Machine Design, Mechanisms, Vibrations and Manufacturing — ESE Mechanical

Weightage: Machine design, theory of machines and manufacturing together are the largest group of topics in the Mechanical papers. Many questions are short formula recalls, so a one-page card per sub-area pays off.

1. Design under static load

Machine design begins with the type of load and the failure mode. For a ductile material, two theories cover most questions:

• Maximum shear stress (Tresca): yield when ${\tau }_{max}={\sigma }_y/2$.
• Distortion energy (von Mises): yield when ${\sigma }_{vm}=\sqrt{{\sigma }_1^2-{\sigma }_1{\sigma }_2+{\sigma }_2^2}={\sigma }_y$.

For brittle materials use the maximum principal stress theory. The factor of safety is $FoS=\text{strength}/\text{working stress}$.

Stress concentration raises local stress at a notch by the factor $K_t$. For static loads in ductile materials it is often ignored, but for brittle and fatigue loading it must be included.

2. Fatigue

A steel member under reversing stress fails below its yield strength. The endurance limit is about half the ultimate strength for steel, reduced by modification factors for size, surface finish and reliability. With mean stress ${\sigma }_m$ and alternating stress ${\sigma }_a$, the safe-design lines are:

Fatigue life is shortened by notches and rough surfaces, and extended by shot peening and surface hardening, which leave compressive residual stress.

3. Shafts, springs, bearings and gears

Shaft. For a solid shaft in torsion, $\tau =16T/(\pi d^3)$. With bending and torsion combined, design is on the equivalent twisting moment $T_e=\sqrt{M^2+T^2}$ (maximum shear theory) or the equivalent bending moment. A key transmits torque and is checked for shear and crushing.

Helical spring. The shear stress is $\tau =K_s\frac{8FD}{\pi d^3}$, with $K_s$ the Wahl factor correcting for curvature. The deflection is $\delta =\frac{8F{D}^{3}n}{G{d}^4}$, so the stiffness is $k=G{d}^4/(8D^{3}n)$. Springs in series add flexibilities and in parallel add stiffnesses.

Rolling bearings. The basic life in millions of revolutions is $L_{10}=(C/P{)}^p$, with $p=3$ for ball bearings and $10/3$ for roller bearings. Doubling the load cuts life to one-eighth for a ball bearing. Plain (journal) bearings depend on hydrodynamic lubrication, described by the Sommerfeld number.

Gears. The module is $m=d/z$. For two meshing gears, the speed ratio is the inverse of the tooth ratio. With a 20-degree involute pressure angle, the minimum number of teeth on a pinion to avoid interference is about 17 (18 in practice). Gear failure modes are tooth bending (Lewis equation) and surface pitting (Hertz contact).

4. Kinematics of mechanisms

The degrees of freedom of a planar mechanism are given by Grubler's equation:

$F=3(n-1)-2j-h$

with $n$ links, $j$ lower pairs and $h$ higher pairs. A four-bar linkage has $n=4$, $j=4$, so $F=1$.

Grashof's law says that a four-bar chain has a fully rotating link if the sum of the shortest and longest links is no more than the sum of the other two. Common inversions of the four-bar chain are the crank-rocker, double-crank (drag link) and double-rocker. The slider-crank gives the engine mechanism and its inversions give the Whitworth quick-return and the oscillating cylinder.

A cam's follower motion is chosen to avoid sudden jumps in velocity or acceleration. Simple harmonic motion gives finite velocity but an acceleration step at the ends, and cycloidal motion removes the step.

5. Flywheels, balancing and gyroscopes

A flywheel stores kinetic energy to smooth speed fluctuation. The energy change over a cycle is $\Delta E=I{\omega }^2{C}_s$, with the coefficient of fluctuation of speed $C_s=({\omega }_{max}-{\omega }_{min})/{\omega }_{mean}$. A heavy rim at a large radius gives the most inertia per unit mass.

Balancing: a rotating mass is balanced by another mass such that the vector sum of centrifugal forces, and of their moments for multi-plane problems, is zero. A reciprocating engine is never fully balanced. A governor controls average speed, whereas a flywheel controls speed within a cycle.

6. Vibrations

A single-degree-of-freedom system has natural frequency ${\omega }_n=\sqrt{k/m}$ and damping ratio $\zeta =c/c_c$, with $c_c=2\sqrt{km}$. The behaviour is under-damped for $\zeta <1$ (oscillation decays), critically damped at 1 and over-damped above.

For a forced response, the resonance peak occurs near $\omega ={\omega }_n$, and transmissibility is greater than 1 for frequency ratios below $\sqrt{2}$ and less than 1 above, so isolation needs a low natural frequency. The critical speed of a shaft is its first natural frequency.

7. Casting and welding

Casting. Molten metal is poured into a mould. A riser feeds metal as the casting shrinks, and Chvorinov's rule gives solidification time $t=C(V/A{)}^2$. A riser must solidify after the casting, so it needs a higher volume-to-area ratio. Defects include porosity, shrinkage cavities and cold shuts.

Welding. Arc welding (MMAW, TIG, MIG/MAG, submerged arc) uses an electric arc, while resistance welding uses $I^{2}Rt$ heat. The heat-affected zone is a region of changed microstructure beside the weld. Gas welding uses an oxy-acetylene flame, with neutral, oxidising and carburising flames.

8. Forming and machining

Bulk forming: rolling, forging, extrusion and drawing. Hot working is above the recrystallisation temperature and gives refined grains but a poorer finish, and cold working gives strength and surface finish with a loss of ductility.

In machining, the chip forms by shear along a shear plane. With chip thickness ratio $r=t_1/t_2$ and rake angle $\alpha$:

$\tan \varphi =\frac{r\cos \alpha }{1-r\sin \alpha }$

Tool life follows Taylor's equation, $V{T}^n=C$.

Worked example. A tool lasts 60 min at 100 m/min with $n=0.25$. At 120 m/min, $T=60\times (100/120{)}^4=28.9$ min.

Non-traditional processes include EDM (spark erosion of conductive materials), ECM (electrochemical dissolution), USM and laser machining. CNC machines move by coded instructions, and CAD/CAM links design to manufacturing. Metrology uses limits and gauges, with the go and no-go principle.

9. Production management and inventory

The economic order quantity minimises the sum of ordering and holding costs:

$EOQ=\sqrt{\frac{2DS}{H}}$

Worked example. With annual demand 1200 units, ordering cost Rs 50 and holding cost Rs 6 per unit per year, $EOQ=\sqrt{2\times 1200\times 50/6}=141$ units.

Other items to know: Pareto (ABC) analysis, just-in-time, MRP, linear programming for allocation, and forecasting by moving average and exponential smoothing.

Common traps

• Using the endurance limit without modification factors.
• Treating Soderberg as less conservative than Goodman. It is more conservative.
• Taking the ball-bearing life exponent as the roller one. Use 3 for ball.
• Forgetting that SHM cams have an acceleration jump at the ends.
• Assuming a reciprocating engine can be fully balanced.

Memory aids

• "Soderberg uses yield, Goodman ultimate": the mean-stress term.
• "Three for ball, ten-thirds for roller": bearing life.
• "Riser must freeze last": casting design.

Summary

Design follows the load type and the failure theory, with fatigue handled by Soderberg, Goodman or Gerber lines and component formulas for shafts, springs, bearings and gears. Mechanisms are counted by Grubler's equation, and flywheels, balancing and vibration deal with dynamic behaviour.

Manufacturing questions cover casting, welding, forming and machining, tied together by Taylor's equation and inventory control by the economic order quantity.

Exam protocol

• State the failure theory before computing a stress.
• Count links and joints before using Grubler's equation.
• Check which fatigue criterion the question names.
• Use consistent units for tool life and speed.