9.3 Spur, Helical, Bevel & Worm Gear Design Calculations

9.3 Spur, Helical, Bevel & Worm Gear Design Calculations

Gears are toothed cylindrical or conical wheel elements that transmit rotary motion and power between shafts with a positive, constant velocity ratio. The PRC Mechanical Engineering Licensure Examination (MELE) thoroughly tests gear terminology, velocity ratio kinematics, tooth bending strength (Lewis equation), AGMA surface fatigue, and force breakdown across spur, helical, bevel, and worm gear configurations.

---

1. Fundamental Gear Nomenclature and Kinematics

Core Parameters and Formulas

Pressure Angle ($\phi$)

Standard pressure angles are $20^\circ$ full depth and $25^\circ$ full depth. The historical $14.5^\circ$ pressure angle requires a minimum of $Z = 32$ teeth to avoid tooth undercutting during hobbing, whereas $20^\circ$ full depth permits pinions down to $Z_{min} = 17$ teeth without undercutting.

---

2. Spur Gear Tooth Bending Strength & Forces

Force Breakdown on Spur Gear Tooth

When transmitting power $P_{power}$ at pitch line velocity $v = \frac{\pi d N}{60}$:

1. Tangential Transmitted Force ($W_t$): $W_t = \frac{2 T}{d} = \frac{P_{power}}{v}$
2. Radial Separating Force ($W_r$): $W_r = W_t \tan\phi$
3. Total Normal Tooth Load ($W$): $W = \frac{W_t}{\cos\phi}$

  Spur Gear Force Diagram:
        Normal Load W
           \   |
            \  | Radial Load Wr = Wt*tan(φ)
             \ |
   -----------v+--------------  Pitch Circle
              |
              | Tangential Load Wt
              v

Lewis Bending Stress Equation

The classical Lewis formula treats a gear tooth as a cantilever beam loaded by tangential force $W_t$ at the tip:

$W_t = \sigma_b \cdot b \cdot Y \cdot m$

Where:

• $\sigma_b$: Allowable bending stress of gear material (MPa).
• $b$: Face width of gear tooth (typically $8m \le b \le 16m$).
• $Y$: Lewis form factor based on tooth count $Z$ and pressure angle $\phi$ ($Y = \pi y$, e.g., $Y = 0.154 - \frac{0.912}{Z}$ for $20^\circ$ full depth).
• $m$: Gear module.

AGMA Bending & Pitting Resistance

Modern AGMA design updates Lewis to include operational stress factors:

$\sigma_b = \frac{W_t K_o K_v K_s K_m K_B}{b \cdot m \cdot J} \le \frac{S_{fb} Y_N}{S_F Y_theta Y_Z}$

Pitting contact stress (Hertzian surface durability):

$\sigma_c = C_p \sqrt{\frac{W_t K_o K_v K_s K_m C_f}{d_1 \cdot b \cdot I}} \le \frac{S_{fc} Z_N C_H}{S_H Y_theta}$

---

3. Helical Gear Design Calculations

Helical gears feature teeth cut at a helix angle ($\psi$) (typically $15^\circ - 30^\circ$). Tooth engagement is gradual, producing smoother, quieter operation at high pitch line velocities.

Geometric Relations

• Normal Module ($m_n$) & Transverse Module ($m_t$): $m_n = m_t \cos\psi$
• Pitch Diameter ($d$): $d = \frac{m_n Z}{\cos\psi}$
• Transverse Pressure Angle ($\phi_t$): $\tan\phi_n = \tan\phi_t \cos\psi$
• Virtual / Formative Number of Teeth ($Z_v$): $Z_v = \frac{Z}{\cos^3\psi}$ Purpose: $Z_v$ is used to select the Lewis form factor $Y$ for helical gear strength evaluation.

Force Breakdown in Helical Gears

1. Tangential Force ($W_t$): $W_t = \frac{2 T}{d}$
2. Radial Force ($W_r$): $W_r = W_t \frac{\tan\phi_n}{\cos\psi} = W_t \tan\phi_t$
3. Axial Thrust Force ($W_a$): $W_a = W_t \tan\psi$

Note: Double-helical (herringbone) gears cancel axial thrust forces internally.

---

4. Bevel Gear Design (Straight Bevel Gears)

Bevel gears transmit power between intersecting shafts (usually at a $90^\circ$ shaft angle).

Geometry & Cone Angles

For a $90^\circ$ shaft angle:

• Pinion Pitch Cone Angle ($\gamma_1$): $\tan\gamma_1 = \frac{Z_1}{Z_2} = \frac{1}{i}$
• Gear Pitch Cone Angle ($\gamma_2$): $\tan\gamma_2 = \frac{Z_2}{Z_1} = i$
• Outer Cone Distance ($R$): $R = \frac{d_1}{2 \sin\gamma_1} = \frac{m}{2} \sqrt{Z_1^2 + Z_2^2}$

Force Vectors on Pinion

$\text{Tangential: } W_t = \frac{2 T_1}{d_{m1}} \quad \text{(using mean pitch diameter } d_{m1} = d_1 - b \sin\gamma_1\text{)}$

$\text{Radial Force: } W_r = W_t \tan\phi \cos\gamma_1$

$\text{Axial Thrust Force: } W_a = W_t \tan\phi \sin\gamma_1$

---

5. Worm Gear Sets

Worm gear sets consist of a threaded worm shaft (1 to 4 threads) meshing with a worm gear wheel. They provide high speed reduction ratios ($i = 10:1$ to $100:1$) in a compact non-intersecting, right-angle shaft layout.

Key Geometry Parameters

• Worm Lead ($L$): $L = p_x \cdot Z_1 = \pi m \cdot Z_1$ (where $Z_1$ is number of worm threads/starts).
• Worm Lead Angle ($\lambda$): $\tan\lambda = \frac{L}{\pi d_1} = \frac{m Z_1}{d_1} = \frac{Z_1}{q}$ (where $q = d_1/m$ is the worm diameter quotient, typically $8 \le q \le 12$).
• Speed Ratio ($i$): $i = \frac{N_1}{N_2} = \frac{Z_2}{Z_1}$

Self-Locking Condition & Efficiency

When driving from worm to worm wheel, sliding friction along threads creates a self-locking state if the lead angle is smaller than the static friction angle $\phi_f = \arctan(f)$:

$\text{Self-Locking Condition: } \tan\lambda < f \quad (\text{typically } \lambda < 5^\circ)$

Worm Set Power Efficiency ($\eta$):

$\eta = \frac{\tan\lambda (1 - f \tan\lambda)}{\tan\lambda + f}$

Thermal Capacity

Due to continuous sliding friction along threads, heat generated ($H_g = P_{in} (1 - \eta)$) must not exceed housing heat dissipation capacity ($H_d = K_{heat} A_{housing} (T_{case} - T_{amb})$).

---

6. Worked Step-by-Step Gear Design Problem

Problem Statement

A standard $20^\circ$ full-depth spur pinion with $Z_1 = 20$ teeth rotates at $N_1 = 1200 \text{ rpm}$ and transmits $15 \text{ kW}$ of power to a driven spur gear with $Z_2 = 60$ teeth. The gear set module is $m = 4 \text{ mm}$ and face width is $b = 40 \text{ mm}$. The Lewis form factor for a 20-tooth $20^\circ$ pinion is $Y_1 = 0.322$.

Calculate:

1. The pitch diameters $d_1, d_2$ and center distance $C$.
2. The pitch line velocity $v$.
3. The tangential transmitted force $W_t$ and radial force $W_r$.
4. The tooth bending stress $\sigma_b$ in the pinion using the Lewis equation.

Step-by-Step Solution

Step 1: Pitch Diameters and Center Distance $d_1 = m \cdot Z_1 = 4 \text{ mm} \times 20 = 80 \text{ mm} = 0.080 \text{ m}$

$d_2 = m \cdot Z_2 = 4 \text{ mm} \times 60 = 240 \text{ mm} = 0.240 \text{ m}$

$C = \frac{d_1 + d_2}{2} = \frac{80 + 240}{2} = 160 \text{ mm}$

Step 2: Pitch Line Velocity $v$ $v = \frac{\pi d_1 N_1}{60} = \frac{\pi \times 0.080 \text{ m} \times 1200 \text{ rpm}}{60} = \frac{301.593}{60} = 5.0265 \text{ m/s}$

Step 3: Tangential Force $W_t$ and Radial Force $W_r$ $W_t = \frac{P_{power}}{v} = \frac{15,000 \text{ W}}{5.0265 \text{ m/s}} = 2984.18 \text{ N}$

$W_r = W_t \tan(20^\circ) = 2984.18 \times 0.36397 = 1086.15 \text{ N}$

Step 4: Pinion Tooth Bending Stress $\sigma_b$ via Lewis Formula $W_t = \sigma_b \cdot b \cdot Y_1 \cdot m$

$\sigma_b = \frac{W_t}{b \cdot Y_1 \cdot m} = \frac{2984.18 \text{ N}}{40 \text{ mm} \times 0.322 \times 4 \text{ mm}}$

$\sigma_b = \frac{2984.18}{51.52} = 57.923 \text{ MPa}$

Conclusion: Pinion tooth bending stress is $57.92 \text{ MPa}$. For standard steel pinion materials ($S_{y} \ge 250 \text{ MPa}$), this stress level provides a generous factor of safety against tooth bending fatigue.