Engineering Mechanics and Strength of Material

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Applied (engineering) mechanics studies forces and their effects on bodies at rest (statics) and in motion (dynamics).

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This section covers idealisations of bodies, force systems and equilibrium, friction, Newton's laws, gravitation, work-energy and impulse-momentum.

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Particles, Rigid Bodies and Deformable Bodies • Particle: body whose size is negligible — only translation considered (mass concentrated at a point). • Rigid body: distance between any two points remains constant under load — no deformation (used in statics/dynamics). • Deformable body: changes shape and size under load — studied in strength of materials. • Principle of transmissibility: a force may be moved anywhere along its line of action without changing its external effect on a RIGID body (not valid for deformable bodies).

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Force Systems and Static Equilibrium • Force systems: collinear (same line), concurrent (meet at a point), coplanar (same plane), parallel, non-concurrent non-parallel (general). • Resultant: parallelogram law R = √(P² + Q² + 2PQ cos θ); triangle and polygon laws of forces. • Moment = force × perpendicular distance.

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Varignon's theorem: moment of the resultant about a point = algebraic sum of moments of the components. • Couple: two equal, opposite, parallel non-collinear forces; moment = F × arm, SAME about every point (free vector); produces pure rotation.

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A force can be replaced by an equal force at another point plus a couple. • Equilibrium conditions (2-D): ΣFx = 0, ΣFy = 0, ΣM = 0 (3 equations).

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6 equations. • Lami's theorem: for three concurrent coplanar forces in equilibrium, each force ∝ sine of the angle between the other two:

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P/sin α = Q/sin β = R/sin γ. • Two-force member: forces equal, opposite, collinear.

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Three-force member: the three forces must be concurrent or parallel. • Free-body diagram (FBD): isolates the body and shows all external forces and reactions. • Support reactions: roller — 1 (normal to surface); hinge/pin — 2 (two components); fixed — 3 (two forces + moment).

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Friction • Laws of dry (Coulomb) friction: friction opposes relative motion; limiting friction F = μN; independent of apparent contact area and (for kinetic) nearly independent of sliding velocity; depends on the nature of surfaces. • Coefficient of friction μ = F/N.

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Static μs > kinetic μk.

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Rolling friction ≪ sliding friction. • Angle of friction φ: tan φ = μ (angle between resultant reaction and normal).

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Angle of repose (max incline on which a body rests without sliding) = angle of friction.

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Cone of friction. • Block on incline at angle θ slides when tan θ > μ.

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Force to pull up an incline, ladder problems, wedges. • Belt friction (capstan):

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T1/T2 = eμθ (θ in radians).

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Newton's Laws of Motion and Gravitation • First law (inertia): a body remains at rest or in uniform motion unless acted upon by an external force. • Second law: rate of change of momentum ∝ applied force:

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1 N = 1 kg·m/s². • Third law: to every action there is an equal and opposite reaction (acting on different bodies). • D'Alembert's principle:

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F − ma = 0 — adding the inertia force (−ma) converts a dynamics problem into static equilibrium (dynamic equilibrium). • Newton's law of gravitation:

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F = G m1m2/r², G = 6.674 × 10−11 N·m²/kg².

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At Earth's surface g = GM/R² ≈ 9.81 m/s²; g decreases with altitude (gh ≈ g(1 − 2h/R)). • Orbital velocity near Earth ≈ √(gR) ≈ 7.9 km/s; escape velocity = √(2gR) ≈ 11.2 km/s. • Kinematics: v = u + at; s = ut + ½at²; v² = u² + 2as.

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Projectile: range R = u² sin 2θ/g (max at 45°), max height = u² sin²θ/2g, time of flight = 2u sin θ/g.

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Circular motion: centripetal acceleration v²/r = ω²r.

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Work-Energy Theorem • Work W = F·s cos θ (J).

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Power = work/time = F·v (W). • Work-energy theorem: net work done on a body = change in its kinetic energy: ΣW = ½mv2² − ½mv1². • Kinetic energy ½mv²; rotational KE ½Iω²; potential energy mgh; spring energy ½kx². • Conservation of mechanical energy:

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KE + PE = constant when only conservative forces (gravity, spring) act.

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Friction is non-conservative.

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Impulse-Momentum Principle • Impulse = F·Δt = change in momentum = m(v − u) (N·s).

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Useful for impacts and short-duration forces. • Conservation of linear momentum: if net external force is zero, total momentum is constant (collisions, explosions, recoil of gun: mbvb = mgvg). • Coefficient of restitution e = (v2 − v1)/(u1 − u2) = relative velocity of separation / approach. e = 1 perfectly elastic (KE conserved); e = 0 perfectly plastic (bodies stick together, maximum KE loss);

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0 < e < 1 real impacts.

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Ball dropped from h rebounds to e²h. • Angular impulse = change in angular momentum (Iω); conservation of angular momentum (ice-skater effect).

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Industrial Applications — Statics, Friction and Material Handling • Friction governs material handling: force to push or pull a load F = μW (μ ≈ 0.02–0.05 for roller conveyors and wheels, ≈ 0.2–0.5 for sliding on steel or concrete); belt conveyors rely on friction between drive pulley and belt (T1/T2 = eμθ), and so do clutches, brakes, screw jacks (self-locking when tan α < μ) and band brakes of hoists. • Static equilibrium is used to find reactions and forces in cranes, hoists, jib and gantry structures, slings and rigging (sling tension rises sharply as the included angle increases), pallet racking and mezzanine supports, and in fixture clamping forces. • Work-energy and impulse-momentum principles appear in conveyor and hoist drive sizing (power = force × velocity), braking of moving loads, drop and impact tests of packaging, and press and hammer operations. • Ergonomics link (Chapter 6): permissible manual pushing, pulling and lifting forces are set from body mechanics and friction — the basis of NIOSH lifting-equation and manual-handling limits.

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This section covers stress and strain, Hooke's law and the elastic constants, deformation of bars, thermal stresses, Poisson's ratio, volumetric strain and bulk modulus, strain energy and impact loading.

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Stress, Strain and Hooke's Law • Stress = internal resisting force per unit area: normal stress σ = P/A (tensile +, compressive −); shear stress τ = P/A (parallel to area).

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1 MPa = 1 N/mm². • Strain: longitudinal ε = δL/L (dimensionless); shear strain φ (radians). • Hooke's law: within the proportional limit stress ∝ strain: σ = Eε; τ = Gφ. • Elongation of a bar δ = PL/(AE).

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Axial stiffness = AE/L. • Tapered circular bar (d1 to d2): δ = 4PL/(πE d1d2).

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Bar hanging under self weight: δ = ρgL²/(2E) = WL/(2AE) — half that of the same load applied at the end. • Composite bars (two materials sharing a load in parallel): strains equal, P = σ1A1 + σ2A2, σ1/σ2 = E1/E2 (modular ratio). • Factor of safety = ultimate (or yield) stress / working (allowable) stress.

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Constant Definition Typical (steel) Young's modulus E σ/ε (tension/compression) ≈ 200–210 GPa (Al ≈ 70, CI ≈ 100, Cu ≈ 110) Modulus of rigidity G τ/φ (shear) ≈ 80 GPa Bulk modulus K p/εv (uniform pressure / volumetric strain) ≈ 160–170 GPa Poisson's ratio μ (ν) −lateral strain / longitudinal strain 0.25–0.30 Thermal Stress • Free thermal expansion: δ = α L ΔT (no stress if free). α(steel) ≈ 12 × 10−6 /°C; α(Al) ≈ 23 × 10−6; α(Cu) ≈ 17 × 10−6. • If expansion is fully prevented: thermal stress σ = E α ΔT (independent of length and area).

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Heating a restrained bar → compressive stress; cooling → tensile. • If support yields by δs: σ = E(αLΔT − δs)/L. • Composite bar heated (e.g., steel bolt in copper tube): material with higher α goes into compression, lower α into tension; forces equal and opposite.

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Poisson's Ratio, Volumetric Strain and Bulk Modulus • Longitudinal strain along the load; lateral strain perpendicular to it (opposite sign).

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Poisson's ratio μ = −εlateral/εlongitudinal; theoretical range 0 to 0.5.

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Cork ≈ 0, concrete ≈ 0.1–0.2, steel ≈ 0.3, rubber ≈ 0.5 (incompressible, μ = 0.5 → volume unchanged). • Volumetric strain εv = δV/V.

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Bar under axial load: εv = ε(1 − 2μ).

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Triaxial stresses: εv = εx + εy + εz = (σx + σy + σz)(1 − 2μ)/E. • Sphere: εv = 3 × diametral strain.

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Cylinder: εv = 2εd + εL. • Bulk modulus K = p/εv (hydrostatic pressure). • Relations between elastic constants:

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E = 9KG/(3K + G); μ = (3K − 2G)/(6K + 2G).

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Only 2 constants are independent for an isotropic material.

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Strain Energy and Impact Loading • Strain energy (resilience): energy stored in a body due to elastic deformation.

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U = σ²/(2E) × volume = P²L/(2AE); shear:

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U = τ²/(2G) × volume. • Proof resilience = maximum strain energy stored up to the elastic limit.

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Modulus of resilience = proof resilience per unit volume = σe²/2E. • Gradually applied load P: σ = P/A.

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Suddenly applied load (h = 0): σ = 2P/A — twice the gradual-load stress (and twice the deformation). • Impact (falling) load from height h: σ = (P/A)[1 + √(1 + 2AEh/(PL))]; if h ≫ δ, σ ≈ √(2EPh/(AL)). • Strain energy in torsion U = T²L/(2GJ); in bending U = ∫M²dx/(2EI).

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Castigliano's theorem: deflection δ = ∂U/∂P.

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Industrial Applications — Allowable Stress, Safety Factor and Reliability • Factor of safety = ultimate (or yield) stress ÷ allowable working stress; typical values 1.5–2 for ductile materials under static load with known conditions, 3–4 for variable loads and 5–10 for shock, brittle materials or lifting equipment (statutory proof load tests on cranes, chains and slings). • Thermal stress (σ = EαΔT when expansion is prevented) explains why steam lines need expansion loops and bellows, why long conveyors and rails need expansion gaps, and why furnace and boiler parts crack in thermal cycling. • Strain energy and impact loading: suddenly applied loads double the stress, and impact loads can raise it many times — important in drop tests, packaging design, press tooling and guards. • Reliability and life: strength and load both vary statistically; designing so that the load distribution and strength distribution barely overlap is the basis of reliability engineering (Chapter 8) — closely related to the process-capability idea in quality control (Chapter 7). • Strain gauges and load cells (based on Hooke's law) are the sensing elements of industrial weighing, force and torque measurement.

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This section covers section properties (centroid, centre of gravity, moments of inertia), shear force and bending moment in beams, bending stresses, deflection of beams, analysis of plane trusses and torsion of circular shafts.

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Centre of Gravity and Centroid • Centre of gravity: point through which the resultant weight of a body acts.

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Centroid: geometric centre of an area/volume (coincides with CG for homogeneous bodies in uniform gravity). • x̄ = ΣAx/ΣA, ȳ = ΣAy/ΣA (composite areas; subtract holes).

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Shape Centroid location Triangle h/3 from base (2h/3 from apex) Semicircle (area) 4r/(3π) from diameter Quarter circle 4r/(3π) from each straight edge Solid hemisphere 3r/8 from flat face Hollow hemisphere (thin shell) r/2 from flat face Solid right circular cone h/4 from base Semicircular arc (wire) 2r/π from diameter Area and Mass Moment of Inertia Section I about centroidal axis Other Rectangle b × d bd³/12 About base: bd³/3;

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Z = bd²/6 Triangle b × h bh³/36 About base: bh³/12; about apex-parallel: bh³/4 Circle (dia d) πd⁴/64 Polar J = πd⁴/32;

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Z = πd³/32 Hollow circle π(D⁴ − d⁴)/64 J = π(D⁴ − d⁴)/32 Semicircle 0.11 r⁴ (about centroid) About diameter: πd⁴/128 = πr⁴/8 • Parallel axis theorem:

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I = IG + Ah² (applies to both area and mass MOI). • Perpendicular axis theorem (plane laminae):

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Iz = Ix + Iy → polar moment J = Ixx + Iyy. • Radius of gyration k = √(I/A) (area) or √(I/m) (mass). • Mass MOI: thin rod about centre mL²/12, about end mL²/3; ring mr²; disc/solid cylinder mr²/2; solid sphere (2/5)mr²; hollow sphere (2/3)mr².

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Shear Force and Bending Moment • Shear force (SF) at a section = algebraic sum of vertical forces on one side.

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Bending moment (BM) = algebraic sum of moments on one side.

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Sagging BM +, hogging BM −. • Relations: dM/dx = V (SF), dV/dx = −w (load intensity).

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BM is maximum where SF is zero or changes sign. • Degree of curves: no load between points → SF constant, BM linear;

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UDL → SF linear, BM parabolic (2nd degree); uniformly varying load → SF parabolic, BM cubic. • Point of contraflexure: point where BM changes sign (BM = 0 within span) — occurs in overhanging and fixed/continuous beams, not in simply supported beams with downward loads.

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Beam & load Max SF Max BM Max deflection Max slope Cantilever, point load W at free end W WL (at fixed end) WL³/3EI WL²/2EI Cantilever, UDL w over L wL wL²/2 (fixed end) wL⁴/8EI wL³/6EI Simply supported, central load W W/2 WL/4 (centre) WL³/48EI WL²/16EI Beam & load Max SF Max BM Max deflection Max slope Simply supported, UDL w wL/2 wL²/8 (centre) 5wL⁴/384EI wL³/24EI Simply supported, load W at a from A (b from B) Wb/L Wab/L (under load) — — Fixed-fixed, central load W W/2 WL/8 (ends & centre) WL³/192EI 0 at ends Fixed-fixed, UDL w wL/2 wL²/12 (ends); wL²/24 (centre) wL⁴/384EI 0 at ends Bending Stress and Shear Stress in Beams • Flexure formula:

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Bending stress varies linearly across depth — zero at the neutral axis (passes through the centroid) and maximum at extreme fibres. • Section modulus Z = I/ymax; σmax = M/Z.

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Rectangle Z = bd²/6; circle Z = πd³/32.

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Strength ∝ Z; stiffness ∝ I. • Assumptions: plane sections remain plane, material homogeneous & isotropic, obeys Hooke's law, same E in tension and compression, pure bending. • Shear stress in beams τ = V·Aȳ/(I·b).

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Rectangle: parabolic, τmax = 1.5 τavg at NA; circle: τmax = (4/3) τavg;

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I-section: web carries most of the shear, flanges most of the bending.

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Deflection of Beams • Differential equation of the elastic curve:

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EI = flexural rigidity. • Methods: double integration, Macaulay's method (discontinuous loads), moment-area method (Mohr's theorems), conjugate beam method, strain energy (Castigliano), superposition. • Mohr's 1st theorem: change in slope between two points = area of M/EI diagram between them.

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2nd theorem: deflection of one point relative to tangent at another = moment of M/EI area. • Deflection ∝ L³ (point load) or L⁴ (UDL) → span has the greatest effect.

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Fixing the ends reduces central deflection by 4× (point load) and 5× (UDL).

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Analysis of Plane Trusses • Assumptions: members connected by frictionless pins; loads applied only at joints; self-weight neglected — every member is a two-force member (pure tension or compression). • Perfect (just rigid) truss: m = 2j − 3 (m = members, j = joints). m < 2j − 3 → deficient (unstable mechanism); m > 2j − 3 → redundant (statically indeterminate). • Methods: method of joints (ΣFx = 0, ΣFy = 0 at each joint — start at a joint with ≤ 2 unknowns); method of sections (Ritter — cut ≤ 3 members, use ΣM = 0 — quick for a particular member); graphical (Maxwell/Cremona diagram). • Zero-force members:

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(1) unloaded joint with two non-collinear members → both zero;

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(2) unloaded joint with three members, two collinear → third member zero.

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Torsion of Circular Shafts • Torsion equation:

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Shear stress varies linearly from zero at centre to maximum at the surface. • Solid shaft: τmax = 16T/(πd³);

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T = (π/16) τ (D⁴ − d⁴)/D. • Power P = 2πNT/60 (W, N in rpm).

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Torsional rigidity = GJ; polar section modulus Zp = J/R. • A hollow shaft is stronger and stiffer than a solid shaft of the same weight (material near centre carries little stress). • Shafts in series: same torque, angles of twist add.

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Shafts in parallel (composite): same angle of twist, torques add. • Combined bending and torsion: equivalent torque Te = √(M² + T²); equivalent moment Me = ½[M + √(M² + T²)]. • Close-coiled helical spring (shear/torsion): deflection δ = 64WR³n/(Gd⁴); stiffness k = Gd⁴/(64R³n).

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1/k = Σ1/ki; in parallel: k = Σki.

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Industrial Applications — Structures, Shafts and Plant Equipment • Beams and trusses: shop floor beams, crane girders, pallet-rack beams (load per level is limited by deflection as well as by strength), mezzanine floors, roof trusses and conveyor support frames; deflection limits (e.g., span/300 to span/500) often govern rather than stress. • Section modulus and moment of inertia guide the choice of standard sections (I, channel, box, tube) — a structural version of standardisation: fewer sections held in stock simplifies fabrication and purchasing. • Torsion of shafts (τ = 16T/πd³; power P = 2πNT/60) is used to size conveyor, mixer, fan, pump and machine-tool shafts, and to check keys and couplings — a common design calculation in plant maintenance and modification work. • Centroid, centre of gravity and moment of inertia also matter in stability of stacked loads, forklift load centres and tipping, tank and hopper supports, and in flywheel design (4.5). • Failure of plant equipment is mostly by fatigue at stress raisers (keyways, shoulders, welds, holes) — which is why shaft fillets, smooth transitions and good weld quality are inspected during maintenance.

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Theory of machines studies the relative motion of machine parts (kinematics) and the forces acting on them (kinetics).

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This section covers kinematic pairs and chains, degrees of freedom, linkage mechanisms and their inversions, velocity and acceleration analysis, force analysis and mechanisms with lower pairs.

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Links, Pairs, Chains and Mechanisms • Link (element): rigid (or resistant) body forming part of a machine.

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4. • Kinematic pair: two links in contact permitting constrained relative motion. • Lower pair — SURFACE contact: turning/revolute (R), sliding/prismatic (P), screw (H), cylindrical, spherical (ball & socket), planar.

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Higher pair — POINT or LINE contact: cam & follower, gear teeth, ball bearing, wheel on rail. • Constrained motion: completely constrained (piston in cylinder), incompletely, successfully constrained (foot-step bearing, cam with spring).

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Pairs may be self-closed (form closure) or force-closed (spring/gravity). • Kinematic chain: links joined so that motion of one gives definite motion of others.

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Mechanism: a chain with one link fixed.

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Machine: a mechanism that transmits force/work.

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Structure: no relative motion. • Inversion: obtaining different mechanisms by fixing different links of the same chain; number of inversions = number of links.

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Degree of Freedom (Mobility) • Kutzbach (Gruebler) criterion for planar mechanisms:

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F = 3(n − 1) − 2j − h (n = links, j = lower pairs/binary joints, h = higher pairs). • Gruebler's equation for constrained motion (F = 1, no higher pairs):

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3n − 2j − 4 = 0. • F = 1 → constrained mechanism;

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F > 1 → unconstrained (needs more inputs);

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F < 0 → pre-loaded/statically indeterminate structure. • Four-bar chain: n = 4, j = 4 → F = 3(3) − 8 = 1.

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A compound joint of k links counts as (k − 1) binary joints.

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A roller on a follower adds a redundant DOF.

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Linkage Mechanisms and Inversions • Grashof's law (four-bar): if s + l ≤ p + q (s = shortest, l = longest), at least one link can make a full revolution relative to the others.

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Fix the link adjacent to shortest → crank-rocker; fix shortest → double-crank (drag link); fix link opposite shortest → double-rocker.

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Chain Inversion (link fixed) Application Four-bar (4R) Crank-rocker (crank & lever) Beam engine, pedal-operated grinder Double crank Coupling rod of locomotive (parallelogram), drag link Double rocker Watt's indicator mechanism, Ackermann steering linkage Single slider-crank (3R-1P) 1st: cylinder/frame fixed Reciprocating engine, reciprocating compressor 2nd: crank fixed Whitworth quick-return mechanism, rotary (Gnome) engine 3rd: connecting rod fixed Crank and slotted lever quick-return (shaper), oscillating cylinder engine 4th: slider fixed Hand pump (pendulum pump / bull engine) Double slider-crank (2R-2P) Frame with two slots fixed Elliptical trammel (draws ellipses) Chain Inversion (link fixed) Application One slider fixed Scotch yoke (converts rotation to SHM) Link joining sliders fixed Oldham's coupling (connects parallel shafts with small offset) Velocity and Acceleration in Mechanisms • Relative velocity method: velocity of B relative to A on a rigid link is perpendicular to AB: vBA = ω·AB. • Instantaneous centre (I-centre): point about which a link has pure rotation at an instant; v = ω × distance from I-centre.

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Number of I-centres N = n(n − 1)/2.

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Kennedy's (three-centres-in-line) theorem: the three I-centres of three bodies lie on a straight line.

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A four-bar has 6 I-centres; slider's I-centre with frame lies at infinity perpendicular to the path. • Rubbing velocity at a pin joint = (ω1 ± ω2)·r (− if same direction, + if opposite). • Acceleration of a point on a rotating link: centripetal (radial) = ω²r = v²/r (towards centre) + tangential = αr. • Coriolis component = 2ωv — appears when a point SLIDES along a ROTATING link (crank & slotted lever, Whitworth, oscillating cylinder engine).

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Direction: v rotated 90° in the sense of ω. • Slider-crank (n = l/r): piston velocity v ≈ ωr(sin θ + sin 2θ/2n); acceleration a ≈ ω²r(cos θ + cos 2θ/n).

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Klein's construction gives velocity and acceleration diagrams graphically.

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Kinetics and Force in Mechanisms • Kinetics: relation between forces and motion (F = ma, T = Iα).

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Inertia force = −ma, inertia torque = −Iα (D'Alembert). • Engine force analysis: piston effort FP = gas force ± inertia force of reciprocating parts; thrust in connecting rod FQ = FP/cos φ; crank effort FT = FQ sin(θ + φ); turning moment T = FT·r; side thrust on cylinder wall FN = FP tan φ. • Equivalent dynamical system: a link replaced by two point masses having same total mass, same CG and same MOI (used for connecting rod). • Static force analysis uses equilibrium of each link (two- and three-force members); virtual work principle also applies.

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Mechanisms with Lower Pairs Mechanism Purpose / key point Pantograph Copies, enlarges or reduces drawings (parallelogram linkage) Exact straight-line mechanisms Peaucellier, Hart, Scott-Russell Approximate straight-line mechanisms Watt, Grasshopper, Tchebicheff, Robert Engine indicators Watt, Richard, Thompson, Crosby — record P-V diagram Davis steering gear Sliding pairs; theoretically exact at all positions; high friction and wear — rarely used Ackermann steering gear Turning pairs only; exact only in 3 positions; widely used in cars Hooke's (universal) joint Connects two INTERSECTING shafts at an angle α (propeller shaft); speed of driven shaft fluctuates; double Hooke's joint gives uniform velocity Oldham's coupling Connects two PARALLEL shafts with small lateral offset • Correct steering condition: cot φ − cot θ = c/b (c = distance between front stub-axle pivots, b = wheelbase; φ = outer wheel angle, θ = inner wheel angle). • Hooke's joint: ωB/ωA = cos α/(1 − cos²θ sin²α); maximum ratio 1/cos α, minimum cos α.

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Driven shaft speed equals driving shaft speed four times per revolution.

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Industrial Applications — Mechanisms in Production Machinery • Degrees of freedom (Grubler/Kutzbach) and linkage types describe the working mechanisms of production machines: quick-return mechanism of a shaper (crank-slotted lever), slider-crank of presses, compressors and injection-moulding clamps, toggle mechanism (huge mechanical advantage) in presses, riveting and clamping devices, and four-bar linkages in packaging and labelling machines. • Velocity and acceleration analysis gives the cycle time of automatic machines and the inertia forces on links — the basis of machine cycle-time estimation used in capacity planning and line balancing (Chapter 6). • Robot arms and pick-and-place units are open kinematic chains: degrees of freedom, work envelope, repeatability and payload are their selection parameters in automation projects. • Lower-pair mechanisms (pantograph for engraving and copying, straight-line mechanisms, Oldham coupling for parallel offset shafts, Hooke's joint for angular drives) appear in plant machinery, and screw pairs in jacks, presses and machine-tool feeds.

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This section covers the dynamics of common machine elements: gyroscopic couple, governors, flywheels, balancing of masses, cams and followers (with standard follower motions), belt, rope and chain drives, and gears and gear trains.

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Gyroscopic Couple and Precessional Motion • When a body spinning about one axis (spin ω) is turned about a perpendicular axis (precession ωp), a gyroscopic couple C = I ω ωp is required, acting about the third mutually perpendicular axis.

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The reactive couple acts on the frame/bearings. • Precessional motion: rotation of the spin axis about another axis. • Aeroplane: propeller spin + turning (left/right) → nose rises or dips. • Ship: steering → pitching effect (bow rises/falls); pitching → steering (yaw) effect; rolling → NO gyroscopic effect (roll axis parallel to spin axis). • Two-wheeler / four-wheeler on a curve: gyroscopic couple of wheels and engine adds to the centrifugal overturning couple; rider leans inward to balance.

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Governors • A governor controls the MEAN speed of an engine over a period by regulating fuel supply according to load.

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(A flywheel controls speed fluctuation WITHIN a cycle — it does not control mean speed.) • Types: centrifugal — pendulum type (Watt), gravity/dead-weight loaded (Porter, Proell), spring-loaded (Hartnell, Hartung, Wilson-Hartnell, Pickering); inertia governors (respond to rate of change of speed). • Watt governor: height h = g/ω² ≈ 895/N² m — suitable only at low speeds.

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Porter: h = [(m + M)/m]·g/ω² (equal arms, central load M).

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Proell is more sensitive than Porter of the same size.

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Hartnell: spring-controlled, compact, high speed. • Sensitiveness: small range of speed between full-load and no-load positions (sensitiveness = 2(N1 − N2)/(N1 + N2) — smaller value = more sensitive). • Stability: radius of rotation increases as speed increases (unique radius for each speed). • Isochronous: same equilibrium speed for all radii — infinitely sensitive but practically unstable (causes hunting). • Hunting: continuous fluctuation of speed above and below mean — due to excessive sensitivity. • Effort = mean force on sleeve for a given % speed change; power = effort × sleeve lift.

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Coefficient of insensitiveness due to friction = (N′ − N″)/N.

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Flywheel • Stores energy when supply > demand and releases it when demand > supply — reduces fluctuation of speed within a cycle.

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Needed in IC engines and punching/shearing presses. • Turning moment diagram (crank-effort diagram): torque vs crank angle; area = work done.

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Multi-cylinder engines have smoother diagrams → smaller flywheel. • Coefficient of fluctuation of speed Cs = (ωmax − ωmin)/ωmean.

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Coefficient of fluctuation of energy CE = ΔE / work done per cycle. • Maximum fluctuation of energy ΔE = I ω² Cs = m k² ω² Cs = 2E Cs (E = mean KE).

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Rim hoop stress σ = ρv² limits rim speed.

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Balancing of Masses • Unbalanced rotating masses produce centrifugal force mω²r → vibration, noise, bearing loads. • Static balancing: Σmr = 0 (CG on axis — no net force; balanced in any angular position).

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Dynamic balancing: Σmr = 0 AND Σmrl = 0 (no net force AND no net couple). • Several masses in different planes need two balancing masses in two planes for complete (dynamic) balancing.

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Static balance does not guarantee dynamic balance. • Reciprocating masses: primary unbalanced force = mω²r cos θ; secondary = mω²r cos 2θ/n (twice crank frequency, n = l/r).

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Cannot be completely balanced by rotating masses — only partial balancing (usually 2/3 to 3/4 of reciprocating mass in locomotives). • Effects of partial balancing in locomotives: hammer blow (vertical force on rails), variation of tractive effort, swaying couple. • Multi-cylinder in-line engines (4-cyl, 6-cyl) can be arranged for good primary balance;

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6-cylinder in-line is fully balanced for primary and secondary forces and couples.

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Cam and Follower • Cams: radial/disc (follower moves ⊥ to cam axis — most common), cylindrical (follower moves ∥ to axis), wedge, spiral, conjugate, globoidal. • Followers by contact: knife-edge (high wear, rarely used), roller (least wear, most common in engines/machines), flat-faced/mushroom (automobile valve trains; no side thrust from pressure angle), spherical.

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By motion: reciprocating or oscillating; by path: radial or offset. • Terms: base circle (smallest circle from cam centre to profile), trace point, prime circle, pitch curve, pressure angle (between normal to pitch curve and direction of follower motion — keep ≤ 30° to avoid jamming), lift/stroke, dwell (follower at rest).

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Follower motion Displaceme nt Velocity / acceleration Remarks Uniform velocity Linear Constant velocity; infinite acceleration at start/end Abrupt; modified with rounded corners; low speeds Simple harmonic motion (SHM) Harmonic (cosine) vmax = πhω/(2θo); amax = π²ω²h/(2θo²) Smooth; finite but sudden change of acceleration at ends Uniform acceleration & retardation Parabolic Constant a = 4hω²/θo²; vmax = 2hω/θo Minimum acceleration for given lift; jerk infinite at transitions Cycloidal Cycloid vmax = 2hω/θo; amax = 2πhω²/θo²; zero acceleration at start and end No abrupt change in acceleration — best for HIGH-SPEED cams Belt, Rope and Chain Drives • Velocity ratio N2/N1 = d1/d2; with belt thickness t:

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(d1 + t)/(d2 + t); with slip s%: × (1 − s/100).

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Creep: relative movement due to unequal stretching of tight and slack sides. • Open belt: shafts rotate in SAME direction;

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L = π(r1 + r2) + 2x + (r1 − r2)²/x.

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L = π(r1 + r2) + 2x + (r1 + r2)²/x — length depends only on (r1 + r2). • Ratio of tensions: flat belt T1/T2 = eμθ;

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V-belt/rope T1/T2 = eμθ cosec β (β = half groove angle) — wedge action gives higher grip. θ = angle of contact on the SMALLER pulley governs. • Power P = (T1 − T2)v.

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Centrifugal tension Tc = mv² (m = mass per metre).

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Maximum power when Tc = Tmax/3, i.e., v = √(Tmax/3m).

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Initial tension T0 = (T1 + T2)/2 (+ Tc). • V-belts: compact, short centre distances, groove angle 30–40°.

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Ropes: long distances (mines, cranes), multiple grooves. • Chain drives:

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POSITIVE drive (no slip), roller chains on sprockets, used for medium distances (bicycles, motorcycles, conveyors).

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Polygonal (chordal) action causes speed fluctuation — reduced by more sprocket teeth.

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Gears and Gear Trains Shaft arrangement Gear type Parallel Spur (straight teeth, noisy at high speed), helical (inclined teeth, quiet, axial thrust), double-helical/herringbone (no net thrust), rack & pinion (rotary ↔ linear) Intersecting Bevel (straight, spiral); mitre gears = equal bevel gears at 90° Non-parallel, non-intersecting Worm & worm wheel (high reduction, often self-locking), crossed/spiral helical, hypoid (automobile differentials) • Terms: pitch circle; module m = d/T (mm); circular pitch p = πm = πd/T; diametral pitch P = T/d; addendum = 1m; dedendum = 1.25m; clearance = 0.25m; standard pressure angle 20° (older 14½°). • Law of gearing: the common normal at the point of contact must always pass through the fixed pitch point → constant velocity ratio.

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Velocity ratio N1/N2 = T2/T1 = d2/d1. • Involute profile: constant pressure angle, easy to cut (single curve), velocity ratio unaffected by small centre-distance changes — universally used.

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Cycloidal profile: no interference, less wear, stronger — used in watches/instruments. • Interference: tip of one tooth digs into the non-involute flank of the other — avoided by minimum number of pinion teeth (≈ 17–18 for 20° full-depth), stub teeth, larger pressure angle, undercutting. • Contact ratio (arc of action/circular pitch) should be > 1 (typically 1.2–1.8) for continuous transmission.

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Backlash = clearance between mating teeth on pitch circle.

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Gear train Feature Use Simple One gear per shaft; idlers change only DIRECTION, not speed ratio Short-distance transmission Compound Two or more gears on intermediate shafts; speed ratio = product of (driven teeth/driver teeth) Large reductions: lathe gearbox Reverted First and last gears on the SAME axis (r1 + r2 = r3 + r4) Clocks (hour & minute hands), lathe back gear Epicyclic (planetary) Axis of some gears (planets) moves around a sun gear; very high ratio in compact space Automobile differential, automatic gearboxes, hoists • Train value = speed of last/speed of first = 1/velocity ratio.

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Epicyclic trains analysed by tabular or algebraic method. • Differential of an automobile (epicyclic bevel train) allows the outer rear wheel to turn faster than the inner wheel on a curve.

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Industrial Applications — Power Transmission and Machine Dynamics in Plants • Belt, rope and chain drives are everywhere in plants: flat and V-belts (cheap, absorb shock, slip protects the machine), timing belts and chains (positive drive for conveyors and hoists), with speed ratio N1/N2 = D2/D1 and belt power P = (T1 − T2)v.

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Maintenance items: tension, alignment, guarding and lubrication;

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V-belt slip wastes energy, so synchronous belts are chosen in energy-saving projects. • Gears and gear trains: speed reducers (helical, worm, planetary) for conveyors, mixers, agitators and cranes; selection by ratio, torque, service factor and efficiency (worm drives ≈ 50–90%, helical ≈ 96–98% per stage — a real energy issue in continuous duty). • Flywheel smooths fluctuating loads in punching, shearing and forging presses: energy per stroke E ≈ ½I(ω1² − ω2²) with coefficient of fluctuation of speed Cs; it allows a much smaller motor to do heavy intermittent work — a classic capacity/economics calculation. • Governors control the speed of engines and generators under varying industrial loads; balancing of rotating masses and vibration monitoring keep fans, blowers, pumps and grinders running smoothly (unbalance is a leading cause of bearing failure, detected in condition-based maintenance, Chapter 8). • Cams and followers drive automatic machines — packaging, filling, textile and machine-tool automats — where a cam profile fixes the motion (uniform, SHM, uniform acceleration, cycloidal) and hence cycle time and smoothness.

1

Mechanics of solids extends strength of materials to general stress states and structures: stress at a point, principal stresses and theories of failure, statically determinate and indeterminate structures, columns, thin and thick cylinders, and torsion of non-circular sections.

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Analysis of a Deformable Body:

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Stress at a Point • Stress at a point is described by the stress tensor:

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9 components (σx, σy, σz, τxy, …) of which 6 are independent (τxy = τyx — complementary shear stresses). • Analysis of a deformable body requires three sets of equations: equilibrium, compatibility (continuity of deformation) and constitutive (stress-strain) relations (Hooke's law). • Plane stress: normal stress on a plane inclined at θ: σθ = (σx + σy)/2 + (σx − σy)/2 cos 2θ + τxy sin 2θ.

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Principal Stresses and Mohr's Circle • Principal planes: planes with ZERO shear stress; normal stresses on them are principal stresses (max and min normal stress). • σ1,2 = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]; location: tan 2θp = 2τxy/(σx − σy).

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The two principal planes are 90° apart. • Maximum shear stress τmax = (σ1 − σ2)/2 on planes at 45° to principal planes (these planes carry normal stress (σ1 + σ2)/2). • Mohr's circle: centre at ((σx + σy)/2, 0), radius = τmax.

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An angle θ on the element = 2θ on the circle. • Special cases: uniaxial tension σ → τmax = σ/2 at 45°; pure shear τ → principal stresses +τ and −τ at 45° (why brittle shafts in torsion fail on 45° helical planes); equal biaxial σx = σy → Mohr's circle is a point, no in-plane shear.

8

Theories of Failure Theory Failure when Suitable for Maximum principal stress (Rankine) σ1 = σyt (or σut) Brittle materials (cast iron) Maximum shear stress (Tresca / Guest) τmax = σy/2 → shear yield τy = 0.5σy Ductile materials — conservative, simple (hexagon) Maximum principal strain (St.

9

Venant) εmax = σy/E Rarely used Total strain energy (Haigh) Strain energy per volume = σy²/2E Ductile (less accurate) Maximum distortion energy (von Mises-Hencky) √(σ1² − σ1σ2 + σ2²) = σy → τy = 0.577σy Ductile materials — most accurate (ellipse) Determinate and Indeterminate Structures • Statically determinate: all reactions and internal forces can be found from equilibrium equations alone (plane:

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Examples: simply supported beam, cantilever, overhanging beam, perfect truss, three-hinged arch. • Statically indeterminate: unknowns exceed equilibrium equations; need compatibility (deformation) conditions.

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Degree of static indeterminacy = unknowns − equations (plane beams: r − 3).

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Examples: fixed-fixed beam (3°;

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2° for vertical loads only), propped cantilever (1°), continuous beam, redundant truss, two-hinged arch. • Methods: consistent deformation, Clapeyron's three-moment theorem (continuous beams), slope-deflection, moment distribution (Hardy Cross), Castigliano's theorem, column analogy, flexibility and stiffness (matrix) methods. • Features of indeterminate structures: smaller moments and deflections, redundancy (alternate load paths), but temperature changes, support settlement and fabrication errors induce stresses. • Useful results: fixed beam, central load — fixed-end moments WL/8;

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UDL — wL²/12 at ends, wL²/24 at centre.

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Propped cantilever with UDL — prop reaction 3wL/8.

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Columns (Buckling) • Euler's critical load Pcr = π²EI/Le² (long columns;

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I = least moment of inertia). • Effective length Le: both ends hinged L; one fixed, other free 2L; both fixed L/2; one fixed, other hinged L/√2.

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So Pcr ratio (free : hinged : fixed-hinged : fixed) = 1/4 :

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4. • Slenderness ratio λ = Le/k (k = least radius of gyration).

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Euler valid for long columns (λ > ≈ 80–100 for mild steel).

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Rankine-Gordon formula for intermediate columns:

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Thin-Walled and Thick-Walled Cylinders • Thin cylinder: t/d < 1/20 (d/t > 20); stresses assumed uniform through the thickness; radial stress neglected. • Hoop (circumferential) stress σh = pd/(2t); longitudinal stress σl = pd/(4t) → σh = 2σl.

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Hence a cylinder tends to burst along a LONGITUDINAL seam; longitudinal joints must be stronger.

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In-plane τmax = pd/(8t). • With joint efficiencies: σh = pd/(2tηl) (longitudinal joint), σl = pd/(4tηc) (circumferential joint). • Thin sphere: σ = pd/(4t) in all directions — a sphere needs half the thickness of a cylinder of same diameter and pressure. • Volumetric strain: cylinder δV/V = (pd/4tE)(5 − 4μ); sphere δV/V = (3pd/4tE)(1 − μ). • Thick cylinder (Lamé's equations): σr = b/r² − a, σh = b/r² + a.

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Hoop stress is maximum at the INNER surface; radial stress = −p at inner surface, 0 at outer surface.

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Stress distribution is non-uniform. • Improving thick cylinders: compound cylinders (shrink fitting one tube over another → initial compressive hoop stress at bore) and autofrettage (pre-yielding the inner layers).

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Torsion of Non-Circular Sections • In non-circular sections plane cross-sections WARP (do not remain plane) — Saint-Venant's theory.

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The Prandtl membrane (soap-film) analogy visualises shear stress (slope of membrane ∝ stress). • Rectangular section b × t (b > t): τmax = T/(αbt²) occurs at the middle of the LONGER side; shear stress at the corners is ZERO.

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Thin rectangle (b/t large): α = 1/3 → τmax = 3T/(bt²), twist θ = 3TL/(Gbt³). • Elliptical section: τmax at ends of the MINOR axis.

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Equilateral triangle: max at middle of sides. • Thin-walled CLOSED tubes (Bredt-Batho): shear flow q = τt = T/(2A) is constant around the wall (A = area enclosed by median line); τ is maximum where t is minimum. • Open thin sections (angle, channel, I-section, slit tube) are very WEAK and flexible in torsion compared with closed sections — slitting a tube longitudinally reduces its torsional stiffness drastically.

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Industrial Applications — Pressure Vessels, Piping and Plant Structures • Thin cylinders (t < D/20): σh = pD/2t, σl = pD/4t — used for air receivers, steam drums, LPG and process vessels, pipelines, silos and tanks; thick cylinders (Lamé's equations) for hydraulic cylinders, high-pressure piping and extrusion dies. • Pressure equipment is governed by codes and statutory inspection (design, material certification, welding qualification, radiography, hydrostatic test at ≈ 1.25–1.5 times design pressure, safety valves, periodic inspection) — a major plant-safety responsibility (Chapter 9). • Statically indeterminate structures (continuous beams, frames, bolted flange joints, redundant supports) are analysed by compatibility methods or FEA — typical in plant structures, piping supports and foundations, where thermal expansion also induces stresses. • Torsion of non-circular sections matters in open channel/angle members of frames (very weak in torsion) versus closed box sections (very stiff) — used in machine bases, conveyor frames and crane girders. • Design of plant equipment always balances safety, weight and cost: standard plate thicknesses, standard pipe schedules and fabrication simplicity are chosen to keep manufacturing cost low.