UA78075C2 - Method for determination of teeth module of involute cylindrical toothed wheels - Google Patents

Abstract

The invention relates to general engineering and can be used to determine the value of the teeth module of involute cylindrical toothed wheels determined under conditions of simultaneous account of bending and contact resistance providing minimal dimensions and mass of transmission.

Description

Description of the invention

The invention relates to the field of general mechanical engineering, namely to the method of determining the value of the module 2 of involute cylindrical gears.

There are known methods of determining the value of the modulus of winding cylindrical gears through the intercenter distance between them (see I. I. Markhel. Machine details. M., Mashinostroenie 1977, p. 104 | t-(0.01 0.02)а,,, where au is the intercenter distance, the value of which is determined from the condition of ensuring the necessary contact endurance of the winding profiles of the wheel teeth (see ibid. transmission ratio, Is) - contact stress for endurance, which is allowed MPa, Mr - calculated torque on the shaft Nm, khra - tooth length coefficient, which depends on the intercenter distance.

The ratio т-(0.01 0.02)a appears in all modern methods of calculating the modulus of winding cylindrical gears, in which the intercenter distance is determined by an expression similar to (1) with small deviations in constant coefficients that depend on the selected units of measurement (see ., example,

Reshetov D.N. Machine details. M., Mashinostroenie, 1975, p. 268 or Iosylevich G.B., Stroganov G.B., Maslov G.S.

Applied mechanics. M., Higher School, 1989, p. 169, etc.|.

The main disadvantages of the proposed methods of determining the value of the modulus of wheels are: c 1) the uncertainty of choosing the value of the modulus, which causes difficulties for the designer in the design of Ge) (the variation interval in this case is up to 5095), 2) an unreasonably large margin of bending strength, which leads to unreasonably large dimensions and metal capacity.

The closest method of determining the modulus of involute gears is the method proposed in (book Iosylevich G.B., Stroganov G.B., Maslov G.S. Prikladnnaya mechanika. M., Vnyisshaya Shkola, 1989, p. 169) che "c) t-- Ka | TaKn o - -----8-t

Tone in the blood Ni so - where Ka is a constant coefficient (Ka-49.5 for spur gears),

T» - torque on the wheel shaft, tva " Coefficient of tooth width, " и - gear ratio,

Yesni is the permissible contact stress, but with Kn is the load factor when calculating the contact endurance.

I" The task of this invention is the development of a new method of determining the modulus of the teeth of involute cylindrical gears, which ensures the minimum dimensions and weight of the transmission while simultaneously meeting the conditions of contact endurance and bending strength.

In the work |Ambartsumyants R.V. About the contact voltages in the winding toothed gear. Bulletin of Podillia Technological University, Mo4, part 2, 2003, p.18-21| it is established that the contact stress is

Go! function not only from the physical quantities included in equations (1), (2), but also from the total number of teeth 2 5-24 725, where 24, 22 are the number of teeth of the gear and wheel. Bending stresses are also a function of the total number of teeth. In view of this conclusion, as well as the equality y - |o | St. |s | Where wed; on - Bending and contact -1 50 nn ET sp tension on the teeth, sn -|sn/, sv: | se | - allowable bending and contact stresses, we will obtain an expression for determining the module of gear wheels

Ф т - Z Rche yu u rapg us vipe - st |on 60 de 2EE" - composite modulus of elasticity of materials of gear wheels, Еи, Ео - modulus of elasticity

Epya U 0---

RE E. gear and wheel materials, 65 p: iDy 1) - coefficient that depends on the transmission ratio c-2o/24, o, - Engaging angle.

The integer value 277 is determined from the inequality with (4) - С с5ипес --- хх Кк - sopvi 25 where

Hello! with pae vip with nap 4 ; Ke-- the load factor when calculating the bending strength, Khe 02544 r|oye Kn - - I 5 256: - sopvi a ueKe|sn - tooth shape factor.

To prove the effectiveness of the application of the new method of calculating the value of the modulus of the winding gear, we will give a numerical example and comparative indicators of the existing method of calculation with the proposed one.

Example. Determine the value of the module of a closed cylindrical spur gear with a gear ratio of i-4, a large torque on the wheel shaft T2-300Nm, wheel materials steel-45 with modulus of elasticity

E EE 5

E41-E»-2.1.10" MPa, allowable contact stress | сн | - 48Б5ММГ|а o allowable bending stress св | - 180MPa for non-reversible transmission, factor Ka-49.5. s

For simplicity of calculations and taking into account that they do not play any role for comparative evaluation, we accept Ke-Kn-1. We also accept in the calculations the coefficient of the width of the tooth of the hooves - 0.3.

Calculation of the value of the module according to the existing method (see expression 1) is . 4 49.5 31071 yuyu zo t - 5 :A3 -3 pkt shi 2536bmMmM sh 10.5 u03.-16.485

We will not round the module value to the nearest standard value, as is recommended in all technical sources, for a more correct comparison. (uh)

Gear parameters:

From The number of gear teeth (see Chernavskyi A.S. et al. Course design of machine parts. M., -

Mashinostroenie, 1988, pp. 361 to 317! pf) vva to t 54025 100 "

C--. Diameters of dividing circles: 44-2,536.25-63.4 mm; 9d2-2,536.100-253, bmm.

Diameters of the top circles: d441-2,536.27-68,472mm, yes"-2,536.102-258,672mm. )" Intercenter distance ahmu-(a:a")/2-158.5 mm. The width of the wheels Bu aa-O, 3.158.5548MM.

Calculation of the value of the module according to the proposed method (see expression 3). We choose the value of the tooth shape coefficient for the worst case calculation of the teeth for bending M 6-4.3 | see Chernavsky A.S. etc. Curriculum and design of machine parts. M., Mashinostroenie, 1988, p. 421). o We calculate the value in . 0.2544-21-107 -180-1 o in ---- 2 - »2377 poi 4. 43.1. (4857

We solve equation (4) numerically and find that at 7 U-197 the left part of equation (4) is larger than the right part (we have 2.3802 22.377). Therefore, the total number of teeth 2y is limited from above by the value 197. The minimum value of 2y; is limited based on the condition of excluding the undercut of the tooth stem. Inequality (4) becomes almost equal at 2U-(2U)tah-196. Then for standard gear wheels at a-29 we obtain: p-:ii1)-4/5-0.8,

F) I-- !

Po) s - u064-1967 5ipi, 203.2: 19654 - in - uZZ072436 - 59,222

Using formula (3), we find b5 theses 02544. 2.10 1 J 03. 08196196. 59222 5ip20 - 3507 2436)4857 - 31313005 - 145bMm

For the number of gear and wheel teeth, see Ambartsumyants R.V. About contact voltages in winding gear meshes. Bulletin of Podillia University of Technology, Mo. 4, 2003, p.18-21|: 7 2-p2U-0,8.196-156,8-4157; 70 214227U(in1)-196/5-39,2239.

The actual transmission ratio is 2-22/24-157/39-4.05. The error of the transmission ratio d-(4.05-4)/4-1.2590, which is assumed | see Chernavsky A.S. etc. Course design of machine parts. M., Mashinostroenie, 1988, p. 37).

Gear parameters:

Diameters of dividing circles: 44-1,456.39-56,784mm; d2-1,456-157-227,136bmm.

Diameters of the top circles: d44-1,456.41-59,696mm, da2-1,456.159-231,504mm.

Intercenter distance amu-tl2u, /250.75.196-147mm.

Width of wheels Btrrvaa-0.3.147а45MM.

To compare the results of both options for calculating the wheel modulus value, we will determine their approximate volume using a simplified formula

Megadamdaadoy.

For the option of calculation according to the existing method

M1-2.158,5.258,672.48-105.39,35953MM. sch rya For the option of calculation using a new method

Mo-2.147.231,504.45-105.30,627978 mm. io)

The ratio of volumes of Duma M2-39.35953/30627978-1.28508.

The obtained numerical value of the volume ratios shows that the volume of the gear transmission calculated by the new method is less than the volume of the gear transmission calculated according to the existing method by almost 2995. This allows us to assert that the mass of the gear transmission calculated by in a new way.

Let's check the strength characteristics of the gears of both options to make sure that the new method of calculation is correct. Av

For an example calculated according to the existing method:

Bending stress | see Chernavsky A.S. etc. Course design of machine parts. M,, so Mashinostroenie, 1988, p. 411. - 2TKE 300-143

CHE E-A B 8VB-o - sot 0.258672-48.2536 « z - 81937MPa no s Contact stress | see Chernavsky A.S. etc. Course design of machine parts. M,, in" Mashinostroenie, 1988, p. 311. I . 270 | yes)

ОН тшВВ--и- -Шжж--8v ryn

She a, By so 2 2790 13-107-1.125 o --- - - 31 - -YAYAYEE- - E - --9 - 558865 5MPa -0.70 158.5 48-16 sl For an example calculated by the proposed method: 2 300-143

CHE A B 8VBoi3 C - s.vt 0.231504-45.1456 (F) - 170.094MPa ime) . 270 |Ttzhtsii 1? bo bno 5 A 5 5 a, Vy 270 /3-107.1-125 -- хх - -Я Е Е Е --olp -8419175mMPa 65 158.5 45-16

From the comparative analysis of the given examples, it follows that both options of gears satisfy the strength conditions, both for bending and contact endurance. However, in the transmission calculated according to the existing method, the actual bending stress is 2.2 times (22095) less than the allowable stress. That is, there is an excessively large margin of strength for bending, which leads to an unreasonable increase in overall dimensions and weight of the transmission. The contact stress is less than the allowable total by 28.89.

In the transmission calculated according to the new method, the actual bending stress is less than the allowable stress by 1.06 times (6905). The contact stress is less than the allowable total by 15.7905, that is, it is approaching its limit value. This is what explains the reduction in the overall dimensions and weight of the transmission while maintaining the strength/o characteristics within the permissible limits.

The given example shows the correctness of the new method of determining the value of the modulus of winding cylindrical gears and its effectiveness in terms of reducing the overall dimensions of the transmission and its weight while maintaining all performance criteria, namely, ensuring contact and bending endurance.

Claims (1)

19 Formulas of the invention The method of determining the modulus of teeth of involute cylindrical gears based on contact and bending stresses that are allowed, also taking into account bending and contact stresses, external torque, modulus of elasticity of the first kind of wheel materials, the relative width of the wheels, the transmission ratio, which is distinguished by the fact that the module of the teeth of gear wheels is determined by the formula: where Y o,2544E pr Tzhnrya rope us IP - S Yesni o 7U,- the integer value of the total number of teeth, where I» 77 ul the number of teeth of the gear and wheel, which is determined from the inequality: И in ) 1 a e - o,eBA4E ris Kn y : 2 ' pr szipa-s duh -- L t 5 5 sopvi o AMEKEsN | (ee) 2 2 2 ' che spy vip se napgu 4 p - 5) « M 1 - sov EE; па ой -- ж 5- - И» ELE 1 v Е, Е - modulus of elasticity of the first type of wheel materials, 10-2 ш- ХЕТ tooth shape coefficient, со - load coefficients when calculating bending and contact endurance, о шк - respectively, the permissible bending and contact endurance stresses, with |se Meni - | Chun - tooth width coefficient, sl Ya re, with. - engagement angle, - torque on the wheel shaft, To gF) c - coefficient depending on the gear ratio and the total number of teeth, ko i - gear ratio, n - coefficient depending on the gear ratio, 60 Epr - modulus of elasticity of gear materials and wheels Official bulletin "Industrial Property". Book 1 "Inventions, useful models, topographies of integrated microcircuits", 2007, M 2, 15.02.2007. State Department of Intellectual Property of the Ministry of Education and Science of Ukraine. b5