Structural Design of a Low-Frequency In-plane Torsional Mode Wheel-Shaped Resonator

Structural Design of a Low-Frequency In-plane Torsional Mode Wheel-Shaped Resonator | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Structural Design of a Low-Frequency In-plane Torsional Mode Wheel-Shaped Resonator Weibing Wang, Dingxin Liu, Wenlong Lv, Jianmao Li, Shitao Chen, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7525253/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In low-frequency vibrating MEMS resonators, device structures typically vibrate in flexural mode and torsional mode. The mechanical stiffness in the motion direction of the bending mode is often strongly correlated with the device’s equivalent stiffness, while torsional mode generally use piezoelectric actuation and usually have a low Q factor. These factors limit the application of low-frequency resonators in some high-reliability scenarios. This paper proposes a low-frequency wheel-shaped resonator vibrating in the in-plane torsional mode. By using the torsional vibration of the wheel-shaped mass instead of translational vibration, and adopting an in-plane capacitive structure for driving and sensing, the device maintains high in-plane stiffness while vibrating at low frequencies. At the same frequency, its in-plane stiffness is 10² orders of magnitude higher than that of the flexural mode. Simulations on the anti-overload performance of the device structure show that under a 100g static load, the frequency drifts in the X-axis and Z-axis directions are 21.67 ppb and 60.39 ppb, respectively. Under 10000g impact loads in the X-axis and Z-axis directions, the maximum in-plane stresses are 24.7 MPa and 45.7 MPa, with maximum displacements of 0.111 µm and 0.137 µm, respectively—values far lower than the maximum allowable stress of silicon material and the minimum line width of the device. This demonstrates that the structure has high frequency stability and anti-overload performance. Physical sciences/Engineering/Electrical and electronic engineering Physical sciences/Nanoscience and technology/Nanoscale devices/Sensors Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction With the rapid development of the Internet of Things (IoT) and 5G communication technologies, the number of various intelligent terminals and interconnected devices has grown explosively. This trend has put forward even more stringent requirements for the miniaturization, integration, low power consumption, and high reliability of electronic devices. Against this backdrop, MEMS (Micro-Electro-Mechanical Systems) technology, leveraging its unique advantages in microstructure design, mass manufacturing, and multi-physics field regulation, has become one of the core technologies supporting the development of new-generation fields such as information communication and intelligent sensing. MEMS resonators are key components capable of achieving precise frequency control and signal filtering. They realize efficient energy conversion and frequency selection by converting electrical signals into mechanical vibrations and utilizing resonance characteristics. Depending on their structural design and dimensions, the resonant frequencies of MEMS resonators typically range from kHz to GHz. MEMS resonators operating in different frequency bands often have distinct application scenarios. For resonators with low-frequency vibrations (below 1 MHz), they play an irreplaceable role in numerous applications, such as high-precision resonant accelerometers (ref.1–2), resonant pressure sensors (ref.3–4), high-precision clocks (ref.5–7) in inertial navigation systems, and trace substance detection in environmental monitoring (ref.8–9). The performance of resonators directly affects the stability, reliability, and integration level of the entire electronic system. For a resonator system, its resonant frequency is related to the physical properties of the system. For a single-degree-of-freedom resonator structure, its natural frequency can be expressed as follows: $$\:{f}_{0}=\frac{1}{2\pi\:}\sqrt{\frac{{k}_{eff}}{{m}_{eff}}}$$ 1 Where f₀ is the natural frequency of the device's vibration, kₑ ff is the equivalent stiffness of the structure in the current vibration mode, and mₑ ff is the equivalent mass of the structure in the current vibration mode. For resonators with low-frequency vibrations, most of their vibration modes are flexural mode, which are characterized by having long or thin flexural geometric structures. For flexural mode, beam structures usually include double-clamped beams, cantilever beams, and double-ended tuning fork beam structures, etc. For membrane structures, the edges of the membrane are often fixed, and the remaining parts undergo out-of-plane vibration. The team from UC Berkeley (ref.10) proposed a 32.796 kHz resonator. Its structure adopts conventional proof mass-spring design, and based on aggressive lithography, the minimum beam width is only 600 nm. The structure is driven by capacitive capacitive comb and performs translational simple harmonic motion in the plane. Sahasrabudhe et al. (ref.11) proposed a resonator structure with a resonant frequency of 497 kHz, which uses a basic differential membrane mode shape of the resonator and is driven by a piezoelectric layer. Shunsuke et al. (ref.12) proposed a torsional cantilever beam structure, vibrating at 31,600 Hz, which combines torsional mode with flexural mode to improve the frequency stability of the structure. Giorgio et al. (ref.13) realized a 541 kHz resonator for clocks using an industrialized epitaxial polysilicon process platform; the structure is a double-clamped beam vibrating in flexural mode, with a length of 400 µm, and is driven and sensed by an in-plane capacitive structure. Among these previous research results, when these low-frequency resonators vibrate in flexural mode, their structural configuration can often be equivalent to a spring-mass system. The device undergoes reciprocating simple harmonic vibration in a linear direction, and the mechanical stiffness in the direction of motion often has a strong correlation with the equivalent stiffness of the device. According to Eq. ( 1 ), to ensure that the device vibrates at low frequencies, it is often necessary to have low equivalent stiffness or large equivalent mass. Thus, the mechanical stiffness of the device in the direction of motion needs to be within a relatively low range, or the mass of the device needs to be increased. However, both of these lead to deficiencies in its stability and anti-overload performance, limiting its application in some high-reliability scenarios. Besides flexural mode, a few structures also adopt torsional mode, with the beam undergoing torsional vibration along its own centerline (ref.14). Torsional vibration is often an out-of-plane vibration mode. Although it can ensure a certain degree of linear stiffness, the arrangement and sensing of out-of-plane electrodes for actuation are relatively difficult. Thus, such structures often use piezoelectric actuation, with a low Q factor. Given the above issues, this paper proposes a low-frequency wheel-type resonator structure with in-plane torsional vibration mode. Its in-plane vibration mode ensures that sensing and measuring electrodes can be easily arranged in-plane, while guaranteeing high linear stiffness of the resonator under low-frequency vibration conditions, thereby improving its anti-overload performance and frequency stability. A theoretical comparison is conducted between the lateral stiffness of the wheel-type structure and that of traditional flexural mode. The vibration principle of the resonator is modeled and derived using the deflection variation of the beam and the Rayleigh energy method, and experiments and analysis of FEM (finite element method) simulation are performed on the anti-overload performance of the resonator. Result Device Structure and Resonant Frequency Derivation The schematic diagram of the device is shown in Fig. 1 a: The device consists of a hollow wheel-type structure in the middle and four surrounding connecting beams. The wheel-type structure is composed of an inner ring, an outer ring, and connecting beams between the inner and outer rings. The connecting beams are connected to the inner ring of the wheel-type structure and distributed along the radius of the wheel-type structure. The outer ring has notches at positions corresponding to the connecting beams, through which the connecting beams can be connected to anchor electrodes located outside the wheel-type structure. This resonator is driven and sensed by capacitive transducers, with capacitive transduction structures distributed within the wheel-type structure. A DC bias voltage is applied to the resonator structure, an AC driving voltage is applied to the driving electrodes, and a load is connected to the sensing electrodes, enabling the output of a sinusoidal current signal with a fixed frequency (ref.15). The anchor electrodes are located outside the wheel-type structure to ensure a larger area for the electrodes, facilitating the application of DC bias voltage to the anchor electrodes during resonator actuation. The number of connecting beams can be multiple, and Fig. 1 b shows a 3D structure of four-connecting-beam. For a general spring-beam-mass structure with in-plane vibration, the mass block undergoes simple harmonic vibration in one direction, providing the system with kinetic energy of translational motion. In the wheel-type resonant structure, the wheel-type mass block mainly performs in-plane simple harmonic torsional vibration during vibration, providing rotational kinetic energy for the entire resonant system. The connecting beams mainly provide elastic strain potential energy, enabling the resonator to exhibit an in-plane torsional mode during vibration. In the wheel-shaped resonator structure, the fixing method of the resonant beam differs from previous research work in papers, and a rotating mass is incorporated. Since it is difficult to directly solve the resonant frequency of the resonator using the traditional method of solving modes via differential equations (ref.16), this paper performs derivation using the static deflection during the beam's motion and combines it with the Rayleigh energy method (ref.17) to model the relationship between the resonator's structure and resonant frequency. The following is the derivation process. To facilitate the explanation of the principle, a schematic diagram of a solid disk connected by four connecting beams is used for illustration in the principle derivation, as shown in Fig. 2a. In the actual device corresponding to Fig. 1 b, it is only necessary to replace the solid disk structure with a hollow wheel-shaped structure. The following is the derivation process. Using the Rayleigh energy method for calculation, it can be obtained that in the resonant system $$\:{\text{E}}_{\text{k}\text{m}\text{a}\text{x}}={\text{E}}_{\text{p}\text{m}\text{a}\text{x}}$$ 2 Where E kmax is the maximum potential energy of the system, and E pmax is the maximum kinetic energy of the system. Since no internal deformation occurs in the wheel-shaped mass block and it only rotates as a whole, the maximum kinetic energy is provided by the vibrational kinetic energy of the beam and that of the mass block, while the maximum potential energy is provided only by the maximum bending strain energy of the beam. The expression is as follows: $$\:{\text{E}}_{\text{k}\text{b}}+{\text{E}}_{\text{k}\text{w}}={\text{E}}_{\text{p}\text{b}}$$ 3 $$\:{\text{E}}_{\text{k}\text{w}}=\frac{1}{2}J{\omega\:}^{2}{\theta\:}^{2}$$ 4 Where \(\:{\text{E}}_{\text{k}\text{b}}\) is the maximum kinetic energy of the beam, \(\:{\text{E}}_{\text{k}\text{w}}\) is the maximum kinetic energy of the wheel-shaped mass, and \(\:{\text{E}}_{\text{p}\text{b}}\) is the maximum strain energy of the beam. J is the moment of inertia of the wheel structure, ω is the natural frequency of the resonator's in-plane torsional mode, and θ is the maximum vibration angle amplitude of the resonator. Next, we derive the deflection of the beam during motion, and obtain its vibrational kinetic energy and bending strain energy from the deflection. It is known that one end of the beam is clamped, and the end connected to the circular wheel undergoes simple harmonic torsion with the disk. Under the conditions of a large length-to-width ratio of the beam and small deflection, it can be approximately considered that during the beam's motion, particles on the beam only move within the wheel plane in the direction perpendicular to the beam, and the beam itself has no tensile strain. Therefore, the Euler-Bernoulli beam theory can be used to solve for the beam deflection. For an Euler-Bernoulli beam (ref.18) without transverse load, the differential equation of the deflection curve is: $$\:EI\frac{{d}^{4}\omega\:}{d{x}^{4}}=0$$ 5 After simplification, Eq. 2 , when integrated four times, yields the general solution of the equation: $$\:\text{w}\left(\text{x},\text{t}\right)=\left({\text{C}}_{1}{\text{x}}^{3}+\:{\text{C}}_{2}{\text{x}}^{2}+\:{\text{C}}_{3}\text{x}\:+\:{\text{C}}_{4}\right)\text{sin}\left(\omega\:t+\phi\:\right)$$ 6 Where \(\:{\text{C}}_{1},\:\:{\text{C}}_{2},\:\:{\text{C}}_{3}\) and \(\:{\text{C}}_{4}\) are undetermined coefficients; EI is the bending stiffness of the beam, which is the product of Young's modulus and the cross-sectional moment of inertia; x is the positional coordinate along the length of the beam; and ω is the frequency at which the beam follows the disk in simple harmonic vibration. Substituting the boundary conditions: One end of the beam is clamped, where both the rotation angle and deflection are zero. When one end of the beam rotates with the disk by an angle of θ, the rotation angle of that end of the beam is -θ, and the deflection at the end is approximately equal to Rθ, $$\:\text{w}\left(0\right)=0,\:{\text{w}}^{{\prime\:}}\left(0\right)=0,\text{w}\left(\text{L}\right)=\text{R}{\theta\:},\:{\text{w}}^{{\prime\:}}\left(\text{L}\right)=-{\theta\:}\:$$ 7 Substituting Eq. 7 into Eq. 6 yields the deflection expression of the beam: $$\:\text{w}\left(\text{x},\text{t}\right)=\theta\:（-\frac{L+2R}{{L}^{3}}{x}^{3}+\frac{L+3R}{{L}^{2}}{x}^{2}）\text{s}\text{i}\text{n}(\omega\:t+\phi\:)\:$$ 8 Where L is the total length of the connecting beam, and R is the radius of the position where the connecting beam is connected to the wheel-shaped structure. After obtaining the deflection expression of the beam, the maximum bending strain energy of the beam can be obtained from the following equation $$\:{\text{E}}_{\text{p}\text{b}}=\frac{1}{2}EI{\int\:}_{0}^{L}{\left(w{\prime\:}{\prime\:}\right(x\left)\right)}^{2}dx\:$$ 9 Substituting into Eq. 8 , where the time term related to simple harmonic vibration takes its maximum value, yields the maximum bending strain energy of a single connecting beam during simple harmonic vibration as: $$\:{\text{E}}_{\text{p}\text{b}}=2EI{\theta\:}^{2}\frac{{L}^{2}+3LR+3{R}^{2}}{{L}^{3}}\:$$ 10 According to Eq. 8 , the maximum kinetic energy of the beam in simple harmonic motion can be obtained from the following expression: $$\:{\text{E}}_{\text{k}\text{b}}=\frac{1}{2}\rho\:A{\omega\:}^{2}{\int\:}_{0}^{L}{w\left(x\right)}^{2}dx\:$$ 11 Substituting into Eq. 8 yields the maximum kinetic energy of a connecting beam as follows: $$\:{\text{E}}_{\text{k}\text{b}}=\frac{1}{2}\rho\:A{\omega\:}^{2}{\theta\:}^{2}[\frac{1}{105}{L}^{3}+{\frac{11}{105}L}^{2}R+\frac{39}{105}{LR}^{2}]\:$$ 12 Let $$\:\alpha\:=\frac{{L}^{2}+3LR+3{R}^{2}}{{L}^{3}},\:\beta\:=\left[\frac{1}{105}{L}^{3}+{\frac{11}{105}L}^{2}R+\frac{39}{105}{LR}^{2}\right]$$ 13 Which are both related to device dimensions. $$\:f=\frac{1}{2\pi\:}\sqrt{\frac{4NEI\alpha\:}{N\rho\:A\beta\:+J}}$$ 14 Where N is the number of connecting beams connected to the wheel-shaped mass; ρ and E are the density and Young's modulus of the device material; I and A are respectively the cross-sectional moment of inertia in the bending direction of the beam and the cross-sectional area; and J is the moment of inertia of the wheel-shaped mass rotating about the center in the plane. Theoretical Calculation of In-plane Stiffness of Structures and Comparison For common low-frequency bending vibration structures, they can be simplified as a structure consisting of two resonant beams with a mass block connected between them, as shown in the Figure.2c. The beams at both ends have one end clamped and the other end connected to the mass block; the length of the beam is L1, the width is b1, and the thickness is h. For ease of comparison, the mass block is also designed to be disk-shaped, with a radius of R and a thickness of h. Figure 2 Schematic diagrams of in-plane vibration mode and flexural vibration mode. a . Schematic diagram of vibration mode of the 4-beam-disc structure. b . Schematic diagram of the in-plane torsional structure under in-plane transverse displacement. c . Schematic diagram of the flexural mode under in-plane transverse displacement. Regarding the calculation of in-plane stiffness during the flexural mode vibration of the beam-mass system, the minimum in-plane stiffness of the structure occurs in the direction where the force is perpendicular to the beam; therefore, what needs to be solved is the transverse stiffness of the structure. When a force of magnitude F is applied to the mass block, the mass block generates a transverse displacement of \(\:{\text{X}}_{0}\) . Since the mass block is located at the midpoint of the structure, the force borne by a single beam is F/2. At this point, one end of the beam is clamped, and the other end translates by a distance of X0 without rotation. Substituting the boundary conditions of the beam yields the deflection expression of the beam as (ref.19): $$\:w\left(x\right)=\frac{F/2}{12EI}(3L{x}^{2}-2{x}^{3})$$ 15 Substituting \(\:x=\) L into Eq. 2 yields the displacement of the mass block \(\:{\text{X}}_{0}\) : $$\:{X}_{0}=\frac{F/2}{12EI}{L}^{3}$$ 16 From the definition of stiffness, the transverse stiffness of the device is yielded as: $$\:{k}_{f}=\frac{F}{{X}_{0}}=\frac{24EI}{{L}^{3}}=\frac{2E{{hb}_{1}}^{3}}{{L}^{3}}$$ 17 When the wheel-shaped structure vibrates in the planar flexural mode, taking the four-connecting-beam structure as an example, the middle wheel-shaped mass block is supported by connecting beams around it, as shown in the Fig. 2b. Therefore, when the structure is subjected to an in-plane force, the force exerted by the mass block on the beams is along the axial direction of the connecting beams. Thus, when calculating the in-plane stiffness of the resonator structure, for the wheel-shaped structure with four connecting beams, it can be considered that an axial force is applied to the beams, while the connecting beams in the vertical direction are subjected to a transverse force. Hence, it is necessary to calculate both the transverse stiffness and longitudinal stiffness of the beams, and the sum of the two is the in-plane stiffness of the device. When a pair of opposite connecting beams are subjected to an axial force, one beam undergoes compression and the other undergoes tension; therefore, its longitudinal stiffness \(\:{k}_{t1}\) can be expressed as: $$\:{k}_{t1}=\frac{2EA}{L}=\frac{2Ebh}{L}$$ 18 Where E is the Young's modulus of the connecting beam, A is the cross-sectional area of the connecting beam, and L is the length of the connecting beam. Since the transverse beams in the in-plane torsional mode provide a transverse stiffness of \(\:{k}_{t2}={k}_{f}\) , the total stiffness is \(\:{k}_{t}={k}_{t1}+{k}_{t2}\) . Thus, the ratio of the in-plane stiffness between the flexural mode and the in-plane torsional mode is $$\:\frac{{k}_{f}}{{k}_{t}}=\frac{{k}_{f}}{{k}_{t1}+{k}_{f}}=\frac{1}{{c}^{2}+1}$$ 19 Where c is the aspect ratio of the beam. For low-frequency resonator structures, the beam is applicable to the Euler-Bernoulli beam theory, and c is generally greater than 10. It can be seen that for the same beam dimensions, the transverse stiffness is much smaller than the longitudinal stiffness. In other words, for the same beam dimensions, the in-plane stiffness of the flexural vibration mode is much smaller than that of the in-plane torsional mode. With the beam width fixed at 4µm, the beam length-stiffness curves under the two modes are shown in Fig. 3 a, Fig. 3 b. It can be seen that under the same beam dimensions, the in-plane stiffness of the two structures differs by 10 2 to 10 4 . Next, comparative verification of the in-plane stiffness of resonators is conducted under the same resonant frequency condition, with the two structures in Fig. 3 c selected for comparison. First, ensure that the two structures have the same mass block dimensions, device thickness, and beam width, and adjust the beam dimensions to achieve different resonant frequencies. Here, the mass block is set to have a radius of 60 µm, the device thickness is 20 µm, and the width of the surrounding connecting beams is set to 4 µm. According to Eq. 14 , the curve of the device's resonant frequency varying with L can be obtained, and then the law of the device's in-plane stiffness varying with resonant frequency can be derived. For the structure shown in Fig. 2c in the flexural vibration mode, since the mass block in the middle section of the beam can be approximated as a rigid body (without deformation, only performing translational motion), the structure can thus be equivalent to a double-ended clamped beam of length 2L1 with a mass block attached. Thus, under the elastic and small deflection theory, the resonant frequency of the first-order transverse bending vibration mode of the beam is given by the following expression: $$\:{f}_{0}=\frac{1}{2\pi\:}\sqrt{\frac{24EI}{(m+{0.236m}_{b}){{L}_{1}}^{3}}}$$ 20 Where m is the mass of the mass block, mb is the mass of the beam, and 0.236 is the equivalent mass coefficient of the beam in the flexural vibration mode. Graphs of stiffness variation for the two structures at characteristic frequencies ranging from 100k to 300k are selected. It can be seen that at the same characteristic frequency, the in-plane stiffness of the in-plane torsional mode is higher than that of the flexural mode by 10². Discussion Simulation of Resonator Stiffness and Frequency drift under Static load Through the previous derivation, it is theoretically demonstrated that wheel-shaped resonators with in-plane torsional mode have advantages in in-plane stiffness. Next, finite element simulation is used to conduct simulation verification on the stiffness of resonators. The simulated device model is shown in Fig. 1 b. Since the electrodes are fixed on the substrate and do not affect the mechanical stiffness of the device itself, for the convenience of simulation, electrodes are not included in the structures for stiffness simulation and subsequent anti-overload related simulations. The key dimensions of the device are shown in Table 1 . Table 1 Key Dimensions of the Device's Simulated Structure Dimensions Value Rin(µm) 80 dRin(µm) 10 Rout (µm) 250 dRout (µm) 15 L beam (µm) 150 b beam (µm) 4 b conncet (µm) 10 h(µm) 20 Simulations were conducted on both the in-plane stiffness and out-of-plane stiffness of the resonator. The simulation material of the device is set to monocrystalline silicon, with Young's modulus E = 169e9 N/m², density ρ = 2330 kg/m³, and Poisson's ratio of 0.28. Fixed constraint conditions are applied to the anchor points of the device. According to the definition of stiffness, loads are applied to the wheel-shaped structure in the X-direction and Z-direction of the device respectively. The relationship between the displacement of the center of the wheel-shaped structure and the force magnitude is simulated in both directions, and the stiffness of the device in this direction is calculated. During the simulation, the magnitude of the total load force varies from 1e-6 N to 1e-5 N, and the force-displacement curve of the wheel-shaped structure is obtained as shown in Fig. 4 a. From the simulation data in Fig. 4 a, it can be calculated that the stiffness of the device in the in-plane X-axis direction is 1.76×10⁵ N/m, and the stiffness in the Z-axis direction is 5297.6 N/m. Through calculation using Eq. 17 , Eq. 18 , the theoretical stiffness values of the device under the current dimensions are 1.802×10⁵ N/m and 5265.9 N/m. The theoretical and simulation results are relatively close, with errors of 2.33% and 0.6%. When the device is subjected to acceleration loads, the beams of the wheel-shaped structure resonator are subjected to axial or transverse stress. Due to the stiffness hardening and softening effects of the beams (ref.20), the resonant frequency of the structure may be affected. Therefore, the characteristic frequency drift under static load conditions can also reflect the stability performance of the device in some high overload scenarios. In this paper, simulations of the frequency drift of the wheel-shaped structure resonator under body loads in the in-plane and out-of-plane directions are conducted. The simulated device parameters are shown in Table 1 . Static body loads are applied in the X-axis and Z-axis directions respectively, ranging from 0 to 100g, and the frequency drift of the device under steady-state conditions is simulated. As shown in Fig. 4 b, the figure presents the frequency shifts of the device when 100g loads are applied in the x-axis and Z-axis directions respectively. It can be seen that the frequency shift under X-axis loading is 21.67 ppb, and that under Z-axis loading is 60.39 ppb, with very small offsets. Since the in-plane transverse stiffness of the device is greater than the out-of-plane stiffness, the frequency shift caused by out-of-plane loads is larger than that caused by in-plane transverse loads. Anti-overload Performance In addition to frequency stability under static loads, in many scenarios, the failure of resonator structures is due to short-term high-intensity impact loads, where the impact g-value can reach nearly 10⁴g in an extremely short time (ref.21). To further verify the anti-overload performance of the device structure, transient simulations of the impact response of the wheel-shaped resonator structure are conducted. Fixed constraint conditions are applied to the anchor points of the device. At time 0, a half-sine shaped body load signal is applied to the entire structure to simulate the impact acceleration load (ref.22), with the peak value of the impact signal being 10000g and the pulse width being 10 ms. Transient simulations of the stress and stress of the resonant structure within the first 100 ms are performed, with the time step set to 0.01 µm, thus obtaining the maximum stress and maximum displacement of the device under the 10000g acceleration impact load. The material of the simulated device is monocrystalline silicon. The simulation results of the 10000g acceleration impact response in the x-direction of the device are shown in Fig. 5 a, Fig. 5 b.It can be seen that under the 10000g impact load in the X-axis direction, the maximum stress of the device appears at 10 ms, corresponding to the peak time of the impact signal. The location is at the junction of the connecting beams and the inner ring of the wheel-shaped structure, with a maximum stress of 24.7 MPa. Analyzing from the maximum displacement of the device, the maximum displacement also occurs at 10 ms, with a displacement of 0.111 µm, located at the y-direction notch of the outer ring of the wheel-shaped structure. The maximum stress occurs at the junction because the wheel-shaped structure is subjected to an in-plane acceleration impact, and the connecting beams in the impact direction bear direct inertial forces. The maximum displacement occurs at the outer ring notch because the outer ring is not directly connected to the connecting beams and has a notch, resulting in uneven impact displacement of each part. During the impact in the X-direction, the inner and outer connecting beams at the y-direction notch are equivalent to being subjected to a transverse impact, thus generating the maximum displacement under the acceleration impact. The simulation results of the 10000g acceleration impact response in the Z-direction of the device are shown in Fig. 5 c, Fig. 5 d. It can be seen that under the 10000g impact load in the Z-axis direction, the maximum stress of the device appears at 10 ms, corresponding to the peak time of the impact signal. The location is at the junction where the connecting beams meet the anchor points, with a maximum stress of 45.7 MPa. Analyzing from the maximum displacement of the device, the maximum displacement also occurs at 10 ms, with a displacement of 0.137 µm, located at the edge of the outer ring of the wheel-shaped structure. For out-of-plane acceleration impact, the wheel-shaped structure can be approximately regarded as a whole connected by connecting beams; thus, the maximum stress occurs at the junction of the connecting beams and the anchor points. The displacement of the wheel-shaped structure is approximately the same at positions with the same radius, and increases as the radius increases. Therefore, the maximum displacement appears at the edge of the outer ring, corresponding to the position with the largest radius of the wheel-shaped structure. Next, the strength of the device is checked. The device material, monocrystalline silicon, is a brittle material, and its yield strength is approximately equal to its fracture strength. Therefore, checking the strength of the device structure requires using the second strength theory to calculate its maximum allowable stress (ref.23): $$\:\sigma\:={\sigma\:}_{1}+\nu\:（{\sigma\:}_{2}+{\sigma\:}_{3}）$$ 21 Where [σ] is the allowable stress of silicon material, typically 700 MPa; (σ = 1, 2, 3) are the principal stresses of silicon material; and v is the Poisson's ratio of silicon material. Combining the maximum allowable stress with the maximum impact displacement and stress obtained from simulations, it can be concluded that whether it is an acceleration impact in the in-plane x-axis direction or an acceleration impact in the out-of-plane z-axis direction, the maximum stress during the device's impact load response is far lower than the maximum allowable stress of the device material (silicon). From the perspective of displacement, the maximum impact displacement in both directions is less than 0.15 µm, which is very small relative to the device scale. This ensures that no connection between device structures or pull-in phenomenon in the capacitive transducing structure will occur. Conclusion In this paper, we propose a low-frequency wheel-shaped resonator with vibration in the in-plane torsional mode. By using the torsional vibration of the wheel-shaped mass block instead of translational vibration, the device maintains a low characteristic frequency while achieving high in-plane stiffness. The frequency characteristics of the device are derived and modeled using beam deflection derivation and the Rayleigh energy method. Theoretical calculations and comparisons are conducted on the in-plane stiffness between the classical flexural -mode beam-mass structure and the in-plane torsional mode structure. Under the same beam dimensions, the in-plane stiffness of the in-plane torsional mode is higher by 10 2 to 10 4 orders of magnitude; under the same frequency, the in-plane stiffness is higher by 10² orders of magnitude. We also conducted simulations on the anti-overload performance of the device structure. By simulating the stiffness and frequency drift under static loads, as well as the maximum stress and displacement under dynamic impact response, it is found that the frequency drifts in the X-axis and Z-axis directions under 100g static load are 21.67 ppb and 60.39 ppb, respectively. Under 10000g impact load, the maximum stresses in the in-plane X-axis and Z-axis directions are 24.7 MPa and 45.7 MPa, with maximum displacements of 0.11 µm and 0.137 µm, respectively, which are far lower than the maximum allowable stress of silicon material and the movable range of the device. Declarations Conflict of interest The authors declare no competing interests. Author contributions D.L. conceived this research, designed the device structure, and established the mathematical and physical models of the device structure. W.L. and J.L. participated in designing the comparative simulations and experiments. L.T. and S.C. participated in the analysis of simulation data. J.L. and W.W. conceived the research and supervised the experiments and analysis. All the authors discussed the device design, simulated results, measured results, and prepared the paper. References Kumar P, Bazaz S A, Saleem M M. Design and Analysis of a Low-g MEMS Accelerometer Utilizing Weakly Coupled Resonators[C]//2023 International Conference on Engineering and Emerging Technologies (ICEET). IEEE, 2023: 1–6. Tu C, Pan Y M, Wang Z, et al. A Bent-TBTF Resonant MEMS Accelerometer using Auxiliary Supporting Beams[J]. IEEE Electron Device Letters, 2024. Yadav C, Paliwal S, Yenuganti S. Design and Simulation of a Differential Resonant Pressure Sensor[C]//2023 16th International Conference on Sensing Technology (ICST). IEEE, 2023: 1–6. Chuai S, Deng J, Li H, et al. Method for Sensitivity Improvement of MEMS Pressure Sensor: Structural Design and Optimization of Concave Resonant Pressure Sensor[J]. IEEE Sensors Journal, 2025. Xu J, Tsai J M. A process-induced-frequency-drift resilient 32 kHz MEMS resonator[J]. Journal of Micromechanics and Microengineering, 2012, 22(10): 105029. Kaajakari V, Pangaro A, Goto Y, et al. A 32.768 kHz MEMS resonator with+/-20 ppm tolerance in 0.9 mm x 0.6 mm chip scale package[C]//2019 Joint Conference of the IEEE International Frequency Control Symposium and European Frequency and Time Forum (EFTF/IFC). IEEE, 2019: 1–4. Zaliasl S, Salvia J C, Hill G C, et al. A 3 ppm 1.5× 0.8 mm 2 1.0 µA 32.768 kHz MEMS-Based Oscillator[J]. IEEE Journal of Solid-State Circuits, 2014, 50(1): 291–302. Yaqoob U, Jaber N, Alcheikh N, et al. Selective multiple analyte detection using multi-mode excitation of a MEMS resonator[J]. Scientific reports, 2022, 12(1): 5297. Eidi A. Design and evaluation of an implantable MEMS based biosensor for blood analysis and real-time measurement[J]. Microsystem Technologies, 2023, 29(6): 857–864. Barrow H G, Naing T L, Schneider R A, et al. A real-time 32.768-kHz clock oscillator using a 0.0154-mm 2 micromechanical resonator frequency-setting element[C]//2012 IEEE International Frequency Control Symposium Proceedings. IEEE, 2012: 1–6. Sahasrabudhe S, Long Y, Liu Z, et al. A Low Phase Jitter MEMS Oscillator with Centrally-Anchored Piezoelectric Resonator for Wide Temperature Range Real Time Clock Applications[J]. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2024. Yamada S, Tanaka S. Temperature-Compensated Pure Silicon Cantilever Resonator with Coupled Torsional Structure at Anchor[C]//2023 22nd International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers). IEEE, 2023: 1817–1820. Mussi G, Bestetti M, Zega V, et al. An outlook on potentialities and limits in using epitaxial polysilicon for MEMS real-time clocks[J]. IEEE Transactions on Industrial Electronics, 2019, 67(8): 6996–7004. Wu J, Song P, Zang S, et al. Limit cycle convergence leads to period-doubling and cyclic-fold bifurcation in internal resonance-induced mechanical frequency combs[J]. Nonlinear Dynamics, 2025: 1–22. Wang S. Hermetically Encapsulated Fully Differential Breathe-mode Ring Resonators for Timing Applications[M]. Stanford University, 2013. Elshurafa A M, Khirallah K, Tawfik H H, et al. Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators[J]. Journal of Microelectromechanical Systems, 2011, 20(4): 943–958. Putty M W, Chang S C, Howe R T, et al. Process integration for active polysilicon resonant microstructures[J]. Sensors and Actuators, 1989, 20(1–2): 143–151. Syed W U, Elfadel I A M. Review of Euler–Bernoulli Rectangular Beam Theory[M]//Tapered Beams in MEMS: A Symbolic Modeling Framework with Applications to Energy Harvesting. Cham: Springer International Publishing, 2024: 17–31. Nashat S E D, AbdelRassoul R, Abd El Bary A E M. Design and simulation of RF MEMS comb drive with ultra-low pull-in voltage and maximum displacement[J]. Microsystem Technologies, 2018, 24(8): 3443–3453. Mestrom R M C, Fey R H B, Phan K L, et al. Simulations and experiments of hardening and softening resonances in a clamped–clamped beam MEMS resonator[J]. Sensors and Actuators A: Physical, 2010, 162(2): 225–234. Peng T, You Z. Reliability of MEMS in shock environments: 2000–2020[J]. Micromachines, 2021, 12(11): 1275. Zhiwei K O U, Xiaoming C U I, Huiliang C A O, et al. Design and analysis of a capacitive MEMS ring wave gyroscope with high-overload[C]//2019 IEEE 3rd Information Technology, Networking, Electronic and Automation Control Conference (ITNEC). IEEE, 2019: 2447–2451. Kou Z, Jin L, Cui X, et al. Optimization and Analysis of Capacitive Ring Vibrating Gyroscope with High-overload[C]//2020 IEEE 9th Joint International Information Technology and Artificial Intelligence Conference (ITAIC). IEEE, 2020, 9: 1871 – 187 Additional Declarations There is no conflict of interest Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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09:25:39","extension":"html","order_by":14,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":91368,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/21a6fc6da88ab2e7d5527f93.html"},{"id":91834184,"identity":"f9ddde30-e22e-4575-8465-9aa40f6dc774","added_by":"auto","created_at":"2025-09-22 09:25:40","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":136291,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDevice structure schematic diagram.\u003c/strong\u003e a. Device 2D schematic diagram, including the capacitive transducing structure. b. Device 3D structural schematic diagram.\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/a4195c50f3b88243dce1c714.jpeg"},{"id":91834157,"identity":"cb6afb38-ef54-44ab-bac1-35895b807d72","added_by":"auto","created_at":"2025-09-22 09:25:35","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":69378,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSchematic diagrams of in-plane vibration mode and flexural vibration mode.\u003c/strong\u003e \u003cstrong\u003ea\u003c/strong\u003e. Schematic diagram of vibration mode of the 4-beam-disc structure. \u003cstrong\u003eb\u003c/strong\u003e. Schematic diagram of the in-plane torsional structure under in-plane transverse displacement. \u003cstrong\u003ec\u003c/strong\u003e. Schematic diagram of the flexural mode under in-plane transverse displacement.\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/2df832d69a1c9ccf02b02002.jpeg"},{"id":91834187,"identity":"48bcac59-17e3-49f9-bf92-d0fd87f8cec8","added_by":"auto","created_at":"2025-09-22 09:25:40","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":205200,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStiffness comparison\u003c/strong\u003e \u003cstrong\u003ecurves of structures with two vibration modes under different conditions. a\u003c/strong\u003e. Comparison of stiffness variation between the two modes under different beam lengths when the beam width is fixed at 4 μm. \u003cstrong\u003eb\u003c/strong\u003e. Comparison of stiffness variation between the two modes under different beam widths when the beam length is fixed at 250 μm. \u003cstrong\u003ec\u003c/strong\u003e. Comparison of stiffness variation between structures with two vibration modes under different vibration frequencies with the beam width fixed at 4 μm. \u003cstrong\u003ed\u003c/strong\u003e. Comparison of beam lengths between the two structures under different frequencies at the same frequency.\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/83925ae523d4288ab960df8a.jpeg"},{"id":91834152,"identity":"a3c2d8be-47c2-46f4-9937-18b13b1caab8","added_by":"auto","created_at":"2025-09-22 09:25:34","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":86064,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eStatic stiffness curves and frequency drift of the wheel-shaped resonator.\u003c/strong\u003e \u003cstrong\u003ea\u003c/strong\u003e. Force-displacement curves of the structure in X-axis and Z-axis directions. \u003cstrong\u003eb\u003c/strong\u003e. Frequency drift curves of the device under static body loads.\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/44942a1b4be2bba1040c0640.jpeg"},{"id":91834183,"identity":"1bfa851b-0acc-4c82-9325-733415257f54","added_by":"auto","created_at":"2025-09-22 09:25:40","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":302471,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFinite element simulation diagrams of the device structure's responses to impact loads in different directions\u003c/strong\u003e. \u003cstrong\u003ea, b\u003c/strong\u003e. Maximum stress and maximum displacement of the device under 10000g impact load in the x-axis direction. \u003cstrong\u003ec, d\u003c/strong\u003e. Maximum stress and maximum displacement of the device under 10000g impact load in the z-axis direction.\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/758670ba9a8ffa899d2e6991.jpeg"},{"id":94987257,"identity":"91af5c21-6c6d-4756-95c7-b00d597625d5","added_by":"auto","created_at":"2025-11-03 07:01:36","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1519053,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7525253/v1/01cc5b4b-03fd-4eb9-b891-46bf6547defd.pdf"}],"financialInterests":"There is no conflict of interest","formattedTitle":"Structural Design of a Low-Frequency In-plane Torsional Mode Wheel-Shaped Resonator","fulltext":[{"header":"Introduction","content":"\u003cp\u003eWith the rapid development of the Internet of Things (IoT) and 5G communication technologies, the number of various intelligent terminals and interconnected devices has grown explosively. This trend has put forward even more stringent requirements for the miniaturization, integration, low power consumption, and high reliability of electronic devices. Against this backdrop, MEMS (Micro-Electro-Mechanical Systems) technology, leveraging its unique advantages in microstructure design, mass manufacturing, and multi-physics field regulation, has become one of the core technologies supporting the development of new-generation fields such as information communication and intelligent sensing. MEMS resonators are key components capable of achieving precise frequency control and signal filtering. They realize efficient energy conversion and frequency selection by converting electrical signals into mechanical vibrations and utilizing resonance characteristics. Depending on their structural design and dimensions, the resonant frequencies of MEMS resonators typically range from kHz to GHz. MEMS resonators operating in different frequency bands often have distinct application scenarios. For resonators with low-frequency vibrations (below 1 MHz), they play an irreplaceable role in numerous applications, such as high-precision resonant accelerometers (ref.1\u0026ndash;2), resonant pressure sensors (ref.3\u0026ndash;4), high-precision clocks (ref.5\u0026ndash;7) in inertial navigation systems, and trace substance detection in environmental monitoring (ref.8\u0026ndash;9). The performance of resonators directly affects the stability, reliability, and integration level of the entire electronic system.\u003c/p\u003e\u003cp\u003eFor a resonator system, its resonant frequency is related to the physical properties of the system. For a single-degree-of-freedom resonator structure, its natural frequency can be expressed as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{f}_{0}=\\frac{1}{2\\pi\\:}\\sqrt{\\frac{{k}_{eff}}{{m}_{eff}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere f₀ is the natural frequency of the device's vibration, kₑ\u003csub\u003eff\u003c/sub\u003e is the equivalent stiffness of the structure in the current vibration mode, and mₑ\u003csub\u003eff\u003c/sub\u003e is the equivalent mass of the structure in the current vibration mode. For resonators with low-frequency vibrations, most of their vibration modes are flexural mode, which are characterized by having long or thin flexural geometric structures. For flexural mode, beam structures usually include double-clamped beams, cantilever beams, and double-ended tuning fork beam structures, etc. For membrane structures, the edges of the membrane are often fixed, and the remaining parts undergo out-of-plane vibration. The team from UC Berkeley (ref.10) proposed a 32.796 kHz resonator. Its structure adopts conventional proof mass-spring design, and based on aggressive lithography, the minimum beam width is only 600 nm. The structure is driven by capacitive capacitive comb and performs translational simple harmonic motion in the plane. Sahasrabudhe et al. (ref.11) proposed a resonator structure with a resonant frequency of 497 kHz, which uses a basic differential membrane mode shape of the resonator and is driven by a piezoelectric layer. Shunsuke et al. (ref.12) proposed a torsional cantilever beam structure, vibrating at 31,600 Hz, which combines torsional mode with flexural mode to improve the frequency stability of the structure. Giorgio et al. (ref.13) realized a 541 kHz resonator for clocks using an industrialized epitaxial polysilicon process platform; the structure is a double-clamped beam vibrating in flexural mode, with a length of 400 \u0026micro;m, and is driven and sensed by an in-plane capacitive structure.\u003c/p\u003e\u003cp\u003eAmong these previous research results, when these low-frequency resonators vibrate in flexural mode, their structural configuration can often be equivalent to a spring-mass system. The device undergoes reciprocating simple harmonic vibration in a linear direction, and the mechanical stiffness in the direction of motion often has a strong correlation with the equivalent stiffness of the device. According to Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), to ensure that the device vibrates at low frequencies, it is often necessary to have low equivalent stiffness or large equivalent mass. Thus, the mechanical stiffness of the device in the direction of motion needs to be within a relatively low range, or the mass of the device needs to be increased. However, both of these lead to deficiencies in its stability and anti-overload performance, limiting its application in some high-reliability scenarios. Besides flexural mode, a few structures also adopt torsional mode, with the beam undergoing torsional vibration along its own centerline (ref.14). Torsional vibration is often an out-of-plane vibration mode. Although it can ensure a certain degree of linear stiffness, the arrangement and sensing of out-of-plane electrodes for actuation are relatively difficult. Thus, such structures often use piezoelectric actuation, with a low Q factor.\u003c/p\u003e\u003cp\u003eGiven the above issues, this paper proposes a low-frequency wheel-type resonator structure with in-plane torsional vibration mode. Its in-plane vibration mode ensures that sensing and measuring electrodes can be easily arranged in-plane, while guaranteeing high linear stiffness of the resonator under low-frequency vibration conditions, thereby improving its anti-overload performance and frequency stability. A theoretical comparison is conducted between the lateral stiffness of the wheel-type structure and that of traditional flexural mode. The vibration principle of the resonator is modeled and derived using the deflection variation of the beam and the Rayleigh energy method, and experiments and analysis of FEM (finite element method) simulation are performed on the anti-overload performance of the resonator.\u003c/p\u003e"},{"header":"Result","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003eDevice Structure and Resonant Frequency Derivation\u003c/h2\u003e\u003cp\u003eThe schematic diagram of the device is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea: The device consists of a hollow wheel-type structure in the middle and four surrounding connecting beams. The wheel-type structure is composed of an inner ring, an outer ring, and connecting beams between the inner and outer rings. The connecting beams are connected to the inner ring of the wheel-type structure and distributed along the radius of the wheel-type structure. The outer ring has notches at positions corresponding to the connecting beams, through which the connecting beams can be connected to anchor electrodes located outside the wheel-type structure. This resonator is driven and sensed by capacitive transducers, with capacitive transduction structures distributed within the wheel-type structure. A DC bias voltage is applied to the resonator structure, an AC driving voltage is applied to the driving electrodes, and a load is connected to the sensing electrodes, enabling the output of a sinusoidal current signal with a fixed frequency (ref.15). The anchor electrodes are located outside the wheel-type structure to ensure a larger area for the electrodes, facilitating the application of DC bias voltage to the anchor electrodes during resonator actuation. The number of connecting beams can be multiple, and Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb shows a 3D structure of four-connecting-beam.\u003c/p\u003e\u003cp\u003eFor a general spring-beam-mass structure with in-plane vibration, the mass block undergoes simple harmonic vibration in one direction, providing the system with kinetic energy of translational motion. In the wheel-type resonant structure, the wheel-type mass block mainly performs in-plane simple harmonic torsional vibration during vibration, providing rotational kinetic energy for the entire resonant system. The connecting beams mainly provide elastic strain potential energy, enabling the resonator to exhibit an in-plane torsional mode during vibration.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn the wheel-shaped resonator structure, the fixing method of the resonant beam differs from previous research work in papers, and a rotating mass is incorporated. Since it is difficult to directly solve the resonant frequency of the resonator using the traditional method of solving modes via differential equations (ref.16), this paper performs derivation using the static deflection during the beam's motion and combines it with the Rayleigh energy method (ref.17) to model the relationship between the resonator's structure and resonant frequency. The following is the derivation process. To facilitate the explanation of the principle, a schematic diagram of a solid disk connected by four connecting beams is used for illustration in the principle derivation, as shown in Fig.\u0026nbsp;2a. In the actual device corresponding to Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb, it is only necessary to replace the solid disk structure with a hollow wheel-shaped structure. The following is the derivation process.\u003c/p\u003e\u003cp\u003eUsing the Rayleigh energy method for calculation, it can be obtained that in the resonant system\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{k}\\text{m}\\text{a}\\text{x}}={\\text{E}}_{\\text{p}\\text{m}\\text{a}\\text{x}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere E\u003csub\u003ekmax\u003c/sub\u003e is the maximum potential energy of the system, and E\u003csub\u003epmax\u003c/sub\u003e is the maximum kinetic energy of the system. Since no internal deformation occurs in the wheel-shaped mass block and it only rotates as a whole, the maximum kinetic energy is provided by the vibrational kinetic energy of the beam and that of the mass block, while the maximum potential energy is provided only by the maximum bending strain energy of the beam. The expression is as follows:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{k}\\text{b}}+{\\text{E}}_{\\text{k}\\text{w}}={\\text{E}}_{\\text{p}\\text{b}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{k}\\text{w}}=\\frac{1}{2}J{\\omega\\:}^{2}{\\theta\\:}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{E}}_{\\text{k}\\text{b}}\\)\u003c/span\u003e\u003c/span\u003e is the maximum kinetic\u003c/p\u003e\u003cp\u003eenergy of the beam, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{E}}_{\\text{k}\\text{w}}\\)\u003c/span\u003e\u003c/span\u003e is the maximum kinetic energy of the wheel-shaped mass, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{E}}_{\\text{p}\\text{b}}\\)\u003c/span\u003e\u003c/span\u003e is the maximum strain energy of the beam. J is the moment of inertia of the wheel structure, ω is the natural frequency of the resonator's in-plane torsional mode, and θ is the maximum vibration angle amplitude of the resonator.\u003c/p\u003e\u003cp\u003eNext, we derive the deflection of the beam during motion, and obtain its vibrational kinetic energy and bending strain energy from the deflection. It is known that one end of the beam is clamped, and the end connected to the circular wheel undergoes simple harmonic torsion with the disk. Under the conditions of a large length-to-width ratio of the beam and small deflection, it can be approximately considered that during the beam's motion, particles on the beam only move within the wheel plane in the direction perpendicular to the beam, and the beam itself has no tensile strain. Therefore, the Euler-Bernoulli beam theory can be used to solve for the beam deflection.\u003c/p\u003e\u003cp\u003eFor an Euler-Bernoulli beam (ref.18) without transverse load, the differential equation of the deflection curve is:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:EI\\frac{{d}^{4}\\omega\\:}{d{x}^{4}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eAfter simplification, Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, when integrated four times, yields the general solution of the equation:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:\\text{w}\\left(\\text{x},\\text{t}\\right)=\\left({\\text{C}}_{1}{\\text{x}}^{3}+\\:{\\text{C}}_{2}{\\text{x}}^{2}+\\:{\\text{C}}_{3}\\text{x}\\:+\\:{\\text{C}}_{4}\\right)\\text{sin}\\left(\\omega\\:t+\\phi\\:\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{C}}_{1},\\:\\:{\\text{C}}_{2},\\:\\:{\\text{C}}_{3}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{C}}_{4}\\)\u003c/span\u003e\u003c/span\u003eare undetermined coefficients; EI is the bending stiffness of the beam, which is the product of Young's modulus and the cross-sectional moment of inertia; x is the positional coordinate along the length of the beam; and ω is the frequency at which the beam follows the disk in simple harmonic vibration.\u003c/p\u003e\u003cp\u003eSubstituting the boundary conditions: One end of the beam is clamped, where both the rotation angle and deflection are zero. When one end of the beam rotates with the disk by an angle of θ, the rotation angle of that end of the beam is -θ, and the deflection at the end is approximately equal to Rθ,\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:\\text{w}\\left(0\\right)=0,\\:{\\text{w}}^{{\\prime\\:}}\\left(0\\right)=0,\\text{w}\\left(\\text{L}\\right)=\\text{R}{\\theta\\:},\\:{\\text{w}}^{{\\prime\\:}}\\left(\\text{L}\\right)=-{\\theta\\:}\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSubstituting Eq.\u0026nbsp;\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e into Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e yields the deflection expression of the beam:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:\\text{w}\\left(\\text{x},\\text{t}\\right)=\\theta\\:（-\\frac{L+2R}{{L}^{3}}{x}^{3}+\\frac{L+3R}{{L}^{2}}{x}^{2}）\\text{s}\\text{i}\\text{n}(\\omega\\:t+\\phi\\:)\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere L is the total length of the connecting beam, and R is the radius of the position where the connecting beam is connected to the wheel-shaped structure.\u003c/p\u003e\u003cp\u003eAfter obtaining the deflection expression of the beam, the maximum bending strain energy of the beam can be obtained from the following equation\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{p}\\text{b}}=\\frac{1}{2}EI{\\int\\:}_{0}^{L}{\\left(w{\\prime\\:}{\\prime\\:}\\right(x\\left)\\right)}^{2}dx\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSubstituting into Eq.\u0026nbsp;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, where the time term related to simple harmonic vibration takes its maximum value, yields the maximum bending strain energy of a single connecting beam during simple harmonic vibration as:\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{p}\\text{b}}=2EI{\\theta\\:}^{2}\\frac{{L}^{2}+3LR+3{R}^{2}}{{L}^{3}}\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eAccording to Eq.\u0026nbsp;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, the maximum kinetic energy of the beam in simple harmonic motion can be obtained from the following expression:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{k}\\text{b}}=\\frac{1}{2}\\rho\\:A{\\omega\\:}^{2}{\\int\\:}_{0}^{L}{w\\left(x\\right)}^{2}dx\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSubstituting into Eq.\u0026nbsp;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e yields the maximum kinetic energy of a connecting beam as follows:\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\:{\\text{E}}_{\\text{k}\\text{b}}=\\frac{1}{2}\\rho\\:A{\\omega\\:}^{2}{\\theta\\:}^{2}[\\frac{1}{105}{L}^{3}+{\\frac{11}{105}L}^{2}R+\\frac{39}{105}{LR}^{2}]\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eLet\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\:\\alpha\\:=\\frac{{L}^{2}+3LR+3{R}^{2}}{{L}^{3}},\\:\\beta\\:=\\left[\\frac{1}{105}{L}^{3}+{\\frac{11}{105}L}^{2}R+\\frac{39}{105}{LR}^{2}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhich are both related to device dimensions.\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$\\:f=\\frac{1}{2\\pi\\:}\\sqrt{\\frac{4NEI\\alpha\\:}{N\\rho\\:A\\beta\\:+J}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere N is the number of connecting beams connected to the wheel-shaped mass; ρ and E are the density and Young's modulus of the device material; I and A are respectively the cross-sectional moment of inertia in the bending direction of the beam and the cross-sectional area; and J is the moment of inertia of the wheel-shaped mass rotating about the center in the plane.\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003eTheoretical Calculation of In-plane Stiffness of Structures and Comparison\u003c/h3\u003e\n\u003cp\u003eFor common low-frequency bending vibration structures, they can be simplified as a structure consisting of two resonant beams with a mass block connected between them, as shown in the Figure.2c. The beams at both ends have one end clamped and the other end connected to the mass block; the length of the beam is L1, the width is b1, and the thickness is h. For ease of comparison, the mass block is also designed to be disk-shaped, with a radius of R and a thickness of h.\u003c/p\u003e\u003cp\u003e\u003cb\u003eFigure\u0026nbsp;2 Schematic diagrams of in-plane vibration mode and flexural vibration mode. a\u003c/b\u003e. Schematic diagram of vibration mode of the 4-beam-disc structure. \u003cb\u003eb\u003c/b\u003e. Schematic diagram of the in-plane torsional structure under in-plane transverse displacement. \u003cb\u003ec\u003c/b\u003e. Schematic diagram of the flexural mode under in-plane transverse displacement.\u003c/p\u003e\u003cp\u003eRegarding the calculation of in-plane stiffness during the flexural mode vibration of the beam-mass system, the minimum in-plane stiffness of the structure occurs in the direction where the force is perpendicular to the beam; therefore, what needs to be solved is the transverse stiffness of the structure. When a force of magnitude F is applied to the mass block, the mass block generates a transverse displacement of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{X}}_{0}\\)\u003c/span\u003e\u003c/span\u003e. Since the mass block is located at the midpoint of the structure, the force borne by a single beam is F/2. At this point, one end of the beam is clamped, and the other end translates by a distance of X0 without rotation. Substituting the boundary conditions of the beam yields the deflection expression of the beam as (ref.19):\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$\\:w\\left(x\\right)=\\frac{F/2}{12EI}(3L{x}^{2}-2{x}^{3})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSubstituting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:x=\\)\u003c/span\u003e\u003c/span\u003eL into Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e yields the displacement of the mass block \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{X}}_{0}\\)\u003c/span\u003e\u003c/span\u003e:\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$$\\:{X}_{0}=\\frac{F/2}{12EI}{L}^{3}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eFrom the definition of stiffness, the transverse stiffness of the device is yielded as:\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$$\\:{k}_{f}=\\frac{F}{{X}_{0}}=\\frac{24EI}{{L}^{3}}=\\frac{2E{{hb}_{1}}^{3}}{{L}^{3}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhen the wheel-shaped structure vibrates in the planar flexural mode, taking the four-connecting-beam structure as an example, the middle wheel-shaped mass block is supported by connecting beams around it, as shown in the Fig.\u0026nbsp;2b. Therefore, when the structure is subjected to an in-plane force, the force exerted by the mass block on the beams is along the axial direction of the connecting beams. Thus, when calculating the in-plane stiffness of the resonator structure, for the wheel-shaped structure with four connecting beams, it can be considered that an axial force is applied to the beams, while the connecting beams in the vertical direction are subjected to a transverse force. Hence, it is necessary to calculate both the transverse stiffness and longitudinal stiffness of the beams, and the sum of the two is the in-plane stiffness of the device. When a pair of opposite connecting beams are subjected to an axial force, one beam undergoes compression and the other undergoes tension; therefore, its longitudinal stiffness \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{t1}\\)\u003c/span\u003e\u003c/span\u003e can be expressed as:\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$$\\:{k}_{t1}=\\frac{2EA}{L}=\\frac{2Ebh}{L}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere E is the Young's modulus of the connecting beam, A is the cross-sectional area of the connecting beam, and L is the length of the connecting beam. Since the transverse beams in the in-plane torsional mode provide a transverse stiffness of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{t2}={k}_{f}\\)\u003c/span\u003e\u003c/span\u003e, the total stiffness is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{t}={k}_{t1}+{k}_{t2}\\)\u003c/span\u003e\u003c/span\u003e. Thus, the ratio of the in-plane stiffness between the flexural mode and the in-plane torsional mode is\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{k}_{f}}{{k}_{t}}=\\frac{{k}_{f}}{{k}_{t1}+{k}_{f}}=\\frac{1}{{c}^{2}+1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere c is the aspect ratio of the beam. For low-frequency resonator structures, the beam is applicable to the Euler-Bernoulli beam theory, and c is generally greater than 10. It can be seen that for the same beam dimensions, the transverse stiffness is much smaller than the longitudinal stiffness. In other words, for the same beam dimensions, the in-plane stiffness of the flexural vibration mode is much smaller than that of the in-plane torsional mode. With the beam width fixed at 4\u0026micro;m, the beam length-stiffness curves under the two modes are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003ea, Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003eb. It can be seen that under the same beam dimensions, the in-plane stiffness of the two structures differs by 10\u003csup\u003e2\u003c/sup\u003e to 10\u003csup\u003e4\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eNext, comparative verification of the in-plane stiffness of resonators is conducted under the same resonant frequency condition, with the two structures in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003ec selected for comparison. First, ensure that the two structures have the same mass block dimensions, device thickness, and beam width, and adjust the beam dimensions to achieve different resonant frequencies. Here, the mass block is set to have a radius of 60 \u0026micro;m, the device thickness is 20 \u0026micro;m, and the width of the surrounding connecting beams is set to 4 \u0026micro;m. According to Eq.\u0026nbsp;\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e, the curve of the device's resonant frequency varying with L can be obtained, and then the law of the device's in-plane stiffness varying with resonant frequency can be derived. For the structure shown in Fig.\u0026nbsp;2c in the flexural vibration mode, since the mass block in the middle section of the beam can be approximated as a rigid body (without deformation, only performing translational motion), the structure can thus be equivalent to a double-ended clamped beam of length 2L1 with a mass block attached. Thus, under the elastic and small deflection theory, the resonant frequency of the first-order transverse bending vibration mode of the beam is given by the following expression:\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$$\\:{f}_{0}=\\frac{1}{2\\pi\\:}\\sqrt{\\frac{24EI}{(m+{0.236m}_{b}){{L}_{1}}^{3}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere m is the mass of the mass block, mb is the mass of the beam, and 0.236 is the equivalent mass coefficient of the beam in the flexural vibration mode. Graphs of stiffness variation for the two structures at characteristic frequencies ranging from 100k to 300k are selected. It can be seen that at the same characteristic frequency, the in-plane stiffness of the in-plane torsional mode is higher than that of the flexural mode by 10\u0026sup2;.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003eSimulation of Resonator Stiffness and Frequency drift under Static load\u003c/h2\u003e\u003cp\u003eThrough the previous derivation, it is theoretically demonstrated that wheel-shaped resonators with in-plane torsional mode have advantages in in-plane stiffness. Next, finite element simulation is used to conduct simulation verification on the stiffness of resonators. The simulated device model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. Since the electrodes are fixed on the substrate and do not affect the mechanical stiffness of the device itself, for the convenience of simulation, electrodes are not included in the structures for stiffness simulation and subsequent anti-overload related simulations. The key dimensions of the device are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eKey Dimensions of the Device's Simulated Structure\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eDimensions\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eValue\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRin(\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e80\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003edRin(\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRout (\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e250\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003edRout (\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eL\u003csub\u003ebeam\u003c/sub\u003e(\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e150\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eb\u003csub\u003ebeam\u003c/sub\u003e(\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e4\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eb\u003csub\u003econncet\u003c/sub\u003e(\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eh(\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSimulations were conducted on both the in-plane stiffness and out-of-plane stiffness of the resonator. The simulation material of the device is set to monocrystalline silicon, with Young's modulus E\u0026thinsp;=\u0026thinsp;169e9 N/m\u0026sup2;, density ρ\u0026thinsp;=\u0026thinsp;2330 kg/m\u0026sup3;, and Poisson's ratio of 0.28. Fixed constraint conditions are applied to the anchor points of the device. According to the definition of stiffness, loads are applied to the wheel-shaped structure in the X-direction and Z-direction of the device respectively. The relationship between the displacement of the center of the wheel-shaped structure and the force magnitude is simulated in both directions, and the stiffness of the device in this direction is calculated. During the simulation, the magnitude of the total load force varies from 1e-6 N to 1e-5 N, and the force-displacement curve of the wheel-shaped structure is obtained as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003ea.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFrom the simulation data in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003ea, it can be calculated that the stiffness of the device in the in-plane X-axis direction is 1.76\u0026times;10⁵ N/m, and the stiffness in the Z-axis direction is 5297.6 N/m. Through calculation using Eq.\u0026nbsp;\u003cspan refid=\"Equ17\" class=\"InternalRef\"\u003e17\u003c/span\u003e, Eq.\u0026nbsp;\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e18\u003c/span\u003e, the theoretical stiffness values of the device under the current dimensions are 1.802\u0026times;10⁵ N/m and 5265.9 N/m. The theoretical and simulation results are relatively close, with errors of 2.33% and 0.6%.\u003c/p\u003e\u003cp\u003eWhen the device is subjected to acceleration loads, the beams of the wheel-shaped structure resonator are subjected to axial or transverse stress. Due to the stiffness hardening and softening effects of the beams (ref.20), the resonant frequency of the structure may be affected. Therefore, the characteristic frequency drift under static load conditions can also reflect the stability performance of the device in some high overload scenarios. In this paper, simulations of the frequency drift of the wheel-shaped structure resonator under body loads in the in-plane and out-of-plane directions are conducted. The simulated device parameters are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Static body loads are applied in the X-axis and Z-axis directions respectively, ranging from 0 to 100g, and the frequency drift of the device under steady-state conditions is simulated.\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003eb, the figure presents the frequency shifts of the device when 100g loads are applied in the x-axis and Z-axis directions respectively. It can be seen that the frequency shift under X-axis loading is 21.67 ppb, and that under Z-axis loading is 60.39 ppb, with very small offsets. Since the in-plane transverse stiffness of the device is greater than the out-of-plane stiffness, the frequency shift caused by out-of-plane loads is larger than that caused by in-plane transverse loads.\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003eAnti-overload Performance\u003c/h3\u003e\n\u003cp\u003eIn addition to frequency stability under static loads, in many scenarios, the failure of resonator structures is due to short-term high-intensity impact loads, where the impact g-value can reach nearly 10⁴g in an extremely short time (ref.21). To further verify the anti-overload performance of the device structure, transient simulations of the impact response of the wheel-shaped resonator structure are conducted. Fixed constraint conditions are applied to the anchor points of the device. At time 0, a half-sine shaped body load signal is applied to the entire structure to simulate the impact acceleration load (ref.22), with the peak value of the impact signal being 10000g and the pulse width being 10 ms. Transient simulations of the stress and stress of the resonant structure within the first 100 ms are performed, with the time step set to 0.01 \u0026micro;m, thus obtaining the maximum stress and maximum displacement of the device under the 10000g acceleration impact load. The material of the simulated device is monocrystalline silicon.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe simulation results of the 10000g acceleration impact response in the x-direction of the device are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003ea, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003eb.It can be seen that under the 10000g impact load in the X-axis direction, the maximum stress of the device appears at 10 ms, corresponding to the peak time of the impact signal. The location is at the junction of the connecting beams and the inner ring of the wheel-shaped structure, with a maximum stress of 24.7 MPa. Analyzing from the maximum displacement of the device, the maximum displacement also occurs at 10 ms, with a displacement of 0.111 \u0026micro;m, located at the y-direction notch of the outer ring of the wheel-shaped structure. The maximum stress occurs at the junction because the wheel-shaped structure is subjected to an in-plane acceleration impact, and the connecting beams in the impact direction bear direct inertial forces. The maximum displacement occurs at the outer ring notch because the outer ring is not directly connected to the connecting beams and has a notch, resulting in uneven impact displacement of each part. During the impact in the X-direction, the inner and outer connecting beams at the y-direction notch are equivalent to being subjected to a transverse impact, thus generating the maximum displacement under the acceleration impact.\u003c/p\u003e\u003cp\u003eThe simulation results of the 10000g acceleration impact response in the Z-direction of the device are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003ec, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003ed. It can be seen that under the 10000g impact load in the Z-axis direction, the maximum stress of the device appears at 10 ms, corresponding to the peak time of the impact signal. The location is at the junction where the connecting beams meet the anchor points, with a maximum stress of 45.7 MPa. Analyzing from the maximum displacement of the device, the maximum displacement also occurs at 10 ms, with a displacement of 0.137 \u0026micro;m, located at the edge of the outer ring of the wheel-shaped structure. For out-of-plane acceleration impact, the wheel-shaped structure can be approximately regarded as a whole connected by connecting beams; thus, the maximum stress occurs at the junction of the connecting beams and the anchor points. The displacement of the wheel-shaped structure is approximately the same at positions with the same radius, and increases as the radius increases. Therefore, the maximum displacement appears at the edge of the outer ring, corresponding to the position with the largest radius of the wheel-shaped structure.\u003c/p\u003e\u003cp\u003eNext, the strength of the device is checked. The device material, monocrystalline silicon, is a brittle material, and its yield strength is approximately equal to its fracture strength. Therefore, checking the strength of the device structure requires using the second strength theory to calculate its maximum allowable stress (ref.23):\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$$\\:\\sigma\\:={\\sigma\\:}_{1}+\\nu\\:（{\\sigma\\:}_{2}+{\\sigma\\:}_{3}）$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere [σ] is the allowable stress of silicon material, typically 700 MPa; (σ\u0026thinsp;=\u0026thinsp;1, 2, 3) are the principal stresses of silicon material; and v is the Poisson's ratio of silicon material.\u003c/p\u003e\u003cp\u003eCombining the maximum allowable stress with the maximum impact displacement and stress obtained from simulations, it can be concluded that whether it is an acceleration impact in the in-plane x-axis direction or an acceleration impact in the out-of-plane z-axis direction, the maximum stress during the device's impact load response is far lower than the maximum allowable stress of the device material (silicon). From the perspective of displacement, the maximum impact displacement in both directions is less than 0.15 \u0026micro;m, which is very small relative to the device scale. This ensures that no connection between device structures or pull-in phenomenon in the capacitive transducing structure will occur.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn this paper, we propose a low-frequency wheel-shaped resonator with vibration in the in-plane torsional mode. By using the torsional vibration of the wheel-shaped mass block instead of translational vibration, the device maintains a low characteristic frequency while achieving high in-plane stiffness. The frequency characteristics of the device are derived and modeled using beam deflection derivation and the Rayleigh energy method. Theoretical calculations and comparisons are conducted on the in-plane stiffness between the classical flexural -mode beam-mass structure and the in-plane torsional mode structure. Under the same beam dimensions, the in-plane stiffness of the in-plane torsional mode is higher by 10\u003csup\u003e2\u003c/sup\u003e to 10\u003csup\u003e4\u003c/sup\u003e orders of magnitude; under the same frequency, the in-plane stiffness is higher by 10\u0026sup2; orders of magnitude. We also conducted simulations on the anti-overload performance of the device structure. By simulating the stiffness and frequency drift under static loads, as well as the maximum stress and displacement under dynamic impact response, it is found that the frequency drifts in the X-axis and Z-axis directions under 100g static load are 21.67 ppb and 60.39 ppb, respectively. Under 10000g impact load, the maximum stresses in the in-plane X-axis and Z-axis directions are 24.7 MPa and 45.7 MPa, with maximum displacements of 0.11 \u0026micro;m and 0.137 \u0026micro;m, respectively, which are far lower than the maximum allowable stress of silicon material and the movable range of the device.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003ch2\u003eConflict of interest\u003c/h2\u003e\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eAuthor contributions\u003c/h2\u003e\u003cp\u003eD.L. conceived this research, designed the device structure, and established the mathematical and physical models of the device structure. W.L. and J.L. participated in designing the comparative simulations and experiments. L.T. and S.C. participated in the analysis of simulation data. J.L. and W.W. conceived the research and supervised the experiments and analysis. All the authors discussed the device design, simulated results, measured results, and prepared the paper.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eKumar P, Bazaz S A, Saleem M M. Design and Analysis of a Low-g MEMS Accelerometer Utilizing Weakly Coupled Resonators[C]//2023 International Conference on Engineering and Emerging Technologies (ICEET). IEEE, 2023: 1\u0026ndash;6.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eTu C, Pan Y M, Wang Z, et al. A Bent-TBTF Resonant MEMS Accelerometer using Auxiliary Supporting Beams[J]. IEEE Electron Device Letters, 2024.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eYadav C, Paliwal S, Yenuganti S. Design and Simulation of a Differential Resonant Pressure Sensor[C]//2023 16th International Conference on Sensing Technology (ICST). IEEE, 2023: 1\u0026ndash;6.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChuai S, Deng J, Li H, et al. Method for Sensitivity Improvement of MEMS Pressure Sensor: Structural Design and Optimization of Concave Resonant Pressure Sensor[J]. IEEE Sensors Journal, 2025.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eXu J, Tsai J M. A process-induced-frequency-drift resilient 32 kHz MEMS resonator[J]. Journal of Micromechanics and Microengineering, 2012, 22(10): 105029.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKaajakari V, Pangaro A, Goto Y, et al. A 32.768 kHz MEMS resonator with+/-20 ppm tolerance in 0.9 mm x 0.6 mm chip scale package[C]//2019 Joint Conference of the IEEE International Frequency Control Symposium and European Frequency and Time Forum (EFTF/IFC). IEEE, 2019: 1\u0026ndash;4.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZaliasl S, Salvia J C, Hill G C, et al. A 3 ppm 1.5\u0026times; 0.8 mm 2 1.0 \u0026micro;A 32.768 kHz MEMS-Based Oscillator[J]. IEEE Journal of Solid-State Circuits, 2014, 50(1): 291\u0026ndash;302.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eYaqoob U, Jaber N, Alcheikh N, et al. Selective multiple analyte detection using multi-mode excitation of a MEMS resonator[J]. Scientific reports, 2022, 12(1): 5297.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eEidi A. Design and evaluation of an implantable MEMS based biosensor for blood analysis and real-time measurement[J]. Microsystem Technologies, 2023, 29(6): 857\u0026ndash;864.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBarrow H G, Naing T L, Schneider R A, et al. A real-time 32.768-kHz clock oscillator using a 0.0154-mm 2 micromechanical resonator frequency-setting element[C]//2012 IEEE International Frequency Control Symposium Proceedings. IEEE, 2012: 1\u0026ndash;6.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSahasrabudhe S, Long Y, Liu Z, et al. A Low Phase Jitter MEMS Oscillator with Centrally-Anchored Piezoelectric Resonator for Wide Temperature Range Real Time Clock Applications[J]. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2024.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eYamada S, Tanaka S. Temperature-Compensated Pure Silicon Cantilever Resonator with Coupled Torsional Structure at Anchor[C]//2023 22nd International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers). IEEE, 2023: 1817\u0026ndash;1820.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMussi G, Bestetti M, Zega V, et al. An outlook on potentialities and limits in using epitaxial polysilicon for MEMS real-time clocks[J]. IEEE Transactions on Industrial Electronics, 2019, 67(8): 6996\u0026ndash;7004.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWu J, Song P, Zang S, et al. Limit cycle convergence leads to period-doubling and cyclic-fold bifurcation in internal resonance-induced mechanical frequency combs[J]. Nonlinear Dynamics, 2025: 1\u0026ndash;22.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWang S. Hermetically Encapsulated Fully Differential Breathe-mode Ring Resonators for Timing Applications[M]. Stanford University, 2013.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eElshurafa A M, Khirallah K, Tawfik H H, et al. Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators[J]. Journal of Microelectromechanical Systems, 2011, 20(4): 943\u0026ndash;958.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePutty M W, Chang S C, Howe R T, et al. Process integration for active polysilicon resonant microstructures[J]. Sensors and Actuators, 1989, 20(1\u0026ndash;2): 143\u0026ndash;151.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSyed W U, Elfadel I A M. Review of Euler\u0026ndash;Bernoulli Rectangular Beam Theory[M]//Tapered Beams in MEMS: A Symbolic Modeling Framework with Applications to Energy Harvesting. Cham: Springer International Publishing, 2024: 17\u0026ndash;31.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNashat S E D, AbdelRassoul R, Abd El Bary A E M. Design and simulation of RF MEMS comb drive with ultra-low pull-in voltage and maximum displacement[J]. Microsystem Technologies, 2018, 24(8): 3443\u0026ndash;3453.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eMestrom R M C, Fey R H B, Phan K L, et al. Simulations and experiments of hardening and softening resonances in a clamped\u0026ndash;clamped beam MEMS resonator[J]. Sensors and Actuators A: Physical, 2010, 162(2): 225\u0026ndash;234.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePeng T, You Z. Reliability of MEMS in shock environments: 2000\u0026ndash;2020[J]. Micromachines, 2021, 12(11): 1275.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZhiwei K O U, Xiaoming C U I, Huiliang C A O, et al. Design and analysis of a capacitive MEMS ring wave gyroscope with high-overload[C]//2019 IEEE 3rd Information Technology, Networking, Electronic and Automation Control Conference (ITNEC). IEEE, 2019: 2447\u0026ndash;2451.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKou Z, Jin L, Cui X, et al. Optimization and Analysis of Capacitive Ring Vibrating Gyroscope with High-overload[C]//2020 IEEE 9th Joint International Information Technology and Artificial Intelligence Conference (ITAIC). IEEE, 2020, 9: 1871\u0026thinsp;\u0026ndash;\u0026thinsp;187\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7525253/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7525253/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn low-frequency vibrating MEMS resonators, device structures typically vibrate in flexural mode and torsional mode. The mechanical stiffness in the motion direction of the bending mode is often strongly correlated with the device\u0026rsquo;s equivalent stiffness, while torsional mode generally use piezoelectric actuation and usually have a low Q factor. These factors limit the application of low-frequency resonators in some high-reliability scenarios. This paper proposes a low-frequency wheel-shaped resonator vibrating in the in-plane torsional mode. By using the torsional vibration of the wheel-shaped mass instead of translational vibration, and adopting an in-plane capacitive structure for driving and sensing, the device maintains high in-plane stiffness while vibrating at low frequencies. At the same frequency, its in-plane stiffness is 10\u0026sup2; orders of magnitude higher than that of the flexural mode. Simulations on the anti-overload performance of the device structure show that under a 100g static load, the frequency drifts in the X-axis and Z-axis directions are 21.67 ppb and 60.39 ppb, respectively. Under 10000g impact loads in the X-axis and Z-axis directions, the maximum in-plane stresses are 24.7 MPa and 45.7 MPa, with maximum displacements of 0.111 \u0026micro;m and 0.137 \u0026micro;m, respectively\u0026mdash;values far lower than the maximum allowable stress of silicon material and the minimum line width of the device. This demonstrates that the structure has high frequency stability and anti-overload performance.\u003c/p\u003e","manuscriptTitle":"Structural Design of a Low-Frequency In-plane Torsional Mode Wheel-Shaped Resonator","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-22 09:23:41","doi":"10.21203/rs.3.rs-7525253/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a20250a7-2543-4587-900f-03a63bf09e0c","owner":[],"postedDate":"September 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":54676736,"name":"Physical sciences/Engineering/Electrical and electronic engineering"},{"id":54676737,"name":"Physical sciences/Nanoscience and technology/Nanoscale devices/Sensors"}],"tags":[],"updatedAt":"2025-11-01T05:22:04+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-22 09:23:41","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7525253","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7525253","identity":"rs-7525253","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}