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Strength calculation of cylindrical gear transmission

1 strength calculation of spur gear transmission

1 Calculation of tooth contact fatigue strength

in order to ensure that the gear does not have pitting failure within the predetermined life, the tooth contact fatigue strength should be calculated. Therefore, the calculation criterion of gear contact fatigue strength is: tooth surface contact stress σ H is less than or equal to the allowable contact stress σ HP, i.e σ H≤ σ HP

Hertz formula

because spur gears are often in the meshing area of a single pair of teeth near the node, and the force on the teeth is large, pitting first occurs near the node. Therefore, the contact fatigue strength of joints is usually calculated

figure a shows the contact of a pair of involute spur gears at the node. In order to simplify the calculation, a pair of cylinders with parallel axes are used to replace it. Radius of two cylinders ρ 1、 ρ 2 is equal to the curvature radius of the two tooth profiles at the node, as shown in Figure B. It can be seen from elasticity that when a pair of cylinders with parallel axes contact and are subjected to pressure, it will change from line contact to surface contact. Its contact surface is a long and narrow rectangle, which generates contact stress on the contact surface, and the maximum contact stress is located on the center line of the contact area, and its value is

in the formula σ H-contact stress (MPA)

fn-normal force (n)

l-contact line length (mm)

rs-comprehensive radius of curvature (mm)

± - positive sign is used for external contact, negative sign is used for internal contact

ze- material elastic coefficient (), where E1 and E2 are respectively the elastic modulus of two cylinder materials (MPA); M1 and M2 are Poisson's ratio of two cylinder materials respectively

the above formula shows that the contact stress varies with the change of the comprehensive curvature radius of each contact point on the tooth profile, and is the largest at the root of the tooth near the node (Fig. C, d). However, in order to simplify the calculation, the contact stress at the node is usually controlled

the parameter at the node

(1) the comprehensive radius of curvature

can be seen from the figure, and is substituted into the re formula to get the

formula:, which is called the tooth number ratio. For reduction drive, u=i; For speed increase transmission, u=1/i

as a result, there are

(2) calculate the normal force

(3) contact line length L

introduce the coincidence coefficient Ze, make the contact line length

substitute the above parameters into the maximum contact stress formula to obtain the contact fatigue strength calculation formula

order, which is called the node area coefficient

then get (1) the check formula of tooth contact fatigue strength

the check formula of tooth contact fatigue strength is

(2) the design formula of tooth contact fatigue strength

set the tooth width coefficient, and substitute it into the above formula, then get the design formula of tooth contact fatigue strength

, in which: D1 - pinion indexing circle diameter (mm)

ze- elastic coefficient of material (), which is checked in the following table

note: Poisson's ratio M1 = M2 = 0.3

zh node area coefficient. Considering the influence of contour curvature at the node on contact stress, it can be found from the lower left figure. For standard spur gears, a=250, zh=2.5

ze- coincidence coefficient. Considering the influence of coincidence on the load per unit tooth width, its value can be obtained from the lower right figure

application description of the contact fatigue strength formula

in the calculation of tooth contact fatigue strength, the contact stress of the mating gear should be equal, that is σ H1= σ H2。 However, the allowable contact stress of the two gears is related to the material, heat treatment and the number of stress cycles of their respective gears, which is generally not equal, that is σ HP1= σ HP2。 Therefore, when using the design formula or check formula, we should take σ HP1 and σ The smaller of HP2 is substituted into the calculation

2. Calculation of tooth root bending fatigue strength

calculation criteria

in order to ensure that tooth fracture failure does not occur within the predetermined life, the tooth root bending fatigue strength should be calculated. The calculation criterion is: tooth root bending stress σ F is less than or equal to the allowable bending stress σ FP, i.e.

simplification of stress

due to the large stiffness of the gear body, the gear teeth can be regarded as cantilever beams. The dangerous section can be determined by the 30 ° tangent method (as shown in the left figure below), that is, two tangents that form a 30 ° angle with the symmetry line of the gear tooth and are tangent to the transition arc of the tooth root. The section passing through the two tangent points and parallel to the gear axis is the dangerous section of the gear tooth

30 ° tangent method to determine the load action point of the maximum bending moment of the dangerous section

theoretical analysis shows that the load action point of the tooth root producing the maximum bending moment should be the external point D of the single pair of teeth meshing area (as shown in the right figure above), but the calculation is relatively complex, and it is usually used for the bending strength calculation of high-precision gear transmission (above grade 6 accuracy). For gear transmission with low manufacturing accuracy (such as grade 7, 8 and 9 accuracy), in order to simplify the calculation, it is usually assumed that all the load acts on the tooth top and is borne by only one pair of teeth. The resulting error is corrected by the coincidence coefficient Ye

as shown in the left figure above, the normal force FN acting on the tooth top can be divided into two components perpendicular to each other: the tangential component fncosaf causes bending stress and shear stress to the tooth root, and the radial component fnsinaf causes compressive stress to the tooth root. Shear stress and compressive stress play a small role, and fatigue cracks often start from the tensile edge of the tooth root. Therefore, only the bending tensile stress that plays a major role is considered, and the tensile side is taken as the basis for the calculation of bending fatigue strength. The influence of shear stress, compressive stress and stress concentration effect of tooth root transition curve is corrected by stress correction coefficient ysa

tooth root fatigue bending strength calculation formula

set the force arm as HF, the width of the dangerous section as SF, and the nominal bending stress of the dangerous section of the tooth root as

, where:, is called the tooth shape coefficient

(1) tooth root bending fatigue strength check formula

if the load coefficient K, coincidence coefficient ye and stress correction coefficient ysa are included, then the check formula of tooth root bending fatigue strength is

will be substituted into the above formula, and the design formula of tooth root bending fatigue strength can be obtained

in which yfa- is the tooth shape coefficient of load acting on the tooth top to consider the influence of tooth profile shape on tooth root bending stress SF. Yfa is a dimensionless quantity, which refers to the parameters that affect the shape of the tooth profile (such as Z, x α Etc.) affect yfa (below and above), which is independent of modulus. Yfa value can be checked from the following figure

ysa- stress correction coefficient, whose value can be checked from the following figure

ye- coincidence coefficient, calculated according to the coincidence EA, according to

σ FP allowable bending stress (MPA), calculated according to the formula

application description of bending strength formula

in the calculation of bending fatigue strength of tooth root, the tooth shape coefficient yfa, stress correction coefficient ysa and allowable bending stress of mating gear σ FP may be different. Therefore, when checking and calculating, the two gears should be carried out separately; When using the design formula, yfa1ysa1 should be taken/σ FP1 and yfa2ysa2/σ The larger one in FP2 is substituted into the calculation

2 allowable stress and design parameter selection of gear transmission

1 Allowable stress

(1) allowable contact stress σ HP

the allowable contact stress is

, where: σ Hlim - when the failure probability is 1%, the contact fatigue limit of the tested tooth surface is determined by the σ Hlim search. Ml, MQ and me in the figure represent the grades required for material quality and heat treatment (ML low, MQ medium and me high), which are generally selected according to MQ σ Hlim。

zn- the life coefficient of contact fatigue strength, whose value can be checked by the contact fatigue strength life coefficient Zn according to the number of stress cycles of the designed gear n=60nkth (n is the gear speed, K is the number of meshing on the same side of the gear per revolution, th is the designed working hours of the gear)

zw- working hardening coefficient; Considering that in the combined transmission process of soft (big gear) and hard (small gear), the small gear will produce cold work hardening on the tooth surface of the big gear, which will increase the allowable contact stress of the big gear, this coefficient is introduced. Its value can be calculated according to the following formula:

where Hb is the Brinell hardness value of the gear tooth surface; When HB ≤ 130hbs, take zw=1.2; When HB ≥ 470hbs, take zw=1

zx- dimension coefficient of contact fatigue strength, considering the coefficient of material strength reduction due to size increase, its value is checked from the diagram

sh- the minimum safety factor of contact fatigue strength can be checked by the reference value of the minimum safety factor

(2) allowable bending stress σ Fp

the allowable bending stress is

, where: σ Flim- when the failure probability is 1%, the bending fatigue limit of the test gear is determined by the bending fatigue limit stress of the gear material σ Flim search. When loaded in both directions, the σ Flim value multiplied by 0.7

yn -- the life coefficient calculated by bending fatigue strength. According to the number of stress cycles n of the designed gear, the life coefficient yn of bending fatigue strength can be used to find

yst- the stress correction coefficient of the experimental gear, taking yst=2.0

yx- dimension coefficient of bending fatigue strength, which is checked from the figure

sf- the safety factor of bending fatigue strength can be checked by the reference value of the minimum safety factor

2. The selection of main parameters of gear transmission

the selection of geometric parameters has a great impact on the structural size and transmission quality of gears,. Under the condition of meeting the strength conditions, it should be selected reasonably

(1) tooth ratio u

in order to avoid the size of gear transmission is too large, the tooth ratio u should not be too large, generally U ≤ 7. When the transmission ratio is required to be large, two-stage or multi-stage gear transmission can be adopted

(2) modulus m and the number of pinion teeth z1

modulus m directly affect the bending strength of the tooth root, but have no direct impact on the contact strength of the tooth surface. Generally, the gear used to transmit power should be m>1.5 ~ 2mm to prevent the gear teeth from breaking suddenly when overloaded

standard gear Zmin ≥ 17, if it is allowed to slightly undercut or use spring tension fixture to modify the gear, Z at the same time, it also has outstanding wear resistance, creep resistance and chemical resistance, min can be as little as 14 or less

for closed soft tooth surface gear transmission, after determining the pinion diameter D1 according to the tooth surface contact strength, on the premise of meeting the bending fatigue strength, it is appropriate to select smaller modulus and more teeth to increase the coincidence, improve the stability of transmission, reduce the tooth height, reduce the gear weight, and reduce the amount of metal cutting. Usually z1=20 ~ 40. For high-speed gear transmission, it can also reduce the relative sliding of the tooth surface and improve the anti scuffing ability

for closed hard tooth surface and open gear transmission, the bearing capacity mainly depends on the bending fatigue strength of the tooth root, and the modulus should not be too small. On the premise of meeting the contact fatigue strength, in order to avoid excessive transmission size, Z1 should be taken as a smaller value, generally Z1 = 17 ~ 20

the number of teeth of the mating gear is better than the prime number, at least not into an integer ratio, so as to make all gears wear evenly and help reduce vibration

(3) tooth width coefficient fd

when the load is certain, choose the larger value of FD, which can reduce the gear diameter and center distance, making the transmission more compact. However, the tooth width will increase, and the uneven distribution of load along the tooth direction will be more serious. Therefore, FD should be reasonably selected. For gear transmission with closed fixed transmission ratio, when the gear accuracy is high and the shaft stiffness is large, a larger value FD can be selected. Generally, it can be selected by referring to the recommended value of tooth width coefficient FD

for the tooth width coefficient fa (= b/a) based on the center distance, the relationship between and is fd=fa (u+1)/2 (external meshing), which can be converted during design

to ensure the contact width after assembly, the pinion tooth width B1 is usually taken as 5 ~ 10mm larger than the gear tooth width B2, and b=b2 is taken as larger in strength calculation

(4) modification coefficient x

the main purpose of using modified gear transmission is to improve the gear strength, improve the transmission quality, avoid undercutting, get close to the center distance, etc. In order to achieve these purposes, the displacement coefficient must be reasonably selected. The following introduces a line diagram method. First, according to the use requirements, select the appropriate total modification coefficient XS (=x1+x) on the selection range a of the modification coefficient of the meshing gear outside the diagram with the number of teeth and ZS (=z1+z2)

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