Nonlinear stiffness dynamic measurement system and method for low-frequency vibration table spring return mechanism

Abstract

The present application belongs to the technical field of nonlinear stiffness identification of vibration excitation device, and particularly relates to a low-frequency vibration table spring piece return mechanism nonlinear stiffness dynamic measurement system and method. The system comprises a program-controlled signal generator, a power amplifier, a low-frequency vibration table installed with a spring piece return mechanism, an eddy current displacement sensor, a data acquisition card, a computer and a software analysis module. The output vibration displacement signals and input standard sinusoidal voltage signals are subjected to FFT analysis by the data processing unit to obtain corresponding signal amplitudes and phases, and then the transfer function of the low-frequency vibration table first-order inertia link form corresponding to the selected amplitude output vibration displacement signals in the working frequency range is calculated by the transfer function calculation unit. Finally, the power series fitting unit is used for power series fitting to obtain the power series expression representing the nonlinear stiffness, thereby completing the dynamic measurement of the low-frequency vibration table spring piece return mechanism nonlinear stiffness. Overall, the measurement system has simple structure, simple measurement method operation process and wide application range, and can realize accurate dynamic measurement of the low-frequency vibration table spring piece return mechanism nonlinear stiffness.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear stiffness identification technology of vibration excitation devices, specifically relating to a dynamic measurement system and method for the nonlinear stiffness of a low-frequency vibration table reed recovery mechanism. Background Technology

[0002] With the development of intelligent technology, vibration sensors are increasingly widely used in industrial production and daily life. To ensure the detection accuracy of vibration sensors, the research and development of vibration calibration devices and related technologies are becoming increasingly important. According to international and domestic vibration calibration standards, the calibration process for vibration sensors requires a vibration table to apply a steady-state sinusoidal excitation signal. Therefore, the distortion of the output vibration waveform is a crucial indicator of the vibration table's performance, significantly affecting the accuracy of vibration calibration. To simplify the structure and save costs, a flexible reed is typically used as the vibration table's return mechanism. To meet the calibration requirements of low-frequency vibration sensors, the stroke of the low-frequency vibration table is usually increased to generate a large displacement vibration signal with a sufficient signal-to-noise ratio. However, with the increase in vibration stroke, the reed return mechanism inevitably develops nonlinear characteristics under large deformation conditions, leading to severe waveform distortion in the low-frequency vibration table's output vibration signal, thus affecting the calibration accuracy of the low-frequency vibration sensor based on this low-frequency vibration table. Therefore, it is urgent to accurately obtain the nonlinear stiffness characteristics of the spring return mechanism under low-frequency large deformation, so as to provide a parameter basis for constructing a motion waveform control system for a low-frequency vibration table, enabling the low-frequency vibration table to generate a low-distortion vibration excitation signal and effectively improve the calibration accuracy of low-frequency vibration.

[0003] Currently, most methods for obtaining the large deformation nonlinear stiffness characteristics of reed return mechanisms employ static testing methods that measure displacement deformation under a specific force. This method typically uses a DC power supply to drive a low-frequency vibration table, causing the reed return mechanism to produce several static displacements along its bending deformation direction. These displacement values ​​are then detected by a displacement sensor and compared with the static force generated by the input current to approximate the nonlinear stiffness characteristic parameters of the reed return mechanism. The drawbacks of this method are that the obtained nonlinear stiffness only reflects the static characteristics of the reed return mechanism and cannot characterize its dynamic nonlinear stiffness characteristics when the low-frequency vibration table generates a vibration excitation signal. Furthermore, this method suffers from cumbersome testing procedures and complex system composition.

[0004] In addition, some scholars have conducted research on the stiffness characteristic testing of other nonlinear structures and systems. A representative method includes Chinese Patent 201310507793.1, which discloses a method for testing the nonlinear stiffness of hard-coated composite structures. First, a pulse excitation device is used to obtain preliminary values ​​of the natural frequencies, modal damping ratios, and linear stiffness of each order of the hard-coated composite structure. Then, based on the criteria for judging the nonlinear stiffness type of the hard-coated composite structure, frequency response curves corresponding to each natural frequency are identified through frequency sweep testing, and the corresponding nonlinear stiffness parameter values ​​are calculated. Finally, by superimposing the nonlinear stiffness values ​​with the linear stiffness values, the strong and weak nonlinear stiffness values ​​of the hard-coated composite structure are calculated. The disadvantages of this method are: the parameter testing process is cumbersome, the system is complex, and the cost is high; because the calculation process depends on the frequency response characteristics corresponding to each natural frequency, the nonlinear stiffness test results of this method require high accuracy in frequency sweep testing and frequency response characteristic calculation, and it cannot accurately characterize the nonlinear stiffness characteristics in frequency bands near non-natural frequencies. Chinese Patent 202110445377.8 discloses a method for identifying the nonlinear stiffness of a multi-degree-of-freedom nonlinear vibration system. First, pulse excitation is applied to the nonlinear vibration system. The transient response is measured, and its characteristic displacements are extracted to obtain the instantaneous frequencies and corresponding frequency-displacement characteristic curves. Then, based on a general dynamic model of the nonlinear vibration system, the relationship between frequency and linear and nonlinear stiffness parameters is analyzed. Finally, by comparing the parameters of the dynamic model with the parameter values ​​corresponding to the measured frequency-displacement characteristic curves, a pattern search algorithm is used to determine the nonlinear stiffness of the system. The drawbacks of this method are: the nonlinear stiffness is obtained by comparing and analyzing a simplified mathematical model and measured data, resulting in low parameter identification accuracy; moreover, this method can only analyze the overall nonlinear stiffness characteristics of the system influenced by multiple parameters, and is not suitable for determining the nonlinear parameters of independent structures such as low-frequency vibration table reed recovery mechanisms. Summary of the Invention

[0005] To address the shortcomings of traditional static measurement methods for the nonlinear stiffness of reed return mechanisms in low-frequency vibration tables, such as low efficiency, complex system composition, inability to characterize dynamic characteristics, and the unsuitability of other methods for determining the nonlinear stiffness of reed return mechanisms, this invention proposes a dynamic measurement system and method for the nonlinear stiffness of reed return mechanisms in low-frequency vibration tables.

[0006] The present invention adopts the following technical solution:

[0007] A dynamic measurement system for the nonlinear stiffness of a low-frequency vibration table with a reed return mechanism includes a programmable signal generator, a power amplifier, a low-frequency vibration table equipped with a reed return mechanism, an eddy current displacement sensor, a data acquisition card, a computer, and a software analysis module. The programmable signal generator generates an input standard sinusoidal voltage signal, which, after amplification by the power amplifier, drives the low-frequency vibration table to generate an output vibration displacement signal. The eddy current displacement sensor is used to detect the output vibration displacement signal in real time. The data acquisition card acquires the output vibration displacement signal and the input standard sinusoidal voltage signal and sends the acquired signals to the computer. The software analysis module, installed in the computer, includes a data processing unit, a transfer function calculation unit, and a nonlinear stiffness fitting unit, which are used to perform FFT analysis, transfer function calculation, and nonlinear stiffness fitting on the output vibration displacement signal and the input standard sinusoidal voltage signal, respectively.

[0008] The method for dynamic measurement of nonlinear stiffness of low-frequency vibration table reed recovery mechanism based on this system includes the following steps:

[0009] Step 1: The programmable signal generator generates the operating frequency range (ω) of the low-frequency vibration table. A ~ω B A certain set frequency ω within) i Input standard sinusoidal voltage signal t is time, U a ω i These represent the amplitude and frequency of the input standard sinusoidal voltage signal, respectively. This input standard sinusoidal voltage signal, after being amplified by a power amplifier, drives a low-frequency vibration table to generate an output vibration displacement signal. X a φ i These represent the amplitude and phase of the output vibration displacement signal, respectively.

[0010] Step 2: The eddy current displacement sensor is installed above the worktable of the low-frequency vibration table to detect the output vibration displacement signal in real time. Then, the data acquisition card collects the output vibration displacement signal and the input standard sinusoidal voltage signal, and sends the collected signal to the computer.

[0011] Step 3: The data processing unit performs FFT analysis on the output vibration displacement signal and the input standard sinusoidal voltage signal to obtain the corresponding signal amplitude and phase.

[0012] Step 4: By changing the frequency to ω i The input standard sinusoidal voltage signal amplitude is used to repeat the test process described in steps one to three to obtain the output vibration displacement signal and the amplitude and phase of the input standard sinusoidal voltage signal corresponding to different amplitudes at the same frequency.

[0013] Step 5: Select other different frequency points within the working frequency range and repeat the test process described in Steps 1 to 4 to obtain the output vibration displacement signal and the amplitude and phase of the input standard sinusoidal voltage signal corresponding to different frequencies and amplitudes.

[0014] Step Six: Based on the low-frequency vibration table output vibration displacement signal of a certain amplitude and the corresponding input standard sinusoidal voltage signal corresponding to each frequency point obtained by the test, the transfer function calculation unit compares the corresponding amplitude and phase, and fits the experimental transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to that amplitude. Then, the experimental transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to other amplitude output vibration displacement signals can be obtained based on the same method.

[0015] Step 7: Based on the electromechanical coupling equation, the theoretical transfer function of the low-frequency vibration table is simplified into a first-order inertial element in the low-frequency range. Then, based on the corner frequencies of the theoretical transfer function and the corner frequencies of the experimental transfer function corresponding to different amplitudes, the nonlinear stiffness values ​​corresponding to different amplitudes are obtained by the nonlinear stiffness fitting unit through comparison calculation. Finally, a power series expression that can accurately characterize the nonlinear stiffness characteristics is obtained by fitting, thereby realizing the dynamic measurement of the nonlinear stiffness of the reed return mechanism of the low-frequency vibration table.

[0016] The specific principle behind the calculation of nonlinear stiffness values ​​corresponding to different amplitudes and the fitting of the power series expression of nonlinear stiffness characteristics by the nonlinear stiffness fitting unit described in step seven is as follows:

[0017] (a) In general, the electromechanical coupling equation of a low-frequency vibration table can be expressed as:

[0018]

[0019] In the formula, m is the mass of the moving parts of the low-frequency vibration table, and k and c are the stiffness and damping of the reed return mechanism, respectively. , and These represent the output vibration displacement signal, output vibration velocity signal, and output vibration acceleration signal of the low-frequency vibration table, respectively. l, L, and R represent the length, inductance, and resistance of the drive coil in the moving part of the low-frequency vibration table, respectively. B is the air gap magnetic induction intensity, and u... a i and i represent the input standard sinusoidal voltage signal and the corresponding driving current of the low-frequency vibration table, respectively.

[0020] (b) Based on equation (1), the voltage-displacement transfer function corresponding to the low-frequency vibration table can be calculated:

[0021] G ( s ) = X a ( s ) U a ( s ) = Bl mLs 3 + ( mR + cL ) s 2 + [ Rc + kL + ( Bl ) 2 ] s + Rk

[0022] In the formula, and The output vibration displacement signal x is respectively a and input standard sinusoidal voltage signal u a The Laplace transform of , where s is the Laplace operator.

[0023] (c) Under normal circumstances, the damping c of the reed return mechanism and the inductance L of the drive coil in the moving parts of the low-frequency vibration table are small and can be ignored. At the same time, in the low-frequency range, the higher-order terms of the transfer function described in equation (2) are also negligible. Based on this, equation (2) can be simplified to a theoretical transfer function in the form of a first-order inertial element:

[0024]

[0025] (d) The corner frequency f of the theoretical transfer function described in equation (3) d It can be represented as:

[0026]

[0027] Since the parameters R, B, l, etc. in equation (4) can all be approximated as constants in the low-frequency range, the theoretical transfer function f described in equation (4) is... d Since the corner frequencies correspond to the corner frequencies of the experimental transfer functions for different amplitudes, the nonlinear stiffness values ​​k corresponding to different amplitudes can be approximately calculated.

[0028] (e) Considering that the stiffness of the low-frequency vibration table reed return mechanism varies with the output vibration displacement signal x a The change in stiffness is periodic and continuous, and the nonlinear stiffness can be fitted using a finite-order power series. Assuming the nonlinear stiffness... Fitting the equation using an nth-order power series, the corresponding expression is:

[0029]

[0030] In the formula, , , , These are the coefficients of each power series. When x a When taking different amplitudes as described in step four, the nonlinear stiffness values ​​corresponding to each different amplitude are calculated based on principle (d). Polynomial fitting is then performed on equation (5) to determine the stiffness. , , , By obtaining the coefficients of each order of the power series, a power series expression characterizing the nonlinear stiffness is obtained, thus completing the dynamic measurement of the nonlinear stiffness of the reed recovery mechanism of the low-frequency vibration table.

[0031] This invention's measurement system accurately measures vibration displacement signals of different amplitudes within the operating frequency range of a low-frequency vibration table. The data processing unit performs FFT analysis on the output vibration displacement signals and the input standard sinusoidal voltage signal to obtain the corresponding signal amplitude and phase. Then, the transfer function calculation unit calculates the transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to all selected amplitude output vibration displacement signals within the operating frequency range. Finally, the nonlinear stiffness fitting unit performs power series fitting to obtain a power series expression characterizing the nonlinear stiffness, thus completing the dynamic measurement of the nonlinear stiffness of the low-frequency vibration table's reed return mechanism.

[0032] Overall, the measurement system of this invention has a simple structure, a convenient operation process, and a wide range of applications. It can achieve accurate dynamic measurement of the nonlinear stiffness of the reed recovery mechanism of a low-frequency vibration table. Attached Figure Description

[0033] Figure 1 This is a diagram showing the composition of the nonlinear stiffness dynamic measurement system of the reed recovery mechanism of the low-frequency vibration table of the present invention.

[0034] Figure 2 This is a flowchart of the nonlinear stiffness dynamic measurement method for the reed recovery mechanism of the low-frequency vibration table of the present invention. Detailed Implementation

[0035] The present invention will now be described in detail with reference to the accompanying drawings:

[0036] like Figure 1 As shown, the nonlinear stiffness dynamic measurement system for the reed return mechanism of the low-frequency vibration table includes a programmable signal generator, a power amplifier, a low-frequency vibration table with the reed return mechanism installed, an eddy current displacement sensor, a data acquisition card, a computer, and a software analysis module.

[0037] The programmable signal generator generates an input standard sinusoidal voltage signal, which, after being amplified by a power amplifier, drives a low-frequency vibration table to generate an output vibration displacement signal. The eddy current displacement sensor is used to detect the output vibration displacement signal in real time. The data acquisition card is used to acquire the output vibration displacement signal and the input standard sinusoidal voltage signal, and sends the acquired signals to a computer. The software analysis module is installed in the computer and includes a data processing unit, a transfer function calculation unit, and a nonlinear stiffness fitting unit, which are used to perform FFT analysis, transfer function calculation, and nonlinear stiffness fitting on the output vibration displacement signal and the input standard sinusoidal voltage signal, respectively.

[0038] like Figure 2 As shown, the specific detection steps of the nonlinear stiffness dynamic measurement method for the reed recovery mechanism based on a low-frequency vibration table are as follows:

[0039] Step 1: Select the operating frequency range of the low-frequency vibration table (ω) A ~ω B Divide it into n segments, using the midpoint frequency ω of the segment division as the dividing point. i As the detection frequency, the frequency range of the i-th segment is:

[0040] [ ω A + ( ω B − ω A ) ( i − 1 ) n ~ ω A + ( ω B − ω A ) i n ]

[0041] Select the first frequency band [ ω A ~ ω A + ( ω B − ω A ) / n ] The intermediate frequency point ω1 is used as the detection frequency, and the low-frequency vibration table operating frequency range (ω) is generated by the programmable signal generator. A ~ω B The input standard sinusoidal voltage signal with a set frequency ω1 is set within the range. t is time, U a ω1 and ω2 represent the amplitude and frequency of the input standard sinusoidal voltage signal, respectively. This input standard sinusoidal voltage signal, after being amplified by a power amplifier, drives a low-frequency vibration table to generate an output vibration displacement signal. X a φ1 and φ2 represent the amplitude and phase of the output vibration displacement signal, respectively.

[0042] Step 2: The eddy current displacement sensor is installed above the worktable of the low-frequency vibration table to detect the output vibration displacement signal in real time. Then, the data acquisition card collects the output vibration displacement signal and the input standard sinusoidal voltage signal, and sends the collected signal to the computer.

[0043] Step 3: The data processing unit performs FFT analysis on the output vibration displacement signal and the input standard sinusoidal voltage signal to obtain the corresponding signal amplitude and phase.

[0044] Step 4: By changing the amplitude of the input standard sinusoidal voltage signal with frequency ω1, repeat the test process described in steps 1 to 3 to obtain the output vibration displacement signal and the amplitude and phase of the input standard sinusoidal voltage signal corresponding to different amplitudes at the same frequency.

[0045] Step 5: Select other different frequency points within the working frequency range and repeat the test process described in Steps 1 to 4 to obtain the output vibration displacement signal and the amplitude and phase of the input standard sinusoidal voltage signal corresponding to different frequencies and amplitudes.

[0046] Step Six: Based on the low-frequency vibration table output vibration displacement signal of a certain amplitude and the corresponding input standard sinusoidal voltage signal corresponding to each frequency point obtained by the test, the transfer function calculation unit compares the corresponding amplitude and phase, and fits the experimental transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to that amplitude. Then, the experimental transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to other amplitude output vibration displacement signals can be obtained based on the same method.

[0047] Step 7: Based on the electromechanical coupling equation, the theoretical transfer function of the low-frequency vibration table is simplified into a first-order inertial element in the low-frequency range. Then, based on the corner frequencies of the theoretical transfer function and the corner frequencies of the experimental transfer function corresponding to different amplitudes, the nonlinear stiffness values ​​corresponding to different amplitudes are obtained by the nonlinear stiffness fitting unit through comparison calculation. Finally, a power series expression that can accurately characterize the nonlinear stiffness characteristics is obtained by fitting, thereby realizing the dynamic measurement of the nonlinear stiffness of the reed return mechanism of the low-frequency vibration table.

[0048] The specific principle behind the calculation of nonlinear stiffness values ​​corresponding to different amplitudes and the fitting of the power series expression of nonlinear stiffness characteristics by the nonlinear stiffness fitting unit described in step seven is as follows:

[0049] (a) In general, the electromechanical coupling equation of a low-frequency vibration table can be expressed as:

[0050]

[0051] In the formula, m is the mass of the moving parts of the low-frequency vibration table, and k and c are the stiffness and damping of the reed return mechanism, respectively. , and These represent the output vibration displacement signal, output vibration velocity signal, and output vibration acceleration signal of the low-frequency vibration table, respectively. l, L, and R represent the length, inductance, and resistance of the drive coil in the moving part of the low-frequency vibration table, respectively. B is the air gap magnetic induction intensity, and u... a i and i represent the input standard sinusoidal voltage signal and the corresponding driving current of the low-frequency vibration table, respectively.

[0052] (b) Based on equation (7), the voltage-displacement transfer function corresponding to the low-frequency vibration table can be calculated:

[0053] G ( s ) = X a ( s ) U a ( s ) = Bl mLs 3 + ( mR + cL ) s 2 + [ Rc + kL + ( Bl ) 2 ] s + Rk

[0054] In the formula, and These are vibration displacement signals x a and standard sinusoidal voltage signal u a The Laplace transform of , where s is the Laplace operator.

[0055] (c) Under normal circumstances, the damping c of the reed return mechanism and the inductance L of the drive coil in the moving parts of the low-frequency vibration table are small and can be ignored. At the same time, in the low-frequency range, the higher-order terms of the transfer function described in equation (8) are also negligible. Based on this, equation (8) can be simplified to a theoretical transfer function in the form of a first-order inertial element:

[0056]

[0057] (d) The theoretical transfer function corner frequency f described in equation (9) d It can be represented as:

[0058] Since the parameters R, B, l, etc. in equation (10) can all be approximated as constants in the low-frequency range, the theoretical transfer function f described in equation (10) is... d Since the corner frequencies correspond to the corner frequencies of the experimental transfer functions for different amplitudes, the nonlinear stiffness values ​​k corresponding to different amplitudes can be approximately calculated.

[0059] (e) Considering that the stiffness of the low-frequency vibration table reed return mechanism varies with the output vibration displacement signal x a The change in stiffness is periodic and continuous, and the nonlinear stiffness can be fitted using a finite-order power series. Assuming the nonlinear stiffness... Fitting the equation using an nth-order power series, the corresponding expression is:

[0060]

[0061] In the formula, , , , These are the coefficients of each power series. When x a When taking different amplitudes as described in step four, the nonlinear stiffness values ​​corresponding to each different amplitude are calculated based on principle (d). Polynomial fitting is then performed on equation (11) to determine the stiffness. , , , By obtaining the coefficients of each order of the power series, a power series expression characterizing the nonlinear stiffness is obtained, thus completing the dynamic measurement of the nonlinear stiffness of the reed recovery mechanism of the low-frequency vibration table.

[0062] This invention's measurement system accurately measures vibration displacement signals of different amplitudes within the operating frequency range of a low-frequency vibration table. The data processing unit performs FFT analysis on the output vibration displacement signals and the input standard sinusoidal voltage signal to obtain the corresponding signal amplitude and phase. Then, the transfer function calculation unit calculates the transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to all selected amplitude output vibration displacement signals within the operating frequency range. Finally, the nonlinear stiffness fitting unit performs power series fitting to obtain a power series expression characterizing the nonlinear stiffness, thus completing the dynamic measurement of the nonlinear stiffness of the low-frequency vibration table's reed return mechanism.

[0063] Overall, the measurement system of this invention has a simple structure, a convenient operation process, and a wide range of applications. It can achieve accurate dynamic measurement of the nonlinear stiffness of the reed recovery mechanism of a low-frequency vibration table.

[0064] The descriptions in the embodiments are merely examples of implementations of the inventive concept. The scope of protection of this invention should not be considered as limited to the specific forms described in the embodiments. The scope of protection of this invention also extends to equivalent technical means that can be conceived by those skilled in the art based on the inventive concept.

Claims

1. A dynamic measurement system for the nonlinear stiffness of a low-frequency vibration table with a reed return mechanism, comprising a programmable signal generator, a power amplifier, a low-frequency vibration table equipped with a reed return mechanism, an eddy current displacement sensor, a data acquisition card, a computer, and a software analysis module. The programmable signal generator generates an input standard sinusoidal voltage signal, which, after amplification by the power amplifier, drives the low-frequency vibration table to generate an output vibration displacement signal. The eddy current displacement sensor is used to detect the output vibration displacement signal in real time. The data acquisition card acquires the output vibration displacement signal and the input standard sinusoidal voltage signal, and sends the acquired signals to the computer. The software analysis module, installed in the computer, includes a data processing unit, a transfer function calculation unit, and a nonlinear stiffness fitting unit, which are used to perform FFT analysis, transfer function calculation, and nonlinear stiffness fitting on the output vibration displacement signal and the input standard sinusoidal voltage signal, respectively.

2. A method for dynamic measurement of nonlinear stiffness of a low-frequency vibration table reed recovery mechanism based on this system, characterized by: It includes the following steps: Step 1: The programmable signal generator generates the operating frequency range (ω) of the low-frequency vibration table. A ~ω B A certain set frequency ω within) i Input standard sinusoidal voltage signal t is time, U a ω i These represent the amplitude and frequency of the input standard sinusoidal voltage signal, respectively. This input standard sinusoidal voltage signal, after being amplified by a power amplifier, drives a low-frequency vibration table to generate an output vibration displacement signal. X a φ i These represent the amplitude and phase of the output vibration displacement signal, respectively. Step 2: The eddy current displacement sensor is installed above the worktable of the low-frequency vibration table to detect the output vibration displacement signal in real time. Then, the data acquisition card collects the output vibration displacement signal and the input standard sinusoidal voltage signal, and sends the collected signal to the computer. Step 3: The data processing unit performs FFT analysis on the output vibration displacement signal and the input standard sinusoidal voltage signal to obtain the corresponding signal amplitude and phase. Step 4: By changing the frequency to ω i The input standard sinusoidal voltage signal amplitude is used to repeat the test process described in steps one to three to obtain the output vibration displacement signal and the amplitude and phase of the input standard sinusoidal voltage signal corresponding to different amplitudes at the same frequency. Step 5: Select other different frequency points within the working frequency range and repeat the test process described in Steps 1 to 4 to obtain the output vibration displacement signal and the amplitude and phase of the input standard sinusoidal voltage signal corresponding to different frequencies and amplitudes. Step Six: Based on the low-frequency vibration table output vibration displacement signal of a certain amplitude and the corresponding input standard sinusoidal voltage signal corresponding to each frequency point obtained by the test, the transfer function calculation unit compares the corresponding amplitude and phase, and fits the experimental transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to that amplitude. Then, the experimental transfer function in the form of the first-order inertial element of the low-frequency vibration table corresponding to other amplitude output vibration displacement signals can be obtained based on the same method. Step 7: Based on the electromechanical coupling equation, the theoretical transfer function of the low-frequency vibration table is simplified into a first-order inertial element in the low-frequency range. Then, based on the corner frequencies of the theoretical transfer function and the corner frequencies of the experimental transfer function corresponding to different amplitudes, the nonlinear stiffness values ​​corresponding to different amplitudes are obtained by the nonlinear stiffness fitting unit through comparison calculation. Finally, a power series expression that can accurately characterize the nonlinear stiffness characteristics is obtained by fitting, thereby realizing the dynamic measurement of the nonlinear stiffness of the reed return mechanism of the low-frequency vibration table. The specific principle behind the calculation of nonlinear stiffness values ​​corresponding to different amplitudes and the fitting of the power series expression of nonlinear stiffness characteristics by the nonlinear stiffness fitting unit described in step seven is as follows: (a) In general, the electromechanical coupling equation of a low-frequency vibration table can be expressed as: In the formula, m is the mass of the moving parts of the low-frequency vibration table, and k and c are the stiffness and damping of the reed return mechanism, respectively. , and These represent the output vibration displacement signal, output vibration velocity signal, and output vibration acceleration signal of the low-frequency vibration table, respectively. l, L, and R represent the length, inductance, and resistance of the drive coil in the moving part of the low-frequency vibration table, respectively. B is the air gap magnetic induction intensity, and u... a i and i represent the input standard sinusoidal voltage signal and the corresponding driving current of the low-frequency vibration table, respectively. (b) Based on equation (1), the voltage-displacement transfer function corresponding to the low-frequency vibration table can be calculated: In the formula, X a (s) and U a (s) represent the output vibration displacement signal x a and input standard sinusoidal voltage signal u a The Laplace transform of , where s is the Laplace operator. (c) Under normal circumstances, the damping c of the reed return mechanism and the inductance L of the drive coil in the moving parts of the low-frequency vibration table are small and can be ignored. At the same time, in the low-frequency range, the higher-order terms of the transfer function described in equation (2) are also negligible. Based on this, equation (2) can be simplified to a theoretical transfer function in the form of a first-order inertial element: (d) The corner frequency f of the theoretical transfer function described in equation (3) d It can be represented as: Since the parameters R, B, l, etc. in equation (4) can all be approximated as constant values ​​in the low-frequency range, the theoretical transfer function f described in equation (4) is... d Since the corner frequencies correspond to the corner frequencies of the experimental transfer functions for different amplitudes, the nonlinear stiffness values ​​k corresponding to different amplitudes can be approximately calculated. (e) Considering that the stiffness of the low-frequency vibration table reed return mechanism varies with the output vibration displacement signal x a The change in stiffness is periodic and continuous, and the nonlinear stiffness can be fitted using a finite-order power series. Assume the nonlinear stiffness k(x) a The approximation is performed using an nth-order power series, and the corresponding expression is: In the formula, These are the coefficients of each power series. When x a When taking different amplitudes as described in step four, the nonlinear stiffness values ​​corresponding to each different amplitude are calculated based on principle (d). Polynomial fitting is then performed on equation (5) to determine the stiffness. By obtaining the coefficients of each order of the power series, a power series expression characterizing the nonlinear stiffness is obtained, thus completing the dynamic measurement of the nonlinear stiffness of the reed recovery mechanism of the low-frequency vibration table.

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