Piezoelectric thruster and method for generating a thrust with said piezoelectric thruster

Abstract

The present invention relates to a piezoelectric thruster comprising: - a set of identical piezoelectric elements by way of a battery (5) of polarised piezoelectric discs (2) alternately arranged relative to conductive electrodes (1), the piezoelectric discs and the electrodes (1) being bonded by an epoxy resin (3); - two parts, a smaller part (6) and a larger part (7) between which the batteries (5) are arranged by applying pressure, and the piezoelectric discs exhibiting a high quality factor Q and a high piezoelectric constant d33; - a power supply source (21) capable of providing an alternating current wave form, wherein the wave form exhibits two sinusoidal components with different frequencies. A piezoelectric thruster is thus obtained, exhibiting very low energy loss due to thermal effects, wherein maximum power is delivered to the piezoelectric material, the piezoelectric materials having a high mechanical excursion.

Description

[0001] PIEZOELECTRIC THRUSTER AND METHOD OF GENERATING A THRUST WITH SAID PIEZOELECTRIC THRUSTER

[0002] DESCRIPTION

[0003] OBJECT OF THE INVENTION

[0004] The subject of the present invention, as the title of the invention states, is a piezoelectric propellant. A piezoelectric material can convert mechanical stress into electric current and vice versa: when its structure is mechanically deformed, an electric stress is produced (direct piezoelectric effect); similarly, when an external electric stress is applied, its structure deforms (inverse piezoelectric effect). The magnitude of the deformation and the intensity of the stress are linearly proportional for most piezoelectric materials.

[0005] Also object of the present invention is a method for generating thrust with the piezoelectric propellant object of the invention.

[0006] The present invention is characterized by the special design and configuration of each and every one of the parts that make up the piezoelectric propellant, as well as the steps of the method of generating a thrust with a piezoelectric material.

[0007] The present invention exploits the inverse piezoelectric effect in certain piezoelectric materials to transform an alternating electrical signal with a specific profile into directed motion.

[0008] Therefore, the present invention is circumscribed within the scope of piezoelectric materials and more precisely among those used in space applications for maneuvering and controlling the attitude of satellites in Earth orbit, or a small but continuous linear thrust for deep space probes.

[0009] BACKGROUND OF THE INVENTION

[0010] Known in the state of the art is patent US2023208321A1, which discloses a method for producing accelerations without expelling any exhaust material, which is based on inducing mass fluctuations in conventional electrical circuit components and combining them with a mechanically coupled periodic thrust to produce propulsive forces without the expulsion of any propellant. Ferroelectric force transducers, in particular piezoelectric transducers, are driven by at least two phase-locked voltage waveforms, such that both the rest mass fluctuation and the mechanical excursion necessary to produce a stationary thrust occur in the transducer itself.Parametric amplification is used to obtain high-amplitude, correctly synchronized voltages at the correct frequencies to drive the transducer by simply adding a DC voltage to the AC voltage of a single-frequency driving signal. The control voltage waveforms can be pulsed to maximize thrust and reduce the duty cycle of the system's active components.

[0011] However, this implementation presents aspects that are susceptible to improvement and that, acting synergistically, result in greater and more effective controlled propulsion. For example, the reduction in heating rate and the activation of vibrational thermal convection effects that generate thrust.

[0012] Therefore, the object of the present invention is to develop a piezoelectric propellant that has a very low energy loss due to thermal effects, where the power delivered to the piezoelectric material is maximum, and the piezoelectric materials have a large mechanical excursion. DESCRIPTION OF THE INVENTION

[0013] The object of the present invention is set forth in its essence in the independent claim and the different embodiments are set forth in the dependent claims.

[0014] The present invention relates to a piezoelectric propellant comprising: a. A set of identical piezoelectric elements, i. In which each piezoelectric element is a stack of polarized piezoelectric discs arranged alternately with respect to conductive electrodes;

[0015] i. Wherein the material of the thin polarized piezoelectric discs has a quality factor Q greater than 1800 and a piezoelectric constant d33 greater than 270*10 -12m / V, which combined conditions lead to a large mechanical excursion of the piezoelectric elements and a negligible amount of energy lost to thermal effects during such excursions; iii. wherein each side of each thin polarized piezoelectric disc in the stack is covered by a conducting electrode; iv. wherein the polarized piezoelectric discs and the conducting electrodes are bonded with a layer of electrically conductive epoxy resin; v. wherein each polarized piezoelectric disc in the stack is oriented so that its positive polarity side faces the positive polarity side of neighboring thin polarized piezoelectric discs, and the end polarized piezoelectric discs in the stack have their positive polarity side facing the center of the stack; vi. wherein the conducting electrodes have projecting flanges to facilitate electrical connections; vii.in which the electrode tabs in contact with the positive polarity of the polarized piezoelectric discs are aligned with each other, on the diametrically opposite side of the stack in relation to the electrode tabs in contact with the negative polarity of the polarized piezoelectric discs; b. two unequal metal pieces, a smaller piece and a larger piece arranged in parallel and placed at the ends of the stacks i. in which a set of screws allows the assembly of piezoelectric elements to be compressed, along their axis of symmetry, between the two unequal pieces and;.

[0016] i. wherein the materials of the unequal metal parts and the screws are such that the necessary preload on the piezoelectric elements can be carried homogeneously and without inducing harmful deformations of the piezoelectric elements, the unequal metal parts or the screws, where the material of the smaller part must be less dense than the material of the larger part c. a power supply capable of providing an alternating current waveform, i. wherein the source comprises an arbitrary waveform generator, an amplifier and a transformer;

[0017] i. in which the positive pole of the source is connected by electric cables to the tabs of the conductive electrodes that are in contact with the positive polarity of the polarized piezoelectric discs of each piezoelectric element, and the negative pole of the source is connected by electric cables to the tabs of the conductive electrodes that are in contact with the negative polarity of the polarized piezoelectric discs of each piezoelectric element

[0018] In a preferred non-limiting embodiment, the total pressure exerted by the preload on the piles should preferably be in the range of ~20 MPa.

[0019] In addition, a phase detector is implemented in the power supply, which calculates the impedance phase angle (IPA) between the voltage and current waveforms, which sets a desired phase angle, and a phase controller varies the drive frequency to maintain it independently of displacements due to thermal effects.

[0020] In a possible embodiment, since the material of the smaller part must be less dense than the material of the larger part, the material of the larger part (7) is brass and that of the smaller part (6) is aluminum.

[0021] The piezoelectric thruster additionally comprises a thermal cooling system to avoid large changes in the resonance condition, where both the larger and smaller pieces have through holes through which a pipe is arranged along the axis of symmetry of the structure, and through which a liquid coolant is pumped from one side of the pipe to the other, passing the pipe close to the set of batteries. Said pipe can be made of aluminum coated externally with black matte paint. Additionally, the thruster would be protected by an external heat shield designed to reflect incident solar radiation and thus reduce heating due to environmental conditions.

[0022] Also the subject of the present invention is a method for generating thrust without expelling propellant, based on the piezoelectric propellant described above, comprising the steps of: a. Compressing the set of piezoelectric elements, along its axis of symmetry, between the two unequal masses; b. Applying an alternating voltage to the set of piezoelectric elements through the tabs of the conductive electrodes, i. where the applied alternating voltage has a waveform comprising two sinusoidal components with different frequencies, the first sinusoidal component with a frequency equal to the thickness resonance frequency (or first harmonic) of the piezoelectric elements, and the second sinusoidal component with a frequency equal to twice the thickness resonance frequency (or second harmonic) of the piezoelectric elements; i.where the amplitude of the second sinusoidal component is not greater than 20% of the amplitude of the first sinusoidal component, and the phase relationship between the two sinusoidal components of the waveform is TT / 2, such that the second derivative of the voltage waveform, which is proportional to the acceleration of the stack, reaches higher positive values ​​than negative ones during each cycle, giving an acceleration during the expansion phase of the piezoelectric elements greater than the acceleration during the contraction phase; iii. wherein, by applying the alternating voltage to the piezoelectric elements, and due to the greater acceleration of the expansion, a thrust is produced along the axis of symmetry of the piezoelectric elements in the direction connecting the piezoelectric elements to the larger metal piece; c.measuring both the impedance of the piezoelectric elements and the phase angle between the voltage and current waveforms supplied to the piezoelectric elements, in order to identify the thickness resonance condition of the piezoelectric element as the frequency range in which both the impedance is at a local minimum and the phase angle between the voltage and current waveforms is zero; the method may further include adding an arbitrary number of higher harmonics of the thickness resonance frequency to the alternating voltage waveform, with amplitudes and relative phase suitably chosen such that the second derivative of the voltage waveform, which is proportional to the acceleration of the stack, reaches positive values ​​greater than negative values ​​during each cycle, giving rise to an acceleration during the expansion phase of the piezoelectric elements greater than the acceleration during the contraction phase.

[0023] In another embodiment, the method may further include one or more systems for stabilizing the thickness resonance condition of the piezoelectric element over extended periods of time, namely: a. a closed-loop system that measures and tracks the thickness resonance condition of the piezoelectric elements throughout the duration of the alternating current supply, and communicates this to the arbitrary waveform generator, so that the arbitrary waveform generator adapts the supplied waveform to the changing resonance condition; b. supplying the alternating voltage in short pulses, to limit the heating of the piezoelectric elements that occurs due to continuous activation, and to limit the consequent drift of the resonance condition,

[0024] . in which the pulse periods are much longer than one cycle of the alternating voltage, thus allowing several acceleration cycles within the same pulse; c. a system for cooling the piezoelectric elements by absorbing their emitted thermal radiation through a material with high absorptivity (greater than 0.9) and high thermal conductivity (greater than 230 W / m*K),

[0025] . in which the cooling system includes a sensor for controlling the temperature of the piezoelectric elements;

[0026] i. wherein the material is in the form of a tube passing close to the piezoelectric elements and allowing a coolant to be pumped and flowed therein. iii. Wherein the tube passing close to the piezoelectric elements would be made of aluminium with a high thermal conductivity (greater than 230 W / m*K), coated externally with black matt paint with a high absorptivity (greater than 0.9).

[0027] Thanks to the claimed characteristics, a piezoelectric propellant is obtained that presents a very low energy loss due to thermal effects, where the power delivered to the piezoelectric material is maximum. Piezoelectric materials have a great mechanical excursion due to the use of higher quality factors, a higher piezoelectric constant, among other characteristic elements. Unless otherwise indicated, all technical and scientific elements used in this specification have the meaning normally understood by a person of ordinary skill in the art to which this invention pertains. In the practice of the present invention, methods and materials similar or equivalent to those described herein may be used.

[0028] Throughout the description and claims, the word "comprises" and its variants are not intended to exclude other technical features, additives, components, or steps. For those skilled in the art, other objects, advantages, and features of the invention will be apparent in part from the description and in part from the practice of the invention.

[0029] EXPLANATION OF THE FIGURES

[0030] To complement the description being made and in order to help better understand the characteristics of the invention, in accordance with a preferred example of practical implementation thereof, a set of drawings is attached as an integral part of said description, in which the following has been represented for illustrative and non-limiting purposes.

[0031] Figure 1 is a schematic of the assembly of a piezoelectric battery.

[0032] Figure 2 is an embodiment of the propellant, containing five identical versions of the stack in Figure 1.

[0033] Figure 3 shows a schematic of the electronic architecture used to activate and control the propellant.

[0034] Figure 4 is an alternative embodiment of the propulsor, including adaptation of the structure for an active cooling system. Figure 5 shows the displacement and acceleration of the piezoelectric stack for different values ​​of the relative phase between the sinusoidal components of the drive signal.

[0035] Figure 6 shows the mean displacements measured by micrometric equipment during propellant activation.

[0036] Figure 7 shows the displacement and acceleration of the piezoelectric stack when the third harmonic of the piezoelectric material resonance is excluded or included in the driving signal.

[0037] PREFERRED EMBODIMENT OF THE INVENTION

[0038] A preferred embodiment of the proposed invention is described below in view of the figures.

[0039] In figure 1 we can see a piezoelectric battery (5) comprising a series of thin piezoelectric discs (2) arranged in an aligned manner and separated from each other by means of a series of conductive electrodes (1) intercalated between the piezoelectric discs (2), the piezoelectric discs (2) being joined to the conductive electrodes (1) by means of an electrically conductive epoxy resin (3), forming what we call a battery.

[0040] The electrodes must have protruding tabs (4) to facilitate alternating electrical connections.

[0041] Piezoelectric discs (2) must have a high value of piezoelectric constant d33 and a high value of quality factor Q (or, equivalently, a low dissipation factor tan(6)).

[0042] Each thin polarized piezoelectric disc in the stack is oriented so that its positive polarity side faces the positive polarity side of neighboring thin polarized piezoelectric discs, and the outermost thin polarized piezoelectric discs in the stack have their positive polarity side facing the center of the stack;

[0043] The electrode tabs in contact with the positive polarity of the polarized thin piezoelectric discs are aligned with each other, on the diametrically opposite side of the stack relative to the electrode tabs in contact with the negative polarity of the polarized piezoelectric discs, as can be seen in Figure 1.

[0044] The piezoelectric coefficient d33 measures the deformation of the stack in the same direction as the applied potential: a high value of this coefficient results in greater deformation compared to materials with a lower d33 subjected to the same stress. This allows for greater mechanical excursion, which facilitates thrust generation.

[0045] The quality factor Q describes the quality of an oscillator under resonant conditions: Piezoelectric materials with higher Q factors exhibit a stronger and more stable resonance, but with a narrower available bandwidth around the resonant frequency, meaning that finer tuning is necessary to locate the resonance.

[0046] Equivalently, a high Q indicates a low rate of energy loss due to thermal effects (low dissipation factor tan(ó)), so that lower propellant heating can be achieved.

[0047] In a possible embodiment, the thin polarized piezoelectric discs have a quality factor Q greater than 1800 and a piezoelectric constant d33 greater than 270*10 -12m / V, combined conditions that lead to a large mechanical excursion of the piezoelectric elements and a negligible amount of energy lost due to thermal effects during such excursions; Figure 2 shows the configuration of the piezoelectric propellant object of the invention, which, as can be seen, comprises a series of cells (5) shaped as described in Figure 1, arranged in parallel and compressed between a smaller piece (6) and a larger piece (7), where the concepts of smaller piece and larger piece refer to their masses; where both pieces are joined by a series of screws (8) since the piezoelectric elements need to be subjected to a preload in order to function correctly and transmit any thrust. The number and arrangement of the preload are such that the pressure exerted on the faces of the cells is as homogeneous as possible.The total pressure exerted by the preload on the piles should preferably be around ~20 MP, which is a necessary and sufficient pressure to allow the piles to perform correctly and at the same time prevent them from buckling.

[0048] The ratio of the radius to the length of the cylindrical piles must be such that buckling does not occur during preloading. The larger piece (7), in addition to acting as a support for the preloading of the piles, constitutes the point of contact between the propellant and the body on which the force is to be applied.

[0049] In a complementary manner, an additional support (9) may be included for the propellant in order to facilitate the assembly process. The choice of materials for the unequal metal parts is governed by both structural requirements (limiting the overall deformation of the parts during preload and activation) and practical considerations (limiting the overall mass of the propellant while providing the preload). In accordance with these requirements, and given the relatively low preload required to compress the stacks, the material of the smaller part (6) must be less dense than the material of the larger part (7). For example, the larger part (7) could be made of brass and the smaller part (6) of aluminum.Figure 3 shows the external source for applying an alternating voltage, which comprises a signal generator (12), an amplifier (13) and a transformer (14) through whose output a modulated alternating voltage is applied through the electrodes of each of the parallel batteries (16), following the alternating pattern shown in Figure 1.

[0050] The simplest voltage waveform consists of two principal frequencies, the thickness resonance frequency w of the piezoelectric element and its twin, with a fixed phase ratio of TT / 2 between them and a possible amplitude ratio of about 0.177. An explicit form of the signal can be given as in equation

[0001] :

[0051] 70

[0052] V = Yo + or 177

[0053] The resonant frequency of a piezoelectric material is the frequency at which it exhibits maximum oscillation amplitude when subjected to an external electric field. The resonance condition corresponds to the point at which the impedance phase angle (IPA) between the voltage and current waveforms is zero, such that the power delivered to the piezoelectric material is maximum, according to equation [2]:

[0054] Where V P he P are the peak voltage and current values, respectively. The addition of a component corresponding to the second harmonic (twice the resonant frequency) induces greater oscillatory displacements, while increasing robustness and stability against the nonlinear effects inherent in the resonance condition.

[0055] The location of the resonant condition of a piezoelectric material in the frequency spectrum can change during operation, due to the temperature changes that inevitably occur as the piezoelectric material converts electrical energy into mechanical energy. Two methods are implemented to prevent this effect from affecting the operation of the thruster. On the one hand, to maintain the zero IPA condition, a phase detector (15) is implemented, which calculates the IPA between the voltage and current waveforms; once the desired phase angle is set (10), a phase controller (11) varies the drive frequency to maintain it independently of displacements due to thermal effects.

[0056] On the other hand, a thermal cooling system can be implemented to avoid large changes in the resonance condition, as shown in Figure 4, where both the larger piece (7) and the smaller piece (6) have through holes (17) that allow the installation of a pipe (18) along the axis of symmetry of the structure. An external system is used to pump liquid coolant through the pipe from one side (19) to the other (20) of the structure, passing the pipe close to the set of batteries (5). In order to facilitate the transfer of heat radiation from the batteries (5) to the pipe (18), the latter must be manufactured with a high absorptivity material (greater than 0.9), so that it can efficiently absorb thermal infrared radiation, and high thermal conductivity (greater than 230 W / m*K), so that it can quickly transfer heat to the outside.For example, the material for the tube is aluminum coated externally with black matte paint, which together present these two properties.

[0057] In Figure 5a it can be observed how the deformation of a piezoelectric material is proportional to the applied voltage. In the case of a simple sinusoidal waveform, the acceleration (i.e., the second derivative of the displacement) of the piezoelectric material during a cycle would have positive and negative peaks with the same amplitude.

[0058] Figure 5b shows how the chosen phase difference between the two sinusoidal components maximizes the acceleration asymmetry, such that it reaches higher positive values ​​than negative ones during each cycle. The asymmetric profile of the excitation signal and the asymmetric arrangement of the metal parts in compression lead to the production of a net acceleration in the direction of the larger part.

[0059] Figures 6a and 6b show the results of experiments designed to measure the displacement induced by the thruster during its activation in a micrometric apparatus. The signal used to activate the thruster is given in equation [1] with Vo=5OV. Each graph is an average of several activations of 20 seconds duration, carried out in both the (conventionally defined) directions forward (a) and backward (b). By calibrating the micrometric apparatus, it is possible to relate the measured displacement to a value of impulse imparted by the thruster: (a) 5.8 pN and (b) 17 pN.

[0060] In principle, amplification of the effect could be obtained by choosing an amplified waveform that also included higher-order harmonics of the fundamental resonant frequency, which are theoretically infinite but of progressively decreasing intensity. The resulting signal would be a sum of terms of the form A sin(nw t+cp), in which the amplitudes A and the phases <p deberían sintonizarse de forma que se satisfaga la condición asimétrica de la aceleración descrita anteriormente. La figura 7 muestra cómo la inclusión del tercer armónico, según la ecuación [3] siguiente, afecta al perfil de aceleración. 0.177 sin(2 co t + n / 2) + 0.06 sin(3 co t + n) ]

[0061] The effect is an increase in asymmetry toward positive values. However, it should be noted that, in some cases, the higher harmonics of the thickness resonance could overlap with frequencies of other types of piezoelectric resonances, such as radial resonances, which would couple and negatively interfere with the required behavior, weakening the net effect. Suppression of unwanted resonances can be achieved by appropriately designing the shape of the stacks, which, however, can also lead to unwanted mechanical problems in the precharge phase. In this sense, the choice of activating the piezoelectric material only with the fundamental frequency and the second harmonic provides better control over the result.

[0062] Having sufficiently described the nature of the present invention, as well as the manner of putting it into practice, it is noted that, within its essence, it may be put into practice in other embodiments that differ in detail from the one indicated as an example, and to which the protection sought will also be achieved, provided that it does not alter, change or modify its fundamental principle.

Claims

CLAIMS 1.- Piezoelectric propellant characterized in that it comprises a. A set of identical piezoelectric elements, i. In which each piezoelectric element is a stack (5) of piezoelectric discs (2) polarized in an alternating arrangement with respect to conductive electrodes (1); i. Wherein the material of the thin polarized piezoelectric discs has a quality factor Q greater than 1800 and a piezoelectric constant d33 greater than 270*10' 12m / V; iii. wherein each side of each thin polarized piezoelectric disc (2) of the stack (5) is covered by a conductive electrode (1); iv. wherein the polarized piezoelectric discs (2) and the conductive electrodes (1) are bonded with a layer of electrically conductive epoxy resin (3); v. wherein each polarized piezoelectric disc (2) of the stack (5) is oriented so that its positive polarity side faces the positive polarity side of neighboring thin polarized piezoelectric discs (2), and the end polarized piezoelectric discs (2) of the stack have their positive polarity side facing the center of the stack; vi. wherein the conductive electrodes (1) have projecting tabs (4) to facilitate electrical connections; vii.in which the electrode tabs (1 ) in contact with the positive polarity of the polarized piezoelectric discs (2) are aligned with each other, on the diametrically opposite side of the battery in relation to the electrode tabs in contact with the negative polarity of the polarized piezoelectric discs (2); b. two unequal metal pieces, a smaller piece (6) and a larger piece (7) in a parallel arrangement and placed at the ends of the batteries (5),. i. in which a set of screws (8) allows the set of piezoelectric discs (2) to be compressed, along their axis of symmetry, between the two unequal pieces (6) and (7); i. wherein the materials of the unequal metal parts and the screws are such that the necessary preload on the piezoelectric elements can be carried homogeneously and without inducing harmful deformations of the piezoelectric elements, the unequal metal parts or the screws, where the material of the smaller part (6) must be less dense than the material of the larger part (7) c. a power supply (21) capable of providing an alternating current waveform, i. wherein the source comprises an arbitrary waveform generator (12), an amplifier (13) and a transformer (14); ¡i. in which the positive pole of the source (21 ) is connected by means of electric cables to the tabs of the conductive electrodes (1 ) that are in contact with the positive polarity of the polarized piezoelectric discs (2) of each piezoelectric element, while the negative pole of the source is connected by means of electric cables to the tabs of the conductive electrodes that are in contact with the negative polarity of the polarized piezoelectric discs (2) of each piezoelectric element. 2.- Piezoelectric propellant according to claim 1, characterized in that the total pressure exerted by the precharge on the batteries must preferably be of the order of ~20 MP 3.- Piezoelectric propellant according to claim 1, characterized in that a phase detector (15) is implemented in the power supply (21), which calculates the impedance phase angle (IPA) between the voltage and current waveforms that sets a desired phase angle (10) and a phase controller (11) varies the drive frequency to maintain it independently of the displacements due to thermal effects. 4.- Piezoelectric propellant according to any of the preceding claims, characterized in that the material of the larger part (7) is brass and that of the smaller part (6) is aluminum. 5.- Piezoelectric propellant according to any of the preceding claims, characterized in that it comprises a thermal cooling system to avoid large changes in the resonance condition, where the larger part (7) as well as the smaller part (6) have through holes (17) through which a pipe (18) is arranged along the axis of symmetry of the structure, and through which a liquid coolant is pumped from one side (19) to the other (20) of the pipe (18), the pipe passing near the set of batteries (5). 6.- Piezoelectric propellant according to claim 5, characterized in that the pipe (18) is made of aluminum coated externally with black matte paint.

7. Method for generating thrust without expelling propellant, based on the piezoelectric propellant according to any one of claims 1 to 6, characterized in that it comprises the steps of: a. Compressing the set of piezoelectric elements, along its axis of symmetry, between the smaller part (6) and the larger part (7); b. Applying an alternating voltage to the set of piezoelectric elements through the tabs of the conductive electrodes, i. where the applied alternating voltage has a waveform comprising two sinusoidal components with different frequencies, the first sinusoidal component with a frequency equal to the thickness resonance frequency or first harmonic of the piezoelectric elements, and the second sinusoidal component with a frequency equal to twice the thickness resonance frequency or second harmonic of the piezoelectric elements; i. where the amplitude of the second sinusoidal component is not greater than 20% of the amplitude of the first sinusoidal component, and the phase relationship between the two sinusoidal components of the waveform is TT / 2. iii. wherein, upon applying the alternating voltage to the piezoelectric elements, and due to the greater acceleration of the expansion, a thrust is produced along the axis of symmetry of the piezoelectric elements in the direction connecting the piezoelectric elements to the larger metal piece; c. measuring both the impedance of the piezoelectric elements and the phase angle between the voltage and current waveforms supplied to the piezoelectric elements, in order to identify the thickness resonance condition of the piezoelectric element and the frequency range in which both the impedance is at a local minimum and the phase angle between the voltage and current waveforms is zero. 8.- Method for generating a thrust without expelling propellant according to claim 7, characterized in that it also includes the addition of a number of higher harmonics of the resonance frequency of the thickness in the alternating voltage waveform, with amplitudes and relative phase chosen so that the second derivative of the voltage waveform, which is proportional to the acceleration of the pile, reaches positive values greater than the negative ones during each cycle.

9. A method for generating thrust without expelling propellant according to claim 7 or 8, characterized in that it further includes one or more systems for stabilizing the thickness resonance condition of the piezoelectric element for periods of time greater than 1 day, namely: a. a closed loop system that measures and tracks the thickness resonance condition of the piezoelectric elements throughout the duration of the alternating current supply, and communicates it to the arbitrary waveform generator, so that the arbitrary waveform generator adapts the supplied waveform to the changing resonance condition; b. supplying the alternating voltage in short pulses, in order to limit the heating of the piezoelectric elements that occurs due to continuous activation, and to limit the consequent drift from the resonance condition, i. in which the pulse periods are much longer than one cycle of the alternating voltage, thus allowing several acceleration cycles within the same pulse; c. a system for cooling the piezoelectric elements by absorbing their emitted thermal radiation by means of a material with an absorptivity greater than 0.9 and a thermal conductivity greater than 230 W / m*K, . in which the cooling system includes a sensor for monitoring the temperature of the piezoelectric elements; i. in which the material with absorptivity greater than 0.9 and thermal conductivity greater than 230 W / m*K has the shape of a tube that passes near the piezoelectric elements and allows a cooling liquid to be pumped and flowed inside it. 10.- Method for generating a thrust without expelling propellant according to claim 7 or 8 or 9 characterized in that the waveform consisting of two main frequencies have a fixed phase relationship of TT / 2 between them and a possible amplitude relationship could be around 0.177, where the explicit form of the signal as in the equation: 70 V = Yo + or 177

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