CN114510795A - Design method of three-level sliding lead screw based on ordinary gear train - Google Patents

Design method of three-level sliding lead screw based on ordinary gear train Download PDF

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CN114510795A
CN114510795A CN202210073142.5A CN202210073142A CN114510795A CN 114510795 A CN114510795 A CN 114510795A CN 202210073142 A CN202210073142 A CN 202210073142A CN 114510795 A CN114510795 A CN 114510795A
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lead screw
stage
sliding lead
sliding
gear train
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胡健
胡良军
陈伟
徐晨晨
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Nanjing University of Science and Technology
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Nanjing University of Science and Technology
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Abstract

The invention discloses a design method of a three-level sliding lead screw based on a fixed-axis gear train, belonging to the technical field of optimal design of the sliding lead screw and comprising the following steps of: firstly, in order to improve the transmission efficiency of the traditional three-level sliding lead screw, an ordinary gear train is added to the traditional three-level sliding lead screw to form the three-level sliding lead screw based on the ordinary gear train; secondly, analyzing the transmission efficiency of the three-level sliding lead screw based on the ordinary gear train and establishing a functional relation; secondly, establishing a mathematical model of the three-level sliding lead screw based on the ordinary gear train, which takes high transmission efficiency and small volume as optimization targets; and finally, completing the optimization design of a mathematical model of the three-level sliding screw based on the ordinary gear train by utilizing a particle swarm optimization algorithm. The method provided by the invention can not only obtain the maximum transmission efficiency of the three-level sliding lead screw based on the ordinary gear train under the condition of meeting the performance requirement, but also reduce the volume of the three-level sliding lead screw. The design method aims to obtain the three-level sliding lead screw based on the fixed-axis gear train, which has high transmission efficiency and small volume.

Description

Design method of three-level sliding lead screw based on fixed-axis gear train
Technical Field
The invention relates to the technical field of multi-stage sliding lead screw optimization design, and mainly relates to a design method of a three-stage sliding lead screw based on an ordinary gear train.
Technical Field
With the progress of modern science and technology and military strength, the screw rod is used as a mechanical transmission device with excellent performance and is applied more and more widely in the fields of precision machine tools, medical instruments, aerospace, weaponry and the like. However, at present, in mechanical equipment using a lead screw as a transmission device at home and abroad, the lead screw is mainly limited to a single-stage lead screw, and the working stroke of the single-stage lead screw is limited by the size of the single-stage lead screw, so that the working conditions of large working stroke and large load, such as a large-load missile erecting mechanism, cannot be met. Compared with a single-stage lead screw, under the conditions of larger stroke and larger load, the multi-stage lead screw with the same stroke has the advantages of smaller volume, lighter weight, larger expansion and contraction, higher speed and the like. Therefore, in many applications where there are severe limitations in volume and weight, a multi-stage lead screw has a better application than a single-stage lead screw. However, the multi-stage screw is more complicated in structure and has more transmission mechanisms, so that the efficiency loss is more serious compared with the single-stage screw.
There are two general types of lead screws: the sliding screw rod and the rolling screw rod are divided into a ball screw rod and a planetary roller screw rod. In the three types of screws, the planetary roller screw has the highest performance and can bear large load and higher allowable rotating speed, but through investigation, under the working conditions of large working stroke and large load, the planetary roller screw has high processing difficulty, and the multi-stage planetary roller screw is easy to have the problem of asynchronism or clamping stagnation when being worn, so that the working times of the multi-stage planetary roller screw are limited. The ball screw cannot meet the requirement of a large-load working condition because the bearing capacity is not large enough, and the application of the ball screw in the large-load working condition is restricted. Because the sliding lead screw has a simple structure, the bearing capacity of the sliding lead screw meets the requirement of a large-load working condition, and the sliding lead screw has the advantage of lower cost in the aspect of heavy-load transmission. As the future industrial development is developed towards high speed and green low energy consumption, the optimization design of the transmission efficiency of the multistage sliding screw has very important theoretical research and practical application values.
Disclosure of Invention
The invention aims to provide a design method of a three-level sliding lead screw based on an ordinary gear train aiming at the limitation of a transmission device in the existing heavy-duty mechanical equipment, and the ordinary gear train is added on the basis of the traditional three-level sliding lead screw to form the three-level sliding lead screw based on the ordinary gear train, so that the transmission efficiency of the three-level sliding lead screw based on the ordinary gear train is improved; meanwhile, the three-level sliding lead screw based on the ordinary gear train is optimally designed by taking high transmission efficiency and small volume as optimization targets. On the premise of meeting the design requirements of wear resistance, lead screw strength and thread strength of all levels of sliding lead screw pairs in the three-level sliding lead screw based on the fixed-axis gear train, the three-level sliding lead screw meets the optimization target of high transmission efficiency and small volume, a mathematical model of the three-level sliding lead screw based on the fixed-axis gear train is established, and then the mathematical model is solved by utilizing a particle swarm optimization algorithm, so that the optimal solution of the three-level sliding lead screw mathematical model based on the fixed-axis gear train is obtained.
The technical solution for realizing the invention is as follows: a design method of a three-level sliding lead screw based on an ordinary gear train comprises the following steps:
s1: the ordinary gear train is added on the basis of the traditional three-level sliding lead screw to form the three-level sliding lead screw based on the ordinary gear train, so that the lead of the third-level sliding lead screw pair is increased under the condition that the motion rule of each-level sleeve is unchanged, the transmission efficiency of the third-level sliding lead screw pair is improved, and the improvement of the overall transmission efficiency of the three-level sliding lead screw based on the ordinary gear train is realized.
S2: the transmission efficiency eta of the three-level sliding screw based on the ordinary gear train is obtained by analyzing the transmission relationship and the mechanical property of the three-level sliding screw based on the ordinary gear trainsThe following were used:
Figure BDA0003482842700000021
in the formula etas1The transmission efficiency of the first-stage sliding lead screw pair is improved. Etas2The transmission efficiency of the second-stage sliding lead screw pair is improved. Etas3The transmission efficiency of the third-stage sliding lead screw pair is improved.
S3: the method is characterized in that a mathematical model of the three-level sliding lead screw based on the ordinary gear train is established on the basis of the structural parameters and the transmission efficiency of the three-level sliding lead screw based on the ordinary gear train, and comprises the following steps:
s3.1: determining design variables:
based on the principle of a transmission structure of a three-level sliding lead screw based on a fixed-axis gear train, determining the pitch diameter d of the thread of a first-level sliding lead screw2s1The pitch P of the first-stage sliding lead screw pairs1Wall thickness b of the second stage sliding screw1Third stage sliding screw wall thickness b2And a clearance c between the second-stage sliding lead screw and the third-stage sliding lead screw is taken as a design variable, and then a vector expression X of the design variable is provided:
X＝[d2s1 Ps1 b1 b2 c]T＝[x1 x2 x3 x4 x5]T
wherein x is1、x2、x3、x4、x5Respectively corresponding to d in the vector expression X2s1、Ps1、b1、b2、c。
S3.2: determining an optimization objective:
the highest transmission efficiency and the minimum volume of the three-level sliding screw based on the ordinary gear train are taken as optimization targets, namely a transmission efficiency objective function F1(X) maximum and radial dimension objective function F2(X) is minimal.
Figure BDA0003482842700000031
Figure BDA0003482842700000032
Wherein eta iss1(X)、ηs2(X)、ηs3And (X) respectively and correspondingly represents transmission efficiency expressions of the first-stage sliding lead screw pair, the second-stage sliding lead screw pair and the third-stage sliding lead screw pair. d1s3And (X) is a thread major diameter expression of the third-stage sliding lead screw.
S3.3: determining a constraint condition:
according to the structural parameters and the stress conditions of the three-level sliding lead screw based on the ordinary gear train, the wear resistance, the lead screw strength and the thread strength of each level of sliding lead screw pair are analyzed, and the constraint conditions of the three-level sliding lead screw mathematical model based on the ordinary gear train are established.
S3.4: according to the above discussion, a mathematical model of a three-level sliding lead screw based on an ordinary gear train is established:
Figure BDA0003482842700000033
wherein, F (X) is the total objective function of the three-stage sliding lead screw based on the ordinary gear train. Omega1A weighting factor that is a linear weighting function of the transmission efficiency objective function. Omega2A weighting factor which is a linear weighting function of the radial dimension objective function, and1+ω2＝1。gh(X) is a constraint, wherein h is the number of the constraint.
According to the established mathematical model of the three-level sliding screw based on the ordinary gear train, aiming at the constraint optimization problem, a penalty function method is adopted to convert the constraint optimization problem into a constraint-free optimization problem, and an optimization function phi (X, r) of the three-level sliding screw mathematical model based on the ordinary gear train is constructed:
Figure BDA0003482842700000034
wherein r is a penalty factor.
S4: and solving phi (X, r) by utilizing a particle swarm optimization algorithm to obtain an optimal solution of a three-level sliding lead screw mathematical model based on the ordinary gear train, namely an optimal vector expression X.
Compared with the prior art, the invention has the remarkable advantages that:
(1) the three-stage sliding lead screw is based on a driven three-stage sliding lead screw, the motion rule of sleeves at all stages is not changed, and the fixed-axis gear train is additionally arranged between the second-stage sliding lead screw and the third-stage sliding lead screw to form the three-stage sliding lead screw based on the fixed-axis gear train, so that the lead of the third-stage sliding lead screw pair is increased, the transmission efficiency of the third-stage sliding lead screw pair is improved, and the improvement of the whole transmission efficiency of the three-stage sliding lead screw based on the fixed-axis gear train is realized;
(2) the invention provides a design method of a three-level sliding lead screw based on an ordinary gear train, which is characterized in that a mathematical model of the three-level sliding lead screw optimization design based on the ordinary gear train is established by taking high transmission efficiency and small volume as optimization design targets, and the solution of the mathematical model is completed by utilizing a particle swarm optimization algorithm, so that the optimal solution of the three-level sliding lead screw mathematical model based on the ordinary gear train is obtained, and the problem that the traditional parameter design is difficult due to more design parameters of the three-level sliding lead screw based on the ordinary gear train is solved.
The invention is further described with reference to the following figures and examples.
Drawings
Fig. 1 is a schematic overall structure diagram of a design method of a three-level sliding lead screw based on an ordinary gear train according to an embodiment of the present invention.
Fig. 2 is a schematic sectional view of the overall structure of a design method of a three-level sliding lead screw based on an ordinary gear train according to an embodiment of the present invention.
Fig. 3 is a schematic structural diagram of a second-stage sliding lead screw of a design method of a three-stage sliding lead screw based on an ordinary gear train according to an embodiment of the present invention.
Fig. 4 is a schematic sectional view of an ordinary gear train based on a design method of a three-level sliding lead screw of the ordinary gear train according to an embodiment of the present invention.
Fig. 5 is a schematic power transmission diagram of a screw transmission system based on a design method of a three-level sliding screw of an ordinary gear train according to an embodiment of the invention.
Fig. 6 is a schematic size diagram of a screw transmission of a design method of a three-level sliding screw based on an ordinary gear train according to an embodiment of the present invention.
Fig. 7 is a schematic cross-sectional view of a lead screw transmission size of a design method of a three-level sliding lead screw based on an ordinary gear train according to an embodiment of the present invention.
Fig. 8 is a schematic diagram of an optimal solution iterative approximation process tracking curve of a design method of a three-level sliding lead screw based on an ordinary gear train according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without inventive step, are within the scope of the present invention.
It should be noted that all the directional indicators (such as upper, lower, left, right, front and rear … …) in the embodiment of the present invention are only used to explain the relative position relationship between the components, the movement situation, etc. in a specific posture (as shown in the drawings), and if the specific posture is changed, the directional indicator is changed accordingly.
In addition, descriptions such as "first", "second", etc. in the present invention are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of the feature. In the description of the present invention, "a plurality" means at least two, e.g., two, three, etc., unless explicitly specifically defined otherwise.
In the present invention, unless otherwise expressly specified or limited, the terms "connected," "secured," and the like are to be construed broadly, e.g., "secured" may be fixedly connected, releasably connected, or integral; "connected" may be mechanically or electrically connected. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
A design method of a three-level sliding lead screw based on an ordinary gear train comprises the following implementation steps:
s1: because lead screw tradition efficiency and lead screw helical pitch direct ratio relation have increased the ordinary gear train on traditional tertiary sliding lead screw basis and have constituted the tertiary sliding lead screw based on the ordinary gear train for under the unchangeable circumstances of sleeve motion law at each level of assurance, increase the vice helical pitch of third level sliding lead screw, improved the vice transmission efficiency of third level sliding lead screw, thereby realized the improvement of the whole transmission efficiency of tertiary sliding lead screw based on the ordinary gear train.
Referring to fig. 1 and 2, the three-stage sliding lead screw based on the ordinary gear train comprises a cylinder body 1, a first-stage sleeve 2, a second-stage sleeve 3, a third-stage sleeve 4, a first-stage sliding lead screw 5, a first-stage nut 6, a second-stage sliding lead screw 7, a second-stage nut 8, a third-stage sliding lead screw 9, a third-stage nut 10 and the ordinary gear train, wherein the ordinary gear train comprises an adapter gear 11, an intermediate wheel 12, a first fixed disk 13, a second fixed disk 14 and an inner gear ring 15. The first-stage sliding lead screw 5 is connected with the first-stage nut 6 through a sliding screw pair, and the first-stage sliding lead screw 5 and the first-stage nut 6 are connected through the sliding screw pair to realize that the first-stage nut 6 makes linear motion when the first-stage sliding lead screw 5 rotates and push the second-stage sliding lead screw pair to make constant-speed linear motion. The first-stage sliding screw 5 drives the second-stage sliding screw 7 to rotate through the hexagonal profile at the right end of the first-stage sliding screw 5, the second-stage sliding screw 7 is driven by the hexagonal profile in the circumferential direction to rotate coaxially with the first-stage sliding screw 5 at the same rotating speed, and the hexagonal profile at the right end of the first-stage sliding screw 5 plays a role in guiding the axial movement of the second-stage sliding screw 7. The second-stage sliding screw 7 rotates to realize the linear motion of the second-stage nut 8 and pushes the ordinary gear train and the third-stage sliding screw pair to do constant-speed linear motion, and at the moment, the linear motion speed v of the second-stage nut 8s2Is the linear movement velocity v of the first-stage nut 6s1And the moving speed v of the second-stage nut 8 relative to the second-stage sliding screw 7s2lsThe sum of (a) and (b). The second-stage sliding lead screw 7 is driven by a fixed-axis gear train so that the rotating speed n of the second-stage sliding lead screw 7s2For the rotational speed n of the third stage sliding screw 9s3Twice of the lead of the third-stage sliding lead screw pair, the lead P of the third-stage sliding lead screw pair is changeds3Is the lead P of the second-stage sliding lead screw pairs2At the time of the third stage nut 10 linear motion velocity vs3Is the linear movement velocity v of the second-stage nut 8s2And third stage nut10 relative to the third stage sliding screw 9s3lsAnd (4) summing. The lead of the third-stage sliding lead screw pair is increased to improve the transmission efficiency of the third-stage sliding lead screw pair, so that the overall transmission efficiency of the three-stage sliding lead screw based on the ordinary gear train is improved.
Referring to fig. 2, the third-stage nut 10 is connected to the third-stage sleeve 4 through a countersunk head screw, and the third-stage nut 10 translates to push the third-stage sleeve 4 to translate. The right end of the third-stage sleeve 4 is fixedly connected with the engine base through a hinge, and the rotation of the second-stage sleeve 3 is limited between the third-stage sleeve 4 and the second-stage sleeve 3 through two symmetrical flat keys. The second-stage nut 8 is connected with the second-stage sleeve 3 through a countersunk screw, the second-stage nut 8 is pushed to translate to push the second-stage sleeve 3 to translate, and the rotation of the first-stage sleeve 2 is limited between the second-stage sleeve 3 and the first-stage sleeve 2 through two symmetrical flat keys. The first-stage nut 6 is connected with the first-stage sleeve 2 through a countersunk head screw, and the first-stage nut 6 pushes the first-stage sleeve 2 to translate.
With reference to fig. 3 and 4, the second-stage sliding screw 7 is provided with four key slots which are uniformly distributed, and the key slots are matched with the rectangular spline of the adapter gear 11, so that the adapter gear 11 and the second-stage sliding screw 7 rotate at the same rotating speed. The rotation of the adaptor gear 11 is transmitted to the inner gear ring 15 through the three middle wheels 12 which are uniformly arranged, and the inner gear ring 15 is connected with the rectangular spline at the left end of the third-stage sliding screw 9 through four key grooves which are uniformly distributed on the inner gear ring 15, so that the inner gear ring 15 and the third-stage sliding screw 9 rotate at the same speed. In the fixed-axis gear train, a first fixed disk 13 is fixedly connected with a second-stage nut 8 through six countersunk head screws, a second fixed disk 14 is connected with the first fixed disk 13 through six countersunk head screws, and the second-stage nut 8 and a second-stage sleeve 3 are fixedly connected together through the countersunk head screws, so that the circumferential rotation of the second-stage nut 8 is limited, and the axis of the intermediate wheel 12 is fixed. In the ordinary gear train, the number of teeth of the ring gear 15 is set to be twice the number of teeth of the adaptor gear 11, thereby realizing that the third stage slide screw 9 rotates at a speed half that of the second stage slide screw 7.
S2: with reference to fig. 5, by analyzing the transmission relationship and mechanical properties of the three-level sliding lead screw based on the ordinary gear train,and establishing the relationship among the thread pitch diameter, the lead and the transmission efficiency of each stage of sliding screw for the three-stage sliding screw based on the ordinary gear train. The transmission efficiency of the three-level sliding lead screw based on the ordinary gear train can be obtained through calculation according to a transmission process schematic diagram, and the transmission efficiency of the three-level sliding lead screw based on the ordinary gear train is in a combination form of series-parallel mixing. Input power P of first-stage sliding lead screwdAnd moves through the first stage and the second stage. After being transmitted, the third-stage sliding lead screw pair and the sleeve sliding pairs at all stages are converted into linear displacement and linear velocity of a bearing end, and the output power of the three-stage sliding lead screw based on the ordinary gear train is PkThe input power and the output power of the ith-stage sliding lead screw pair are respectively Pdi、PkiWherein i is 1,2, 3. Transmission efficiency eta of three-level sliding lead screw based on ordinary gear trains：
Figure BDA0003482842700000071
In the formula (I), the compound is shown in the specification,
Figure BDA0003482842700000072
output power P of i-th stage sliding lead screw pairki：
Pki＝Fzvi＝FznsiSsi
In the formula, FzThe axial load is borne by a three-level sliding lead screw based on an ordinary gear train. v. ofiThe linear motion speed of the ith-stage sleeve is shown. n issiIs the rotational speed of the i-th stage sliding screw, and ns1＝ns2＝2ns3。SsiIs the lead of the i-th sliding screw, Ssi＝NPsiN is the number of threads of the screw, and N is 6, PsiIs the pitch of the i-th sliding screw, and Ps1＝Ps2＝P s32, i.e. Ss1＝Ss2＝Ss3/2. Then:
Pk1＝Pk2＝Pk3
from the figure5, the input power P of the i-th stage sliding lead screw pair can be obtaineddiAs follows:
Figure BDA0003482842700000073
in the formula etaziTaking eta as the transmission efficiency of the supporting bearing sets at the two ends of the i-th sliding lead screw pairzi＝0.985。ηsiThe transmission efficiency of the screw transmission in the i-th stage sliding screw rod pair is improved. EtatiFor the i-th sleeve sliding connection transmission efficiency, take etati＝0.970。η1For the transmission efficiency of the hexagonal surface at the right end of the first-stage sliding lead screw, as the sliding speed of the sliding lead screw is lower and the hexagonal surface is mostly friction loss, the eta is taken1＝1。η2For the sliding efficiency of the gear of the adapter relative to the second-stage sliding screw rod, the eta is taken similarly2＝1。ηxlIs the transmission efficiency of the ordinary gear train, and etaxl＝0.98520.985, where the ordinary gear train includes two gears and a set of rolling bearings.
Transmission efficiency eta of screw transmission in i-th-stage sliding screw rod pairsi：
Figure BDA0003482842700000081
Wherein mu is the friction factor of the material selected by the sliding lead screw pair. Gamma rayiIs the lead angle of the i-th sliding lead screw pair, and
Figure BDA0003482842700000082
d2sithe pitch diameter of the screw thread of the i-th stage sliding screw rod. The transmission efficiency eta of the screw transmission in the i-th stage sliding screw rod pairsiCan be expressed as:
Figure BDA0003482842700000083
then the transmission efficiency eta of the three-level sliding lead screw based on the ordinary gear trainsCan be expressed as:
Figure BDA0003482842700000084
s3: the method is characterized in that a mathematical model of the three-level sliding lead screw based on the ordinary gear train is established on the basis of the structural parameters and the transmission efficiency of the three-level sliding lead screw based on the ordinary gear train, and comprises the following steps:
s3.1: with reference to fig. 6 and 7, design variables are determined: starting from the transmission structure principle of the three-stage sliding lead screw based on the ordinary gear train, the size parameter to be solved is the thread major diameter d of the i-th stage sliding lead screw1siPitch diameter d of the thread2siMinor diameter of thread d3siPitch of thread Psi. Diameter d of right hexagonal profile inscribed circle of first-stage sliding lead screwxnDiameter d of circumscribed circlexw. Wall thickness b of second stage sliding screw1Third stage sliding screw wall thickness b2And a clearance c between the second and third stage sliding lead screws. Wherein, the structural design of the three-level sliding lead screw based on the ordinary gear train shows that dxnThe value of the major diameter d is equal to that of the first-stage sliding lead screw1s1
The first-stage sliding lead screw pair has the following size parameters:
Figure BDA0003482842700000085
in the formula, acIs the crest clearance of the screw thread in the sliding lead screw pair.
The second-stage sliding lead screw pair has the following size parameters:
Figure BDA0003482842700000091
the size parameters of the third-stage sliding lead screw pair are as follows:
Figure BDA0003482842700000092
according to step S2, P is knowns1＝Ps2＝Ps3Per 2, in summary, take d2s1、Ps1、b1、b2And c is taken as a design variable, and a vector expression X of the design variable is provided:
X＝[d2s1 Ps1 b1 b2 c]T＝[x1 x2 x3 x4 x5]T
wherein x is1、x2、x3、x4、x5Respectively corresponding to d in the vector expression X2s1、Ps1、b1、b2、c。
S3.2: determining an optimization objective: the highest transmission efficiency and the smallest volume of the three-stage sliding lead screw based on the ordinary gear train are taken as optimization targets. In engineering applications, the optimization problem is often not a single objective optimization, and the present invention requires volume optimization and material savings while seeking to maximize transmission efficiency. Because the volume of the three-stage sliding lead screw based on the ordinary gear train is determined by the radial dimension of the lead screw, the axial dimension only needs to meet the requirement of the motion stroke of the lead screw, and the smaller the external dimension is, the more compact the overall structure is, the more saved the material is, and the lower the production cost is. Therefore, the three-level sliding lead screw optimization design based on the ordinary gear train belongs to multi-objective optimization design.
The value range of the transmission efficiency is [0,1 ]]And maximum is solved, while optimization usually makes minimum problem, so a transmission efficiency objective function F is obtained1(X) is:
Figure BDA0003482842700000093
wherein eta iss1(X)、ηs2(X)、ηs3And (X) respectively corresponds to transmission efficiency expressions of the first-stage sliding lead screw pair, the second-stage sliding lead screw pair and the third-stage sliding lead screw pair.
The value range of the transmission efficiency objective function is [0,1 ]]Should do the radial dimension objective functionThe processing with uniform magnitude order needs to set the range of the radial size objective function within the range of (0,1), thereby eliminating the difference of the magnitude order of the time division objective function by adopting the linear weighted combination method. Because the sizes of all stages of sliding lead screws in the three stages of sliding lead screws based on the fixed-axis gear train have certain correlation, the overall radial size can be limited by restricting the thread major diameter of the third stage of sliding lead screw, and therefore the following radial size objective function F is established2(X)：
Figure BDA0003482842700000101
In the formula (d)1s3And (X) is a thread major diameter expression of the third-stage sliding lead screw.
Applying linear weighting combination method in evaluation function method to obtain target function F1(X)、F2(X) is converted into a single objective function optimization problem for solving a unified objective function. The linear weighted total objective function F (X) of the three-level sliding screw based on the ordinary gear train is as follows:
F(X)＝ω1F1(X)+ω2F2(X)
in the formula, ω1Weighting factors, ω, for linear weighting functions of the transmission efficiency objective function2A weighting factor which is a linear weighting function of the radial dimension objective function, and1+ω 21 is ═ 1; the optimization target of the embodiment is mainly transmission efficiency, and omega is taken1＞ω2And ω is1＝0.8、ω2＝0.2。
S3.3: determining a constraint condition: according to the structural parameters and the stress conditions of the three-level sliding lead screw based on the ordinary gear train, the wear resistance, the lead screw strength and the thread strength of each level of sliding lead screw pair are analyzed, and the constraint conditions of the three-level sliding lead screw mathematical model based on the ordinary gear train are established.
S3.3.1: determining the wear resistance constraint conditions of each stage of sliding lead screw pair, including the wear resistance constraint condition g of the first stage of sliding lead screw pair1(X) and the wear resistance constraint condition g of the second-stage sliding lead screw pair2(X) andwear resistance constraint condition g of third-stage sliding lead screw pair3(X)。
Figure BDA0003482842700000102
In the formula, the working height h of the screw thread of the i-th sliding lead screw pairi＝0.5Psi(ii) a Screw thread screwing number of turns z of i-th-stage sliding lead screw pairi＝Hi/PsiHeight H of nut in i-th sliding screw pairi＝(1.2～2.5)d2si；ppThe allowable specific pressure of the material selected for the sliding lead screw pair.
S3.3.2: determining the strength constraint conditions of the three-stage sliding screw rod, including the first-stage sliding strength constraint condition g4(X), second stage sliding screw Strength Beam Condition g5(X) and third-stage sliding screw strength constraint condition g6(X)。
Figure BDA0003482842700000111
In the formula, the cross-sectional area A of the first stage sliding screw1＝πd3s1 2/4, the second stage sliding screw section area A2＝π(d3s2 2-dxw 2) /4, the cross-sectional area A of the third stage sliding screw3＝π(d3s3 2-(d1s2+2c)2)/4，dxwThe diameter of a circumscribed circle of a hexagonal molded surface at the right end of the first-stage sliding screw rod is the diameter of a circumscribed circle of the hexagonal molded surface at the right end of the first-stage sliding screw rod; thread friction torque T of i-th stage sliding lead screw1si＝d2siFztan(γi+ρv)/2，γiThe lead angle of the i-th sliding screw pair and the equivalent friction angle rho of the threads of the sliding screwvα is the thread profile angle of the sliding screw thread, and α is 30 ° for a trapezoidal screw; torsional section coefficient W of first-stage sliding lead screwt1＝πd3s1 3/16 torsional section coefficient of second stage sliding screw
Figure BDA0003482842700000112
Torsional section coefficient of third-stage sliding lead screw
Figure BDA0003482842700000113
σpAllowable stress of the material selected for the sliding lead screw.
S3.3.3: and determining the strength constraint condition of the thread of the three-level sliding screw based on the ordinary gear train, and only calculating the thread strength of the nut because the thread strength of the steel screw is higher than that of the bronze nut. The strength constraint conditions of the three-level sliding lead screw thread based on the fixed-axis gear train comprise shear strength constraint conditions g of the thread screw thread in the first-level sliding lead screw pair7(X) flexural Strength constraint Condition g8(X); shear strength constraint condition g of thread screw in second-stage sliding lead screw pair9(X) flexural Strength constraint Condition g10(X); shear strength constraint condition g of thread screw in third-stage sliding lead screw pair11(X) flexural Strength constraint Condition g12(X)。
The strength constraint conditions of the thread in the first-stage sliding lead screw pair are as follows:
Figure BDA0003482842700000121
the strength constraint conditions of the thread in the second-stage sliding lead screw pair are as follows:
Figure BDA0003482842700000122
the strength constraint conditions of the thread in the third-stage sliding lead screw pair are as follows:
Figure BDA0003482842700000123
in the above formulas, biThe width of the root of the thread in the i-th sliding screw rod pair is equal to that of the trapezoidal thread bi＝0.65Psi；τpAllowable shear stress for the nut material; sigmabbpThe allowable bending stress of the nut material.
S3.3.4: determining boundary condition constraints, wherein the following boundary conditions exist according to relevant industry standards:
Figure BDA0003482842700000124
wherein, g13(X)、g14(X)、g15(X) respectively corresponding to the pitch constraint conditions of the first-stage sliding lead screw pair, the second-stage sliding lead screw pair and the third-stage sliding lead screw pair; g16(X) is a lower bound condition of the pitch in the sliding lead screw pair; g17(X) is the upper bound condition of the pitch in the sliding lead screw pair; g18(X) is a lower bound condition of the pitch diameter of the first-stage sliding lead screw thread; g19(X) is the upper bound condition of the pitch diameter of the third-stage sliding lead screw thread; g is a radical of formula20(X) is the lower bound constraint condition of the wall thickness of the second-stage sliding screw, g21(X) is the lower bound constraint condition of the wall thickness of the third-stage sliding lead screw, g22(X) is a lower bound on the gap between the second and third stage sliding screws.
S3.4: from the above discussion, a mathematical model of an ordinary gear train based three-level sliding screw is established as follows:
Figure BDA0003482842700000131
in the formula, gh(X) is a constraint, and h is the number of the constraint
According to the established mathematical model of the three-level sliding screw based on the ordinary gear train, aiming at the constraint optimization problem, a penalty function method is adopted to convert the constraint optimization problem into a constraint-free optimization problem, and an optimization function phi (X, r) of the three-level sliding screw mathematical model based on the ordinary gear train is constructed:
Figure BDA0003482842700000132
wherein r is a penalty factor.
S4: and solving phi (X, r) by utilizing a particle swarm optimization algorithm to obtain an optimal solution of a three-level sliding lead screw mathematical model based on the ordinary gear train, namely an optimal vector expression X.
Example 1
Taking a certain type of heavy-load missile erection experiment as an example, the erection performance parameters are as follows: axial load F of three-stage sliding lead screw based on fixed-axis gear train is 408100N, lead screw and nut materials in each stage of sliding lead screw pair are respectively selected to be 45 steel and cast aluminum bronze, and allowable specific pressure p of the materials selected by the sliding lead screw pairpAllowable stress sigma of sliding screw material of 15MPapAllowable shear stress tau of nut material under 90MPapAllowable bending stress sigma of nut material under 35MPabbp＝50MPa。
In the particle swarm optimization algorithm, a target search space dimension D is set to be 5, the number N of particle swarm is set to be 100, the iteration frequency T is set to be 300, and a maximum inertia weight omega is setmax0.9, minimum inertia weight ωmax0.4, individual learning factor c1Group learning factor c ═ 222. The inertia weight ω is updated according to the following formula:
Figure BDA0003482842700000141
in the formula, t is the current iteration step number.
The position and velocity of the individual particles are updated according to the formula shown below:
Figure BDA0003482842700000142
in the formula, vm,n(t)、vm,n(t +1) respectively representing the speeds of the particle individuals at the current moment and the next moment; x is the number ofm,n(t)、xm,n(t) respectively representing the positions of the particle individuals at the current moment and the next moment; r is1、r2Is [0,1 ]]A uniform random number within a range;pm,nsearching the optimal solution for the particle individual at present; p is a radical ofg,nSearching the optimal solution for the whole particle population at present; where the subscript m denotes the m-th particle in the population and n denotes the nth dimension of that particle, the nth corresponding intermediate variable X in the vector expression XnThe subscript g indicates the particles of the best solution currently searched for throughout the population of particles.
An optimal solution iterative approximation process tracking curve of a three-level sliding screw mathematical model based on an ordinary gear train is shown in fig. 8, and the optimized and corrected parameter value of the three-level sliding screw based on the ordinary gear train is compared with the initial parameter value before the optimization of the traditional three-level sliding screw:
Figure BDA0003482842700000143
the results in the table show that the optimized and corrected parameter values of the three-level sliding screw based on the ordinary gear train are compared with the initial parameter values before the optimization of the traditional three-level sliding screw: the radial size is reduced by 14.31%, and the transmission efficiency is improved by 27.84%. Therefore, the optimized parameter combination obtained by the design method of the three-level sliding screw based on the ordinary gear train really achieves the optimization effect.

Claims (3)

1. A design method of a three-level sliding lead screw based on an ordinary gear train is characterized by comprising the following steps:
s1: the fixed-axis gear train is added on the basis of the traditional three-level sliding lead screw to form the three-level sliding lead screw based on the fixed-axis gear train, so that the lead of the third-level sliding lead screw pair is increased under the condition of ensuring that the motion rule of sleeves at all levels is unchanged, the transmission efficiency of the third-level sliding lead screw pair is improved, and the overall transmission efficiency of the three-level sliding lead screw based on the fixed-axis gear train is improved;
s2: the transmission efficiency eta of the three-level sliding lead screw based on the ordinary gear train is obtained by analyzing the transmission relationship and the mechanical property of the three-level sliding lead screw based on the ordinary gear trainsAs follows：
Figure FDA0003482842690000011
In the formula etas1The transmission efficiency of the first-stage sliding lead screw pair is improved; etas2The transmission efficiency of the second-stage sliding lead screw pair is improved; etas3The transmission efficiency of the third-stage sliding lead screw pair is improved;
s3: the method is characterized in that a mathematical model of the three-level sliding lead screw based on the ordinary gear train is established on the basis of the structural parameters and the transmission efficiency of the three-level sliding lead screw based on the ordinary gear train, and comprises the following steps:
s3.1: determining design variables:
based on the principle of a transmission structure of a three-level sliding lead screw based on a fixed-axis gear train, determining the pitch diameter d of the thread of a first-level sliding lead screw2s1The pitch P of the first-stage sliding lead screw pairs1Wall thickness b of the second stage sliding screw1Third stage sliding screw wall thickness b2And a clearance c between the second-stage sliding lead screw and the third-stage sliding lead screw is used as a design variable, and then a vector expression X of the design variable is provided:
X＝[d2s1 Ps1 b1 b2 c]T＝[x1 x2 x3 x4 x5]T
wherein x is1、x2、x3、x4、x5Respectively corresponding to the vector expression X2s1、Ps1、b1、b2、c；
S3.2: determining an optimization objective:
the highest transmission efficiency and the minimum volume of the three-level sliding screw based on the ordinary gear train are taken as optimization targets, namely a transmission efficiency objective function F1(X) maximum and radial dimension objective function F2(X) minimum;
Figure FDA0003482842690000021
Figure FDA0003482842690000022
wherein eta iss1(X)、ηs2(X)、ηs3(X) respectively and correspondingly representing transmission efficiency expressions of a first-stage sliding lead screw pair, a second-stage sliding lead screw pair and a third-stage sliding lead screw pair; d1s3(X) is a thread major diameter expression of a third-stage sliding lead screw;
s3.3: determining a constraint condition:
analyzing the wear resistance, the screw strength and the thread strength of each sliding screw pair according to the structural parameters and the stress condition of the three-level sliding screw based on the ordinary gear train, and establishing the constraint condition of the three-level sliding screw mathematical model based on the ordinary gear train;
s3.4: according to the above discussion, a mathematical model of a three-level sliding lead screw based on an ordinary gear train is established:
Figure FDA0003482842690000023
in the formula, F (X) is a total objective function of a three-level sliding lead screw based on an ordinary gear train; omega1A weighting factor that is a linear weighting function of the transmission efficiency objective function; omega2A weighting factor which is a linear weighting function of the radial dimension objective function, and1+ω2＝1；gh(X) is a constraint condition, wherein h is the serial number of the constraint condition;
according to the established mathematical model of the three-level sliding screw based on the ordinary gear train, aiming at the constraint optimization problem, a penalty function method is adopted to convert the constraint optimization problem into a constraint-free optimization problem, and an optimization function phi (X, r) of the three-level sliding screw mathematical model based on the ordinary gear train is constructed:
Figure FDA0003482842690000024
in the formula, r is a penalty factor;
s4: and solving phi (X, r) by utilizing a particle swarm optimization algorithm to obtain an optimal solution of a three-level sliding lead screw mathematical model based on the ordinary gear train, namely an optimal vector expression X.
2. The design method of the three-level sliding lead screw based on the ordinary gear train as claimed in claim 1, wherein: the fixed-axis gear train in the S1 comprises an adapter gear, an intermediate wheel and an inner gear ring gear, wherein the adapter gear is connected with the second-stage sliding lead screw, and the inner gear ring gear is connected with the third-stage sliding lead screw in a matched mode through key grooves.
3. The design method of the three-stage sliding lead screw with high transmission efficiency as claimed in claim 2, wherein: in S3.3, the established constraint conditions of the three-level sliding lead screw mathematical model based on the ordinary gear train are as follows:
s3.3.1: determining three-level sliding lead screw wear resistance constraint conditions based on the ordinary gear train, including a first-level sliding lead screw wear resistance constraint condition g1(X) second-stage sliding lead screw wear resistance constraint condition g2(X) and third-stage sliding lead screw wear resistance constraint condition g3(X)：
Figure FDA0003482842690000031
In the formula, FzThe axial load is borne by a three-level sliding lead screw based on a fixed-axis gear train; d2siThe pitch diameter of the screw thread of the i-th stage sliding lead screw; working height h of i-th-stage sliding lead screw pair threadi＝0.5Psi，PsiThe screw pitch of the i-th sliding screw pair; screw thread screwing number of turns z of i-th-stage sliding lead screw pairi＝Hi/PsiHeight H of nut in i-th sliding screw pairi＝(1.2～2.5)d2si；ppAllowable specific pressure of materials selected for the sliding lead screw pair; wherein i is 1,2, 3;
s3.3.2: determining gear train based on ordinary axisThe three-stage sliding lead screw strength constraint condition comprises a first-stage sliding lead screw strength constraint condition g4(X), second stage sliding screw Strength Beam Condition g5(X) and third-stage sliding screw strength constraint condition g6(X)：
Figure FDA0003482842690000041
In the formula, the cross-sectional area A of the first stage sliding screw1＝πd3s1 2Second stage sliding screw section area A2＝π(d3s2 2-dxw 2) /4, the cross-sectional area A of the third stage sliding screw3＝π(d3s3 2-(d1s2+2c)2)/4，dxwThe diameter of a circumscribed circle of a hexagonal molded surface at the right end of the first-stage sliding screw rod is the diameter of a circumscribed circle of the hexagonal molded surface at the right end of the first-stage sliding screw rod; thread friction torque T of i-th stage sliding lead screw1si＝d2siFztan(γi+ρv)/2，γiThe lead angle of the i-th sliding screw pair and the equivalent friction angle rho of the threads of the sliding screwvμ is the friction factor of the selected material of the sliding screw pair, and α is the thread form angle of the sliding screw thread; torsional section coefficient W of first-stage sliding lead screwt1＝πd3s1 3/16 torsional section coefficient of second stage sliding screw
Figure FDA0003482842690000042
Torsion-resistant section coefficient of third-stage sliding lead screw
Figure FDA0003482842690000043
d3siThe thread diameter of the i-th level sliding screw rod is small; d1siThe diameter of the screw thread of the i-th stage sliding screw rod is large; sigmapAllowable stress of the material selected for the sliding lead screw; c is a gap between the second-stage sliding lead screw and the third-stage sliding lead screw;
s3.3.3: determining the strength constraint conditions of three-stage sliding lead screw thread threads based on the ordinary gear train, including the first-stage slidingShear strength constraint condition g of thread in screw pair7(X) flexural Strength constraint Condition g8(X); shear strength constraint condition g of thread screw in second-stage sliding lead screw pair9(X) flexural Strength constraint Condition g10(X); shear strength constraint condition g of thread screw in third-stage sliding lead screw pair11(X) flexural Strength constraint Condition g12(X)；
The strength constraint conditions of the thread in the first-stage sliding lead screw pair are as follows:
Figure FDA0003482842690000051
the strength constraint conditions of the thread in the second-stage sliding lead screw pair are as follows:
Figure FDA0003482842690000052
the strength constraint conditions of the thread in the third-stage sliding lead screw pair are as follows:
Figure FDA0003482842690000053
in the above formulas, biThe width of the root of the thread in the i-th sliding screw rod pair is equal to that of the trapezoidal thread bi＝0.65Psi；τpAllowable shear stress for the nut material; sigmabbpAllowable bending stress of the nut material;
s3.3.4: determining boundary condition constraints, wherein the following boundary conditions exist according to relevant industry standards:
Figure FDA0003482842690000054
wherein, g13(X)、g14(X)、g15(X) corresponding to the first stage, the second stage and the third stage of sliding wires respectivelyThe screw pitch constraint condition of the lever pair; g16(X) is a lower bound condition of the pitch in the sliding lead screw pair; g17(X) is the upper bound condition of the pitch in the sliding lead screw pair; g18(X) is a lower bound condition of the pitch diameter of the first-stage sliding lead screw thread; g19(X) is the upper bound condition of the pitch diameter of the third-stage sliding lead screw thread; g20(X) is the lower bound constraint condition of the wall thickness of the second-stage sliding screw, g21(X) is the lower bound constraint condition of the wall thickness of the third-stage sliding lead screw, g22(X) is a lower bound on the gap between the second and third stage sliding screws.
CN202210073142.5A 2022-01-21 2022-01-21 Design method of three-level sliding lead screw based on ordinary gear train Pending CN114510795A (en)

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