Borges, lector del Quijote
5. Afinidad de Cervantes y Borges
As was said in Section 7.2, the contact stress value is obtained using Hertz’s formula derived for two round cylinders pressed against each other with no motion. Equation 7.1 and Equation 7.2 are valid for perfectly elastic and smooth cylinders. In reality, the teeth are involute cylinders (not round), and they have deviations from their theoretical shape, such as surface roughness and waviness. Therefore, force Fn is never distributed evenly along the line of contact.
Perpendicular to the line of contact, the stress σH is not distributed across the contact area in a semielliptic form as shown in Figure 7.8b. This theoretical form is distorted because of form deviations, friction forces, hydrodynamic effects, and the effect of time delay in deformation. As the gears rotate, the new surfaces come into contact, and the pattern of the surface stress permanently changes, quickly and unpredictably.
Nobody knows definitely what the contact stress pattern is because there is no possibility to measure it. But there is one more problem. For the strength evaluation not only the stress, but also the metal condition is important. In the course of rolling and sliding under load, the surface layers are plastically deformed and the structure of metal is changed. In addition, chemical reactions and electric effects take place in the contact area, so the processes that affect the contact strength are very complex.
Because of this complexity, calculation of the contact strength of gears is based on experimental data and operating experience with gears of similar purpose and duty. As a rule, the Hertzian stress σH is used as a crucial factor for evaluation of the stress level and service life. That means the admissible value of σH may be safely chosen relying on the operating experience with gears of the same purpose and service conditions. If these data are not available, general recommendations may be used for design, but thereafter experimental testing should be carried out.
To derive the relationship between the gear dimensions, torque transmitted, and stress σH, we shall return to the geometry of a spur gear (Figure 7.3 and Figure 7.4). A pair of teeth come into contact at point b2 and disengage at point b1. In each point on this path, the radii of curvature and sliding velocities are different. For contact strength calculations, the pitch point p is usually considered because it is convenient. In addition, in spur gears, the pitch point lies in the one-pair zone, which is the weak place. The radius of curvature of the pinion tooth in the pitch point
The radius of curvature for the gear,
The equivalent radius of curvature is
Remember that the “plus” in the denominator is valid for external gearing and the “minus” for internal gearing.
If i = 1 in internal gearing, the equivalent radius ρ = ∞. But don’t be delighted with this fact:
it is not a gear; it is just a gear coupling.
The length of the contact line L = W, where W is the face width.
ρ1 1 1 ϕ
Force Fn is given by
where
T1= nominal torque applied to the pinion (N·mm)
KWH = factor of load distribution across the face width (for contact strength calculation) Kd = factor of dynamic load
Now we have all the needed factors to insert into Equation 7.1. Raising this equation to the second power gives
or, after substitution of Fn, ρe, and L,
(7.3)
In Equation 7.3 parameter E (modulus of elasticity) is nearly constant for all kinds of steel, and angle ϕ doesn’t change much (usually 20–22°). Therefore, several variables in Equation 7.3 may be substituted by one variable CH, which conforms to the square of Hertzian stress:
There is a linear dependence between the load and the stress factor CH. The following formula derived from Equation 7.3 yields an admissible value of CH by calculation of tested and field-proven gears made of similar materials:
The factors KWH and Kd should be determined taking into account the design features, speed, and load of the prototype gear. If the gear that is to be designed is similar to the field-proven prototype, factors KWH and Kd can also be eliminated because they are nearly the same. In this case, the formula for comparative calculation becomes simpler:
(7.4)
It follows that two gears of similar function and design, and having the same KH value, will have nearly the same Hertzian stress σH and supposedly the same service life.
Approximately, the admissible value of KH for general-purpose gears may be defined as
F T
where
K0 = contact stress factor admissible for infinite service life (MPa) KHL= service life factor for the contact strength calculations
For the preliminary calculation of gears, the following formulas for K0 can be used:
1. For helical gears with case-hardened teeth: K0 = 0.126HRC − 1.78 MPa 2. For spur gears with case-hardened teeth: K0 = 0.099HRC − 1.22 MPa
3. For helical gears with surface-hardened pinion and tempered gear: K0 = 0.0127HB − 1.1 MPa
4. For helical gears with both pinion and gear tempered: K0 = 0.0107HB − 1.1 MPa 5. For spur gears with both pinion and gear tempered: K0 = 0.008HB − 0.87 MPa
Here, HRC and HB are the surface hardness of the larger gear (wheel) in Rockwell-C and Brinell hardness units, respectively.
In cases 4 and 5 it is recommended to make the pinion harder than the gear by approximately 40 HB. The increased hardness of the pinion shall compensate for its greater number of cycles and lower bending strength of its teeth as compared to the gear (compare teeth N = 17 and N = 50 in Figure 7.2).
Factor KHL is made to express the dependence of admissible load on the run time that is given as a number of loading cycles. It is clear that the less the specified number of cycles, the greater the load can be applied to the part (within certain limits) with the same probability of failure during that number of cycles. Nowadays, this idea may seem trivial, but it was first thoroughly investigated by A. Wöhler only 150 years ago (see Chapter 12). The experimental points look as shown in Figure 7.10a. Each point represents one experiment and is characterized by a certain stress amplitude, σa, and number of cycles to failure N. (Points with arrows belong to experiments in which the part did not fail.) As is seen from the chart, the parts tested at the same stress amplitude have scattering number of cycles to failure, though they have been made of the same material and by the same drawing. Fatigue life line 1 is usually curved so that 90 or 95% of the experimental points remain above it. This is to decrease the probability of failure when the part is calculated for durability using the fatigue life curve. Please have a sense of humor and don’t count the number of points above and below line 1 in Figure 7.10a!
The reasons for that scattering are numerous, but in experiments for bending or torsional stress, we at least know exactly the number of cycles to failure. In experiments for contact strength, the point of failure is a matter of opinion. In fact, a gear with initial pitting continues working as if nothing has happened. You can’t detect any negative changes in its behavior; the temperature is the same, and the noise even decreases (because of running-in). With time, the pitting may become smaller and even disappear, but it may also grow and finally cover most of the working surfaces of the teeth. It depends on the hardness of these surfaces, evenness of load distribution along the
FIGURE 7.10 The dependence between stress amplitude and the number of cycles to failure N.
N log N
σa
(a) (b)
σH, lim A
1 2
3
Nlim log σH
contact lines, and the rim speed. Therefore, the fatigue lines for contact strength of the same gear may differ, depending on the experience and criteria of the investigator.
When the dependence σ vs. N is plotted in double-log or semilog coordinates, the fatigue life curve looks nearly as a broken line, (Figure 7.10b). (Because we are speaking here about the contact strength, the stress σa is replaced by Hertzian stress σH.) Its breaking point A is characterized by fatigue limit σH,lim and number of cycles Nlim.
Inclined part 2 of this broken line is approximated by this equation:
For pitting deterioration of the teeth surfaces, the value of exponent m is approximately equal to 6 (to be more accurate, m = 6 ± 2). That is,
(7.5) Horizontal part 3 of the fatigue life curve has two meanings. First, it means that if the stress is equal to or less than fatigue limit σH,lim, the fatigue failure is not likely to occur. The second meaning is that if, during the number of cycles Nlim, the fatigue failure didn’t occur, the part can be considered as passing the test without failure. As mentioned in the preceding text, the number of cycles in this kind of experiment can’t be estimated exactly. One of the most reliable sources5 recommends the following formula for Nlim:
(7.6) where HB is the Brinell hardness of the working surfaces.
On the basis of Equation 7.5 we can write the following:
(7.7)
Here
Ni= specified number of cycles that is less than Nlim σH,i = admissible Hertzian stress for Ni cycles of loading
Inasmuch as the admissible torque, as well as the admissible KH value, is proportional to the square of the Hertzian stress (see Equation 7.3 and Equation 7.4), the expression for KHL derived from Equation 7.7 looks as follows:
(7.8)
When determining the service life factor KHL using Equation 7.8, the following remarks should be taken into account:
• The maximal working load should not exceed the magnitude characterized by KHL = 6–7 for teeth hardness of 350 HB or less and KHL = 3–3.5 for harder teeth (40 HRC and more).7
• Infrequent momentary overloads should not be considered, but their amount is limited by the maximum admissible KHL value: for hardness 350 HB and less KHL,max = 10–12, and for harder teeth surfaces, KHL,max = 4–5.7
N⋅σHm=const
• When the hardness of the pinion and gear is nearly the same, determination of the factor KHL should be based on the rotational speed of the pinion. When the pinion teeth are much harder (for example, surface-hardened pinion teeth and high-tempered gear wheel), the rotational speed of the gear counts.
• If a gear is engaged with two or more others (such as sun wheel of a planetary gear), its rotational speed should be multiplied by the number of mating gears.
If a gear works at a variable load and speed and it is known that the maximal torque T1 is applied during n1 hours in all, the lesser torque T2, during n2, T3, during n3, etc., the equivalent number of cycles ne at the maximal torque
(7.9)
This equation is based on the hypothesis about the linear summation of damaging effects (see Chapter 12, Subsection 12.1.3).
Remark:
The working surfaces of non-surface-hardened teeth continue to deteriorate slowly even if the stress is less than the fatigue limit. This is reflected in the fatigue life line for contact strength of such teeth. The line has no horizontal part and should look as shown in Figure 7.11. It is recommended5 that the KHL value, should be obtained at Ni > Nlim by the formula
but not less than KHL = 0.8.
EXAMPLE 7.2
A helical gear works 1 hour per day, 200 days a year, and the needed service life is 20 years. The pinion teeth are induction-hardened to 55 HRC, and the gear is tempered to 300 HB. The rotational frequencies of the pinion and gear are n1= 1000 r/min, n2= 150 r/min. The gear works 10% of the time at full load, 30% at 70% of the full load, and 60% at half-load.
FIGURE 7.11 The fatigue life line for non–surface-hardened teeth.
n n n T
T n T
e= + T
+
+
1 2 2
1 3
3 3
1 3
...
K N
HL n
i
=12 lim
log N σH
σH, lim A
Nlim
From the equation for helical gears with surface-hardened pinion and tempered gear (see the preceding text),
The total working time is 20 (years) × 200 (days) × 1 (hour) = 4000 h. The gear set works 400 h at full load, 1200 h at 70% load, and 2400 h at half-load. Now, we can determine the needed equivalent service life at full load:
The number of cycles of the gear wheel equals 1112 · 150 = 1.67 · 105. From Equation 7.6,
From Equation 7.8,
Thus, the admissible contact stress factor KH= 2.71 · 5.4 = 14.6 MPa. In this case, the bending strength of the teeth may limit the load capacity of the gear; it must be checked.
Using the KH value, the permissible pinion torque T1H limited by its contact strength is given by
For bevel gears,
If a gear should be designed to transmit the given torque T1, the diameter of the pinion of spur or helical gear can be found from the following equation:
(7.10)
Here ψ = W/d1, the ratio of face width to pinion diameter, usually, 0.8–1.2.
For bevel gears,
The ψb value usually equals 0.25–0.30.
As soon as the pinion diameter is determined, all other dimensions can be easily calculated using the gear ratio and factor ψ (ψk for a bevel gear).
When using these equations it is necessary to be careful with the dimensions of the variables.
Sometimes, the gear dimensions and load data are given in different systems, and they must be converted to be compatible. If the K0 values are given in MPa (that is, in N/mm2), it is convenient (in this case) to express all the linear dimensions in millimeters (mm), and the torque in newton millimeters (N·mm).
EXAMPLE 7.3
Design a speed reducer to run at speeds from 2000 to 285 r/min. The transmitted power is 1000 kW. The assumed service life is indefinite, meaning that KHL = 1 for surface-hardened teeth or KHL
= 0.8 for tempered teeth.
First calculate the pinion torque T1:
The gear ratio is
Let’s take ψ = W/d1= 1. Now, we have to choose the hardness of the teeth. To decrease the dimensions and weight of the gear, we choose a helical gear with case-hardened teeth, surface hardness ≥56 HRC. For this gear, K0 = 0.126 · 56 − 1.78 = 5.28 MPa. Because KHL = 1, KH = K0
= 5.28 MPa. Now we have all the needed variables for Equation 7.10:
The other dimensions of the gears are as follows:
Face width W =ψ · d1= 128 mm
This calculation is approximate, and it is not obligatory. The strength of the predesigned gear should be checked using methods recommended in the respective branch of industry. To fulfill some requirements of standards, the center distance C can be expressed in round numbers: 500 mm or 20 in. (depending on accepted system of units). The other dimensions can be also changed to meet the designer’s requirements.