UPSC ESE Mechanical Engineering Machine Design Syllabus

State the basic equation of fatigue (S-N) behaviour and define the endurance limit.

Fatigue is failure under fluctuating stress below the static yield. The S-N curve plots stress amplitude vs cycles to failure $N$. The endurance limit $\sigma_e$ (or $S_e$) is the stress amplitude below which a steel specimen survives infinitely many cycles (knee at about $10^{6}$ cycles).

Define mean stress, stress amplitude, stress ratio $R$, and amplitude ratio $A$ for a fluctuating load.

$\sigma_m = \frac{\sigma_{max}+\sigma_{min}}{2}$, $\sigma_a = \frac{\sigma_{max}-\sigma_{min}}{2}$, stress ratio $R = \frac{\sigma_{min}}{\sigma_{max}}$, amplitude ratio $A = \frac{\sigma_a}{\sigma_m}$.

Write the Goodman equation for fatigue design with a factor of safety $n$.

$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_{ut}} = \frac{1}{n}$, where $\sigma_e$ is the endurance limit, $\sigma_{ut}$ the ultimate tensile strength, $\sigma_a$ the alternating and $\sigma_m$ the mean stress component.

Write the Soderberg equation and state how it differs from the Goodman line.

$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_{yt}} = \frac{1}{n}$. Soderberg uses the yield strength $\sigma_{yt}$ in the mean-stress term (guards against yielding) and is more conservative than Goodman, which uses ultimate strength $\sigma_{ut}$.

Write the Gerber parabolic relation for fatigue and note its accuracy.

$\frac{n\,\sigma_a}{\sigma_e} + \left(\frac{n\,\sigma_m}{\sigma_{ut}}\right)^{2} = 1$. The Gerber parabola fits experimental ductile-material data best (least conservative), lying above the Goodman line.

List the main factors that modify the endurance limit of a real machine part versus a polished test specimen.

Surface finish factor $k_a$, size factor $k_b$, load (type) factor $k_c$, temperature factor $k_d$, reliability factor $k_e$, and miscellaneous/stress-concentration factor $k_f$. Corrected endurance limit $\sigma_e' = k_a k_b k_c k_d k_e\,\sigma_e$.

Define the theoretical (geometric) stress concentration factor $K_t$.

$K_t = \frac{\sigma_{max}}{\sigma_{nom}}$, the ratio of the maximum localized stress at a discontinuity (hole, fillet, notch) to the nominal stress computed from the net section. It depends only on geometry, not material.

For a small circular hole in a wide plate under uniaxial tension $\sigma$, what is the maximum stress at the hole edge?

The stress concentration factor is $K_t = 3$, so $\sigma_{max} = 3\sigma$ at the edge of the hole (perpendicular to the load). This is the classic Kirsch solution result.

Define the fatigue stress concentration (notch) factor $K_f$ and the notch sensitivity $q$.

$K_f = \frac{\text{endurance limit of notch-free specimen}}{\text{endurance limit of notched specimen}}$. Notch sensitivity $q = \frac{K_f - 1}{K_t - 1}$, so $K_f = 1 + q(K_t - 1)$, with $0 \le q \le 1$.

What are the main methods to reduce stress concentration in machine parts?

Provide generous fillet radii, use multiple/relief notches, drill relief holes, undercutting, remove material to make stress flow smoother (e.g., gradual transitions), and avoid sharp corners. The goal is to make stress lines (force flow) less crowded.

Define creep and the three stages of a creep curve.

Creep is the slow, time-dependent permanent deformation of a material under constant load at high temperature (above ~$0.4\,T_m$). Stages: (1) Primary — decreasing creep rate, (2) Secondary — constant minimum creep rate (longest, design-relevant), (3) Tertiary — accelerating rate leading to rupture.

What is the homologous temperature and at roughly what value does creep become significant?

Homologous temperature is the ratio $\frac{T}{T_m}$ of operating absolute temperature to the absolute melting temperature. Creep becomes significant above about $0.4\,T_m$ to $0.5\,T_m$.

Define creep strength and creep rupture strength.

Creep strength (creep limit) is the maximum stress that produces less than a specified creep strain in a given time at a given temperature. Creep rupture strength is the stress that causes fracture after a specified time at a given temperature.

For a pair of spur gears in mesh, define module $m$ and write its relation to pitch circle diameter $D$ and number of teeth $T$.

Module $m = \frac{D}{T}$ (mm/tooth), so $D = m\,T$. The module must be equal for both meshing gears. Circular pitch $p_c = \pi m$ and diametral pitch $P_d = \frac{1}{m}$ (in inch units).

State the Lewis equation for the beam (bending) strength of a spur gear tooth.

$W_t = \sigma_o\,b\,p_c\,y = \sigma_o\,b\,m\,\pi\,y = \sigma_o\,b\,m\,Y$, where $\sigma_o$ is allowable bending stress, $b$ face width, $y$ Lewis form factor (circular pitch), $Y=\pi y$ (module-based form factor).

How does the Lewis form factor $y$ vary with the number of teeth, and what does it physically represent?

The Lewis form factor $y$ increases with the number of teeth (gear teeth become stronger). It accounts for the tooth profile geometry, treating the tooth as a cantilever beam of uniform strength (parabola inscribed in the tooth).

Why are helical gears generally smoother and quieter than spur gears, and what is the penalty?

Helical gear teeth engage gradually along the helix, so contact begins at a point and spreads — giving smoother, quieter operation and higher load capacity at high speed. The penalty is an axial thrust force $W_a = W_t\tan\psi$ requiring thrust bearings.

For a helical gear with helix angle $\psi$, relate the normal module $m_n$ to the transverse module $m_t$ and write the formula for the number of teeth.

$m_n = m_t \cos\psi$. Pitch diameter $D = \frac{m_n\,T}{\cos\psi}$. The formative (virtual) number of teeth used in the Lewis equation is $T_v = \frac{T}{\cos^{3}\psi}$.

State the dynamic load and wear load considerations (Buckingham) used in gear design.

Dynamic load $W_D = W_t + W_i$ where $W_i$ is the increment due to tooth errors and velocity. Limiting wear load $W_w = D_p\,b\,Q\,K$, where $Q = \frac{2 T_G}{T_G + T_P}$ (ratio factor) and $K$ is the load-stress (wear) factor. Design requires $W_D \le W_b$ (beam strength) and $W_D \le W_w$.

Define the standard pressure angles used in involute gear systems and a key advantage of higher pressure angle.

Standard pressure angles are $14.5^\circ$ (old composite), $20^\circ$ (full-depth, most common), and $25^\circ$. A higher pressure angle gives stronger teeth (wider base), reduces undercutting and the minimum number of teeth to avoid interference, but increases bearing loads.

Classify rolling contact (anti-friction) bearings into the two main families with examples.

(1) Ball bearings: deep-groove, angular-contact, self-aligning, thrust ball bearings. (2) Roller bearings: cylindrical, tapered, spherical, and needle roller bearings. Choice depends on radial/axial load and speed requirements.

State the basic dynamic load rating relation (life equation) for rolling contact bearings.

$L = \left(\frac{C}{P}\right)^{k}$, where $L$ is rated life in millions of revolutions, $C$ the basic dynamic capacity, $P$ the equivalent dynamic load, and $k=3$ for ball bearings, $k=\frac{10}{3}$ for roller bearings.

Define the rating life $L_{10}$ of a rolling bearing.

$L_{10}$ is the life (in millions of revolutions, or hours) that $90\%$ of an identical group of bearings will reach or exceed before the first evidence of fatigue failure. Equivalently, only $10\%$ are expected to fail before this life.

Write the equivalent dynamic load equation for a rolling bearing under combined radial $F_r$ and axial $F_a$ loads.

$P = X\,V\,F_r + Y\,F_a$, where $V$ is the rotation factor ($1$ for inner-race rotation, $1.2$ for outer), and $X$, $Y$ are radial and thrust factors that depend on $\frac{F_a}{F_r}$ relative to the parameter $e$.

What is the function of a lubricant and the three principal regimes of lubrication?

A lubricant reduces friction and wear, removes heat, and seals out contaminants. Regimes: (1) Thick-film/hydrodynamic — surfaces fully separated by a fluid film, (2) Thin-film/boundary — partial metal contact, additives carry load, (3) Mixed/elastohydrodynamic — between the two, common in gears/bearings.

State the Petroff's equation for the coefficient of friction in a lightly loaded journal bearing.

$\mu = 2\pi^{2}\,\frac{\eta\,N}{P}\,\frac{r}{c}$, where $\eta$ is absolute viscosity, $N$ speed (rev/s), $P$ bearing pressure, $r$ journal radius, and $c$ radial clearance. It assumes a concentric (no-load) journal.

Define the Sommerfeld number and its significance in journal bearing design.

$S = \left(\frac{r}{c}\right)^{2}\frac{\eta\,N}{P}$, a dimensionless bearing characteristic number. It groups the governing parameters; bearing performance variables (friction, flow, eccentricity, temperature rise) are plotted against $S$ for design.

State Newton's law of viscosity and define dynamic (absolute) viscosity.

$\tau = \eta\,\frac{du}{dy}$, where $\tau$ is shear stress, $\frac{du}{dy}$ the velocity gradient, and $\eta$ the dynamic (absolute) viscosity. SI unit is Pa·s; the CGS unit is the poise ($1\ \text{Pa·s} = 10\ \text{poise}$).

Distinguish between dynamic viscosity and kinematic viscosity, including units.

Dynamic viscosity $\eta$ relates shear stress to shear rate (units Pa·s or poise). Kinematic viscosity $\nu = \frac{\eta}{\rho}$ is dynamic viscosity divided by density (units $\text{m}^{2}/\text{s}$, or stokes in CGS, where $1\ \text{stoke} = 10^{-4}\ \text{m}^{2}/\text{s}$).

What conditions are required to establish hydrodynamic (thick-film) lubrication in a journal bearing?

A converging wedge-shaped clearance space, sufficient relative sliding velocity, adequate supply of a viscous lubricant, and a load low enough that the generated film pressure can fully separate the surfaces. The journal runs eccentric to the bearing, forming the wedge.