A low-frequency optimization design method for a small-size cantilever beam vibration absorber with magnetic nonlinearity

Abstract

The present invention belongs to the technical field of dynamic vibration absorbers, and discloses a low-frequency optimization design method for a magnetic nonlinear small-size cantilever beam vibration absorber, including: constructing a magnetically coupled nonlinear cantilever beam dynamic vibration absorber model, using the lateral vibration model under the first-order mode of the dynamic vibration absorber to describe the actual dynamic characteristics, using the nonlinear magnetic force as a part of the external excitation, simplifying it into a concentrated parameter model, and obtaining an equivalent simplified model of the magnetically coupled nonlinear cantilever beam dynamic vibration absorber; frequency optimization design of the dynamic vibration absorber, including: geometric parameter design combined with the material parameters of the dynamic vibration absorber, optimizing the shape boundary, and vibration fatigue analysis after optimization; equivalent parameter identification of the dynamic vibration absorber. Based on the cantilever beam dynamic vibration absorber design method and combined with the magnetic nonlinear correlation principle, the present invention performs frequency optimization design and parameter identification on the dynamic vibration absorber, providing a useful reference and theoretical basis for the lightweight miniaturization and low-frequency vibration reduction design of the dynamic vibration absorber.

Description

Technical Field

[0001] The invention belongs to the technical field of computer-aided design, and in particular to the technical field of optimized design of dynamic vibration absorbers, and specifically to a low-frequency optimized design method of a magnetic nonlinear small-size cantilever beam vibration absorber. Background Art

[0002] The continuous development of science and technology has led to increasingly stringent requirements for precision in industrial production. The accuracy requirements for precision instruments, mechanical equipment, and medical devices and their use standards have also been continuously improved. In order to meet the use requirements, it is particularly important to control and reduce the generation of vibration.

[0003] Dynamic vibration absorbers have been widely studied and applied as a simple and efficient passive vibration control device. However, in terms of low-frequency vibration reduction, their structural design is limited by mass and volume, making it difficult to achieve good vibration reduction effects in small sizes. In addition, the vibration reduction frequency band of passive dynamic vibration absorbers is relatively narrow, which limits the development of vibration absorbers. Therefore, it is of great significance to seek a vibration absorber design method that can achieve broadband low-frequency vibration reduction in a small-size structure. Summary of the invention

[0004] The present invention aims to address the technical problems existing in the background technology and provide a low-frequency optimization design method for a small-size cantilever beam vibration absorber with magnetic nonlinearity. Based on the design method of the cantilever beam dynamic vibration absorber and combined with the relevant principles of magnetic nonlinearity, the dynamic vibration absorber is subjected to frequency optimization design and parameter identification, providing a useful reference and theoretical basis for the lightweight and miniaturization and low-frequency vibration reduction design of the dynamic vibration absorber.

[0005] In order to achieve the above technical objectives, the present invention adopts the following technical solutions:

[0006] A low-frequency optimization design method for a magnetic nonlinear small-size cantilever beam vibration absorber, the method comprising the following steps:

[0007] Step S1: constructing a simplified equivalent model of a magnetically coupled nonlinear cantilever beam dynamic vibration absorber;

[0008] Step S11: constructing a magnetically coupled nonlinear cantilever beam dynamic vibration absorber model;

[0009] Step S12: using the lateral vibration model of the dynamic vibration absorber in the first-order mode to describe the actual dynamic characteristics, using the nonlinear magnetic force as a part of the external excitation, simplifying the nonlinear cantilever beam dynamic vibration absorber into a lumped parameter model, and obtaining an equivalent simplified model of the magnetically coupled nonlinear cantilever beam dynamic vibration absorber;

[0010] Step S2: Frequency optimization design of the dynamic vibration absorber, including:

[0011] Step S21: designing geometric parameters of the dynamic vibration absorber in combination with material parameters of the dynamic vibration absorber;

[0012] Step S22: performing shape optimization on the cantilever beam boundary of the dynamic vibration absorber;

[0013] Step S23: performing vibration fatigue analysis on the optimized dynamic vibration absorber;

[0014] Step S3: Identification of equivalent parameters of the dynamic vibration absorber, including:

[0015] Based on the particle swarm algorithm, the dynamic vibration absorber after frequency optimization in step S2 is simplified into a lumped parameter model through the system parameter identification method. , equivalent damping and equivalent stiffness parameter.

[0016] Further, in step S12, based on the assumption of the Euler-Bernoulli beam, the dynamic equation of the cantilever beam dynamic vibration absorber in the first-order mode is obtained as follows:

[0017]

[0018]

[0019] In the above formula, 、 、 are the equivalent mass, equivalent damping and equivalent stiffness of the dynamic vibration absorber respectively; is the relative displacement of the dynamic vibration absorber, ,in, is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system; is the external nonlinear magnetic force; , , , …, are the coefficients of each order of the nonlinear magnetic expression, which can be measured by experiments or simulations.

[0020] Furthermore, step S21 specifically includes:

[0021] Step S211: designing geometric parameters of the cantilever beam and the end magnet in combination with the material parameters of the cantilever beam and the end magnet, and calculating the first-order natural frequency of the cantilever beam dynamic vibration absorber according to the support width of the root of the cantilever beam;

[0022] Among them, material parameters include: material, density, Young's modulus and Poisson's ratio parameters, and geometric parameters include: length, width and thickness parameters;

[0023] Step S212: applying a simple harmonic excitation force of a certain magnitude to the cantilever beam dynamic vibration absorber, analyzing the mechanical characteristics of the dynamic vibration absorber at this time, and obtaining the maximum bending stress of the dynamic vibration absorber;

[0024] Compare the maximum bending stress of the dynamic vibration absorber with the allowable stress of the material used for the cantilever beam at room temperature to determine whether the maximum bending stress of the dynamic vibration absorber meets the stress condition:

[0025] If the maximum bending stress of the dynamic vibration absorber is less than the allowable stress of the material used for the cantilever beam at room temperature, the stress condition is met;

[0026] Otherwise, the stress condition is not met, and the process returns to step S21 to adjust the geometric parameters of the cantilever beam and the end magnet until the stress condition is met.

[0027] Furthermore, step S22 specifically includes:

[0028] Step S221: Taking into account the vibration characteristics of the dynamic vibration absorber, the actual structure of the cantilever beam is parameterized using the Bernstein polynomial. Since one end of the cantilever beam is a fixed end and the other end is a suspended end, the left and right sides of the suspended end are the suspended long sides, and the function DT ( x ) is used to describe the boundary deformation of the suspended long side on one side of the cantilever beam, and the boundary deformation of the suspended long side on the other side is symmetrical to it;

[0029] Step S222: Taking the maximum bending stress of the dynamic vibration absorber in the resonant state and the first-order natural frequency of the dynamic vibration absorber as the objective functions, and the allowable stress of the stainless steel cantilever beam as the constraint condition, and under the premise of meeting the stress requirements, using COMSOL simulation analysis software to optimize the parameters of the cantilever beam boundary function, taking the initial value of the optimization variable as 0, and determining the value range of the optimization variable in combination with the cantilever beam parameters to obtain the optimized design results.

[0030] Furthermore, in step S221, according to the Weierstrass approximation theorem, the Bernstein polynomial constructor DT( x ).

[0031] Furthermore, in step S221, a fifth-order Bernstein polynomial is used to construct the function DT ( x ) is:

[0032]

[0033] In the above formula, C 0. C 1. C 2. C 3.C 4. C 5 is the function DT( x )The coefficients of each order of the expression; ,in, is the distance from any point on the suspended long side of the cantilever beam to the fixed end, is the total length of the cantilever beam.

[0034] Furthermore, step S23 specifically includes:

[0035] Step S231: establishing a random vibration model of the dynamic vibration absorber through ANSYS software, the random vibration results of the dynamic vibration absorber conform to statistical principles and satisfy normal distribution, and analyzing the random vibration characteristics of the dynamic vibration absorber according to the random vibration results;

[0036] Step S232: Based on the random vibration analysis, the ANSYS simulation software and the nCode DesignLife plug-in are used to perform random vibration fatigue analysis on the dynamic vibration absorber.

[0037] Furthermore, in step S3, the fitness function of the particle swarm algorithm is:

[0038]

[0039] In the above formula, J is the fitness function of the particle swarm algorithm; is the lateral vibration displacement of the end of the cantilever beam dynamic vibration absorber under acceleration excitation; is the theoretical solution for the lateral vibration displacement of the cantilever beam; Refers to the total number of analyzed frequencies.

[0040] Furthermore, after step S3, the following steps are also included:

[0041] Step S4: Simulation analysis of vibration reduction performance of nonlinear dynamic vibration absorber, specifically including:

[0042] Step S41: establishing a vibration control model of a nonlinear dynamic vibration absorber;

[0043] Step S42: analyzing the vibration reduction characteristics of the nonlinear dynamic vibration absorber.

[0044] The purpose of step S4 is to analyze the vibration reduction performance of the designed dynamic vibration absorber and its influence on the object to be damped, so as to provide a theoretical reference for the subsequent further application of the dynamic vibration absorber.

[0045] Furthermore, in step S41, the vibration equation of the nonlinear dynamic vibration absorber system is expressed as:

[0046]

[0047] In the above formula, 、 are the equivalent mass and equivalent damping of the dynamic vibration absorber respectively; and are the acceleration responses of the dynamic vibration absorber and the main vibration system respectively; and are the velocity responses of the dynamic vibration absorber and the main vibration system respectively; It represents the linear stiffness coefficient of the dynamic vibration absorber; It represents the nonlinear stiffness coefficient of the dynamic vibration absorber; is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system; , and are the equivalent mass, equivalent damping and equivalent stiffness of the main vibration system respectively; is the excitation force amplitude;

[0048] Non-dimensionalizing the above formula yields:

[0049]

[0050] In the dimensionless formula above, ; ; ; ; ; ; ; ; ; ; , .

[0051] Further, step S42 includes:

[0052] Step S421: Calculate the elastic stiffness of the main vibration system, use Maxwell electromagnetic simulation software to calculate the longitudinal static magnetic force on the end magnets of the magnetically coupled nonlinear dynamic vibration absorber during the vibration process, process the nonlinear magnetic force simulation data by a parameter fitting method, and obtain the coefficients of the external nonlinear magnetic force Fr;

[0053] Step S422: adjusting the linear stiffness and nonlinear stiffness of the dynamic vibration absorber by changing the size of the external magnet in the dynamic vibration absorber and the relative position between the external magnet and the end magnet to achieve different vibration reduction effects;

[0054] The Runge-Kutta method is used to calculate the bifurcation diagram of the amplitude of the main vibration system changing with the excitation force amplitude, and the motion state of the main vibration system containing the nonlinear dynamic vibration absorber under different excitation amplitudes is analyzed to determine the feasibility of the nonlinear dynamic vibration absorber in vibration reduction applications.

[0055] The phase trajectory, time domain and frequency domain results of the main vibration system under different excitation amplitudes are calculated, and the vibration reduction effect of the nonlinear vibration absorber and the nonlinear motion characteristics of the main vibration system during the excitation process are analyzed.

[0056] Furthermore, in step S421, the elastic stiffness calculation formula of the main vibration system is as follows:

[0057]

[0058] In the above formula, is the equivalent stiffness of the main vibration system, is the natural frequency of the main vibration system, is the equivalent mass of the main vibration system.

[0059] Compared with the prior art, the beneficial effects of the present invention are:

[0060] (1) The present invention provides a low-frequency optimization design method for a small-size cantilever beam vibration absorber with magnetic nonlinearity. Based on the design method of the cantilever beam dynamic vibration absorber and combined with the magnetic nonlinearity related principle, the dynamic vibration absorber is subjected to frequency optimization design and parameter identification. When performing the frequency optimization design, a boundary optimization method is proposed to reduce the natural frequency of the dynamic vibration absorber while ensuring the structural strength. The subsequent fatigue analysis results prove the feasibility of the design under random loads. When performing parameter identification, based on the particle swarm algorithm, the dynamic vibration absorber after frequency optimization is equivalently simplified into a lumped parameter model through the system parameter identification method, and the equivalent mass of the dynamic vibration absorber is obtained. , equivalent damping and equivalent stiffness The present invention realizes the lightweight and small-size design of the magnetically coupled nonlinear cantilever beam dynamic vibration absorber structure, and provides a useful reference and theoretical basis for the lightweight and miniaturization and low-frequency vibration reduction design of the dynamic vibration absorber;

[0061] (2) The frequency optimization design method of the dynamic vibration absorber of the present invention is proposed for lightweight and miniaturized dynamic vibration absorbers. The difference from the existing optimization method is that the shape boundary optimization of the dynamic vibration absorber of the present invention rationally utilizes Bernstein polynomials to parameterize the actual structure of the dynamic vibration absorber. According to the order taken, the cantilever beam boundary of any shape in theory can be described, which solves the limitations of the existing lightweight and miniaturized dynamic vibration absorbers in low-frequency design. Therefore, the optimization design method proposed by the present invention is more in line with the structural design of small-sized nonlinear cantilever beam dynamic vibration absorbers;

[0062] (3) Based on the structural design of a lightweight, small-sized magnetically coupled nonlinear cantilever beam vibration absorber, the present invention simulates and analyzes the vibration reduction performance of the designed dynamic vibration absorber and its influence on the object to be damped, providing a theoretical reference for the further application of the subsequent dynamic vibration absorber; BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 It is a flow chart of a low-frequency optimization design method of a magnetic nonlinear small-size cantilever beam vibration absorber according to an embodiment of the present invention;

[0064] Figure 2 A model diagram of a magnetically coupled nonlinear cantilever beam dynamic vibration absorber according to an embodiment of the present invention;

[0065] Figure 3 This is a simplified equivalent model diagram of a magnetically coupled nonlinear cantilever beam dynamic vibration absorber according to an embodiment of the present invention;

[0066] Figure 4 A stress distribution cloud diagram of a dynamic vibration absorber according to an embodiment of the present invention;

[0067] Figure 5 A schematic diagram of the boundary deformation of a cantilever beam according to an embodiment of the present invention;

[0068] FIG6 (a) is a stress distribution cloud diagram of the optimized cantilever beam dynamic vibration absorber in an embodiment of the present invention;

[0069] FIG6( b ) is a cantilever beam boundary shape diagram of the cantilever beam dynamic vibration absorber after optimization in an embodiment of the present invention;

[0070] FIG. 7 ( a ) is a schematic diagram of a random excitation signal according to an embodiment of the present invention;

[0071] FIG7( b ) is a power spectrum density diagram of a random excitation signal according to an embodiment of the present invention;

[0072] Figure 8 This is a diagram showing fatigue analysis results of the optimized dynamic vibration absorber in an embodiment of the present invention;

[0073] Fig. 9 The end displacement curve of the optimized dynamic vibration absorber and its parameter identification result diagram in the embodiment of the present invention;

[0074] Fig.10 A vibration control model diagram of a nonlinear dynamic vibration absorber according to an embodiment of the present invention;

[0075] Fig.11 A comparison diagram of simulation data and fitting curve of nonlinear magnetic force of the end magnet of an embodiment of the present invention;

[0076] Fig.12A comparison diagram of vibration reduction effects of a linear dynamic vibration absorber and a nonlinear dynamic vibration absorber according to an embodiment of the present invention;

[0077] Fig.13 This is a result diagram of the effect of nonlinear stiffness on vibration reduction performance of an embodiment of the present invention;

[0078] Fig.14 is a bifurcation diagram of the amplitude of the main vibration system of the embodiment of the present invention changing with the amplitude of the excitation force;

[0079] Figures 15(a) to 15(c) are the time domain diagram, phase trajectory diagram and frequency domain diagram of the single-cycle motion under the condition of 0.01N excitation amplitude and frequency = 1.3; Figures 15(d) to 15(f) are the time domain diagram, phase trajectory diagram and frequency domain diagram of the chaotic motion under the condition of 0.13N excitation amplitude and frequency = 1.3; Figures 15(g) to 15(i) are the time domain diagram, phase trajectory diagram and frequency domain diagram of the multi-cycle motion under the condition of 0.16N excitation amplitude and frequency = 1.3. DETAILED DESCRIPTION

[0080] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0081] In the description of the present invention, it is necessary to understand that the terms "coaxial", "bottom", "one end", "top", "middle", "the other end", "upper", "one side", "top", "inside", "front", "center", "both ends" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as a limitation on the present invention.

[0082] Example 1

[0083] The embodiment of the present invention provides a low-frequency optimization design method for a magnetic nonlinear small-size cantilever beam vibration absorber, combining Figure 1 As shown, the method comprises the following steps:

[0084] Step S1: constructing a simplified equivalent model of a magnetically coupled nonlinear cantilever beam dynamic vibration absorber, specifically including:

[0085] Step S11: constructing a magnetically coupled nonlinear cantilever beam dynamic vibration absorber model;

[0086] Cantilever beam dynamic vibration absorber is a common form of vibration absorber. Its vibration mode is linear vibration. In order to broaden the vibration reduction frequency band of the vibration absorber, it is considered to apply a magnetic field outside the vibration absorber to provide nonlinear magnetic force for the vibration absorber. Figure 2 The dynamic model shown in the figure, in which the end magnet is fixed to the free end of the cantilever beam, and the distance from the fixed end of the cantilever beam is l , the external magnet is fixed at a distance of d When d When it is large enough, there is no force between the end magnet and the external magnet, and the system exhibits linear vibration. d When reduced to a certain range, the external magnet exerts a nonlinear force on the end magnet, and the system exhibits nonlinear vibration.

[0087] Step S12: using the lateral vibration model of the dynamic vibration absorber in the first-order mode to describe the actual dynamic characteristics, using the nonlinear magnetic force as a part of the external excitation, simplifying the nonlinear cantilever beam dynamic vibration absorber into a lumped parameter model, and obtaining an equivalent simplified model of the magnetically coupled nonlinear cantilever beam dynamic vibration absorber;

[0088] Under low-frequency excitation, the vibration of the cantilever beam dynamic vibration absorber is mainly concentrated in the first-order mode. The lateral vibration model of the dynamic vibration absorber in the first-order mode can be used to describe its actual dynamic characteristics. Taking the nonlinear magnetic force as part of the external excitation, the nonlinear cantilever beam dynamic vibration absorber can be simplified to a lumped parameter model, such as Figure 3 shown.

[0089] Figure 3 In the equation, the equivalent mass of the dynamic vibration absorber is , the equivalent damping is , the equivalent stiffness is (linear spring stiffness), assuming the relative displacement of the dynamic absorber for: ,in, is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system.

[0090] According to the Euler-Bernoulli beam assumption, the dynamic equation of the cantilever beam dynamic vibration absorber in the first-order mode is as follows:

[0091] (1)

[0092] (2)

[0093] In the above formulas (1)-(2), 、 、 are the equivalent mass, equivalent damping and equivalent stiffness of the dynamic vibration absorber respectively; is the relative displacement of the dynamic vibration absorber, ,in, is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system; is the external nonlinear magnetic force; , , , …, are the coefficients of each order of the nonlinear magnetic expression, which can be measured by experiments or simulations.

[0094] Step S2: frequency optimization design of the dynamic vibration absorber, including: geometric parameter design of the dynamic vibration absorber, shape boundary optimization of the dynamic vibration absorber, and vibration fatigue analysis of the optimized dynamic vibration absorber;

[0095] In order to further realize the lightweight, miniaturization and low-frequency design of dynamic vibration absorbers and promote the application of dynamic vibration absorbers in the field of low-frequency vibration and noise reduction engineering, it is necessary to optimize the design of dynamic vibration absorbers under the premise of meeting structural mechanics conditions.

[0096] Therefore, without considering the external magnetic field, combined with the calculation case of the cantilever beam dynamic vibration absorber, the implementation steps of step S2 are explained as follows:

[0097] Step S21: Designing geometric parameters of the dynamic vibration absorber in combination with the material parameters of the dynamic vibration absorber, specifically including:

[0098] Step S211: designing geometric parameters of the cantilever beam and the end magnet in combination with the material parameters of the cantilever beam and the end magnet, and calculating the first-order natural frequency of the cantilever beam dynamic vibration absorber according to the support width of the root of the cantilever beam;

[0099] Among them, the material parameters include: material, density, Young's modulus and Poisson's ratio parameters, and the geometric parameters include: length, width and thickness parameters.

[0100] In the calculation case of the present invention:

[0101] The cantilever beam is made of 304 stainless steel, and its material parameters are: density 7850 kg / m 3 , Young's modulus 2.0×10 11 Pa, Poisson's ratio 0.3;

[0102] The end magnet is a neodymium iron boron magnet, and its material parameters are: density 7450 kg / m 3 , Young's modulus 1.6×10 11 Pa, Poisson's ratio 0.3.

[0103] The geometric parameters of the cantilever beam and the end magnet are shown in Table 1. Considering that the support width of the root of the cantilever beam is 2 mm, the first-order natural frequency of the cantilever beam dynamic vibration absorber is calculated to be 65.65 Hz.

[0104] Table 1 Geometric parameters of cantilever beam and end magnet

[0105]

[0106] Step S212: a certain magnitude of simple harmonic excitation force is applied to the cantilever beam dynamic vibration absorber, and the mechanical characteristics of the dynamic vibration absorber are analyzed to obtain the maximum bending stress of the dynamic vibration absorber; the maximum bending stress of the dynamic vibration absorber is compared with the allowable stress of the material used for the cantilever beam at room temperature to determine whether the maximum bending stress of the dynamic vibration absorber meets the stress condition:

[0107] If the maximum bending stress of the dynamic vibration absorber is less than the allowable stress of the material used for the cantilever beam at room temperature, the stress condition is met;

[0108] Otherwise, the stress condition is not met, and the process returns to step S21 to adjust the geometric parameters of the cantilever beam and the end magnet until the stress condition is met.

[0109] Assuming that a simple harmonic excitation force of 0.1N is applied to the cantilever beam dynamic vibration absorber, the mechanical characteristics of the dynamic vibration absorber are analyzed and the maximum bending stress of the dynamic vibration absorber is obtained to be 82.1MPa. The stress distribution cloud diagram of the dynamic vibration absorber is shown in the figure below: Figure 4 As shown. Figure 4 It can be seen that the maximum stress of the dynamic vibration absorber occurs at the connection between the cantilever beam and the root support. It is known that the allowable stress of 304 stainless steel at room temperature is 137MPa, which meets the stress condition at this time.

[0110] Step S22: performing shape optimization on the cantilever beam boundary of the dynamic vibration absorber, specifically comprising:

[0111] Step S221: Taking into account the vibration characteristics of the dynamic vibration absorber, the actual structure of the cantilever beam is parameterized using the Bernstein polynomial. Since one end of the cantilever beam is a fixed end and the other end is a suspended end, the left and right sides of the suspended end are the suspended long sides, and the function DT ( x ) is used to describe the boundary deformation of the suspended long side on one side of the cantilever beam, and the boundary deformation of the suspended long side on the other side is symmetrical to it;

[0112] Specifically, Figure 5 As shown, Figure 5 middle l 0 and b 0 represents the total length and total width of the cantilever beam, l s represents the length of the fixed support portion of the cantilever beam,l 1 represents the length of the end magnet, and the length of the suspended portion of the cantilever beam between the fixed support portion and the end magnet is given by l 0- l s - l 1 is calculated.

[0113] According to Weierstrass approximation theorem, function DT( x ) can be constructed by Bernstein polynomials. In order to better describe complex continuous functions, the fifth-order Bernstein polynomial constructor DT( x ), which is involved in calculating the cantilever beam boundary shape optimization.

[0114] Function DT( x The specific expression of ) is shown in formula (3):

[0115] (3)

[0116] In the above formula (3), C 0. C 1. C 2. C 3. C 4. C 5 is the function DT( x )The coefficients of each order of the expression; ,in, is the distance from any point on the suspended long side of the cantilever beam to the fixed end, is the total length of the cantilever beam.

[0117] Since the cantilever beam boundary function DT( x ) has a starting point and an end point, respectively, of the fixed support part and the end magnet position, so C 0 and C 5 The value is 0.

[0118] When performing shape optimization of the cantilever beam boundary of the dynamic vibration absorber, the embodiment of the present invention rationally utilizes the Bernstein polynomials to parameterize the actual structure of the dynamic vibration absorber in view of the lightweight and miniaturized characteristics of the dynamic vibration absorber. The cantilever beam boundary of any shape in theory can be described according to the taken order, which can solve the limitations of the existing lightweight and miniaturized dynamic vibration absorbers in low-frequency design. Therefore, the optimization design method proposed in the embodiment of the present invention is more in line with the structural design of small-size nonlinear cantilever beam dynamic vibration absorbers.

[0119] Step S222: Taking the maximum bending stress of the dynamic vibration absorber in the resonant state and the first-order natural frequency of the dynamic vibration absorber as the objective functions, and the allowable stress of the stainless steel cantilever beam as the constraint condition, and under the premise of meeting the stress requirements, using COMSOL simulation analysis software, optimize the parameters of the cantilever beam boundary function, take the initial value of the optimization variable as 0, and determine the value range of the optimization variable in combination with the cantilever beam parameters to obtain the optimized design results.

[0120] Specifically, assuming that the dynamic vibration absorber vibrates under a simple harmonic excitation force of 0.1 N, the maximum bending stress of the dynamic vibration absorber in the resonant state is and the first-order natural frequency of the dynamic vibration absorber as the objective function, and the allowable stress of the stainless steel cantilever beam As a constraint condition, COMSOL simulation analysis software is used to analyze the cantilever beam boundary function DT( x )middle C 1. C 2. C 3. C 4 parameters are optimized, the initial value of the optimization variable is taken as 0, and the value range of the optimization variable is determined to be ≤18 in combination with the cantilever beam parameters. The optimization design results are shown in Table 2:

[0121] Table 2 Cantilever beam parameter optimization results

[0122]

[0123] After optimization, the first-order natural frequency of the cantilever beam dynamic vibration absorber is reduced from 65.65 Hz to 42.455 Hz, a decrease of 35.33%. The maximum bending stress is 134 MPa, which meets the stress condition. The stress distribution cloud diagram of the optimized cantilever beam dynamic vibration absorber and the cantilever beam boundary shape are shown in Figure 6 (a) and Figure 6 (b), respectively.

[0124] Step S23: Performing vibration fatigue analysis on the optimized dynamic vibration absorber to determine the reliability of the dynamic vibration absorber, specifically including:

[0125] Step S231: establishing a random vibration model of the dynamic vibration absorber through ANSYS software, the random vibration results of the dynamic vibration absorber conform to statistical principles and satisfy normal distribution, and analyzing the random vibration characteristics of the dynamic vibration absorber according to the random vibration results;

[0126] Step S232: Based on the random vibration analysis, the ANSYS simulation software and the nCode DesignLife plug-in are used to perform random vibration fatigue analysis on the dynamic vibration absorber.

[0127] Specifically, in order to ensure that the optimized cantilever beam dynamic vibration absorber can meet the actual use requirements, it is also necessary to perform vibration fatigue analysis on the dynamic vibration absorber, that is: when the applied load is lower than the structural strength limit or even the yield limit, the dynamic vibration absorber is caused to fracture and damage in order to determine the reliability of the dynamic vibration absorber.

[0128] Considering that the dynamic vibration absorber mainly undergoes elastic deformation, the stress intensity is low (far lower than the ultimate strength of the material). To simulate the actual low-frequency working condition, it is assumed that the dynamic vibration absorber is excited by random acceleration, and the excitation signal is 50 Hz and its half frequency and 3 / 2 multiple frequency as the main frequency and accompanied by a noise signal. The expression of the random excitation signal S is shown in formula (4), the random excitation signal is shown in Figure 7 (a), and the power spectral density (PSD) of the random excitation signal is shown in Figure 7 (b).

[0129] (4)

[0130] In the above formula (4), S is a random excitation signal; f 1. f 2. f 3. Amp They are the excitation amplitudes of 25Hz, 50Hz, 75Hz and noise signals, all of which are 0.25g; noise ( t ) is the noise signal.

[0131] The random vibration model of the dynamic vibration absorber was established by ANSYS software, and its random vibration characteristics were analyzed. The random vibration results of the dynamic vibration absorber conformed to the statistical principles and satisfied the normal distribution. The root mean square values ​​of the surface stress and end amplitude of the dynamic vibration absorber during random vibration were calculated to be 17.299MPa and 0.46mm respectively, and the confidence value of the normal distribution was defined by σ. The probability that the stress of the dynamic vibration absorber was ≤17.299MPa and the end amplitude was ≤0.46mm under random excitation was 68.269%. According to the statistical law of 3σ, the probability that the stress of the dynamic vibration absorber was ≤51.896MPa and the end amplitude was ≤1.39mm was 99.73%. The analysis results can guide the design and reliability evaluation of the dynamic vibration absorber.

[0132] Based on the random vibration analysis, the ANSYS simulation software and the nCode DesignLife plug-in are used to perform random vibration fatigue analysis on the dynamic vibration absorber. The conditions of each part of the dynamic vibration absorber when damaged are as follows: Figure 8 As shown. Figure 8 It can be seen that the narrow part of the cantilever beam of the dynamic vibration absorber is most likely to fail due to fatigue fracture. The maximum number of vibrations that the dynamic vibration absorber can normally operate is calculated to be 9.088×10 6 Second-rate.

[0133] Step S3: Identification of equivalent parameters of the dynamic vibration absorber;

[0134] Based on the particle swarm algorithm, the dynamic vibration absorber after frequency optimization in step S2 is simplified into a lumped parameter model through the system parameter identification method, and the equivalent mass of the dynamic vibration absorber is obtained. , equivalent damping and equivalent stiffness .

[0135] In order to analyze the vibration characteristics of the cantilever beam dynamic vibration absorber, it is necessary to obtain the equivalent parameters of the dynamic vibration absorber more accurately.

[0136] Without considering the external magnetic force of the cantilever beam dynamic vibration absorber, that is, F r = 0, and the lateral vibration displacement of the end of the cantilever beam dynamic vibration absorber under acceleration excitation is calculated using ANSYS finite element simulation software: X i , and the theoretical solution of the lateral vibration displacement of the cantilever beam is calculated by formula (1), which is expressed as Y i .

[0137] The fitness function of the particle swarm algorithm is defined as follows:

[0138] (5)

[0139] In the above formula (5), J is the fitness function of the particle swarm algorithm; is the lateral vibration displacement of the end of the cantilever beam dynamic vibration absorber under acceleration excitation; is the theoretical solution for the lateral vibration displacement of the cantilever beam; Refers to the total number of analyzed frequencies.

[0140] Assume that the external acceleration excitation is , is the external acceleration excitation amplitude, ignoring the external magnetic force of the dynamic vibration absorber, then the external nonlinear magnetic force , the relative displacement of the dynamic vibration absorber It can be expressed as , substituting into formula (1), we can get by the harmonic balance method:

[0141] (6)

[0142] Convert equation (6) into a matrix equation:

[0143] (7)

[0144] By solving the matrix equation (7), the relative vibration displacement amplitude of the dynamic vibration absorber can be obtained as follows: .

[0145] In the above formulas (6)-(7), 、 、 are the equivalent mass, equivalent damping and equivalent stiffness of the dynamic vibration absorber respectively; is the relative vibration displacement amplitude of the dynamic vibration absorber, is the external acceleration excitation amplitude; A 1. A 2 is the sine component coefficient and cosine component coefficient of the relative position of the dynamic vibration absorber; is the excitation angular frequency.

[0146] Taking the optimized model of the cantilever beam dynamic vibration absorber in step S2 as an example, the dynamic vibration absorber vibrates under external excitation. Combined with the statistical results of random vibration of the dynamic vibration absorber, it is assumed that the amplitude of the dynamic vibration absorber is 0.4 g The equivalent stiffness of the dynamic vibration absorber is calculated using the particle swarm algorithm. , equivalent mass 、 And the equivalent damping Dimensionless damping ratio After parameter identification, it is calculated that the total mass of the cantilever beam and the end magnet of the dynamic vibration absorber is 8.4407×10 -4 kg, the target frequency band is 0-100 Hz, and the value range of the identification parameters is shown in Table 3.

[0147] Table 3 Identification parameter value range

[0148]

[0149] The parameter identification result is: equivalent mass = 7.3598 × 10 -4 kg, equivalent stiffness = 52.6353 N / m, damping ratio =0.0203.

[0150] The end displacement curve of the dynamic vibration absorber and its parameter identification results are shown in Fig. 9 As shown in the figure, the finite element results of the end displacement of the dynamic vibration absorber are consistent with the parameter identification results, which proves the correctness of the parameter identification results.

[0151] The embodiment of the present invention is based on the design method of the cantilever beam dynamic vibration absorber, combined with the nonlinear correlation principle of magnetic force, and proposes a nonlinear structural model of a magnetically coupled small-size cantilever beam dynamic vibration absorber, and performs frequency optimization design and parameter identification on the dynamic vibration absorber; when performing frequency optimization design, a boundary optimization method for reducing the natural frequency of the dynamic vibration absorber while ensuring structural strength is proposed, and subsequent fatigue analysis results prove the feasibility of the design under random loads; when performing parameter identification, based on the particle swarm algorithm, the frequency-optimized dynamic vibration absorber is equivalently simplified into a lumped parameter model through a system parameter identification method, and the equivalent mass of the dynamic vibration absorber is obtained. , equivalent damping (The dimensionless damping ratio ) and equivalent stiffness , which facilitates the subsequent analysis of the vibration characteristics of the designed cantilever beam dynamic vibration absorber; the embodiment of the present invention realizes the lightweight and small-size design of the magnetically coupled nonlinear cantilever beam dynamic vibration absorber structure, and provides a useful reference and theoretical basis for the lightweight, miniaturization and low-frequency vibration reduction design of the dynamic vibration absorber.

[0152] The embodiment of the present invention can limit the overall size of the dynamic vibration absorber to a range of 30 mm in length×5 mm in width×3 mm in height, and the mass of the dynamic vibration absorber is only 0.844 g.

[0153] Example 2

[0154] Based on Example 1, the embodiment of the present invention further carries out a subsequent step S4 of establishing a vibration reduction system to analyze the vibration reduction performance, the purpose of which is to analyze the vibration reduction performance of the designed dynamic vibration absorber and its influence on the object to be damped, so as to provide a theoretical reference for the further application of the subsequent dynamic vibration absorber.

[0155] Specifically, combined Figure 1 The method further includes, after step S1 to step S3 in embodiment 1:

[0156] Step S4: Simulation analysis of vibration reduction performance of nonlinear dynamic vibration absorber, specifically including:

[0157] Step S41: establishing a vibration control model of a nonlinear dynamic vibration absorber;

[0158] The vibration control model of nonlinear dynamic vibration absorber is established as follows Fig.10 As shown, among which a The variables with subscripts represent the parameters of the dynamic vibration absorber. s The variables with subscripts represent the parameters of the main vibration system. F is the simple harmonic load acting on the main vibration system, specifically: , represents the linear stiffness coefficient of the dynamic vibration absorber, represents the nonlinear stiffness coefficient of the dynamic vibration absorber. It is known that the equivalent stiffness of the cantilever beam dynamic vibration absorber is , combining equation (1) and equation (2) we can get: , ,in, , The external nonlinear magnetic force is The first-order coefficients and third-order coefficients of are shown in formula (2).

[0159] The vibration equation of the nonlinear dynamic vibration absorber system is expressed as:

[0160] (8)

[0161] In the above formula (8), 、 are the equivalent mass and equivalent damping of the dynamic vibration absorber respectively; and are the acceleration responses of the dynamic vibration absorber and the main vibration system respectively; and are the velocity responses of the dynamic vibration absorber and the main vibration system respectively; It represents the linear stiffness coefficient of the dynamic vibration absorber; It represents the nonlinear stiffness coefficient of the dynamic vibration absorber; is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system; , and are the equivalent mass, equivalent damping and equivalent stiffness of the main vibration system respectively; is the excitation force amplitude;

[0162] By non-dimensionalizing equation (8), we can obtain:

[0163] (9)

[0164] In the dimensionless formula (9): ; ; ; ; ; ; ; ; ; ; , ;in, x 0 can be 0.001m; F 0 can be 1N.

[0165] Step S42: analyzing the vibration reduction characteristics of the nonlinear dynamic vibration absorber; including:

[0166] Step S421: Calculate the elastic stiffness of the main vibration system, use Maxwell electromagnetic simulation software to calculate the longitudinal static magnetic force on the end magnet of the magnetic coupling nonlinear dynamic vibration absorber during the vibration process, and process the nonlinear magnetic force simulation data through parameter fitting method to obtain the external nonlinear magnetic force. F r The coefficients of ;

[0167] Among them, the elastic stiffness calculation formula of the main vibration system is as follows:

[0168] (10)

[0169] In the above formula (10), is the equivalent stiffness of the main vibration system, is the natural frequency of the main vibration system, is the equivalent mass of the main vibration system;

[0170] Assume that the natural frequency of the main vibration system is 50Hz and the mass ratio is , the equivalent stiffness of the main vibration system can be calculated for: .

[0171] The length, width and thickness of the external magnet are 3mm, 5mm and 1mm respectively, and the distance between the external magnet and the end magnet is d =4mm. In order to study the nonlinear vibration characteristics of the dynamic vibration absorber, the Maxwell electromagnetic simulation software is used to calculate the static magnetic force in the longitudinal direction of the dynamic vibration absorber. The simulation data of the nonlinear magnetic force on the dynamic vibration absorber when the end magnet amplitude is -6mm to 6mm are obtained as follows: Fig.11 As shown. Fig.11 It can be seen that the magnetically coupled cantilever beam dynamic vibration absorber is different from the traditional linear system. Due to the introduction of the magnetic field, its restoring force is no longer linear, but is approximated to a polynomial curve. This shows that the magnetically coupled cantilever beam dynamic vibration absorber has nonlinear dynamic characteristics. The parameter fitting method is used to obtain Fig.11 The coefficients of the longitudinal external nonlinear magnetic force in the middle are obtained, and the specific expression of the external nonlinear magnetic force is:

[0172] (11)

[0173] In the above formula (11), Y is the relative displacement of the dynamic vibration absorber; is the external nonlinear magnetic force.

[0174] Combining equation (1) and equation (2), we can get: k1= k 0-s1=72.6367, k 3= s 3 = -5.9913 × 10 8 .

[0175] Assuming the excitation force amplitude is 0.01N, the dimensionless parameters of the structure are calculated as shown in Table 4.

[0176] Table 4 Structural dimensionless parameters

[0177]

[0178] Considering the excitation frequency of 0-100 Hz, the fourth-order Runge-Kutta method is used to solve the nonlinear equation system (9). The initial value of the system is taken as [0 0 0 0], and the variables are declared as in formula (12). The vibration response curves of the linear dynamic vibration absorber and the nonlinear dynamic vibration absorber are calculated respectively. Fig.12 shown.

[0179] Declare variables:

[0180] (12)

[0181] The solution is:

[0182] (13)

[0183] In the above formulas (12)-(13), , , , They are , , , Declare variables; , They are , The meanings of the other variables are the same as those in equations (8)-(9) and will not be repeated here.

[0184] Depend on Fig.12 The comparison of the vibration reduction performance of the linear dynamic vibration absorber and the nonlinear dynamic vibration absorber in the figure shows that the dynamic vibration absorber causes a resonance peak to appear on both sides of the target frequency in the main vibration system, and at the resonance frequency of the dynamic vibration absorber, the amplitude response of the main vibration system is reduced from 14.83dB to -7.64dB. When the nonlinear dynamic vibration absorber is installed, the amplitude-frequency response curve of the main vibration system shows an obvious jump phenomenon at the resonance peak on the right side of the target frequency, and the resonance peak is deflected to the right, which causes the vibration reduction bandwidth of the nonlinear dynamic vibration absorber to be larger than that of the linear dynamic vibration absorber.

[0185] Step S422: adjusting the linear stiffness and nonlinear stiffness of the dynamic vibration absorber by changing the size of the external magnet in the dynamic vibration absorber and the relative position between the external magnet and the end magnet to achieve different vibration reduction effects;

[0186] The Runge-Kutta method is used to calculate the bifurcation diagram of the amplitude of the main vibration system changing with the excitation force amplitude, and the motion state of the main vibration system containing the nonlinear dynamic vibration absorber under different excitation amplitudes is analyzed to determine the feasibility of the nonlinear dynamic vibration absorber in vibration reduction applications.

[0187] The phase trajectory, time domain and frequency domain results of the main vibration system under different excitation amplitudes are calculated, and the vibration reduction effect of the nonlinear vibration absorber and the nonlinear motion characteristics of the main vibration system during the excitation process are analyzed.

[0188] To further study the effect of the nonlinear stiffness of the vibration absorber on the vibration reduction performance, keep the formula (9) Unchanged and Changing The linear stiffness of the dynamic vibration absorber can be adjusted by changing the size of the external magnet in the dynamic vibration absorber and the relative position between the external magnet and the end magnet. k 1 and nonlinear stiffness k 3.

[0189] Calculate separately When the values ​​are 1.65, 4.00, and 8.00, the frequency response curve of the main vibration system is as follows: Fig.13 As shown. Fig.13 It can be seen that with the increase of the nonlinear stiffness of the dynamic vibration absorber, the position of the resonance peak of the main vibration system on the left side of the target frequency moves slightly to the right, the right side resonance peak deflects more to the right, the resonance peak value decreases and the vibration reduction bandwidth increases.

[0190] Depend on Fig.13 From the analysis of the results, it can be seen that the jumping phenomenon of the resonance peak on the right side of the target frequency is caused by the nonlinear characteristics of the dynamic vibration absorber, which is also the main factor in widening the vibration reduction frequency band.

[0191] In order to further study the influence of excitation amplitude on the vibration reduction performance of the vibration absorber, the excitation frequency =1.3 (dimensionless), nonlinear stiffness =1.65, the bifurcation diagram of the amplitude of the main vibration system changing with the amplitude of the excitation force is calculated as follows Fig.14 shown.

[0192] Depend on Fig.14 It can be seen that when the excitation amplitude is low, the main vibration system exhibits a single-cycle motion. FWhen 0=0.01N, the amplitude jump occurs. As the excitation amplitude increases, the main oscillation system enters a chaotic motion state. When the excitation amplitude exceeds 0.15N, the main oscillation system is in a multi-period motion state. When the excitation amplitude continues to increase to 0.17N, the main oscillation system continues to enter a chaotic motion state. When the excitation amplitude exceeds 0.25N, the main oscillation system enters a single-period motion state.

[0193] Consider the excitation frequency =1.3, the time domain, phase trajectory and frequency domain results of the main vibration system under the excitation amplitude of 0.01N, 0.13N and 0.16N are calculated respectively as shown in Figure 15 (a) - Figure 15 (i), where Figure 15 (a) - Figure 15 (c) are the time domain, phase trajectory and frequency domain results of the main vibration system under the excitation amplitude of 0.01N, 0.13N and 0.16N, respectively. =1.3 condition; Figure 15 (d) - Figure 15 (f) are respectively the time domain diagram, phase trajectory diagram and frequency domain diagram of the single cycle motion under the condition of 0.13N excitation amplitude and frequency =1.3 condition; Figure 15 (g) - Figure 15 (i) are respectively the time domain diagram, phase trajectory diagram and frequency domain diagram of chaotic motion under the condition of 0.16N excitation amplitude and frequency =1.3 for multi-period motion in the time domain, phase trajectory, and frequency domain.

[0194] As shown in Figures 15(a) to 15(c), the motion state of the main vibration system under the excitation amplitude of 0.01N is a single-cycle motion with 0mm as the equilibrium point. The maximum dimensionless amplitude is 0.052, the maximum dimensionless velocity is 0.067, and the frequency spectrum of the main vibration system is concentrated at =1.3 position; from Figure 15 (d) to Figure 15 (f), it can be seen that the motion state of the main vibration system under the excitation amplitude of 0.13N is a chaotic motion state, the displacement change of the main vibration system is disordered, and the spectrum is mainly concentrated at several frequencies; Figure 15 (g) shows the time domain response of the main vibration system under the excitation amplitude of 0.16N. It can be seen that the main vibration system is a multi-periodic motion at this time, and the spectrum of the main vibration system is concentrated =0.87, =1.3 and =1.73.

[0195] From the analysis results of the vibration characteristics of the main vibration system, it can be seen that in order to widen the vibration reduction frequency band and improve the vibration reduction performance of the nonlinear dynamic vibration absorber, the excitation level should be comprehensively considered and a reasonable nonlinear dynamic vibration absorber should be designed to avoid the chaotic motion of the main vibration system, thereby improving the robustness of the system.

[0196] The above description is only an embodiment of the present invention and is not intended to limit the present invention. Any modification, equivalent replacement and improvement made within the application scope of the present invention shall be included in the protection scope of the present invention.

Claims

1. A low-frequency optimization design method for a magnetic nonlinear small-size cantilever beam vibration absorber, characterized in that: The method comprises the following steps: Step S1: constructing a simplified equivalent model of a magnetically coupled nonlinear cantilever beam dynamic vibration absorber; Step S11: constructing a magnetically coupled nonlinear cantilever beam dynamic vibration absorber model; Step S12: using the lateral vibration model of the dynamic vibration absorber in the first-order mode to describe the actual dynamic characteristics, using the nonlinear magnetic force as a part of the external excitation, simplifying the nonlinear cantilever beam dynamic vibration absorber into a lumped parameter model, and obtaining an equivalent simplified model of the magnetically coupled nonlinear cantilever beam dynamic vibration absorber; In step S12, based on the assumption of the Euler-Bernoulli beam, the dynamic equation of the cantilever beam dynamic vibration absorber in the first-order mode is obtained as follows: ； ； In the above formula, , , are the equivalent mass, equivalent damping and equivalent stiffness of the dynamic vibration absorber respectively; is the relative displacement of the dynamic vibration absorber, ,in, is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system; is the external nonlinear magnetic force; , , , …, are coefficients of each order of the nonlinear magnetic expression, which can be measured by experiment or simulation; Step S2: Frequency optimization design of the dynamic vibration absorber, including: Step S21: designing geometric parameters of the dynamic vibration absorber in combination with material parameters of the dynamic vibration absorber; Step S21 specifically includes: Step S211: designing geometric parameters of the cantilever beam and the end magnet in combination with the material parameters of the cantilever beam and the end magnet, and calculating the first-order natural frequency of the cantilever beam dynamic vibration absorber according to the support width of the root of the cantilever beam; Among them, material parameters include: material, density, Young's modulus and Poisson's ratio parameters, and geometric parameters include: length, width and thickness parameters; Step S212: applying a simple harmonic excitation force of a certain magnitude to the cantilever beam dynamic vibration absorber, analyzing the mechanical characteristics of the dynamic vibration absorber at this time, and obtaining the maximum bending stress of the dynamic vibration absorber; Compare the maximum bending stress of the dynamic vibration absorber with the allowable stress of the material used for the cantilever beam at room temperature to determine whether the maximum bending stress of the dynamic vibration absorber meets the stress condition: If the maximum bending stress of the dynamic vibration absorber is less than the allowable stress of the material used for the cantilever beam at room temperature, the stress condition is met; Otherwise, the stress condition is not met, and the process returns to step S21 to adjust the geometric parameters of the cantilever beam and the end magnet until the stress condition is met; Step S22: performing shape optimization on the cantilever beam boundary of the dynamic vibration absorber; Step S22 specifically includes: Step S221: Taking into account the vibration characteristics of the dynamic vibration absorber, the actual structure of the cantilever beam is parameterized using Bernstein polynomials. Since one end of the cantilever beam is a fixed end and the other end is a suspended end, and the left and right sides of the suspended end are suspended long sides, a function DT(x) is defined to describe the boundary deformation of the suspended long side on one side of the cantilever beam, and the boundary deformation of the suspended long side on the other side is symmetrical to it. Step S222: taking the maximum bending stress of the dynamic vibration absorber in the resonant state and the first-order natural frequency of the dynamic vibration absorber as the objective function, taking the allowable stress of the stainless steel cantilever beam as the constraint condition, and optimizing the parameters of the cantilever beam boundary function by using COMSOL simulation analysis software on the premise of meeting the stress requirements, taking the initial value of the optimization variable as 0, and determining the value range of the optimization variable in combination with the cantilever beam parameters to obtain the optimization design result; Step S23: performing vibration fatigue analysis on the optimized dynamic vibration absorber; Step S3: Identification of equivalent parameters of the dynamic vibration absorber, including: Based on the particle swarm algorithm, the dynamic vibration absorber after frequency optimization in step S2 is simplified into a lumped parameter model through the system parameter identification method. , equivalent damping and equivalent stiffness parameter.

2. The low-frequency optimization design method of the magnetic nonlinear small-size cantilever beam vibration absorber according to claim 1 is characterized in that: In step S221, according to the Weierstrass approximation theorem, the Bernstein polynomial constructor DT( x ).

3. The low-frequency optimization design method of the magnetic nonlinear small-size cantilever beam vibration absorber according to claim 2 is characterized in that: In step S221, a fifth-order Bernstein polynomial is used to construct the function DT ( x ) is: ； In the above formula, C 0. C 1. C 2. C 3. C 4. C 5 is the function DT( x )The coefficients of each order of the expression; ,in, is the distance from any point on the suspended long side of the cantilever beam to the fixed end, is the total length of the cantilever beam.

4. The low-frequency optimization design method of the magnetic nonlinear small-size cantilever beam vibration absorber according to claim 1 is characterized in that: Step S23 specifically includes: Step S231: establishing a random vibration model of the dynamic vibration absorber through ANSYS software, the random vibration results of the dynamic vibration absorber conform to statistical principles and satisfy normal distribution, and analyzing the random vibration characteristics of the dynamic vibration absorber according to the random vibration results; Step S232: Based on the random vibration analysis, the ANSYS simulation software and the nCode DesignLife plug-in are used to perform random vibration fatigue analysis on the dynamic vibration absorber.

5. The low-frequency optimization design method of the magnetic nonlinear small-size cantilever beam vibration absorber according to claim 1 is characterized in that: In step S3, the fitness function of the particle swarm algorithm is: ； In the above formula, J is the fitness function of the particle swarm algorithm; is the lateral vibration displacement of the end of the cantilever beam dynamic vibration absorber under acceleration excitation; is the theoretical solution for the lateral vibration displacement of the cantilever beam; Refers to the total number of analyzed frequencies.

6. The low-frequency optimization design method of the magnetic nonlinear small-size cantilever beam vibration absorber according to claim 1 is characterized in that: After step S3, the following steps are also included: Step S4: Simulation analysis of vibration reduction performance of nonlinear dynamic vibration absorber, specifically including: Step S41: establishing a vibration control model of a nonlinear dynamic vibration absorber; Step S42: analyzing the vibration reduction characteristics of the nonlinear dynamic vibration absorber.

7. The low-frequency optimization design method of a magnetic nonlinear small-size cantilever beam vibration absorber according to claim 6, characterized in that: In step S41, the vibration equation of the nonlinear dynamic vibration absorber system is expressed as: ； In the above formula, 、 are the equivalent mass and equivalent damping of the dynamic vibration absorber respectively; and are the acceleration responses of the dynamic vibration absorber and the main vibration system respectively; and are the velocity responses of the dynamic vibration absorber and the main vibration system respectively; It represents the linear stiffness coefficient of the dynamic vibration absorber; It represents the nonlinear stiffness coefficient of the dynamic vibration absorber; is the displacement response of the dynamic vibration absorber, is the displacement response of the main vibration system; , and are the equivalent mass, equivalent damping and equivalent stiffness of the main vibration system respectively; is the excitation force amplitude; Non-dimensionalizing the above formula yields: ； In the dimensionless formula above, ; ; ; ; ; ; ; ; ; ; , .

8. The low-frequency optimization design method of a magnetic nonlinear small-size cantilever beam vibration absorber according to claim 6, characterized in that: Step S42 includes: Step S421: Calculate the elastic stiffness of the main vibration system, use Maxwell electromagnetic simulation software to calculate the longitudinal static magnetic force on the end magnets of the magnetically coupled nonlinear dynamic vibration absorber during the vibration process, process the nonlinear magnetic force simulation data by a parameter fitting method, and obtain the coefficients of the external nonlinear magnetic force Fr; Step S422: adjusting the linear stiffness and nonlinear stiffness of the dynamic vibration absorber by changing the size of the external magnet in the dynamic vibration absorber and the relative position between the external magnet and the end magnet to achieve different vibration reduction effects; The Runge-Kutta method is used to calculate the bifurcation diagram of the amplitude of the main vibration system changing with the excitation force amplitude, and the motion state of the main vibration system containing the nonlinear dynamic vibration absorber under different excitation amplitudes is analyzed to determine the feasibility of the nonlinear dynamic vibration absorber in vibration reduction applications. The phase trajectory, time domain and frequency domain results of the main vibration system under different excitation amplitudes are calculated, and the vibration reduction effect of the nonlinear vibration absorber and the nonlinear motion characteristics of the main vibration system during the excitation process are analyzed.

9. The low-frequency optimization design method of the magnetic nonlinear small-size cantilever beam vibration absorber according to claim 8, characterized in that: In step S421, the elastic stiffness calculation formula of the main vibration system is as follows: ； In the above formula, is the equivalent stiffness of the main vibration system, is the natural frequency of the main vibration system, is the equivalent mass of the main vibration system.

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