COREN Mechanical cheat sheet

Stress: σ = F/A (normal), τ = V/A (average shear); strain ε = ΔL/L; Hooke's law σ = Eε holds below proportional limit.

Bending: σ = Mc/I with I = bd³/12 for a rectangle; maximum bending stress occurs at the greatest distance c from the neutral axis.

Torsion of a solid round shaft: τ_max = Tc/J with J = πd⁴/32; angle of twist θ = TL/(GJ).

Thermal expansion: ΔL = αLΔT; restrained expansion develops σ = EαΔT (fully constrained member).

Fatigue: endurance limit applies to steels; S-N curves are plotted on log-log axes; stress concentration Kt reduces fatigue life but not static ductile strength.

Bolted joints: preload Fi ≈ 0.75·Sp·At for reused connections; joint separation occurs when external load exceeds preload divided by the joint stiffness fraction.

Bearings: rated life L10 = (C/P)^p million revolutions, p = 3 for ball bearings and 10/3 for roller bearings.

Tolerance stack-up: worst-case = sum of tolerances; statistical (RSS) = square root of the sum of squared tolerances.

Welding symbols: arrow side vs other side of the reference line; the tail carries process or specification references.

First law for steady flow: q − w = Δh + ΔKE + ΔPE per unit mass; for a nozzle or diffuser w = 0 and q ≈ 0.

Carnot efficiency: η = 1 − T_L/T_H with absolute temperatures; no real cycle between the same reservoirs exceeds it.

Otto cycle efficiency: η = 1 − r^(1−k); Diesel cycle adds the cut-off ratio term and is lower than Otto at equal compression ratio.

Heat conduction: q = kAΔT/L for a plane wall; for a composite wall, resistances R = L/(kA) add in series.

Convection: q = hAΔT; the Nusselt number Nu = hL/k relates convection to conduction across the same fluid layer.

Radiation: q = εσA(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/(m²·K⁴); radiation dominates at high absolute temperature.

Pipe flow: Reynolds number Re = ρVD/μ; laminar below 2300, transitional to 4000, turbulent above 4000.

Darcy–Weisbach head loss: h_f = f(L/D)(V²/2g); friction factor f from the Moody chart depends on Re and relative roughness ε/D.

Pump systems: operating point is the intersection of the pump curve and the system curve; NPSH available must exceed NPSH required to avoid cavitation.

Kinematics: v = u + at, s = ut + ½at², v² = u² + 2as apply only for constant acceleration in a straight line.

Newton's second law in rotation: ΣM = Iα; for a body about a parallel axis, I = I_cm + md² (parallel-axis theorem).

Work–energy: U = ΔKE = ½m(v₂² − v₁²); conservative forces have potential energy and path-independent work.

Impulse–momentum: FΔt = Δ(mv); for impact, coefficient of restitution e = (v₂ − v₁)/(u₁ − u₂).

Free vibration: natural frequency ω_n = √(k/m); damping ratio ζ = c/(2√(km)); underdamped when ζ < 1.

Forced vibration: resonance occurs near ω = ω_n; transmissibility peaks at resonance and falls at high frequency ratios.

Balancing: static balance requires the center of mass on the axis of rotation; dynamic balance additionally requires zero product of inertia.

Four-bar linkage Grashof condition: s + l ≤ p + q ensures at least one link can fully rotate.

Tolerance: total permissible variation; bilateral tolerance is expressed as ±; allowance is the intentional minimum clearance (or maximum interference) between mating parts.

Hole-basis system: hole is the basic size with the letter H; shaft letters indicate position of the tolerance zone relative to basic size.

Surface roughness Ra is the arithmetic average deviation; Rz is the ten-point height; lower Ra values indicate smoother surfaces and higher cost.

Geometric tolerancing: flatness, straightness, and circularity are form controls and need no datum; position and profile usually reference datums.

GD&T modifiers: MMC (maximum material condition) permits bonus tolerance as the feature departs from MMC; LMC applies the opposite rule.

Casting: shrinkage allowance is added to pattern dimensions; draft angle is required on vertical faces for pattern withdrawal.

Machining: cutting speed V = πDN/1000 (m/min with D in mm); feed rate is expressed per revolution or per tooth.

Sheet metal: bend allowance = θ(R + K·T) with θ in radians; K-factor typically 0.3–0.5 depending on material and radius-to-thickness ratio.

Joining: rivet and bolt hole clearance affects bearing stress; interference fits transmit torque by friction generated by radial pressure.

Factor of safety: n = S_y/σ for ductile materials using yield strength; for brittle materials use ultimate strength and consider stress concentration.

Failure theories: von Mises (distortion energy) for ductile materials; maximum principal stress (Rankine) for brittle materials; Tresca is conservative.

Soderberg, Goodman, and Gerber criteria relate mean and alternating stresses to fatigue limits; Goodman is the common linear design line.

Stress concentration: Kt from geometry charts; for ductile static loading Kt is often ignored, for fatigue use Kf = 1 + q(Kt − 1).

Shaft design: combine bending and torsion using equivalent von Mises stress σ' = √(σ² + 3τ²).

Welded joints: throat thickness = 0.707 × leg size for a fillet weld; shear stress on the throat governs design.

Threaded fasteners: tensile stress area A_t ≈ 0.7854(d − 0.9743/n)²; stripping strength depends on engaged thread length.

Springs: spring rate k = Gd⁴/(8D³N_a); Wahl factor corrects for curvature and direct shear stress in helical springs.

Gears: Lewis bending equation σ = W_t/(F·m·Y); contact stress follows Hertzian theory and limits pitting life.

Measurement uncertainty: total uncertainty combines Type A (statistical) and Type B (instrument) contributions as the root sum of squares.

Calibration traces to national or international standards; a calibration certificate states the measured value, uncertainty, and reference standard.

Gauge R&R: repeatability is variation of one operator on one part; reproducibility is variation between operators; both should be small relative to tolerance.

Control charts: X-bar and R charts monitor mean and variation; a process is in statistical control when points fall within control limits without patterns.

Process capability: Cp = (USL − LSL)/(6σ); Cpk accounts for centering and equals the lesser of (USL − μ)/(3σ) and (μ − LSL)/(3σ).

Sampling: random sampling avoids bias; sample size affects the width of confidence intervals, not the location of the mean.

Dimensional inspection: go/no-go gauges check maximum and minimum material limits; a go gauge must pass every conforming part.

Reliability: for constant failure rate, R(t) = e^(−λt) and MTBF = 1/λ; the bathtub curve has infant mortality, useful life, and wear-out regions.

Design review and verification: V&V distinguishes verification (built right) from validation (right product for the user need).