Mechanical Design

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Machine design is the creation of new or improved machines to satisfy a need, using scientific principles, technical information and imagination.

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This section covers the design process, types of design, design requirements and considerations, factor of safety, preferred numbers, and the need for codes and standards including mechanical engineering standards and the ISO series.

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The Design Process • Shigley's design process: recognition of need → definition of problem → synthesis → analysis and optimisation → evaluation → presentation.

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The process is iterative (feedback loops). • Systematic (Pahl & Beitz) phases: task clarification (specification) → conceptual design (functions, working principles, concepts) → embodiment design (layout, form, materials) → detail design (drawings, tolerances, documentation). • Steps in designing a machine element: select mechanism → determine forces → select material → determine size (strength and rigidity) → modify for manufacture and assembly → prepare detailed drawings.

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Classification of Design Basis Type Description Novelty Adaptive design Existing design adapted with minor modifications (most common in practice) Development (variant) design Considerable modification of existing design using new ideas/materials/processes New (original/innovative) design Entirely new product; needs research, creativity and invention Method Rational design Based on mathematical formulae of mechanics and strength of materials Empirical design Based on empirical formulae derived from practice and past experience Industrial design Considers production aspects, ergonomics and aesthetics for mass manufacture Optimum design Best design for a chosen criterion (min. weight, min. cost) under constraints System design / element design Design of complete system vs design of individual elements Computer-aided design Design using CAD/CAE tools (modelling, FEA, simulation) Design Requirements and Considerations • Design requirements (what the product must satisfy): function/performance, strength and rigidity, reliability and life, safety, cost, weight and size, manufacturability and assembly, maintainability, ergonomics and aesthetics, environmental friendliness, compliance with codes and standards. • Design considerations (factors the designer must take into account): type of load and stresses, motion of parts (kinematics), selection of materials, form and size, frictional resistance and lubrication, convenient and economical features, use of standard parts, safety of operation, workshop facilities, number to be manufactured, cost of construction, assembly, stress concentration, wear, corrosion, thermal effects, noise and vibration. • Factor of safety (FS) = failure stress / allowable (working) stress.

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FS based on yield strength; brittle materials: based on ultimate strength.

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1.5–2 for ductile static loads;

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3–4 for brittle; higher for shock, uncertain loads or where failure is dangerous. • Factors affecting FS: uncertainty in loads and material properties, type of load (static, fatigue, shock), consequences of failure, accuracy of analysis, manufacturing quality, service environment. • DFM / DFA (design for manufacture and assembly): fewer parts, standard components, simple shapes, self-locating features, generous tolerances where possible.

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Preferred Numbers • Standard sizes follow Renard (geometric) series — ISO 3:

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R5 (ratio 101/5 ≈ 1.58), R10 (≈ 1.25), R20 (≈ 1.12), R40 (≈ 1.06), R80 (≈ 1.03). • R10 series:

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1, 1.25, 1.6, 2, 2.5, 3.15, 4, 5, 6.3, 8, 10.

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Used for motor powers, shaft sizes, speeds of machine-tool gearboxes, load capacities — reduce variety and inventory.

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Codes and Standards:

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Needs and Benefits • Standard: a set of specifications for parts, materials or processes intended to achieve uniformity, efficiency, interchangeability and specified quality (usually voluntary unless adopted by law). • Code: a set of specifications for the analysis, design, manufacture and construction of something, intended to achieve a specified degree of safety; often legally binding (e.g., ASME Boiler and Pressure Vessel Code, building codes). • Benefits: interchangeability and easy replacement of parts, reduced cost and inventory through mass production of standard items, assured quality and safety, reduced design time, common technical language between designer, manufacturer and user, legal compliance, easier international trade.

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Organisation Scope ISO (1947, Geneva) International standards in all fields except electrical IEC International electrical and electronic standards ASME Boilers & pressure vessels, piping (B31), GD&T;

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(Y14.5) ASTM Material specifications and testing methods ANSI (USA), BSI (UK), DIN (Germany), JIS (Japan) National standards bodies BIS (IS — India) Indian Standards widely used in Nepal;

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IBR — Indian Boiler Regulations NBSM (NS — Nepal) Nepal Bureau of Standards and Metrology — Nepal Standards (NS mark) SAE, AISI Automotive standards, oil grades; steel designations (AISI 1040, 4140) AGMA, AWS, API, TEMA Gears; welding; petroleum equipment; heat exchangers • Mechanical engineering standards (examples):

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ISO 286 (limits & fits), ISO 68/261 (metric threads), ISO 128 (technical drawings), ISO 1101 (GD&T;), ISO 1302/21920 (surface texture), ISO 281 (rolling bearing life), ISO 6336 (gear strength), ISO 3 (preferred numbers). • Steel designation:

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IS 40C8 = 0.40% C, 0.8% Mn (plain carbon);

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AISI 1040 = plain carbon steel with 0.40% C;

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AISI 41xx = Cr-Mo steels.

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The ISO Series (Management System Standards) Standard Subject ISO 9001 (2015) Quality management system — the only certifiable standard of the ISO 9000 family ISO 9000 / ISO 9004 QMS fundamentals & vocabulary / guidance for sustained success ISO 14001 Environmental management system ISO 45001 Occupational health & safety (replaced OHSAS 18001) ISO 50001 Energy management system ISO/IEC 27001 Information security management ISO 22000 Food safety management ISO 31000 Risk management — guidelines ISO 26000 Social responsibility — guidance only (not certifiable) ISO/IEC 17025 Competence of testing and calibration laboratories ISO 13485 / IATF 16949 / AS9100 Medical devices / automotive / aerospace quality systems • ISO 9001 is built on seven quality management principles: customer focus, leadership, engagement of people, process approach, improvement, evidence-based decision making, relationship management — and the PDCA (Plan-Do-Check-Act) cycle with risk-based thinking. • Certification process: gap analysis → documentation → implementation → internal audit → management review → certification audit by accredited body → surveillance audits (3-year certificate).

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This section covers the design and selection of standard machine elements: shafts, couplings, bearings, bolts, springs and dampers, power screws, brakes, clutches, gears and belt-pulley drives.

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(Kinematics of gears, belts and bolt/thread geometry are in Chapters 1.1 and 4.5.) Shaft • Shaft: rotating member transmitting power (torsion + bending).

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Axle: supports rotating elements, carries no torque (bending only).

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Spindle: short shaft in machine tools.

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Counter-shaft, line shaft. • Materials: plain carbon steels 30C8, 40C8, 45C8; alloy steels (Ni-Cr, Cr-Mo) for high strength. • Strength: pure torsion d³ = 16T/(πτ); combined bending + torsion (max shear theory) d³ = (16/πτ)√(M² + T²).

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ASME code with shock/fatigue factors: d³ = (16/πτ)√((KbM)² + (KtT)²). • Rigidity: angle of twist limit (≈ 0.25°/m for line shafts, ≈ 1° in 20d); lateral deflection limits for gears and bearings. • Critical (whirling) speed: speed at which shaft becomes dynamically unstable — equals natural frequency of transverse vibration: ωc = √(g/δ).

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Operate well away from it.

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Couplings Type Examples Use Rigid (for accurately aligned, collinear shafts) Sleeve/muff, clamp (split-muff), flange coupling (protected type) Line shafts; no misalignment or shock absorption Flexible (tolerate misalignment, absorb shock) Bushed-pin flange (rubber bushes), jaw, gear, disc, tyre coupling Motor–pump, motor–gearbox connections Special Oldham (parallel offset shafts), Hooke's/universal (intersecting shafts), fluid coupling Automobile drives, misaligned shafts • Flange coupling bolts in shear:

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T = n·(π/4)db²·τ·(Dp/2) (n bolts on pitch circle Dp).

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Also check key, hub and flange crushing/shear.

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Bearings Feature Sliding contact (journal/plain) Rolling contact (anti-friction) Friction Higher starting friction; low when full film forms Low starting and running friction Load / speed Heavy loads, very high speeds, shock loads Moderate loads; speed limited Space Small radial space Larger radial space, small axial length Lubrication Needs careful oil supply Simple (grease) Noise / life Quiet, long life if well lubricated Noisier at high speed; finite fatigue life Examples Crankshaft & big-end bearings, turbines Motors, gearboxes, vehicle wheels, machine tools • Sliding bearings: hydrodynamic (film pressure from rotation, wedge action), hydrostatic (externally pressurised), boundary lubrication.

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Materials: babbitt (white metal), bronze, Al alloys, PTFE, nylon, porous sintered bronze.

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Sommerfeld number S = (r/c)²·μN/p;

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Petroff's law for lightly loaded bearings. • Rolling bearings: deep-groove ball (radial + moderate axial — most common), angular-contact ball (combined loads), self-aligning ball, thrust ball (axial only), cylindrical roller (heavy radial), taper roller (heavy combined radial + axial — vehicle wheel hubs, gearboxes), spherical roller (heavy load, self-aligning), needle roller (small radial space — gudgeon pins, universal joints). • Bearing life L10 = (C/P)p million revolutions — p = 3 for ball, 10/3 for roller bearings;

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C = basic dynamic load rating;

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L10 = life that 90% of bearings reach (rating life).

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Equivalent load P = XVFr + YFa. • Designation e.g.

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6 = deep-groove ball;

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05 → bore = 05 × 5 = 25 mm (00 = 10, 01 = 12, 02 = 15, 03 = 17 mm).

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Bolts (Threaded Fasteners) • Bolt in tension: σt = P/At, where tensile-stress area At ≈ (π/4)[(dp + dc)/2]².

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Initial tightening load (empirical) Pi ≈ 2840d N (d in mm). • Bolt of uniform strength: reduce shank diameter to core diameter (or drill axial hole) so that shank and thread have equal stress — better for shock loads (more strain energy absorbed). • Preloaded joint: external load P shared — bolt load Fb = Fi + C·P, joint constant C = kb/(kb + km).

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A stiff joint (small C) protects the bolt from fatigue. • Locking devices and thread forms — see 1.1.

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Eccentrically loaded bolted joints — see 6.4.

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Springs and Dampers • Types: helical compression/extension (close-coiled), torsion springs, leaf (laminated) springs (vehicle suspensions; nipping = pre-stressing leaves by different radii), Belleville (disc) springs (high load, small deflection), spiral/clock springs, garter springs. • Helical spring: spring index C = D/d (4–12 practical).

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Shear stress τ = K·8WD/(πd³), with Wahl factor K = (4C − 1)/(4C − 4) + 0.615/C (accounts for curvature and direct shear).

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Deflection δ = 8WD³n/(Gd⁴); stiffness k = Gd⁴/(8D³n). • Springs in series:

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1/k = Σ1/ki; in parallel: k = Σki.

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End types: plain, plain-ground, squared, squared-and-ground (best seating).

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Surge: resonance of spring coils when excitation frequency equals spring's natural frequency. • Materials: patented cold-drawn carbon steel, oil-tempered wire, music wire, chrome-vanadium, stainless steel, phosphor bronze. • Dampers dissipate vibration energy: viscous (dashpot, F = cv — hydraulic shock absorbers), Coulomb (friction), hysteretic/structural (material), eddy-current, tuned mass dampers.

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Rubber mounts combine spring and damping.

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Power Screws • Convert rotary to linear motion with force amplification: screw jack, lathe lead screw, presses, vices.

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Threads: square (highest efficiency), Acme/trapezoidal, buttress. • Torque to raise load:

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T = W·tan(φ + α)·dm/2; to lower:

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T = W·tan(φ − α)·dm/2 (α = helix angle, tan φ = μ).

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Add collar friction μcWRc. • Efficiency η = tan α / tan(α + φ); maximum efficiency (1 − sin φ)/(1 + sin φ) at α = 45° − φ/2. • Self-locking condition: φ > α (friction angle > helix angle) — load does not lower by itself.

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Efficiency of a self-locking screw is less than 50%. • Differential screw (two threads of different pitch, same hand) gives very fine motion; compound screw (opposite hand) gives rapid motion.

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Brakes • Brakes absorb kinetic/potential energy and dissipate it as heat.

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Energy to absorb E = ½mv² + ½Iω² (+ mgh).

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Temperature rise ΔT = E/(mbc). • Types: block (shoe) brake (single/double), band brake (T1/T2 = eμθ; simple and differential — differential can be made self-locking), band-and-block, internal expanding shoe (drum) brake (automobiles), disc brake (caliper — better heat dissipation and fade resistance), electromagnetic, hydraulic/pneumatic actuation. • Self-energising: friction moment assists the applied force.

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Self-locking: brake applies with zero (or negative) effort — usually undesirable except in hoists/back-stops. • Lining materials: moulded organic/semi-metallic, sintered metal, ceramic, woven (asbestos now banned).

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Brake fade: loss of friction at high temperature.

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Clutches • Clutch connects/disconnects driving and driven shafts while running.

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Positive (jaw/claw, dog — no slip, engage at rest or low speed) vs friction clutches: single-plate (cars), multi-plate (motorcycles, wet type; compact), cone, centrifugal (automatic engagement with speed — mopeds), plus fluid coupling, electromagnetic, and overrunning (freewheel/one-way) clutches. • Uniform pressure theory (new clutch):

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T = (2/3)·μWn·(R1³ − R2³)/(R1² − R2²). • Uniform wear theory (worn clutch, p·r = constant):

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T = μWn·(R1 + R2)/2 — gives LOWER torque, hence safer and used in design. • n = number of pairs of friction surfaces (single plate, both sides effective: n = 2); multi-plate n = n1 + n2 − 1.

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T = μW Rm/sin α (semi-cone angle α).

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Gears and Belt-Pulley Drives (Design) • Lewis equation (bending strength of gear tooth, as cantilever):

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Fb = σb·b·m·Y (Y = Lewis form factor, depends on number of teeth and pressure angle).

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Face width b ≈ 10m (9.5m–12.5m). • Dynamic effects:

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Barth velocity factor Cv = 3/(3 + v) (ordinary cut gears);

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Buckingham dynamic load.

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Wear (surface) strength — Buckingham:

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Fw = dp·b·Q·K. • Gear tooth failures: bending fatigue (tooth breakage), pitting (surface contact fatigue), scoring/scuffing (lubricant film breakdown), abrasive wear, corrosive wear.

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Materials: cast iron (quiet), steels (case-hardened), bronze (worm wheels), nylon/phenolic (silent, light duty). • Worm gears: large speed reduction in one stage, compact, often self-locking, but lower efficiency and heat generation. • Belt drive design: select belt from catalogue using design power = rated power × service factor;

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V-belt sections Z, A, B, C, D, E (increasing size); check number of belts, centre distance, angle of contact on smaller pulley (≥ 120°). • Pulleys: cast iron or steel; flat-belt pulleys are crowned (convex rim) to keep the belt centred; arms elliptical in section; stepped (cone) pulleys for speed changes; fast and loose pulleys to start/stop driven shaft; idler (jockey) pulley increases angle of contact and tension.

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This section covers the forces and moments acting on machine elements (types of loads and stress concentration) and the design features of production tooling: cutting tools, press tools, drills and forging dies, together with lubrication and coolants.

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Forces, Moments and Types of Loads • By nature: static (dead) — constant; dynamic (live) — varying; impact/shock — suddenly applied; fatigue (cyclic) — fluctuating, repeated (σmin = 0), completely reversed (σm = 0); thermal loads. • By effect: axial (tension/compression), direct shear, bending moment, torsional moment (torque), and combined loading.

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A moment = force × perpendicular distance; a couple produces pure rotation. • Free-body diagrams identify all forces and moments; equilibrium ΣF = 0, ΣM = 0 gives reactions, then internal SF, BM and torque diagrams locate critical sections. • Stress concentration: local rise in stress at discontinuities (holes, fillets, keyways, grooves, threads).

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Theoretical factor Kt = σmax/σnominal (depends only on geometry; small circular hole in a wide plate in tension:

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Fatigue factor Kf = 1 + q(Kt − 1), q = notch sensitivity (0–1). • Stress concentration matters for brittle materials and fatigue loading; for ductile materials under static load it is often ignored (local yielding redistributes stress). • Reduce stress concentration: generous fillet radii, relief grooves, gradual changes of section, avoid sharp corners, additional smaller holes (to smooth stress flow), shot peening/rolling.

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Cutting Tools • Single-point tools: lathe, shaper, planer, boring tools.

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Multi-point tools: milling cutters, drills, reamers, taps, broaches, saws, grinding wheels. • Rake angle: positive rake → lower cutting force, better finish, weaker edge (HSS, soft ductile materials); negative rake → stronger edge for carbide/ceramic tools, hard materials and interrupted cuts.

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Clearance (relief) angle (≈ 6–10°) prevents rubbing of flank on work.

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Nose radius improves finish and edge strength (too large → chatter). • ASA tool signature order: back rake – side rake – end relief – side relief – end cutting-edge angle – side cutting-edge angle – nose radius (e.g., 8-14-6-6-6-15-1). • Cutting forces (turning): tangential/cutting force Fc (largest — determines power P = Fc·V), feed (axial) force Ff, radial (thrust) force Fr.

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Measured with a tool dynamometer. • Carbide inserts (ISO designation e.g.

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CNMG 120408: shape, clearance, tolerance, type, size, thickness, nose radius) are clamped in tool holders; chip breakers curl and break continuous chips. • Tool materials and wear — see 5.4.

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Press Tools • Die-set elements: punch (male), die (female), punch holder and die shoe, guide pins and bushes (alignment), stripper (strips sheet off the punch — fixed or spring-loaded), pilots (locate strip accurately in progressive dies), stock guides and stops, backing plate, shank. • Press force for blanking/piercing:

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Reduce force by shear on punch/die or stepping (staggering) punches.

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Stripping force ≈ 10–20% of cutting force. • Centre of pressure: point where resultant cutting force acts — die set should be positioned so that the press ram axis passes through it (avoids tilting and uneven wear). • Strip layout: arrangement of blanks to maximise material utilisation with adequate scrap bridge/margin. • Clearance (5–10% t per side): blanking — clearance on punch; piercing — clearance on die.

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Die types (simple, compound, progressive, combination, transfer) — see 5.4. • Materials: high-carbon high-chromium tool steel (D2), oil-hardening (O1), air-hardening (A2); carbides for long production runs.

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Drill (Twist Drill) Element / angle Description / typical value Shank Straight (small drills, chuck) or Morse taper (self-holding taper, larger drills) with tang Flutes Two helical grooves — form cutting lips, remove chips, admit coolant Web Central core; thickens towards the shank for strength; web thinning reduces thrust Chisel edge Formed at web; does not cut properly (extrudes metal) — causes high thrust Lips (cutting edges), margin/land Lips do the cutting; margin guides and sizes the hole Point angle 118° for general work (≈ 90° soft materials/plastics;

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135–140° hard steels) Helix angle ≈ 20–35° (≈ 30° standard); low for brass/hard materials, high for Al/soft materials Lip clearance (relief) angle 8–15° Chisel-edge angle 120–135° • Drilling time T = (L + 0.3D)/(f·N) (0.3D approach allowance for the point).

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Related tools: reamers (straight or helical flutes, finish holes), counterbores, countersinks (82°/90°), centre drills, spade drills, gun drills (deep holes), trepanning tools. • Jigs locate, hold and GUIDE the tool (drill bushes) — drill jigs.

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Fixtures locate and hold the work but do not guide the tool — milling, turning, welding fixtures.

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3-2-1 location principle:

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3 points on primary datum, 2 on secondary, 1 on tertiary; clamps complete the restraint.

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Forging Dies • Parting line: preferably at the largest cross-section, flat where possible, to ease metal flow and die making. • Draft angles: external 5–7°, internal 7–10° (for removal of forging).

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Fillet and corner radii: ensure smooth metal flow, prevent laps/cold shuts and die cracking. • Flash and gutter: the thin flash land restricts outward flow so that the cavity fills completely; the gutter accommodates excess metal.

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Flash is later removed in a trimming die. • Allowances: shrinkage, machining, die wear, mismatch tolerances. • Die impressions in multi-impression dies: fullering → edging → bending → blocking (pre-form) → finishing → trimming. • Die materials: hot-work die steels (H11, H13 — Cr-Mo-V) with high hot hardness, toughness and resistance to thermal fatigue (heat checking).

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Dies are preheated (≈ 150–300 °C) and lubricated (graphite). • Die failure: wear/erosion, thermal fatigue (heat checking), plastic deformation, mechanical fatigue cracking.

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Lubrication • Functions: reduce friction and wear, carry away heat, remove wear debris, prevent corrosion, seal, damp shock and noise. • Regimes (Stribeck curve — friction vs μN/p): boundary (thin adsorbed layers, metal contact), mixed, hydrodynamic (full fluid film, lowest wear), elastohydrodynamic (EHL) (gears, rolling bearings — elastic deformation + pressure-viscosity), hydrostatic (externally pressurised). • Lubricant types: liquid (mineral oils, synthetic oils — PAO, esters, silicones), semi-solid (greases) = oil + soap thickener (lithium — multipurpose, most common; calcium — water resistant, low temperature; sodium — higher temperature, not water resistant), solid (graphite, MoS2, PTFE — high temperature, vacuum, very high loads), gaseous (air bearings). • Properties: viscosity (most important); viscosity index (VI) — resistance to viscosity change with temperature (high VI is better); flash and fire point; pour point (lowest temperature at which oil flows); cloud point; oiliness; oxidation stability;

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SAE (engine oils — multigrade e.g.

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20W-50: 'W' = winter low-temperature grade) and ISO VG (kinematic viscosity at 40 °C in cSt). • Additives:

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VI improvers, detergents/dispersants, anti-wear (ZDDP), extreme-pressure (EP) additives (S, P, Cl compounds — hypoid gears), antioxidants, rust inhibitors, pour-point depressants, anti-foam agents. • Methods: splash, drip feed, wick, ring oiling, forced (pressure) circulation, oil mist, grease cups/guns.

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Coolants (Cutting Fluids) • Functions: cooling (dominant at high speeds), lubrication (dominant at low speeds — reduces BUE), flushing chips, corrosion protection, better finish and tool life. • Types: straight (neat) cutting oils — mineral oil ± EP additives — heavy, low-speed operations (broaching, threading, gear cutting); soluble oils (emulsions) — oil + water + emulsifier, milky (≈ 1 :

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20) — general turning, milling, drilling; semi-synthetic and synthetic fluids (chemical solutions — grinding); gaseous/cryogenic (air, CO2, LN2);

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MQL (minimum quantity lubrication); dry machining. • Cast iron and brass are usually machined dry (graphite in CI acts as lubricant; chips are discontinuous).

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Aluminium: soluble oil or kerosene. • Engine coolants: water + ethylene glycol (antifreeze — lowers freezing point, raises boiling point) + corrosion inhibitors.

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Static analysis designs components against failure under steady loads.

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This section covers design for static strength, design equations for transmission components, fasteners and connections (bolted, welded, cotter and knuckle joints) and load-carrying members.

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Design for Static Strength • Failure modes under static load: yielding (ductile) and fracture (brittle); also excessive deflection and buckling. • Allowable stress: σall = σy/FS (ductile), σu/FS (brittle).

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For ductile shear: τy = 0.5σy (Tresca) or 0.577σy (von Mises). • Combined normal σ and shear τ:

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Tresca σeq = √(σ² + 4τ²); von Mises σeq = √(σ² + 3τ²); max principal (brittle) σ1 = σ/2 + √((σ/2)² + τ²). • Direct + bending: σ = P/A ± My/I.

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Eccentric axial load on rectangular section: σ = (P/A)(1 ± 6e/b) → no tension if e ≤ b/6 (middle-third rule / kern); for circular section e ≤ d/8. • Curved beams (crane hooks, C-frames): neutral axis shifts towards the centre of curvature; maximum stress at the INNER fibre (Winkler-Bach theory).

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Trapezoidal section with wider inner side is economical for hooks. • Design procedure: define loads → FBD and reactions → find critical sections → select material and FS → size the section → check deflection, stability and stress concentration → standardise sizes.

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Transmission Components (Design Equations) Component Design equation Shaft (torsion) τ = 16T/(πd³); combined: d³ = (16/πτ)√(M² + T²) Key (width w, height h, length l) Shear: τ = 2T/(d·w·l); crushing: σc = 4T/(d·h·l); square key (w = h) equally strong if σc = 2τ; key length ≈ 1.5d Flange coupling bolts T = n·(π/4)db²·τ·Dp/2 Gear tooth (Lewis) Ft ≤ σb·Cv·b·m·Y Belt drive P = (T1 − T2)v;

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T1/T2 = eμθ Power & torque P = 2πNT/60 (W) Design of Fasteners and Connections • Bolt in tension: σt = P/At (tensile-stress/core area).

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Bolt in shear: τ = P/(n·πd²/4) using shank diameter. • Eccentric load in the plane of bolts (bracket bolted to column face, load parallel to plane): each bolt carries primary (direct) shear P/n + secondary (torsional) shear F″ = P·e·ri/Σr² (∝ distance from bolt-group centroid).

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Resultant = vector sum; most distant bolt is critical. • Eccentric load perpendicular to bolt axis (bracket tending to tilt about an edge): bolts carry tensile loads ∝ their distance from the tilting edge:

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Fi = P·e·li/Σl²; plus direct shear P/n. • Welded joints: fillet weld P = 0.707·s·l·τ; butt weld P = t·l·σt.

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Eccentric welded joints: primary shear + secondary shear from torsion using polar moment of weld group (weld treated as a line). • Riveted joints: tearing, shearing and crushing strengths; efficiency — see 1.1. • Cotter joint: connects two co-axial rods subjected to AXIAL tension/compression (not rotation) — socket and spigot with a tapered cotter (taper ≈ 1:48 to 1:24).

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Checked for tension in rod, spigot and socket; shear and crushing of cotter, spigot and socket collar.

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Uses: piston rod to crosshead, tie rods, foundation bolts. • Knuckle joint: connects two rods under tensile load where small angular movement is needed — eye and fork with a knuckle pin in double shear.

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Uses: tie rods of roof trusses, valve rods, links of bicycle chain, suspension links. • Pins: check double shear, bearing and bending.

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Design of Load-Carrying Members • Beams: select section with required section modulus Z = M/σall; then check deflection and shear.

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I-sections are most economical in bending (material placed far from neutral axis); hollow tubes best for torsion. • Columns/struts: design against buckling — Euler Pcr = π²EI/Le² (long), Rankine-Gordon or Johnson's parabolic formula for intermediate columns; use least radius of gyration; tubular sections are efficient. • Tension members:

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A = P/σall on net section (after holes). • Thin pressure vessels: wall thickness t = pD/(2σtη) + C (η = joint efficiency, C = corrosion allowance); heads (hemispherical — thinnest, torispherical, ellipsoidal, flat — thickest). • Levers: mechanical advantage = effort arm / load arm; first class (fulcrum between — bell-crank, rocker arm), second class (load between — wheel-barrow), third class (effort between — tweezers).

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Lever arm designed for bending at the section near the fulcrum boss; rectangular or elliptical cross-section with depth ≈ 2–3 × width. • Frames and brackets: combined direct and bending stresses;

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C-frames of presses and crane hooks treated as curved beams.

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Vibration is oscillatory motion of a body about its equilibrium position.

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This section covers generalised coordinates and degrees of freedom, free undamped and damped responses, forced harmonic response of discrete systems, natural and resonant frequency, transmissibility and vibration control.

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Basic Terms and Classification • Period T (time per cycle), frequency f = 1/T (Hz), angular frequency ω = 2πf, amplitude, phase. • Generalised coordinates: minimum set of independent coordinates required to completely describe the configuration of a system.

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Degree of freedom (DOF) = number of such independent coordinates.

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Simple pendulum, spring-mass:

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1 DOF; double pendulum:

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2 DOF; rigid body in space:

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6 DOF; continuous (beam, shaft) systems: infinite DOF. • Classification: free (no external force after initial disturbance) vs forced (external periodic force); undamped vs damped; linear vs non-linear; deterministic vs random; longitudinal, transverse and torsional vibrations. • Discrete (lumped-parameter) systems: masses, springs and dampers as separate elements — finite DOF.

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Free Undamped Vibration (1 DOF) • Equation: m ẍ + kx = 0 → natural frequency ωn = √(k/m) rad/s; fn = (1/2π)√(k/m). • Using static deflection δ = mg/k: fn = (1/2π)√(g/δ) ≈ 0.4985/√δ Hz (δ in m) ≈ 15.76/√δ Hz (δ in mm). • Methods: equilibrium (Newton / D'Alembert), energy method (KE + PE = constant; d/dt = 0), Rayleigh's method (KEmax = PEmax). • Springs: series 1/k = Σ1/ki; parallel k = Σki.

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Beam stiffness: cantilever k = 3EI/L³; simply supported (central mass) k = 48EI/L³; fixed-fixed k = 192EI/L³.

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Axial rod k = AE/L. • Simple pendulum ωn = √(g/L); compound pendulum ωn = √(mgh/IO); torsional ωn = √(kt/I) with kt = GJ/L. • Adding mass lowers natural frequency; adding stiffness raises it.

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Free Damped Vibration • Equation: m ẍ + c ẋ + kx = 0.

13

Critical damping cc = 2√(km) = 2mωn.

14

Damping ratio ζ = c/cc. • Damped natural frequency ωd = ωn√(1 − ζ²) (always less than ωn). • Logarithmic decrement δ = ln(xn/xn+1) = 2πζ/√(1 − ζ²) ≈ 2πζ (small damping) — used to measure damping from a decaying record. • Coulomb (dry friction) damping: amplitude decays LINEARLY (by 4F/k per cycle); viscous damping: decay is EXPONENTIAL.

15

Damping ratio Case Response ζ = 0 Undamped Oscillates forever at ωn 0 < ζ < 1 Under-damped Oscillatory with exponentially decaying amplitude at ωd (most mechanical systems) ζ = 1 Critically damped Returns to equilibrium in the SHORTEST time WITHOUT oscillation (instruments, door closers, gun recoil) ζ > 1 Over-damped Non-oscillatory, slow creeping return Forced Harmonic Response • Equation: m ẍ + c ẋ + kx = F0 sin ωt.

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Complete solution = transient (decays) + steady-state (at forcing frequency ω). • Frequency ratio r = ω/ωn.

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Steady-state amplitude X = (F0/k)/√[(1 − r²)² + (2ζr)²].

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Magnification factor MF = X/(F0/k). • Phase angle tan φ = 2ζr/(1 − r²): φ ≈ 0° for r ≪ 1 (stiffness controlled), φ = 90° at resonance (r = 1), φ → 180° for r ≫ 1 (mass controlled). • At resonance (r = 1):

19

MF = 1/(2ζ); undamped → infinite amplitude.

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Peak MF actually occurs at r = √(1 − 2ζ²) (slightly below 1) with MFmax = 1/(2ζ√(1 − ζ²)). • Rotating unbalance excitation:

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F0 = m0eω²; base (support) excitation similar in form.

22

Natural Frequency vs Resonant Frequency • Natural frequency: frequency at which a system vibrates freely after disturbance — a property of the system (depends on mass and stiffness), independent of excitation. • Resonant frequency: excitation frequency at which the response amplitude is maximum.

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For an undamped system it equals ωn; with damping, displacement resonance occurs at ωn√(1 − 2ζ²) (slightly lower).

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Damped natural frequency ωd = ωn√(1 − ζ²). • Resonance causes very large amplitudes → fatigue failure, noise, loosening (Tacoma Narrows Bridge collapse, 1940, is a famous example of wind-induced aeroelastic oscillation). • Avoid resonance: change mass or stiffness (detuning), add damping, use a dynamic vibration absorber (tuned mass damper — tuned so √(k2/m2) = forcing frequency, making main-mass amplitude zero), operate at least ±20% away from natural frequency, pass quickly through resonance during start-up.

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Transmissibility and Isolation • Transmissibility TR = transmitted force / applied force = √[1 + (2ζr)²] / √[(1 − r²)² + (2ζr)²]. • Isolation (TR < 1) is possible only when r > √2.

26

For r < √2, TR > 1 (force amplified). • In the isolation region (r > √2), more damping INCREASES transmissibility — so isolators use soft springs (low ωn) with small damping (some damping still needed to pass through resonance safely). • Isolation materials: steel springs, rubber pads, cork, felt, air springs.

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Machine foundations are designed so that ωn ≪ operating frequency.

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Multi-Degree-of-Freedom Systems • An n-DOF system has n natural frequencies and n principal (normal) modes.

29

2-DOF: two natural frequencies; in each mode all masses vibrate at the same frequency with a fixed amplitude ratio. • Two-rotor torsional system: one node (point of zero twist) and one non-zero natural frequency (the other is zero — rigid-body rotation).

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Three-rotor: two nodes. • Approximate methods for fundamental frequency:

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Dunkerley (1/ω1² ≈ Σ1/ωi² — gives lower bound), Rayleigh (upper bound), Holzer (torsional systems), matrix iteration. • Vibration measurement: vibrometer/seismometer (low natural frequency — measures displacement, r ≫ 1), accelerometer (high natural frequency — measures acceleration, r ≪ 1), velocity pick-ups;

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FFT analysers for condition monitoring (see 7.3).

1

Engineering is fundamentally problem solving.

2

This section covers the problem-solving process, writing a problem statement, invention and creativity (preparation, incubation, inspiration, verification), brainstorming and other idea-generation techniques, and decision tools including the decision matrix and decision tree.

3

The Problem-Solving Process • General steps:

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(1) identify and define the problem → (2) gather information and analyse causes → (3) generate alternative solutions → (4) evaluate alternatives → (5) select the best (decide) → (6) implement → (7) monitor and evaluate results (feedback). • Models:

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Polya — understand the problem, devise a plan, carry out the plan, look back.

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IDEAL — Identify, Define, Explore strategies, Act, Look back.

7

Engineering design process (6.1) is itself a problem-solving loop. • Root-cause tools:

8

5 Whys, fishbone (Ishikawa, cause-and-effect) diagram, Pareto analysis (80/20 rule), flowcharts, check sheets. • Barriers to problem solving: poorly defined problem, jumping to solutions, functional fixedness, fear of failure, lack of information, group-think.

9

Problem Statement • A problem statement is a clear, concise description of the issue to be solved — it defines what the problem is, where/when it occurs, its extent and impact, and the goal — WITHOUT prescribing a particular solution. • A good statement includes objectives, constraints (cost, time, size, regulations) and criteria for judging solutions; it is specific, measurable and solution-neutral (e.g., 'design a way to transport 50 kg loads up 3 floors' rather than 'design a better lift'). • 'A problem well stated is a problem half solved' (C.

10

The 5W1H questions (What, Why, Where, When, Who, How) help frame it. • The problem statement leads to a design specification (list of requirements: demands and wishes).

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Invention, Innovation and the Creative Process • Discovery: finding something that already exists but was unknown (a natural law).

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Invention: creating a new device, process or product that did not exist before — patentable if novel, non-obvious (inventive step) and useful (industrially applicable).

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Innovation: successful commercial or practical application of an invention or idea (can be incremental or radical). • Graham Wallas' four stages of creativity (1926):

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Stage What happens 1.

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Preparation Conscious study of the problem — gathering information, defining the problem, trying known methods 2.

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Incubation Stepping away; the subconscious mind works on the problem while attention is elsewhere 3.

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Inspiration (Illumination) Sudden insight — the 'Aha!' or 'Eureka' moment when an idea emerges 4.

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Verification Conscious testing, evaluation, refinement and elaboration of the idea to prove it works • Barriers to creativity: perceptual (seeing the problem narrowly), emotional (fear of failure/criticism), cultural (tradition, 'not done here'), environmental (distractions, unsupportive boss), intellectual (lack of knowledge or wrong strategy).

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Brainstorming and Idea-Generation Techniques • Brainstorming (Alex Osborn): group technique (≈ 5–10 people with a facilitator and recorder) to generate a LARGE number of ideas in a short time. • Rules of brainstorming:

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(1) defer judgement — no criticism during idea generation;

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(2) aim for quantity — quantity breeds quality;

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(3) welcome wild and unusual ideas;

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(4) combine and build on others' ideas (piggy-backing/hitch-hiking).

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Evaluation is done in a separate later session. • Brainwriting / 6-3-5 method:

25

6 people write 3 ideas each in 5 minutes, then pass sheets on.

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Nominal group technique: silent individual idea generation → round-robin sharing → discussion → voting/ranking.

27

Delphi technique: anonymous expert opinions in rounds until consensus. • SCAMPER:

28

Substitute, Combine, Adapt, Modify/Magnify, Put to other uses, Eliminate, Reverse/Rearrange.

29

Morphological chart (Zwicky): functions vs possible means → combine.

30

Synectics (Gordon): use of analogies.

31

40 inventive principles to resolve contradictions.

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Lateral thinking and Six Thinking Hats (Edward de Bono — white: facts, red: feelings, black: caution, yellow: benefits, green: creativity, blue: process control).

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Mind mapping, attribute listing, checklists.

34

Decision Making • Decisions: programmed (routine, rule-based) vs non-programmed (novel, unstructured); strategic, tactical and operational; individual vs group. • Decision environments: certainty (outcomes known); risk (probabilities known — use expected value); uncertainty (probabilities unknown) — criteria: maximax (optimistic), maximin (pessimistic, Wald), minimax regret (Savage), Hurwicz (coefficient of optimism α), Laplace (equally likely). • Herbert Simon's bounded rationality: managers 'satisfice' (choose a satisfactory option) rather than optimise. • Other tools: payoff tables, cost-benefit analysis, SWOT, break-even analysis, Analytic Hierarchy Process (AHP) (pairwise comparisons), sensitivity analysis.

35

Decision Matrix • A decision (weighted-scoring) matrix lists alternatives in rows and criteria in columns.

36

Each criterion gets a weight (importance, often summing to 1 or 100%); each alternative is scored (e.g., 1–10) against each criterion; weighted score = Σ(weight × score); the alternative with the highest total is preferred. • Pugh concept-selection matrix: each concept compared with a reference (datum) concept using + (better), − (worse), S (same); net score ranks concepts. • Example: criteria cost (0.4), efficiency (0.35), maintenance (0.25).

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Design A scores 8, 6, 7 → 0.4×8 + 0.35×6 + 0.25×7 = 3.2 + 2.1 + 1.75 = 7.05;

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Design B scores 6, 9, 8 → 2.4 + 3.15 + 2.0 = 7.55 → choose B.

39

Decision Tree • A decision tree is a graphical model of sequential decisions and uncertain events.

40

Symbols: square □ = decision node, circle ○ = chance (event) node with probabilities on branches, end points/triangles = payoffs (outcomes). • Solution by rolling back (backward induction) from right to left: at each chance node compute Expected Monetary Value EMV = Σ(probability × payoff); at each decision node choose the branch with the best EMV and prune the others. • Example:

41

60% chance of high demand (profit Rs 50 lakh), 40% low demand (loss Rs 10 lakh) → EMV = 0.6×50 + 0.4×(−10) = 26 lakh.

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Do not expand: sure profit Rs 20 lakh.

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Best decision: expand (26 > 20). • Expected Value of Perfect Information (EVPI) = expected value with perfect information − best EMV without it — the maximum worth paying for information (e.g., market survey). • Advantages: clear visual structure, handles sequential decisions, quantifies risk.

44

Limitations: probabilities may be subjective, trees become large and complex.