MEC-00 · laboratory
Shaft torsion, Soderberg, Hertz contact, heat engines and bearings.
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39 governing sheets
θ = T L /(G J) for a circular shaft.
Shafts1/n = σm/Sy + σa/Se, a conservative Goodman variant.
Fatiguea = (¾ F R / E*)^{1/3} for two elastic spheres.
Contactη = 1 − Tc/Th between two thermal reservoirs.
Heat enginesf = 2π² (μ N / P) (r/c) for a lightly loaded journal.
Bearingsσa = σf' (2N)^b in high-cycle fatigue.
Fatigueσ = Wt /(F m Y) treating the tooth as a cantilever.
GearsΔE = I ω² Cs with Cs = (ωmax−ωmin)/ω.
Energy storage1/n = σm/Sy + σa/Se. The conservative mean–amplitude line through yield.
Fatiguea = (3 F R / 4 E*)^{1/3}, p0 = 3F /(2π a²). Elastic spheres.
Contactσa = σf' (2N)^b in high-cycle fatigue.
Fatigue1/n = σm/Sut + σa/Se. Mean stress referred to ultimate, not yield.
Fatigueθ = T L /(G J) with J = π d⁴/32 for a circular shaft.
ShaftsL10 = (C/P)^p million revolutions. p = 3 ball, 10/3 roller.
Bearingsη = Wnet / Qin = 1 − Qout/Qin. First-law bound of a cyclic engine.
Heat enginesη = (wt − wp)/qin with wt = h3−h4, wp = h2−h1, qin = h3−h2.
Heat enginesΔE = I ω² Cs with Cs = (ωmax−ωmin)/ω. Stored kinetic buffer.
Energyω2/ω1 = N1/N2, T2/T1 = η N2/N1. Teeth count sets the reduction.
Gearsωn = √(k/m), fn = ωn/2π. One-mass, one-spring, no damper.
Vibrationωcr = √(g/δst). Whirl when rotation meets the bending natural frequency.
Vibrationσh = p r / t, σℓ = p r /(2 t). Membrane theory, t/r < 0.1.
Pressure vesselsσθ,i = p (ro²+ri²)/(ro²−ri²) at the inner face of a pressurised cylinder.
Pressure vesselsU = P² L³ /(6 EI) for a tip-loaded cantilever. Castigliano δ = ∂U/∂P.
Energy(σa/Se)² + σm/Sut = 1/n. A parabola through Se and Sut.
FatigueD = Σ ni/Ni. Failure is predicted at D = 1 under block loading.
Fatigueδ = ∂U/∂P = P L /(A E) for an axial bar. U = P² L /(2 A E).
EnergyLightly loaded long journal.
MachinesHelical-spring curvature and shear.
MachinesSpur-tooth bending with form factor Y.
MachinesUniform-pressure disk clutch.
MachinesEuler capstan formula.
MachinesRigid-body rotational kinetic energy.
MachinesInstantaneous mechanical power.
MachinesSecond-moment shift to a parallel axis.
MachinesFully constrained thermal expansion.
MachinesLateral contraction in uniaxial tension.
MachinesEmpirical strut between Euler and crush.
MachinesSteady rotor precession.
Machinesτ = T r / J, θ = T L / (G J). Elastic circular shafts.
Strength