Atomic force microscope vibration isolation method based on X-shaped nonlinear local resonant beam

Abstract

The invention discloses an atomic force microscope vibration isolation method based on an X-shaped nonlinear local resonant beam, relates to the technical field of vibration analysis and metamaterials, and aims to solve the problem that transverse micro-vibration in a probe scanning process cannot be effectively suppressed in the prior art. Transverse vibration of the probe in the scanning process is suppressed, and 50-500 Hz environment vibration interference of an AFM probe cantilever beam is suppressed through nonlinear band gap broadening. According to the technical scheme, the problem of transverse micro-vibration in the probe scanning process can be effectively solved.

Description

Technical Field

[0001] The present invention relates to the technical fields of vibration analysis and metamaterials, and specifically to a vibration isolation method for an atomic force microscope based on an X-shaped nonlinear local resonance beam. Background Art

[0002] In the field of vibration isolation of an atomic force microscope (AFM), a local resonance beam structure is used to suppress the environmental vibration interference of the probe cantilever beam. When the AFM scans the surface topography of a sample through a nanoscale probe, micro-vibrations in the range of 50 - 500 Hz will cause the probe to deflect, resulting in blurred images or distorted data. In view of the nanoscale accuracy requirements of the AFM probe cantilever beam, traditional linear metamaterial beams have the problem of too narrow a bandgap in the 50 - 500 Hz frequency band, thus resulting in the problem that the lateral micro-vibrations during the probe scanning cannot be effectively suppressed. Summary of the Invention

[0003] The purpose of the present invention is to provide a vibration isolation method for an atomic force microscope based on an X-shaped nonlinear local resonance beam to solve the problem that the existing technology cannot effectively suppress the lateral micro-vibrations during the probe scanning.

[0004] The technical solution adopted by the present invention to solve the above technical problems is as follows:

[0005] A vibration isolation method for an atomic force microscope based on an X-shaped nonlinear local resonance beam includes the following steps:

[0006] Step 1: Establish a model of an Euler beam, and periodically distribute X-shaped nonlinear oscillators on the beam in the model. Then, obtain the relationship between the equivalent nonlinear stiffness and the structural parameters through the deformation geometric relationship of the X-shaped nonlinear oscillators;

[0007] Step 2: Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, obtain the bandgap position, and obtain the corresponding frequency range according to the bandgap position. Finally, adjust the atomic force microscope according to the frequency range.

[0008] Further, the relationship between the equivalent nonlinear stiffness and the structural parameters is expressed as:

[0009]

[0010] where k is the horizontal spring stiffness, al is the length of the X-shaped structure rod, α is the initial installation angle, a is the unit cell length, k r is the stiffness coefficient, y is the variable, O(y 3 ) is the high-order term, b0 is the zero-order stiffness term, b1 is the first-order stiffness coefficient, and b2 is the second-order stiffness coefficient.

[0011] Further, the specific steps of Step 2 are as follows:

[0012] Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, the assumed mode method is adopted to obtain the vibration differential equation, and the vibration differential equation is discretized by Galerkin to obtain the ordinary differential equation. Then, the fourth-order Runge-Kutta method and the harmonic balance method are used to solve the ordinary differential equation to obtain the bandgap position, and the corresponding frequency range is obtained according to the bandgap position. Finally, the atomic force microscope is adjusted according to the frequency range.

[0013] Further, the specific steps of adopting the assumed mode method to obtain the vibration differential equation and discretizing the vibration differential equation by Galerkin to obtain the ordinary differential equation are as follows:

[0014] The vibration differential equation of the nonlinear local resonance beam is:

[0015]

[0016] where EI is the bending stiffness of the beam, A is the cross-sectional area, ρ is the fluid density, f q (x, t) is the distributed load, m r is the effective mass of the probe, y(x, t) is the dynamic topography or displacement of the sample surface, z(x, t) is the dynamic displacement of the probe, is the second derivative of z(x, t);

[0017] According to the Galerkin method, the lateral displacement of the beam at any point x is:

[0018] u(x, t) = ξ i (x)·p(t)

[0019] where i is the order of the eigenfunction, ξ i (x) is the generalized coordinate vector, and p(t) is the time modulation function;

[0020] Under the free boundary condition, the displacement shape function is:

[0021]

[0022] where β i is the dimensionless parameter, and the eigenvalue β i l is obtained by solving the characteristic equation cosβ i lchβ i l = 1. Substituting the above expressions into the vibration differential equation, the control equation of the local resonance beam under external excitation is obtained:

[0023]

[0024] where M and K represent the mass matrix and the stiffness matrix respectively, is the first derivative of p(t), is the second derivative of p(t), ξ(x0) is the derivative of the modal shape function at the position x0, and f q (t) is the concentrated external excitation force, and ξ(x r ) is the derivative of the modal shape function at the position x r ; f d (x r , t) and f r (x r , t) are the distributed damping force and the external disturbance force respectively, is the second time derivative of the local displacement;

[0025] The restoring forces f r (x d , t) and f r (x r , t) caused by the oscillator at the point x r on the beam are expressed as:

[0026]

[0027] f r (x r , t) = b0[w r (t) - ξ(x r )p(t)] + b1[w r (t) - ξ(x r )p(t)] 2 + b2[w r (t) - ξ(x r )p(t)] 3

[0028] where C1 and C2 are the structural damping of the beam and the oscillator respectively, ζ is the modal damping ratio, ζ0 is the damping ratio of the harmonic oscillator, and the values of ζ and ζ0 are taken as 0.02 and 0.01 respectively.

[0029] Furthermore, the specific steps of step 2 are as follows:

[0030] Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, and combined with the equivalent medium method, analyze the equivalent mass density and wave number characteristics of the nonlinear local resonance beam under the long-wave approximation. Then, according to the equivalent mass density and wave number characteristics, obtain the bandgap position, the unit cell length a, and the initial angle α. After that, obtain the corresponding frequency range according to the bandgap position. Finally, adjust the atomic force microscope according to the frequency range, the unit cell length a, and the initial angle α.

[0031] Furthermore, the specific steps of analyzing the equivalent mass density and wave number characteristics of the nonlinear local resonance beam under the long-wave approximation by combining the equivalent medium method are as follows:

[0032] Based on the effective medium theory, in the long-wave approximation, the wave equation of a linear metamaterial beam in a homogeneous material is usually expressed as:

[0033]

[0034] D0 = EI

[0035] ρ eff = ρA

[0036] where u is the instantaneous amplitude of the wave, L is the linear partial differential oscillator, f is the external load, D0 is an intermediate variable. For a metamaterial beam, let u = Ue i(ωt-kx) , where U is the complex amplitude of the wave, ω is the angular frequency, x is the spatial coordinate, and t is the time variable. Then the characteristic equation is:

[0037]

[0038] Considering the nonlinear metamaterial local resonance beam structure, for the unit cell model, denote w0 and w r as the transverse displacements of the fixed point m0 and the oscillator on the beam respectively. Then the relative displacement u between the spring-deformed beam, i.e., the oscillator mass and the base beam, r = w r - w0. Then, under the excitation of the external force f q (t), the equation of motion of the unit cell with a linear oscillator is:

[0039]

[0040] where, is the second derivative of u r , is the second derivative of w0, and m0 is the mass of the main system;

[0041] Based on the HBM assumption, its solution form is f q (t) = q0sin(ωt), u r = U1sin(ωt), w0 = W1sin(ωt). Then its equivalent mass can be expressed as:

[0042]

[0043] where, And when this unit cell is equivalent to a homogeneous material in the long-wave approximation, its equivalent mass density is:

[0044]

[0045] Then, according to Equation (8), the wave number of the linear metamaterial is:

[0046]

[0047] When the unit cell has a Duffing - type oscillator, its equation of motion is expressed as:

[0048]

[0049] Assuming the solution form as above is \(f\) q (t) = \(q_0\sin(\omega t)\), \(u\) r = \(U_1\sin(\omega t)\), \(w_0 = W_1\sin(\omega t)\), the equivalent mass is expressed as:

[0050]

[0051] where \(q_0\) is the generalized force amplitude, then the wave number of the metamaterial beam structure with a Duffing - type oscillator is:

[0052]

[0053] Similarly, the equation of motion of the X - shaped nonlinear unit cell under external excitation force is:

[0054]

[0055] Considering the fundamental wave, for the nonlinear equivalent mass density characteristic of the X - shaped unit cell, based on the HBM, we have:

[0056] \(f\) q (t) = \(q_0\sin(\omega t)\)

[0057] \(u\) r = \(U_0+U_1\sin(\omega t)+U_2\cos(\omega t)\)

[0058] \(w_0 = W_1\sin(\omega t)+W_2\cos(\omega t)\)

[0059] where \(U_0\) is the static displacement component, \(U_1\) and \(U_2\) are the amplitude coefficients of dynamic vibration, \(W_1\) and \(W_2\) are the amplitude coefficients of dynamic vibration,

[0060] Then the equivalent mass density of the unit cell under long - wave approximation is:

[0061]

[0062] The equivalent mass density of the unit cell is obtained by solving the following system of equations under the given external force amplitude \(q_0\) or displacement amplitude \(w_0\):

[0063]

[0064] \(W_2m\) r \(\omega\) 2 +\(U_2m\) r \(\omega\) 2+W2aρω 2 = 0

[0065] W1m r ω 2 +U1m r ω 2 +W1aρω 2 +q0 = 0。

[0066] The beneficial effects of the present invention are as follows:

[0067] In this application, by integrating the X-shaped non-linear oscillator into the support frame of the AFM probe cantilever beam, the lateral vibration of the probe during scanning is suppressed, and the environmental vibration interference of 50 - 500 Hz of the AFM probe cantilever beam is suppressed by broadening the non-linear bandgap. The technical solution of this application can effectively suppress the problem of lateral micro-vibration during the probe scanning process. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a schematic diagram of the vibration isolation support beam model for an atomic force microscope (AFM) (including the X-shaped non-linear oscillator unit and the probe cantilever beam structure);

[0069] Figure 2 It is a schematic diagram of the unit cell model and its characteristic frequencies Figure 1 ;

[0070] Figure 3 It is a schematic diagram of the unit cell model and its characteristic frequencies Figure 2 ;

[0071] Figure 4 It is a finite element model diagram;

[0072] Figure 5 It is a verification diagram of the bandgap characteristics of the AFM support beam;

[0073] Figure 6 It is a modal response diagram of the AFM support beam at different frequencies;

[0074] Figure 7 It is a frequency response curve diagram of the AFM support beam at different initial angles α;

[0075] Figure 8 It is a frequency response curve diagram of the oscillator unit of the AFM support beam at different initial angles α;

[0076] Figure 9 It is a comparison diagram of the real part and the imaginary part when the assembly angle is π / 3;

[0077] Figure 10 It is the real part and the imaginary part when the assembly angle is different. DETAILED DESCRIPTION OF THE INVENTION

[0078] It should be noted specifically that, without conflict, the various embodiments disclosed in this application can be combined with each other.

[0079] Specific Embodiment 1: A vibration isolation method for an atomic force microscope based on an X-shaped nonlinear local resonance beam according to this embodiment includes the following steps:

[0080] Step 1: Establish a model of an Euler beam, and periodically distribute X-shaped nonlinear oscillators on the beam in the model. Then, obtain the relationship between the equivalent nonlinear stiffness and the structural parameters through the deformation geometric relationship of the X-shaped nonlinear oscillators;

[0081] Step 2: Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, obtain the bandgap position, and obtain the corresponding frequency range according to the bandgap position. Finally, adjust the atomic force microscope according to the frequency range.

[0082] In this application, a metamaterial structure is constructed by periodically distributing X-shaped nonlinear oscillators on a continuous beam, and its horizontal spring is equivalent to a nonlinear spring.

[0083] S1: Establish a nonlinear local resonance beam model, and obtain its nonlinear stiffness according to the deformation geometric relationship of the X-shaped structure.

[0084] S2: Analyze the vibration characteristics by using the equivalent medium method and the assumed mode method, and verify the bandgap position through finite element simulation.

[0085] S3: Compare the results of different methods and analyze the influence of parameters. Further analyze the influence of different parameters on the vibration characteristics, considering structural parameters, the initial angle of the X-shaped structure, etc. At the same time, compare and summarize the influence of vibration characteristics on the vibration reduction effect and the influence of structural parameters on the vibration reduction effect.

[0086] Next is the expansion of the specific content of different parts. For part S1, for the periodically arranged single-oscillator metamaterial beam structure, the unit cell length (lattice constant) is a, the cross-sectional area of the matrix beam element is A (b is the width of the beam, h is the thickness of the beam), the linear spring stiffness coefficient is k, the mass of the additional mass block is m r , the length and initial angle of the X-shaped structure are represented by al and α, the density of the beam is ρ, and EI is the flexural rigidity. According to the deformation geometric relationship of the X-shaped structure, its nonlinear stiffness is:

[0087]

[0088] Since the expression of the nonlinear stiffness is relatively complex, a third-order Taylor expansion is performed at y = 0:

[0089]

[0090] The expression for the equivalent nonlinear spring stiffness is obtained as:

[0091]

[0092] Wherein:

[0093]

[0094] After the above model is established, the expression of the nonlinear stiffness is pre - processed, making the expression of the nonlinear stiffness more convenient for subsequent analysis and calculation, laying a foundation for studying the vibration characteristics of the structure. Next is the analysis method for the characteristics of the S2 nonlinear local resonance beam:

[0095] When the beam is an Euler beam with a constant cross - section, the vibration differential equation of the local resonance beam model is as follows:

[0096]

[0097] Wherein, y(x, t) and z(x, t) are the displacement field functions of the Euler beam and the oscillator mass block m r respectively. When there is no X - shaped nonlinear structure, k r = k. At this time, the structure is a linear local resonance metamaterial beam. For this linear model, the following analysis methods are commonly used:

[0098] Equivalent medium method: Based on the effective medium theory, in the long - wave approximation, the wave equation of the linear metamaterial beam in a homogeneous material is usually expressed as:

[0099]

[0100] Wherein, D0 = EI, ρ eff = ρA, L is the linear partial differential oscillator, f is the external load, For the metamaterial beam, let u = Ue i(ωt-kx) , then the characteristic equation is:

[0101]

[0102] The equivalent mass density of the unit cell is:

[0103]

[0104] Where U0, U1, U2, W1, W2 are obtained from the amplitude - frequency relationship expression of the unit cell. Substituting this equivalent mass density into the characteristic equation, the wave number expression of the nonlinear local resonance beam can be obtained. When analyzing the unit cell structure by the equivalent medium method, not only can the dynamic nature of the stiffness coefficient be considered, but also the nonlinear frequency response of the oscillator unit cell can be obtained simultaneously, so as to analyze the influence of the nonlinear structure on the structure bandgap and elastic wave propagation.

[0105] Assumed mode method: The vibration differential equation of the nonlinear local resonance beam is:

[0106]

[0107] According to the Galerkin method, the lateral displacement of the beam at any point x is:

[0108] u(x, t) = ξ i (x)·p(t)

[0109] where i is the order of the eigenfunction, and ξ i (x) is the generalized coordinate vector. Under the free boundary condition, the displacement shape function is:

[0110]

[0111] The eigenvalue β i l can be obtained by solving the characteristic equation cosβ i lchβ i l = 1. Substituting the above expressions into the vibration differential equation, the control equation of the locally resonant beam under external excitation is obtained. To solve this nonlinear ordinary differential equation, the fourth-order Runge-Kutta method and the HBM are used for comparative verification. Taking an X-shaped locally resonant beam with 5 periods as the model, assuming the form of the harmonic solution is:

[0112] p(t) = p0 + p1 sin(ωt) + p2 cos(ωt)

[0113] y i = y i_0 + y i_1 sin(ωt) + y i_2 cos(ωt), i = 1, 2, 3, 4, 5

[0114] Finite element simulation: The vibration transmission of the nonlinear locally resonant beam structure is calculated using Comsol Multiphysics 5.6. First, a nonlinear spring structure equivalent to the X-shaped structure is established, and its correctness is verified by comparing the linearized natural frequencies, thereby establishing the overall periodic structure model. Four rectangular rods are defined as rigid domains, and the other parts are defined as linearly elastic materials, and hinges are realized through connectors to complete the model creation and form an assembly. Calculate the characteristic frequencies of the harmonic oscillator system and compare them with the theoretical calculation values to verify the correctness of the model. On the premise of model equivalence, the spring stiffness in the Comsol spring damper module is expressed in the same form as the nonlinear stiffness of the X-shaped structure, and a metamaterial beam model with 20 periods is established, and displacement constraints are imposed on it, allowing only longitudinal displacement to occur. A unit force excitation is applied at the left end of the model, and the frequency response results of the finite-length locally resonant beam are calculated based on the frequency response module, and the midpoint of the other end is selected as the pick-up point to draw the bending vibration transmission diagram of this point.

[0115] Considering the bandgap characteristics of the system, the corresponding frequency is calculated from the wave vector in the formula obtained by the equivalent medium method. To verify the position of the bandgap, the vibration transmission of the locally resonant beam structure is calculated by the assumed mode method and Comsol simulation software.

[0116] From the bandgap regions of the dispersion diagram, the results of the assumed mode method, and the finite element simulation, it can be seen that the finite element method, the assumed mode method, and the equivalent medium method, in combination with the harmonic balance method, are consistent and correct in calculating the band structure of the nonlinear locally resonant beam. Next, the content of S3 is expanded to further analyze the influence of different parameters on the vibration characteristics.

[0117] S3: Result Verification and Comparison

[0118] Bandgap Characteristic Verification: The bandgap range predicted by the equivalent medium method is 157.1 - 325.3 Hz, which is consistent with the results of the assumed mode method and the finite element method. The finite element model shows that the vibration amplitude attenuation in the bandgap exceeds 20 dB.

[0119] Nonlinear Dispersion Relationship Analysis: The initial angle α regulates the nonlinear intensity: soft nonlinearity occurs when α = π / 5, and it turns into hard nonlinearity when α = π / 3. The real part Re(k) of the wave number characterizes the phase change, and the imaginary part Im(k) characterizes the attenuation rate.

[0120] Taking the vibration isolation support beam of an atomic force microscope (AFM) as an example, the X-shaped nonlinear oscillator is integrated into the support frame of the AFM probe cantilever beam to suppress the lateral vibration of the probe during scanning. By broadening the nonlinear bandgap, the environmental vibration interference of the AFM probe cantilever beam in the range of 50 - 500 Hz is suppressed, covering the typical working frequency band (200 - 400 Hz) of the AFM. The method of this application enriches the content of the transition from a linear model to a nonlinear model in the equivalent medium method and compares the results of the two types of models.

[0121] The structural modeling of this application includes the following steps:

[0122] S1: Springs are installed in the horizontal direction of the X-shaped structure. Such a composite structure can be equivalent to a nonlinear spring. For the periodically arranged single-oscillator metamaterial beam structure, its nonlinear stiffness is:

[0123]

[0124] Due to the complexity of f(y), its third-order Taylor expansion is carried out at y = 0, and the form is as follows:

[0125]

[0126] Then the equivalent nonlinear spring stiffness of the X-shaped structure is expressed as:

[0127]

[0128] Among them:

[0129]

[0130] Taking the support structure of the atomic force microscope (AFM) probe cantilever beam as an example, the unit cell length (lattice constant) is a, the cross-sectional area of the matrix beam element is A (b is the width of the beam, h is the thickness of the beam), the linear spring stiffness coefficient is k, and the mass of the additional mass block is m r , the length and initial angle of the X-shaped structure are represented by al and α, the density of the beam is ρ, and EI is the flexural rigidity. S2: When the beam is an Euler beam with a constant cross-section, the vibration differential equation of the local resonance beam model is as follows:

[0131]

[0132]

[0133] Among them, y(x,t) and z(x,t) are the displacement field functions of the Euler beam and the oscillator mass block m r respectively. When there is no X-shaped nonlinear structure, k r = k, and at this time the structure is a linear local resonance metamaterial beam. For this linear model, the following analysis methods are commonly used:

[0134] Equivalent medium method: Based on the effective medium theory, in the long-wave approximation, the wave equation of the linear metamaterial beam in a homogeneous material is usually expressed as:

[0135]

[0136] Among them, D0 = EI, ρ eff = ρA, L is the linear partial differential oscillator, f is the external load, For the metamaterial beam, let u = Ue i(ωt-kx) , then the characteristic equation is:

[0137]

[0138] Considering the nonlinear metamaterial local resonance beam structure, for the unit cell model, see Figure 2 , denote w0 and w r as the transverse displacements of the fixed point m0 on the beam and the oscillator respectively, then the relative displacement u r between the spring deformation beam, that is, the oscillator mass and the base beam, is u r = w q - w0. Then, under the excitation of the external force f

[0139]

[0140] Based on the HBM, assume its solution form is f q (t) = q0sin(ωt), u r = U1sin(ωt), w0 = W1sin(ωt). Then its equivalent mass can be expressed as:

[0141]

[0142] where When the unit cell is equivalent to a homogeneous material under the long - wave approximation, its equivalent mass density is:

[0143]

[0144] Then, according to Equation (8), the wave number of the linear metamaterial is:

[0145]

[0146] When the unit cell has a Duffing - type oscillator, its equation of motion can be expressed as:

[0147]

[0148]

[0149] As assumed above, the solution form is f q (t) = q0sin(ωt), u r = U1sin(ωt), w0 = W1sin(ωt). The equivalent mass can be expressed as:

[0150]

[0151] Then the wave number of the metamaterial beam structure with a Duffing - type oscillator is:

[0152]

[0153] Similarly, the equation of motion of the X - shaped nonlinear unit cell under external excitation force is:

[0154]

[0155] Considering the fundamental wave, for the nonlinear equivalent mass density characteristic of the X - shaped unit cell, based on the HBM, we have:

[0156] f q (t) = q0sin(ωt) (20)

[0157] u r = U0 + U1sin(ωt)+U2cos(ωt) (21)

[0158] w0 = W1sin(ωt) + W2cos(ωt) (22)

[0159] Substituting Eqs. (20), (21), and (22) into (18) and (19), the equivalent mass density of the unit cell under the long - wave approximation is as follows:

[0160]

[0161] which is obtained from the amplitude - frequency relationship expression of the unit cell described by the following equation.

[0162]

[0163] W2m r ω 2 + U2m r ω 2 + W2aρω 2 =0(24d)

[0164] W1m r ω 2 + U1m r ω 2 + W1aρω 2 + q0=0(24e)

[0165] Substituting Eq. (23) into (8) gives the wave - number expression of the nonlinear local resonance beam. From Eq. (23), it can be seen that when analyzing the unit - cell structure using the equivalent - medium method, not only the dynamic nature of the stiffness coefficient is considered, but also the nonlinear frequency response of the oscillator unit cell can be obtained synchronously. The equivalent mass density of the unit cell can be obtained by solving the system of equations (24) under a given external - force amplitude q0 or displacement amplitude w0. Thus, the influence of the nonlinear structure on the structural bandgap and elastic - wave propagation can be analyzed.

[0166] Assumed - mode method: The vibration differential equation of the nonlinear local resonance beam is:

[0167]

[0168] According to the Galerkin method, the transverse displacement of the beam at any point x is:

[0169] u(x,t) = ξ i (x)·p(t) (27)

[0170] where i is the order of the eigen - function, ξ i (x) is the generalized - coordinate vector, and under the free - boundary condition, the displacement shape function is:

[0171]

[0172] The eigenvalue βi l can be obtained by solving the characteristic equation cosβ i lchβ i l = 1. Substituting the above expressions into the vibration differential equation, the control equation of the locally resonant beam under external excitation is obtained:

[0173]

[0174] where M and K represent the mass matrix and the stiffness matrix respectively. The restoring forces f r (x d (x r , t) and f r (x r , t) caused by the oscillator at the point x on the beam are:

[0175]

[0176] f r (x r , t) = b0[w r (t) - ξ(x r )p(t)] + b1[w r (t) - ξ(x r )p(t)] 2 + b2[w r (t) - ξ(x r )p(t)] 3 (30b)

[0177] where C1 and C2 are the structural damping of the beam and the oscillator respectively, which can make the transient response converge quickly. ζ is the modal damping ratio, and ζ0 is the damping ratio of the harmonic oscillator. The values of ζ and ζ0 are taken as 0.02 and 0.01 respectively.

[0178] To solve this nonlinear ordinary differential equation, the fourth-order Runge-Kutta method and the HBM are used for comparative verification. Taking an X-shaped locally resonant beam with 5 periods as the model, assuming the form of the harmonic solution is:

[0179] p(t) = p0 + p1 sin(ωt) + p2 cos(ωt)(31a)

[0180] y i = y i_0 + y i_1 sin(ωt) + y i_2 cos(ωt), i = 1, 2, 3, 4, 5(31b)

[0181] Finite element simulation: The vibration transmission of the nonlinear local resonance beam structure was calculated using Comsol Multiphysics 5.6. First, an equivalent nonlinear spring structure to the X-shaped structure was established, and its correctness was verified by comparing the linearized natural frequencies, thereby establishing an overall periodic structure model. The structural parameters are shown in Table 1 below.

[0182] Table 1 Model parameters of the local resonance beam structure

[0183]

[0184] The specific process is as follows:

[0185] (1) Operating steps: An equivalent nonlinear spring model (parameters shown in Table 1) was established in COMSOL, displacement constraints were applied, and the frequency response was calculated.

[0186]

[0187] In the formula: k - the horizontal spring stiffness of the oscillator;

[0188] al - the rod length of the oscillator structure;

[0189] α - the initial installation angle;

[0190] A metamaterial beam model with 20 periods was established, and its structural parameters were the same as those in Table 1. Displacement constraints were applied to ensure that only longitudinal displacement occurred. The beam structure in the established single-oscillator Euler beam model was meshed using quadrilateral elements, and a unit force excitation perpendicular to the beam plane was applied at the key point on the left end. The finite element model is shown in Figure 3 . There are massless connection points above the mass block and below the corresponding beam unit cell for connecting the spring damper.

[0191] (2) Extract the response results

[0192] After applying a unit force excitation at the left end of the model, the frequency response results of this finite-length local resonance beam can be calculated based on the frequency response module, and the midpoint at the other end is selected as the pick-up point to plot the bending vibration transmission diagram of this point.

[0193] S3: In this section, the bandgap characteristics of the nonlinear local resonance beam were verified by three methods: the equivalent medium method, the assumed mode method, and finite element simulation, as follows:

[0194] Analysis by the equivalent medium method: Based on Eqs. (25) and (26), the wave vector was calculated, and the bandgap range was obtained as 157.1 Hz to 325.3 Hz, as shown in Figure 5 (a)( Figure 5(a) Equivalent medium method, (b) assumed mode method, and (c) comparison with finite element results. The bandgap range of 157.1 - 325.3 Hz covers the sensitive frequency band of the AFM probe cantilever beam. This bandgap is formed by the Taylor expansion of the equivalent stiffness of the X-shaped nonlinear oscillator in Equation (2) and its coupling with the matrix beam, verifying the influence of nonlinear effects on the bandgap position under the long-wave approximation.

[0195] Verification of the assumed mode method: The vibration equation (29) is established using the Galerkin method and solved by combining the Runge - Kutta method and the harmonic balance method (HBM). The results show that the vibration transmission curve of the 5 - period beam model is shown in Figure 5 (b), showing significant attenuation within the bandgap range of 157.1 - 325.3 Hz. The nonlinear restoring force in Equation (30b) affects the frequency response characteristics through quadratic and cubic terms, which is consistent with the results of the equivalent medium method.

[0196] Verification of finite element simulation: A 20 - period beam model is established through Comsol, as shown in Figure 3 , where the spring stiffness is defined by Equation (1). The simulation results are shown in Figure 5 (c), indicating that the bandgap frequency interval is in complete agreement with the theoretical prediction. Finite element modal analysis Figure 5 shows that the bending vibration of the beam within the bandgap is strongly suppressed, and the energy localization phenomenon is significant.

[0197] Method comparison and conclusion: Consistency: The equivalent medium method predicts a bandgap range of 157.1 - 325.3 Hz, covering the typical working frequency band (200 - 400 Hz) of the AFM probe cantilever beam, which is consistent with the results of the assumed mode method and finite element method. Nonlinear effects: Both the equivalent medium method and the assumed mode method show that the nonlinear stiffness coefficient can dynamically adjust the equivalent mass density, thereby affecting the bandgap characteristics. Engineering applicability: The finite element simulation steps (1) - (2) prove the modelability of complex nonlinear metamaterial structures, providing a reliable tool for practical design.

[0198] Analysis of nonlinear dispersion relations under different assembly angles:

[0199] In this section, through the external force regulation method (given amplitude \(q_0 = 0.2\ N\)), the influence of the initial assembly angle \(\alpha\) of the X - shaped oscillator on the elastic wave propagation characteristics of the semi - infinite nonlinear metamaterial beam is studied, where:

[0200] Frequency response characteristics: By solving the nonlinear equation set (24), the frequency response curve of the beam - oscillator unit cell is obtained, as shown in Figure 7 . The frequency response curve of the oscillator unit, i.e., Figure 7 (a), shows a trend of changing from soft characteristics to hard characteristics with the increase of the initial angle \(\alpha\), and multi - valued intervals appear at \(\alpha=\pi / 5\) and \(\pi / 3\), resulting in the beam unit, i.e., Figure 7The frequency response curve of (b) also exhibits a multi-valued phenomenon. The anti-resonance peaks correspond to the linearized natural frequencies of the oscillator, which increase with the increase of α (when α = π / 5, π / 4, π / 3, the natural frequencies are 114.1 rad / s, 157.05 rad / s, and 272.02 rad / s respectively).

[0201] Nonlinear dispersion relation: The dispersion curves corresponding to different initial angles are shown by Figure 8 It shows that the real part of the wave number Re(k) characterizes the phase change in the passband, and the imaginary part |Im(k)| reflects the attenuation rate. The multi-valued interval appears outside the bandgap (corresponding to the frequency band where Im(k) = 0), avoiding the interference of nonlinear bifurcation on the propagation of elastic waves. The negative equivalent mass density region (ρ eff / ρ0 < 0) and the corresponding bandgap range of the AFM probe cantilever ensure that the probe is free from environmental vibration interference during the scanning process.

[0202] This application is applicable to scenarios with strict requirements for wide-frequency micro-vibration suppression, such as the vibration isolation support beam of an atomic force microscope (AFM). By dynamically regulating the nonlinear bandgap, the environmental vibration interference of the AFM probe cantilever in the range of 50 - 500 Hz is suppressed, providing a theoretical basis for the design and optimization of the vibration damping structure of the AFM probe cantilever, and improving the imaging resolution of the AFM in a complex vibration environment. The composite spring structure designed based on the X-shaped structure is a nonlinear unit that can be integrated into the AFM support beam. When it deforms, it provides both quadratic and cubic nonlinear stiffness, and dynamically matches the working frequency band of the AFM probe cantilever by adjusting the initial angle α. The bandgap position is calculated based on the equivalent medium method and the assumed mode method. On this basis, finite element simulation is used for modeling to verify its bandgap characteristics, ensuring the correctness of the results.

[0203] It should be noted that the specific implementation manners are only explanations and illustrations of the technical solutions of the present invention, and the scope of the right protection cannot be limited thereby. Those that are only partial changes made according to the claims and the description of the present invention should still fall within the protection scope of the present invention.

Claims

1. An atomic force microscope vibration isolation method based on an X-shaped non-linear local resonance beam, characterized in that It includes the following steps: Step 1: Establish a model of an Euler beam, and set X-shaped nonlinear oscillators on the beam in the model. The X-shaped nonlinear oscillators are distributed periodically. Then, obtain the relationship between the equivalent nonlinear stiffness and the structural parameters through the deformation geometric relationship of the X-shaped nonlinear oscillators; Step 2: Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, obtain the bandgap positions, and obtain the corresponding frequency ranges according to the bandgap positions. Finally, adjust the atomic force microscope according to the frequency ranges; 2. The atomic force microscope vibration isolation method based on an X-shaped non-linear local resonance beam according to claim 1, characterized in that The relationship between the equivalent nonlinear stiffness and the structural parameters is expressed as: Among them, k is the horizontal spring stiffness, al is the length of the X-shaped structure rod, α is the initial installation angle, a is the unit cell length, k r is the stiffness coefficient, y is the variable, O(y 3 ) is the high-order term, b0 is the zero-order stiffness term, b1 is the first-order stiffness coefficient, and b2 is the second-order stiffness coefficient.

3. A vibration isolation method for an atomic force microscope based on an X-shaped non-linear local resonance beam according to claim 2, characterized in that The specific steps of Step 2 are as follows: Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, adopt the assumed mode method to obtain the vibration differential equation, and discretize the vibration differential equation through Galerkin to obtain the ordinary differential equation. Then, use the fourth-order Runge-Kutta method and the harmonic balance method to solve the ordinary differential equation to obtain the bandgap positions, and obtain the corresponding frequency ranges according to the bandgap positions. Finally, adjust the atomic force microscope according to the frequency ranges; 4. A vibration isolation method for an atomic force microscope based on an X-shaped non-linear local resonance beam according to claim 3, characterized in that The specific steps of adopting the assumed mode method to obtain the vibration differential equation and discretizing the vibration differential equation through Galerkin to obtain the ordinary differential equation are as follows: The vibration differential equation of the nonlinear local resonance beam is: where, EI is the flexural rigidity of the beam, A is the cross-sectional area, ρ is the fluid density, f q (x,t) is the distributed load, m r is the effective mass of the probe, y(x,t) is the dynamic topography or displacement of the sample surface, z(x,t) is the dynamic displacement of the probe, is the second derivative of z(x,t); According to the Galerkin method, the transverse displacement of the beam at any point x is: u(x,t) = ξ i (x)·p(t) where \(i\) is the order of the characteristic function, \(\xi\) i (x) is the generalized coordinate vector, and \(p(t)\) is the time modulation function; The displacement shape function under the free boundary condition is: Among them, β i is a dimensionless parameter, and the eigenvalue β i l is obtained by solving the characteristic equation cosβ i lchβ i l = 1. Substituting the above expression into the vibration differential equation, the control equation of the locally resonant beam under external excitation is obtained: where M and K represent the mass matrix and the stiffness matrix respectively, is the first derivative of p(t), is the second derivative of p(t), ξ(x0) is the derivative of the modal shape function at the position x0, f q (t) is the concentrated external excitation force, ξ(x r ) is the derivative of the modal shape function at x r position, f d (x r ,t) and f r (x r ,t) are the distributed damping force and the external disturbance force respectively, is the second time derivative of the local displacement; The restoring force f r induced by the oscillator at point x d (x r , t) and f r (x r , t) are expressed as: f r (x r ,t) = b0[w r (t) - ξ(x r )p(t)] + b1[w r (t) - ξ(x r )p(t)] 2 + b2[w r (t) - ξ(x r )p(t)] 3 where C1 and C2 are the structural damping of the beam and the oscillator respectively, ζ is the modal damping ratio, ζ0 is the damping ratio of the harmonic oscillator, and the values of ζ and ζ0 are taken as 0.02 and 0.01 respectively.

5. A vibration isolation method for an atomic force microscope based on an X-shaped non-linear local resonance beam according to claim 2, characterized in that The specific steps of Step 2 are as follows: Based on the relationship between the equivalent nonlinear stiffness and the structural parameters, and combined with the equivalent medium method, analyze the equivalent mass density and wave number characteristics of the nonlinear local resonance beam under the long-wave approximation. According to the equivalent mass density and wave number characteristics, obtain the bandgap positions, the unit cell length a, and the initial angle α. Then, obtain the corresponding frequency ranges according to the bandgap positions. Finally, adjust the atomic force microscope according to the frequency ranges, the unit cell length a, and the initial angle α; 6. The atomic force microscope vibration isolation method based on an X-shaped non-linear local resonance beam according to claim 5, characterized in that The specific steps of combining the equivalent medium method and analyzing the equivalent mass density and wave number characteristics of the nonlinear local resonance beam under the long-wave approximation are as follows: Based on the effective medium theory, under the long-wave approximation, the wave equation of the linear metamaterial beam in the homogeneous material is usually expressed as: D0 = EI ρ eff = ρA where, u is the instantaneous amplitude of the fluctuation, L is the linear partial differential oscillator, and f is the external load, D0 is an intermediate variable. For the metamaterial beam, let u = Ue i(ωt-kx) , U is the complex amplitude of the wave, ω is the angular frequency, x is the spatial coordinate, and t is the time variable. Then the characteristic equation is: Considering the local resonance beam structure of nonlinear metamaterials, for the unit cell model, let \(w_0\) and \(w\) r be the transverse displacements of the fixed point \(m_0\) and the oscillator on the beam respectively. Then the relative displacement \(u\) r between the spring-deformed beam, i.e., the oscillator mass, and the base beam is \(u =\) r \(w -\) q \(w_0\). Under the excitation of the external force \(f(t)\), the motion equation of the unit cell with a linear oscillator is: Among them, is the second derivative of u r , is the second derivative of w0, and m0 is the mass of the main system; Based on the HBM, assume that its solution form is f q (t) = q0sin(ωt), u r = U1sin(ωt), w0 = W1sin(ωt), Then its equivalent mass can be expressed as: Among them, When this unit cell is equivalent to a homogeneous material under the long-wave approximation, its equivalent mass density is: Then, according to Equation (8), the wave number of the linear metamaterial is obtained as: When the unit cell has a Duffing-type oscillator, its motion equation is expressed as: The assumed solution form is as follows: f q (t) = q0sin(ωt), u r = U1sin(ωt), w0 = W1sin(ωt), and the equivalent mass is expressed as: Among them, q0 is the generalized force amplitude, then the wave number of the metamaterial beam structure with a Duffing-type oscillator is: Similarly, the motion equation of the X-shaped nonlinear unit cell under the external excitation force is: Considering the fundamental wave, the nonlinear equivalent mass density characteristics of the X-shaped unit cell are obtained based on HBM as: f q (t) = q0sin(ωt) u r = U0 + U1sin(ωt) + U2cos(ωt) w0 = W1sin(ωt) + W2cos(ωt) Among them, U0 is the static displacement component, U1 and U2 are the amplitude coefficients of the dynamic vibration, and W1 and W2 are the amplitude coefficients of the dynamic vibration. Then, the equivalent mass density of the unit cell under the long-wave approximation is: The equivalent mass density of the unit cell is obtained by solving the following system of equations when the given external force amplitude is q0 or the displacement amplitude is w0: W2m r ω 2 +U2m r ω 2 +W2aρω 2 =0 W1m r ω 2 +U1m r ω 2 +W1aρω 2 +q0 = 0.

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