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Optimization design of screw jack screw pair

Abstract: using the non-traditional optimization design method such as force value shaking at the yielding point, taking the minimum volume of screw jack screw pair as the objective function, the optimization design is carried out with MATLAB language optimization tool, and a calculation example is given

key words: optimize the design of import growth for the first time in the second half of the year in October; lifting jack; The establishment of mathematical model of transmission thread pair

manual screw jack mainly includes base, ratchet, bevel gear pair, cup, transmission thread pair and so on. The maximum lifting capacity of Jack is one of its main performance indicators. During the working process of the jack, the transmission thread pair bears the main working load, and the working life of the thread pair determines the service life of the jack. Therefore, the design of the transmission thread pair is the most critical, which is related to the maximum lifting weight, the material of the thread pair, the thread profile and the number of thread heads

1.1 objective function and design variables

on the premise of meeting the design performance and requirements, the manual screw jack is considered from the aspects of compact structure, weight reduction, material saving and cost reduction. Given the basic design requirements such as the maximum lifting capacity of the jack, the material of the transmission thread pair and its yield stress, the number of thread heads, and the number of bevel gear pairs, we can consider from the design of the thread pair to minimize the material used in the thread pair, that is, under the condition of meeting the design performance, the volume of the transmission screw and nut is the least

volume of screw: v1= π d22l/2

volume of nut: v2= π (D ′ 2-d22) h/4

where: D2 - pitch diameter of screw, mm

d '- outer diameter of nut (virtual), mm

d2 - pitch diameter of nut, mm

l - total length of screw, mm

h - nut height, mm

considering the higher requirements of transmission efficiency and the greater stress on the thread, the jack generally adopts serrated thread transmission, and the relationship between its major diameter, pitch diameter and minor diameter is as follows:

d2=d-0.75p

d1=d-1.736p

and the relationship between internal and external threads is as follows: D = D; D2=d2;

where D2 and D2 are the pitch diameters of internal and external threads; P is the pitch; D. D is the major diameter of internal and external threads; D1 is the minor diameter of internal thread

then the objective function (that is, the sum of the volume of the transmission thread pair) is:

v=v1+v2= π L (d-0.75p) 2-/4+ π [D ′ 2-- (d-0.75p) 2-] h/4

from the expression of the objective function, it can be seen that l and d 'are constants, and although the value of the pitch P is an integer, its value changes with the nominal diameter of the thread, which is taken as a variable here. Therefore, there are three variables D, h and P, which are recorded as:

x= [x1, X2, X3] t= [D, h, P] t

the expression of the objective function is:

V (x) = π L (x.75x3) 3/4 + π [D ′ 2- (x.75x3) 2] x2/4

1.2 optimization constraint conditions

1.2.1 constraint condition analysis

(1) wear resistance condition

.working height of serrated thread h:h=0.75p

according to the low sliding speed of the screw pair driven by the manual screw jack, And nut and screw material, check the allowable specific pressure [P]:

the calculated specific pressure is: p=fp/[(d-0.75p) π hh] <[P]

(2) thread self-locking condition

spiral rise angle ψ：ψ= Arctanp/π d2=arctanp/[π (d-0.75p)]

equivalent friction angle ρ V=arctanuv, UV is the equivalent friction coefficient of the thread pair

self locking conditions are: ψ＜ρ V - (1 ° ~ 1.5 °), i.e.

arctanp/[π (d-0.75p)] <ρ V - (1 ° ~ 1.5 °)

(3) strength condition of screw

dangerous cross-sectional area of thread a: a= π (d-1.736p) 2/4

torque of screw t:t=f · Tan（ ψ+ρ v) (D － 0.75p)/2

equivalent stress is:

where f is the maximum lifting capacity of the jack, and the unit is n.

check the table to determine the allowable stress [ σ］.

the equivalent stress should be less than the allowable stress, that is: σ ca＜［ σ］

(4) the shear strength condition of thread teeth

is calculated according to the nut material with weak mechanical properties:

the outer diameter of the nut D is equal to the outer diameter of the screw d:d=d

the root thickness of thread teeth b:b=0.75p

the number of thread turns z:z=h/p

check the table to obtain the allowable shear stress [ τ］.

calculated by shear strength: τ= F/(πDbz)=F/(πd·0.75P·H/P)， τ＜［τ］.

(5) bending strength condition of thread teeth

similarly, take the nut material with weak mechanical properties for calculation

calculated according to bending strength: σ b=3F(D-D2)/(πDb2z)=3F(d-d2)/(πDb2z)=3F［d-(d-0.75P)］/(πDb2z)

σ b＜［ σ b］.

for static load, the allowable stress should be taken as a larger value

(6) stability condition of screw

determine the flexibility of screw λ Value: λ=μ L/i

therefore, the price is generally higher than that of two components. In many formulas, μ Is the length coefficient of the screw, l is the total length of the screw, I is the inertia radius of the dangerous section of the screw, i=d1/4

the length coefficient of the screw is taken according to the fixed form of the screw pair

λ The value is less than the allowable value λ］， Namely: λ＜［λ］.

(7) range of nominal diameter of screw

check the mechanical design manual, and the range of D value is: 20mm ≤ D ≤ 650mm

(8) maximum nut height (thread engagement length) range: 30 mm ≤ h ≤ 280 mm

(9) value range of thread pitch

check the mechanical design manual, and the range of P value is 2 mm ≤ P ≤ 24 mm

1.2.2 constraints

the constraint expression is as follows:

g1 (x) =f- [0.75 π (x.75x3) x2] [P] ≤ 0

g2 (x) =arctanx3/[π (x.75x3)]- ρ v+(1°～1.5°)≤0

g4(x)=F/(0.75πx1x2)-［ τ］ ≤0

g5(x)=3F/(0.75πx1x2)-［ σ b］≤0

g6(x)=4 μ L/(x.736x3)-［ λ］ ≤ 0

g7 (x) =30-x2 ≤ 0

g8 (x) =x ≤ 0

g9 (x) =20-x1 ≤ 0

g10 (x) =x ≤ 0

g11 (x) =2-x3 ≤ 0

g12 (x) =x ≤ 0

2 optimization method

this problem has three variables and 12 constraints, and MATLAB optimization tool is used to optimize it

3 optimization design example

a factory produces a manual screw jack with a maximum design lifting capacity of 40 kn, serrated thread, 40Cr screw material, heat treatment hrc45 ~ 50, σ S=785mpa, ZCuAl10Fe3 for nut, equivalent friction coefficient of thread pair is μ V=0.13, the maximum lifting height of Jack is 130 mm, the thickness of bevel gear is 30 mm, and the fixed end of bearing l0/d0=7/18. Try to design the transmission thread pair, so that its structure is compact and the material used is the most economical

according to the previous mathematical modeling, we first obtain the relevant parameters of the constraint conditions by looking up the table or calculation, and then substitute them into the constraint conditions of the above modeling, so as to obtain the optimal design scheme of the screw pair

comparing the results of the original design and the optimized design (Table 1), it can be seen that the volume of the thread pair after the optimized design is reduced by 12.51% compared with the original design. Adopting the optimization design method can not only save materials, reduce factory production costs, but also save design time. This helps to reform the traditional design methods and provides a strong basis for the development and improvement of new products. Table 1 original design and optimized design results

References:

[1] joint compilation group of mechanical design manual Mechanical Design Manual Volume II Second edition [M] Beijing: Chemical Industry Press, 1987 2 〕 fan Mingyu, Zhang Ying Fundamentals of optimization technology [M] Beijing: Tsinghua University Press, 1982

[3] Qiu Xuanhuai, Wu Zongze, Guo Keqian, et al Mechanical design Fourth edition M 〕 Beijing: Higher Education Press, 1987

[4] EVA part － anger and TLAB 5 manual [M] Beijing: China Machine Press, 2000

[5] Gao Junbin Matlab 5.0 language and program design M 〕 Wuhan: Huazhong University of science and Technology Press, 1998.. [ 6 〕 Gong Xiaoping, Zhang Danfeng Reliability optimization design of high speed gear transmission [J] Mechanical design and manufacturing, 2000, (4):

[7] Wang buying A review of modern design methods [M] Beijing: Higher Education Press, 1985

［8］ Rao S sription and Optimum Design of Fuzzy Mechanical System［J］. Transactions o position it against the circular axis f ASME, 1987, (109): (end)

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