The aviation industry is a strategic high-technology sector whose development level directly reflects national defense capability and economic strength. High reliability, high efficiency and lightweight design are essential for modern aircraft engines, and gear transmission is a key subsystem that largely determines the overall performance of the engine. Most aviation gear drives operate under heavy load and high linear velocity conditions. The relatively high sliding speed on the tooth surface and heavy transmitted load cause severe vibration, noise and additional dynamic loading, which threatens the safety and stability of the whole transmission train. Long-term experience in both theoretical and engineering practice has shown that gear failures are one of the most frequent causes of engine malfunction. Therefore, a reliable prediction of the dynamic factor is crucial in the design of aviation spur gears, especially when the gear pair runs beyond the critical speed region.
In the classical bending strength verification of spur gears, the dynamic factor \(K_v\) is used to account for internal additional dynamic loads. The ISO standard provides both a precise method (method B) and a simplified method (method C). However, when the classical B-method formula is applied to aviation spur gears running at supercritical speeds, the calculated dynamic factor often exceeds reasonable values by a large margin. This discrepancy indicates that the classical formulation is not fully suitable for heavy-load high-speed involute spur gears. The objective of this study is to investigate the mechanism of internal dynamic loading, compare classical formulas, and propose a revised dynamic factor formula specifically applicable to aviation spur gears.
Dynamic Load Mechanism of Spur Gear Pairs
The dynamic load in spur gear meshing originates mainly from two sources: gear manufacturing errors and periodic variation of mesh stiffness. These excitations produce a variable load on the tooth surface even when the externally applied torque is constant. The vibration of the gear system can be represented by a torsional model, as shown conceptually in the following equations. Let \(I_1\) and \(I_2\) be the moments of inertia of the driving and driven gears, \(r_1\) and \(r_2\) their base radii, \(\theta_1\) and \(\theta_2\) their torsional displacements, and \(W_d\) the dynamic normal load. The rotational equations of motion are:
$$
I_1 \ddot{\theta}_1 = W_d r_1
$$
$$
I_2 \ddot{\theta}_2 = -W_d r_2
$$
When several tooth pairs are in contact, the dynamic load is expressed as
$$
W_d = \sum_i k_i (x_1 – x_2 – e_i)
$$
where \(x_1 = r_1 \theta_1\), \(x_2 = r_2 \theta_2\), \(k_i\) is the stiffness of the \(i\)-th tooth pair, and \(e_i\) is the combined error at the contact point. The summation is performed over all pairs instantaneously in contact. If the relative displacement becomes non-positive, the tooth surfaces separate and the dynamic load becomes zero except for the static load. The total mesh stiffness is \(k(t) = \sum_i k_i(t)\), and the equivalent mass is defined as
$$
\frac{1}{m_e} = \frac{r_1^2}{I_1} + \frac{r_2^2}{I_2}
$$
Taking damping into account, the relative vibration along the line of action can be written as
$$
m_e \ddot{x} + 2 \zeta \sqrt{m_e k(t)} \dot{x} + k(t) x = W + \sum_i k_i e_i
$$
where \(\zeta\) is the damping ratio, \(W\) is the static transmitted load, and \(x = x_1 – x_2\). The natural frequency of the equivalent system is approximately
$$
f_e = \frac{1}{2\pi} \sqrt{\frac{k_m}{m_e}}
$$
where \(k_m\) is the average mesh stiffness. The primary excitation frequency is the mesh frequency
$$
f_m = \frac{n z}{60}
$$
where \(n\) is the rotational speed in revolutions per minute and \(z\) is the number of teeth. The ratio \(f_m / f_e\) determines the dynamic response regime.
The impact theory, originally developed by Buckingham, considers the tooth profile error as the main cause of impact loading. The acceleration force due to a step change in the tooth profile leads to a separation and subsequent re-establishment of contact. The maximum impact force can be approximated from momentum and energy considerations. If the effective gear mass is \(m\) and the applied load is \(W\), the acceleration of the gear is \(a = W/m\). Assuming the effective error \(e\) and the single-pair contact duration, the critical speed at which the impact force becomes maximum is
$$
n_{1,\text{crit}} = \frac{30}{\pi z_1} \sqrt{\frac{W}{m e}}
$$
This equation was one of the first attempts to quantify the speed dependence of dynamic gear loads. It also revealed that after the critical speed is exceeded, the dynamic load tends to decrease again, which is analogous to the shaft vibration above the critical whirling speed. This behavior implies that a supercritical operating region may be acceptable for high-speed spur gears, which is important for aviation applications.
From the vibration point of view, the gear pair can be modeled as a single-degree-of-freedom system with stiffness \(k\) and equivalent mass \(m_{\text{red}}\). The undamped natural frequency is
$$
\omega_n = \sqrt{\frac{k_{\gamma}}{m_{\text{red}}}}
$$
where \(k_{\gamma}\) is the average mesh stiffness. The excitation frequency caused by the gear error is equal to the mesh frequency
$$
\omega = \frac{\pi n z}{30}
$$
Resonance occurs when \(\omega = \omega_n\). The corresponding critical rotational speed is
$$
n_E = \frac{30}{\pi z} \sqrt{\frac{k_{\gamma}}{m_{\text{red}}}}
$$
The dimensionless frequency ratio, or critical speed ratio, is defined as
$$
N = \frac{n}{n_E}
$$
When \(N < 1\), the gear pair operates in the subcritical region; when \(N > 1\), it operates in the supercritical region; when \(N = 1\), resonance occurs. In the supercritical region, the dynamic response may become smaller than in the main resonance region, provided that the fluctuating excitation is not able to maintain a large amplitude.
Classical Dynamic Factor Calculation Methods
Several national and international standards present formulas for the dynamic factor. The Japanese Society of Mechanical Engineers uses a direct dynamic load expression rather than a factor:
$$
W_d = \frac{W \, f_{pb}}{B} \, b \, k \, f_b
$$
The Russian standard ГОСТ defines the dynamic factor as
$$
K_v = 1 + \frac{1}{200} \frac{v \, w_b \, d}{T_1 \, K_\alpha K_\beta}
$$
where \(v\) is the pitch line velocity, \(w_b\) is the unit load, \(d\) is the pitch diameter, \(T_1\) is the transmitted torque, and \(K_\alpha\), \(K_\beta\) are load distribution factors.
The American Gear Manufacturers Association (AGMA) standard provides a chart-based method where the dynamic factor is read as a function of pitch line velocity and an accuracy grade. This graphic method is not very reliable for heavy-load high-speed aviation gears.
The German standard DIN and the ISO standard use a more sophisticated approach based on the critical speed ratio \(N\). The ISO method recognizes three major excitation sources: pitch error \(f_{pb}\), profile error \(f_f\), and time-varying mesh stiffness \(k_m\). The total internal dynamic load is decomposed into three parts:
$$
W_d = W_p + W_f + W_k
$$
and the dynamic factor is expressed as
$$
K_v = \frac{W + W_p + W_f + W_k}{W}
$$
In the ISO standard, the following interval formulas are provided:
| Operating region | Dynamic factor formula |
|---|---|
| Subcritical \(N \le 0.85\) | \(K_v = N (C_{v1} B_p + C_{v2} B_f + C_{v3} B_k) + 1\) |
| Main resonance \(0.85 < N \le 1.15\) | \(K_v = C_{v1} B_p + C_{v2} B_f + C_{v4} B_k + 1\) |
| Transition \(1.15 < N < 1.5\) | \(K_v = K_{v(N=1.15)} + \frac{K_{v(N=1.5)} – K_{v(N=1.15)}}{0.35} (N – 1.15)\) |
| Supercritical \(N \ge 1.5\) | \(K_v = C_{v5} B_p + C_{v6} B_f + C_{v7}\) |
Here, \(C_{v1}\) and \(C_{v2}\) are influence factors for pitch deviation and profile error; \(C_{v3}\) and \(C_{v4}\) account for mesh stiffness variation and resonance; \(C_{v5}\), \(C_{v6}\), and \(C_{v7}\) are used in the supercritical region. The factors \(B_p\), \(B_f\), and \(B_k\) depend on gear accuracy, tooth profile modification, and stiffness fluctuation.
For the simplified method C, a chart is used with the precision coefficient \(C\) given by:
$$
C = \frac{(z)^{0.5048} (\ln f_{pt})}{…}
$$
Actually, the ISO chart is curve-based and depends on the accuracy grade and pitch line velocity. Method C is generally valid for low-speed applications; when the pitch line velocity exceeds a certain value, the chart cannot be used.
To demonstrate the inadequacy of method B for high-speed aviation spur gears, we selected a representative gear pair used in an aero-engine transmission. The basic parameters are listed in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 20 | 26 |
| Module / mm | 1.5 | 1.5 |
| Pressure angle / ° | 25 | 25 |
| Addendum coefficient | 1 | 1 |
| Tip clearance coefficient | 0.2 | 0.2 |
| Face width / mm | 4 | 3 |
| Rotational speed / (r/min) | 54600 | 42000 |
| Density / (kg/mm³) | 7.8e-6 | 7.8e-6 |
| Poisson’s ratio | 0.26 | 0.26 |
| Elastic modulus / MPa | 206000 | 206000 |
Using the classical B-method, we calculated the dynamic factor for a range of speeds, as shown in Figure (not depicted). The B-method values increased beyond 3 when the speed was above the critical region. These values would suggest severe dynamic overload and tooth breakage risk, but in reality the gear pair operated without failure. After crossing the supercritical threshold, the actual dynamic load should diminish rather than keep increasing. Therefore, the B-method formula is not suitable for this regime.
Dynamic Meshing Force Analysis of an Aviation Spur Gear Pair
To obtain a more realistic dynamic factor, a single-degree-of-freedom torsional dynamic model was established. The model includes time-varying mesh stiffness, static transmission error, damping, and backlash. The equation of motion for the relative displacement \(x(t)\) along the line of action is:
$$
m_e \ddot{x}(t) + c_m \dot{x}(t) + k(t) f(x(t)) = F_m – m_e \ddot{e}(t)
$$
where \(m_e\) is the equivalent mass, \(c_m\) is the mesh damping coefficient, \(k(t)\) is the time-varying mesh stiffness, \(e(t)\) is the static transmission error, \(F_m\) is the mean mesh force, and \(f(x)\) is the backlash function. The backlash function is approximated by
$$
f(x) = \begin{cases}
x – b, & x > b \\
0, & |x| \le b \\
x + b, & x < -b
\end{cases}
$$
where \(b\) is half of the backlash. The equivalent mass is
$$
m_e = \frac{I_1 I_2}{I_1 r_2^2 + I_2 r_1^2}
$$
The mesh force is computed from the solved displacement:
$$
\text{DMF}(t) = k(t) f(x(t)) + c_m \dot{x}(t)
$$
The time-varying mesh stiffness was obtained by the Weber energy method, which includes bending, shear, base deformation, and local contact deformation. The calculated stiffness variation for one mesh cycle is presented in Figure (conceptual). In the numerical solution, the stiffness was approximated by a Fourier series:
$$
k(t) = k_m + \sum_{j=1}^{N} \left( a_j \cos(j \omega_m t) + b_j \sin(j \omega_m t) \right)
$$
where \(\omega_m = 2\pi f_m\) is the mesh angular frequency. The static transmission error was also represented as a Fourier series:
$$
e(t) = e_0 + \sum_{j=1}^{N} e_j \cos(j \omega_m t + \varphi_j)
$$
For a gear with accuracy grade 5 and speed 54600 r/min, the simulated transmission error amplitude is about 0.08 mm.
The dynamic equation was solved numerically using a fourth-order Runge–Kutta method implemented in MATLAB. For the gear pair described in Table 1, the dynamic factor at the operating speed was found to be
$$
K_v = \frac{W_{\text{max}} + F}{F} \approx 1.16
$$
where \(F = T_1 / (r_1 \cos \alpha)\) is the static tangential load on the line of action. This value is considerably smaller than the value given by the ISO B-method. The variation of \(K_v\) with speed is shown in Figure (conceptual). The dynamic factor reaches a peak near the resonance region and then drops in the supercritical region. This confirms the existence of a safe supercritical operating zone.
The influence of the transmission error amplitude was also studied. When the error amplitude increased from 10 to 150 μm, the dynamic factor increased almost linearly from 1.35 to 1.94. Therefore, a higher manufacturing accuracy (smaller errors) is beneficial for reducing the dynamic factor.
An independent simulation was carried out with the multi-body dynamics software Adams. The contact force model used the IMPACT function, which is based on nonlinear Hertzian contact and damping. The contact stiffness was calculated as
$$
K = \frac{4}{3} R^{1/2} E
$$
with the combined radius
$$
\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}
$$
and the equivalent elastic modulus
$$
\frac{1}{E} = \frac{1-\mu_1^2}{E_1} + \frac{1-\mu_2^2}{E_2}
$$
For the steel gears, the contact stiffness was \(K = 5.76 \times 10^5 \, \text{N/mm}\). The simulation parameters are listed in Table 2.
| Parameter | Value |
|---|---|
| Force exponent | 1.5 |
| Damping / (N·s/mm) | 50 |
| Static friction coefficient | 0.3 |
| Dynamic friction coefficient | 0.1 |
| Penetration depth / mm | 0.1 |
During the simulation, the angular velocity of the driving gear was accelerated smoothly from zero to 54600 r/min using a step function, and the driven gear torque was also smoothly applied. The steady-state contact force was extracted. The maximum contact force in a time window was 106 N. Using the static tangential load, the dynamic factor was calculated as
$$
K_v = \frac{106}{T_1/r_1} \approx 1.22
$$
Both the numerical integration and the Adams simulation produced dynamic factor values less than 1.5, which is in line with the practical observation of normal operation. This result reinforces the idea that the ISO B-method is overly conservative for aviation spur gears operating at high speeds.
Natural Vibration Characteristics of Aviation Spur Gears
The natural frequencies and mode shapes of the gear pair are fundamental to understanding resonance and dynamic loads. The gear structure was modeled as a three-dimensional solid, and finite element modal analysis was performed in ANSYS. The equation for free vibration is
$$
[ M ] \{\ddot{X}\} + [ K ] \{X\} = 0
$$
The eigenvalues satisfy
$$
\left( [K] – \omega_i^2 [M] \right) \{X_i\} = 0
$$
where \(\omega_i\) is the \(i\)-th natural frequency and \(\{X_i\}\) is the corresponding mode shape. For a gear pair in contact, the modal analysis should include the contact constraint and the prestress due to centrifugal loading. In the prestressed contact modal analysis, the equation is
$$
[ M ] \{\ddot{X}\} + \left( [K] + [K_c] \right) \{X\} = 0
$$
where \([K_c]\) is the stress-stiffening matrix generated by the internal centrifugal force field. The static analysis was first performed with an angular velocity of 54600 r/min, and the resulting stress field was included in the subsequent modal analysis.
First, the non-prestressed contact modal analysis was performed. The first 20 natural frequencies are listed in Table 3.
| Mode | Frequency / Hz | Mode | Frequency / Hz |
|---|---|---|---|
| 1 | 9002.7 | 11 | 24946 |
| 2 | 9009.5 | 12 | 25111 |
| 3 | 9166.1 | 13 | 25880 |
| 4 | 9453.6 | 14 | 28657 |
| 5 | 10296 | 15 | 28728 |
| 6 | 10744 | 16 | 31664 |
| 7 | 21224 | 17 | 40843 |
| 8 | 21396 | 18 | 40892 |
| 9 | 23063 | 19 | 41114 |
| 10 | 24876 | 20 | 41811 |
The prestressed modal analysis gave the natural frequencies and mode shapes summarized in Table 4.
| Mode | Frequency / Hz | Main mode shape |
|---|---|---|
| 1 | 9020.6 | Rigid body rotation about Z axis |
| 2 | 9162.6 | Large gear tilting about X axis |
| 3 | 9828.2 | Large gear first-order nodal-diameter vibration |
| 4 | 10959 | Large gear second-order nodal-diameter vibration |
| 5 | 15804 | Large gear bending and pinion tilting |
| 6 | 21392 | Pinion tilting about X axis |
| 7 | 22156 | Pinion tilting about Y axis |
| 8 | 23773 | Pinion first-order nodal-diameter vibration |
| 9 | 25044 | Pinion second-order, large gear third-order nodal-diameter |
| 10 | 25899 | Pinion second-order, large gear third-order nodal-diameter |
| 11 | 26574 | Pinion second-order, large gear third-order nodal-diameter |
| 12 | 28922 | Large gear radial vibration along X-axis 45° |
| 13 | 30264 | Large gear radial vibration along Y axis |
| 14 | 32409 | Large gear radial vibration along X axis |
| 15 | 34630 | Small and large gears nodal-diameter vibration |
| 16 | 39186 | Large gear rotation and pinion meshing bending |
| 17 | 41137 | Pinion third-order, large gear fourth-order nodal-diameter |
| 18 | 41896 | Pinion third-order, large gear fourth-order nodal-diameter |
| 19 | 42956 | Pinion third-order, large gear fourth-order nodal-diameter |
| 20 | 50222 | Pinion fourth-order, large gear fifth-order nodal-diameter |
Comparing Tables 3 and 4, the prestressed contact natural frequencies are generally higher than the non-prestressed ones because the centrifugal stress increases the apparent stiffness of the gear body. The mode shapes reveal that the main vibration forms include rigid body swing, nodal-diameter vibration, and radial breathing. The large gear tends to vibrate at lower frequencies than the pinion because of its larger mass.
The mesh frequency of the gear pair is
$$
f_m = \frac{n_1 z_1}{60} = \frac{54600 \times 20}{60} = 18200 \, \text{Hz}
$$
This frequency is well separated from the natural frequencies listed in Table 4. Therefore, the gear pair does not encounter any resonance at the operating speed. This explains why the gear can run smoothly without excessive dynamic load.
The effect of design parameters on the natural frequency was investigated by changing one parameter at a time. The pinion model was used as a baseline. The results are summarized in Figure (conceptual). Key findings are as follows:
- Increasing the pressure angle from 20° to 25° increases the natural frequency by a few percent.
- Increasing the module from 1 mm to 3 mm causes a strong increase in the natural frequency. This is because the tooth thickness and the overall stiffness increase more than the mass.
- Increasing the number of teeth from 20 to 30 decreases the natural frequency, since the gear body becomes larger and more massive while the overall stiffness does not increase at the same rate.
- Increasing the addendum coefficient from 0.8 to 1.0 slightly decreases the natural frequency.
- Increasing the face width from 3 mm to 10 mm first increases the natural frequency and then flattens out.
These trends allow a designer to adjust the natural frequency and avoid resonance by changing the basic spur gear parameters. In particular, the module has the largest influence on the natural frequency.
Revised Dynamic Factor Formula for Supercritical Speeds
Based on the analysis of dynamic load mechanisms and the observation that classical formulas are not suitable for high-speed aviation spur gears, we derived a new statistical formula. The main sensitivity factors considered are manufacturing accuracy, rotational speed, and mesh stiffness. The internal additional dynamic load can be decomposed as
$$
D = D_n + D_C + D_k
$$
where \(D_n\) is the speed-related contribution, \(D_C\) is the accuracy-related contribution, and \(D_k\) is the stiffness-related contribution. The dynamic factor can then be expressed as
$$
K_v = 1 + \frac{D_n}{W} + \frac{D_C}{W} + \frac{D_k}{W}
$$
After introducing dimensionless coefficients \(B_n\), \(B_C\), and \(B_k\), the general form is
$$
K_v = 1 + B_n + B_C + B_k
$$
Using data from multiple spur gear pairs of an aero-engine transmission, we performed nonlinear regression with the 1stOpt software. The resulting formula for the subcritical region is
$$
K_v = 1 + \frac{n^2}{79219^2} + 3 \left( \frac{E_1}{100} \right)^2 + \left( \frac{S_1}{5} \right)^2
$$
$$
E_1 = C_{v1} + C_{v2}, \quad S_1 = C_{v3}
$$
and for the supercritical region
$$
K_v = 1 + \frac{n^2}{79219^2} + 3 \left( \frac{E_2}{100} \right)^2 + \left( \frac{S_2}{5} \right)^2
$$
$$
E_2 = C_{v5} + C_{v6}, \quad S_2 = C_{v7}
$$
Here, \(n\) is the driving gear speed in revolutions per minute, \(E\) is a factor that combines the effects of pitch error and profile error on manufacturing precision, and \(S\) is a factor accounting for mesh stiffness variation. The coefficients \(C_{v1}, C_{v2}, C_{v3}, C_{v5}, C_{v6}, C_{v7}\) are taken from the ISO amplitude-frequency curves:
$$
C_{v1}=0.32,\; C_{v2}=0.34,\; C_{v3}=0.23,\; C_{v5}=0.47,\; C_{v6}=0.75,\; C_{v7}=0.47
$$
The new formula is valid under the following conditions:
- Manufacturing accuracy grade between 4 and 8
- Number of teeth between 17 and 500
- Contact ratio between 1 and 2
- Normal module between 1.5 and 10 mm
- No resonance at the operating speed
Three gear pairs were used for validation. Gear pair 1 operated in the subcritical region at 14788 r/min, gear pair 2 in the supercritical region at 42000 r/min, and gear pair 3 in the supercritical region at 54600 r/min. The parameters of gear pair 1 and 2 are listed in Table 5.
| Parameter | Pair 1 (Pinion/Gear) | Pair 2 (Pinion/Gear) |
|---|---|---|
| Number of teeth | 13/16 | 26/32 |
| Module / mm | 1.5/1.5 | 1.5/1.5 |
| Pressure angle / ° | 25 | 25 |
| Addendum coefficient | 1 | 1 |
| Face width / mm | 4/3 | 6/3 |
| Pinion speed / (r/min) | 14788 | 42000 |
| Mesh stiffness / (N/(mm·μm)) | 8.24 | 10.21 |
| Accuracy grade | 5 | 5 |
For gear pair 1, the ISO B-method gives
$$
K_{v,B1} = 1 + N (C_{v1} B_p + C_{v2} B_f + C_{v3} B_k) = 1.2109
$$
and the proposed formula gives
$$
K_{v,N1} = 1 + \frac{14788^2}{79219^2} + 3 \left( \frac{0.66}{100} \right)^2 + \left( \frac{0.23}{5} \right)^2 = 1.1611
$$
The relative difference is 4.11%, which is acceptable.
For gear pair 2, the ISO C-method gives
$$
K_{v,C2} = 1.4530
$$
The proposed formula gives
$$
K_{v,N2} = 1.3730
$$
The relative difference is 5.5%. These deviations are within ±10%, indicating that the new formula is applicable to both subcritical and supercritical regions, but it is especially intended for supercritical high-speed aviation spur gears.
Further verification was performed by varying the manufacturing accuracy grade from 4 to 8 while keeping other parameters constant for gear pair 2. The results are shown in Table 6.
| Accuracy grade | Kv (method C) | Kv (proposed) | Relative error / % |
|---|---|---|---|
| 4 | 1.2477 | 1.3291 | 6.52 |
| 5 | 1.4530 | 1.4774 | 1.68 |
| 6 | 1.6777 | 1.7907 | 6.73 |
| 7 | 2.2408 | 2.3130 | 3.22 |
| 8 | 2.6083 | 3.0441 | 16.71 |
For the commonly used accuracy grade 5, the error is very small. For grade 8, the discrepancy grows, indicating that the new formula is best suited for the accuracy levels typical of aviation spur gears.
The influence of mesh stiffness on \(K_v\) was also evaluated for gear pair 1 in the subcritical region. The results are shown in Table 7.
| Mesh stiffness / (N/(mm·μm)) | Kv (method B) | Kv (proposed) | Relative error / % |
|---|---|---|---|
| 13.89 | 1.1952 | 1.1735 | 1.82 |
| 15.16 | 1.2070 | 1.1794 | 2.29 |
| 16.42 | 1.2181 | 1.1853 | 2.69 |
| 17.68 | 1.2287 | 1.1913 | 3.04 |
| 18.95 | 1.2390 | 1.1972 | 3.37 |
| 20.21 | 1.2488 | 1.2032 | 3.65 |
Finally, the speed dependency was examined. For gear pair 1, between 10000 and 20000 r/min, the proposed formula agrees with both B-method and C-method within 10%. For gear pair 3, between 30000 and 56000 r/min, the B-method gives values above 2.5 and even 3.29, while the proposed formula stays between 1.24 and 1.60. The Russian standard formula gives values similar to the proposed formula; the average error between them was 2.26%. This confirms that the new formula is more reasonable for supercritical aviation spur gears.
Conclusion
This study analyzed the dynamic factor of heavy-load high-speed involute spur gears used in aviation applications. The main conclusions are as follows:
- The dynamic load in spur gear pairs is caused by a combination of gear errors and time-varying mesh stiffness. When the gear runs above the critical speed, the dynamic load tends to decrease rather than increase continuously, contradicting the classical ISO B-method prediction.
- A single-degree-of-freedom torsional model was solved numerically and validated by Adams contact simulation. The dynamic factor at supercritical speeds was found to be between 1.1 and 1.5 for the investigated aviation spur gears, which is much lower than the values obtained by the ISO B-method.
- Finite element modal analysis revealed the natural frequencies and mode shapes of the gear pair. The mesh frequency was well separated from the natural frequencies, so resonance was avoided. The module has the strongest influence on the natural frequency.
- A revised dynamic factor formula was proposed based on manufacturing accuracy, rotational speed, and mesh stiffness. The formula was verified for several spur gear pairs and showed good agreement with both the ISO simplified method and the Russian standard, while avoiding the excessive conservatism of the B-method.
The proposed formula provides a more realistic estimate of the dynamic factor for heavy-load high-speed involute spur gears. It can be used in gear design and strength verification procedures for aviation transmission systems, contributing to lower cost and improved reliability.