ME 423: Mechanical Design 2
Estimated study time: 10 minutes
Table of contents
Sources and References
Primary texts: Shigley’s Mechanical Engineering Design by Budynas and Nisbett; Machine Design: An Integrated Approach by Norton; Fundamentals of Machine Component Design by Juvinall and Marshek.
Supplementary texts: Theory of Machines and Mechanisms by Uicker, Pennock, and Shigley; Roark’s Formulas for Stress and Strain by Young, Budynas, and Sadegh; Engineering Design by Pahl, Beitz, Feldhusen, and Grote.
Online resources: MIT OpenCourseWare 2.72 Elements of Mechanical Design; AGMA gear design standards; ISO 281 and 76 rolling bearing life; SKF, Timken, and NSK bearing catalogues.
Chapter 1: Advanced Design Philosophy
1.1 From Component to System
Where introductory mechanical design focuses on single components, advanced design integrates components into systems. Shafts, bearings, gears, clutches, brakes, and couplings share loads, alignments, and environments, and must be synthesised jointly rather than sized independently.
1.2 Reliability
Reliability is the probability of survival to a specified mission time. For \( n \) independent components in series, system reliability is \( R_s = \prod_i R_i \); in parallel with full redundancy, \( R_s = 1 - \prod_i(1 - R_i) \). Probabilistic design sizes components for target reliabilities, typically 90–99.9% depending on consequence of failure.
1.3 Optimisation
Design optimisation seeks the configuration that minimises or maximises an objective (mass, cost, energy) subject to constraints (strength, deflection, geometry). Formally:
\[ \min_{\mathbf{x}}\; f(\mathbf{x})\quad \text{subject to}\quad g_i(\mathbf{x}) \leq 0,\quad h_j(\mathbf{x}) = 0. \]Lagrange multipliers, sequential quadratic programming, and evolutionary algorithms solve the problem at varying levels of rigor.
Chapter 2: Clutches and Brakes
2.1 Function and Classification
Clutches transmit torque between rotating members; brakes absorb kinetic energy to decelerate them. Friction clutches and brakes include disc, drum, cone, and band configurations. Magnetic, fluid (torque converters), and electromagnetic clutches use non-friction principles.
2.2 Disc-Type Analysis
For a disc clutch under uniform wear, the torque capacity is
\[ T = \tfrac{1}{2}\,\mu F (D_o + D_i)/2, \]while under uniform pressure,
\[ T = \tfrac{1}{3}\,\mu F \frac{D_o^3 - D_i^3}{D_o^2 - D_i^2}. \]Uniform-wear analysis yields lower, more conservative torque predictions for used clutches.
2.3 Thermal Capacity
Energy absorption during engagement raises temperature: \( \Delta T = E/(m c_p) \). Repeated engagements must dissipate heat between cycles. Brake fade, where coefficient of friction drops at high temperature, is a critical service concern in vehicle design.
Chapter 3: Couplings
3.1 Rigid, Flexible, and Universal
Rigid couplings (sleeve, flange) join shafts in precise alignment. Flexible couplings (jaw, gear, chain, Oldham, disc, diaphragm) accommodate misalignment, damp torsional oscillations, and isolate shocks. Universal joints transmit torque through significant angular misalignment, at the cost of non-constant velocity unless used in cancelling pairs.
3.2 Torsional Vibration
Shaft-coupling systems exhibit torsional natural frequencies
\[ \omega_n = \sqrt{\frac{k}{J_{eq}}}. \]Design avoids forcing frequencies (engine firing orders, gear mesh) near natural frequencies. Flexible couplings with calibrated torsional stiffness and damping shift frequencies and dissipate energy.
Chapter 4: Journal Bearings
4.1 Hydrodynamic Lubrication
A journal bearing supports a shaft on a film of lubricant whose pressure is generated by the viscous shearing action of the rotating shaft. The Reynolds equation describes the pressure field; the Sommerfeld number
\[ S = \left(\frac{r}{c}\right)^2 \frac{\mu N}{P} \]characterises operating conditions, with radius \( r \), clearance \( c \), viscosity \( \mu \), speed \( N \), and bearing pressure \( P \).
4.2 Performance Parameters
Eccentricity ratio, minimum film thickness, friction coefficient, and flow rate follow from Sommerfeld analysis. Design charts derived from Raimondi–Boyd solutions simplify selection. Temperature rise sets lubricant viscosity, often requiring iteration.
4.3 Boundary and Mixed Regimes
At low speed or high load, film thickness drops below surface asperity scale. Boundary and mixed lubrication involve asperity contact and rely on additives that form protective films. Wear rises sharply in these regimes; magnetic or hydrostatic bearings are alternatives for demanding applications.
Chapter 5: Rolling-Element Bearings
5.1 Types and Capabilities
Ball bearings, cylindrical roller bearings, tapered roller bearings, spherical roller bearings, and needle bearings differ in load capacity, speed, and misalignment tolerance. Sealed versus open, lubrication method, and preload significantly affect life and noise.
5.2 Life Equations
Basic dynamic capacity \( C \) and basic rating life \( L_{10} \) follow
\[ L_{10} = \left(\frac{C}{P}\right)^a \text{ million revolutions}, \]with \( a = 3 \) for ball bearings and \( 10/3 \) for roller bearings. Reliability factors, lubrication factors, and contamination factors modify the rating. For combined radial and thrust loads, an equivalent load \( P = X F_r + Y F_a \) is used.
5.3 Selection and Installation
Bearing selection begins with load type and magnitude, speed, required life, alignment, and operating environment. Shaft and housing fits, preload, lubrication, and sealing complete the installation design.
Chapter 6: Gears
6.1 Involute Geometry
Standard spur-gear teeth have involute profiles, which maintain conjugate action under varying centre distance. Pitch diameter \( d = m z \), with module \( m \) and tooth count \( z \); addendum, dedendum, and clearance follow standards. Contact ratio, ideally above 1.2, determines how smoothly teeth engage.
6.2 Tooth Bending Stress
The Lewis equation captures tooth bending stress; the AGMA equation augments it with dynamic, size, load-distribution, rim-thickness, and geometry factors:
\[ \sigma = W^t K_o K_v K_s \frac{P_d}{F}\frac{K_m K_B}{J}. \]Tooth surface durability is governed by a parallel pitting equation balancing contact stress against allowable Hertzian stress, accounting for hardness, surface finish, and lubrication.
6.3 Gear Types
Helical gears reduce noise and allow higher speeds at the cost of axial thrust. Bevel gears transmit power between intersecting shafts. Worm gears provide large speed reductions in small packages at typically lower efficiency. Planetary gears combine compactness, torque splitting, and variable ratios, used in automotive transmissions and speed reducers.
Chapter 7: Innovation, Optimisation, and Synthesis
7.1 Techniques for Innovation
Creativity tools such as TRIZ, biomimicry, morphological analysis, and function decomposition stimulate novel concept generation. Scientific literature, patent databases, and cross-disciplinary analogies enrich the solution space.
7.2 Computer-Aided Optimisation
Finite-element analysis, topology optimisation, and surrogate modelling enable designers to explore large parameter spaces. Topology optimisation produces material layouts that minimise compliance subject to volume constraints; the resulting geometry often requires manufacturability filtering, or is produced by additive processes that tolerate complex shapes.
7.3 Integrated Machine Design Project
A capstone-style project integrates shafts, bearings, gears, clutches, brakes, couplings, and housings into a working subsystem such as a speed reducer, hoist, or automated manipulator. Design proceeds through concept, detailed design, tolerancing, and drawing preparation, with reliability, manufacturability, and cost as explicit objectives. The outcome teaches how component-level decisions interact and how a robust machine is synthesised from rigorous analyses and creative judgement.
