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1. When the structure design of the multi-locking point is adopted, the size of the mounting hardware of the door and window fan profile is narrow, and the transmission locker of the rack-and-pinion transmission structure is installed at the handle, and the outer dimensions thereof are greatly limited. Therefore, in the design of the corresponding product, in order to reduce the outer circle of the gear without damaging the bending strength of the straight teeth of the gear (without reducing the modulus), the number Z of the gear can only be reduced, so Z<12 is used in most occasions. For Z<Zmin=17, it is easy to cause interference between teeth and even jam. In order to make the Z<12 gears do not interfere, the author has compiled the mechanical principle, the relevant information in the mechanical parts, and the experience of measuring the related products abroad, and explained the minimum number of teeth that can be processed by the rack tool without the undercut condition. With ultra-short tooth tip height coefficient h
Analytical formula for the data and geometric dimensions of a, and give Zmin and h without rooting
a calculation table and computer examples.
Second, the rack and pinion does not produce the undercut geometry 1. Several basic concepts of the rack and pinion (1) The process of machining the gear with the rack tool (using the Fan method) is equivalent to the transmission process of the gear and the rack.
(2) It is well known that when the number of gear teeth is less than the minimum number of teeth, Zmin<17; when the displacement coefficient X is smaller than the minimum displacement coefficient Xmin, interference (root cut) occurs.
Namely: Xmin≤17-Z17=0.29
(3) Regarding the names of the rack cutters, regardless of the position of the rack line, the circumference of the gear pitch circle must be equal to the circumference of the gear shaping cutter, and the pitch circle of the cut gear is rolling. In the process of the system, the circumference of the tooth cutter is divided into Z equal parts.
Therefore, the pitch line of the rack insert is called the index line (machine line) and the pitch circle of the cut gear is called the index circle.
1 is a rack cutter. The modulus line is the average line of the rack insert, and the rack tool is also called the center line. The gear that is rolled by the center line of the rack knife is called a standard gear, and its tooth thickness S is equal to the tooth width W is equal to πm/2.
For the same reason, the tooth thickness is not equal to the tooth between the teeth, which is called a non-standard gear or a correction gear.
(4) When Z < 17, a root cut phenomenon occurs. Decrease Zmin without rooting using the following three methods:
1 does not produce root cutting can reduce the tool tip height coefficient ha (because Zmin = 2h
The disadvantage of a/Sin2a is that the overlap coefficient is reduced, which is detrimental to the smooth operation, but it is not suitable for manual rack and pinion transmission.
(Reducing Zmin and not cutting it can also use double modulus gear, not elaborated)
2 does not produce root cutting can increase the tool tooth angle a, (Z = 2h
a/Sin2a) has the disadvantage of high power consumption. Usually a = 15 °, 20 ° is the standard.
3 The root cutting is not performed by the distance correction method (displacement gear), which can improve the pinion strength and transmission quality.
2. Reduce the height coefficient of the crest to prevent the undercut mechanism 2 is the geometric relationship between the rack tool and the gear to be machined. The rack-cutting tool produces a mechanism for undercutting and no undercutting, which can be analyzed by the geometric relationship between the normal tool and the gear being machined.
2P-P is (ha
=1, α=20°) The center line of the rack cutter is the tangent PW of the base circle (R0=Rcosa) through the intersection point P of the gear vertical center line and the gear index circle; the KK side line of the rack is regarded as The cutting edge of the rack tool. During the cutting process of the gear by the rack, the rack cutter performs a reciprocating cutting motion perpendicular to the drawing surface. The cut gear rotates around the center O, which is equivalent to the pure rolling of the tool KK edge along the base circle 2R0 (same motion as PW-). It is now described in two cases.
(1) When the tool tooth tip HK is below the base circle tangent point N1, it is the tool solid line aNN1K state, or the tool b-N1-K state. At this time, the involute tooth surface S outside the base circle 2R0 has been completely cut out. This point is the limit contact point. When the tool b-N1-K continues to move, it has left the tooth profile S, and no undercut occurs.
(2) When the tool tooth top HK is higher than the N1 point (tool solid line a state, aHK), when the cutting edge KK is also moved to the N1 point, the tooth profile other than the base circle is also cut out, but the cutting edge KK is Without leaving the cut tooth profile S, it is possible to produce an undercut. The analysis is as follows: Assume that when cutting, the blade is moved to the right by a distance R ψ, KK is moved to K1-K1 (dashed line), and the gear tooth profile S has moved to the S1 position by the condition that the index circle and the tool center line PP are rolled. The blade K1-K1 is about to be separated from the tooth profile S1, and no undercut occurs. We assume that the tooth profile does not roll to the S1 position and only rolls to the virtual tooth line at the S2 position. It is obvious that the tooth profile must interfere with the cutting edge K1-K1 to produce an undercut. If the K1-K1 corresponding gear center angle is set to ψ', it is regarded as the angle at which no undercut occurs. If the gear actually turns, the angle is ψ. It can be seen that when ψ>ψ' (K1-K1 leaves early), it is not cut, otherwise ψ<ψ', K1-K1 does not leave S2, it must produce undercut.
(3) The geometric relationship between the rack tool and the cut gear can be seen that when the tooth tip HK of the blade exceeds the contact trajectory and the base circle tangent point N1, the cut gear should be rooted. When the rack tool is meshed with the gear as a rack, interference or even jamming of the tooth tip and the tooth profile occurs, which is attributed to the fact that the number of teeth of the gear is less than a certain minimum number of teeth. In today's 2, the distance between the center line PP to N-N1 of the rack (tool) is set to NP, and the height of the NP must be much smaller than the height h of the normal tooth, so that it is not cut.
That is, NP<<ha=hamNP=ha-Xm(1)ha
―Normal (tool) tooth top coefficient, ha
=1; m - the modulus of the rack tool; X - the coefficient of displacement.
According to Figure 1: NP=R-R0
Cosα=R-R
Cos2α=R(1-cos2α)=m
Zmin2(1-cos2α)(2) where: m-rack tool modulus; Zmin - the minimum number of teeth of the cut gear; R - gear index circle radius; α = 200 - rack tool angle.
It is known from equation (2) that the smaller the number of gear teeth, the smaller the tooth height ha'=NP of the tooth tip to the center line of the rack, so the gear does not cause undercut, that is, the number of teeth must not be less than the minimum number of teeth Zmin. , namely: Note: After using the positive displacement coefficient x, the tool is removed from the center distance, and the size of the machined pinion becomes larger, which is not in line with the purpose of reducing the gear size, and the ultra short gear coefficient ha
The calculation reduces the outer diameter of the gear.
Door and window fittings ha'=ha
m<Zmin2 (1-cos2α)Zmin≥2ha'(1-cos2α)=2ha
Mm(1-cos2α)(3)
In 3, the normal rack tool is moved away from the center of the gear by a distance xm (x is the shift coefficient). The physical meaning of ha in this case is the "ultra-short tooth coefficient" in which the tooth tip becomes short.
Known by the above formula: ha' ≤ Zmin
Sin2α2(4)
The top of the tooth in 3 is ha'=ha
m is the length of the tooth on the top of the normal rack tool is cut X
The crest of the abnormal tooth (ultra-short tooth) after m is high.
The PP line is the a-a1-a2-a3 tooth shape of the abbreviated a-type tooth of Fig. 3, and the original tool center line PP is meshed with the gear indexing circle as the ultra-short toothed rack modulus line (index line). Called the tool pitch line; if the HN is not cut in 3, but HN is moved down a section, then XN=x
The m, N-a2 line is the top line of the tooth, and the height of the tip is ha'=m
Ha
. The center line of the original PP also becomes the ultra-short tooth cutter pitch line at the meshing circle of the gear; ha' is the pitch of the pitch of the ultra-short rack tool to the top of the tooth, and the amount of the original midline is shifted to x.
m. Ultra short and short rack:
Ha'=ha m(5)
It should be noted that there is a pitch circle when the two gears mesh, and there is no pitch circle in the single gear. The pitch circle and the index circle may overlap or may not coincide. When the center distance of the two wheels is changed, the pitch diameter is also Change, but the index circle diameter D does not change. See 3. The calculation formula of the tooth thickness S, the inter-tooth W, the circumference P and the pitch circle diameter DJ and the number of gear teeth on the pitch circle is: P=S W=Ï€
D/Z(6)
3 original tooth top height cut to x
After m becomes a-a1-a2-a3 tooth profile P is the modulus line; down shift x
After a period of m, it becomes the bN-a2-a3 ultra-short tooth cutter tooth shape, P is the pitch line, and p' is the modulus line.
According to (4), the ultra-short tooth height coefficient ha of the rack tool when the gear is not cut can be calculated.
(The values ​​of the ultra-short tooth tip coefficient and the number of teeth are listed in Table 1).
Based on the above analysis, it can be seen that there are two cases in which Zmin ≤ 17 does not cause interference.
(1) 3 cut the top of the tool x
After a period of m, the new tooth height of the tool is lower than NN1. Cut the rack of the full tooth height Ha size of the rack to HN=x
After m, it becomes an ultra-short rack aa1a2a3.
PP becomes the modulus line for designing ultra-short toothed racks, and its pitch P=Ï€m is used to design other geometric parameters of ultrashort gears. The tooth shape of a-a1-a2-a3 in Fig. 3a is called an "a" shaped rack. Its pitch line coincides with the modulus line.
(2) In Figure 3, the standard rack is moved down for a period of time to make HN=m
After x, the rack becomes the tooth shape at the shadow of bN-a3. When this is used as the rack tool, the original tool center line position PP becomes the rack line line that meshes with the gear indexing circle, and the tooth thickness on the meshing circle of the machined gear is thickened, which is called narrower. The rack is called "b"
A rack (see Figure 3b), P'-P' is a modulus line.
Summary is: use ha
The ultra-short tooth profile design of the rack and pinion gear, that is, the a-shaped and b-shaped rack-and-pinion tools can be used to machine gears that do not produce undercut.
Note: Why not use the shift to design a gear that processes Z=17 teeth? Because the pinion uses Z in the 8 ~ 17 is a compact need. The outer diameter of the pinion is not allowed to increase. The gear shifting with the displacement coefficient must be positively displaced, and the tool is used to evacuate the center of the gear, so that the outer circle of the gear is increased. The method of the present invention is to reduce the gear height of the gear into an ultra-short gear, satisfying the pinion. Size requirements.
4. The calculation of the tooth profile angle α=20°, modulus m, tip height ha′, tooth root height hf′ of the non-standard rack gear.
(1) The tooth height of the rack cutter is ha', the root height is hf', and the pressure angle α=20°. The tooth profile of the rack cutter is a-shaped and b-shaped.
The tooth tip cutting distance H of 3 is a short tooth a tooth shape, a tooth shape: ha'=m
Ha; (or a maximum of 1.25m); the midline tooth thickness S = π
m/2.
The tool of Fig. 3 is moved to HN and is a short tooth b-shaped tooth cutter tooth (shaded portion).
b tooth shape: ha'=m
Ha,hf'=m midline tooth thickness S=Ï€
m/2.
(2) Gear tooth tip height ha, tooth root height hf, full tooth height H index circle radius R, and tip circle radius Ra.
Ha1=m hahf=ha 0.25m (or =m)R= Zmin/2R=mZmin2 haH=ha hf pitch: P=Ï€. m
(3) The geometric parameters of the ultra-short-toothed rack and pinion tool (the parameters are unified in the table) P = π. m Note: Ha see 1 with zinc alloy die-casting, powder metallurgy molding, when punching the rack and pinion, the mold can be processed according to specific parameters.
(4) Backlash of the tooth, radial clearance of the rack and pinion, and guiding center distance.
See parameter 4 for the parameters in the figure.
2Ra-gear tip diameter, 2R-gear index circle diameter, H(=R)- is the equivalent center distance, A'-guide center distance (assembly reference), A' should use positive deviation â–³.
The positive value of the backlash is ξ = 0.2 ∽ 0.3, and the tooth thickness S is controlled by the deviation under the deviation of the tooth thickness. Radial clearance of gear rack meshing C=mh
Am can fully meet the requirements of the tooth backlash tolerance according to the positive and negative deviation of the tooth thickness. In addition, to repair the gear, its rounding radius r = 0.3 ~ 0.6.
Third, the application example 1.
Multi-point lock rack and pinion transmission mapping and geometric ruler door and window fittings Figure 3a tooth shape (I recommend the design of the Z<17 tooth gears used in the rack shape) Figure 3b Toothed table 2 ultra short tooth rack and pinion Tool geometry parameter table Parameter name Gear a-shaped rack b-shaped rack index circle diameter 2RZm tooth tip height ha (rounded R=0.6) m tooth root height hfm tooth tip circle diameter 2Ra2R 2ha tooth root circle diameter 2Rfm
Z-2hf base circle diameter 2R02Rcosα tooth full height Hha hf center line tooth thickness Sπm2πm2πm2 door and window fittings inch calculation.
Surveying and drawing dimensions: gear outer circle 2Ra=21.3; tooth number Z=12, rack pitch P=5; tooth angle α=20°; calculation of 1 modulus calculated by a-shaped tooth:
Then m=PÏ€=53.14=1.59
2 Gear index circle 2R and tip circle diameter 2Ra calculation: 2R=Zm=12×1.59=19 Check the super tooth tooth height coefficient ha=0.702 tooth tip height: ha=mha=1.59×0.702=1.116 tooth Root height: hf=m=1.59 molar top diameter 2Ra=2R 2ha=19 2.32=21.232 (rounded R=0.6) 3 rack line position, tooth height, root height: tooth height ha= Mha=1.59×0.702=1.116 (rounding R=0.6) root height hf=m=1.59 tooth full height h=ha hf=1.116 1.59=2.706 tooth side gap ξ0.2-0.3 rack and pinion simulation movement on computer It can be seen that no interference occurs and the physical operation is normal after installation.
5, 6 is the result of computer design.
Example 2.
Designed with CAXA electronic board 2005, Z=12 pinion. The diameter of the pinion tip circle is limited to da23.52 mm, and the root circle df is 17.43 mm. According to Table 1 of this paper, the short tooth height coefficient ha
And the tooth shape of Fig. 3a is used to design the pinion gear, and the formula is: da=Zm 2h
Amh=ha hf=h
Am m1 select the appropriate h according to Table 1.
a=0.72, a=20° input to the computer, see if the computer shows the integrity of the root shape; 2h
a = 0.72, a = 25 ° input to the computer to show the integrity of the root shape.
For Zmin=2ha
/sin2α, where Zmin is ha
A function with a.
Figure 7 is the computer input a = 20 °, ha
=0.72, m = 1.75 calculated results and tooth profile.
Figure 8 is the computer input a = 25 °, h
a=0.72, m=1.75 calculated results and tooth profile.
7 and 8 are not cut, and one of them can be used.
Rack design: After using CAXA electronic version 2005 gear design software to convert to the more skilled AutoCAD2007 version, the formula: the top of the tooth is ha=h
Am = 0.72 × 1.75 = 1.26; the modulus line (center line) tooth thickness S = πm / 2 = 2.75, a = 200 drawn rack and gear mesh as shown in Figure 9, no interference. The tooth thickness tolerance of gears and racks is s-0.165.
IV. Conclusion Example 1: The gear parameters obtained by the analysis of the rack and pinion are as shown in Table 2 of Table 1, and the ultra-short tooth coefficient of Table 1 is applied.
Regardless of the calibration of the surveying dimensions and the redesign of the rack and pinion dimensions, no interference occurs. It is very simple to use and input the relevant parameters into the computer to complete the design. This is a non-standard tooth height rack and is a non-standard complete involute ultra-short gear. Figure 5 and Figure 6 show that the non-standard tooth top coefficient gear parts are not generated by CAXA software, and the full-tooth gear diagram can be drawn according to the given geometrical parameters. The tooth thickness should be marked with the upper and lower deviations. For example, the shape of the teeth is drawn by CAXA. As a result, the two methods of drawing have a tooth tip diameter of 0.09.
Figure 5 shows the meshing of the rack and pinion of the computer according to the parameters. Figure 6 is a simulation diagram of Figure 5.
Example 2: a = 20 °, ha
= 0.72, m = 1.75 drawing Figure 9 rack without interference, using a computer simulation diagram as shown in Figure 10. Because the tooth thickness tolerance is not included, there is a slight friction between the tooth profiles. Actually, the S-0.15 thickness is not used for friction. If the center distance tolerance is included, the A â–³ tolerance will not cause friction at the pitch circle. Estimated may be a software problem). Therefore, according to the above computer design is reasonable and available.
In summary, the design of the gear rack for the transmission mechanism of the door and window hardware multi-point lock is extremely critical. At present, the hardware fittings for the irrational rack and pinion structure design used on the market are not easy to appear in the short term, but the time is Long, the comprehensive mechanical effect of the transmission trajectory is not effective, and serious quality problems will inevitably occur.