CN112597636A - Multistage planetary roller screw motion and stress state analysis method - Google Patents

Multistage planetary roller screw motion and stress state analysis method Download PDF

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CN112597636A
CN112597636A CN202011432780.9A CN202011432780A CN112597636A CN 112597636 A CN112597636 A CN 112597636A CN 202011432780 A CN202011432780 A CN 202011432780A CN 112597636 A CN112597636 A CN 112597636A
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screw
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kth
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CN112597636B (en
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刘更
李欣
付晓军
马尚君
周勇
张力
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Northwestern Polytechnical University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H25/00—Gearings comprising primarily only cams, cam-followers and screw-and-nut mechanisms
    • F16H25/18—Gearings comprising primarily only cams, cam-followers and screw-and-nut mechanisms for conveying or interconverting oscillating or reciprocating motions
    • F16H25/20—Screw mechanisms
    • F16H25/22—Screw mechanisms with balls, rollers, or similar members between the co-operating parts; Elements essential to the use of such members
    • F16H25/2247—Screw mechanisms with balls, rollers, or similar members between the co-operating parts; Elements essential to the use of such members with rollers
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H25/00—Gearings comprising primarily only cams, cam-followers and screw-and-nut mechanisms
    • F16H25/18—Gearings comprising primarily only cams, cam-followers and screw-and-nut mechanisms for conveying or interconverting oscillating or reciprocating motions
    • F16H25/20—Screw mechanisms
    • F16H25/22—Screw mechanisms with balls, rollers, or similar members between the co-operating parts; Elements essential to the use of such members
    • F16H25/2247—Screw mechanisms with balls, rollers, or similar members between the co-operating parts; Elements essential to the use of such members with rollers
    • F16H25/2252—Planetary rollers between nut and screw
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H25/00—Gearings comprising primarily only cams, cam-followers and screw-and-nut mechanisms
    • F16H25/18—Gearings comprising primarily only cams, cam-followers and screw-and-nut mechanisms for conveying or interconverting oscillating or reciprocating motions
    • F16H25/20—Screw mechanisms
    • F16H25/24—Elements essential to such mechanisms, e.g. screws, nuts
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Abstract

The invention discloses a method for analyzing the motion and stress state of a multistage planetary roller screw, which relates to the technical field of screw drive dynamics and comprises the following steps: according to the motion and stress characteristics of the multistage planetary roller screw, the generalized force corresponding to the rotational freedom of the screw and the retainer at each stage is deduced, and a Lagrange equation of the mechanism is established; and (3) considering the friction force between each stage of screw rod and each roller and the friction force between each stage of screw rod, supplementing and establishing a rotating motion equation of each stage of screw rod and an axial movement motion equation of each stage of screw rod, each roller and each nut, and completing simultaneous solution of each motion equation to obtain the motion and stress states of the multistage planetary roller screw rod. The invention realizes the analysis of the motion and stress states of each part of the multi-stage planetary roller screw under different working conditions, and provides a basis for the structural design and performance optimization of the mechanism.

Description

Multistage planetary roller screw motion and stress state analysis method
Technical Field
The invention relates to the technical field of screw drive dynamics, in particular to a method for analyzing the motion and stress state of a multi-stage planetary roller screw.
Background
The multi-stage planetary roller screw is a transmission mechanism which couples a plurality of sets of planetary roller screws to realize conversion from rotary motion to long-stroke linear drive. At present, single-stage planetary roller screws are mostly used as research objects by researchers, and the researches mainly focus on the research directions of meshing characteristics, load distribution and rigidity calculation, lubricating friction and motion characteristics, transmission precision and the like. Compared with a single-stage planetary roller screw, the multi-stage planetary roller screw has more complex motion and stress states. Firstly, in the multi-stage planetary roller screw, the motion and the stress of each part in each stage are different and mutually influenced. Secondly, the multi-stage planetary roller screw contains more parts. Finally, friction and structural parameters at the connection of the planetary roller screws of all levels have great influence on the movement and stress of the planetary roller screws of all levels.
However, no analysis method related to the motion and stress of the multistage planetary roller screw exists in the prior art. And the analysis of the motion and stress under different working conditions is the basis of the structural design and performance optimization of the multistage planetary roller screw and is also the basis of the drive, control and motor design of the electromechanical servo system of the multistage planetary roller screw.
Therefore, in order to avoid the complicated analysis process, it is necessary to establish a more efficient analysis method. Aiming at the problems in the prior art, the application provides a method for analyzing the motion and stress states of a multi-stage planetary roller screw, which is used for analyzing the motion and stress states of all parts in the mechanism under different use working conditions. The method is convenient for strength and rigidity check, friction and wear analysis and structural design of the multi-stage planetary roller screw.
Disclosure of Invention
The invention aims to provide a method for analyzing the motion and stress states of a multi-stage planetary roller screw, which is used for analyzing the motion and stress states of all parts in a mechanism under different use conditions. The method is convenient for strength and rigidity check, friction and wear analysis and structural design of the multi-stage planetary roller screw.
The invention provides a method for analyzing the motion and stress state of a multistage planetary roller screw, which comprises the following steps:
step 1: collecting motion data of the multistage planetary roller screw, analyzing stress characteristics of the multistage planetary roller screw, and acquiring total kinetic energy of the multistage planetary roller screw in the operation process;
step 2: establishing a Lagrange model of the multistage planetary roller screw to obtain generalized force corresponding to the rotational freedom degrees of the screw and the retainer at each stage;
and step 3: performing stress analysis on the motion of the screw rod, the roller and the nut, additionally establishing a motion equation according to a Newton second law, and combining a Lagrange model with the motion equation to establish a dynamic model of the multistage planetary roller screw rod;
and 4, step 4: and analyzing the motion and stress states of the multistage planetary roller screw based on a dynamic model.
Further, the calculation mode of the total kinetic energy in the operation process of the multistage planetary roller screw in the step 1 is as follows:
Figure BDA0002825484000000021
in the formula:
Figure BDA0002825484000000022
-the rotation speed of the screw,
Figure BDA0002825484000000023
Rotational speed n of the kth cageTOverall number of stages of multi-stage planetary roller screws, NRkThe number of k-th stage rollers, JSkAnd JPkMoment of inertia, m, of the k-th screw and the cageSk、mPk、mNkAnd mPkQuality of screw, roller, nut and cage of the kth stage, LSiLead of i-th stage screw, nSkThe number of thread heads of the kth-stage screw rod, rSkAnd rRk-nominal radius of the k-th order screw and roller.
Further, in the lagrangian model in step 2, the generalized force calculation method corresponding to the rotational degrees of freedom of the lead screw and the retainer is as follows:
the axial movement speed of the kth stage nut can be expressed as:
Figure BDA0002825484000000031
the drive torque of the k-th stage screw and roller can be expressed as:
Figure BDA0002825484000000032
in the formula: f. ofSrxk、fSrykAnd fSrzk——fSrkComponent of x-, y-and z-, muSRCoefficient of friction between screw and roller, FSrk-the contact force between the k-th stage screw and the roller,
Figure BDA0002825484000000033
-the sliding speed between the k-th stage screw and the roller;
Figure BDA0002825484000000034
in the formula: r isSrk、
Figure BDA0002825484000000035
rRskAnd
Figure BDA0002825484000000036
-the engagement radius and the engagement declination angle, L, of the k-th stage screw and the rollerSk-lead of the kth stage lead screw;
the k-th order rotation matrix may be expressed as:
Figure BDA0002825484000000037
in the formula: thetaPk-angle of rotation of the kth stage cage;
the speed of movement of the k-th stage screw contact point can be expressed as:
Figure BDA0002825484000000038
the friction between the kth stage screw and the kth-1 stage screw can be expressed as:
Figure BDA0002825484000000039
in the formula: mu.sSS-coefficient of friction, r, between the kth and the kth-1 stage screwSSk-equivalent radius, M, of the connection of the kth screw and the kth-1 screwSk-drive torque of the kth stage screw;
the axial movement speed of the k-th stage screw can be expressed as:
Figure BDA0002825484000000041
in the formula:
Figure BDA0002825484000000042
-speed of movement of the k-1 th nut;
in the lagrangian model in the step 2, the generalized force corresponding to the rotational degree of freedom of the screw is expressed as:
Figure BDA0002825484000000043
in the formula:
Figure BDA0002825484000000044
n thTThe external force applied to the secondary nut,
Figure BDA0002825484000000045
N thTAxial moving speed M of step nutS1Drive torque f of the 1 st stage screwSrk-friction between the k-th lead screw and the roller,
Figure BDA0002825484000000046
-speed of movement of k-th stage screw contact point, HPkThe kth order rotation matrix, fSkThe friction force between the kth-level screw rod and the kth-1-level screw rod,
Figure BDA0002825484000000047
-axial displacement speed of the kth stage screw;
according to the lagrange method, the generalized force corresponding to the kth stage cage rotational degree of freedom can be expressed as:
Figure BDA0002825484000000048
in the formula:
Figure BDA0002825484000000049
-speed of the k-th stage roller contact point;
Figure BDA00028254840000000410
the lagrangian equation for a multistage planetary roller screw can be expressed as:
Figure BDA00028254840000000411
using kinetic energy TtotalThe lagrange equation may be further expressed as:
Figure BDA0002825484000000051
in the formula: gS-generalized force, G, corresponding to the degree of freedom of rotation of the screwPk-generalized forces corresponding to the rotational degrees of freedom of the kth stage cage,
Figure BDA0002825484000000052
-the rotational acceleration of the screw,
Figure BDA0002825484000000053
-rotational acceleration of the kth stage cage.
Further, in the step 3, the process of performing stress analysis on the motion of the screw, the roller and the nut and additionally establishing a motion equation is as follows:
the kinetic equation corresponding to the kth stage screw rotation is:
Figure BDA0002825484000000054
in the formula: j. the design is a squareSkMoment of inertia, M, of the k-th stage screwSkAnd MS(k+1)-drive torque of the k-th and k + 1-th leadscrews;
the dynamic equation corresponding to the axial movement of the screw in the kth stage (k >1) is as follows:
Figure BDA0002825484000000055
in the formula: m isSkThe mass of the kth-stage screw rod,
Figure BDA0002825484000000056
-axial movement acceleration of the kth lead screw, FN(k-1)Load of the k-1 st nut, fSkAnd fS(k+1)Friction force f experienced at the connection of the k-th and k + 1-th lead screwsSrzk-axial component of friction force of screw and roller of the kth stage, FSrzk-the axial component of the k-th stage screw-to-roller contact force;
Figure BDA0002825484000000057
in the formula: fSrk-contact force, β, of screw and roller of the k-th stageSkFlank angle of the kth stage lead screw;
the kinetic equation for the axial movement of the kth stage roller and nut is:
Figure BDA0002825484000000058
in the formula: m isRkMass of the kth roller, mNkThe mass of the kth-stage nut,
Figure BDA0002825484000000061
-axial movement acceleration of the kth nut, FNk-load of the kth stage nut.
Further, step 4 is based on a dynamic model, and the method for analyzing the motion and stress state of the multistage planetary roller screw comprises the following steps:
the Lagrange equation and the complementary equation of motion are jointly contained (4 n)T-1) nonlinear equations with (4 n)T-1) unknowns, directly solving by using a nonlinear equation solving function fsolve provided by Matlab software after the rotation speed of the screw and the load of the nut are known, and analyzing and obtaining screws, rollers, nuts and protectors at all stages by applying a multi-stage planetary roller screw Lagrange equation, a screw rotation motion equation and screw, roller and nut axial movement motion equationsThe movement and the stress state of the holder.
Compared with the prior art, the invention has the following remarkable advantages:
the method for analyzing the motion and stress state of the multi-stage planetary roller screw does not need to repeatedly consider motion equations of the retainers of all stages, moving motion equations of the rollers of all stages along the tangential direction and the radial direction and rotating motion equations of the rollers of all stages, greatly simplifies a dynamic model and improves the solving efficiency. The motion and stress state calculation of each part of the multistage planetary roller screw under different use conditions can be completed under the condition of avoiding solving a large number of nonlinear equations, and a basis is provided for strength and rigidity check, friction and wear analysis and structural design of the mechanism.
Drawings
FIG. 1 is a flow chart of a method for analyzing rigid motion and stress states of a multi-stage planetary roller screw according to an embodiment of the present invention;
FIG. 2 is a structural composition and motion analysis diagram of a multi-stage planetary roller screw provided by an embodiment of the invention;
FIG. 3 is a force analysis diagram of a kth-stage planetary roller screw according to an embodiment of the present invention;
fig. 4 is a diagram illustrating a calculation result of the motion and stress states of the two-stage planetary roller screw according to the embodiment of the present invention.
Detailed Description
The technical solutions of the embodiments of the present invention are clearly and completely described below with reference to the drawings in the present invention, and it is obvious that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, shall fall within the scope of protection of the present invention.
Referring to fig. 1-4, the invention provides a method for analyzing the motion and stress state of a multistage planetary roller screw, which comprises the following steps:
step 1: collecting motion data of the multistage planetary roller screw, analyzing stress characteristics of the multistage planetary roller screw, and acquiring total kinetic energy of the multistage planetary roller screw in the operation process;
step 2: establishing a Lagrange model of the multistage planetary roller screw to obtain generalized force corresponding to the rotational freedom degrees of the screw and the retainer at each stage;
and step 3: performing stress analysis on the motion of the screw rod, the roller and the nut, additionally establishing a motion equation according to a Newton second law, and combining a Lagrange model with the motion equation to establish a dynamic model of the multistage planetary roller screw rod;
and 4, step 4: and analyzing the motion and stress states of the multistage planetary roller screw based on a dynamic model.
Example 1
As shown in fig. 2, the multi-stage planetary roller screw mainly comprises a screw, a nut, rollers, a retainer and an inner gear ring. Symbol k denotes a k-th planetary roller screw, where k is 1,2, …, nT，nTRepresenting the total number of stages of the multi-stage planetary roller screw. In FIG. 1, nT2. Since the screw # (k-1) and the screw # k are connected by the sliding spline, the screws of the respective stages have the same rotational speed.
Figure BDA0002825484000000071
In the formula:
Figure BDA0002825484000000072
-rotational speed of the kth stage screw;
as shown in fig. 2, the nut # (k-1) and the screw # k are connected through a thrust bearing, so that both have the same axial moving speed;
Figure BDA0002825484000000073
in the formula:
Figure BDA0002825484000000074
and
Figure BDA0002825484000000075
-axial displacement speed of the kth stage screw and the kth-1 stage nut;
if the screw, the nut and the roller are all right-handed threads, the axial moving speed of the k-th nut can be expressed as:
Figure BDA0002825484000000081
in the formula: l isSi-lead of the i-th stage lead screw (i ═ 1,2, …, k);
the autorotation speed of the kth-stage roller and the rotating speed of the retainer have the following relations:
Figure BDA0002825484000000082
in the formula:
Figure BDA0002825484000000083
and
Figure BDA0002825484000000084
-the rotation speed of the kth roller and the rotation speed of the cage, nSk-number of screw heads of kth stage.
The calculation mode of the total kinetic energy in the operation process of the multistage planetary roller screw in the step 1 is as follows:
Figure BDA0002825484000000085
in the formula:
Figure BDA0002825484000000086
-the rotation speed of the screw,
Figure BDA0002825484000000087
Rotational speed n of the kth cageTOverall number of stages of multi-stage planetary roller screws, NRkThe number of k-th stage rollers, JSkAnd JPkThe moment of inertia of the k-th spindle and cage (k ═ 1,2, …, nT)、mSk、mRk、mNkAnd mPkQuality of screw, roller, nut and cage of the kth stage, LSi-lead (i ═ 1,2, …, k) of i-th stage lead screw, nSkThe number of thread heads of the kth-stage screw rod, rSkAnd rRk-nominal radius of the k-th order screw and roller.
Example 2
The force analysis of the kth stage planetary roller screw is shown in fig. 3. In FIG. 3, MSkAnd fSkRespectively the driving torque and the friction force acting on the k-th stage screw connection. FNkIs the kth stage nut load, fSRkIs the friction between the k-th stage screw and the roller. oPk-xPkyPkzPkO-XYZ is a global coordinate system for a local coordinate system fixed on the kth-stage retainer. ThetaPkIs the angle of rotation of the kth cage. The force analysis shows that the friction force at the k-th stage screw rod connection part is as follows:
Figure BDA0002825484000000091
in the formula: mu.sSS-coefficient of friction, r, between the kth and the kth-1 stage screwSSk-the equivalent radius of the connection of the kth stage screw and the kth-1 stage screw;
according to the coulomb friction model, the friction force between the k-th stage screw and the roller can be expressed as:
Figure BDA0002825484000000092
in the formula: f. ofSrxk、fSrykAnd fSrzk——fSrkComponent of x-, y-and z-, muSRCoefficient of friction between screw and roller, FSrk-the contact force between the k-th stage screw and the roller,
Figure BDA0002825484000000093
-the sliding speed between the k-th stage screw and the roller;
Figure BDA0002825484000000094
in the formula: r isSrk、
Figure BDA0002825484000000095
rRskAnd
Figure BDA0002825484000000096
-the engagement radius and the engagement declination angle, L, of the k-th stage screw and the rollerSk-lead of the kth stage lead screw;
local coordinate system o of the k-th orderPk-xPkyPkzPkThe rotation matrix transformed to the global coordinate system O-XYZ is:
Figure BDA0002825484000000097
in the formula: thetaPk-angle of rotation of the kth stage cage;
in the lagrangian model in the step 2, the generalized force corresponding to the rotational degree of freedom of the screw is expressed as:
Figure BDA0002825484000000098
in the formula:
Figure BDA00028254840000000910
n thTThe external force applied to the secondary nut,
Figure BDA0002825484000000099
N thTAxial moving speed M of step nutS1Drive torque f of the 1 st stage screwSrk-friction between the k-th lead screw and the roller,
Figure BDA0002825484000000101
-speed of movement of k-th stage screw contact point, HPkThe kth order rotation matrix, fSkFriction between the kth screw and the kth-1 screw (k)>1)、
Figure BDA0002825484000000102
-axial displacement speed of the kth stage screw;
Figure BDA0002825484000000103
the generalized force corresponding to the rotational degree of freedom of the kth stage cage can be expressed as:
Figure BDA0002825484000000104
in the formula:
Figure BDA0002825484000000105
-the speed of the k-th order roller contact point.
Figure BDA0002825484000000106
By utilizing the kinetic energy and the generalized force expression of the multistage planetary roller screw, the lagrangian equation of the multistage planetary roller screw can be expressed as follows:
Figure BDA0002825484000000107
using kinetic energy TtotalThe lagrange equation may be further expressed as:
Figure BDA0002825484000000108
in the formula: gS-generalized force, G, corresponding to the degree of freedom of rotation of the screwPk-generalized forces corresponding to the rotational degrees of freedom of the kth stage cage,
Figure BDA0002825484000000109
-the rotational acceleration of the screw,
Figure BDA00028254840000001010
-rotational acceleration of the kth stage cage.
Example 3
Lagrange's equation of multi-stage planetary roller screw is contained (n)T+1) nonlinear equations. Due to the contact force F between the k-th order screw and the roller in the Lagrange's equationSRkAnd k (k)>1) Drive torque M of a stepped spindleSkUnknown, so additional equations of motion need to be supplemented to complete the equation solution. The step 4 is based on a dynamic model, and the steps of analyzing the motion and stress state of the multistage planetary roller screw are as follows:
the kinetic equation corresponding to the kth stage screw rotation is:
Figure BDA0002825484000000111
in the formula: j. the design is a squareSkMoment of inertia, M, of the k-th stage screwSkAnd MS(k+1)-drive torque of the k-th and k + 1-th leadscrews;
the dynamic equation corresponding to the axial movement of the screw in the kth stage (k >1) is as follows:
Figure BDA0002825484000000112
in the formula: m isSkThe mass of the kth-stage screw rod,
Figure BDA0002825484000000113
-axial movement acceleration of the kth lead screw, FN(k-1)Load of the k-1 st nut, fSkAnd fS(k+1)Friction force f experienced at the connection of the k-th and k + 1-th lead screwsSrzk-axial component of friction force of screw and roller of the kth stage, FSrzk-the axial component of the k-th stage screw-to-roller contact force;
Figure BDA0002825484000000114
in the formula: fSrk-contact force, β, of screw and roller of the k-th stageSkFlank angle of the kth stage lead screw;
the kinetic equation for the axial movement of the kth stage roller and nut is:
Figure BDA0002825484000000115
in the formula: m isRkMass of the kth roller, mNkThe mass of the kth-stage nut,
Figure BDA0002825484000000116
-axial movement acceleration of the kth nut, FNk-load of the kth stage nut.
The Lagrange equation and the complementary equation of motion are jointly contained (4 n)T-1) nonlinear equations with (4 n)T-1) unknowns, when the rotation speed of the screw and the load of the nut are known, solving by directly using a nonlinear equation solving function fsolve provided by Matlab software, and analyzing and obtaining the motion and stress states of the screw, the roller, the nut and the retainer at each stage by applying a multi-stage planetary roller screw Lagrange equation, a screw rotation motion equation and a screw, roller and nut axial movement motion equation.
Example 4
The analysis method established by the invention is illustrated by taking a double-stage planetary roller screw as an example. TheThe mechanical parameters of the double-stage planetary roller screw are shown in table 1, and the mass parameters are shown in tables 2 and 3. And calculating the motion and stress state of the mechanism when the rotating speed of the screw rod is step input 955r/min and the external load of the nut is FN2 to 5000N all the time. Taking u from friction coefficient of screw and roller at each stageSR0.1, the coefficient of friction at the screw connection is uSS＝0.2。
Calculated k-th-stage retainer and screw rotating speed ratio zetaPSk(k 1 or 2) drive torque M of the k-th stage screwSkAxial component F of the contact force between the screw and the roller of the k-th orderSrzkAnd the friction force f at the joint of the transmission efficiency eta and the 2 nd-stage screw rodS2As shown in fig. 4(a) - (d), respectively. Wherein, the ratio of the retainer to the rotating speed of the screw is ζPSkAnd the transmission efficiency η is a dimensionless quantity, and is calculated by the following formulas, respectively.
Figure BDA0002825484000000121
Figure BDA0002825484000000122
Under the working condition of step input, the 1 st-stage retainer and the rotating speed of the screw rod reach a steady-state value earlier. The drive torque of the 1 st stage screw is significantly higher than that of the 2 nd stage screw, although the nominal radius of the 1 st stage screw is smaller. The axial component of the contact force between the stage 1 screw and the rollers increases first and then decreases to 818N. The axial component of the contact force between the stage 2 screw and the roller gradually increases to 794N. The initial efficiency of the double-stage planetary roller screw is less than 0.45, gradually increases along with simulation time, and reaches 0.80 in a steady state. The change of the friction force at the connection part of the 2 nd-stage screw rod is the same as the change trend of the driving torque of the 2 nd-stage screw rod, and is reduced from 300N to 150N.
TABLE 1 structural parameters of a two-stage planetary roller screw
Figure BDA0002825484000000123
TABLE 2 quality parameters of the 1 st planetary roller screw
Figure BDA0002825484000000131
TABLE 3 quality parameters of the 2 nd stage planetary roller screw
Figure BDA0002825484000000132
The above disclosure is only for a few specific embodiments of the present invention, however, the present invention is not limited to the above embodiments, and any variations that can be made by those skilled in the art are intended to fall within the scope of the present invention.

Claims (5)

1. A method for analyzing the motion and stress state of a multi-stage planetary roller screw is characterized by comprising the following steps:
step 1: collecting motion data of the multistage planetary roller screw, analyzing stress characteristics of the multistage planetary roller screw, and acquiring total kinetic energy of the multistage planetary roller screw in the operation process;
step 2: establishing a Lagrange model of the multistage planetary roller screw to obtain generalized force corresponding to the rotational freedom degrees of the screw and the retainer at each stage;
and step 3: performing stress analysis on the motion of the screw rod, the roller and the nut, additionally establishing a motion equation according to a Newton second law, and combining a Lagrange model with the motion equation to establish a dynamic model of the multistage planetary roller screw rod;
and 4, step 4: and analyzing the motion and stress states of the multistage planetary roller screw based on a dynamic model.
2. The method for analyzing the motion and stress state of the multistage planetary roller screw according to claim 1, wherein the total kinetic energy of the multistage planetary roller screw in the step 1 in the operation process is calculated as follows:
Figure FDA0002825483990000011
in the formula:
Figure FDA0002825483990000012
-the rotation speed of the screw,
Figure FDA0002825483990000013
Rotational speed n of the kth cageTOverall number of stages of multi-stage planetary roller screws, NRkThe number of k-th stage rollers, JSkAnd JPkMoment of inertia, m, of the k-th screw and the cageSk、mRk、mNkAnd mPkQuality of screw, roller, nut and cage of the kth stage, LSiLead of i-th stage screw, nSkThe number of thread heads of the kth-stage screw rod, rSkAnd rRk-nominal radius of the k-th order screw and roller.
3. The method for analyzing the motion and stress state of the multistage planetary roller screw according to claim 1, wherein the Lagrangian model in the step 2 has the following generalized force calculation mode corresponding to the rotational degrees of freedom of each stage of the screw and the retainer:
the axial movement speed of the kth stage nut can be expressed as:
Figure FDA0002825483990000021
the drive torque of the k-th stage screw and roller can be expressed as:
Figure FDA0002825483990000022
in the formula: f. ofSrxk、fSrykAnd fSrzk——fSrkComponent of x-, y-and z-, muSRCoefficient of friction between screw and roller, FSrk-the contact force between the k-th stage screw and the roller,
Figure FDA0002825483990000023
-the sliding speed between the k-th stage screw and the roller;
Figure FDA0002825483990000024
in the formula: r isSrk、
Figure FDA0002825483990000025
rRskAnd
Figure FDA0002825483990000026
-the engagement radius and the engagement declination angle, L, of the k-th stage screw and the rollerSk-lead of the kth stage lead screw;
the k-th order rotation matrix may be expressed as:
Figure FDA0002825483990000027
in the formula: thetaPk-angle of rotation of the kth stage cage;
the speed of movement of the k-th stage screw contact point can be expressed as:
Figure FDA0002825483990000028
the friction between the kth stage screw and the kth-1 stage screw can be expressed as:
Figure FDA0002825483990000029
in the formula: mu.sSS-coefficient of friction, r, between the kth and the kth-1 stage screwSSk-equivalent radius, M, of the connection of the kth screw and the kth-1 screwSk-drive torque of the kth stage screw;
the axial movement speed of the k-th stage screw can be expressed as:
Figure FDA0002825483990000031
in the formula:
Figure FDA0002825483990000032
-speed of movement of the k-1 th nut;
in the lagrangian model in the step 2, the generalized force corresponding to the rotational degree of freedom of the screw is expressed as:
Figure FDA0002825483990000033
in the formula:
Figure FDA0002825483990000034
n thTThe external force applied to the secondary nut,
Figure FDA0002825483990000035
N thTAxial moving speed M of step nutS1Drive torque f of the 1 st stage screwSrk-friction between the k-th lead screw and the roller,
Figure FDA0002825483990000036
-speed of movement of k-th stage screw contact point, HPkThe kth order rotation matrix, fSkThe friction force between the kth-level screw rod and the kth-1-level screw rod,
Figure FDA0002825483990000037
-axial displacement speed of the kth stage screw;
according to the lagrange method, the generalized force corresponding to the kth stage cage rotational degree of freedom can be expressed as:
Figure FDA0002825483990000038
in the formula:
Figure FDA0002825483990000039
-speed of the k-th stage roller contact point;
Figure FDA00028254839900000310
the lagrangian equation for a multistage planetary roller screw can be expressed as:
Figure FDA00028254839900000311
using kinetic energy TtotalThe lagrange equation may be further expressed as:
Figure FDA0002825483990000041
in the formula: gS-generalized force, G, corresponding to the degree of freedom of rotation of the screwPk-generalized forces corresponding to the rotational degrees of freedom of the kth stage cage,
Figure FDA0002825483990000042
-the rotational acceleration of the screw,
Figure FDA0002825483990000043
-rotational acceleration of the kth stage cage.
4. The method for analyzing the motion and stress state of the multistage planetary roller screw according to claim 1, wherein the process of analyzing the stress of the motion of the screw, the roller and the nut and additionally establishing a motion equation in the step 3 is as follows:
the kinetic equation corresponding to the kth stage screw rotation is:
Figure FDA0002825483990000044
in the formula: j. the design is a squareSkMoment of inertia, M, of the k-th stage screwSkAnd MS(K+1)-drive torque of the k-th and k + 1-th leadscrews;
the kinetic equation corresponding to the axial movement of the screw in the kth stage (k >1) is as follows:
Figure FDA0002825483990000045
in the formula: m isSkThe mass of the kth-stage screw rod,
Figure FDA0002825483990000046
-axial movement acceleration of the kth lead screw, FN(k-1)Load of the k-1 st nut, fSkAnd fS(k+1)Friction force f experienced at the connection of the k-th and k + 1-th lead screwsSrzk-axial component of friction force of screw and roller of the kth stage, FSrzk-the axial component of the k-th stage screw-to-roller contact force;
Figure FDA0002825483990000047
in the formula: fSrk-contact force of screw and roller of the kth stage、βSkFlank angle of the kth stage lead screw;
the kinetic equation for the axial movement of the kth stage roller and nut is:
Figure FDA0002825483990000051
in the formula: m isRkMass of the kth roller, mNkThe mass of the kth-stage nut,
Figure FDA0002825483990000052
-axial movement acceleration of the kth nut, FNk-load of the kth stage nut.
5. The method for analyzing the motion and stress state of the multistage planetary roller screw according to claim 1, wherein the step 4 is based on a dynamic model, and the method for analyzing the motion and stress state of the multistage planetary roller screw is as follows:
the Lagrange equation and the complementary equation of motion are jointly contained (4 n)T-1) nonlinear equations with (4 n)T-1) unknowns; when the rotation speed of the screw rod and the load of the nut are known, a nonlinear equation solving function fsolve provided by Matlab software is directly used for solving, and the motion and stress states of the screw rod, the roller, the nut and the retainer at all levels are obtained through analysis by applying a multi-level planetary roller screw Lagrange equation, a screw rod rotation motion equation and a screw rod, roller and nut axial movement motion equation.
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