Mechanical System) or NEMS (Nano-Electro-Mechanical System) resonant structure. The method can for example be used for tuning a resonance frequency of a vibration energy harvester or a sensor. 
     The disclosed methods can be used for electrically tuning a resonance frequency of double-clamped resonant structures, i.e. resonant structures that are clamped or fixed at two opposite sides. For example, the resonant structure may comprise a double-clamped beam (also called a bridge) or a plurality of double-clamped beams. In such case, the beam(s) is (are) clamped in longitudinal direction, for example by means of a suspension element at each extremity in longitudinal direction of the bridge. The resonance frequency of a mechanical structure in general is derived from its geometry and material parameters. If the structure comprises multiple beams, each beam may have its own distinct resonance frequency, and the resonance frequency of the complete structure is a function of the distinct resonance frequencies of the beams. In alternative embodiments, the movable structure is not just a bridge, but has a brick-shape, e.g. a rectangular shape with a pre-determined volume, for example a rectangular shape having a width and length of comparable size. In such case it is preferred to have at least four suspension elements, e.g. two suspension elements at each of two opposite sides. It is preferred that suspension elements opposite to each other have the same longitudinal axis. It is the in-line stress, i.e. the stress along the longitudinal axis of the suspension elements that determines the change in stiffness of the suspension elements and thus the change in resonance frequency of the resonant structure. Stress in other directions has a negligible effect. In some embodiments, the suspension elements can be made from semiconductor material, e.g. from silicon. Silicon can be strained by 0.5% of its length. This is sufficient for tuning. 
     In some embodiments, the resonant structure may comprise a seismic mass that is clamped by at least two suspension elements at two opposite sides of the mass, wherein both suspension elements are fixed to a substrate. It is an advantage of these embodiments that the suspension elements are straight elements, because if they would be bent (e.g. L- or U-shaped) the axial stress applied for tuning the resonant structure could relax by twisting the joints. In general, a tuning method according to some embodiments can be used for configurations where the restoring force, i.e. the force arising from the mechanical stiffness of the suspension, or worded differently the ratio between exciting force and resulting displacement, is generated by a mechanical suspension. 
     In a configuration with a double-clamped resonant structure in accordance with some embodiments, providing an axial force (tensile or compressive) on the suspension elements or clamps leads to a change in the effective stiffness of the suspension. It is known that the resonance frequency of a spring suspended mass is defined through the inertia of the seismic mass and the stiffness of the suspension. 
     In some embodiments, a piezoelectric tuning actuator is used for adapting the stiffness of the suspension of a resonant structure, thereby adapting or tuning the resonance frequency of the resonant structure. The resonant structure is physically attached to a substrate by means of the suspension elements. A method according to such embodiments comprises applying a force (tensile or compressive) on the suspension elements by means of the substrate, thus changing the stiffness of the suspension elements, hence tuning the resonance frequency of the resonant structure. 
     In some embodiments, the substrate adapted for applying a force on the suspension elements may be a piezoelectric substrate. In the presence of an electrical field, the dimensions of the piezoelectric substrate change, which generates stress in the suspension elements attached thereto. Piezoelectricity is the ability of some materials, notably crystals and certain ceramics, to generate an electric potential in response to applied mechanical stress. The piezoelectric effect is reversible in that materials exhibiting the direct piezoelectric effect (the generation of an electrical field, resp. electrical charge, when stress is applied) also exhibit the reverse piezoelectric effect (the production of stress and/or strain when an electric field is applied). It is this reverse piezoelectric effect which may be used in accordance with the disclosed embodiments. For example, lead zirconate titanate crystals will exhibit a maximum shape change of about 0.1% of the original dimension. 
     In some embodiments, the effective mechanical stiffness of the double-clamped resonant structure, which is the superposition of the stiffness of the unstressed suspension elements and the stiffness change induced by stretching the suspension elements, is thus electrically controlled by a voltage source which biases the piezoelectric actuator. 
     It is an advantage of using a piezoelectric substrate as a tuning actuator that the resonant structure and the tuning actuator can be fabricated separately, allowing optimization of the fabrication process for both the resonant structure and the tuning actuator. Piezoelectric bulk material is available in a variety of shapes, e.g. flat squares or disks, with lateral sizes up to 6 cm. This material can for example be based on lead zirconate titanate (PZT) with modifications to the stochiometry, e.g. to optimize piezoelectric properties, temperature stability or mechanical stiffness. It is an advantage of using a piezoelectric substrate, that material with well-defined and stable properties is used. Furthermore, piezoelectric material can be integrated in the processing of MEMS or NEMS devices. 
     It is another advantage of at least some of the disclosed methods that the power consumption for tuning can be very low. Piezoelectric static actuation requires a DC voltage, and due to the capacitance characteristic of the actuator (the actuator comprises two electrodes with piezoelectric material in between) only very low power levels are needed, e.g. 1 μW or lower. Such a piezoelectric capacitor is biased with a DC voltage, and thus the steady-state power consumption is determined by the parasitic (leakage) resistance of the capacitor. The intrinsic resistivity of piezoelectric materials is very high (e.g. higher than 10 12  Ohm cm), which leads to a very low leakage current. Only parasitic shunt resistances might give rise to power consumption. 
     It is another advantage of at least some of the disclosed methods that the resonance frequency can be tuned continuously. This allows adapting at any time the resonance frequency to input vibration frequencies which may vary over time. 
     In another aspect, the disclosed systems and methods may include a microstructure comprising a substrate, a resonant structure having a resonance frequency, and suspension elements for suspending the resonant structure at two opposite sides onto the substrate. According to such embodiments, the substrate is adapted for functioning as a tuning actuator adapted for applying stress onto the suspension elements, thus changing the stiffness of the suspension elements and hence changing the resonance frequency of the resonant structure. In particular embodiments, the substrate is made from piezoelectric material. Both piezoelectric material with in-plane extension and the more efficient shear actuation of a piezoelectric substrate may be used. 
     In use, a double-clamped resonant structure will be excited by a vibration of certain frequency and displacement amplitude. This driving frequency makes the resonant structure move with a different amplitude at the same frequency, and this movement generates an output power by converting the mechanical energy from vibrations into usable electrical energy. As stated above, the conversion may be performed either by the resonator integrating piezoelectric material on the deformed sections of the suspensions elements, in which case the stress in the deformed area generates electrical field/charge, or by the resonator integrating a biased variable capacitor, in which case electrical power can be obtained from the capacitance variation. If the resonance frequency of the double-clamped resonant structure is tuned so as to be equal to or closely matching the driving frequency, then the output power is maximal. Preferably the frequency mismatch should be less than a few percent, e.g. less than 3%, less than 2%, less than 1%. A frequency mismatch of only a few percent already results in a significant decrease in output power. 
     Additionally, the disclosed systems may include devices, e.g. micromachined devices, comprising a resonant structure (e.g. sensor, energy harvester) and a tuning actuator for electrically tuning a resonance frequency of the resonant structure. The resonant structure and the tuning actuator may be fabricated separately. Their fabrication may be integrated in one process flow. 
     In the further description, the frequency tuning method and device are illustrated for double-clamped energy harvesters. However, the disclosed methods can also be used for tuning a resonance frequency of other devices, such as for example resonators used in resonant sensors (e.g. accelerometers, pressure sensors, mass sensors, . . . ) or in oscillators. 
     Two embodiments of a method and device for resonance frequency tuning of a resonant device or structure are illustrated in  FIG. 1(   a ) and  FIG. 1(   b ), and  FIG. 2 , respectively. 
       FIG. 1(   a ) shows a top view and  FIG. 1(   b ) shows a cross section of a microstructure in the form of a vibration energy harvester  10 , comprising a piezoelectric substrate  11  and a resonant structure  100  comprising a mass  12  physically attached to the piezoelectric substrate  11  by means of suspension elements  13 , wherein the piezoelectric substrate  11  can be used as an actuator for tuning the resonance frequency of the resonant structure  100 .  FIG. 1(   a ) and  FIG. 1(   b ) show an embodiment with longitudinal piezoelectric actuation, i.e. an embodiment wherein the actuator is fabricated from piezoelectric bulk material with in-plane extension. Such a piezoelectric substrate  11  is also called a piezoelectric disc. When applying a voltage over this substrate  11 , it is not bending but it is stretching in all directions. If the substrate would be a circular substrate, it would remain circular but with a larger diameter. In case of a rectangular substrate the sides become a little bit rounded, but this effect is very small. Stretching of the substrate  11  leads to an outwards movement of the four anchor points  14  of the suspension elements  13  of the suspended structure. The forces inside the piezoelectric substrate are the same in all directions, as illustrated by arrows  18 . 
       FIG. 2  shows a top view of a microstructure according to one embodiment, in the form of a vibration energy harvester  20 , comprising a piezoelectric substrate under the form of two piezoelectric shear actuators  21  and a resonant structure  200  comprising a mass  22  physically attached to the piezoelectric substrate  21  by means of suspension elements  23 , wherein the piezoelectric substrate  21  can be used as an actuator for tuning the resonance frequency of the resonant structure  200 .  FIG. 5(   a ) and  FIG. 5(   b ) show cross sections of this microstructure along line V-V′, in non-actuated state ( FIG. 5(   a )) and in actuated state ( FIG. 5(   b )). Contrary to the embodiment illustrated in  FIG. 1(   a ) and  FIG. 1(   b ), the embodiment of  FIG. 2  works in shearing mode. In this embodiment, the piezoelectric crystal is cut in a different way as compared to the embodiment of  FIG. 1(   a ) and  FIG. 1(   b ). The behavior of a piezoelectric bulk material depends on the orientation of the crystals inside this structure. By using a different cut direction when cutting the material, a different behavior is obtained. Piezoelectric materials with different crystal directions are commercially available. 
     In some embodiments, the resonant structure  100 ,  200  is physically attached to the substrate  11 ,  21  by means of suspension elements  13 ,  23  at least two anchor points  14 ,  24 , for example at four anchor points. The number of anchor points actually used depends on the dimensions of the resonant structure  100 ,  200 . An example of a resonant structure  300  comprising a mass  30  and only two suspension elements  31  and two anchor points  32  is illustrated in  FIG. 3 . The suspension elements  31  are provided in the longitudinal direction of the resonant structure  300 . For the devices as shown in  FIG. 1  and  FIG. 2 , it is preferred that the resonant structure  100 ,  200  (including or consisting of a seismic mass  12 ,  22 ) is attached to four anchor points  14 ,  24 , as this allows preventing rotational motion of the resonant structure  100 ,  200 . The suspension elements  13 ,  23  are physically connected at one end to the anchor point  14 ,  24  on the substrate  11 ,  21  and at the other end to the seismic mass  12 ,  22 . As illustrated, the suspension elements  13 ,  23 ,  31  comprise straight beams  15 ,  25 ,  33  with a pre-determined height, width and length. In some embodiments, the seismic mass  12 ,  22 ,  30 , the beams  15 ,  25 ,  33  of the suspension elements  13 ,  23 ,  31  and optionally the anchor points  14 ,  24 ,  32  may be fabricated from a same layer of material, for example semiconductor material such as silicon or silicon germanium. Therefore there is a continuous volume of material along this arrangement. At the anchor point  14 ,  24 ,  32  the suspension beam  15 ,  25 ,  33  widens to form an anchor pad—such anchor point  14 ,  24 ,  32  is connected to a suspension beam  15 ,  25 ,  33  which is called its related suspension beam. The pad&#39;s top or bottom side is mechanically connected to the substrate  11 ,  21 , for example by means of bonding or by means of e.g. glue  50 . 
     In the embodiment illustrated in  FIG. 1(   a ) and  FIG. 1(   b ), the mass  12  of the resonant structure  100  is attached at four anchor points  14  to a single piezoelectric substrate  11 , acting as a tuning actuator. The mass  12 , in the embodiment illustrated, is a rectangular structure in top view, having two opposite major sides. Two anchor points  14  and corresponding suspension beams  15  are provided at one major side of the rectangular structure and perpendicular thereto in the plane of the substrate, and two other anchor points and corresponding suspension beams are provided at the other major side of the rectangular structure and perpendicular thereto in the plane of the substrate. In embodiments of the present invention, the suspension beams are provided at the extremities of the major sides, or close thereto. By providing the suspension beams at the extremities of the major sides or close thereto, the displacement of the mass  12  is limited to one direction. Also rotations are restricted if the connections to the suspension beams are done at the outermost corners of the mass. In the example shown, the tuning actuator, substrate  11 , is fabricated from piezoelectric bulk material with in-plane extension, i.e. a material that changes its lateral dimensions (i.e. in a plane parallel to the substrate surface) in the presence of a vertical electrical field (i.e. an electrical field that is substantially orthogonal to the substrate surface). In this embodiment, an electric field is applied parallel to the polarization direction of the piezoelectric material. The actuators&#39; top and bottom surface are covered with a top electrode  16  and a bottom electrode  17 , respectively; for example electrodes made from platinum, aluminum, silver. Tuning the resonance frequency of the resonant structure  100  comprises applying a DC voltage between these electrodes  16 ,  17  so that a suitable electrical field is obtained over the piezoelectric substrate  11  so as to deform the substrate  11  under the piezoelectric effect. Such a suitable electrical field strength may be for example 10 kV/m, which, as an example only, corresponds to a voltage of 10 V over a substrate thickness of 1 mm. It is an advantage of the embodiment illustrated in  FIG. 1(   a ) and  FIG. 1(   b ) that it has a better power efficiency as compared to other actuation mechanisms which use current-based schemes. 
     In an alternative embodiment, illustrated in  FIG. 2 , the seismic mass  22  is attached at four anchor points  24  to a substrate comprising two piezoelectric shear actuators  21 . In this embodiment, contrary to the embodiment illustrated in  FIG. 1(   a ) and  FIG. 1(   b ), an electric field is applied perpendicular to the polarization direction of the piezoelectric material. In the illustration of  FIG. 2 , the two piezoelectric shear actuators  21  are provided on a further bottom substrate  26  which is not necessarily piezoelectrically active. The seismic mass  22 , in the embodiment illustrated, is a rectangular structure in top view, having two opposite major sides. Two anchor points  24  and corresponding suspension beams  25  are provided at one major side of the rectangular structure and perpendicular thereto in the plane of the substrate  21 , and two other anchor points  24  and corresponding suspension beams  25  are provided at the other major side of the rectangular structure and perpendicular thereto in the plane of the substrate  21 . In some embodiments, the suspension beams  25  are provided at the extremities of the major sides, or close thereto. By providing the suspension beams at the extremities of the major sides or close thereto, the displacement of the mass  22  is limited to one direction. Also rotations are restricted if the connections to the suspension beams are done at the outermost corners of the mass. The shear actuators  21  are located at two opposite sides of the seismic mass  22 , each underneath the anchor points  24  provided at one major side of the rectangular structure forming the seismic mass  22 . 
     In the presence of an electric field a shear moment is induced in the piezoelectric material  21 . In order to obtain this, electrodes  27 ,  28  are provided. In  FIG. 2  the electrodes  27 ,  28  are shown as dotted lines because in the example shown the electrodes  27 ,  28  are provided in between the substrate and the piezoelectric shear actuators  21 . It is an advantage of the embodiment illustrated in  FIG. 2  that the electrodes  27 ,  28  may be provided in a plane, both at one side of the piezoelectric material  21 , such that double-side processing of the piezoelectric substrate material  21  can be avoided. Biasing the piezoelectric material  21  leads to a shear deformation thereof (as illustrated in  FIG. 5(   b )), hence to a lateral displacement of the piezoelectric material surface onto which the resonant structure  200  is attached. This lateral deformation is indicated in  FIG. 2  by arrows  29   a . This deformation of the piezoelectric material  21 , if in opposing directions for each of the piezoelectric shear actuators, as illustrated in  FIG. 2 , applies forces to the anchor points  24 ; the forces are indicated by the arrows  29   b . These forces indicated by arrows  29   b  strain the suspension elements  23 . Actuating the piezoelectric material  21  thus leads to a tensile stress in the double-clamped suspension. 
     The piezoelectric actuators  21  in the embodiment shown in  FIG. 2  cannot be sheared with an electrical field perpendicular to its plane. Therefore, the stray field of interdigitated electrodes  27 ,  28  below the actuator  21  as shown in  FIG. 2  or in more detail in  FIG. 4  may be used. 
     Actuation of the interdigitated electrodes  27 ,  28  generates a net electrical field which is in the plane of the shear actuator. The use of the stray field also simplifies the electrode scheme. Here, two electrodes  27 ,  28 , e.g. interdigitated electrodes having electrode fingers, can be arranged on one side of the piezoelectric crystal  21  only, so there is no need to contact two opposite sides of the piezoelectric crystal  21 . In the embodiment illustrated in  FIG. 4 , grounded lines  40  are provided in between each pair of electrode fingers. These grounded lines  40  function as shield electrodes, preventing the field lines to stretch to neighboring actuation electrodes. 
       FIG. 5(   b ), when compared to  FIG. 5(   a ) illustrates the effect of actuating the piezoelectric substrate  21  by means of applying an actuation signal to the electrodes  27 ,  28  (not illustrated in  FIG. 5(   a ) or  FIG. 5(   b )). The resonant structure  200  is connected by means of the suspension elements  23  to the piezoelectric substrate  21 , e.g. by means of anchor pads  24  which are mechanically attached to the piezoelectric substrate  21 , for example by means of glue  50  or another bonding mechanism. The deformation of the piezoelectric substrate  21  when actuated is illustrated in  FIG. 5(   b ). This deformation of the piezoelectric substrate  21  leads to stress on the suspension elements  23 , hence to a change of the resonance frequency of the resonant structure  200 . 
     In some embodiments, stiffening of a suspension element  13 ,  23 ,  31  is used for frequency tuning. The stiffening effect causes the spring constant of a suspension element or flexure to increase under application of an applied tensile stress. The restoring force as a function of the mass deflection x and the total normal force F N  is given by: 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       x 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         
                           F 
                           N 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             k 
                             0 
                           
                           + 
                           
                             
                               12 
                                
                               
                                 F 
                                 N 
                               
                             
                             
                               5 
                                
                               
                                   
                               
                                
                               l 
                             
                           
                         
                         ) 
                       
                        
                       x 
                        
                       
                           
                       
                        
                       with 
                        
                       
                           
                       
                        
                       
                         k 
                         0 
                       
                     
                     = 
                     
                       
                         12 
                          
                         
                             
                         
                          
                         
                           EI 
                           
                             l 
                             3 
                           
                         
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         I 
                       
                       = 
                       
                         
                           
                             hw 
                             3 
                           
                           12 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Here, k 0  is the spring constant of the non-tuned system, E is the Young&#39;s modulus, I the moment of inertia and l the length of the (double-clamped) flexure. 
     The tuned resonance frequency is then given by 
     
       
         
           
             
               
                 
                   
                     f 
                     res 
                   
                   = 
                   
                     
                       
                         1 
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                         
                       
                        
                       
                         
                           
                             k 
                             t 
                           
                           m 
                         
                       
                        
                       
                           
                       
                        
                       with 
                        
                       
                           
                       
                        
                       
                         k 
                         t 
                       
                     
                     = 
                     
                       
                         k 
                         0 
                       
                       + 
                       
                         
                           
                             12 
                              
                             
                                 
                             
                              
                             
                               F 
                               N 
                             
                           
                           
                             5 
                              
                             
                                 
                             
                              
                             l 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In order to verify the feasibility of the tuning approach according to embodiments of the present invention, the stiffness term (induced by tensile stress) can be compared with the mechanical spring constant k 0 . This gives: 
     
       
         
           
             
               
                 
                   
                     
                       k 
                       t 
                     
                     
                       k 
                       0 
                     
                   
                   = 
                   
                     
                       1 
                       + 
                       
                         
                           12 
                           5 
                         
                          
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             ll 
                           
                           
                             w 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         with 
                          
                         
                             
                         
                          
                         
                           F 
                           N 
                         
                       
                     
                     = 
                     
                       
                         wh 
                          
                         
                             
                         
                          
                         σ 
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         σ 
                       
                       = 
                       
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             l 
                           
                           l 
                         
                          
                         
                           E 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     This can be illustrated for an example wherein the following assumptions are made: 
     desired resonance frequency, f res ≈300 Hz, 
     seismic mass, m=20 mg (density, ρ=2330 kg m −3 , dimensions 3×6×0.5 mm 3 ), 
     width of suspension element, w=10 μm 
     height of suspension element, h=100 μm 
     required length of suspension element, l=1 mm (four suspension elements in total) 
     In order to achieve a shift in resonance frequency of 5% in this case, a change in stiffness of 10.3% is needed. Therefore a change in suspension element length of approximately 4.3 nm is needed to induce the required strain. The tensile stress (σ=0.73 MPa) is still far below the fracture strength of silicon (σ crit =30 GPa). 
     Next a block of piezoelectric material, e.g. lead zirconate titanate (PZT-5A), is considered. Its piezoelectricity is described mathematically within the material&#39;s constitutive equation, which defines how the piezoelectric material&#39;s stress (T), strain (S) and electric field (E) interact. 
         S=s   E     T +d   T   {right arrow over (E)}   (4)
 
     Here, only strain induced by the presence of an electrical field is considered. The piezoelectric coupling factor d of PZT-5A is given by the tensor 
     
       
         
           
             
               
                 
                   d 
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               584 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               584 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 - 
                                 171 
                               
                             
                             
                               
                                 - 
                                 171 
                               
                             
                             
                               374 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       · 
                       
                         10 
                         
                           - 
                           12 
                         
                       
                     
                      
                     
                       
                         C 
                         m 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The feasibility of the approach can be analyzed based on the lateral extension of the piezoelectric substrate. In order to extend the flexure length by 4.3 nm the substrate has to extend laterally by that value over a length of 2.5 mm (1 mm flexure length+1.5 mm as half of the mass width). This relates to a strain inside the PZT material of 1.72 10 −6 . This strain can be achieved by an electrical field of 10 kV/m. This field can be provided by a moderate voltage of 10 V applied over a PZT-thickness of 1 mm. 
     As an example, energy harvesting structures according to embodiments of the present invention can be made, with a footprint of around 1 cm 2 . Such devices can have a frequency change of up to 5%, and an output power of about 100 μW. 
     The foregoing description details certain embodiments. It will be appreciated, however, that no matter how detailed the foregoing appears in text, the invention may be practiced in many ways. It should be noted that the use of particular terminology when describing certain features or aspects of the invention should not be taken to imply that the terminology is being re-defined herein to be restricted to include any specific characteristics of the features or aspects of the invention with which that terminology is associated.