A dynamic modeling method for planetary roller screw based on generalized finite element method

Abstract

The present invention relates to a planetary roller screw dynamic modeling method based on the generalized finite element method, which belongs to the field of dynamic characteristic analysis of planetary roller screws. The method comprises the following steps: dividing the planetary roller screw finite element model into units; constructing dynamic models of each type of unit: establishing the mass matrix, stiffness matrix, damping matrix, and load column vector for the shaft segment unit, the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller tooth-ring gear meshing unit; and assembling the system dynamics model to complete the planetary roller screw dynamic modeling. The present invention can accurately analyze the dynamic characteristics of real physical models and provide an effective means for analyzing the vibration response of the system.

Description

Technical Field

[0001] The present invention belongs to the field of dynamic characteristic analysis of planetary roller screws, and in particular relates to a planetary roller screw dynamics modeling method based on a generalized finite element method. Background Art

[0002] Planetary roller screws primarily consist of a screw, rollers, a nut, an internal gear ring, and a cage. Force and motion are transmitted through the threaded engagement between multiple rollers, the screw, and the nut. As a new type of actuator, they feature multi-pair (threaded and geared), multi-body (multiple rollers simultaneously contact the screw and nut), and multi-point (multiple roller teeth and thread teeth simultaneously engage each other), resulting in advantages such as high precision, high load capacity, high rigidity, and long life.

[0003] Currently, electromechanical servo technology based on planetary roller screws has been widely used in a wide range of fields, including aircraft control surfaces, aircraft brakes, rocket launcher erection, and humanoid robot servo joints. To improve the dynamic integrated transmission performance of planetary roller screws, it is essential to conduct research on the dynamic contact characteristics of multiple meshing pairs. Establishing a complete and accurate dynamic model of a planetary roller screw system is the foundation for its dynamic characteristics and vibration analysis.

[0004] However, the dynamic models used in existing research are often based on certain simplifying assumptions, such as the use of lumped mass methods to model the system. While computationally efficient, these methods struggle to accurately reflect the complex connections between components, particularly for precision transmission components such as planetary roller screws. The lumped mass method discretizes a part into point masses and rigid connections, failing to characterize the distribution characteristics of shaft segments. This leads to inherent limitations in the model's analysis of vibration or contact nonlinear behavior, further impacting the accuracy of dynamic characteristic predictions. Therefore, a more refined dynamic model is urgently needed to address these shortcomings. Summary of the Invention

[0005] Technical issues to be solved:

[0006] To overcome the shortcomings of existing technologies, the present invention provides a planetary roller screw dynamics modeling method based on the generalized finite element method (GFE). This method aims to establish a dynamics modeling approach that simultaneously considers the synchronous meshing of thread pairs and gear pairs and characterizes the distribution characteristics of shaft segment units. This method uses the GFE method to divide the system into different types of units (such as shaft segment units, thread meshing units, and gear meshing units), assembling their mass, stiffness, damping matrices, and load vectors to ultimately form system-level dynamic equations. This method accurately analyzes the dynamic characteristics of real physical models and provides an effective means for analyzing the system's vibration response.

[0007] The technical solution of the present invention is: a planetary roller screw dynamics modeling method based on the generalized finite element method, the specific steps are as follows:

[0008] The finite element model of the planetary roller screw is divided into units: the screw, nut, roller and inner ring of the planetary roller screw are discretized into nodes, and divided into shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units and roller tooth-inner ring gear meshing units;

[0009] Construct dynamic models of various types of units: establish the mass matrix, stiffness matrix, damping matrix, and load column vector of the shaft segment unit, the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller gear-ring gear meshing unit respectively;

[0010] System dynamics model assembly: According to the mapping relationship between unit nodes and global nodes, the unit matrices are assembled into the system-level overall mass matrix, overall stiffness matrix and overall damping matrix, thus completing the dynamic modeling of the planetary roller screw.

[0011] A further technical solution of the present invention is: the shaft segment unit is modeled using the Timoshenko beam unit, the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller tooth-internal ring gear meshing unit are modeled based on the relative displacement relationship of the contact points, and each contact unit node coincides with the shaft segment unit node.

[0012] A further technical solution of the present invention is: the dynamic model expression of the shaft segment unit is as follows:

[0013]

[0014] Where q T The generalized coordinates of the two nodes of the shaft segment element; M T is the consistent mass matrix of the shaft segment element; K T is the stiffness matrix of the shaft segment element; C T is the damping matrix of the shaft segment element;

[0015] Mass matrix M T The expression is as follows:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] Where, A is the cross-sectional area of the shaft segment; ρ is the material density; l is the length of the shaft segment; I is the polar moment of inertia; I x For yz The moment of inertia of the area in the coordinate plane, I y For xz moment of inertia of the area in the coordinate plane;

[0022] Stiffness matrix K T The expression is as follows:

[0023]

[0024]

[0025]

[0026]

[0027]

[0028] Where, E is the elastic modulus of the material; G is the shear elastic modulus of the material; k The correction factor introduced to take into account that the actual shear strain and shear stress are not uniformly distributed is the shear coefficient of the material;

[0029] Damping Matrix C T The expression is as follows:

[0030]

[0031] Where, α 0. α 1 is the mass proportional coefficient and stiffness proportional coefficient in Rayleigh damping.

[0032] A further technical solution of the present invention is that the relative displacement relationship between the screw-roller contact side thread engagement unit, the nut-roller contact side thread engagement unit, and the roller gear-ring gear engagement unit is as follows:

[0033] The relative displacement of the screw-roller contact side thread engagement unit is:

[0034]

[0035]

[0036]

[0037]

[0038] Where, The screw is Vibration displacement in the direction; The screw is Torsional displacement in direction; The rollers are Vibration displacement in the direction of The rollers are Torsional displacement in direction; is the nominal radius of the screw; is the nominal radius of the roller; Normal force exist The projection angle of the surface; Normal force exist The projection angle of the surface; Normal force exist Projection of the surface and x The angle between the axes; is the roller phase angle, j Indicates the serial number of the roller, is the total number of rollers; n is the number of thread starts; Indicates the coordinate system of the roller part around which z The angle through which the axis rotates; symbol " "and" The upper part of the screw rotates counterclockwise, and the lower part of the screw rotates clockwise;

[0039] The relative displacement of the threaded engagement unit on the nut-roller contact side is:

[0040]

[0041]

[0042] Where, The nuts are Vibration displacement in the direction of The nuts are Torsional displacement in direction; The rollers are Vibration displacement in the direction of The rollers are Torsional displacement in direction; is the nominal radius of the nut; is the nominal radius of the roller; Normal force exist The projection angle of the surface; Normal force exist The projection angle of the surface; Normal force exist Projection of the surface and x The angle between the axes;

[0043] The relative displacement of the roller gear-ring gear meshing unit is:

[0044]

[0045]

[0046] Where, The inner ring gear is Vibration displacement in the direction of The inner ring gear is Torsional displacement in direction; The roller teeth are Vibration displacement in the direction of The roller teeth are Torsional displacement in direction; is the pitch circle radius of the inner gear ring; is the roller gear pitch circle radius; is the gear helix angle; is the gear pressure angle; is the sum of the gear pressure angle and the roller phase angle.

[0047] A further technical solution of the present invention is that the dynamic model expression of the screw-roller contact side thread engagement unit is as follows:

[0048]

[0049] Where, The screw-roller contact side q The mass matrix of the thread engagement element; The screw-roller contact side q The displacement matrix of two nodes of a threaded meshing element; The screw-roller contact side q The stiffness matrix of the threaded meshing element; The screw-roller contact side q The damping matrix of the threaded meshing element;

[0050] Mass Matrix The expression is as follows:

[0051]

[0052] Where, For the screw q Thread quality; 、 and Screw No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; For roller q Thread quality; 、 and Roller No. q Thread teeth in x 、 y 、 z moment of inertia in the direction;

[0053] Displacement Matrix The expression is as follows:

[0054]

[0055] Where, 、 and Screw No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Screw No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Torsional displacement in direction;

[0056] Stiffness matrix The expression is as follows:

[0057]

[0058] Where, V is the thread engagement stiffness of the screw-roller contact side; S is the screw-roller side projection matrix, which is expressed as follows:

[0059]

[0060] Damping Matrix The expression is as follows:

[0061]

[0062] Where, The thread engagement damping on the screw-roller contact side.

[0063] A further technical solution of the present invention is that the dynamic model expression of the thread engagement unit on the nut-roller contact side is as follows:

[0064]

[0065] Where, The nut-roller contact side q The mass matrix of the thread engagement element; The nut-roller contact side q The displacement matrix of two nodes of a threaded meshing element; The nut-roller contact side q The stiffness matrix of the threaded meshing element; The nut-roller contact side q The damping matrix of the threaded meshing element;

[0066] Mass Matrix The expression is as follows:

[0067]

[0068] Where, For nut q Thread quality; 、 and Nut No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; For roller q Thread quality; 、 and Roller No.q Thread teeth in x 、 y 、 z moment of inertia in the direction;

[0069] Displacement Matrix The expression is as follows:

[0070]

[0071] Where, 、 、 Nut No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Nut No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Torsional displacement in direction;

[0072] Stiffness matrix The expression is as follows:

[0073]

[0074] Where, V is the thread engagement stiffness of the nut-roller contact side; N is the nut-roller side projection matrix, expressed as:

[0075]

[0076] Damping Matrix The expression is as follows:

[0077]

[0078] Where, is the thread engagement damping on the nut-roller contact side.

[0079] A further technical solution of the present invention is that the dynamic model expression of the roller gear-ring gear meshing unit is as follows:

[0080]

[0081] Where, Roller gear-internal gear q The mass matrix of each gear meshing unit; Roller gear-internal gear q The displacement matrix of two nodes of a gear meshing unit; Roller gear-internal gear q Gear mesh element stiffness matrix; Roller gear-internal gear q Gear meshing unit damping matrix; It is the column vector formed by the components of the normal impact force of the gear pair in each degree of freedom;

[0082] Mass Matrix The expression is as follows:

[0083]

[0084] Where, For the inner ring gear q The quality of each gear tooth; 、 and The inner ring gear q The gear teeth x 、 y 、 z moment of inertia in the direction; For roller q The mass of roller teeth; 、 and Roller No. q Roller teeth in x 、 y 、 z moment of inertia in the direction;

[0085] Displacement Matrix The expression is as follows:

[0086]

[0087] Where, 、 、 The inner ring gear q The gear teeth x 、 y 、z Vibration displacement in the direction; 、 、 The qth gear tooth of the inner ring is x 、 y 、 z Torsional displacement in direction; 、 、 The first q Roller teeth in x 、 y 、 z Vibration displacement in the direction; 、 、 Roller No. q Roller teeth in x 、 y 、 z Torsional displacement in direction;

[0088] Stiffness matrix The expression is as follows:

[0089]

[0090] Where, V is the meshing stiffness of the roller gear-ring gear contact side; r is the projection matrix of the roller gear-ring contact side, and its expression is:

[0091]

[0092] Damping Matrix The expression is as follows:

[0093]

[0094] Where, is the meshing damping of the roller gear-ring gear contact side gear pair;

[0095] Loading column vector The expression is as follows:

[0096]

[0097] Where, is the normal impact force of the gear pair.

[0098] A further technical solution of the present invention is that: when assembling the unit matrices, the sub-blocks of the unit stiffness matrix are superimposed on the corresponding positions of the global matrix according to the correspondence between the local number and the global number of the unit node;

[0099] The expression of the differential equation of the planetary roller screw dynamic motion is as follows:

[0100]

[0101] Where, is the overall mass matrix of the planetary roller screw system; is the overall damping matrix of the planetary roller screw system; is the overall stiffness matrix of the planetary roller screw system; where,

[0102]

[0103]

[0104]

[0105] Where, 、 、 and 、 、 The first and second shaft segments are q The mass matrix, damping matrix and stiffness matrix of each shaft segment unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix and stiffness matrix of the screw-roller contact side thread engagement unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix and stiffness matrix of the roller gear-ring contact side gear meshing unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix, and stiffness matrix of the threaded meshing element on the nut-roller contact side.

[0106] A further technical solution of the present invention is: the dynamic response of the planetary roller screw dynamic motion differential equation is solved by numerical integration method, specifically including:

[0107] Divide the total load into several load steps and apply them gradually;

[0108] Update the contact state after each calculation step, and dynamically adjust the stiffness matrix and adjoint matrix;

[0109] The convergence is judged by the residual error, and if it does not converge, it is repeated.

[0110] A planetary roller screw dynamics modeling system based on the generalized finite element method includes the following modules:

[0111] Structural discretization module: used to discretize the screw, nut, roller and internal gear ring of the planetary roller screw into global nodes, and divide them into shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units and roller tooth-internal gear ring gear meshing units;

[0112] Unit dynamics modeling module: Based on the generalized finite element method, it generates the mass matrix, stiffness matrix, damping matrix and load vector of each unit;

[0113] System matrix assembly module: Based on the global number mapping relationship of unit nodes, each unit matrix is superimposed into the system-level overall matrix according to the degree of freedom;

[0114] Dynamic response solution module: Based on the overall mass matrix, overall stiffness matrix and overall damping matrix, the system dynamics differential equation is established and the dynamic response is solved;

[0115] Model verification module: compares the relative error between the dynamic contact force calculation results and the theoretical value to ensure that the model accuracy error is less than the threshold.

[0116] Beneficial effects

[0117] The beneficial effects of the present invention are:

[0118] 1. This paper uses the generalized finite element method to model the dynamics of a planetary roller screw system. High-order shape functions and a consistent mass matrix accurately describe the distribution characteristics of the shaft segment elements between the screw, roller, and nut nodes. The influence of the distributed mass is coupled to all relevant nodes via the off-diagonal terms of the stiffness and mass matrices, enabling high-precision prediction of the shaft segment's dynamic behavior while avoiding the model distortion caused by simplification in traditional lumped mass methods. This approach lays an important foundation for the accuracy of subsequent dynamic characteristic predictions and vibration analysis of planetary roller screw systems.

[0119] 2. The relative errors of the steady-state values of the dynamic model calculated based on the generalized finite element method in the present invention and the theoretical formula calculation results and finite element simulation results are all within 5%, verifying the accuracy and effectiveness of the constructed planetary roller screw dynamic model, which can effectively describe the dynamic characteristics of the planetary roller screw.

[0120] 3. The dynamic model constructed in this invention generates various unit matrices for the system by simply inputting the required basic system parameters. This modeling method is also suitable for dynamic modeling of other types of planetary roller screw transmission systems, effectively reducing repetitive modeling. Compared with finite element simulation analysis, it can significantly shorten calculation time and improve analysis efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0121] Figure 1 A flow chart of establishing a dynamic model of a planetary roller screw system in an embodiment of the present invention;

[0122] Figure 2 This is a force analysis diagram of the planetary roller screw transmission system in an embodiment of the present invention;

[0123] Figure 3 : This is a finite element model diagram of a planetary roller screw transmission system in an embodiment of the present invention;

[0124] Figure 4 This is a diagram of a shaft segment unit of a planetary roller screw transmission system in an embodiment of the present invention;

[0125] Figure 5 : A diagram showing the relative displacement relationship between the contact position of the screw-roller contact side thread engagement unit in an embodiment of the present invention;

[0126] Figure 6 : A diagram showing the relative displacement relationship between the contact position of the threaded engagement unit on the nut-roller contact side in an embodiment of the present invention;

[0127] Figure 7 : A diagram showing the relative displacement relationship of the contact position of the meshing unit of the roller gear-ring gear contact side in an embodiment of the present invention;

[0128] Figure 8 Schematic diagram of the assembly of the stiffness matrix of the planetary roller screw transmission system according to an embodiment of the present invention;

[0129] Figure 9 : is a dynamic contact force curve diagram of the screw-roller contact side in an embodiment of the present invention;

[0130] Figure 10 : is a dynamic contact force curve diagram of the nut-roller contact side in an embodiment of the present invention;

[0131] Figure 11 A comparison diagram of the dynamic contact force on the contact side of the screw and roller in an embodiment of the present invention;

[0132] Figure 12 2 is a comparison diagram of the dynamic contact force on the nut-roller contact side in an embodiment of the present invention. DETAILED DESCRIPTION

[0133] The embodiments described below with reference to the accompanying drawings are exemplary and are intended to explain the present invention, but should not be construed as limiting the present invention.

[0134] At present, the research methods and dynamic models of planetary roller screw dynamics include Jones MH, Velinsky SA, Lasky TA Dynamics of the planetary roller screw mechanism. Journal of Mechanisms and Robotics, 2016, 8(1): 014503, which established the component motion differential equation based on the Lagrangian method; Ma SJ, Zhang T., Liu G., et al. Bond Graph-based dynamicmodel of planetary roller screw mechanism with consideration of axial clearance and friction. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2018, 232(16):2899-2911, which established the bond graph model; Guo Jianan, He Peng, Huang Hongyan, et al. Analysis and experimental study on the dynamic characteristics of planetary roller screw. Propulsion Technology, 2018, 39(8): 841-848, which established the dynamic model using the discrete element method; Wu LP, Ma SJ, Wan Q., et al. Dynamic modal of planetary roller screw mechanism with considering torsional degree of freedom. The 6th International Conference on Mechatronics and Mechanical Engineering, Japan, Osaka, Sep. 27-30, 2019; and Fu XJ, Liu G.,Tong RT, et al. A nonlinear six degrees of freedom dynamic model of planetary roller Screw mechanism.Mechanism and Machine Theory, 2018, 119:22-36. The dynamic models used in existing research are often based on certain simplifying assumptions, such as using the lumped mass method to model the system. While computationally efficient, these methods struggle to accurately reflect the complex connections between components, especially for precision transmission components such as planetary roller screws. The lumped mass method discretizes a part into point masses and rigid connections, failing to characterize the distribution characteristics of the shaft segment units. This leads to inherent limitations in the model's analysis of vibration or contact nonlinear behavior, further impacting the accuracy of dynamic characteristic predictions. Therefore, the present invention provides a planetary roller screw dynamic modeling method based on the generalized finite element method, including the following steps:

[0135] The finite element model of the planetary roller screw is divided into units: the screw, nut, roller and inner ring of the planetary roller screw are discretized into nodes, and divided into shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units and roller tooth-inner ring gear meshing units;

[0136] Construct dynamic models of various types of units: establish the mass matrix, stiffness matrix, damping matrix, and load column vector of the shaft segment unit, the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller gear-ring gear meshing unit respectively;

[0137] System dynamics model assembly: According to the mapping relationship between unit nodes and global nodes, the unit matrices are assembled into the system-level overall mass matrix, overall stiffness matrix and overall damping matrix, thus completing the dynamic modeling of the planetary roller screw.

[0138] The present invention provides a planetary roller screw dynamics modeling system based on the generalized finite element method, comprising the following modules:

[0139] Structural discretization module: used to discretize the screw, nut, roller and internal gear ring of the planetary roller screw into global nodes, and divide them into shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units and roller tooth-internal gear ring gear meshing units;

[0140] Unit dynamics modeling module: Based on the generalized finite element method, it generates the mass matrix, stiffness matrix, damping matrix and load vector of each unit;

[0141] System matrix assembly module: Based on the global number mapping relationship of unit nodes, each unit matrix is superimposed into the system-level overall matrix according to the degree of freedom;

[0142] Dynamic response solution module: Based on the overall mass matrix, overall stiffness matrix and overall damping matrix, the system dynamics differential equation is established and the dynamic response is solved;

[0143] Model verification module: compares the relative error between the dynamic contact force calculation results and the theoretical value to ensure that the model accuracy error is less than the threshold.

[0144] The above technical solution is further described below with reference to the accompanying drawings:

[0145] In one embodiment, referring to Figure 1-8 This embodiment provides a planetary roller screw dynamics modeling method based on the generalized finite element method, comprising the following steps:

[0146] S1: If Figure 3 As shown in the figure, the finite element model of the planetary roller screw is divided according to its structural characteristics. The planetary roller screw is discretized into a series of nodes and composed of different types of units: shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units, and roller tooth-ring gear meshing units. Because the gear pair nodes and thread pair nodes coincide with the shaft nodes, the shaft nodes in the substructures (screw, roller, nut, ring gear) are numbered. The position and numbering relationship between the substructures are used to obtain the numbering of the shaft nodes and each unit in the planetary roller screw.

[0147] S2: If Figure 4 As shown, the dynamic modeling of the planetary roller screw shaft segment unit: Timoshenko beam unit is used to establish the motion equation of the shaft segment unit to complete the dynamic modeling of the shaft segment unit. The matrix form of the dynamic motion differential equation of the planetary roller screw shaft segment unit is shown in formula (1):

[0148] (1)

[0149] Where q T The generalized coordinates of the two nodes of the axis segment are q T = { x 1, y 1, z 1, θ x1 , θ y1 , θ z1 , x 2, y 2, z 2, θ x2 , θ y2 , θ z2},in:x q 、 y q 、 z q ( q =1, 2) are nodes q Displacement along the local coordinate direction, θ xq 、 θ yq 、 θ zq ( q =1, 2) are nodes q The rotation angle of the section around the three coordinate axes; M T is the consistent mass matrix of the shaft segment unit, as shown in formula (2-6); K T is the stiffness matrix of the shaft segment element, which is represented by the element stiffness matrix of the two-node Timoshenko spatial beam element, as shown in Equation (7-11); C T is the damping matrix of the shaft segment unit. The Rayleigh damping calculation commonly used in engineering is shown in formula (12).

[0150] Mass matrix M T The expression is as follows:

[0151] (2)

[0152] (3)

[0153] (4)

[0154] (5)

[0155] (6)

[0156] Where, A is the cross-sectional area of the shaft segment; ρ is the material density; l is the length of the shaft segment; I is the polar moment of inertia; I x and I y Respectively in yz Coordinate plane and xz Moment of inertia of the area in the coordinate plane.

[0157] Stiffness matrix K T The expression is as follows:

[0158] (7)

[0159] (8)

[0160] (9)

[0161] (10)

[0162] (11)

[0163] Where, E is the elastic modulus of the material; G is the shear elastic modulus of the material; k The correction factor introduced to take into account the fact that the actual shear strain and shear stress are not uniformly distributed is the shear coefficient of the material. For a circular cross section, it can be taken as k =0.9.

[0164] (12)

[0165] Where, α 0. α 1 is the mass proportional coefficient and stiffness proportional coefficient in Rayleigh damping.

[0166] S3: If Figure 5 As shown in the figure, the dynamic modeling of the threaded meshing unit on the screw-roller contact side is as follows: According to the force analysis diagram of the screw-roller contact side, the relative displacement of the screw-roller contact side is derived, and the dynamic motion differential equation of the threaded meshing unit on the screw-roller contact side is obtained.

[0167] The relative displacement of the screw-roller contact side is:

[0168]

[0169] (14)

[0170] (15)

[0171] (16)

[0172] Where, The screw is Vibration displacement in the direction; The screw is Torsional displacement in direction; The rollers are Vibration displacement in the direction of The rollers are Torsional displacement in direction; is the nominal radius of the screw; is the nominal radius of the roller; Normal force exist The projection angle of the surface; Normal force exist The projection angle of the surface; Normal force exist Projection of the surface and x The angle between the axes; is the roller phase angle, j Indicates the serial number of the roller, is the total number of rollers; n is the number of thread starts; Indicates the coordinate system of the roller part around which z The angle through which the axis rotates; symbol " "and" The upper part of the screw rotates counterclockwise, and the lower part of the screw rotates clockwise;

[0173] The differential equation of dynamic motion of the screw-roller side thread engagement unit is shown in Equation (17), and its matrix form is shown in Equation (18):

[0174]

[0175] (18)

[0176] Where, The screw-roller contact side q The mass matrix of the thread engagement element; The screw-roller contact side q The displacement matrix of two nodes of a threaded meshing element; The screw-roller contact side q The stiffness matrix of the thread engagement element; The screw-roller contact side q The damping matrix of the threaded meshing unit. The specific expressions of the mass matrix, displacement matrix, stiffness matrix and damping matrix are shown in formula (19-26):

[0177] (19)

[0178] Where, For the screw q Thread quality; 、 and Screw No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; For roller q Thread quality; 、 and Roller No. q Thread teeth in x 、 y 、 z moment of inertia in the direction;

[0179] (20)

[0180] Where, 、 and Screw No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Screw No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Torsional displacement in direction;

[0181] (twenty one)

[0182] (twenty two)

[0183] (twenty three)

[0184]

[0185]

[0186] Stiffness matrix of the threaded engagement element on the screw-roller contact side and the damping matrix The form is complex, but by carefully studying the structure of the matrix, a simple calculation form can be obtained:

[0187] (27)

[0188] (28)

[0189] Where, is the thread engagement stiffness of the screw-roller contact side, which can be calculated by formulas (29 to 31); is the thread engagement damping on the screw-roller contact side, which can be calculated by formula (32); V S is the screw-roller side projection matrix, as shown in formula (33):

[0190] (29)

[0191] (30)

[0192] (31)

[0193] (32)

[0194]

[0195] Where, is the normal contact force on the screw-roller contact side; is the normal contact deformation of the screw-roller contact side; is the axial contact force on the screw-roller contact side; is the screw-roller contact side tooth angle; is the roller helix angle; k ( e ) is the elliptic integral of the first kind, e is the eccentricity; is the equivalent elastic modulus, as shown in formula (34); is the principal curvature function F ( ρ ) related coefficients can be obtained by looking up the table, and the principal curvature function is calculated by formulas (35-37); η is the meshing damping ratio coefficient, ranging from 0.03 to 0.17; and are the masses of the two meshing nodes on the screw-roller contact side.

[0196] (34)

[0197] (35)

[0198]

[0199] (37)

[0200] Where, E S and E R are the elastic moduli of the screw and roller respectively; μ S and μ R are the Poisson's ratios of the screw and roller, respectively; R is the equivalent spherical radius; d S is the middle diameter of the screw; d R is the roller median diameter; α is the thread flank angle.

[0201] S4: As Figure 6 As shown in the figure, the dynamic modeling of the threaded meshing unit on the nut-roller contact side is as follows: Based on the force analysis diagram of the nut-roller contact side, the relative displacement of the nut-roller contact side is derived, and the dynamic motion differential equation of the threaded meshing unit on the nut-roller contact side is obtained. The relative displacement of the nut-roller contact side is:

[0202]

[0203] (39)

[0204] Where, The nuts are Vibration displacement in the direction of The nuts are Torsional displacement in direction; The rollers are Vibration displacement in the direction of The rollers are Torsional displacement in direction; is the nominal radius of the nut; is the nominal radius of the roller; Normal force exist The projection angle of the surface; Normal force exist The projection angle of the surface; Normal force exist Projection of the surface and x The angle between the axes;

[0205] The differential equation of dynamic motion of the nut-roller side thread engagement unit is shown in Equation (40), and its matrix form is shown in Equation (41):

[0206]

[0207] (41)

[0208] Where, The nut-roller contact side q The mass matrix of the thread engagement element; is the displacement matrix of the two nodes of the threaded meshing element on the nut-roller contact side; is the stiffness matrix of the thread engagement element on the nut-roller contact side; is the damping matrix of the threaded meshing element on the nut-roller contact side. The specific expressions of the mass matrix, displacement matrix, stiffness matrix, and damping matrix are shown in Equations (42-45):

[0209]

[0210] Where, For nut q Thread quality; 、 and Nut No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; For roller q Thread quality; 、 and Roller No. q Thread teeth in x 、 y 、 z moment of inertia in the direction;

[0211]

[0212] Where, 、 、 Nut No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Nut No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; 、 and Roller No. qThread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Torsional displacement in direction;

[0213] (44)

[0214] (45)

[0215] Where, is the thread engagement stiffness on the nut-roller contact side, which can be calculated by formula (46); is the thread engagement damping on the nut-roller contact side, which can be calculated by formula (47); V N is the nut-roller side projection matrix, as shown in formula (48):

[0216] (46)

[0217] (47)

[0218]

[0219] Where, and are the normal contact force and normal contact deformation on the nut-roller contact side, respectively. Their calculation method is the same as that on the screw-roller side. is the axial contact force on the nut-roller contact side; α N is the flank angle of the nut-roller contact side; and are the masses of the two meshing nodes on the nut-roller contact side respectively.

[0220] S5: If Figure 7 As shown in the figure, the dynamic modeling of the roller gear-internal gear meshing unit is as follows: According to the force analysis diagram of the roller gear-internal gear, the relative displacement of the roller gear-internal gear is derived, and the dynamic motion differential equation of the roller gear-internal gear meshing unit is obtained. The relative displacement of the roller gear-internal gear is:

[0221]

[0222] (50)

[0223] Where, The inner ring gear is Vibration displacement in the direction of The inner ring gear is Torsional displacement in direction; The roller teeth are Vibration displacement in the direction of The roller teeth are Torsional displacement in direction; is the pitch circle radius of the inner gear ring; is the roller gear pitch circle radius; is the gear helix angle; is the gear pressure angle; is the sum of the gear pressure angle and the roller phase angle.

[0224] The differential equation of dynamic motion of the roller gear-ring gear meshing unit is shown in Equation (51), and its matrix form is shown in Equation (52):

[0225] (52)

[0226] Where, Roller gear-internal gear q The mass matrix of each gear meshing unit; Roller gear-internal gear q The displacement matrix of two nodes of a gear meshing unit; Roller gear-internal gear q Gear mesh element stiffness matrix; Roller gear-internal gear q Gear meshing unit damping matrix; is the column vector formed by the components of the normal impact force of the gear pair on each degree of freedom. Among them, the specific expressions of the mass matrix, displacement matrix, stiffness matrix and damping matrix are shown in formulas (53-57):

[0227] (53)

[0228] Where, For the inner ring gear q The quality of each gear tooth; 、 and The inner ring gear q The gear teeth x 、 y 、 z moment of inertia in the direction; For roller q The mass of roller teeth; 、 and Roller No.q Roller teeth in x 、 y 、 z moment of inertia in the direction;

[0229] (54)

[0230] Where, 、 、 The inner ring gear q Vibration displacement of each gear tooth in the x, y, and z directions; 、 、 The inner ring gear q Torsional displacement of each gear tooth in the x, y, and z directions; 、 、 Roller No. q Vibration displacement of each roller tooth in the x, y, and z directions; 、 、 Roller No. q Torsional displacement of each roller tooth in the x, y, and z directions;

[0231] (55)

[0232] (56)

[0233] (57)

[0234] Where, is the average meshing stiffness of the roller gear-ring gear pair on the contact side. Its calculation method can be found in the reference [Compilation of Components and Related Standards, Gears and Gear Transmission Volume (Volume 1). Beijing: China Standards Press, 2012.] is the meshing damping of the roller gear-ring gear contact side gear pair, which can be calculated by formula (58); is the normal impact force of the gear pair; V r is the roller gear-ring contact side projection matrix, as shown in formula (59):

[0235] (58)

[0236]

[0237] Where, and are the masses of the two meshing nodes on the contact side of the roller gear and the internal gear ring.

[0238] S6: As Figure 8 As shown in Figure 6, the dynamic model of the planetary roller screw system is based on the unit numbering, which assembles the mass, stiffness, damping matrix, and load column vector of the shaft segment unit, the screw-roller contact side thread pair meshing unit, the nut-roller contact side thread pair meshing unit, and the roller tooth-internal gear ring meshing unit. The assembly process of the system's overall stiffness matrix is the same as the assembly process of the finite element overall stiffness matrix in structural mechanics analysis. That is, according to the correspondence between the local number of each unit node and the overall number of the system node, the sub-matrix corresponding to each degree of freedom of the unit matrix is superimposed on the corresponding position of the overall matrix. The matrix form of the differential equation of dynamic motion of the planetary roller screw system is shown in Equation (60):

[0239] (60)

[0240] Where, is the overall mass matrix of the planetary roller screw system; is the overall damping matrix of the planetary roller screw system; is the overall stiffness matrix of the planetary roller screw system; its specific expression is shown in Equations (61-63).

[0241] (61)

[0242] (62)

[0243] (63)

[0244] Where, 、 、 and 、 、 The first and second shaft segments are q The mass matrix, damping matrix and stiffness matrix of each shaft segment unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix and stiffness matrix of the screw-roller contact side thread engagement unit; 、 、 and 、 、 The first and q Mass matrix, damping matrix, and stiffness matrix of a roller tooth-ring gear meshing unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix and stiffness matrix of the thread engagement element on the nut-roller contact side;

[0245] S7: In order to verify the accuracy of the planetary roller screw dynamic model based on the generalized finite element method proposed in this paper, two methods are used for comparative verification: first, the static contact force is calculated by the theoretical formula, and secondly, the literature [Dynamic contact load characteristics of synchronous meshing of planetary roller screw thread pair and gear pair, Journal of Xi'an Jiaotong University, 2022, 56(10), 11-21.] uses finite element simulation to obtain the steady-state value of the dynamic contact force. Finally, the calculation results of the two methods are compared and analyzed with the steady-state value of the dynamic contact force calculated by the dynamic model in this paper.

[0246] According to the structural parameters and load conditions of the planetary roller screw given in Table 1, the stiffness parameters of the planetary roller screw are calculated and shown in Table 2. During the calculation process, the number of rollers is set to 3, and the system is discretized into 18 nodes. The dynamic model thus established has 108 degrees of freedom. Based on this model, the time domain response of the dynamic contact force on the screw-roller contact side and the nut-roller contact side of the planetary roller screw system is solved, and its change rules are as follows: Figure 9 and Figure 10 shown.

[0247] On this basis, the static contact forces on the screw-roller contact side and the nut-roller contact side are calculated using Equations (29) to (31) and (46), respectively, and compared with the steady-state values of the dynamic contact forces of the threaded pair obtained by the proposed model. The results are shown in Table 3. The comparative analysis shows that the relative error between the steady-state values of the dynamic contact forces calculated by the proposed model and the static contact forces does not exceed 2%, verifying the correctness of the proposed dynamic model.

[0248] According to the research in the literature [Dynamic contact load characteristics of synchronous meshing of planetary roller screw thread pair and gear pair, Journal of Xi'an Jiaotong University, 2022, 56(10), 11-21], the planetary roller screw was simulated and analyzed using the finite element method under the working conditions of 3 rollers, 5000N axial load, and 30rad / s screw speed. The dynamic model established in this invention numerically solves the dynamic contact force of the screw-roller contact side and the nut-roller contact side while maintaining the same structural parameters, material parameters, and working conditions. The results are compared in the figure below. Figure 11 、 Figure 12The comparative analysis of the results shown in Table 4 shows that the relative errors between the steady-state values of the dynamic contact force of the threaded pair calculated by the dynamic model of the present invention and the finite element simulation results in the literature are 0.65% and 0.33%, respectively. The two are highly consistent, further confirming the correctness and reliability of the established dynamic model.

[0249] In summary, the planetary roller screw dynamic modeling method proposed in this paper, based on the concept of generalized finite element analysis, can accurately describe the distribution characteristics of shaft segment units and accurately reflect the complex connection relationships between components, making the dynamic model more closely aligned with the physical model. Furthermore, this modeling method effectively reduces repetitive modeling, significantly shortens calculation time, and improves analysis efficiency compared to finite element software analysis.

[0250] Table 1 Planetary roller screw structural parameters

[0251]

[0252] Table 2 Static contact force and stiffness parameters of planetary roller screw

[0253]

[0254] Table 3 Comparison of steady-state values of static contact force and dynamic contact force

[0255]

[0256] Table 4 Comparison of finite element simulation results and steady-state dynamic contact force results in the literature

[0257]

[0258] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and purpose of the present invention.

Claims

1. A planetary roller screw dynamics modeling method based on the generalized finite element method, characterized in that The specific steps are as follows: The finite element model of the planetary roller screw is divided into units: the screw, nut, roller and inner ring of the planetary roller screw are discretized into nodes, and divided into shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units and roller tooth-inner ring gear meshing units; Construct dynamic models of various types of units: establish the mass matrix, stiffness matrix, damping matrix, and load column vector of the shaft segment unit, the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller gear-ring gear meshing unit respectively; System dynamics model assembly: According to the mapping relationship between unit nodes and global nodes, the unit matrices are assembled into the system-level overall mass matrix, overall stiffness matrix, and overall damping matrix, thus completing the dynamic modeling of the planetary roller screw; The shaft segment unit is modeled using the Timoshenko beam unit, the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller tooth-ring gear meshing unit are modeled based on the relative displacement relationship of the contact points, and the nodes of each contact unit coincide with the nodes of the shaft segment unit; The dynamic model expression of the shaft segment unit is as follows: Where q T The generalized coordinates of the two nodes of the shaft segment element; M T is the consistent mass matrix of the shaft segment element; K T is the stiffness matrix of the shaft segment element; C T is the damping matrix of the shaft segment element; Mass matrix M T The expression is as follows: Where, A is the cross-sectional area of the shaft segment; ρ is the material density; l is the length of the shaft segment; I is the polar moment of inertia; I x For yz The moment of inertia of the area in the coordinate plane, I y For xz moment of inertia of the section in the coordinate plane; Stiffness matrix K T The expression is as follows: Where, E is the elastic modulus of the material; G is the shear elastic modulus of the material; k The correction factor introduced to take into account that the actual shear strain and shear stress are not uniformly distributed is the shear coefficient of the material; Damping Matrix C T The expression is as follows: Where, α 0. α 1 is the mass proportional coefficient and stiffness proportional coefficient in Rayleigh damping.

2. The planetary roller screw dynamics modeling method based on the generalized finite element method according to claim 1, characterized in that: The relative displacement relationship between the screw-roller contact side thread meshing unit, the nut-roller contact side thread meshing unit, and the roller gear-ring gear meshing unit is as follows: The relative displacement of the screw-roller contact side thread engagement unit is: Where, The screw is Vibration displacement in the direction; The screw is Torsional displacement in direction; The rollers are Vibration displacement in the direction of The rollers are Torsional displacement in direction; is the nominal radius of the screw; is the nominal radius of the roller; Normal force exist The projection angle of the surface; Normal force exist The projection angle of the surface; Normal force exist Projection of the surface and x The angle between the axes; is the roller phase angle, j Indicates the serial number of the roller, is the total number of rollers; n is the number of thread starts; Indicates the coordinate system of the roller part around which z The angle through which the axis rotates; symbol " "and" The upper part of the screw rotates counterclockwise, and the lower part of the screw rotates clockwise; The relative displacement of the threaded engagement unit on the nut-roller contact side is: Where, The nuts are Vibration displacement in the direction of The nuts are Torsional displacement in direction; The rollers are Vibration displacement in the direction of The rollers are Torsional displacement in direction; is the nominal radius of the nut; is the nominal radius of the roller; Normal force exist The projection angle of the surface; Normal force exist The projection angle of the surface; Normal force exist Projection of the surface and x The angle between the axes; The relative displacement of the roller gear-ring gear meshing unit is: Where, The inner ring gear is Vibration displacement in the direction of The inner ring gear is Torsional displacement in direction; The roller teeth are Vibration displacement in the direction of The roller teeth are Torsional displacement in direction; is the pitch circle radius of the inner gear ring; is the roller gear pitch circle radius; is the gear helix angle; is the gear pressure angle; is the sum of the gear pressure angle and the roller phase angle.

3. The planetary roller screw dynamics modeling method based on the generalized finite element method according to claim 2, characterized in that: The dynamic model expression of the screw-roller contact side thread engagement unit is as follows: Where, The screw-roller contact side q The mass matrix of the thread engagement element; The screw-roller contact side q The displacement matrix of two nodes of a threaded meshing element; The screw-roller contact side q The stiffness matrix of the threaded meshing element; The screw-roller contact side q The damping matrix of the threaded meshing element; Mass Matrix The expression is as follows: Where, For the screw q Thread quality; 、 and Screw No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; For roller q Thread quality; 、 and Roller No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; Displacement Matrix The expression is as follows: Where, 、 and Screw No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Screw No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; Stiffness matrix The expression is as follows: Where, V is the thread engagement stiffness of the screw-roller contact side; S is the screw-roller side projection matrix, which is expressed as follows: Damping Matrix The expression is as follows: Where, The thread engagement damping on the screw-roller contact side.

4. The planetary roller screw dynamics modeling method based on the generalized finite element method according to claim 3, characterized in that: The dynamic model expression of the thread engagement unit on the nut-roller contact side is as follows: Where, The nut-roller contact side q The mass matrix of the thread engagement element; The nut-roller contact side q The displacement matrix of two nodes of a threaded meshing element; The nut-roller contact side q The stiffness matrix of the threaded meshing element; The nut-roller contact side q The damping matrix of the threaded meshing element; Mass Matrix The expression is as follows: Where, For nut q Thread quality; 、 and Nut No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; For roller q Thread quality; 、 and Roller No. q Thread teeth in x 、 y 、 z moment of inertia in the direction; Displacement Matrix The expression is as follows: Where, 、 、 Nut No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Nut No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Vibration displacement in the direction; 、 and Roller No. q Thread teeth in x 、 y 、 z Torsional displacement in direction; Stiffness matrix The expression is as follows: Where, V is the thread engagement stiffness of the nut-roller contact side; N is the nut-roller side projection matrix, expressed as: Damping Matrix The expression is as follows: Where, is the thread engagement damping on the nut-roller contact side.

5. The planetary roller screw dynamics modeling method based on the generalized finite element method according to claim 4, characterized in that: The dynamic model expression of the roller gear-ring gear meshing unit is as follows: Where, Roller gear-internal gear q The mass matrix of each gear meshing unit; Roller gear-internal gear q The displacement matrix of two nodes of a gear meshing unit; Roller gear-internal gear q Gear mesh element stiffness matrix; Roller gear-internal gear q Gear meshing unit damping matrix; It is the column vector formed by the components of the normal impact force of the gear pair in each degree of freedom; Mass Matrix The expression is as follows: Where, For the inner ring gear q The quality of each gear tooth; 、 and The inner ring gear q The gear teeth x 、 y 、 z moment of inertia in the direction; For roller q The mass of roller teeth; 、 and Roller No. q Roller teeth in x 、 y 、 z moment of inertia in the direction; Displacement Matrix The expression is as follows: Where, 、 、 The inner ring gear q The gear teeth x 、 y 、 z Vibration displacement in the direction; 、 、 The inner ring gear q The gear teeth x 、 y 、 z Torsional displacement in direction; 、 、 Roller No. q Roller teeth in x 、 y 、 z Vibration displacement in the direction; 、 、 Roller No. q Roller teeth in x 、 y 、 z Torsional displacement in direction; Stiffness matrix The expression is as follows: Where, V is the meshing stiffness of the roller gear-ring gear contact side; r is the projection matrix of the roller gear-ring contact side, and its expression is: Damping Matrix The expression is as follows: Where, is the meshing damping of the roller gear-ring gear contact side gear pair; Loading column vector The expression is as follows: Where, is the normal impact force of the gear pair.

6. The planetary roller screw dynamic modeling method based on the generalized finite element method according to claim 5, characterized in that: When assembling the unit matrices, the sub-blocks of the unit stiffness matrix are superimposed on the corresponding positions of the global matrix according to the correspondence between the local number and the global number of the unit node; The expression of the differential equation of the planetary roller screw dynamic motion is as follows: Where, is the overall mass matrix of the planetary roller screw system; is the overall damping matrix of the planetary roller screw system; is the overall stiffness matrix of the planetary roller screw system; where, Where, 、 、 and 、 、 The first and second shaft segments are q The mass matrix, damping matrix and stiffness matrix of each shaft segment unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix and stiffness matrix of the screw-roller contact side thread engagement unit; 、 、 and 、 、 The first and q Mass matrix, damping matrix, and stiffness matrix of a roller tooth-ring gear meshing unit; 、 、 and 、 、 The first and q The mass matrix, damping matrix, and stiffness matrix of the threaded meshing element on the nut-roller contact side.

7. The planetary roller screw dynamic modeling method based on the generalized finite element method according to claim 6, characterized in that: The numerical integration method for the differential equation of dynamic motion of the planetary roller screw is used to solve the dynamic response, specifically including: Divide the total load into several load steps and apply them gradually; Update the contact state after each calculation step, and dynamically adjust the stiffness matrix and adjoint matrix; The convergence is judged by the residual error, and if it does not converge, it is repeated.

8. A planetary roller screw dynamics modeling system based on the generalized finite element method, used to implement the planetary roller screw dynamics modeling method based on the generalized finite element method according to any one of claims 1 to 7; characterized in that: Includes the following modules: Structural discretization module: used to discretize the screw, nut, roller and internal gear ring of the planetary roller screw into global nodes, and divide them into shaft segment units, screw-roller contact side thread meshing units, nut-roller contact side thread meshing units and roller tooth-internal gear ring gear meshing units; Unit dynamics modeling module: Based on the generalized finite element method, it generates the mass matrix, stiffness matrix, damping matrix and load vector of each unit; System matrix assembly module: Based on the global number mapping relationship of unit nodes, each unit matrix is superimposed into the system-level overall matrix according to the degree of freedom; Dynamic response solution module: Based on the overall mass matrix, overall stiffness matrix and overall damping matrix, the system dynamics differential equation is established and the dynamic response is solved; Model verification module: compares the relative error between the dynamic contact force calculation results and the theoretical value to ensure that the model accuracy error is less than the threshold.

Images