A stiffness identification calculation method for an electric cylinder based on natural frequency

Abstract

The application discloses a kind of electric cylinder stiffness identification calculation methods based on inherent frequency, comprising the following steps: establishing electric cylinder system dynamics model, based on theoretical mechanics, material mechanics and Hertz theory calculation push rod stiffness, ball nut restoring force, screw stiffness and bearing restoring force and establish dynamics differential equation;Simulation solves the inherent frequency of electric cylinder system under different axial loads;Correct electric cylinder model, using improved Berman-Baruch method is based on inherent frequency and modal identification to carry out dynamics inverse solution, and identify stiffness and other physical parameters.The application can effectively improve the accuracy and convenience of electric cylinder stiffness identification, and provide support for electric cylinder stiffness optimization design.

Description

Technical Field

[0001] This invention belongs to the field of electric cylinder dynamics modeling and analysis technology, and in particular, it is a method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency. Background Technology

[0002] Electric cylinders are widely used in automated production lines, machine tools and processing equipment, automated packaging lines, robotics, the automotive industry, and medical equipment. Stiffness is one of the most important indicators for evaluating their overall performance. Existing research on electric cylinders mainly focuses on the precision control of electric cylinders in servo systems, while research on the calculation and identification of electric cylinder stiffness is relatively limited. Electric cylinders mainly consist of servo motors, reduction mechanisms, ball screw pairs, bearings, etc., and are typical nonlinear systems, making it difficult to accurately derive their stiffness through theoretical modeling. Therefore, it is of great significance to develop an algorithm that can accurately and conveniently derive the actual stiffness of the coupled components of an electric cylinder. Summary of the Invention

[0003] The purpose of this invention is to provide a method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency, addressing the problems existing in the prior art.

[0004] The technical solution to achieve the objective of this invention is as follows: On the one hand, a method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency is provided, the method comprising the following steps:

[0005] Step 1: Establish the differential equations of dynamics for the electric cylinder;

[0006] Step 2: Simulate and solve for the natural frequencies of the electric cylinder system under different axial loads;

[0007] Step 3: Solve for the orthogonal modes of the electric cylinder system using the assumed modal method;

[0008] Step 4: Identify and correct stiffness parameters based on the natural frequency identification using the improved Berman-Baruch method.

[0009] Furthermore, the differential equation for the dynamics of the electric cylinder mentioned in step 1 is specifically as follows:

[0010]

[0011] In the formula, m s m N and m T These are the masses of the lead screw, nut, and push rod, respectively. and (x) S y S , z S These represent the lead screw acceleration, lead screw velocity, and lead screw displacement, respectively. and (x) w yw , z w These represent the velocity and displacement of the screw's center of mass, respectively. and (x) N y N , z N These represent the nut acceleration, nut velocity, and nut displacement, respectively. and (x) T y T , z T ) represent the acceleration of the push rod's center of mass, the velocity of the push rod's center of mass, and the displacement of the push rod's center of mass, respectively; (F Nx F Ny ), (F Ax F Ay F Az ) and (F Dx F Dy ) represent the restoring force of the lead screw nut, the restoring force of the angular contact ball bearing, and the restoring force of the deep groove ball bearing, respectively; (k Sx k Sy k Sz ) and (k Tx k Ty k Tz ) represent the stiffness of the lead screw and the push rod, respectively; C represents the viscous damping caused by the ball screw, ball bearings, and push rod; F represents the stiffness of the lead screw and the push rod, respectively. X (t), F Y (t) and F Z (i) are the loads on the x, y, and z axes, respectively, and their expressions are F. X (t)=F Y (t)=F Z (t)=F0sinΩt, where F0 is the excitation amplitude and Ω is the excitation frequency.

[0012] Furthermore, step 1 also includes calculating the push rod stiffness, ball nut restoring force, screw stiffness, and bearing restoring force based on theoretical mechanics, mechanics of materials, and Hertz contact theory, specifically including:

[0013] (1) When the push rod of the servo electric cylinder is subjected to an external axial load Fa, it undergoes elastic deformation in the axial direction. Neglecting the influence of the bolt group on the axial displacement of the push rod, the total deformation is the sum of the axial deformations of the three segments l1, l2 and l3 of the push rod, and the push rod stiffness K is obtained. l for:

[0014]

[0015] In the formula, D1, D2, D3, and D4 are the diameters of each section of the push rod, and a and b are the flange thicknesses; E l E is the elastic modulus of the push rod. l=2.05×10⁵ MPa; A1, A2, and A3 are the cross-sectional areas of segments l1, l2, and l3, respectively;

[0016] (2) Calculate the distances between the center of curvature of the screw raceway and the center of curvature of the nut raceway at the ball contact point before and after the ball screw pair is loaded. Based on Hertz contact theory, the restoring force functions of the ball and nut in the x, y, and z directions are as follows:

[0017]

[0018] In the formula, K is the contact stiffness between the ball and the raceway of the lead screw or nut. The position angle of the ball. The contact angle of the ball. γ is the deformation of the ball, and γ is the lead angle of the ball screw;

[0019] (3) When a radial force is applied to the lead screw shaft, the formula for calculating the displacement of the lead screw's center of mass caused by the elastic deformation of the lead screw shaft along the x and y directions is:

[0020]

[0021]

[0022] In the formula, w x Let θ represent the axial and bending deformation of the lead screw shaft at a certain moment. i Let l be the gradient of the i-th element of the lead screw axis, l be the lead screw length, A be the axial cross-sectional area of ​​the lead screw, and ρ be the density of the lead screw material.

[0023] Based on the force-displacement relationship of the Euler-Bernoulli beam, the formulas for calculating the stiffness of the lead screw shaft in the x and y directions at a certain position on the push rod are as follows:

[0024]

[0025]

[0026] (4) Considering the nonlinear contact behavior between the ball and the raceway, the restoring force calculation formula for the left and right angular contact ball bearings is as follows:

[0027]

[0028] in, and The contact forces θ and θ are the left and right angular contact ball bearings, respectively. Ai It is the position angle of the i-th ball in the angular contact ball bearing. and These are left and right angular contact ball bearings, respectively.

[0029] The formulas for calculating the restoring force function of a deep groove ball bearing in the x and y directions are as follows:

[0030]

[0031] Among them, K D Let θ be the stiffness of the deep groove sphere, where θ is the stiffness of the sphere. Di Let be the position angle of the i-th ball in the deep groove ball bearing. This is the radial distance between the centers of the raceways of the deep groove ball bearing.

[0032] Furthermore, step 2 involves simulating and solving for the natural frequencies of the electric cylinder system under different axial loads, specifically including:

[0033] The matrix form of the differential equation of the electric cylinder's dynamics is decomposed into a mass matrix M and a stiffness matrix K:

[0034] M = diag(m) S ,0,m N ,m T ,m S ,0,m N ,m T ,m S ,0,m N ,m T )

[0035]

[0036] Solve its characteristic equation (K-ω) 2 The non-zero solutions of M)u=0 are:

[0037] Δ(ω 2 )=det(K-ω 2 M)=0

[0038] Expanding it yields ω 2 nth degree algebraic equation:

[0039] ω 2n +a1ω 2(n-1) +a2ω 2(n-2) +…+a n-1 ω 2 +a n =0

[0040] This nth-degree algebraic equation has n roots. These roots are called eigenvalues;

[0041] The square root of all eigenvalues ​​ω r As the inherent frequency of the system.

[0042] Furthermore, step 3, which involves using the assumed modal method to solve for the orthogonal modes of the electric cylinder system, specifically includes the following process:

[0043] Step 3-1, construct the function that the eigenvalues ​​satisfy:

[0044]

[0045] In the formula, M, K, and C are the corrected system mass, stiffness, and damping matrices, respectively; ne is the measured complex mode number of the system, and ne≤n, where n is the degree of freedom of the structure; λ i Φ i These are the eigenvalue of order ith and the identified complex mode, respectively.

[0046] Step 3-2: Obtain higher-order eigenvalues ​​using the generalized eigenvalue solution method, such that the higher-order eigenvalue λ... ai and complex modes Φ ai satisfy:

[0047]

[0048] Step 3-3: Combining eigenvalues ​​and higher-order eigenvalues, use the least squares method to substitute the values ​​into the function constructed in step 3-1 to solve for M. -1 K;

[0049] Steps 3-4, using the solved M -1 K is obtained by the following formula for the real mode. and eigenvalue matrix Λ:

[0050]

[0051] in, For the ith-th order real mode,

[0052] Furthermore, step 4 specifically includes:

[0053] Step 4-1: Establish the objective function ε, which is based on the system quality matrix. M :

[0054]

[0055] In the formula, M A Let I be the finite element mass matrix, where I is the unit diagonal matrix and M is the corrected mass matrix.

[0056] Step 4-2: Minimize the objective function using the Lagrange multiplier method, and establish the Lagrange function L. M ;

[0057]

[0058] In the formula, α ijIt is a Lagrange mass multiplier. The real modes obtained above;

[0059] Step 4-3: Take the partial derivatives with respect to each element of the mass matrix M. Then we have:

[0060]

[0061] In the formula, α is α ij The Lagrange multiplier matrix formed;

[0062] Step 4-4: Substitute the formula from Step 4-3 into the constraint conditions. Then the corrected mass matrix is:

[0063]

[0064] in,

[0065] Step 4-5: Establish the objective function based on the system mass matrix and stiffness matrix. Repeat steps 4-2 and 4-3 to obtain:

[0066]

[0067] Step 4-6: Substitute the formula from step 4-5 into the constraint conditions. and The corrected stiffness matrix is ​​then:

[0068]

[0069] Furthermore, step 3 also includes:

[0070] Steps 3-5 involve normalizing the real modes.

[0071] Furthermore, steps 3-5 specifically include the following processes:

[0072] Step 3-5-1, define the unknown matrix Λ2;

[0073] Step 3-5-2, Define the modified real modes

[0074] Step 3-5-3, establish the objective function ε composed of the unknown matrix Λ2:

[0075]

[0076] in,

[0077] Steps 3-5-4: Minimize the objective function ε to obtain the normalized result.

[0078] Then repeat steps 4-1 to 4-6 to obtain the corrected mass matrix and stiffness matrix.

[0079] Compared with the prior art, the significant advantages of this invention are:

[0080] 1) This invention considers the coupling effect of the functional and structural components of the electric cylinder, performs dynamic modeling of the system, and then solves for the relevant modal parameters that reflect the characteristics of the system.

[0081] 2) This invention introduces experimental data to correct physical parameters such as model stiffness, so that the dynamic characteristics of the corrected dynamic model are consistent with the experimental data.

[0082] 3) This invention performs dynamic modeling and stiffness parameter identification of the electric cylinder system, which is of great significance for optimizing the stiffness and improving the load-bearing performance of the electric cylinder.

[0083] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0084] Figure 1 This is a flowchart of a method for calculating the stiffness of an electric cylinder based on its natural frequency, as shown in one embodiment.

[0085] Figure 2 This is a schematic diagram of the dynamic model of an electric cylinder system in one embodiment.

[0086] Figure 3 This is a comparison chart of push rod stiffness results in one embodiment.

[0087] Figure 4 This is a comparison chart of the stiffness results of the lead screw pair in one embodiment.

[0088] Figure 5 This is a comparison chart of the overall system stiffness results in one embodiment. Detailed Implementation

[0089] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0090] It should be noted that if the embodiments of the present invention involve directional indicators (such as up, down, left, right, front, back, etc.), the directional indicators are only used to explain the relative positional relationship and movement of the components in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indicators will also change accordingly.

[0091] Furthermore, if the embodiments of this invention involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

[0092] In one embodiment, combined Figure 1 A method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency is provided, the method comprising:

[0093] Step 1: Establish the differential equations of dynamics for the electric cylinder;

[0094] Step 2: Simulate and solve for the natural frequencies of the electric cylinder system under different axial loads;

[0095] Step 3: Solve for the orthogonal modes of the electric cylinder system using the assumed modal method;

[0096] Step 4: Identify and correct stiffness parameters based on the natural frequency identification using the improved Berman-Baruch method.

[0097] Furthermore, in one embodiment, the dynamic model of the electric cylinder system in step 1 is as follows: Figure 2 As shown, the specific differential equation for the dynamics of the electric cylinder is:

[0098]

[0099] In the formula, m S m N and m T These are the masses of the lead screw, nut, and push rod, respectively. and (x) S y S , z S These represent the lead screw acceleration, lead screw velocity, and lead screw displacement, respectively. and (x) w y w , z w These represent the velocity and displacement of the screw's center of mass, respectively. and (x) N y N , z N These represent the nut acceleration, nut velocity, and nut displacement, respectively. and (x) T yT , z T ) represent the acceleration of the push rod's center of mass, the velocity of the push rod's center of mass, and the displacement of the push rod's center of mass, respectively; (F Nx F Ny ), (F Ax F Ay F Az ) and (F Dx F Dy ) represent the restoring force of the lead screw nut, the restoring force of the angular contact ball bearing, and the restoring force of the deep groove ball bearing, respectively; (k Sx k Sy k Sz ) and (k Tx k Ty k Tz ) represent the stiffness of the lead screw and the push rod, respectively; C represents the viscous damping caused by the ball screw, ball bearings, and push rod; F represents the stiffness of the lead screw and the push rod, respectively. X (t), F Y (t) and F Z (t) represent the loads along the x, y, and z axes, respectively, and the load expression is F. X (t)=F Y (t)=F Z (t)=F0sinΩt, where F0 is the excitation amplitude and Ω is the excitation frequency.

[0100] Furthermore, step 1 also includes calculating the push rod stiffness, ball nut restoring force, screw stiffness, and bearing restoring force based on theoretical mechanics, mechanics of materials, and Hertz contact theory, specifically including:

[0101] When the push rod of the servo electric cylinder is subjected to an external axial load Fa, it will undergo elastic deformation in the axial direction. Ignoring the influence of the bolt group on the axial displacement of the push rod, the axial deformations δ1, δ2, and δ3 of push rod segments l1, l2, and l3 are respectively:

[0102]

[0103] In the formula, E l E is the elastic modulus of the push rod. l = 2.05 × 10⁵ MPa, where A1, A2, and A3 are the cross-sectional areas of segments l1, l2, and l3, respectively. The total deformation is the sum of the axial deformations of the three segments, from which the axial tensile and compressive stiffness of the push rod can be obtained:

[0104]

[0105] In the formula, D1, D2, D3 and D4 are the diameters of each section of the push rod, and a and b are the flange thicknesses.

[0106] Furthermore, when the ball screw pair is unloaded, the distance O between the center of curvature of the screw raceway and the center of curvature of the nut raceway at the i-th ball contact point is... N O S The calculation formula is:

[0107] O N O S =r N +r S -2r b

[0108] Where, r N r S r b The raceways are the radii of curvature of the nut raceway, the radii of curvature of the screw raceway, and the ball radius. When the ball screw pair is under load, the distance between the two centers of curvature on the left side of the screw and nut becomes 0. N O′ Sleft Its expression is:

[0109]

[0110] Where α0 represents the initial contact angle of the nut, Y N Let represent the displacement of the nut in the y-direction. The position angle of a certain ball. σ is the angle between adjacent balls, and σ is the position angle of the first ball. δ represents the radial displacement of the i-th ball in the nut. 扭 This represents the lateral displacement in the normal plane caused by the torsional bending deformation of the ball screw under load. This indicates the actual lateral displacement of the nut's center of mass.

[0111] radial displacement of the inner ball of the nut The expression is:

[0112]

[0113] Where, ΔL=ZL P / N is the axial length of adjacent balls, Z is the number of revolutions in the nut, N is the number of balls, and the screw pitch is L. P .

[0114] The total contact deformation of the i-th ball on the left side of the lead screw nut in the raceway contact direction The expression is:

[0115]

[0116] In addition, the formula for calculating the contact angle of the ball bearings on the left side of the lead screw nut is:

[0117]

[0118] Based on geometric relationships and Hertzian contact theory, the restoring force functions of the lead screw nut along the x, y, and z axes can be expressed as:

[0119]

[0120] In the formula, K is the contact stiffness between the ball and the raceway of the lead screw or nut. The position angle of the ball. and These are the contact angles of the left and right corner contact balls, respectively. and These represent the deformation of the left and right angular contact balls, respectively, and γ is the lead angle of the ball screw.

[0121] The formula for calculating the displacement of the screw's center of mass caused by the elastic deformation of the screw shaft along the x and y directions when a radial force is applied is as follows:

[0122]

[0123]

[0124] In the formula, x wi y wi These represent the displacements of the screw's center of mass caused by the elastic deformation of the screw shaft along the x and y directions, respectively; w x Let w represent the axial deformation and bending deformation of the lead screw shaft in the x-direction at a certain moment. y Let θ represent the axial deformation and bending deformation of the lead screw shaft in the y-direction at a certain moment. i Let l be the gradient of the i-th element of the lead screw axis, l be the lead screw length, A be the axial cross-sectional area of ​​the lead screw, and ρ be the density of the lead screw material.

[0125] From the force-displacement relationship of the Euler-Bernoulli beam, the formulas for calculating the stiffness of the lead screw shaft in the x and y directions at a certain position of the push rod can be obtained as follows:

[0126]

[0127]

[0128] Considering the nonlinear contact behavior between the ball and the raceway, the formula for calculating the restoring force of angular contact ball bearings (both left and right contact) is as follows:

[0129]

[0130] in, These are the restoring forces in the x, y, and z directions, respectively. and The contact forces θ and θ are the left and right angular contact ball bearings, respectively. AiIt is the position angle of the i-th ball in the angular contact ball bearing. and These are left and right angular contact ball bearings, respectively; Nb is the number of angular contact balls.

[0131] The formulas for calculating the restoring force function of a deep groove ball bearing in the x and y directions are as follows:

[0132]

[0133] in, K represents the restoring force of the deep groove ball bearing in the x and y directions, respectively. D For the stiffness of the deep groove sphere, θ Di Let be the position angle of the i-th ball in the deep groove ball bearing. This is the radial distance between the centers of the raceways of the deep groove ball bearing.

[0134] Furthermore, in one embodiment, step 2 involves solving for the natural frequency of the electric cylinder system. The matrix form of the power consists of mass and stiffness matrices, and its characteristic equation (K-ω) is solved. 2 The non-zero solutions of M)u=0, i.e.

[0135] Δ(ω 2 )=det(K-ω 2 M)=0

[0136] In the formula Δ(ω) 2 This is called the characteristic determinant, where:

[0137] M = diag(m) s ,0,m N m T m s ,0,m N m T m S ,0,m N m T )

[0138]

[0139] Expanding the differential equation yields ω. 2 nth degree algebraic equation:

[0140] ω 2n +a1ω 2(n-1) +a2ω 2(n-2) +…+a n-1 ω 2 +a n =0

[0141] This nth-degree algebraic equation has n roots. These roots are called eigenvalues, and their square roots ωr (r = 1, 2, ..., n) are called the natural frequencies of the system.

[0142] Furthermore, in one embodiment, step 3, which involves solving for the orthogonal modes of the electric cylinder system using the assumed modal method, specifically includes the following process:

[0143] Step 3-1, construct the function that the eigenvalues ​​satisfy:

[0144]

[0145] In the formula, M, K, and C are the corrected system mass, stiffness, and damping matrices, respectively; ne is the measured complex mode number of the system, and ne≤n, where n is the degree of freedom of the structure; λ i Φ i These are the eigenvalue of order ith and the identified complex mode, respectively.

[0146] Step 3-2: Obtain higher-order eigenvalues ​​using the generalized eigenvalue solution method, such that the higher-order eigenvalue λ... ai and complex modes Φ ai satisfy:

[0147]

[0148] Step 3-3: Combining eigenvalues ​​and higher-order eigenvalues, use the least squares method to substitute the values ​​into the function constructed in step 3-1 to solve for M. -1 K;

[0149] Steps 3-4, using the solved M -1 K is obtained by the following formula for the real mode. and eigenvalue matrix Λ:

[0150]

[0151] in, For the ith-th order real mode,

[0152] Furthermore, in one embodiment, step 4 specifically includes:

[0153] Step 4-1: Establish the objective function ε, which is based on the system quality matrix. M :

[0154]

[0155] In the formula, M A Let I be the finite element mass matrix, where I is the unit diagonal matrix and M is the corrected mass matrix.

[0156] Step 4-2: Minimize the objective function using the Lagrange multiplier method, and establish the Lagrange function L. M ;

[0157]

[0158] In the formula, α ij It is a Lagrange mass multiplier. The real modes obtained above;

[0159] Step 4-3: Take the partial derivatives with respect to each element of the mass matrix M. Then we have:

[0160]

[0161] In the formula, α is α ij The Lagrange multiplier matrix formed;

[0162] Step 4-4: Substitute the formula from Step 4-3 into the constraint conditions. Then the corrected mass matrix is:

[0163]

[0164] in,

[0165] Step 4-5: Establish the objective function based on the system mass matrix and stiffness matrix. Repeat steps 4-2 and 4-3 to obtain:

[0166]

[0167] Step 4-6: Substitute the formula from step 4-5 into the constraint conditions. and The corrected stiffness matrix is ​​then:

[0168]

[0169] Furthermore, in one embodiment, step 3 further includes:

[0170] Steps 3-5 involve normalizing the real modes.

[0171] Furthermore, in one embodiment, steps 3-5 specifically include:

[0172] Step 3-5-1: Define the unknown matrix Λ2, where Λ2 is n×n and n is the degree of freedom of the structure;

[0173] Step 3-5-2, Define the modified real modes

[0174] Step 3-5-3, establish the objective function ε composed of the unknown matrix Λ2:

[0175]

[0176] in,

[0177] Steps 3-5-4: Minimize the objective function ε to obtain the normalized result.

[0178] Then repeat steps 4-1 to 4-6 to obtain the corrected mass matrix and stiffness matrix.

[0179] In one embodiment, a system for identifying and calculating the stiffness of an electric cylinder based on its natural frequency is provided, the system comprising:

[0180] The first module is used to establish the differential equations of electric cylinder dynamics;

[0181] The second module is used to simulate and solve the natural frequencies of the electric cylinder system under different axial loads.

[0182] The third module is used to solve the orthogonal modes of the electric cylinder system using the assumed mode method.

[0183] The fourth module is used to identify and correct stiffness parameters based on the natural frequency identification using the improved Berman-Baruch method.

[0184] Specific limitations regarding the electric cylinder stiffness identification and calculation system based on natural frequency can be found in the limitations of the electric cylinder stiffness identification and calculation method based on natural frequency mentioned above, and will not be repeated here. Each module in the above-mentioned electric cylinder stiffness identification and calculation system based on natural frequency can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device in hardware form, or stored in the memory of a computer device in software form, so that the processor can call and execute the corresponding operations of each module.

[0185] As a specific example, the invention will be further described in detail in one embodiment.

[0186] This embodiment verifies the electric cylinder stiffness identification method based on natural frequency of the present invention by comparing it with the traditional electric cylinder stiffness calculation model. The results are compared by selecting the key functional component of the electric cylinder, the lead screw pair, the structural component, the push rod, and the overall system. The results are as follows: Figure 3 , Figure 4 and Figure 5As shown in the figure, compared with the finite element simulation results, the maximum errors of the three stiffness parameters of this invention are 6.7%, 28.6%, and 39.7%, respectively, and the errors are greatest in the range of 5 to 40 kN. Various errors will occur in the manufacturing and assembly process of the lead screw pair. Typical geometric errors include pitch error, profile error, ball radius error, and waviness. The deflection error generated during the assembly of the nut and the lead screw shaft will also affect the stiffness modeling. Therefore, this invention introduces dynamic modeling and experimental data to correct the stiffness matrix. As shown in the figure, compared with the finite element results, the maximum errors of the three stiffness parameters are 3.4%, 13%, and 20%, respectively. Thus, this invention greatly improves the accuracy of stiffness calculation for the electric cylinder system.

[0187] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention without departing from its spirit and scope should be included within the protection scope of the present invention.

Claims

1. A method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency, characterized in that, The method includes the following steps: Step 1: Establish the differential equations of dynamics for the electric cylinder; Step 2: Simulate and solve for the natural frequencies of the electric cylinder system under different axial loads; Step 3: Solve for the orthogonal modes of the electric cylinder system using the assumed modal method; Step 4: Identify and correct stiffness parameters based on the natural frequency identification using the improved Berman-Baruch method; The specific differential equation of the electric cylinder dynamics mentioned in step 1 is as follows: In the formula, , and The masses of the lead screw, nut, and push rod are respectively; , , ), ( , , )and( , , These represent the lead screw acceleration, lead screw velocity, and lead screw displacement, respectively. , , )and( , , These represent the velocity and displacement of the screw's center of mass, respectively. , , ), ( , , )and( , , These represent the nut acceleration, nut velocity, and nut displacement, respectively. , , ), ( , , )and( , , These represent the acceleration, velocity, and displacement of the push rod's center of mass, respectively. , ), ( , , )and( , These represent the restoring force of the lead screw nut, the restoring force of the angular contact ball bearing, and the restoring force of the deep groove ball bearing, respectively. , , )and( , , These represent the stiffness of the lead screw and the push rod, respectively. Viscous damping caused by the ball screw, ball bearings, and push rod; , and The loads are respectively applied to the x, y, and z axes, and their expressions are as follows: ,in, To excite the amplitude, For excitation frequency; Step 4 specifically includes: Step 4-1: Establish the objective function with the system quality matrix as the main component. : In the formula, Let I be the finite element mass matrix, and let I be the unit diagonal matrix. This is the corrected mass matrix; Step 4-2: Minimize the objective function using the Lagrange multiplier method, and establish the Lagrange function L. M ; In the formula, It is a Lagrange mass multiplier. The real modes obtained above; Step 4-3: Take the partial derivatives with respect to each element of the mass matrix M. Then we have: In the formula, for The Lagrange multiplier matrix formed; Step 4-4: Substitute the formula from Step 4-3 into the constraint conditions. Then we have the corrected mass matrix: in, ; Step 4-5: Establish the objective function based on the system mass matrix and stiffness matrix. Repeat steps 4-2 and 4-3 to obtain: Step 4-6: Substitute the formula from step 4-5 into the constraint conditions. and The corrected stiffness matrix is: 。 2. The method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency according to claim 1, characterized in that, Step 1 also includes calculating the push rod stiffness, ball nut restoring force, screw stiffness, and bearing restoring force based on theoretical mechanics, mechanics of materials, and Hertz contact theory. Specifically, this includes: (1) When the push rod of the servo electric cylinder is subjected to an external axial load Fa, it undergoes elastic deformation in the axial direction. Ignoring the influence of the bolt group on the axial displacement of the push rod, the total deformation is three segments of the push rod. , and The sum of axial deformations gives the stiffness of the push rod. for: In the formula, , , and These are the diameters of each section of the push rod. and Flange thickness; Let be the elastic modulus of the push rod. =2.05×10⁵ MPa; , and They are respectively , and The cross-sectional area of ​​the segment; (2) Calculate the distances between the center of curvature of the screw raceway and the center of curvature of the nut raceway at the ball contact point before and after the ball screw pair is loaded. Combined with Hertz contact theory, the restoring force functions of the ball and nut in the x, y, and z directions are respectively , , : In the formula, This refers to the contact stiffness between the balls and the raceway of the lead screw or nut. The position angle of the ball. and These are the contact angles of the left and right corner contact balls, respectively. and These represent the deformation of the left and right corner contact balls, respectively. The lead angle of the ball screw; (3) When a radial force is applied to the lead screw shaft, the formulas for calculating the displacement of the center of mass of the lead screw caused by the elastic deformation of the lead screw shaft along the x and y directions are as follows: In the formula, , These represent the displacements of the screw's center of mass caused by the elastic deformation of the screw shaft along the x and y directions, respectively. This refers to the axial deformation and bending deformation of the lead screw shaft in the x-direction at a certain moment. This refers to the axial deformation and bending deformation of the lead screw shaft in the y-direction at a certain moment. Let be the gradient of the i-th element of the leadscrew axis, l be the length of the leadscrew, and A be the axial cross-sectional area of ​​the leadscrew. The density of the lead screw material; Based on the force-displacement relationship of the Euler-Bernoulli beam, the formulas for calculating the stiffness of the lead screw shaft in the x and y directions at a certain position on the push rod are as follows: (4) Considering the nonlinear contact behavior between the ball and the raceway, the formula for calculating the restoring force of the left and right angular contact ball bearings is: in, , , These are the restoring forces in the x, y, and z directions, respectively. and These represent the contact forces of the left and right angular contact ball bearings, respectively. It is the position angle of the i-th ball in the angular contact ball bearing. and These are left and right angular contact ball bearings, respectively; Nb is the number of angular contact balls. The formulas for calculating the restoring force function of a deep groove ball bearing in the x and y directions are as follows: in, , These represent the restoring forces of the deep groove ball bearing in the x and y directions, respectively. For deep groove ball stiffness, Let be the position angle of the i-th ball in the deep groove ball bearing. This is the radial distance between the centers of the raceways of the deep groove ball bearing.

3. The method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency according to claim 2, characterized in that, Step 2 involves simulating and solving for the natural frequencies of the electric cylinder system under different axial loads, specifically including: The matrix form of the differential equation of the electric cylinder's dynamics is decomposed into a mass matrix M and a stiffness matrix K: Solve its characteristic equation The non-zero solution is: in, Let u be the eigenvalue of the equation and u be the eigenvector in linear algebra. Expanding u, we can obtain... nth degree algebraic equation: in, The constant term indicates that the nth degree algebraic equation has n roots. , r=1,2,…,n, these roots are called eigenvalues; The square root of all eigenvalues As the inherent frequency of the system.

4. The method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency according to claim 3, characterized in that, Step 3, which involves using the assumed modal method to solve for the orthogonal modes of the electric cylinder system, specifically includes the following steps: Step 3-1, construct the function that the eigenvalues ​​satisfy: In the formula, M, K, and C are the corrected system mass, stiffness, and damping matrices, respectively; ne is the measured complex mode number of the system, and n is the degree of freedom of the structure; , These are the eigenvalue of order ith and the identified complex mode, respectively. Step 3-2: Obtain higher-order eigenvalues ​​using the generalized eigenvalue solving method, so that the higher-order eigenvalues... and complex modes satisfy: Step 3-3: Combining eigenvalues ​​and higher-order eigenvalues, use the least squares method to substitute them into the function constructed in step 3-1 to solve the problem. ; Steps 3-4, using the solution... The real mode is obtained by the following formula. and eigenvalue matrix : in, , For the ith-th order real mode, .

5. The method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency according to claim 4, characterized in that, Step 3 also includes: Steps 3-5 involve normalizing the real modes.

6. The method for identifying and calculating the stiffness of an electric cylinder based on its natural frequency according to claim 5, characterized in that, Steps 3-5 include the following specific processes: Step 3-5-1, Define the unknown matrix , The structure is n×n, where n is the degree of freedom of the structure; Step 3-5-2, Define the modified real modes ; Step 3-5-3, establish the unknown matrix The objective function : in, ; Steps 3-5-4: Minimize the objective function After normalization ; Then repeat steps 4-1 to 4-6 to obtain the corrected mass matrix and stiffness matrix.

7. A system for identifying and calculating the stiffness of an electric cylinder based on its natural frequency, according to the method described in any one of claims 1 to 6, characterized in that, The system includes: The first module is used to establish the differential equations of electric cylinder dynamics; The second module is used to simulate and solve the natural frequencies of the electric cylinder system under different axial loads. The third module is used to solve the orthogonal modes of the electric cylinder system using the assumed mode method. The fourth module is used to identify and correct stiffness parameters based on the natural frequency identification using the improved Berman-Baruch method.

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