10.3 Design of Shafts, Keys, Couplings & Rolling/Sliding Bearings

Design of Shafts, Keys, Couplings & Rolling/Sliding Bearings

Rotating mechanical drivetrains form the core of industrial material handling, mining hoists, pumps, and heavy gearboxes. A complete shaft assembly includes the transmission shaft subjected to combined bending and twisting moments, keys and couplings for torque transmission and shaft alignment, and bearings (hydrodynamic sliding journal bearings or rolling contact bearings) to support radial and axial thrust loads with minimal friction.

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1. ASME Transmission Shaft Design Code

Transmission shafts are rotating machine elements subjected to simultaneously acting bending moments $(M)$, torsional twisting moments $(T)$, and occasional axial thrust loads $(F)$.

1.1 Combined Stress Formulation

For a solid circular shaft of diameter $d$ under bending moment $M$ and torque $T$:

• Bending stress at outer surface: $\sigma_b = \frac{32 M}{\pi d^3}$
• Torsional shear stress at outer surface: $\tau = \frac{16 T}{\pi d^3}$
• Applying the Maximum Shear Stress Theory (Guest's Theory):

$$\tau_{\max} = \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau^2} = \sqrt{\left(\frac{16 M}{\pi d^3}\right)^2 + \left(\frac{16 T}{\pi d^3}\right)^2} = \frac{16}{\pi d^3}\sqrt{M^2 + T^2}$$

1.2 ASME Code Shock and Fatigue Factors

In actual machinery, bending moments fluctuate due to shaft rotation (producing completely reversed cyclic bending stresses, $R = -1$), while torque may fluctuate with load changes. The ASME Shaft Design Code introduces shock and fatigue combined factors:

• $K_m$ (Combined shock and fatigue factor for bending)
• $K_t$ (Combined shock and fatigue factor for torsion)

1.3 Equivalent Design Moments

1. Equivalent Twisting Moment $(T_e)$: The pure torsional moment that produces the same maximum shear stress as the combined bending and twisting actions:

$$T_e = \sqrt{(K_m M)^2 + (K_t T)^2} = \frac{\pi}{16} d^3 \tau_{\text{allow}} \implies d = \left[ \frac{16 T_e}{\pi \tau_{\text{allow}}} \right]^{1/3}$$

1. Equivalent Bending Moment $(M_e)$: The pure bending moment that produces the same maximum normal tensile/compressive stress (Rankine criterion) as the combined loading:

$$M_e = \frac{1}{2}\left[ K_m M + \sqrt{(K_m M)^2 + (K_t T)^2} \right] = \frac{1}{2}[K_m M + T_e] = \frac{\pi}{32} d^3 \sigma_{b, \text{allow}}$$ $$d = \left[ \frac{32 M_e}{\pi \sigma_{b, \text{allow}}} \right]^{1/3}$$

1.4 ASME Allowable Stress Limits

According to the ASME code for commercial steel transmission shafting without keyways:

$$\tau_{\text{allow}} = \min(0.30 S_{yt}, 0.18 S_{ut})$$

• Keyway Effect: When keyways are cut into the shaft, the allowable shear stress must be reduced by $18\%$:

$$\tau_{\text{allow, keyway}} = 0.82 \cdot \tau_{\text{allow}}$$

1.5 Torsional Rigidity Criterion

Shafts supporting precision gears or machine tool spindles must satisfy torsional deflection limits:

$$\theta = \frac{T L}{G J} = \frac{584 T L}{G d^4} \quad (\text{degrees})$$

Standard industrial practice limits torsional twist to $\theta \le 1^{\circ}$ per 20 shaft diameters or $0.3^{\circ} \text{ to } 1.0^{\circ} \text{ per meter}$ of shaft length.

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2. Design of Sunk Keys & Shaft Couplings

2.1 Sunk Key Geometry & Failure Analysis

A key is a demountable fastener fitted into matching keyways machined in the shaft and hub to transmit torque.

• Square Sunk Key: Width $w = d/4$, Height $h = d/4$.
• Rectangular Sunk Key (Flat Key): Width $w = d/4$, Height $h = 2d/3 \approx d/6$.
• Key Length $(l)$: Sized based on torque transmission capacity $T$.

              +---------------+     ^ 
              |   Hub Recess  |     | h/2 (in Hub)
        ------+---------------+-----v-------
        Shaft |   Key (w x h) |     | h/2 (in Shaft)
        ------+---------------+-----v-------
              |  Shaft Body   |
              +---------------+
              <------- l ----->

1. Shearing Failure of Key: Shear force acts across the longitudinal pitch plane $A_s = w \cdot l$ at radius $d/2$:

$$T = F_s \cdot \frac{d}{2} = (l \cdot w \cdot \tau) \cdot \frac{d}{2} \implies \tau = \frac{2 T}{d \cdot w \cdot l} \le \tau_{\text{allow}}$$

1. Crushing / Compressive Bearing Failure of Key: Crushing force acts on half the key height $A_c = \frac{h}{2} \cdot l$:

$$T = F_c \cdot \frac{d}{2} = \left(l \cdot \frac{h}{2} \cdot \sigma_c\right) \cdot \frac{d}{2} \implies \sigma_c = \frac{4 T}{d \cdot h \cdot l} \le \sigma_{c, \text{allow}}$$

1. Condition for Key of Equal Strength in Shear and Crushing:

$$\frac{2 T}{d w l \tau} = \frac{4 T}{d h l \sigma_c} \implies \frac{w}{h} = \frac{\sigma_c}{2\tau}$$

For ductile mild steel, compressive yield strength is approximately twice the shear yield strength ($\sigma_c \approx 2\tau$). Therefore: $w = h$ (A square key provides equal strength in shearing and crushing). 4. Key Length for Equal Strength with Shaft: Equating key shear torque to solid shaft torsional capacity $\left(T = \frac{\pi}{16} d^3 \tau_s\right)$:

$$l \cdot w \cdot \tau_k \cdot \frac{d}{2} = \frac{\pi}{16} d^3 \tau_s$$

Assuming key and shaft have equal shear strength $(\tau_k = \tau_s)$ and $w = d/4$:

$$l \left(\frac{d}{4}\right) \frac{d}{2} = \frac{\pi}{16} d^3 \implies l = \frac{\pi}{2} d \approx 1.571 d$$

2.2 Shaft Couplings

1. Rigid Flange Coupling:
   • Consists of two cast iron flanges keyed to shaft ends and bolted together with $n$ precision reamed fitted bolts along pitch circle diameter $D_p = 3d$.
   • Standard proportions: Hub outer diameter $D_h = 2d$, hub length $L_h = 1.5d$, flange thickness $t_f = 0.5d$, protective circumferential rim thickness $t_p = 0.25d$.
   • Bolt Shear Stress: $\tau_b = \frac{8 T}{\pi d_b^2 n D_p} \le \tau_{\text{allow}}$
   • Bolt Crushing Stress: $\sigma_{cb} = \frac{2 T}{n d_b t_f D_p} \le \sigma_{c, \text{allow}}$
2. Oldham Flexible Coupling: Accommodates pure lateral (parallel) shaft misalignment using a floating central disc with two orthogonal tongues sliding in flange slots.
3. Bush-Pin Flexible Coupling: Accommodates angular, axial, and minor parallel misalignment while absorbing torsional vibrations and shocks via rubber/leather bushes fitted over steel driving pins.

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3. Sliding Contact Bearings & Hydrodynamic Lubrication

In a sliding contact (journal) bearing, a rotating cylindrical shaft (journal) is supported by a stationary bushing separated by a viscous lubricant film.

3.1 Lubrication Regimes & The Stribeck Curve

   Coefficient of
   Friction (f)
   ^
   | \  Boundary Lubrication (Metal-to-metal contact)
   |  \ 
   |   \  Mixed / Thin-Film Lubrication
   |    \__
   |       \___  Bearing Modulus (K)
   |           \________________ Hydrodynamic / Thick-Film Lubrication
   |                            (f proportional to mu * N / p)
   +------------------------------------------------------------->
   0                                Bearing Characteristic (mu * N / p)

• Bearing Characteristic Number: $\frac{\mu N}{p}$, where $\mu$ is dynamic viscosity $(\text{Pa}\cdot\text{s})$, $N$ is rotational speed $(\text{rpm})$, and $p = \frac{W}{d \cdot l}$ is bearing unit projected pressure $(\text{N/mm}^2)$.
• Bearing Modulus $(K)$: The transition point between mixed and full hydrodynamic lubrication. To ensure stable hydrodynamic operation without danger of metal seizure under operating fluctuations:

$$\text{Operating } \frac{\mu N}{p} \ge 3 K \quad (\text{Design stability rule})$$

3.2 Hydrodynamic Wedge Mechanism & Petroff's Law

Hydrodynamic pressure generation requires three essential conditions: (1) relative sliding motion, (2) viscous fluid, and (3) a converging geometric wedge created by radial clearance $c = r_b - r_j$.

• Eccentricity $(e)$ and Minimum Film Thickness $(h_0)$: Under radial load $W$, the journal shifts eccentrically by distance $e$, establishing attitude angle $\phi$. The eccentricity ratio is $\epsilon = e/c$.

$$h_0 = c - e = c(1 - \epsilon)$$

• Petroff's Law (Lightly Loaded Concentric Bearing): For a concentric journal running at speed $n_s = N/60\text{ rev/s}$:
  • Viscous shear force: $F = \mu A \frac{U}{c} = \mu (\pi d l) \frac{\pi d n_s}{c}$
  • Frictional torque: $T_f = F \cdot \frac{d}{2} = \frac{2 \pi^2 \mu n_s r^3 l}{c}$
  • Coefficient of friction: $f = \frac{F}{W} = 2 \pi^2 \left(\frac{\mu n_s}{p}\right)\left(\frac{r}{c}\right)$

3.3 The Sommerfeld Number $(S)$

The dimensionless Sommerfeld number is the fundamental operating parameter in hydrodynamic bearing design:

$$S = \left(\frac{r}{c}\right)^2 \left(\frac{\mu n_s}{p}\right)$$

where $r$ is journal radius, $c$ is radial clearance, $\mu$ is lubricant viscosity, $n_s$ is speed in rev/sec, and $p = W/(2rl)$ is projected pressure. Sommerfeld number directly determines eccentricity ratio $\epsilon$, friction variable, oil flow rate, and temperature rise via Raimondi and Boyd design charts.

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4. Rolling Contact Bearings: Selection & Life Rating

Rolling contact bearings (anti-friction bearings) substitute sliding friction with rolling friction using hardened balls or cylindrical/tapered/spherical rollers between inner and outer raceways.

4.1 Static and Dynamic Load Ratings

• Basic Static Load Rating $(C_0)$: Radial (or thrust) static load producing a total permanent plastic deformation of rolling element and raceway equal to $0.0001 \times$ rolling element diameter at the most heavily stressed contact zone.
• Basic Dynamic Load Rating $(C)$: The constant radial load that a group of apparently identical rolling bearings can withstand for a rating life of 1 million revolutions $(10^6\text{ rev})$ with a $90\%$ probability of survival without developing fatigue flaking (spalling).

4.2 Equivalent Dynamic Load $(P)$

When a bearing is subjected to combined radial load $F_r$ and axial thrust load $F_a$:

$$P = X \cdot V \cdot F_r + Y \cdot F_a$$

where:

• $V$ = Rotation factor ($V = 1.0$ when inner ring rotates relative to load; $V = 1.2$ when outer ring rotates relative to load).
• $X$ = Radial load factor, $Y$ = Axial thrust load factor (determined from manufacturer tables based on $F_a / (V F_r)$ relative to limiting parameter $e$).

4.3 Bearing Rating Life Equation ($L_{10}$ Life)

The rating life $L_{10}$ (in millions of revolutions) at $90\%$ reliability is:

$$L_{10} = \left(\frac{C}{P}\right)^p \quad (\text{million revolutions})$$

where the life exponent is:

• $p = 3$ for ball bearings (point contact, Hertzian stress $\sigma \propto P^{1/3}$)
• $p = \frac{10}{3} \approx 3.333$ for roller bearings (line contact, Hertzian stress $\sigma \propto P^{1/2}$)

4.4 Life in Operating Hours ($L_{10h}$)

$$L_{10h} = \frac{L_{10} \times 10^6}{60 \cdot N} = \frac{10^6}{60 \cdot N} \left(\frac{C}{P}\right)^p \quad (\text{hours})$$

where $N$ is shaft rotational speed in rpm.

4.5 Life at Modified Reliability ($L_R$)

Based on the two-parameter Weibull distribution for bearing fatigue life:

$$\frac{L_R}{L_{10}} = \left( \frac{\ln(1/R)}{\ln(1/0.90)} \right)^{1/b} = a_1$$

where $b \approx 1.17$ (Weibull slope parameter for ball bearings) and $a_1$ is the reliability adjustment factor ($a_1 = 1.0$ for $90\%$, $a_1 = 0.62$ for $95\%$, $a_1 = 0.21$ for $99\%$ survival).

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5. Comprehensive Summary Matrix