Sources and References

• Budynas, R. G. and Nisbett, J. K. Shigley’s Mechanical Engineering Design, McGraw-Hill, 11th edition and later.
• Norton, R. L. Machine Design: An Integrated Approach, Pearson, 6th edition and later.
• Juvinall, R. C. and Marshek, K. M. Fundamentals of Machine Component Design, Wiley.
• Hamrock, B. J., Schmid, S. R., and Jacobson, B. Fundamentals of Machine Elements, CRC Press.
• Hughes, A. and Drury, B. Electric Motors and Drives: Fundamentals, Types and Applications, Newnes.
• Slocum, A. H. Precision Machine Design, Prentice Hall, foundational material also used in MIT 2.72 Elements of Mechanical Design.
• MIT OpenCourseWare 2.72 (Elements of Mechanical Design) and 2.008 (Design and Manufacturing II) public notes on shafts, bearings, and power transmission.
• Stanford ME 112 (Mechanical Systems Design) public materials on gear trains and actuator selection.
• University of Cambridge Engineering Tripos Part IIA Mechanics modules on fatigue and machine elements.

Chapter 1: Foundations of Machine Element Design

Mechanical design for motion transmission is a discipline of informed compromise. Every load-bearing element must carry forces, torques, and moments reliably while respecting limits on deflection, vibration, wear, thermal growth, cost, and manufacturability. In electromechanical systems the designer inherits one additional constraint: the mechanical hardware must match the inertia, speed, and torque envelope of an electric actuator and must behave well inside a closed-loop controller. The mindset of this course is therefore to treat the machine as a single transmission chain from supply voltage to useful motion at a payload, with each link in that chain sized by the physics of its weakest failure mode.

The classical failure modes encountered across the course repeat with minor variations. Static ductile failure is handled by distortion-energy (von Mises) or maximum-shear theories; brittle failure uses a modified Mohr criterion. Fatigue under fluctuating loads is evaluated with a Goodman, modified Goodman, Gerber, ASME-elliptic, or Soderberg diagram, built on an endurance limit \( S_e \) that is derated from the rotating-beam value \( S_e' \approx 0.5\,S_{ut} \) by Marin factors for surface, size, load type, temperature, reliability, and miscellaneous effects. Contact between mating surfaces is governed by Hertzian stress, which dominates gear teeth and rolling bearings. Wear, fretting, and lubrication effects appear as empirical coefficients in catalog selection formulas. A running theme is that the designer’s first job is to decide which failure mode controls, then to size the element against that mode with an explicit factor of safety \( n \).

Chapter 2: Spur Gears I – Geometry and Kinematics

Gears transmit motion through conjugate surfaces that satisfy the fundamental law of gearing: the common normal at the contact point must pass through a fixed point called the pitch point, so that the angular velocity ratio is constant. For circular gears this condition leads to involute tooth profiles, because an involute rolled on its base circle produces a straight line of action tangent to the base circles of the mating pair. The pressure angle \( \phi \) is the angle between the line of action and the common tangent at the pitch point, and in North America it is almost always 20 degrees (25 degrees for heavy-duty power gearing).

Key parameters include the number of teeth \( N \), the diametral pitch \( P = N/d \) (in teeth per inch) or the metric module \( m = d/N \) (in mm per tooth), the circular pitch \( p = \pi d / N \), and the addendum and dedendum that define tooth height. The base circle diameter is

\[ d_b = d \cos\phi. \]

The contact ratio, defined as the average number of tooth pairs in mesh, should satisfy \( m_c \ge 1.2 \) for smooth operation and is computed from the length of the line of action divided by the base pitch \( p_b = \pi d \cos\phi / N \).

Chapter 3: Spur Gears II – Interference and Gear Trains

Interference (undercut) occurs when the tip of one gear contacts its mate below the base circle, where involute action is undefined. To avoid interference on 20 degree full-depth teeth, a pinion meshing with a rack needs at least 18 teeth, and a pinion meshing with an equal gear needs at least 13 teeth; these counts fall if profile-shifted (addendum-modified) teeth are used. Speed ratio across a simple train is \( m_G = N_G / N_P \); for a compound train it is the product of individual ratios. Planetary (epicyclic) trains superpose relative motions and are analyzed by the tabular or formula method, giving compact high-ratio reductions used in robot joints and automotive transmissions.

Chapter 4: Helical, Bevel, and Worm Gears

Helical gears place teeth on a helix of angle \( \psi \) so that tooth engagement is gradual. This yields quieter meshing and higher load capacity at the cost of an axial thrust \( F_a = F_t \tan\psi \) that must be reacted by the bearings. The normal and transverse pressure angles are related by \( \tan\phi_n = \tan\phi_t \cos\psi \). Bevel gears transmit power between intersecting shafts on conical pitch surfaces; the Tredgold approximation treats the tooth as an equivalent spur gear on the back cone. Worm gearing delivers very large ratios in one stage by meshing a helical screw (the worm) with a conjugate wheel; sliding dominates the contact so efficiency is modest and lubrication is critical. The self-locking property, which depends on the lead angle and friction coefficient, makes worm drives popular in lifting applications.

Chapter 5: Gear Loads – AGMA Bending Analysis

Load analysis on a gear tooth starts from the transmitted tangential force \( W_t = T/(d/2) \), with radial and axial components following from the pressure and helix angles. The AGMA bending stress equation (USCS form) is

\[ \sigma = W_t K_o K_v K_s \,\frac{P_d}{F}\, \frac{K_m K_B}{J}, \]

where \( K_o \) is an overload factor, \( K_v \) a dynamic factor accounting for velocity-induced impact, \( K_s \) a size factor, \( K_m \) a load distribution factor for misalignment, \( K_B \) a rim-thickness factor, \( F \) the face width, and \( J \) a geometry factor combining the Lewis form factor with a stress-concentration correction. The allowable bending stress is

\[ \sigma_{\text{all}} = \frac{S_t}{S_F}\,\frac{Y_N}{K_T K_R}, \]

where \( S_t \) is the tabulated material bending strength, \( Y_N \) a stress-cycle (life) factor, \( K_T \) a temperature factor, \( K_R \) a reliability factor, and \( S_F \) the bending safety factor. Protecting the tooth from fatigue fracture at the root requires \( \sigma \le \sigma_{\text{all}} \).

Chapter 6: Gear Loads – AGMA Surface Durability

A second, independent failure mode is surface pitting from contact fatigue. The AGMA contact stress equation is

\[ \sigma_c = C_p \sqrt{W_t K_o K_v K_s \,\frac{K_m C_f}{d_P F\, I}}, \]

where \( C_p \) is an elastic coefficient that depends on the Young’s modulus and Poisson’s ratio of the two materials, and \( I \) is a geometry factor for pitting resistance that encodes the radii of curvature at the contact and the load sharing between tooth pairs. The allowable contact stress is

\[ \sigma_{c,\text{all}} = \frac{S_c}{S_H}\,\frac{Z_N C_H}{K_T K_R}, \]

with \( Z_N \) a pitting cycle factor and \( C_H \) a hardness ratio factor used when the pinion and gear have different hardnesses. Because \( \sigma_c \propto \sqrt{W_t} \), doubling the load only multiplies the contact stress by \( \sqrt{2} \), but the allowable contact strength is typically much lower than the bending strength on a comparable scale, so pitting life is often the binding constraint on hardened steel gears.

Chapter 7: Gear Strength, Materials, and Design Workflow

The complete gear design loop iterates on number of teeth, module, face width, and material until both bending and wear safety factors meet targets. Typical starting face widths lie in the range \( 3\pi/P_d \le F \le 5\pi/P_d \) to keep the load distribution factor manageable. Common materials include through-hardened medium-carbon steel for low-to-moderate duty and case-hardened alloy steel (carburized or nitrided) for the highest contact strengths; brass, bronze, and polymers serve light-duty and quiet applications. Heat treatment choices trade case hardness (good for pitting) against core toughness (good against impact). For efficiency, a well-cut, well-lubricated spur or helical stage reaches 98–99 percent, while a single worm stage may be 40–90 percent depending on lead angle. The MIT 2.72 and Cambridge IIA treatments of gears emphasize that the bending calculation bounds the worst-case overload event while the contact calculation bounds the long-term survival; both must be checked at the rated torque.

Chapter 8: Motors I – Torque, Speed, and Inertia Matching

Electromechanical design begins with the load: what torque-versus-speed trajectory must the actuator drive, and what is the reflected inertia through the transmission? The load torque curve typically combines constant friction, viscous drag, gravity or preload, and an inertial term \( J_L\,\dot\omega \). At any instant the motor must supply \( T_m = T_{\text{load, reflected}} + (J_m + J_{L,\text{refl}})\dot\omega_m \). Reflecting load inertia through a reduction \( n = \omega_m/\omega_L \) gives \( J_{L,\text{refl}} = J_L / n^2 \), while reflected torque scales as \( T_{L,\text{refl}} = T_L / n \). The inertia-match condition \( J_m = J_{L,\text{refl}} \) maximizes acceleration per unit motor torque and is a standard rule of thumb for servo selection.

Chapter 9: Motors II – DC and Brushless DC Machines

The brushed permanent-magnet DC motor is the didactic baseline: armature voltage \( V = R_a I + K_e \omega \) and torque \( T = K_t I \), with \( K_t = K_e \) in SI. Eliminating current gives the straight-line torque-speed curve \( T(\omega) = (K_t/R_a)(V - K_e\omega) \), with no-load speed \( \omega_0 = V/K_e \) and stall torque \( T_s = K_t V/R_a \). Mechanical power is \( P_m = T\omega \), electrical input is \( VI \), and the difference is ohmic loss \( I^2 R_a \) plus friction and iron losses. Continuous operation is bounded by thermal limits on the winding, while peak operation is bounded by demagnetization, commutation, or amplifier current. Brushless DC (BLDC) and permanent-magnet synchronous motors remove the mechanical commutator, replacing it with electronic commutation from Hall sensors or a resolver/encoder; they dominate modern servo drives for their torque density, efficiency, and low maintenance.

Chapter 10: Motors III – Induction, Stepper, and Gearmotors

Three-phase induction motors generate torque through slip between the rotating stator field and the rotor, and their torque-speed curve has a characteristic peak (pull-out torque) near 20 percent slip. Variable-frequency drives deliver wide-range speed control by modulating the electrical frequency while keeping \( V/f \) approximately constant to preserve flux. Stepper motors move in discrete angular increments by sequentially energizing phases; they provide open-loop position control at low cost but suffer resonance, step loss at high acceleration, and fall-off of torque with speed. Planetary and harmonic-drive gearmotors combine a servo or stepper with a high-ratio reducer to multiply torque, reduce reflected inertia, and simplify controller gain selection — at the cost of backlash or hysteresis that affects precision applications.

Chapter 11: Motors IV – Drive Sizing and Thermal Limits

Duty cycle analysis converts a motion profile into an equivalent continuous torque using the RMS torque

\[ T_{\text{rms}} = \sqrt{\frac{1}{t_{\text{cyc}}}\int_0^{t_{\text{cyc}}} T^2(t)\,dt}. \]

The motor is acceptable when \( T_{\text{rms}} \le T_{\text{cont}} \) and the peak of \( T(t) \) stays within \( T_{\text{peak}} \), while the peak speed stays below the safe mechanical limit. Winding temperature is governed by a thermal time constant \( \tau_{\text{th}} \), so short overloads well above the continuous rating are permissible, but extended duty above \( T_{\text{cont}} \) causes insulation failure. A complete sizing also verifies bus voltage headroom so that back-EMF plus \( I R \) plus \( L\,dI/dt \) remains below the drive supply at peak speed.

Chapter 12: Motors V – Motion Control Integration

From a controls perspective the motor, drive, transmission, and load together form the plant of a cascaded current-velocity-position loop. The current loop is typically the fastest (kHz bandwidth) and is tuned for a low-order response to a torque command. The velocity loop closes around the current loop and must tolerate the mechanical resonance set by the torsional stiffness of shafts and couplings and the reflected load inertia; above this resonance the loop gain must roll off to preserve stability. The outer position loop sets trajectory tracking bandwidth. Backlash, compliance, and friction all degrade achievable bandwidth, which is why MTE 322 emphasizes that machine element choices are a controls decision: a stiffer coupling, a preloaded bearing, or a lower-backlash reducer directly expand the usable control bandwidth.

Chapter 13: Shafts – Stresses, Fatigue, and Deflection

Shafts are the central structural elements of rotating machinery, carrying torque between gears, pulleys, or couplings while withstanding bending moments from transverse loads. The dominant failure mode on a steel shaft at moderate speed is fatigue from the bending moment that rotates relative to a fixed fiber, producing fully reversed bending, together with a largely steady torque. The DE-Goodman criterion gives the minimum diameter at a cross section as

\[ d = \left(\frac{16 n}{\pi}\left\{\frac{1}{S_e}\left[4(K_f M_a)^2 + 3(K_{fs} T_a)^2\right]^{1/2} + \frac{1}{S_{ut}}\left[4(K_f M_m)^2 + 3(K_{fs} T_m)^2\right]^{1/2}\right\}\right)^{1/3}, \]

where \( M_a, T_a \) are alternating components and \( M_m, T_m \) are mean components, with \( K_f \) and \( K_{fs} \) the fatigue stress-concentration factors at keyways, fillets, or shoulders. Shafts must also satisfy deflection limits that protect gear-tooth alignment and bearing life — typical ceilings are 0.001 in per foot of shaft for general machinery — and must operate safely below their first bending critical speed, found from Rayleigh’s method as

\[ \omega_1 \approx \sqrt{\frac{g \sum w_i y_i}{\sum w_i y_i^2}}. \]

Chapter 14: Rolling-Element Bearings I – Types and Geometry

Rolling bearings replace sliding with rolling contact between hardened raceways and balls or rollers. Deep-groove ball bearings carry radial and modest axial loads in both directions; angular-contact ball bearings carry heavier thrust in one direction and are often mounted in preloaded pairs (back-to-back or face-to-face) to give a stiff, zero-backlash support for spindles. Cylindrical roller bearings carry large radial loads with negligible thrust; tapered roller bearings handle combined loads and are ubiquitous in automotive wheel hubs; spherical rollers accommodate misalignment; needle bearings offer high radial capacity in limited radial space. Bearing nomenclature uses a bore-code system (ISO), and cages, seals, and lubricant retention are integral to the performance claim.

Chapter 15: Rolling-Element Bearings II – Life and Load Rating

Fatigue of rolling contacts follows a Weibull distribution of the number of cycles to first spall. The catalog dynamic load rating \( C \) is defined so that a population of identical bearings reaches one million revolutions at 90 percent survival under radial load \( C \). For a different load \( P \) the Lundberg-Palmgren (L10) life is

\[ L_{10} = \left(\frac{C}{P}\right)^a, \]

with exponent \( a = 3 \) for ball bearings and \( 10/3 \) for roller bearings, expressed in millions of revolutions. Reliability scaling to \( L_R \) at reliability \( R \) uses a Weibull correction. Combined radial and thrust loads are reduced to an equivalent radial \( P = X F_r + Y F_a \) where the factors \( X, Y \) depend on the ratio \( F_a/(C_0 e) \) against a load-dependent threshold \( e \). Required life is typically stated in hours and converted via shaft speed.

Chapter 16: Rolling-Element Bearings III – Mounting, Preload, and Lubrication

A bearing must be mounted so that only one race is fixed axially, allowing thermal expansion to be absorbed by a floating race at the other end of the shaft — the classical fixed-float layout. Interference fits follow ISO shaft and housing tolerance tables selected for load type and direction. Preload on angular-contact pairs increases stiffness and removes internal clearance at the cost of elevated contact stress and heat generation; it is set by grinding the spacer rings, by spring washers, or by tightening a nut through a calibrated displacement. Grease lubrication suffices below a \( n \cdot d_m \) (speed times mean diameter) limit specific to the grease and bearing type; above it, oil mist or jet lubrication becomes necessary. The Cambridge IIA treatment notes that bearing failure is rarely the catalog fatigue life — it is contamination, poor lubrication, misalignment, or installation damage, so seals and installation tooling deserve as much attention as the selection math.

Chapter 17: Journal Bearings – Hydrodynamic Lubrication

Plain (journal) bearings replace rolling contact with a pressurized fluid film and, when properly designed, have effectively infinite fatigue life. The Sommerfeld number

\[ S = \left(\frac{r}{c}\right)^2 \frac{\mu N}{P} \]

collapses geometry (radius \( r \), clearance \( c \)), viscosity \( \mu \), rotational speed \( N \), and average pressure \( P \) into a single parameter that maps to friction coefficient, minimum film thickness, flow rate, and temperature rise via Raimondi-Boyd charts. Operation on the rising side of the Stribeck curve ensures hydrodynamic separation; operation through zero speed (startup) passes through boundary lubrication, so hydrostatic lift pockets or hardened coatings are used in heavily loaded applications. Journal bearings are quieter and better damped than rolling bearings, which is why they dominate in large turbomachinery.

Chapter 18: Couplings, Belts, and Chains

Couplings connect collinear shafts while accommodating small misalignments. Rigid couplings demand precise alignment and offer full torsional stiffness; flexible couplings (jaw, disc, bellows, Oldham) absorb angular, parallel, and axial misalignment and isolate torsional vibration. Servo applications favor bellows or disc couplings for their zero backlash and high torsional stiffness. Belt drives (flat, V, and synchronous) transmit power between parallel shafts with modest center-distance tolerance; V-belts rely on friction amplified by the wedge angle, while synchronous (timing) belts use meshing teeth and do not slip. Belt tension is sized to prevent slip at peak torque while limiting bearing preload; the initial tension required is \( T_0 = (F_1 + F_2)/2 \) with the capstan relation \( F_1/F_2 = e^{\mu\theta} \) at impending slip. Roller chains transmit high torques at low speeds, following a similar catalog-based selection with power correction factors for chain speed, number of teeth, and service class.

Chapter 19: Power Screws I – Geometry and Kinematics

Power screws convert rotation into linear motion for presses, jacks, lead screws, and linear actuators. The three thread profiles are square (highest efficiency), Acme (good strength and manufacturability), and buttress (unidirectional thrust). Key quantities are the nominal diameter \( d \), the pitch \( p \) (axial distance between threads), and the lead \( L \) (axial advance per revolution), with \( L = n_{\text{starts}} \cdot p \). The lead angle is

\[ \lambda = \arctan\!\left(\frac{L}{\pi d_m}\right), \]

where \( d_m \) is the mean diameter. Multi-start threads achieve high lead (fast motion) without sacrificing tooth depth.

Chapter 20: Power Screws II – Torque, Efficiency, and Self-Locking

The torque required to raise a load \( F \) on a square-thread screw is

\[ T_R = \frac{F d_m}{2}\,\frac{L + \pi \mu d_m}{\pi d_m - \mu L}, \]

and to lower it

\[ T_L = \frac{F d_m}{2}\,\frac{\pi \mu d_m - L}{\pi d_m + \mu L}. \]

Efficiency of raising is \( e = FL/(2\pi T_R) \). When \( T_L > 0 \), the screw is self-locking — the load cannot back-drive the screw, which is desirable in jacks and clamps but precludes back-driven applications such as ball-screw servo stages, which therefore use low-friction rolling elements and separate brakes. Acme threads add a factor of \( \sec(\alpha/2) \) with \( \alpha = 29^\circ \) on the friction terms because the thread face is inclined. A collar friction term \( \tfrac{1}{2} F \mu_c d_c \) is added when the screw thrust is taken against a plain collar.

Chapter 21: Power Screws III – Stress, Buckling, and Ball Screws

A power screw is simultaneously compressed, twisted, and loaded in bearing on the thread flanks. Axial-plus-torsional combined stress is evaluated at the thread root with the distortion-energy criterion; bearing pressure on the thread is kept below an allowable \( p_b \) that depends on material pair (typical values: 18 MPa steel-on-bronze for lead screws). Long slender screws must be checked against Euler or Johnson column buckling using the effective length set by the end conditions. Ball screws replace sliding with recirculating balls, raising efficiency above 90 percent, eliminating self-locking, and providing the stiffness and repeatability required by CNC machines and robot linear axes; they are selected from manufacturer catalogs using \( L_{10} \) life formulas analogous to rolling bearings.

Chapter 22: Fasteners and Bolted Joints I – Geometry and Preload

Threaded fasteners are the single most common machine element and, when preloaded correctly, carry fluctuating loads almost indefinitely. Thread standards (Unified and ISO metric) specify major, pitch, and minor diameters and the tensile stress area \( A_t \) used to compute bolt stress. The bolt stiffness \( k_b = A_t E / L \) and the effective member stiffness \( k_m \) (from a frustum-of-cone approximation or from Wileman’s exponential fit) combine into the joint stiffness constant

\[ C = \frac{k_b}{k_b + k_m}, \]

which is typically 0.2–0.3 for steel bolts in steel members. A proof-strength-limited preload \( F_i \approx 0.75\,F_p \) for reusable connections or \( 0.90\,F_p \) for permanent ones, with \( F_p = S_p A_t \), clamps the joint solidly.

Chapter 23: Fasteners and Bolted Joints II – External Loads and Fatigue

When an external tensile load \( P \) is applied per bolt, the bolt load becomes \( F_b = C P + F_i \) and the clamp force on the member becomes \( F_m = (1 - C) P - F_i \). Joint separation occurs when \( F_m = 0 \), so the load factor against separation is \( n_0 = F_i / [(1-C) P] \). For fluctuating external load the mean and alternating bolt stress components are built from \( F_b \) and evaluated on a Goodman or ASME-elliptic diagram using the bolt endurance limit with rolled-thread factors. The geometric effect of preload is that the alternating stress seen by the bolt is multiplied by \( C \), which is small, so a properly preloaded joint is remarkably fatigue-resistant — provided the preload is actually achieved and maintained. Torque control using the elementary formula \( T = K F_i d \) (with nut factor \( K \approx 0.2 \)) is only approximate; angle-of-turn or direct tension indicators give much tighter preload scatter.

Chapter 24: Fasteners and Bolted Joints III – Shear, Gaskets, and Practice

Bolts in shear-loaded joints either transmit load by friction (when preload is sufficient) or bear on the fastener shank. Shear bolts are checked for shank shear, bearing on the hole wall, and tearout/net-section failure in the plate. Eccentric shear loads create a combined primary-plus-secondary shear on each fastener, distributed by the bolt pattern’s polar moment of inertia about the centroid. Gasketed joints (flanges, pressure vessels) require enough preload to seat the gasket plus the hydrostatic end load, with an additional margin for relaxation. Good fastener practice — matched material grades, controlled surface condition, hardened washers under rotating elements, correct torque sequence — matters as much as the stress calculation and is the recurring lesson of Shigley’s, Norton’s, and the MIT 2.008 treatments.

Chapter 25: Integrated Machine Design

The MTE 322 synthesis is that every mechanical choice — module and face width of a gear, diameter and fillet radius of a shaft, bore and preload of a bearing, preload on a bolt, lead of a power screw, type and coupling of a motor — must be justified against a specific failure mode with a factor of safety, and must be compatible with the rest of the transmission chain and with the closed-loop controller above it. A robot joint, a lead-screw table, a conveyor drive, and a precision spindle all draw on the same short list of machine elements, assembled in different proportions. The designer’s discipline is to write down the motion profile, reflect inertias and torques through the transmission, size the actuator against RMS and peak limits, then cascade through couplings, shafts, bearings, gears, and fasteners, checking each against static, fatigue, wear, buckling, and deflection criteria. The measure of a good design is that no single element is dramatically overbuilt relative to its neighbors, that the assembly can be manufactured within tolerance, and that the controller achieves its bandwidth without exciting the mechanical resonances the transmission introduces.