Finite element simulation method for hot-pressing and hot-deformation process of neodymium-iron-boron magnet

Abstract

This invention provides a finite element simulation method for the hot pressing and hot deformation process of NdFeB magnets, comprising: establishing a geometric model of the NdFeB magnet hot pressing and hot deformation process components, wherein the hot pressing and hot deformation process components include a mold, a punch, and a NdFeB blank; discretizing the geometric model into a finite element mesh; assigning material properties and contact properties to the geometric model to obtain a simulation model; and applying loads and boundary conditions to the simulation model for calculation to obtain the variation law of the NdFeB magnet hot pressing and hot deformation process. The method provided by this invention is based on finite element simulation, utilizing finite element simulation to establish an accurate model and discretization technology, accurately simulating the hot pressing and hot deformation process, exploring the influence of manufacturing process parameters on NdFeB magnets, evaluating process quality from multiple dimensions, providing theoretical guidance for the hot pressing and hot deformation process, improving production efficiency, and reducing process costs.

Description

Technical Field

[0001] This invention belongs to the field of neodymium iron boron magnet manufacturing technology, and relates to a simulation method for neodymium iron boron magnet manufacturing, particularly a finite element simulation method for the hot pressing-hot deformation process of neodymium iron boron magnets. Background Technology

[0002] Neodymium iron boron permanent magnets have high coercivity, energy product, remanence, and cost-effectiveness. They are the magnets with the strongest comprehensive magnetic properties known to date. Their theoretical maximum energy product can reach 64 MGOe. They are widely used in aerospace, wind power generation, automotive industry, medical devices, mechanical and electronic fields.

[0003] Common fabrication processes for NdFeB magnets include bonding, sintering, and hot pressing-hot deformation. Among these, hot pressing-hot deformation is one of the most effective methods for preparing high-performance, full-density NdFeB permanent magnets. This process offers advantages such as a simple flow, no need for magnetic field orientation, low cost, small magnet grain size, and good magnet stability. Furthermore, this method can produce arc-shaped NdFeB magnets with unique toroidal magnetic fields, thus attracting considerable attention from researchers. The fabrication process mainly involves cold pressing, hot pressing, and hot deformation of NdFeB powder. During the process, changes in powder characteristics and fabrication parameters can affect the quality of the NdFeB compact and the magnetic properties of the final product, potentially leading to structural inhomogeneity and cracks. Therefore, selecting a suitable fabrication process is crucial. Currently, research on hot pressing-hot deformation primarily focuses on experiments. This involves controlling variables by varying process parameters such as temperature, pressure, and magnetic powder particle size to obtain empirical parameters, and repeatedly testing the quality of the green compact to analyze the impact of different parameters on the green compact's forming characteristics. This approach not only increases costs but also requires a long development cycle.

[0004] Therefore, a numerical simulation method is introduced to provide a simulation method based on finite element simulation for the hot pressing-hot deformation process of NdFeB magnets, so as to achieve efficient research on the NdFeB magnet manufacturing process. Summary of the Invention

[0005] The purpose of this invention is to provide a finite element simulation method for the hot pressing and hot deformation process of neodymium iron boron magnets. By using the finite element simulation method, the hot pressing and hot deformation process of neodymium iron boron magnets can be simulated, avoiding a large number of repetitive experiments and improving production efficiency.

[0006] To achieve this objective, the present invention adopts the following technical solution:

[0007] This invention provides a finite element simulation method for the hot pressing-hot deformation process of neodymium iron boron magnets, the finite element simulation method comprising the following steps:

[0008] A geometric model of a neodymium iron boron magnet hot-pressing-hot-deformation process component is established, the component including a mold, a punch, and a neodymium iron boron blank;

[0009] Discretize the geometric model into a finite element mesh;

[0010] By assigning material and contact properties to the geometric model, a simulation model is obtained;

[0011] By applying loads and boundary conditions to the simulation model, the variation law of the hot pressing-hot deformation process of NdFeB magnets was obtained.

[0012] The simulation method provided by this invention is based on finite element simulation. It uses finite element simulation to establish an accurate model and discretization technology to accurately simulate the hot pressing-hot deformation process. It can characterize the stress deformation behavior of the NdFeB blank during preparation due to the high temperature and large plastic deformation of the hot pressing-hot deformation process, explore the influence of the preparation process parameters on NdFeB magnets, and evaluate the process quality from multiple dimensions. Thus, it provides theoretical guidance for the hot pressing-hot deformation process, improves production efficiency, and reduces process costs.

[0013] Preferably, the finite element simulation method further includes: preparing neodymium iron boron magnets using a hot pressing-hot deformation process for testing, comparing and verifying the simulation model, correcting the simulation model parameters based on the test data, and optimizing the simulation model.

[0014] Preferably, discretizing the geometric model into a finite element mesh includes:

[0015] The neodymium iron boron blank model is segmented;

[0016] Perform global seeding, dividing the area into a hexahedral mesh.

[0017] Preferably, the fineness of the finite element mesh is ≥15, where fineness is the ratio of the diameter of the NdFeB particles to the diameter of the mesh cells.

[0018] Preferably, the material properties include plasticity, density, Poisson's ratio, and Young's modulus.

[0019] Preferably, the plasticity model is described using the Johnson-Cook model, the formula of which is as follows:

[0020]

[0021] Where σ is the yield stress, A is the yield strength at room temperature, B is the strain hardening constant, ε is the strain, n is the hardening exponent, and C is the strain rate sensitivity index. For equivalent plastic strain rate, T represents the plastic strain rate at or below the transformation temperature. room and T melt , respectively, represent room temperature and solidus lines of the material, and m is the temperature softening coefficient.

[0022] Preferably, the calculation method of the Johnson-Cook model includes: determining the stress-strain values ​​of the NdFeB material through the Brazilian disc splitting experiment, and calculating the parameters of the Johnson-Cook model using the inversion method based on the obtained stress-strain values.

[0023] Preferably, the contact properties include:

[0024] Friction is described using the Mohr Coulomb model, with the normal direction representing hard contact and the tangential direction set to the Lagrange multiplier method.

[0025] Preferably, the formula for the Mohr Coulomb model is as follows:

[0026]

[0027] Where f is the frictional force between the contacting objects; σ n The normal stress is t; the tangent vector is t in the direction of the relative sliding displacement; v α v is the relative slip velocity. β denoted as 1, representing the relative sliding velocity at the instant the two contacting objects begin to slide; μ is the coefficient of friction.

[0028] Preferably, the load includes: applying a gravity load and a temperature load to the NdFeB blank, and applying pressure to the punch.

[0029] Preferably, the boundary conditions include: setting degrees of freedom for the mold, punch, and NdFeB blank, setting the mold as a rigid body, and setting the particles of the NdFeB blank as Lagrange deformable entities.

[0030] Preferably, the calculation employs an explicit dynamic-displacement coupling algorithm.

[0031] The explicit dynamic-displacement coupling algorithm employs an explicit central difference time integration rule. The calculation satisfies the dynamic equilibrium equation at the start of the increment, time t. The acceleration calculated at time t is used to solve for the velocity at time t+Δt / 2, and the displacement from t to t+Δt is also calculated. The explicit central difference time integration rule is as follows:

[0032]

[0033] Among them, u N Let i be a variable with one degree of freedom, and the subscript i refers to the increment number in an explicit dynamic step.

[0034] Compared with the prior art, the present invention has the following beneficial effects:

[0035] The method provided by this invention is based on finite element simulation. It uses finite element simulation to establish an accurate model and discretization technology to accurately simulate the hot pressing-hot deformation process, explore the influence of the manufacturing process parameters on neodymium iron boron magnets, evaluate the process quality from multiple dimensions, provide theoretical guidance for the hot pressing-hot deformation process, improve production efficiency, and reduce process costs. Attached Figure Description

[0036] Figure 1 This is a flowchart of the finite element simulation method for the hot pressing-hot deformation process of neodymium iron boron magnets provided in Example 1;

[0037] Figure 2 This is a schematic diagram of the plasma discharge sintering furnace structure for the finite element simulation method of the hot pressing-hot deformation process of NdFeB magnets provided in Example 2.

[0038] Figure 3 This is a temperature-pressure cloud map of the NdFeB particle densification process, obtained using the finite element simulation method of the NdFeB magnet hot pressing-hot deformation process provided in Example 2. Detailed Implementation

[0039] The technical solution of the present invention will be further illustrated below through specific embodiments. Those skilled in the art should understand that the embodiments described are merely illustrative of the present invention and should not be construed as limiting the invention.

[0040] Example 1

[0041] This embodiment provides a method such as Figure 1 The finite element simulation method for the hot pressing-hot deformation process of neodymium iron boron magnets, shown below, uses the finite element software Abaqus. The finite element simulation method includes the following steps:

[0042] (1) Establish the geometric model of the neodymium iron boron magnet hot pressing-hot deformation process component.

[0043] The components used in the hot pressing-hot deformation process include neodymium iron boron blanks, punches, and dies.

[0044] The neodymium iron boron blank is designed as a cylinder.

[0045] The punch is divided into an upper punch and a lower punch, both of which are cylindrical.

[0046] The mold is divided into a hot deformation mold and a reverse extrusion mold. The hot deformation mold is a hollow cylinder with a certain wall thickness, and the reverse extrusion mold is a cylindrical barrel with a certain wall thickness and a cover at one end.

[0047] Establish geometric models of the aforementioned components separately, and combine all components into an assembly.

[0048] (2) Discretize the geometric model into a finite element mesh.

[0049] The axial direction of the cylindrical NdFeB blank is set as the X direction, and the X, Y and Z directions are perpendicular to each other.

[0050] (2.1) The neodymium iron boron blank is cut along the XZ and YZ planes to form four identical quarter-column components.

[0051] (2.2) Perform global seeding and divide the area into a hexahedral mesh.

[0052] The mesh fineness is ≥15 to ensure that the finite element model has good convergence.

[0053] (3) Assign material properties and contact properties to the model to obtain the simulation model.

[0054] For neodymium iron boron magnets, the physical parameters of the material include plasticity and temperature-related thermal conductivity, density, specific heat capacity, Poisson's ratio, and Young's modulus.

[0055] (3.1) Plasticity is described using the Johnson-Cook model. The specific process and method are as follows:

[0056] ① The stress-strain data of neodymium iron boron magnets were determined using the Brazilian splitting test.

[0057] Specifically, the neodymium iron boron magnet samples were cut into standard samples with a diameter of 30 mm. Using a universal testing machine, the samples were placed on a loading device and axial load was applied. Tests were conducted at different temperatures and strain rates, and the corresponding stress and strain values ​​were recorded.

[0058] ②Based on the stress-strain data obtained in ①, the Johnson-Cook model of the neodymium iron boron magnet was established by fitting the parameters of the Johnson-Cook model of the neodymium iron boron magnet using the inversion method.

[0059] The formula for the Johnson-cook model is as follows:

[0060]

[0061] Where σ is the yield stress, A is the yield strength at room temperature, B is the strain hardening constant, ε is the strain, n is the hardening exponent, and C is the strain rate sensitivity index. For equivalent plastic strain rate, T represents the plastic strain rate at or below the transformation temperature. room and T melt , respectively, represent room temperature and solidus lines of the material, and m is the temperature softening coefficient.

[0062] The specific fitting method is as follows:

[0063] The value of A is determined:

[0064] A is the yield strength at room temperature, determined based on the stress-strain curve of NdFeB magnets at room temperature.

[0065] Determining B and n:

[0066] When the effects of strain rate and temperature are not considered, the formula for the Johnson-Cook model can be simplified to:

[0067] σ=A+Bε n

[0068] Taking the logarithm of both sides simultaneously yields:

[0069] ln(σ-A)=ln(Bε n )=n lnε+ln B

[0070] Substituting the measured stress and strain values ​​into the above equation, we obtain the relationship between ln(σ-A) and lnε. By performing linear fitting, we obtain B and n.

[0071] Determining the value of C:

[0072] At room temperature, the formula for the Johnson-Cook model can be simplified to:

[0073]

[0074] By taking different strain values ​​and performing linear fitting, the C value is obtained.

[0075] Determining the value of m:

[0076] The Johnson-Cook model formula is modified as follows:

[0077]

[0078] Where A, B, C, and n are constants obtained from the above calculations, then α is defined as:

[0079]

[0080] The Johnson-Cook model formula can then be rewritten in the following form:

[0081]

[0082] By substituting the stress values ​​at different temperatures and strains into the above formula, we can obtain m through fitting.

[0083] (3.2) Temperature-related thermal conductivity, density, specific heat capacity, Poisson's ratio and Young's modulus data were obtained from existing research reports.

[0084] (3.3) In the model, friction is described by the Mohr Coulomb model, tangential contact is set as the Lagrange multiplier method, the friction coefficient is set, and normal contact is set as hard contact.

[0085] The formula for the Mohr Coulomb model is as follows:

[0086]

[0087] Where f is the frictional force between the contacting objects; σ n The normal stress is t; the tangent vector is t in the direction of the relative sliding displacement; v α v is the relative slip velocity. β denoted as 1, representing the relative sliding velocity at the instant the two contacting objects begin to slide; μ is the coefficient of friction.

[0088] (4) Neodymium iron boron magnets were prepared by hot pressing-hot deformation process and tested. The model was subjected to load and boundary conditions according to the actual parameters of hot pressing-hot deformation process. The test data and simulation data were cross-compared. The parameters of the simulation model were corrected according to the test data and the simulation model was optimized.

[0089] Specifically, the standard for model validation can be achieved by modifying the load settings or boundary conditions of the mold, such as the surface energy of the particles, model parameters, or contact parameters.

[0090] (5) Calculate the simulation model to obtain the variation law of the hot pressing-hot deformation process of neodymium iron boron magnet.

[0091] Specifically, the calculation employs an explicit dynamic-displacement coupling algorithm, setting the analysis step duration for the calculation.

[0092] The explicit dynamic-displacement coupling algorithm adopts the explicit central difference time integration rule. The operation satisfies the dynamic equilibrium equation at the time t at the beginning of the increment. The acceleration calculated at time t is used to solve for the velocity at time t+Δt / 2, and to solve for the displacement from t to t+Δt.

[0093] The calculation formula for the explicit central difference time integration rule is as follows:

[0094]

[0095] Among them, u N Let i be a variable with one degree of freedom, and the subscript i refers to the increment number in an explicit dynamic step.

[0096] The parameter variation law of NdFeB magnets during the hot pressing-hot deformation process was calculated and used to characterize and analyze the influence of the preparation process on NdFeB magnets.

[0097] For example, by visualizing the data, relative density-pressure curves, equivalent Mises stress-pressure curves, equivalent strain-pressure curves, or deformation contour maps can be calculated.

[0098] The equivalent Mises stress is defined as follows:

[0099]

[0100] in, σ1, σ2, and σ3 represent the Mises stress, and σ1, σ2, and σ3 represent the Cauchy principal stresses in the X, Y, and Z dimensions, respectively.

[0101] Equivalent strain is defined as:

[0102]

[0103] in, ε1, ε2, and ε3 represent the equivalent plastic strain, and ε3 represent the principal strains in the X, Y, and Z dimensions, respectively.

[0104] Example 2

[0105] This embodiment provides a finite element simulation method for the hot pressing-hot deformation process of neodymium iron boron magnets. Based on the finite element simulation method provided in Embodiment 1, a specific simulation process is provided using the finite element software Abaqus to illustrate the technical solution of the present invention.

[0106] The finite element simulation method includes the following steps:

[0107] (1) Establish the geometric model of the neodymium iron boron magnet hot pressing-hot deformation process component.

[0108] Specifically, the NdFeB blank is set as a cylinder with a diameter of 13 and a height of 10.

[0109] The upper punch and lower punch are set as cylinders with diameters of 25 and 23, respectively.

[0110] The hot deformation mold is set as a hollow cylinder with a wall thickness of 1 and an internal inner diameter of 25.

[0111] The reverse extrusion mold is set as a cylindrical barrel with a cap at one end, with a wall thickness of 1 and an inner diameter of 25.

[0112] After establishing the geometric model, all components are assembled into an assembly.

[0113] (2) Discretize the geometric model into a finite element mesh.

[0114] The axial direction of the cylindrical NdFeB blank is set as the X direction, and the X, Y and Z directions are perpendicular to each other.

[0115] (2.1) The neodymium iron boron blank is cut along the XZ and YZ planes to form four identical quarter-column components.

[0116] (2.2) Perform global seeding and divide the area into a hexahedral mesh.

[0117] The fineness of the finite element mesh is 15.

[0118] (3) Assign material properties and contact properties to the model to obtain the simulation model.

[0119] Specifically, material properties include plasticity, thermal conductivity, density, specific heat capacity, Poisson's ratio, and Young's modulus.

[0120] (3.1) Plasticity is described using the Johnson-Cook model, and the formula for the Johnson-Cook model is as follows:

[0121]

[0122] Where σ is the yield stress, A is the yield strength at room temperature, B is the strain hardening constant, ε is the strain, n is the hardening exponent, and C is the strain rate sensitivity index. For equivalent plastic strain rate, T represents the plastic strain rate at or below the transformation temperature. room and T melt , respectively, represent room temperature and solidus lines of the material, and m is the temperature softening coefficient.

[0123] The stress-strain data of NdFeB magnets were determined using the Brazilian splitting test. Experiments were conducted at different temperatures and strain rates to obtain the stress-strain values ​​of NdFeB magnets.

[0124] The fitting method described in Example 1 fits the specific parameters of the Johnson-Cook model based on the measured stress and strain values.

[0125] (3.2) The data on thermal conductivity, density, specific heat capacity, Poisson's ratio and Young's modulus related to temperature are from "Numerical Simulation and Experimental Study on Laser Cutting of NdFeB Magnetic Materials" (Ren Ning. Numerical Simulation and Experimental Study on Laser Cutting of NdFeB Magnetic Materials [D]. Taiyuan, Shanxi: North University of China, 2020). The specific data are shown in Table 1.

[0126] Table 1

[0127]

[0128] (3.3) Tangential contact is set as Lagrange multiplier method, friction is described by Mohr Coulomb model, and friction coefficient is 0.2; normal contact is set as hard contact.

[0129] The formula for the Mohr Coulomb model is as follows:

[0130]

[0131] Where f is the frictional force between the contacting objects; σ n The normal stress is t; the tangent vector is t in the direction of the relative sliding displacement; v α v is the relative slip velocity. β denoted as 1, representing the relative sliding velocity at the instant the two contacting objects begin to slide; μ is the coefficient of friction.

[0132] (4) Neodymium iron boron magnets were prepared by hot pressing-hot deformation process and tested. The model was subjected to load and boundary conditions according to the actual parameters of hot pressing-hot deformation process. The test data and simulation data were cross-compared. The parameters of the simulation model were corrected according to the test data and the simulation model was optimized.

[0133] (4.1) The preparation process of neodymium iron boron magnets by hot pressing-hot deformation process.

[0134] Adopting such Figure 2 The plasma discharge sintering furnace shown has a hot pressing mold made of a hollow tungsten steel cylinder with a diameter of 13 mm and a hot deformation mold made of a hollow tungsten steel cylinder with a diameter of 25 mm.

[0135] To enhance conductivity and facilitate demolding, graphite paper is laid inside the hot press mold to form a paper cage. 10g of self-fluxing NdFeB powder is then filled into the paper cage, and the upper and lower punches are assembled.

[0136] To prevent particle oxidation, a two-stage vacuum pump is used to evacuate the sintering furnace to a vacuum level of 10. -3 Pa.

[0137] Then, the powder particles were heated to a specified temperature. Under each heating temperature condition, linear pressure was applied to the mold using a punch. After pressure application, the pressure was held for 5 minutes, and then the mold was demolded after cooling to room temperature. The experimental values ​​of temperature, displacement, pressure, and voltage were derived to form a density-temperature / pressure graph, as shown below. Figure 3 As shown.

[0138] Then, the hot-pressed blank is placed in the center of the mold for hot deformation process, and the corresponding parameters are recorded.

[0139] (4.2) Apply loads and boundary conditions to the model according to the process parameters in the actual preparation process.

[0140] Specifically, a gravity load of -9800 is applied to the NdFeB billet in the direction of -Z.

[0141] During the hot pressing process, a temperature load is applied to the NdFeB blank, and a pressure is applied to the punch in the -Z direction.

[0142] Set the pressure application rate of the punch.

[0143] Specifically, all degrees of freedom of the mold are set to 0; the punch is not constrained along the Z direction of the NdFeB blank, and the remaining degrees of freedom are set to 0; the NdFeB blank is not constrained; the mold is set as a rigid body, and the particles are set as Lagrange deformable solids.

[0144] The hot pressing results of NdFeB particles were obtained by simulation based on the set load and boundary conditions.

[0145] (4.3) Cross-compare the experimental data with the simulation data, correct the parameters of the simulation model, and optimize the simulation model.

[0146] Specifically, modify the load settings or boundary conditions of the hot press mold, such as the surface energy of the particles, model parameters, or contact parameters.

[0147] (5) Calculate the simulation model to obtain the variation law of the hot pressing-hot deformation process of neodymium iron boron magnet.

[0148] The calculation was performed using an explicit dynamic-displacement coupling algorithm to obtain the results of the influence of different temperatures and pressures on the NdFeB hot pressing-hot deformation process.

[0149] The applicant declares that the above description is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Those skilled in the art should understand that any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention fall within the protection and disclosure scope of the present invention.

Claims

1. A finite element simulation method for the hot pressing-hot deformation process of neodymium iron boron magnets, characterized in that, The finite element simulation method includes the following steps: A geometric model of a neodymium iron boron magnet hot-pressing-hot-deformation process component is established, the component including a mold, a punch, and a neodymium iron boron blank; Discretize the geometric model into a finite element mesh; By assigning material and contact properties to the geometric model, a simulation model is obtained; By applying loads and boundary conditions to the simulation model, the variation law of the hot pressing-hot deformation process of NdFeB magnets was obtained.

2. The finite element simulation method according to claim 1, characterized in that, The finite element simulation method further includes: preparing neodymium iron boron magnets using a hot pressing-hot deformation process for testing, comparing and verifying the simulation model, correcting the parameters of the simulation model based on the test data, and optimizing the simulation model.

3. The finite element simulation method according to claim 1 or 2, characterized in that, Discretizing the geometric model into a finite element mesh includes: The neodymium iron boron blank model is segmented; Perform global seeding, dividing the area into a hexahedral grid; The fineness of the finite element mesh is ≥15, where fineness is the ratio of the diameter of the NdFeB particles to the diameter of the mesh element.

4. The finite element simulation method according to any one of claims 1-3, characterized in that, The material properties include plasticity, thermal conductivity, density, specific heat capacity, Poisson's ratio, and Young's modulus.

5. The finite element simulation method according to claim 4, characterized in that, The plasticity model is described using the Johnson-Cook model, and the formula of the Johnson-Cook model is as follows: Where σ is the yield stress, A is the yield strength at room temperature, B is the strain hardening constant, ε is the strain, n is the hardening exponent, and C is the strain rate sensitivity index. For equivalent plastic strain rate, T represents the plastic strain rate at or below the transformation temperature. room and T melt , respectively, represent room temperature and solidus lines of the material, and m is the temperature softening coefficient.

6. The finite element simulation method according to claim 5, characterized in that, The calculation method of the Johnson-Cook model includes: determining the stress and strain values ​​of the neodymium iron boron material through the Brazilian disk splitting experiment, and calculating the parameters of the Johnson-Cook model using the inversion method based on the obtained stress and strain values.

7. The finite element simulation method according to any one of claims 1-6, characterized in that, The contact properties include: Friction is described using the Mohr Coulomb model, with the normal direction representing hard contact and the tangential direction set to the Lagrange multiplier method.

8. The finite element simulation method according to claim 7, characterized in that, The formula for the Mohr Coulomb model is as follows: Where f is the frictional force between the contacting objects; σ n The normal stress is t; the tangent vector is t in the direction of the relative sliding displacement; v α v is the relative slip velocity. β denoted as 1, representing the relative sliding velocity at the instant the two contacting objects begin to slide; μ is the coefficient of friction.

9. The finite element simulation method according to any one of claims 1-8, characterized in that, The loads include: applying a gravitational load and a temperature load to the NdFeB blank, and applying pressure to the punch; The boundary conditions include: setting degrees of freedom for the mold, punch, and NdFeB blank; setting the mold as a rigid body; and setting the particles of the NdFeB blank as Lagrange deformable entities.

10. The finite element simulation method according to any one of claims 1-9, characterized in that, The calculation employs an explicit dynamic-displacement coupling algorithm; The explicit dynamic-displacement coupling algorithm employs the explicit central difference time integration rule. The calculation satisfies the dynamic equilibrium equation at the start of the increment, time t. The acceleration calculated at time t is used to solve for the velocity at time t+Δt / 2, and the displacement from t to t+Δt is also calculated. The calculation formula for the explicit central difference time integration rule is as follows: Among them, u N For a single degree of freedom variable, the subscript i refers to the increment number in an explicit dynamic step.

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