Planetary roller screw gear-thread coupling excitation dynamics modeling method

Abstract

The invention discloses a planetary roller screw gear-thread coupling excitation dynamics modeling method, which comprises the following steps: step (1), establishing a planetary roller screw stress model, and analyzing thread meshing elastic deformation; (2) establishing a calculation model of a gear pair system, and analyzing gear engagement excitation and time-varying engagement stiffness excitation; (3) establishing a mechanical equation set of a planetary roller screw gear pair and a thread pair, and analyzing the influence of gear meshing excitation on the thread dynamic contact force; (4) establishing a nonlinear dynamic differential equation set of the planetary roller screw transmission system; (5) reasonable parameters are selected to solve nonlinear response of the system; the method has the beneficial effects that the nonlinear dynamic response of the planetary roller screw considering the gear meshing and thread meshing coupling excitation can be solved, the current vacancy of the nonlinear dynamic modeling method of the planetary roller screw considering the gear meshing and thread meshing coupling excitation is filled, and the method has good universality.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear dynamics, and in particular to a planetary roller screw gear-thread coupling excitation dynamics modeling method. Background Art

[0002] A planetary roller screw is a high-precision transmission device that converts rotational and linear motion through threaded engagement. It boasts advantages such as small size, low noise, strong load-bearing capacity, high speed, and long life. It has broad application prospects in aircraft control surfaces, high-precision machine tools, and humanoid robots, playing a particularly key role in the linear joints of humanoid robots. However, due to its complex structure and significant nonlinear motion characteristics, its dynamic response and vibration characteristics are highly sensitive to changes in operating conditions, which has a direct impact on its transmission efficiency and service life. Existing technologies do not consider the vibration characteristics of planetary roller screws under the coupled excitation of gear meshing and thread meshing. Therefore, in-depth research on the nonlinear dynamic characteristics of PRSMs is an important topic for achieving precise robot control and system optimization.

[0003] In order to solve the above problems, the present invention proposes a planetary roller screw gear-thread coupling excitation dynamic modeling method, which accurately and effectively reveals the vibration characteristics of the planetary roller screw under the coupling excitation of gear meshing and thread meshing; fills the technical gap in the coupling dynamic modeling of planetary roller screw transmission systems for humanoid robots, promotes the development of engineering technology, and can generate greater social and economic benefits. Summary of the Invention

[0004] To overcome the problem that current research on planetary roller screws focuses primarily on thread contact characteristics and structural parameter design, with little consideration of the effects of thread elastic deformation and gear pair meshing excitation on load distribution and system dynamics, making it difficult for existing research to accurately describe the nonlinear dynamic behavior of planetary roller screws under the combined action of gear pair meshing excitation and thread pair meshing excitation in actual applications, the present invention provides a planetary roller screw gear-thread coupled excitation dynamic modeling method. This method derives a thread pair dynamic load distribution model including gear pair meshing excitation, establishes a system vibration equation, and reveals its dynamic response characteristics through numerical analysis, providing theoretical support for accurately describing nonlinear dynamic behavior under complex excitations.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a planetary roller screw gear-thread coupling excitation dynamic modeling method, characterized by comprising the following steps:

[0006] 1. A planetary roller screw gear-thread coupling excitation dynamics modeling method, characterized by comprising the following steps:

[0007] Step (1): Establish a force model for the planetary roller screw and analyze the elastic deformation of the thread engagement; when the planetary roller screw is under load, the normal contact force between the thread teeth can be decomposed into the axial component force F ak , radial force F ck and the tangential force F rk , respectively expressed as follows:

[0008]

[0009] Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and F L is the external load, λ r is the roller helix angle, β t is the contact angle;

[0010] Deformation of the screw or roller thread caused by radial force and deformation of nut thread teeth They are:

[0011]

[0012] Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and F r is the radial force, F L is the external load, β r is the roller tooth flank angle, d ak is the equivalent thread major diameter, μ is the Poisson's ratio of the thread material, E is the elastic modulus, p is the thread pitch, D nm is the major diameter of the nut thread, D ns is the nut thread diameter, β n is the nut flank angle, β t is the contact angle;

[0013] Axial force F a Total axial deformation caused by Can be calculated by thread profile parameters:

[0014]

[0015] in, Deformation caused by bending or shear force, Deformation caused by root shearing and is the deformation caused by the inclination of the tooth root, F a is the axial force, μ is the Poisson's ratio of the thread material, p is the thread pitch, E is the elastic modulus, k is s, r, n, representing the screw, roller, and nut respectively, a k Indicates the width of the thread bottom, b k Indicates the thread thickness, c k is the thread crest width, β kis the thread flank angle;

[0016] Normal contact deformation of the thread contact points on the screw-roller contact side and the nut-roller contact side for:

[0017]

[0018] Where k is s and n represents the screw and nut respectively, F(φ) represents the first kind of elliptic integral, E' is the equivalent elastic modulus, ε represents the correlation coefficient with the curvature difference function, F T is the contact force, Σρ is the sum of the principal curvatures of the contact points, μ r and E r is the Poisson's ratio and elastic modulus of the roller, μ k and E k Poisson's ratio and elastic modulus respectively; ρ kk is the curvature of the corresponding curve of contact body 1 in plane I, ρ rk is the curvature of the corresponding curve of contact body 1 in plane II, ρ kr and ρ rr is the curvature of contact body 2 in planes I and II, which is expressed as follows:

[0019]

[0020] Among them, r rp is the radius of the roller thread arc profile, β r is the roller tooth flank angle, d s and d r are the nominal diameters of the screw and roller respectively;

[0021] Thread axial contact deformation and radial contact deformation It can be calculated by the following formula:

[0022]

[0023] Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and β t is the normal contact angle, λ r is the roller helix angle, is the normal contact deformation, k T is the contact stiffness, F L is the external load;

[0024] Step (2): Establish a calculation model of the gear pair system and analyze the gear meshing excitation and time-varying meshing stiffness excitation; at the meshing point outside the line between the roller gear and the inner ring gear, the linear speeds of the roller gear and the inner ring gear are v and v respectively. G and v R, decompose these two velocities in the direction of the instantaneous meshing line to obtain the impact velocity v along this direction m :

[0025]

[0026] Among them, v sG 、v sR v G 、v R The component on the instantaneous meshing line, ω R 、ω G are the angular velocities of the ring gear teeth and the roller gear teeth, r bG is the base circle radius of the internal gear ring teeth, r' bR is the instantaneous value of the base circle radius of the inner ring gear, a is the center distance, and α is the engagement angle;

[0027] When the gear pair is impacted, the gear tooth surface will produce a certain elastic deformation, and this deformation reaches the maximum value δ m When the impact force on the gear pair increases, it will increase until it reaches its peak value, that is, the maximum impact force F m ; Maximum deformation δ between internal meshing gear pairs m , Maximum impact force F m and impact energy E k The relationship between them is shown as follows:

[0028]

[0029] Where b is the tooth width, v m is the impact velocity, l m is the comprehensive flexibility of the roller gear teeth and the inner ring gear teeth at the impact point, l1 and l2 are the bending deformation flexibility of the roller gear teeth and the inner ring gear teeth at the impact point, l t is the contact flexibility of the gear teeth at the impact point, m q1 、m q2 are the equivalent masses of the roller gear and the internal gear ring on the instantaneous meshing line;

[0030] The maximum impact force F when the roller gear teeth mesh with the internal gear ring teeth m for:

[0031]

[0032] Among them, v m is the impact velocity, b is the tooth width, I1 and I2 are the moments of inertia of the roller and the inner gear ring respectively, r br 、r bgare the base circle radii of the roller gear and the inner ring gear, l1 and l2 are the bending deformation flexibility of the roller gear and the inner ring gear at the impact point, respectively. t is the contact compliance of the gear teeth at the impact point;

[0033] The collision impact process is a semi-sine pulse of impact force and impact time, so the impact force F of the roller gear and the inner gear ring is m (t) is:

[0034]

[0035] Among them, F m is the maximum impact force, t is the engagement time, t m is the meshing impact time;

[0036] The meshing overlap between the roller gear and the inner gear ring is e>1, and there is a state of alternating single and double teeth meshing. The time-varying meshing stiffness k of the gear pair is considered by correcting the matrix stiffness of the double tooth meshing area. m for:

[0037]

[0038] Among them, ξ g and ξ r are the base correction coefficients of the roller gear and the internal gear ring, k bg and k br are the base stiffness of the roller gear and the inner ring gear, k t is the meshing stiffness when meshing with corresponding overlap;

[0039]

[0040] in, is the stiffness of the j-th gear pair, and are the stiffness of the jth pair of rollers and the inner gear ring, is the nonlinear Hertzian contact stiffness, and are the bending stiffness, shear stiffness and axial compression stiffness of the roller and the j-th tooth of the inner ring respectively;

[0041] Maximum meshing excitation between roller gear teeth and internal gear ring teeth for:

[0042]

[0043] Among them, k m (t) is the gear ring meshing stiffness, Δy is the relative displacement of the gear ring, y R (t) and y G(t) are the linear vibration displacements of a point on the base circle of the inner ring gear and the roller gear respectively;

[0044] Step (3): Establish the mechanical equations of the planetary roller screw gear pair and the thread pair, and analyze the influence of gear meshing excitation on the dynamic contact force of the thread; according to the normal vector analysis results of the screw and roller spiral surface at the contact point, the contact force F ts It can be expressed as:

[0045]

[0046] Among them, F ts is the contact force amplitude, λ s and β s They represent the helix angle and flank angle at the contact point between the screw and the roller, θ sr It represents the meshing angle of the roller on one side of the screw, F L is the external load, β t is the contact angle;

[0047] Roller-nut side contact force F tn :

[0048]

[0049] Among them, F tn is the static contact force amplitude on the roller-nut side, λ n and β n They represent the helix angle and flank angle at the contact point between the nut and the roller, θ nr It represents the meshing angle of the roller on the nut side, F L is the external load, β t is the contact angle;

[0050] Contact force between the roller and the screw considering the effect of gear meshing and the contact force between the roller and the nut side It can be expressed as:

[0051]

[0052] Among them, F ts is the contact force amplitude between roller and screw, F tn is the contact force amplitude between roller and nut, F m is the maximum impact force, F G is the meshing excitation, t s is the time of the impact of the outside bite, t e The time when a pair of gear teeth meshing ends;

[0053] Step (4): Establish the nonlinear dynamic equations of the planetary roller screw transmission system; consider the vibration displacement of the roller and cage in the x, y directions and around their own axis, and the vibration displacement of the screw in the z-axis direction according to the concentrated mass method.

[0054] X=[z s ,x pn ,y pn ,z pn ,u pn ,x c ,y c ,u c ,x nr ,y nr ,u nr ](n=1,2,3,…,n roller ),

[0055] Among them, pn represents the nth roller shaft end gear, n roller represents the number of rollers, s represents the screw, c represents the cage, nr represents the inner gear ring at the nut shaft end, x, y, z represent the vibration displacement along the axis of their respective coordinates, and u represents the vibration angular displacement around the rotary axis;

[0056] From the above model, the vibration differential equation of the planetary roller screw transmission system and the screw dynamics differential equation can be derived:

[0057]

[0058] Among them, m s is the mass of the screw, z s is the vibration displacement of the screw, k sz is the support stiffness of the screw thread in the z-axis direction, c sz is the support damping of the screw thread in the z direction, F sr is the contact force between roller and screw, F fsqz is the friction force between the screw and the roller thread in the z direction;

[0059] Differential equation for roller dynamics:

[0060]

[0061] Among them, m q is the mass of the roller, ω q is the angular velocity of the roller, x pn 、y pn 、z pn are the vibration displacements of the cylindrical gear in the x, y, and z directions, respectively, and k qx 、k qy 、k qz are the support stiffness of the roller in the x, y, and z directions, δ rnis the projection of the displacement of the gear relative to the nut along the meshing line, φ rn is the angle between the line connecting the screw center and the roller center and the X-axis, δ cnx , δ cny are the vibration displacements of the cage in the x and y directions, c qx 、c qy are the support damping of the roller in the x and y directions, δ xn , δ yn , δ zn are the vibration displacements of the roller in the x, y, and z directions, respectively, and I q is the moment of inertia of the roller, r q is the pitch circle radius of the roller, u pn is the torsional displacement of the cylindrical gear around the axis, k qu is the torsional stiffness of the roller, c qu is the torsional damping of the roller, F sqx 、F sqy are the resultant forces of the contact force between the screw and roller thread and the meshing force of the gear pair in the x and y directions respectively, rqx 、F rqy are the resultant forces of the contact force between the nut and the roller thread and the meshing force of the gear pair in the x and y directions, respectively, fsqx 、F fsqy are the resultant forces of the friction between the roller and the screw thread and the friction between the gear pair in the x and y directions, respectively. frqx 、F frqy are the resultant friction between the roller and the nut thread and the gear pair in the x and y directions, respectively, and F srz 、F nrz are the components of the contact force between the roller and the screw and between the roller and the nut in the z direction, T in is the input torque of the roller;

[0062] The cage vibration differential equation is expressed as:

[0063]

[0064] Among them, m c is the mass of the cage, x c 、y c are the two vibration displacements of the cage in the x and y directions, ω q is the angular velocity of the roller, k qx 、k qy are the support stiffness of the roller in the x and y directions, k cx 、k cy are the support stiffness of the cage in the x and y directions, δ cnx , δ cny are the vibration displacements of the cage in the x and y directions, ccx 、c cy are the support damping of the cage in the x and y directions, I c is the moment of inertia of the cage, r c is the pitch circle radius of the cage, u c k is the torsional displacement of the cage around the axis, qu is the torsional stiffness of the roller, k cu is the torsional stiffness of the cage, c cu is the torsional damping of the cage, δ cnu is the projection of the cage's displacement relative to the roller along the cage's tangent direction;

[0065] The differential equation of nut vibration is expressed as:

[0066]

[0067] Among them, m nr is the mass of the nut, x nr 、y nr 、z nr are the vibration displacements of the nut in the x, y, and z directions, ω q is the roller rotation angular velocity, F rqx 、F rqy are the resultant forces of the contact force between the nut and the roller thread and the meshing force of the gear pair in the x and y directions, respectively, frqx 、F frqy are the resultant forces of the friction between the nut and roller thread and the friction between the gear pair in the x and y directions, respectively, k rx 、k ry 、k rz are the support stiffness of the nut in the x, y, and z directions, δ rn is the projection of the displacement of the gear relative to the nut along the meshing line, φ rn c is the angle between the line connecting the center of the screw and the center of the roller and the X axis, rx 、c ry 、c rz are the support damping of the inner gear ring in the x, y, and z directions, respectively, and F nrz is the component of the contact force between the roller and the nut in the z direction, I nr is the moment of inertia of the nut, r nr is the pitch circle radius of the nut, u nr is the torsional displacement of the nut around the axis, k ru is the torsional stiffness of the nut, c ru is the torsional damping of the nut, T out is the output torque of the nut;

[0068] Step (5): Select reasonable parameters to calculate the vibration response of the system, and establish the evaluation index of the stability of the planetary roller screw transmission system based on the calculation results to guide the selection of system parameters; the planetary roller screw system exhibits rich nonlinear behavior under external excitation, and the influence of the excitation frequency on the vibration characteristics of the planetary roller screw transmission system is studied through the bifurcation diagram.

[0069] Compared with the existing technology, the beneficial effects of the present invention are: under the premise of considering the influence of thread elastic deformation and gear pair meshing excitation on load distribution and system dynamics, a nonlinear dynamic model of the planetary roller screw is established, and the nonlinear response characteristics of the system under external excitation are obtained, filling the technical gap in nonlinear dynamic modeling of planetary roller screw transmission systems for humanoid robots under the coupled excitation of gear meshing and thread meshing, and providing theoretical support for vibration reduction, noise reduction and performance improvement of planetary roller screw transmission. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 It is a flow chart of the planetary roller screw gear-thread coupling excitation dynamics modeling method;

[0071] Figure 2 It is the dynamic model of the planetary roller screw transmission system;

[0072] Figure 3 This is the dynamic load distribution curve of the thread pair of the planetary roller screw transmission system without considering the gear pair meshing excitation;

[0073] Figure 4 This is a curve diagram of the dynamic load distribution of the thread pair of the planetary roller screw transmission system considering the gear pair meshing excitation;

[0074] Figure 5 It is the bifurcation diagram of the vibration response of the planetary roller screw transmission system. DETAILED DESCRIPTION

[0075] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1-Figure 5 The specific embodiments of the present invention are described in detail.

[0076] Figure 1 The flowchart of a planetary roller screw gear-thread coupling excitation dynamics modeling method is as follows: Figure 1 As shown, the specific steps are as follows:

[0077] Step (1): Establish a force model for the planetary roller screw and analyze the elastic deformation of the thread engagement; when the planetary roller screw is under load, the normal contact force between the thread teeth can be decomposed into the axial component force F ak , radial force F ck and the tangential force F rk , respectively expressed as follows:

[0078]

[0079] Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and F L is the external load, λ r is the roller helix angle, β t is the contact angle;

[0080] Due to the difference in the structure of internal and external threads, the deformation of thread teeth caused by radial force is also different. For external thread parts, it can be equivalent to a solid cylinder with a diameter equal to the major diameter of the thread. For internal thread parts, it can be equivalent to a small diameter D ns , the major diameter is D nm The radial force causes the screw or roller thread to deform. and deformation of nut thread teeth They are:

[0081]

[0082] Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and F r is the radial force, F L is the external load, β r is the roller tooth flank angle, d ak is the equivalent thread major diameter, μ is the Poisson's ratio of the thread material, E is the elastic modulus, p is the thread pitch, D nm is the major diameter of the nut thread, D ns is the nut thread diameter, β n is the nut flank angle, β t is the contact angle;

[0083] Axial force F a Total axial deformation caused by It can be calculated from the thread profile parameters and expressed as the sum of deformations caused by bending, shear, root tilting and root shearing:

[0084]

[0085] in, Deformation caused by bending or shear force, Deformation caused by root shearing and is the deformation caused by the inclination of the tooth root, F a is the axial force, μ is the Poisson's ratio of the thread material, p is the thread pitch, E is the elastic modulus, k is s, r, n, representing the screw, roller, and nut respectively, a k Indicates the width of the thread bottom, b k Indicates the thread thickness, c k is the thread crest width, βk is the thread flank angle;

[0086] Normal contact deformation of the thread contact points on the screw-roller contact side and the nut-roller contact side for:

[0087]

[0088] Where k is s and n represents the screw and nut respectively, F(φ) represents the first kind of elliptic integral, E' is the equivalent elastic modulus, ε represents the correlation coefficient with the curvature difference function, F T is the contact force, Σρ is the sum of the principal curvatures of the contact points, μ r and E r is the Poisson's ratio and elastic modulus of the roller, μ k and E k Poisson's ratio and elastic modulus respectively; ρ kk is the curvature of the corresponding curve of contact body 1 in plane I, ρ rk is the curvature of the corresponding curve of contact body 1 in plane II, ρ kr and ρ rr is the curvature of contact body 2 in planes I and II, which is expressed as follows:

[0089]

[0090] Among them, r rp is the radius of the roller thread arc profile, β r is the roller tooth flank angle, d s and d r are the nominal diameters of the screw and roller respectively;

[0091] Thread axial contact deformation and radial contact deformation It can be calculated by the following formula:

[0092]

[0093] Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and β t is the normal contact angle, λ r is the roller helix angle, is the normal contact deformation, k T is the contact stiffness, F L is the external load;

[0094] Step (2): Establish a calculation model of the gear pair system and analyze the gear meshing excitation and time-varying meshing stiffness excitation; at the meshing point outside the line between the roller gear and the inner ring gear, the linear speeds of the roller gear and the inner ring gear are v and v respectively. G and v R, decompose these two velocities in the direction of the instantaneous meshing line to obtain the impact velocity v along this direction m :

[0095]

[0096] Among them, v sG 、v sR v G 、v R The component on the instantaneous meshing line, ω R 、ω G are the angular velocities of the ring gear teeth and the roller gear teeth, r bG is the base circle radius of the internal gear ring teeth, r' bR is the instantaneous value of the base circle radius of the inner ring gear, a is the center distance, and α is the engagement angle;

[0097] When the gear pair is impacted, the gear tooth surface will produce a certain elastic deformation, and this deformation reaches the maximum value δ m When the impact force on the gear pair increases, it will increase until it reaches its peak value, that is, the maximum impact force F m During the entire impact process, the energy absorbed and released by the system, namely the impact energy E k , is closely related to the deformation and impact force of the gear pair. The maximum deformation δ generated between the internal meshing gear pairs m , Maximum impact force F m and impact energy E k The relationship between them is shown as follows:

[0098]

[0099] Where b is the tooth width, v m is the impact velocity, l m is the comprehensive flexibility of the roller gear teeth and the inner ring gear teeth at the impact point, l1 and l2 are the bending deformation flexibility of the roller gear teeth and the inner ring gear teeth at the impact point, l t is the contact flexibility of the gear teeth at the impact point, m q1 、m q2 are the equivalent masses of the roller gear and the internal gear ring on the instantaneous meshing line;

[0100] The maximum impact force F when the roller gear teeth mesh with the internal gear ring teeth m for:

[0101]

[0102] Among them, v m is the impact velocity, b is the tooth width, I1 and I2 are the moments of inertia of the roller and the inner gear ring respectively, r br 、rbg are the base circle radii of the roller gear and the inner ring gear, l1 and l2 are the bending deformation flexibility of the roller gear and the inner ring gear at the impact point, respectively. t is the contact compliance of the gear teeth at the impact point;

[0103] The collision impact process is a semi-sine pulse of impact force and impact time, so the impact force F of the roller gear and the inner gear ring is m (t) is:

[0104]

[0105] Among them, F m is the maximum impact force, t is the engagement time, t m is the meshing impact time;

[0106] The meshing overlap between the roller gear and the inner gear ring is e>1, and there is a state of alternating single and double teeth meshing. The time-varying meshing stiffness k of the gear pair is considered by correcting the matrix stiffness of the double tooth meshing area. m for:

[0107]

[0108] Among them, ξ g and ξ r are the base correction coefficients of the roller gear and the internal gear ring, k bg and k br are the base stiffness of the roller gear and the inner ring gear, k t is the meshing stiffness when meshing with corresponding overlap;

[0109]

[0110] in, is the stiffness of the j-th gear pair, and are the stiffness of the jth pair of rollers and the inner gear ring, is the nonlinear Hertzian contact stiffness, and are the bending stiffness, shear stiffness and axial compression stiffness of the roller and the j-th tooth of the inner ring respectively;

[0111] The meshing process between the roller gear and the internal gear ring can be simplified as a time-varying spring system acting along the meshing line, with the spring stiffness set to k. m (t), maximum meshing excitation between the roller gear teeth and the internal gear ring teeth for:

[0112]

[0113] Among them, km (t) is the gear ring meshing stiffness, Δy is the relative displacement of the gear ring, y R (t) and y G (t) are the linear vibration displacements of a point on the base circle of the inner ring gear and the roller gear respectively;

[0114] Step (3): Establish the mechanical equations of the planetary roller screw gear pair and the thread pair, and analyze the influence of gear meshing excitation on the dynamic contact force of the thread; according to the normal vector analysis results of the screw and roller spiral surface at the contact point, the contact force F ts It can be expressed as:

[0115]

[0116] Among them, F ts is the contact force amplitude, λ s and β s They represent the helix angle and flank angle at the contact point between the screw and the roller, θ sr It represents the meshing angle of the roller on one side of the screw, F L is the external load, β t is the contact angle;

[0117] Roller-nut side contact force F tn :

[0118]

[0119] Among them, F tn is the static contact force amplitude on the roller-nut side, λ n and β n They represent the helix angle and flank angle at the contact point between the nut and the roller, θ nr It represents the meshing angle of the roller on the nut side, F L is the external load, β t is the contact angle;

[0120] Since the roller is simultaneously excited by the gear pairs on both sides, two different types of excitation factors need to be considered when analyzing the meshing process between the roller and the inner ring gear: meshing excitation and time-varying meshing excitation. Based on these effects, the contact force between the roller and the screw side considering the influence of gear meshing is and the contact force between the roller and the nut side It can be expressed as:

[0121]

[0122] Among them, F ts is the contact force amplitude between roller and screw, F tn is the contact force amplitude between roller and nut, F m is the maximum impact force, FG is the meshing excitation, t s is the time of the impact of the outside bite, t e The time when a pair of gear teeth meshing ends;

[0123] Step (4): Establish the nonlinear dynamic equations of the planetary roller screw transmission system; Figure 2 The figure shows the system dynamics model. To simplify the calculation, only the Coulomb friction between the screw, roller, and nut thread is considered when establishing the system dynamics model. The friction between the cage and roller, as well as between the cage and the inner gear ring, is ignored. The force and motion state of each roller are the same. According to the concentrated mass method, the vibration displacement of the roller and cage in the x, y direction and around its own axis, as well as the vibration displacement of the screw in the z-axis direction, are considered.

[0124] X=[z s ,x pn ,y pn ,z pn ,u pn ,x c ,y c ,u c ,x nr ,y nr ,u nr ](n=1,2,3,…,n roller ),

[0125] Among them, pn represents the nth roller shaft end gear, n roller represents the number of rollers, s represents the screw, c represents the cage, nr represents the inner gear ring at the nut shaft end, x, y, z represent the vibration displacement along the axis of their respective coordinates, and u represents the vibration angular displacement around the rotary axis;

[0126] From the above model, the vibration differential equation of the planetary roller screw transmission system and the screw dynamics differential equation can be derived:

[0127]

[0128] Among them, m s is the mass of the screw, z s is the vibration displacement of the screw, k sz is the support stiffness of the screw thread in the z-axis direction, c sz is the support damping of the screw thread in the z direction, F sr is the contact force between roller and screw, F fsqz is the friction force between the screw and the roller thread in the z direction;

[0129] Differential equation for roller dynamics:

[0130]

[0131] Among them, m q is the mass of the roller, ω q is the angular velocity of the roller, x pn 、y pn 、z pn are the vibration displacements of the cylindrical gear in the x, y, and z directions, respectively, and k qx 、k qy 、k qz are the support stiffness of the roller in the x, y, and z directions, δ rn is the projection of the displacement of the gear relative to the nut along the meshing line, φ rn is the angle between the line connecting the screw center and the roller center and the X-axis, δ cnx , δ cny are the vibration displacements of the cage in the x and y directions, c qx 、c qy are the support damping of the roller in the x and y directions, δ xn , δ yn , δ zn are the vibration displacements of the roller in the x, y, and z directions, respectively, and I q is the moment of inertia of the roller, r q is the pitch circle radius of the roller, u pn is the torsional displacement of the cylindrical gear around the axis, k qu is the torsional stiffness of the roller, c qu is the torsional damping of the roller, F sqx 、F sqy are the resultant forces of the contact force between the screw and roller thread and the meshing force of the gear pair in the x and y directions respectively, rqx 、F rqy are the resultant forces of the contact force between the nut and the roller thread and the meshing force of the gear pair in the x and y directions, respectively, fsqx 、F fsqy are the resultant forces of the friction between the roller and the screw thread and the friction between the gear pair in the x and y directions, respectively. frqx 、F frqy are the resultant friction between the roller and the nut thread and the gear pair in the x and y directions, respectively, and F srz 、F nrz are the components of the contact force between the roller and the screw and between the roller and the nut in the z direction, T in is the input torque of the roller;

[0132] The cage vibration differential equation is expressed as:

[0133]

[0134] Among them, m c is the mass of the cage, x c 、yc are the two vibration displacements of the cage in the x and y directions, ω q is the angular velocity of the roller, k qx 、k qy are the support stiffness of the roller in the x and y directions, k cx 、k cy are the support stiffness of the cage in the x and y directions, δ cnx , δ cny are the vibration displacements of the cage in the x and y directions, c cx 、c cy are the support damping of the cage in the x and y directions, I c is the moment of inertia of the cage, r c is the pitch circle radius of the cage, u c k is the torsional displacement of the cage around the axis, qu is the torsional stiffness of the roller, k cu is the torsional stiffness of the cage, c cu is the torsional damping of the cage, δ cnu is the projection of the cage's displacement relative to the roller along the cage's tangent direction;

[0135] The differential equation of nut vibration is expressed as:

[0136]

[0137] Among them, m nr is the mass of the nut, x nr 、y nr 、z nr are the vibration displacements of the nut in the x, y, and z directions, ω q is the roller rotation angular velocity, F rqx 、F rqy are the resultant forces of the contact force between the nut and the roller thread and the meshing force of the gear pair in the x and y directions, respectively, frqx 、F frqy are the resultant forces of the friction between the nut and roller thread and the friction between the gear pair in the x and y directions, respectively, k rx 、k ry 、k rz are the support stiffness of the nut in the x, y, and z directions, δ rn is the projection of the displacement of the gear relative to the nut along the meshing line, φ rn c is the angle between the line connecting the center of the screw and the center of the roller and the X axis, rx 、c ry 、c rz are the support damping of the inner gear ring in the x, y, and z directions, respectively, and F nrz is the component of the contact force between the roller and the nut in the z direction, I nris the moment of inertia of the nut, r nr is the pitch circle radius of the nut, u nr is the torsional displacement of the nut around the axis, k ru is the torsional stiffness of the nut, c ru is the torsional damping of the nut, T out is the output torque of the nut;

[0138] Step (5): Select reasonable parameters to calculate the vibration response of the system, and establish the evaluation index of the stability of the planetary roller screw transmission system based on the calculation results to guide the selection of system parameters; the planetary roller screw system exhibits rich nonlinear behavior under external excitation, and the influence of the excitation frequency on the vibration characteristics of the planetary roller screw transmission system is studied through the bifurcation diagram. In the planetary roller screw mechanism, as the dimensionless excitation frequency increases in ω e ∈(0.7, 1.6), the system will experience periodic motion, period-doubling motion, chaos and other motion states. e ∈(1.19, 1.6), that is, the screw speed ω h When ∈(73, 98)rad / s, the system is in a stable periodic motion state.

[0139] In this example, the parameters selected for the system are shown in Table 1. Using the above method, the nonlinear response of the system is calculated and the results are as follows: Figure 3 and Figure 4 shown.

[0140] Table 1 Planetary roller screw parameters

[0141]

[0142] Figure 3 This is a dynamic load distribution curve of the thread pair of the planetary roller screw transmission system without considering the gear pair meshing excitation. The contact force of the roller-nut side thread teeth increases with the increase of the thread tooth number, and the contact force of the roller screw side thread teeth decreases with the increase of the thread tooth number. Figure 4 The dynamic load distribution curve of the thread pair considering the gear pair meshing excitation for the planetary roller screw transmission system is shown in Figure 2. Figure 3 The load distribution pattern for the threaded pair shown without considering gear mesh excitation is the same. When considering gear mesh excitation, the contact forces on each thread on the roller-screw and roller-nut sides fluctuate over time in the same manner, with their magnitudes depending on the magnitude of the gear mesh excitation. The dynamic load distribution calculation results for the threaded pair that consider gear mesh excitation and inertia forces, compared to those without, better reflect the load distribution characteristics of the planetary roller screw in actual operation.

[0143] Figure 5The bifurcation diagram of the vibration response of the planetary roller screw transmission system is shown in Figure 2. The influence of the excitation frequency on the vibration characteristics of the planetary roller screw transmission system is studied through the bifurcation diagram. In the planetary roller screw mechanism, as the dimensionless excitation frequency increases in ω e ∈(0.7, 1.6), the system will experience periodic motion, period-doubling motion, chaos and other motion states. e ∈(1.19, 1.6), that is, the screw speed ω h When ∈(73, 98)rad / s, the system is in a stable periodic motion state.

[0144] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the present invention shall still fall within the scope of protection of the technology of the present invention.

Claims

1. A planetary roller screw gear-thread coupling excitation dynamics modeling method, characterized in that: The following steps are involved: Step (1): Establish a force model for the planetary roller screw and analyze the elastic deformation of the thread engagement; when the planetary roller screw is under load, the normal contact force between the thread teeth can be decomposed into the axial component force F ak , radial force F ck and the tangential force F rk , respectively expressed as follows: Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and F L is the external load, λ r is the roller helix angle, β t is the contact angle; Deformation of the screw or roller thread caused by radial force and deformation of nut thread teeth They are: Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and F r is the radial force, F L is the external load, β r is the roller tooth flank angle, d ak is the equivalent thread major diameter, μ is the Poisson's ratio of the thread material, E is the elastic modulus, p is the thread pitch, D nm is the major diameter of the nut thread, D ns is the nut thread diameter, β n is the nut flank angle, β t is the contact angle; Axial force F a The total axial deformation caused by Can be calculated by thread profile parameters: in, Deformation caused by bending or shear force, Deformation caused by root shearing and is the deformation caused by the inclination of the tooth root, F a is the axial force, μ is the Poisson's ratio of the thread material, p is the thread pitch, E is the elastic modulus, k is s, r, n, representing the screw, roller, and nut respectively, a k Indicates the width of the thread bottom, b k Indicates the thread thickness, c k is the thread crest width, β k is the thread flank angle; Normal contact deformation of the thread contact points on the screw-roller contact side and the nut-roller contact side for: Where k is s and n represents the screw and nut respectively, F(φ) represents the first kind of elliptic integral, E' is the equivalent elastic modulus, ε represents the correlation coefficient with the curvature difference function, F T is the contact force, Σρ is the sum of the principal curvatures of the contact points, μ r and E r is the Poisson's ratio and elastic modulus of the roller, μ k and E k Poisson's ratio and elastic modulus respectively; ρ kk is the curvature of the corresponding curve of contact body 1 in plane I, ρ rk is the curvature of the corresponding curve of contact body 1 in plane II, ρ kr and ρ rr is the corresponding curvature of contact body 2 in planes I and II; Thread axial contact deformation and radial contact deformation It can be calculated by the following formula: Among them, k is s, r, and n, representing the screw, roller, and nut respectively, and β t is the normal contact angle, λ r is the roller helix angle, is the normal contact deformation, k T is the contact stiffness, F L is the external load; Step (2): Establish a calculation model of the gear pair system and analyze the gear meshing excitation and time-varying meshing stiffness excitation; at the meshing point outside the line between the roller gear and the inner ring gear, the linear speeds of the roller gear and the inner ring gear are v and v respectively. g and v r , decompose these two velocities in the direction of the instantaneous meshing line to obtain the impact velocity v along this direction m : Among them, v sg 、v sr v g 、v r The component on the instantaneous meshing line, ω r 、ω g are the angular velocities of the ring gear teeth and the roller gear teeth, r bg is the base circle radius of the internal gear ring teeth, r' br is the instantaneous value of the base circle radius of the inner ring gear, a is the center distance, and α is the engagement angle; When the gear pair is impacted, the gear tooth surface will produce a certain elastic deformation, and this deformation reaches the maximum value δ m When the impact force on the gear pair increases, it will increase until it reaches its peak value, that is, the maximum impact force F m ; Maximum deformation δ between internal meshing gear pairs m , Maximum impact force F m and impact energy E k The relationship between them is shown as follows: Where b is the tooth width, v m is the impact velocity, l m is the comprehensive flexibility of the roller gear teeth and the inner ring gear teeth at the impact point, l1 and l2 are the bending deformation flexibility of the roller gear teeth and the inner ring gear teeth at the impact point, l t is the contact flexibility of the gear teeth at the impact point, m q1 、m q2 are the equivalent masses of the roller gear and the internal gear ring on the instantaneous meshing line; The maximum impact force F when the roller gear teeth mesh with the internal gear ring teeth m for: Among them, v m is the impact velocity, b is the tooth width, I1 and I2 are the moments of inertia of the roller and the inner gear ring respectively, r br 、r bg are the base circle radii of the roller gear and the inner ring gear, l1 and l2 are the bending deformation flexibility of the roller gear and the inner ring gear at the impact point, respectively. t is the contact compliance of the gear teeth at the impact point; The collision impact process is a semi-sine pulse of impact force and impact time, so the impact force F of the roller gear and the inner gear ring is m (t) is: Among them, F m is the maximum impact force, t is the engagement time, t m is the meshing impact time; The meshing overlap between the roller gear and the inner gear ring is e>1, and there is a state of alternating single and double teeth meshing. The time-varying meshing stiffness k of the gear pair is considered by correcting the matrix stiffness of the double tooth meshing area. m for: Among them, ξ g and ξ r are the base correction coefficients of the roller gear and the internal gear ring, k bg and k br are the base stiffness of the roller gear and the inner ring gear, k t is the meshing stiffness when meshing with corresponding overlap; Maximum meshing excitation between roller gear teeth and internal gear ring teeth for: Among them, k m (t) is the gear ring meshing stiffness, Δy is the relative displacement of the gear ring, y R (t) and y G (t) are the linear vibration displacements of a point on the base circle of the inner ring gear and the roller gear respectively; Step (3): Establish the mechanical equations of the planetary roller screw gear pair and the thread pair, and analyze the influence of gear meshing excitation on the dynamic contact force of the thread; according to the normal vector analysis results of the screw and roller spiral surface at the contact point, the contact force F ts It can be expressed as: Among them, F ts is the contact force amplitude, λ s and β s They represent the helix angle and flank angle at the contact point between the screw and the roller, θ sr It represents the meshing angle of the roller on one side of the screw, F L is the external load, β t is the contact angle; Roller-nut side contact force F tn : Among them, F tn is the static contact force amplitude on the roller-nut side, λ n and β n They represent the helix angle and flank angle at the contact point between the nut and the roller, θ nr It represents the meshing angle of the roller on the nut side, F L is the external load, β t is the contact angle; Contact force between the roller and the screw considering the effect of gear meshing and the contact force between the roller and the nut side It can be expressed as: Among them, F ts is the contact force amplitude between roller and screw, F tn is the contact force amplitude between roller and nut, F m is the maximum impact force, F G is the meshing excitation, t s is the time of the impact of the outside bite, t e The time when a pair of gear teeth meshing ends; Step (4): Establish the nonlinear dynamic equations of the planetary roller screw transmission system; consider the vibration displacement of the roller and cage in the x, y directions and around their own axis, and the vibration displacement of the screw in the z-axis direction according to the concentrated mass method. X=[z s ,x pn ,y pn ,z pn ,u pn ,x c ,y c ,u c ,x nr ,y nr ,u nr ](n=1,2,3,…,n roller ), Among them, pn represents the nth roller shaft end gear, n roller represents the number of rollers, s represents the screw, c represents the cage, nr represents the inner gear ring at the nut shaft end, x, y, z represent the vibration displacement along the axis of their respective coordinates, and u represents the vibration angular displacement around the rotary axis; From the above model, the vibration differential equation of the planetary roller screw transmission system and the screw dynamics differential equation can be derived: Among them, m s is the mass of the screw, z s is the vibration displacement of the screw, k sz is the support stiffness of the screw thread in the z-axis direction, c sz is the support damping of the screw thread in the z direction, F sr is the contact force between roller and screw, F fsqz is the friction force between the screw and the roller thread in the z direction; Differential equation for roller dynamics: Among them, m q is the mass of the roller, ω q is the angular velocity of the roller, x pn 、y pn 、z pn are the vibration displacements of the cylindrical gear in the x, y, and z directions, respectively, and k qx 、k qy 、k qz are the support stiffness of the roller in the x, y, and z directions, δ rn is the projection of the displacement of the gear relative to the nut along the meshing line, φ rn is the angle between the line connecting the screw center and the roller center and the X-axis, δ cnx , δ cny are the vibration displacements of the cage in the x and y directions, c qx 、c qy are the support damping of the roller in the x and y directions, δ xn , δ yn , δ zn are the vibration displacements of the roller in the x, y, and z directions, respectively, and I q is the moment of inertia of the roller, r q is the pitch circle radius of the roller, u pn is the torsional displacement of the cylindrical gear around the axis, k qu is the torsional stiffness of the roller, c qu is the torsional damping of the roller, F sqx 、F sqy are the resultant forces of the contact force between the screw and roller thread and the meshing force of the gear pair in the x and y directions respectively, rqx 、F rqy are the resultant forces of the contact force between the nut and the roller thread and the meshing force of the gear pair in the x and y directions, respectively, fsqx 、F fsqy are the resultant forces of the friction between the roller and the screw thread and the friction between the gear pair in the x and y directions, respectively. frqx 、F frqy are the resultant friction between the roller and the nut thread and the gear pair in the x and y directions, respectively, and F srz 、F nrz are the components of the contact force between the roller and the screw and between the roller and the nut in the z direction, T in is the input torque of the roller; The cage vibration differential equation is expressed as: Among them, m c is the mass of the cage, x c 、y c are the two vibration displacements of the cage in the x and y directions, ω q is the angular velocity of the roller, k qx 、k qy are the support stiffness of the roller in the x and y directions, k cx 、k cy are the support stiffness of the cage in the x and y directions, δ cnx , δ cny are the vibration displacements of the cage in the x and y directions, c cx 、c cy are the support damping of the cage in the x and y directions, I c is the moment of inertia of the cage, r c is the pitch circle radius of the cage, u c k is the torsional displacement of the cage around the axis, qu is the torsional stiffness of the roller, k cu is the torsional stiffness of the cage, c cu is the torsional damping of the cage, δ cnu is the projection of the cage's displacement relative to the roller along the cage's tangent direction; The differential equation of nut vibration is expressed as: Among them, m nr is the mass of the nut, x nr 、y nr 、z nr are the vibration displacements of the nut in the x, y, and z directions, ω q is the angular velocity of the roller, F rqx 、F rqy are the resultant forces of the contact force between the nut and the roller thread and the meshing force of the gear pair in the x and y directions, respectively, frqx 、F frqy are the resultant forces of the friction between the nut and roller thread and the friction between the gear pair in the x and y directions, respectively, k rx 、k ry 、k rz are the support stiffness of the nut in the x, y, and z directions, δ rn is the projection of the displacement of the gear relative to the nut along the meshing line, φ rn c is the angle between the line connecting the center of the screw and the center of the roller and the X axis, rx 、c ry 、c rz are the support damping of the inner gear ring in the x, y, and z directions, respectively, and F nrz is the component of the contact force between the roller and the nut in the z direction, I nr is the moment of inertia of the nut, r nr is the pitch circle radius of the nut, u nr is the torsional displacement of the nut around the axis, k ru is the torsional stiffness of the nut, c ru is the torsional damping of the nut, T out is the output torque of the nut; Step (5): Select reasonable parameters to calculate the vibration response of the system, and establish the evaluation index of the stability of the planetary roller screw transmission system based on the calculation results to guide the selection of system parameters; the planetary roller screw system exhibits rich nonlinear behavior under external excitation, and the influence of the excitation frequency on the vibration characteristics of the planetary roller screw transmission system is studied through the bifurcation diagram.

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