Nano-structure-based micro-electro-mechanical resonator and frequency stability control system thereof

Abstract

The invention relates to the technical field of resonators, in particular to a micro-electro-mechanical resonator based on a nano structure and a frequency stability control system of the micro-electro-mechanical resonator. The system comprises a modeling part, an analysis part and a self-adaptive control part, the modeling part is used for establishing a structural characteristic equation based on a periodic nanorod array on the surface of a resonator body and then solving a frequency response function based on a resonator kinetic equation; the analysis part is used for establishing a relation model of the temperature and the frequency deviation based on the frequency response function to obtain the frequency deviation caused by the temperature; and the adaptive control part is used for constructing a frequency-based adaptive control model based on frequency deviation to obtain control output and then converting the control output into driving voltage. The frequency deviation can be effectively reduced, and the frequency control precision and the system stability are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of resonators, and particularly relates to a microelectromechanical resonator based on nanostructures and its frequency stabilization control system. Background Art

[0002] With the rapid development of microelectromechanical system (MEMS) technology, MEMS-based resonators have been widely used in many fields, such as sensors, filters, oscillators, etc. Microelectromechanical resonators have become the core components of many precision devices due to their advantages of miniaturization, low power consumption, and high frequency stability. However, with the continuous changes in the application environment and functional requirements, the frequency stability problem of microelectromechanical resonators in actual operation has gradually become a key factor restricting their further development. Especially in complex environmental conditions, such as temperature fluctuations and external force disturbances, these factors can cause unpredictable frequency shifts of microelectromechanical resonators, thereby affecting the overall performance of the system.

[0003] Traditional MEMS resonators mostly adopt a silicon-based cantilever beam structure and are driven and detected by electrostatic or piezoelectric methods. The dynamic characteristics of such resonators are usually determined by their geometric dimensions, material properties, and external excitation conditions. In the prior art, the control methods for frequency stability mainly rely on the optimization of mechanical structure design and external feedback control strategies. For example, by introducing different coating layers on the surface of the cantilever beam to increase the stiffness of the resonator, thereby enhancing its anti-disturbance ability. However, the improvement of this method is limited. Especially in the case of drastic temperature changes, the thermal expansion effect of the material will directly cause the frequency of the resonator to shift. In order to address this frequency shift problem, some prior arts attempt to detect and adjust the resonant frequency through a closed-loop feedback control circuit. However, traditional feedback control circuits have certain limitations in dealing with the nonlinear effects caused by microstructural features and are difficult to achieve fine adjustment of frequency stability.

[0004] In addition, most of the existing microelectromechanical resonators rely on macroscopic mechanical models to analyze and design resonance characteristics, such as cantilever beams, tuning fork structures, etc. Although these macroscopic models can effectively describe the dynamic behavior of resonators under specific conditions, under the influence of high-frequency vibrations and nanoscale structural features, it is difficult for traditional models to accurately capture the regulatory effect of microscopic structures on frequency stability. This is mainly because the existing technologies have not effectively incorporated microscopic geometric features on the surface of the resonator, such as surface roughness, nanostructure arrangement, etc., into the dynamic model. Therefore, frequency offset remains an inevitable problem for traditional microelectromechanical resonators when facing complex external disturbances (such as temperature, pressure, etc.). To solve these problems, in recent years, researchers have begun to attempt to introduce nanostructures on the surface of the resonator, such as nanowire arrays, nanopore structures, etc. The introduction of these nanostructures can significantly change the energy distribution and stiffness characteristics on the surface of the resonator, thereby achieving fine regulation of vibration characteristics. However, there are still many challenges in the modeling and control of nanostructures in the existing technologies. The traditional dynamic equations do not fully consider the periodic arrangement characteristics of surface nanostructures, nor can they accurately reflect the influence of microscopic surface features on the frequency response of the resonator. In addition, the existing feedback control circuits usually ignore the surface feature effects at the nanoscale, making them insufficient in high-precision frequency control. Summary of the Invention

[0005] The main object of the present invention is to provide a microelectromechanical resonator based on nanostructures and its frequency stabilization control system, which can effectively reduce frequency offset and improve frequency control accuracy and system stability.

[0006] To solve the above problems, the technical solution of the present invention is implemented as follows:

[0007] On the one hand, the present invention provides a microelectromechanical resonator based on nanostructures, which includes: a silicon substrate, a resonator body, a nanowire array, anchor points, and driving electrodes; the resonator body is connected to the silicon substrate through the anchor points at both ends; the driving electrodes are directly integrated on the silicon substrate and are located below the resonator body; the resonator body is a double-ended fixed cantilever beam structure, and a nanowire array is uniformly distributed along the length direction on the upper surface, fixed at both ends through the anchor points, and the middle part is suspended, maintaining a gap with the driving electrode below to form a capacitive structure required for electrostatic driving; the nanowire array is arranged periodically to form a regular matrix structure, each nanowire is perpendicular to the surface of the resonator body, and a fixed spacing of 100 - 400 nm is maintained between adjacent nanowires; the anchor points are located at both ends of the resonator body, the upper end is integrated with the resonator body, and the lower end is connected to the silicon substrate; the driving electrodes are embedded or deposited on the silicon substrate, located directly below the resonator body, and are connected to an external circuit through metal leads.

[0008] Further, in the nanocolumn array, the height of each nanocolumn ranges from 100 to 500 nm; the diameter of each nanocolumn ranges from 50 to 200 nm; and the spacing between each nanocolumn is 100 to 400 nm.

[0009] On the other hand, the present invention also provides a frequency stabilization control system for a microelectromechanical resonator based on nanostructures, the system comprising: a modeling part, an analysis part, and an adaptive control part; the modeling part is configured to establish a structural characteristic equation based on the periodic nanocolumn array on the surface of the resonator body, obtain a surface characteristic function, introduce the surface characteristic function into the resonator dynamics equation, and then solve the frequency response function based on the resonator dynamics equation; the analysis part is configured to establish a relationship model between temperature and frequency offset based on the frequency response function to obtain the frequency offset caused by temperature; the adaptive control part is configured to construct a frequency-based adaptive control model based on the frequency offset to obtain a control output, and then convert the control output into a driving voltage.

[0010] Further, the formula of the structural characteristic equation is:

[0011]

[0012] where, F nano (x, y) represents the surface characteristic function; x is the abscissa of the surface; y is the ordinate of the surface; A 0 is the amplitude coefficient of each nanocolumn in the nanocolumn array; (x i , y j ) is the coordinate position of the nanocolumn located in the i-th row and the j-th column, representing the position of the nanocolumn in the two-dimensional plane, x i is the abscissa; y j is the ordinate; d ij is the characteristic size of the nanocolumn in the i-th row and the j-th column, which determines the width affected by each nanocolumn in the two-dimensional plane; h ij is the height of the nanocolumn in the i-th row and the j-th column; N is the total number of nanocolumns in the horizontal axis direction; M represents the total number of nanocolumns in the vertical axis direction; both i and j are integer subscript indexes.

[0013] Further, the formula of the resonator dynamics equation is:

[0014]

[0015] where, m represents the equivalent mass of the resonator body, which reflects the inertial characteristics exhibited by the resonator body during vibration. The larger the mass, the greater the inertia and the slower the response speed; represents the acceleration of the resonator body, describing the acceleration behavior of the displacement changing with time during vibration; c represents the damping coefficient of the resonator body, which reflects the resistance or energy loss suffered by the resonator body during vibration; represents the velocity of the resonator body; k represents the elastic coefficient of the resonator body; z represents the displacement of the resonator body, which reflects the offset of the resonator body relative to the equilibrium position; F ext (t) represents the external force applied to the resonator body at time t, which is any form of excitation force, including electromagnetic force or mechanical force; α represents the coupling coefficient between the nano-column array and the vibration of the resonator body, which reflects the influence intensity of the nano-column array on the vibration behavior of the resonator body. The larger the coupling coefficient, the more obvious the influence of the nano-column array on the vibration of the resonator body.

[0016] Furthermore, through the following formula, the frequency response function is solved based on the resonator dynamics equation:

[0017]

[0018] where ω represents the angular frequency; s represents the imaginary unit; A(ω) represents the amplitude-frequency characteristic; φ(ω) represents the phase-frequency characteristic of the system; where,

[0019]

[0020] Furthermore, through the following formula, based on the frequency response function, a relationship model between temperature and frequency offset is established:

[0021] Δf T =f 0 [1 + β 1 (T(t) - T 0 ) + β 2 (T(t) - T 0 ) 2 ·H(ω);

[0022] where Δf T is the frequency offset caused by temperature; f 0 is the initial frequency; T 0 is the set reference temperature; β 1 is the first-order temperature coefficient; β 2 is the second-order temperature coefficient; T(t) is the temperature at time t.

[0023] Furthermore, through the following formula, based on the frequency offset, an adaptive control model based on frequency is constructed to obtain the control output:

[0024]

[0025] where u(t) is the control output; Kp represents the proportional gain; K i represents the integral gain; K d represents the derivative gain; η represents the adaptive gain; is the Planck constant; m * is the average mass of the nanocolumns; is the Planck constant; k z is the wave vector in the Z-axis direction; Δ s is the average surface energy gap of the nanocolumns.

[0026] Furthermore, the control output is converted into a driving voltage through the following formula:

[0027]

[0028] where, V drive (t) represents the driving voltage at time t; C 0 represents the static capacitance; ΔF ext (t) represents the change in the external force applied to the resonator body at time t compared to time t - 1; C(t) is the capacitance at time t; τ RC represents the time constant, which reflects the charge and discharge characteristics. The larger the time constant, the longer the response time; ΔT(t) represents the change in temperature at time t compared to time t - 1.

[0029] The nanostructure-based microelectromechanical resonator and its frequency stabilization control system of the present invention have the following beneficial effects:

[0030] By introducing a periodically arranged nanocolumn array on the resonator surface, the present invention realizes precise adjustment of the resonator surface characteristics. The geometric characteristics (such as height, diameter, spacing) of the nanocolumn array can significantly change the energy density and stiffness distribution on the resonator surface, thereby adjusting the overall dynamic characteristics of the resonator. By establishing a surface characteristic function, the geometric parameters of the nanocolumns can be closely associated with the surface characteristics and introduced into the dynamic equation of the resonator. Compared with the traditional resonator design method, the method of the present invention can more accurately describe the adjustment effect of the nanoscale surface structure on the resonator frequency characteristics, especially under high-frequency vibration conditions, the adjustment effect brought by this nanostructure is particularly significant.

[0031] The present invention realizes a refined modeling of the frequency characteristics of a resonator by constructing a kinetic equation containing surface features. Traditional kinetic equations usually only consider the inertial, damping, and elastic characteristics of the resonator, ignoring the influence of surface structures on the resonant characteristics. However, the present invention combines the surface feature function with the kinetic equation by introducing a coupling term, which can reflect the influence of the nanocolumn array on the vibration behavior of the resonator. Specifically, the introduction of the surface feature function can correct the stiffness term in the kinetic equation, enabling the kinetic equation to better capture the surface effects of the nanocolumns. In this way, the kinetic equation can not only describe the basic vibration characteristics of the resonator but also reflect the adjustment effect of the surface microstructure on the frequency response, thereby improving the accuracy and stability of frequency control.

[0032] By establishing the frequency response function, the present invention can accurately analyze the response characteristics of the resonator at different excitation frequencies. The frequency response function can comprehensively describe the amplitude and phase characteristics of the resonator under different frequency conditions by combining the surface feature function, external excitation force, and the influence of temperature changes. This frequency response analysis method enables the present invention to more accurately predict and correct frequency offsets. Compared with traditional methods, the frequency response analysis method of the present invention can take into account the microscopic characteristics of the nanocolumn array and external temperature fluctuations, thereby effectively improving the frequency control ability of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 is a schematic structural diagram of a nanostructure-based microelectromechanical resonator provided by an embodiment of the present invention;

[0034] Figure 2 is an enlarged view of the nanocolumn array structure of the nanostructure-based microelectromechanical resonator provided by an embodiment of the present invention;

[0035] Figure 3 is a schematic structural diagram of a frequency stabilization control system of the nanostructure-based microelectromechanical resonator provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0036] In order to enable those skilled in the art to better understand the solution of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0037] Example 1, refer to Figure 1 and Figure 2: A nanostructure-based microelectromechanical resonator, which comprises: a silicon substrate, a resonator body, a nano-column array, anchors, and drive electrodes; the resonator body is connected to the silicon substrate through the anchors at both ends; the drive electrodes are directly integrated on the silicon substrate and are located below the resonator body; the resonator body is a double-ended fixed cantilever beam structure, and a nano-column array is uniformly distributed along the length direction on the upper surface, fixed at both ends through the anchors, with the middle part suspended, and a gap is maintained with the drive electrodes below to form a capacitive structure required for electrostatic drive; the nano-column array is arranged periodically to form a regular matrix structure, each nano-column is perpendicular to the surface of the resonator body, and a fixed spacing of 100-400 nm is maintained between adjacent nano-columns; the anchors are located at both ends of the resonator body, the upper end is integrated with the resonator body, and the lower end is connected to the silicon substrate; the drive electrodes are embedded or deposited on the silicon substrate, located directly below the resonator body, and are connected to an external circuit through metal leads.

[0038] Specifically, the present invention adopts a double-ended fixed cantilever beam structure, and a nano-column array is uniformly distributed along the length direction on its upper surface. The periodic arrangement of the nano-columns forms a regular matrix structure. The design concept of the entire resonator system is to integrate a nano-column array with specific geometric parameters on the cantilever beam of the resonator to achieve the effects of improving the resonance frequency stability and enhancing the resonance performance. The introduction of the nano-column array not only changes the mechanical properties of the cantilever beam surface, but also forms a new "surface energy gap", which has a significant impact on the frequency selection and vibration characteristics of the resonator. During the actual operation of the resonator, these nano-columns perpendicular to the surface of the resonator body can play a role in stabilizing the frequency and adjusting the resonance mode during mechanical vibration. The resonator body is connected to the silicon substrate through the anchors at both ends. The design of the anchors takes into account the mechanical coupling characteristics between the resonator and the silicon substrate to ensure that the resonator body can maintain a stable mechanical connection under high-frequency vibration conditions. The upper end of the anchor is integrated with the resonator body, and a high-strength connection method is adopted between the lower end and the silicon substrate, so that the resonator will not loosen or displace under high-frequency conditions, effectively avoiding vibration distortion caused by mechanical connection. At the same time, the geometric shape and material selection of the anchors have also been precisely designed to minimize their absorption of vibration energy as much as possible, so that more vibration energy is concentrated in the resonator body for propagation, thereby improving the energy utilization efficiency of the system. The resonator body is connected to the silicon substrate through the anchors at both ends. The design of the anchors takes into account the mechanical coupling characteristics between the resonator and the silicon substrate to ensure that the resonator body can maintain a stable mechanical connection under high-frequency vibration conditions. The upper end of the anchor is integrated with the resonator body, and a high-strength connection method is adopted between the lower end and the silicon substrate, so that the resonator will not loosen or displace under high-frequency conditions, effectively avoiding vibration distortion caused by mechanical connection. At the same time, the geometric shape and material selection of the anchors have also been precisely designed to minimize their absorption of vibration energy as much as possible, so that more vibration energy is concentrated in the resonator body for propagation, thereby improving the energy utilization efficiency of the system.

[0039] The nanocolumns are arranged at specific periodic intervals, and each nanocolumn grows vertically on the surface of the resonator body. This periodic array structure forms a regular energy transfer path. When the resonator vibrates under the electrostatic drive of the external drive electrode, the nanocolumn array adjusts the overall resonance frequency by changing the surface energy density on the resonator surface. This surface-energy-based adjustment mechanism can further optimize the frequency response characteristics of the resonator by controlling the geometric parameters of the nanocolumns (such as height, diameter, and spacing). This microscopic mechanical coupling between the nanocolumns and the cantilever beam makes the frequency adjustment of the system more flexible and precise. At the same time, the electrostatic drive model of the resonator has also undergone precise capacitance design and mathematical modeling. In the middle part of the cantilever beam, a capacitive structure with a gap from the lower drive electrode is designed. The formation of this capacitive structure is to generate sufficient electrostatic force to drive the resonator to vibrate by adjusting the voltage difference between the electrode and the resonator body when the resonator vibrates. As the cantilever beam vibrates periodically, the capacitance value between the electrode and the resonator also changes periodically, thus affecting the magnitude of the electrostatic force between the electrode and the resonator. Through this capacitive coupling mechanism, the resonator can vibrate at a stable frequency under the precise control of the external circuit.

[0040] Example 2: In the nanocolumn array, the height of each nanocolumn ranges from 100 to 500 nm; the diameter of each nanocolumn ranges from 50 to 200 nm; the spacing between each nanocolumn is 100 to 400 nm.

[0041] Specifically, the height of the nanocolumns determines their influence on the energy distribution on the surface of the resonator. The height range is set between 100 - 500 nm to enhance the surface stiffness and mechanical response characteristics of the resonator without affecting the overall structural stability of the resonator. As the height increases, the "stiffness enhancement effect" formed by the nanocolumns on the resonator surface gradually strengthens, thus being more remarkable in terms of the stability of the resonance frequency and the response speed. However, too high a height may cause mechanical stability problems of the resonator. The diameter range of the nanocolumns is set between 50 - 200 nm, which directly affects the mechanical strength and the distribution of surface energy of the nanocolumns. A smaller diameter makes the nanocolumns more likely to generate local mechanical deformation during vibration, thus forming dynamic mechanical response characteristics on the surface of the resonator. While a larger diameter enhances the stiffness of the nanocolumns, making the resonator exhibit better frequency stability under high-frequency vibration conditions. By controlling the diameter of the nanocolumns, an optimized balance can be achieved between frequency stability and flexible response, and this design provides diverse choices for the application of microelectromechanical resonators. The diameter range of the nanocolumns is set between 50 - 200 nm, which directly affects the mechanical strength and the distribution of surface energy of the nanocolumns. A smaller diameter makes the nanocolumns more likely to generate local mechanical deformation during vibration, thus forming dynamic mechanical response characteristics on the surface of the resonator. While a larger diameter enhances the stiffness of the nanocolumns, making the resonator exhibit better frequency stability under high-frequency vibration conditions. By controlling the diameter of the nanocolumns, an optimized balance can be achieved between frequency stability and flexible response, and this design provides diverse choices for the application of microelectromechanical resonators.

[0042] Example 3, refer to Figure 3 ,: A frequency stabilization control system for a nanostructure-based microelectromechanical resonator, the system includes: a modeling part, an analysis part, and an adaptive control part; the modeling part is used to establish a structural characteristic equation based on the periodic nanocolumn array on the surface of the resonator body, obtain a surface characteristic function, introduce the surface characteristic function into the resonator dynamics equation, and then solve the frequency response function based on the resonator dynamics equation; the analysis part is used to establish a relationship model between temperature and frequency offset based on the frequency response function, and obtain the frequency offset caused by temperature; the adaptive control part is used to construct a frequency-based adaptive control model based on the frequency offset, obtain a control output, and then convert the control output into a driving voltage.

[0043] Specifically, the presence of the periodic nano-pillar array greatly affects the surface energy distribution and stiffness characteristics of the resonator. The height, diameter, and arrangement spacing of the nano-pillars will change the local surface stiffness of the resonator, thereby affecting the dynamic behavior of the entire system. In the modeling part, by introducing the surface characteristic function, these microstructural characteristics are transformed into macroscopic system parameters, so as to accurately describe the geometric and mechanical characteristics of the resonator surface. The construction of the surface characteristic function is based on the analysis of the geometric parameters of the nano-pillars, and it can reflect how the nano-pillar array changes the surface energy density and vibration characteristics. This function not only simplifies the complex geometric structure analysis, but also provides accurate data input for subsequent dynamic modeling. Next, the surface characteristic function is introduced into the dynamic equation of the resonator to describe the vibration behavior of the system under external excitation. The dynamic equation mainly describes the motion characteristics of the resonator under the action of an external driving force, specifically including the elastic modulus, resonant frequency of the resonator, and the electrostatic effect of the driving electrode. In the traditional design of microelectromechanical resonators, the dynamic equation is often modeled based on the basic mechanical model of a cantilever beam, ignoring the influence of surface microstructures on the overall dynamic behavior. However, in the present invention, by combining the surface characteristic function with the dynamic equation, the influence of the nano-pillar array on the surface energy distribution can be accurately considered in the vibration analysis. This modeling method improves the prediction accuracy of the vibration behavior, making the response of the resonator under high-frequency vibration conditions more accurate and stable. In addition, in the modeling part, by solving the dynamic equation, the frequency response function of the resonator is obtained. The frequency response function is the key to the design of the entire system, and it describes the vibration amplitude and phase characteristics of the resonator at different excitation frequencies. By introducing the surface characteristic function, the frequency response function can better reflect how the surface microstructures affect the frequency characteristics of the resonator. Especially under high-frequency excitation conditions, the geometric characteristics of the nano-pillar array will have a significant impact on the resonant frequency of the system, and this impact can be accurately quantified by the frequency response function. The solution of the frequency response function provides an important basis for subsequent analysis and control. Especially in the process of adaptive control, the frequency response function can provide accurate feedback information for the frequency adjustment of the system.

[0044] Example 4: The formula of the structural characteristic equation is:

[0045]

[0046] where F nano (x, y) represents the surface characteristic function; x is the abscissa of the surface; y is the ordinate of the surface; A 0 is the amplitude coefficient of each nano-pillar in the nano-pillar array; (x i , y j ) is the coordinate position of the nano-pillar located in the i-th row and the j-th column, indicating the position of the nano-pillar in the two-dimensional plane, x iis the abscissa; y j is the ordinate; d ij is the characteristic size of the nanocolumn at the i-th row and j-th column, which determines the width affected by each nanocolumn on the two-dimensional plane; h ij is the height of the nanocolumn at the i-th row and j-th column; N is the total number of nanocolumns in the horizontal axis direction; M represents the total number of nanocolumns in the vertical axis direction; both i and j are integer subscript indices.

[0047] Specifically, the formula of this surface feature function essentially superimposes the local effects generated by all nanocolumns on the two-dimensional plane, and represents the feature value at any position on the surface through the coordinates (x, y). In the specific expression, the formula describes the coordinate position (x i , y j ) of each nanocolumn and its geometric parameters (such as the height h ij and the characteristic size d ij ) to achieve the precise modeling of the entire surface micro-features. For each nanocolumn, its effect is described as a Gaussian distribution form centered at the coordinate center (x i , y j ), and this Gaussian function reflects the local feature attenuation effect generated by this nanocolumn around it. The attenuation effect is represented by an exponential term in the formula, and its core idea is to simulate the gradual influence of each nanocolumn on the surrounding area, so that the area closer to the nanocolumn is more affected, while the influence on the area farther away gradually weakens. In this formula, the parameter A 0 acts as the amplitude coefficient, characterizing the overall vibration intensity of each nanocolumn. Through this coefficient, the characteristics of all nanocolumns can be uniformly normalized, so that the amplitudes of each nanocolumn can be uniformly represented in the feature function. This processing method can effectively simplify the calculation in the subsequent solution of the kinetic equation, making the expression of the overall model more concise. The geometric parameters (x i , y j ) in the formula determine the position of each nanocolumn, which is not only the core component of the surface feature function, but also the basis for describing the periodic arrangement of the entire array. By introducing these position parameters, the formula can accurately depict the periodic characteristics of the nanocolumns, and on the two-dimensional plane, through the change of the coordinate position, the microscopic geometric changes in different regions can be captured.

[0048] The height parameter h ijIt is an important factor that determines the influence degree of each nanocolumn in the surface feature function. As the height of the nanocolumn changes, the contribution of the nanocolumn to the surface feature also changes significantly. The nanocolumn with a higher height usually has a stronger influence on the surface, thus showing a higher feature value in the overall surface feature. The change in this geometric height directly reflects the contribution degree of each nanocolumn to the surface feature through the product term in the formula. In addition, the feature size d ij The introduction in the formula is to describe the local influence range of the nanocolumn. Under the assumption of the Gaussian distribution model, the feature size determines the diffusion degree of the influence of the nanocolumn on the surrounding surface area. A larger feature size means a larger influence range of the nanocolumn, while a smaller feature size indicates that the influence of the nanocolumn is more concentrated. In this way, the formula can dynamically adjust and optimize the surface feature according to the specific size characteristics of the nanocolumn. The formula realizes the global feature description of the entire nanocolumn array through a double summation operation. The essence of the summation operation is to superimpose the local influences of all nanocolumns, thereby forming a comprehensive surface feature expression. This global summation method enables the entire surface feature function to reflect both the local influence of each nanocolumn and the periodic characteristics of the overall array in the description. Through this global description, the surface feature function can comprehensively depict the geometric features of the entire resonator surface, providing an important mathematical basis for subsequent dynamic analysis. This modeling method based on the surface feature function provides a new path for the frequency stability control of the resonator. By introducing the surface feature into the dynamic equation, the influence of the nanocolumn array on the dynamic characteristics of the resonator can be analyzed more accurately. In the frequency stability control system, the surface feature function serves as the modeling basis and can reflect how the periodic arrangement of the nanocolumn array affects the resonance frequency and vibration characteristics of the resonator. Especially under the conditions of high-frequency vibration and external temperature changes, the geometric changes of the nanocolumn array can dynamically compensate for the influence of environmental changes on frequency stability through the adjustment of the surface feature function.

[0049] Example 5: The formula of the resonator dynamic equation is as follows:

[0050]

[0051] Among them, m represents the equivalent mass of the resonator body, which reflects the inertial characteristics shown by the resonator body during vibration. The larger the mass, the greater the inertia and the slower the response speed; represents the acceleration of the resonator body, which describes the acceleration behavior of the displacement changing with time during vibration; c represents the damping coefficient of the resonator body, which reflects the resistance or energy loss suffered by the resonator body during vibration; represents the velocity of the resonator body; k represents the elastic coefficient of the resonator body; z represents the displacement of the resonator body, which reflects the offset of the resonator body relative to the equilibrium position; F ext (t) represents the external force applied to the resonator body at time t, which is an excitation force in any form, including electromagnetic force or mechanical force; α represents the coupling coefficient between the nanocolumn array and the vibration of the resonator body, which reflects the influence intensity of the nanocolumn array on the vibration behavior of the resonator body. The larger the coupling coefficient, the more obvious the influence of the nanocolumn array on the vibration of the resonator body.

[0052] Specifically, the inertial term in the dynamic equation reflects the influence of the equivalent mass m of the resonator on the dynamic response of the system. The concept of equivalent mass simplifies the mass of the actual resonator body, and it covers the inertial characteristics of the resonator during vibration. According to classical mechanics, inertia is the property of an object to resist changes in acceleration when subjected to an external force. Therefore, a larger equivalent mass means that the resonator has a greater inertia and its response speed is relatively slower. This inertial term describes the acceleration behavior of the resonator changing with time during vibration through the modeling of acceleration . This term can not only reflect the basic dynamic behavior of the resonator under external force excitation but also provide a basis for the dynamic control of the system. At the same time, the damping term in the equation is used to describe the energy loss generated by the resonator during vibration due to damping. The damping coefficient c is a key parameter describing the damping characteristics of the resonator during vibration. It reflects the degree of internal resistance or external energy dissipation suffered by the system during vibration. In practical applications, the magnitude of the damping coefficient directly affects the vibration amplitude and decay speed of the resonator. A larger damping coefficient means that the energy loss of the system is larger and the vibration stability is better; but at the same time, it may also reduce the response sensitivity of the system. Therefore, reasonably selecting and optimizing the damping coefficient can ensure the stability of the resonator under external interference while maintaining the good dynamic performance of the system.

[0053] The elastic term kz represents the elastic restoring force of the resonator. When the resonator is subjected to an external excitation, it shows a displacement deviating from the equilibrium position, and this displacement will cause a restoring force to be generated inside the system, making the resonator tend to return to its initial equilibrium position. The elastic coefficient k reflects the stiffness characteristics of the resonator, which represents the ability of the resonator to resist deformation under an external force. The larger the elastic coefficient, the higher the stiffness of the resonator. Therefore, under the same external force excitation, the displacement change of the system is smaller. This elastic term is a key factor in maintaining the balance of the entire dynamic equation. It not only describes the mechanical characteristics of the resonator but also provides an important guarantee for the frequency stability of the system. The external excitation term F ext(t) plays a driving role in the kinetic equation. The form of this external excitation force can be diverse, including electromagnetic force, mechanical force, or other forms of external acting forces. The introduction of the external excitation force is to simulate the external interference or artificial control that the resonator receives under actual operating conditions. Through this excitation term, the kinetic equation can accurately describe the vibration response of the resonator under different excitation conditions. The intensity and form of the external excitation directly affect the amplitude and frequency of the resonator, enabling the kinetic equation to adapt to different application scenarios. In traditional kinetic models of resonators, usually only three factors, namely inertia, damping, and elasticity, are considered, while the influence of surface microstructure on the overall kinetic characteristics is ignored. In the kinetic equation of the present invention, by introducing the coupling term αF nano (x, y)·z, the coupling relationship between the nanocolumn array and the vibration behavior of the resonator is realized. The F nano (x, y) in this coupling term is the surface feature function, which represents the characteristic effects generated by the nanocolumn array on the surface. The introduction of the surface feature function enables the kinetic equation to accurately reflect the geometric characteristics of the nanocolumn array and its influence on the vibration characteristics of the resonator. In the coupling term of the kinetic equation, the coupling coefficient α is a key parameter, which represents the influence intensity of the nanocolumn array on the vibration of the resonator. The magnitude of the coefficient α directly determines the contribution of the nanocolumn array in the kinetic equation. The larger the coupling coefficient, the stronger the regulation effect of the nanocolumn array on the surface energy of the resonator. The introduction of this coupling relationship enables the kinetic equation to comprehensively capture how the surface microstructure of the nanocolumn affects the macroscopic vibration behavior of the resonator. This coupling relationship between surface microstructure and vibration is an important innovation of the present invention. Specifically, the surface feature function F nano (x, y) describes the periodic arrangement and geometric characteristics of the nanocolumn array, reflecting its contribution to the surface energy at different positions. Through the introduction of this function, the kinetic equation can more precisely depict how the microscopic changes of the nanocolumn array affect the overall vibration of the resonator. Especially when the temperature changes or other external conditions change, the geometric characteristics of the nanocolumn may undergo slight changes, and these changes are dynamically reflected in the kinetic equation through the correction of the surface feature function, thereby realizing the frequency adjustment and stable control of the resonator.

[0054] Example 6: Solve the frequency response function based on the resonator kinetic equation through the following formula:

[0055]

[0056] where, ω represents the angular frequency; s represents the imaginary unit; A(ω) represents the amplitude-frequency characteristic; φ(ω) represents the phase-frequency characteristic of the system; where,

[0057]

[0058] Specifically, the frequency response function is to reveal the dynamic behavior of the resonator under different frequency excitations. In micro-electromechanical systems, the vibration characteristics of the resonator are directly affected by its structural parameters (such as mass, stiffness, damping, etc.) and the external excitation frequency. Traditional dynamic models usually describe these vibration behaviors through simplified second-order differential equations, while in the present invention, the introduction of nanostructures brings new complexity. The coupling between the nanocolumn array and the resonator surface leads to additional surface energy effects, which are reflected in the frequency response as a key factor in frequency modulation. Therefore, the frequency response function is not only a tool to describe the inherent dynamic characteristics of the resonator, but also includes an accurate characterization of the influence of the surface microstructure. The angular frequency ω is the core parameter in the frequency response formula, which represents the periodic changes of the system under external excitation. As the angular frequency increases, the inertia term mω of the system 2 The impact on the frequency response becomes more significant because the inertial effect resists the vibration acceleration of the system. At the same time, the elastic term k reflects the stiffness characteristics of the system. At low frequencies, the stiffness dominates the response of the system and the resonator exhibits a relatively stable vibration amplitude. However, when the frequency increases, the damping term cω begins to play a greater role, especially when approaching the resonant frequency, the impact of damping on the energy loss of the system will be aggravated, thereby suppressing the vibration amplitude of the system.

[0059] The coupling effect of the nanopillar array is expressed through the surface characteristic function F nano (x, y) is introduced into the frequency response function. The surface characteristic function describes the geometric characteristics of the nanopillars and their distribution on the surface of the resonator, and expresses the interaction between the nanopillars and the resonator body through the coupling coefficient α. This coupling effect changes the surface stiffness of the resonator, thereby affecting the overall frequency response characteristics of the system. The existence of the coupling term enables the frequency response function to not only reflect the traditional vibration characteristics of the resonator, but also to describe how the nanopillar array adjusts the frequency response of the system. Specifically, the height, spacing and arrangement of the nanopillars will affect the stiffness and damping effect of the system through the surface characteristic function, thereby adjusting the vibration amplitude and phase response of the resonator. The frequency response function represents the dynamic behavior of the system in complex form, and the introduction of the imaginary unit s enables the function to simultaneously describe the amplitude-frequency characteristics and the phase-frequency characteristics. The amplitude-frequency characteristic A(ω) represents the vibration amplitude of the resonator at different frequencies, while the phase-frequency characteristic φ(ω) represents the phase change or time lag effect of the system at different frequencies. As the frequency changes, the phase response of the resonator will change significantly, especially when it is close to the resonant frequency, the phase change may be more drastic, which reflects the delay in the system's response to external excitation. Among them, the denominator of the frequency response function It reflects the competitive relationship between the stiffness, inertia and damping of the system at different frequencies. Stiffness k and inertia term mω 2The difference reflects the vibration resistance of the system at different frequencies, while the damping term cω describes the energy dissipation characteristics of the system. The surface characteristic function F nano (x, y) is added to further correct the frequency response of the system. By adjusting the stiffness and inertial effects of the system, it affects the amplitude and phase characteristics of the resonator at different frequencies. Traditional analysis methods for microelectromechanical resonators usually only consider macroscopic mechanical characteristics and ignore the influence of microscopic surface structures on the system. In the present invention, by introducing the surface characteristics of the nanocolumn array, the frequency response function can comprehensively capture the adjustment effect of the microscopic structural changes on the resonator surface on the vibration behavior of the system. This adjustment is not only manifested in the amplitude of the frequency response of the system, but also affects the phase response and frequency stability of the system. Especially when vibrating at high frequencies or when the external environment changes, the adjustment effect of this surface characteristic is particularly important.

[0060] Example 7: Based on the frequency response function, establish a relationship model between temperature and frequency offset through the following formula:

[0061] Δf T = f 0 [1 + β 1 (T(t) - T 0 ) + β 2 (T(t) - T 0 ) 2 ·H(ω);

[0062] where, Δf T is the frequency offset caused by temperature; f 0 is the initial frequency; T 0 is the set reference temperature; β 1 is the first-order temperature coefficient; β 2 is the second-order temperature coefficient; T(t) is the temperature at time t.

[0063] Specifically, the frequency offset Δf in the formula T directly reflects the frequency offset of the resonator caused by temperature change. The initial frequency f 0 represents the reference frequency at the reference temperature T 0 , that is, the design frequency of the system in the ideal state. By modeling the temperature change, the system can accurately capture the frequency change behavior of the resonator at different ambient temperatures. The influence of temperature change on the resonator is reflected in multiple aspects such as the thermal expansion of materials, the structural changes of nanocolumns, and the changes in capacitance coupling. Therefore, the quantitative analysis of this influence is crucial. The temperature correction term 1 + β 1 (T(t) - T 0 ) +

[0064] β 2 (T(t) - T0 ) 2 Capture the influence of temperature on the resonator frequency through two temperature coefficients β 1 and β 2 . Here, β 1 represents the first-order temperature coefficient, which is used to describe the influence of linear temperature changes on frequency, while β 2 represents the second-order temperature coefficient, which is used to characterize the nonlinear temperature effect. The introduction of these two temperature coefficients enables this model to not only handle simple linear temperature changes but also cope with complex nonlinear effects. In reality, the materials and structures of microelectromechanical systems usually exhibit complex nonlinear expansion behaviors under temperature changes. Therefore, considering the second-order temperature effect is the key to improving the accuracy of frequency stability analysis. In this model, the temperature T(t) is introduced as a function of time, representing the actual temperature of the system at a specific time t. The advantage of doing this is that it allows for real-time analysis of the frequency changes of the resonator under dynamically changing temperature conditions. Temperature changes not only cause thermal expansion of the resonator material but also change the geometric characteristics of the nanocolumn array, thereby affecting the surface energy distribution. With temperature fluctuations, the height, diameter, and spacing of the nanocolumns may undergo small changes, and these changes are introduced into the frequency response function H(ω) through the correction of the surface characteristic function F nano (x, y).

[0065] The frequency response function H(ω) plays a bridging role in this relational model. It links the structural characteristic changes caused by temperature changes with the dynamic response of the resonator. Specifically, the surface characteristic function F nano (x, y) reflects the regulating effect of the geometric characteristics of the nanocolumns on frequency, while the frequency response function further describes the coupling effect between the nanocolumns and the external excitation. Therefore, in the temperature and frequency offset model, by varying the temperature correction term, the dynamic characteristics of the coupled frequency response function can accurately reflect the influence of temperature changes on the system frequency. Through this model, it is possible to understand how temperature changes affect the frequency stability of the resonator through the complex interactions among the thermal expansion effect of the material, surface characteristic changes, and the dynamic response of the resonator. When the temperature rises, the thermal expansion of the material causes changes in the geometric parameters of the nanocolumns, and the surface characteristic function changes accordingly, resulting in a correction of the frequency response function. This correction is further mapped to the frequency offset Δf T , thereby causing an actual change in the system frequency. By introducing the initial frequency f 0 , this model can ensure that at the reference temperature T 0 , the frequency offset is zero, that is, the system is in an ideal operating state. The design of the temperature correction term takes into account the complex response behaviors of the resonator under different temperature change conditions. The linear term β 1 (T(t) - T 0)It is mainly used to describe simple linear temperature effects, which is reasonable within most common temperature change ranges. However, in order to more accurately reflect complex temperature changes, the introduction of the second-order term β 2 (T(t)-T 0 ) 2 is necessary. This second-order term effectively captures the nonlinear frequency response caused by the nonlinear thermal expansion of the material and changes in surface characteristics, thereby improving the accuracy of the entire model.

[0066] Example 8: Through the following formula, based on the frequency offset, construct a frequency-based adaptive control model to obtain the control output:

[0067]

[0068] where u(t) is the control output; K p represents the proportional gain; K i represents the integral gain; K d represents the derivative gain; η represents the adaptive gain; is the Planck constant; m * is the average mass of the nanocolumn; is the Planck constant; k z is the wave vector in the Z-axis direction; Δ s is the average surface energy gap of the nanocolumn.

[0069] Specifically, in the control output u(t), the role of the PID part is to perform basic regulation of the frequency using traditional control methods. The proportional control term K p Δf T is directly proportional to the frequency offset Δf T , which means that when the system detects a frequency offset, it immediately applies a correction output to quickly cancel the offset, ensuring the immediate response ability of the system. And the integral control term accumulates the error of the frequency offset over a period of time, and corrects the long-term frequency drift by adjusting the accumulated error, so that the system maintains a long-term stable state. The derivative control term It is adjusted according to the change rate of the frequency offset, which provides a predictive adjustment for the dynamic changes of the system. By responding to the rate of frequency change, it suppresses the drastic fluctuations of the system and maintains the stability of the resonator. The combination of these three control terms enables the PID control to achieve a balance among the response speed, steady-state error, and dynamic characteristics. However, relying solely on PID control may be insufficient to cope with the frequency offset caused by the microstructure. Therefore, the present invention introduces an adaptive adjustment mechanism based on the characteristics of the nanocolumn array. The introduction of the adaptive adjustment term is based on the geometric arrangement and quantum mechanical properties of the nanocolumns. Through the number of nanocolumns arranged in the two-dimensional plane, N and M, the model can reflect the overall impact of the periodic arrangement of the nanocolumn array on the frequency characteristics of the system. These arrangement characteristics macroscopically determine the characteristic energy distribution on the surface of the resonator and further affect the vibration response of the system. The wave vector k z and the Planck constant are introduced to capture the effects of the nanocolumns at the quantum scale. The wave vector k z reflects the quantum characteristics in the Z-axis direction, and the Planck constant, as a fundamental quantum mechanical parameter, is used to describe the quantum behavior of the nanocolumns in the surface vibration of the resonator. At the nanoscale, quantum effects cannot be ignored. Especially under high-frequency vibration conditions, these quantum characteristics may significantly affect the frequency response of the resonator. Through these quantum parameters, the control model can more accurately describe the adjustment effect of the surface characteristics of the nanocolumn array on the frequency offset. Another key parameter is the average mass m * of the nanocolumns, which reflects the inertial characteristics of the nanocolumns during vibration. A larger average mass means that the nanocolumns respond more strongly to the surface vibration, and this inertial effect may have a significant impact on the dynamic response of the resonator under high-frequency conditions. In addition, the surface energy gap Δ s in the formula represents the surface energy change of the nanocolumn array, which is an important parameter for describing the energy adjustment characteristics of the system under temperature changes. When the temperature changes, the size of the surface energy gap will change accordingly, and this change directly affects the distribution of the surface energy, thereby changing the frequency response of the system through the coupling effect. By considering this energy gap, the model can adaptively adjust the influence of external temperature changes on the frequency and thus maintain the stability of the system. The external excitation force F ext(t) and the product term of displacement z reflect the direct perturbation of the external force on the resonator. The external excitation may be applied in different forms at different times, and its intensity and direction will fluctuate over time. Incorporating this external force term into the control model can form a real-time feedback mechanism during the control process to cancel out the frequency shift caused by the external force and enhance the anti-interference ability of the system. Through this combination of traditional PID and adaptive control based on microscopic feature adjustment, the present invention can maintain the frequency stability of the resonator in a complex and changing environment. This method not only solves the frequency shift problem caused by temperature changes, but also realizes fine adjustment of the frequency response by considering the quantum effect and surface energy gap change of the nanowire array. Especially in high-precision application scenarios, this adaptive control can greatly improve the frequency control accuracy of the system and reduce the adverse effects brought by external perturbations.

[0070] Example 9: Convert the control output into a driving voltage through the following formula:

[0071]

[0072] where, V drive (t) represents the driving voltage at time t; C 0 represents the static capacitance; ΔF ext (t) represents the change in the external force applied to the resonator body at time t compared to time t - 1; C(t) is the capacitance at time t; τ RC represents the time constant, which reflects the charge and discharge characteristics. The larger the time constant, the longer the response time; ΔT(t) represents the change in temperature at time t compared to time t - 1.

[0073] Specifically, in the formula for the driving voltage, the control output u(t) comes from the previous adaptive control model, which is the direct result of adaptive control. It synthesizes various parts of PID control and nanostructure adjustment and represents the correction signal for frequency shift. However, simply using the control output is not enough because the system also needs to respond to changes in the externally applied force. Therefore, the external force change term ΔF ext (t) is specifically introduced into the formula. This external force change term is used to describe the change in the external force at the current time t compared to the previous time t - 1. This is because changes in the external force will directly affect the dynamic characteristics of the resonator and may thus cause frequency shift. Therefore, considering this change in the control output can further improve the adjustment accuracy of the driving voltage. The denominator part of the formula includes the static capacitance C 0 and the derivative term related to the capacitance change and The static capacitance C 0 represents the basic capacitance value of the resonator without external changes, while the derivative term of the capacitance It reflects the rate of change of capacitance with displacement z. The existence of this derivative term is because the vibration of the resonator directly changes the gap between the electrodes, thus affecting the magnitude of the capacitance. For a microelectromechanical resonator, the dynamic change of capacitance is crucial for the response characteristics of the driving voltage. Therefore, this change needs to be precisely considered in the formula. The derivative term further refines the response to changes over time. The time constant τ in the formula RC is used to describe the charge and discharge characteristics of the system. The time constant is a key parameter of the capacitive circuit, which reflects the response speed of the circuit to changes in the driving signal. The larger the time constant, the slower the system responds to the driving signal, while a smaller time constant indicates that the system can adjust the voltage faster. Therefore, the choice of the time constant has a direct impact on the regulation performance of the driving voltage. By considering the time constant τ RC , the formula can more accurately describe the dynamic behavior of the resonator during the frequency adjustment process. The exponential decay term introduced in the formula further reflects the influence of temperature change on the driving voltage. Here, ΔT(t) represents the change in temperature at the current time t compared to the previous moment, and T 0 is the reference temperature. Temperature change will cause thermal expansion of the resonator material, and this expansion effect will change the surface characteristics and capacitance characteristics of the resonator. Therefore, through the consideration of temperature change, the exponential decay term enables the formula to adaptively adjust the driving voltage when the ambient temperature fluctuates. This exponential form of design aims to ensure that the system can maintain precise regulation of the driving signal when external conditions change. Temperature change not only affects the material properties of the resonator but also changes the geometric parameters of the nanorod array, thus affecting the overall frequency response. By introducing the temperature correction term, the formula can dynamically adapt to temperature fluctuations and reduce the influence of temperature on the frequency stability of the system.

[0074] In several embodiments provided in this application, it should be understood that the disclosed systems, devices, and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the division of the units is only a logical function division. In actual implementation, there can be other division methods. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed couplings or direct couplings or communication connections to each other can be through some interfaces. The indirect couplings or communication connections of the devices or units can be in electrical, mechanical, or other forms.

[0075] As described above, the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A nanostructured micro-electromechanical resonator, characterized in that: It includes: Silicon substrate, resonator body, nanopillar array, anchor and drive electrodes; The resonator body is connected to the silicon substrate through anchor points at both ends; the driving electrode is directly integrated on the silicon substrate and is located below the resonator body; the resonator body is a cantilever beam structure with double ends fixed, and a nanocolumn array is evenly distributed on the upper surface along the length direction. The two ends are fixed by anchor points, and the middle part is suspended, maintaining a gap with the driving electrode below to form a capacitor structure required for electrostatic drive; the nanocolumn array is arranged periodically to form a regular matrix structure, each nanocolumn is perpendicular to the surface of the resonator body, and a fixed spacing of 100-400nm is maintained between adjacent nanocolumns; the anchor points are located at both ends of the resonator body, the upper end is integrated with the resonator body, and the lower end is connected to the silicon substrate; the driving electrode is embedded or deposited on the silicon substrate, located directly below the resonator body, and connected to the external circuit through a metal lead.

2. The nanostructured micro-electromechanical resonator according to claim 1, characterized in that: In the nanocolumn array, the height of each nanocolumn ranges from 100 to 500 nm; the diameter of each nanocolumn ranges from 50 to 200 nm; and the spacing between each nanocolumn ranges from 100 to 400 nm.

3. A frequency stabilization control system for a nanostructured micro-electromechanical resonator according to any one of claims 1 to 2, characterized in that: The system includes: a modeling part, an analysis part and an adaptive control part; the modeling part is used to establish a structural characteristic equation based on the periodic nanocolumn array on the surface of the resonator body, obtain a surface characteristic function, introduce the surface characteristic function into the resonator dynamics equation, and then solve the frequency response function based on the resonator dynamics equation; the analysis part is used to establish a relationship model between temperature and frequency offset based on the frequency response function, and obtain the frequency offset caused by temperature; the adaptive control part is used to construct a frequency-based adaptive control model based on the frequency offset, obtain a control output, and then convert the control output into a driving voltage.

4. The frequency stabilization control system based on nanostructured micro-electromechanical resonator according to claim 3, characterized in that: The formula of the structural characteristic equation is: Among them, F nano (x, y) represents the surface characteristic function; x is the horizontal coordinate of the surface; y is the vertical coordinate of the surface; A0 is the amplitude coefficient of each nanopillar in the nanopillar array; (x i ,y j ) is the coordinate position of the nanorod located in the i-th row and j-th column, indicating the position of the nanorod on the two-dimensional plane, x i is the horizontal axis; j is the vertical axis; d ij is the characteristic size of the nanopillar in the i-th row and j-th column, which determines the width of each nanopillar in the two-dimensional plane; h ij is the height of the nanopillar in the i-th row and j-th column; N is the total number of nanopillars in the horizontal direction; M represents the total number of nanopillars in the vertical direction; i and j are both integer subscript indices.

5. The frequency stabilization control system based on nanostructured micro-electromechanical resonator according to claim 4, characterized in that: The formula for the resonator dynamics equation is: Among them, m represents the equivalent mass of the resonator body, which reflects the inertial characteristics of the resonator body during vibration. The larger the mass, the greater the inertia and the slower the response speed. represents the acceleration of the resonator body, which describes the acceleration behavior of the displacement changing with time during the vibration process; c represents the damping coefficient of the resonator body, which reflects the resistance or energy loss encountered by the resonator body during the vibration process; represents the velocity of the resonator body; k represents the elastic coefficient of the resonator body; z represents the displacement of the resonator body, which reflects the displacement of the resonator body relative to the equilibrium position; F ext (t) represents the external force applied to the resonator body at time t, which is any form of excitation force, including electromagnetic force or mechanical force; α represents the coupling coefficient between the nanocolumn array and the vibration of the resonator body, which reflects the intensity of the influence of the nanocolumn array on the vibration behavior of the resonator body. The larger the coupling coefficient, the more obvious the influence of the nanocolumn array on the vibration of the resonator body.

6. The frequency stabilization control system based on nanostructured micro-electromechanical resonator according to claim 5, characterized in that: The frequency response function is solved based on the resonator dynamics equation using the following formula: Where ω represents the angular frequency; s represents the imaginary unit; A(ω) represents the amplitude-frequency characteristic; φ(ω) represents the phase-frequency characteristic of the system; where 7. The frequency stabilization control system based on nanostructured micro-electromechanical resonator according to claim 6, characterized in that: The relationship model between temperature and frequency offset is established based on the frequency response function through the following formula: Δf T =f0[1+β1(T(t)-T0)+β2(T(t)-T0) 2 ]·H(ω); Where Δf T is the frequency deviation caused by temperature; f0 is the initial frequency; T0 is the set reference temperature; β1 is the first-order temperature coefficient; β2 is the second-order temperature coefficient; T(t) is the temperature at time t.

8. The frequency stabilization control system based on nanostructured micro-electromechanical resonator according to claim 7, characterized in that: Based on the frequency offset, a frequency-based adaptive control model is constructed through the following formula to obtain the control output: Where, u(t) is the control output; K p Represents proportional gain; K i Indicates the integral gain; K d represents the differential gain; η represents the adaptive gain; is Planck's constant; m * is the average mass of the nanorods; h is Planck's constant; k z is the wave vector in the Z-axis direction; Δ s is the average surface energy gap of the nanorods.

9. The frequency stabilization control system based on nanostructured micro-electromechanical resonator according to claim 8, characterized in that: The control output is converted into a drive voltage using the following formula: Among them, V drive (t) represents the driving voltage at time t; C0 represents the static capacitance; ΔF ext (t) represents the change of the external force applied to the resonator body at time t compared to time t-1; C(t) is the capacitance at time t; τ RC It represents the time constant, which reflects the charge and discharge characteristics. The larger the time constant, the longer the response time. ΔT(t) represents the change in temperature at time t compared to time t-1.

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