1. What is the fundamental law of tooth profile meshing? What is the fundamental law of tooth profile meshing with a constant transmission ratio? What is the function of the fundamental law of tooth profile meshing?
A: In a pair of meshing gears, where the tooth profiles contact at any point, the transmission ratio is equal to the inverse ratio of the two segments divided by the common normal at the contact point of the line connecting the centers of the two gears. This law is called the fundamental law of tooth profile meshing. If the common normals at all contact points of the tooth profiles intersect the line connecting the centers at a fixed point, then it is the fundamental law of constant transmission ratio tooth profile meshing. Its function is to establish requirements for the tooth profile curve based on whether the transmission ratio is constant.
2. What are pitch nodes, pitch lines, and pitch circles? What kind of gear is a gear whose pitch node traces a circle on its surface?
A: The intersection of the common normal to the contact point of the tooth profiles and the line connecting their centers is called the pitch point. The trajectory of the pitch point on the gear during the meshing of a pair of tooth profiles is called the pitch line. If the pitch line is circular, it is called the pitch circle. Gears with a pitch circle are called circular gears; otherwise, they are non-circular gears.
3. What is a conjugate dental arch?
A: A pair of tooth profiles that satisfy the basic law of tooth profile meshing are called conjugate tooth profiles.
4. How is an involute formed? What are its properties?
A: The generating line rolls purely on the base circle, and the trajectory of any point on the generating line is called an involute.
nature:
(1) The length of the straight line that the generating line rolls over is equal to the length of the arc that is rolled over on the base circle.
(2) The normal to any point on the involute must be tangent to the base circle.
(3) The closer the point on the involute is to the base circle, the smaller the radius of curvature, and vice versa. The involute is also straighter.
(4) The distance between the normal directions of two involutes on the same base circle is equal.
(5) The shape of the involute depends on the size of the base circle. When the expansion angle is the same, the smaller the base circle, the greater the curvature of the involute. The larger the base circle, the smaller the curvature. When the base circle is infinitely large, the involute becomes a straight line.
(6) There is no involute within the base circle.
5. Please write the polar equation of the involute.
Answer: rk = rb/cosαk
θk=invαk=tgαk一αk
6. What is the reason why involute tooth profiles satisfy the fundamental law of tooth profile meshing?
answer:
(1) According to the properties of involute, the normal to any point of the involute must be tangent to the base circle.
(2) There is only one common internal tangent on the same side of the two circles, and the common normal at the contact point of the involute profiles of the two gears must be tangent to the two base circles. Therefore, there is only one node, namely:
i12＝ω1/ω2＝O2P/O1P＝r2′/r1′=rb2/rb1=constant
7. What is the line of engagement?
A: The trajectory of the contact point between the tooth profiles of the two wheels.
8. What are the characteristics of involute tooth profile meshing, and why?
answer:
(1) The transmission ratio is constant because i12＝ω1/ω2＝r2′/r1′. Since there is only one common tangent on the same side of the two base circles, and it is the common normal and meshing line of the contact point of the two tooth profiles, there is only one intersection point with the line connecting the centers. Therefore, the transmission ratio is constant.
(2) The center distance is separable and the rotation ratio remains unchanged. Since i12＝ω1/ω2＝rb2/rb1, the transmission ratio is determined after a pair of gears is machined and is independent of the center distance.
(3) The direction of the normal pressure between the tooth profiles remains unchanged because the direction of the normal pressure between the tooth profiles is along the common normal of the contact point. This common normal is also the common tangent on the same side of the two base circles, and there is only one such tangent. Therefore, the direction of the normal pressure between the tooth profiles remains unchanged.
(4) The meshing angle α varies with the center distance because aCOSα=a′COSα′.
(5) Four lines in one: The meshing line is the common tangent on the same side of the two base circles; it is the common normal of the tooth profile contact point; the trajectory of the contact point is the meshing line; it is the line of action of the normal pressure between the tooth profiles and the sum of the curvature radii of the contact point.
9. What are module and pitch circle?
A: The circle in which m = p/π is the modulus and m and α are the standard values is called the pitch circle.
10. What are pitch, tooth thickness, and tooth space width?
A: The arc length between the tooth profiles of two adjacent teeth on the same side of a circle is called the circumferential pitch. The arc length occupied by the tooth thickness is called the tooth thickness, and the arc length occupied by the tooth groove is called the tooth groove width.
11. What is a standard gear?
A: The gear has standard values for m, α, h*a, and c*, and s=e=p/2.
12. What are the characteristics of a rack and pinion?
answer:
(1) The pitch of each straight line parallel to the tooth tip line is equal, and its module and pressure angle are standard values.
(2) The straight line parallel to the tooth tip line and the tooth groove width is equal to the tooth thickness is called the center line, which is the reference line for determining the size of the rack.
13. What are the theoretical line of engagement, the actual line of engagement, and the working section of the tooth profile?
answer:
Theoretical line of engagement: The common tangent on the same side of the two base circles, which is theoretically the trajectory of the meshing point of the tooth profile, and the two tangent points are the meshing limit points.
Actual line of action: The line segment between the intersection of the two tooth tip circles and the theoretical line of action.
Working section of tooth profile: The part of the tooth profile that participates in meshing during gear transmission.
14. What are the correct meshing conditions and continuous meshing transmission conditions for involute spur gears?
A: The correct meshing condition is
m1=m2=m
α1=α2=α
Continuous meshing condition: εα=B1B2/Pb≥1
15. What is the essential meaning of overlap? What is overlap related to?
A: The contact ratio indicates the number of gear pairs that mesh simultaneously on the meshing line during a gear transmission. The contact ratio is an important indicator of the gear's load-bearing capacity and smoothness. The contact ratio is independent of m and increases with Z1 and Z2. The larger α′ is, the smaller εα is. α′ varies with the center distance; the larger a is, the larger α′ is, and the smaller εα is.
16. What is the standard mounting center distance for standard gears? What are the characteristics of standard mounting?
A: The center distance for mounting standard gears according to the toothless side clearance is called the standard mounting center distance for standard gears. The center distance for mounting standard gears according to the standard tip clearance is also called the standard mounting center distance. In standard installation, a = a′, r = r′, a = r1 + r2
17. What is a non-standard installation center distance? What are the characteristics of non-standard installation?
A: When a pair of meshing gears are installed in a way that the pitch circle and the index circle do not coincide, it is called a non-standard installation, and the center distance between them is called the non-standard installation center distance.
Features:
r≠r′, a≠a′, a′=r1′+r′2=(r1+r2)cosα/cosα′
That is, a′≠aα′≠α
r1′≠r1r2′≠r2c′≠c
Tooth flank clearance causes impact, reduces overlap, and results in poor stability.
18. What are the characteristics of gear and rack meshing transmission?
answer:
(1) The position of the meshing line does not change due to the relative position between the gear and the rack. It is always a fixed straight line tangent to the base circle and perpendicular to the straight tooth profile of the rack.
(2) r=r′α′=α=rack tooth profile angle
19. What are the characteristics of machining standard gears with standard rack and pinion cutters?
A: The pitch circle of the wheel blank is tangent to the centerline of the rack cutter and rolls purely. The number of teeth of the gear being machined is ensured by the cutting speed and the angular velocity of the wheel blank: V_cutter = rω_blank.
20. What is the root-cutting phenomenon of involute tooth profile? What are the causes?
A: When gears are machined using the generating method, the phenomenon where the machined involute tooth profile is cut off is called undercutting.
Cause: The intersection of the tool's tooth tip line and the line of engagement exceeds the meshing limit point of the gear being cut. The reason why the tool's tooth tip line exceeds the meshing limit point is that the number of teeth of the gear being machined is too small, the pressure angle is too small, and the tooth tip height coefficient is too large.
21. How is the minimum number of teeth required to prevent undercutting in a standard external meshing gear determined?
Answer: It is determined by Zmin = 2h*a/sin2α.
22. What is a modified gear?
A: Gears whose pitch circle tooth thickness is not equal to the tooth space width, and gears whose addendum is not a standard value, are called modified gears. Gears in which the centerline of the rack cutter is not tangent to the pitch circle of the gear being machined are also called modified gears.
23. What are displacement, displacement coefficient, and minimum displacement coefficient?
answer:
Displacement: The vertical distance by which the centerline of the cutting tool is shifted from the position where the standard gear is being machined.
Displacement coefficient: The coefficient required to express the displacement using a standard modulus.
Minimum displacement coefficient: The minimum displacement coefficient required to prevent undercutting during machining involute gears.
xmin = h * a(Zmin - Z) / Zmin
24. Compared with a standard gear, which dimensions of a modified gear with the same number of teeth have changed and which have remained the same? Why?
A: The number of teeth, module, pressure angle, pitch circle, base circle, pitch circle circumferential pitch, and total tooth height remain unchanged, while the addendum circle, dedendum circle, pitch circle tooth thickness, and tooth space width have changed.
Reason: When machining modified gears with standard gear cutters, the machining method remains unchanged, meaning the correct meshing conditions remain unchanged, so the pitch circle module and pressure angle remain unchanged. Therefore, according to the formula, the pitch circle and base circle remain unchanged. Furthermore, the calculation of tooth root height, tooth addendum, tooth root circle, and tooth tip circle is based on the pitch circle. When machining modified gears, if the centerline of the standard cutter moves outward from the pitch circle, the tooth root height decreases, and the tooth root circle increases. However, to ensure that the total tooth height remains unchanged, the tooth addendum increases, and the tooth addendum circle increases. Because the cutter moves outward at the gear's pitch circle, the cutter tooth thickness decreases, meaning the machined tooth grooves become smaller. Also, because the pitch circle circumferential pitch remains unchanged, the tooth thickness increases.
25. How is the involute helical tooth profile of a helical gear formed?
A: When the involute generating surface rolls purely on the base cylinder, a line on the generating surface forms an angle βb with the generatrix of the base circle. Its trajectory forms the involute helical surface of the helical gear teeth.
26. Are the helix angles of the helical lines on the coaxial cylindrical surfaces containing the tooth profile of a helical gear the same? Why?
A: The helix angles are different because the helix angle βi is determined by the lead L and the diameter di of the cylinder. Even with the same lead, different cylinder diameters result in different helix angles. The relationship is: tgβi = L/πdi
27. What are the characteristics of helical gear meshing?
answer:
(l) The tooth profiles of the two gears start to contact from a point, the contact line changes from short to long, then to short again, until point contact, and then disengages. Unlike the sudden contact and sudden disengagement along the entire tooth width of spur gear transmission, the two gears gradually enter and gradually disengage, resulting in less impact, less noise, and smooth transmission.
(2) The degree of coincidence is large ε=εα+εβ
28. Which surface is the standard parameter surface of a helical gear, and which surface is the standard involute? Explain why.
A: The normal surface is a standard parametric surface. Theoretically, the end face is a standard involute because the involute is formed by the generating surface rolling purely on the base cylindrical surface, and the trajectory of the oblique straight line on the generating surface is the involute. From a machining perspective, the normal surface is a standard involute because machining the tooth profile of helical gears uses the standard tool used for machining spur gears, and its cutting motion direction is along the helical tangent. The tool face is on its normal surface; therefore, the normal surface is a standard involute.
29. What is the relationship between the geometric parameters of the end face and the normal face of a helical gear? Why are end face parameters necessary?
A: mn=mtcosβ, tgαn=tgαtcosβb, h*at=h*ancosβ, c*t=c*ancosβ, because the geometric dimensions are end face dt, dbt, dat, dft, pt, pbt.
30. What are the correct meshing conditions and continuous transmission conditions for a pair of helical gears?
A: Correct meshing conditions: mn1=mn2=mαn1=αn2=α
External meshing β1=-β2, internal meshing β1=β2
Continuous transmission condition: ε = εα + εβ ≥ 1
31. What are the equivalent gear and equivalent number of teeth of a helical gear? What is the purpose of the equivalent number of teeth?
A: A spur gear with a tooth profile equivalent to that of a helical gear is called an equivalent gear. The number of teeth on the equivalent gear is called the equivalent number of teeth. The equivalent number of teeth is an important basis for selecting the tooth profile of the cutting tool in the profile machining method, and it is also a major basis for gear strength design.
32. What are the characteristics of a worm gear mechanism?
answer:
(1) Transmitting motion and power between spatially intersecting axes, i.e., spatial mechanisms.
(2) When the worm gear meshes, theoretically the tooth profile contact is a point contact, but the worm gear is made by the hob of the worm gear that meshes with the worm gear, so in reality it is a spatial curve contact.
(3) The transmission ratio of the worm gear is calculated using the number of worm threads.
(4) The pitch circle diameter of the worm is not the number of threads multiplied by the module, but the characteristic coefficient multiplied by the module, i.e., d1=qm
(5) The center distance of the worm gear is also calculated using the characteristic coefficient.
a = m(q + Z²)/2
(6) It can achieve a large transmission ratio and is self-locking when the worm gear is driven.
33. What is the standard parameter surface of a worm gear? What are the conditions for achieving correct meshing?
answer:
(1) is the main section, that is, the cross section parallel to the end face of the worm wheel and passing through the axis of the worm is called the main section.
(2) Correct meshing conditions: ma1=mt2=mαa1=αt2=αβ1+β2=90° Same rotation direction
34. Why is the characteristic coefficient q of the worm gear determined to be a standard value?
answer:
(1) It is conducive to the standardization of worm gears and reduces the number of worm gears.
(2) The number of worm hobs used to machine worm gears has been reduced.
35. How to determine the direction of rotation during worm gear meshing transmission?
A: First, determine the direction of rotation of the worm or worm wheel: With the axis of the worm or worm wheel vertical, the helix with the right side higher is right-handed, and the left side higher is left-handed. Then determine the direction of rotation: For right-handed worms, use the right-hand rule. For the driving worm, grasp the worm with four fingers following its direction of rotation, with your thumb pointing in the opposite direction to the node velocity of the worm wheel.
36. What are the characteristics of a spur bevel gear mechanism?
answer:
(1) Transmitting motion and power between two intersecting axes.
(2) The gears are distributed on the cone and shrink from the large end to the small end.
(3) The large end face is the standard parameter face.
(4) The tooth profile curve is a spherical involute.
37. What are the correct meshing conditions for spur bevel gears?
A: For the large end face, m1=m2=m, α1=α2=αR1=R2 (R is the cone distance).
38. What are the back cone, equivalent gear, and equivalent number of teeth of a bevel gear?
A: The cone tangent to the pitch circle on the large end spherical surface of the bevel gear is called the back cone of the bevel gear. The sector formed by projecting the tooth profile of the large end face of the bevel gear onto the back cone parallel to the generatrix of the cone is called a sector gear. The spur gear that corresponds to the tooth profile of the large end face of the bevel gear is called the equivalent gear of the bevel gear, and its number of teeth is called the equivalent number of teeth.
39. What are the uses of equivalent gears and equivalent number of teeth?
A: The equivalent gear of a pair of bevel gears is used to study the meshing principle of bevel gears, such as the overlap ratio and the correct meshing conditions. A single equivalent gear is used to calculate the minimum number of teeth without undercutting and to select the tool number and calculate the bending strength of bevel gears when machining bevel gears using the profile method.
40. How are the pitch circle diameter, transmission ratio, and equivalent number of teeth of a bevel gear calculated?
Answer: d=2Rsinδ; i12=ω1/ω2=z2/z1=d2/d1=sinδ2/sinδ1; zv=z/cosδ
Typical Examples and Answers
Example 1: Given a pair of externally meshing involute spur gears with the following parameters: m=5mm, ha*=1, c*=0.25, α=200, z1=10, z2=20, x1=0.4249, x2=0;
(1) Use calculation to determine whether undercutting will occur when machining the pinion; (4 points)
(2) Calculate the base circle radii rb1 and rb2, the addendum circle radii ra1 and ra2, and the base pitch Pb of the two gears; (12 points) (3) If the center distance a' = 76.95 mm when the two gears are meshing without tooth clearance, use a graphical method to find their overlap coefficient ε. (Specified: Take the length scale μL = 1 mm/mm)
untie:
(1)∵When z1=10,
xmin=（17-z1）/17=（17-10）/17=0.418
Since x1=0.429>xmin=0.418, root cutting will not occur.
(2)
rb1=r1cosα=mz1/2cosα=5×10/2×cos200=23.49mm
rb2=r2cosα=mz2/2cosα=5×20/2×cos200=46.98mm
ra1=m(z1/2+h*a+x1)=5×(10/2+1+0.4249)=32.12mm
ra2=m(z2/2+h*a+x2)=5×(20/2+1+0)=55mm
pb=pcosα=πmcosα=5×π×cos200=14.76mm
(3) Correctly draw the center distance a'=76.95mm and the base circle and addendum circle of the two gears;
Accurately draw the theoretical line of engagement N1N2, find the points B1 and B2 on the actual line of engagement, and measure B1B2 = 20.5 mm.
Substituting into the formula: ε = B1B2/pb = 20.5/1.4.76 = 1.39
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