Engineer General Machinery cheat sheet

Static equilibrium requires ΣF = 0 and ΣM = 0 about any point; for a 2D body this gives three independent equations, so a free-body diagram can solve at most three unknowns directly.

Centroid of a composite area: x̄ = Σ(Aᵢx̄ᵢ)/ΣAᵢ; the centroid of a semicircle lies at 4r/3π from its diameter, not at the geometric center.

Parallel-axis theorem: I = Ī + Ad², where Ī is about the centroidal axis; moment of inertia has units of length⁴ and is always smallest about the centroid.

Kinematics with constant acceleration: v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as; these hold only when acceleration is constant, not for general motion.

Newton's second law in rotation: ΣM = Iα; for a rolling body without slip, v = ωr and a = αr, so friction does no work.

Work-energy: U₁₋₂ = ΔKE = ½m(v₂² − v₁²); for a conservative system, KE + PE = constant, with PE = mgh or ½kx².

Impulse-momentum: ∫F dt = Δ(mv); coefficient of restitution e = (v₂ − v₁)/(u₁ − u₂), with e = 1 elastic and e = 0 perfectly plastic.

A truss is statically determinate when m + r = 2j; zero-force members are found at unloaded two-member joints and at joints where three members meet with two collinear.

Shaft diameter for torsion: τ = 16T/(πd³); for combined bending and torsion, use the equivalent torque Tₑ = √(M² + T²) for solid shafts under maximum-shear theory.

Soderberg, Goodman, and Gerber criteria relate mean and alternating stresses to yield and endurance limits; Goodman is the straight-line σₐ/Se + σₘ/Sut = 1/n.

Endurance limit Se = kₐk_bk_ck_dk_ek_f·S′e, where S′e ≈ 0.5Sut for steels with Sut ≤ 1400 MPa (200 ksi) and S′e = 700 MPa above that.

Marin surface factor kₐ = a·Sut^b; ground finishes give higher kₐ than machined or hot-rolled surfaces, so surface finish directly raises fatigue life.

For a bolted joint, the stiffness ratio C = k_b/(k_b + k_m) determines the fraction of external load taken by the bolt: F_b = F_i + C·P.

Ball bearing rating life L₁₀ = (C/P)^p million revolutions, with p = 3 for ball bearings and p = 10/3 for roller bearings; C is the basic dynamic load rating.

Key shear stress: τ = 2T/(d·w·L) for a square key of width w and length L on shaft diameter d; keys fail in shear or crushing, whichever is lower.

Gear geometry: module m = d/N, circular pitch p = πm, and the involute system requires the same module and pressure angle (typically 20°) for meshing gears.

Hooke's law: σ = Eε in the elastic range; Poisson's ratio ν = −ε_lateral/ε_axial, typically 0.3 for steel and 0.33 for aluminum.

Thermal stress in a fully constrained bar: σ = −EαΔT; a bar free to expand develops no thermal stress regardless of ΔT.

Principal stresses in 2D: σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]; maximum shear τmax = (σ₁ − σ₂)/2.

Von Mises (distortion energy) equivalent stress: σ′ = √(σx² − σxσy + σy² + 3τxy²); it predicts yielding of ductile metals better than maximum principal stress.

Stress concentration factor Kt multiplies nominal stress; it applies to brittle materials directly but is reduced by notch sensitivity q for ductile fatigue: Kf = 1 + q(Kt − 1).

Paris law da/dN = C(ΔK)^m describes fatigue crack growth; the threshold ΔKth below which no growth occurs and KIC (fracture toughness) set the safe design window.

Hall-Petch relation σy = σ₀ + k/√d: finer grain size raises yield strength; grain refinement strengthens without reducing ductility.

Jominy end-quench test measures hardenability (depth of hardness vs distance); it is not a measure of hardness itself, which is measured by Rockwell or Brinell.

First law for a steady-flow open system: q − w = Δh + ΔKE + ΔPE per unit mass; for a closed system at constant volume, q = Δu.

Carnot efficiency η = 1 − T_L/T_H with absolute temperatures in kelvin; no real cycle between the same reservoirs can exceed it.

Entropy change for an ideal gas: Δs = cp ln(T₂/T₁) − R ln(P₂/P₁); irreversible processes increase total entropy of the universe.

Continuity: ρ₁A₁V₁ = ρ₂A₂V₂; for incompressible flow, Q = AV is constant along a stream tube.

Bernoulli along a streamline (incompressible, inviscid, steady): P/γ + V²/2g + z = constant; add head loss hL for real fluids.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below about 2300 and turbulent above about 4000 in circular pipes.

Darcy-Weisbach head loss hf = f(L/D)(V²/2g); in laminar pipe flow f = 64/Re, independent of roughness.

Fourier's law: q = −kA dT/dx; for a composite plane wall, resistances R = L/(kA) add in series, like electrical resistances.

Newton's law of cooling: q = hA(Ts − T∞); the Biot number Bi = hL/k compares internal conduction resistance with surface convection resistance.

Pump affinity laws: Q ∝ N, H ∝ N², P ∝ N³ for the same impeller diameter; trimming diameter gives Q ∝ D, H ∝ D², P ∝ D³.

Specific speed Ns = N√Q/H^(3/4) classifies impellers: low Ns favors radial (centrifugal), high Ns favors axial flow.

NPSH available must exceed NPSH required to avoid cavitation; NPSHA = (Patm − Pv)/(ρg) + z − hf, with z positive when the pump is below the source.

Best efficiency point (BEP) is where a pump should operate; running far left or right of BEP increases vibration, radial thrust, and recirculation damage.

System curve H = Hstatic + kQ² intersects the pump curve at the operating point; throttling or speed change moves the operating point along the curve.

Compressor work for isentropic compression of an ideal gas: w = cpT₁[(P₂/P₁)^((k−1)/k) − 1]; intercooling between stages reduces total work.

For multi-stage compression, equal pressure ratio per stage minimizes work: r = (Pfinal/Pinitial)^(1/n) for n stages.

Reciprocating compressor volumetric efficiency drops with clearance: ηv = 1 + C − C(P₂/P₁)^(1/n), where C is clearance fraction.

For turbines, stage efficiency and reheat factor matter; the Rankine cycle improves with reheat and regenerative feedwater heating.

Cutting speed V = πDN/1000 (m/min with D in mm); increasing speed raises tool temperature and tool wear, following the Taylor tool-life equation VT^n = C.

In orthogonal machining, shear angle φ relates to rake α and friction angle β by 2φ + β − α = 90° (Merchant's relation).

Casting: risers must freeze after the casting; Chvorinov's rule t = B(V/A)² shows that a high volume-to-surface ratio feeds a low one.

Welding symbol: the arrow side is below the reference line and the other side above; the tail carries process or specification references.

For a fillet weld, throat thickness = 0.707 × leg size; weld capacity depends on throat area, not leg area.

Preheat reduces hydrogen-assisted cold cracking in steels; carbon equivalent CE = C + Mn/6 + (Cr+Mo+V)/5 + (Ni+Cu)/15 above about 0.4 raises cracking risk.

Arc welding current and heat input: HI = (V·I·60)/(1000·v) kJ/mm, where v is travel speed in mm/min; excessive heat input causes distortion and HAZ softening.

GMAW uses a consumable wire electrode with shielding gas; GTAW uses a non-consumable tungsten electrode and requires separate filler, giving cleaner welds.

Powder metallurgy and forging change grain flow; forging aligns grain flow with stress direction, improving fatigue strength over machining from bar stock.

Machine guarding must prevent access to the point of operation; guards are fixed, interlocked, adjustable, or self-adjusting, and must not create new hazards.

Safety distance for presence-sensing devices: D = K·T + C, where K ≈ 1.6 m/s (hand speed) and T is total stopping time including response delay.

Lockout/tagout requires isolating all energy sources, verifying zero energy, and applying a personal lock; each worker applies their own lock.

ASME BPVC Section VIII governs pressure vessel design; Section IX covers welding procedure and performance qualification (WPS/PQR/WPQ).

ASME B31.3 covers process piping; B31.1 covers power piping; the applicable code is chosen by service, not by convenience.

Non-destructive testing methods: PT finds surface-breaking defects, MT finds surface and near-surface in ferromagnetic parts, UT and RT find internal defects.

Radiographic sensitivity is measured by IQI (penetrameter) wire or hole size; ultrasonic testing uses calibration blocks and detects planar flaws better than RT.

Time value of money: F = P(1 + i)^n; present worth P = F/(1 + i)^n; for annuities use the capital recovery factor A = P·i(1+i)^n/[(1+i)^n − 1].

Net present value decision rule: accept independent projects with NPV > 0; for mutually exclusive alternatives choose the highest NPV at a common analysis period.

Internal rate of return is the i that makes NPV = 0; IRR can give multiple roots when cash flows change sign more than once, so use NPV instead.

Depreciation: straight-line D = (B − S)/n; MACRS uses declining-balance with a half-year convention and ignores salvage value.

Critical path method: the critical path has zero total float; crashing shortens the project at the least cost per unit time along the critical path.

Earned value: CV = EV − AC, SV = EV − PV, CPI = EV/AC, SPI = EV/PV; CPI below 1 indicates cost overrun.

Professional ethics: engineers must hold paramount public safety, health, and welfare; they sign and seal only work they are competent in and disclose conflicts of interest.

Bidding ethics: the engineer may not accept compensation from both owner and contractor for the same project without full disclosure and consent of all parties.

Static equilibrium requires ΣF = 0 and ΣM = 0 about any point; for a 2D body this gives three independent equations, so a free-body diagram can solve at most three unknowns directly.

Centroid of a composite area: x̄ = Σ(Aᵢx̄ᵢ)/ΣAᵢ; the centroid of a semicircle lies at 4r/3π from its diameter, not at the geometric center.

Parallel-axis theorem: I = Ī + Ad², where Ī is about the centroidal axis; moment of inertia has units of length⁴ and is always smallest about the centroid.

Kinematics with constant acceleration: v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as; these hold only when acceleration is constant, not for general motion.

Newton's second law in rotation: ΣM = Iα; for a rolling body without slip, v = ωr and a = αr, so friction does no work.

Work-energy: U₁₋₂ = ΔKE = ½m(v₂² − v₁²); for a conservative system, KE + PE = constant, with PE = mgh or ½kx².

Impulse-momentum: ∫F dt = Δ(mv); coefficient of restitution e = (v₂ − v₁)/(u₁ − u₂), with e = 1 elastic and e = 0 perfectly plastic.

A truss is statically determinate when m + r = 2j; zero-force members are found at unloaded two-member joints and at joints where three members meet with two collinear.

Shaft diameter for torsion: τ = 16T/(πd³); for combined bending and torsion, use the equivalent torque Tₑ = √(M² + T²) for solid shafts under maximum-shear theory.

Soderberg, Goodman, and Gerber criteria relate mean and alternating stresses to yield and endurance limits; Goodman is the straight-line σₐ/Se + σₘ/Sut = 1/n.

Endurance limit Se = kₐk_bk_ck_dk_ek_f·S′e, where S′e ≈ 0.5Sut for steels with Sut ≤ 1400 MPa (200 ksi) and S′e = 700 MPa above that.

Marin surface factor kₐ = a·Sut^b; ground finishes give higher kₐ than machined or hot-rolled surfaces, so surface finish directly raises fatigue life.

For a bolted joint, the stiffness ratio C = k_b/(k_b + k_m) determines the fraction of external load taken by the bolt: F_b = F_i + C·P.

Ball bearing rating life L₁₀ = (C/P)^p million revolutions, with p = 3 for ball bearings and p = 10/3 for roller bearings; C is the basic dynamic load rating.

Key shear stress: τ = 2T/(d·w·L) for a square key of width w and length L on shaft diameter d; keys fail in shear or crushing, whichever is lower.

Gear geometry: module m = d/N, circular pitch p = πm, and the involute system requires the same module and pressure angle (typically 20°) for meshing gears.

Hooke's law: σ = Eε in the elastic range; Poisson's ratio ν = −ε_lateral/ε_axial, typically 0.3 for steel and 0.33 for aluminum.

Thermal stress in a fully constrained bar: σ = −EαΔT; a bar free to expand develops no thermal stress regardless of ΔT.

Principal stresses in 2D: σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]; maximum shear τmax = (σ₁ − σ₂)/2.

Von Mises (distortion energy) equivalent stress: σ′ = √(σx² − σxσy + σy² + 3τxy²); it predicts yielding of ductile metals better than maximum principal stress.

Stress concentration factor Kt multiplies nominal stress; it applies to brittle materials directly but is reduced by notch sensitivity q for ductile fatigue: Kf = 1 + q(Kt − 1).

Paris law da/dN = C(ΔK)^m describes fatigue crack growth; the threshold ΔKth below which no growth occurs and KIC (fracture toughness) set the safe design window.

Hall-Petch relation σy = σ₀ + k/√d: finer grain size raises yield strength; grain refinement strengthens without reducing ductility.

Jominy end-quench test measures hardenability (depth of hardness vs distance); it is not a measure of hardness itself, which is measured by Rockwell or Brinell.

First law for a steady-flow open system: q − w = Δh + ΔKE + ΔPE per unit mass; for a closed system at constant volume, q = Δu.

Carnot efficiency η = 1 − T_L/T_H with absolute temperatures in kelvin; no real cycle between the same reservoirs can exceed it.

Entropy change for an ideal gas: Δs = cp ln(T₂/T₁) − R ln(P₂/P₁); irreversible processes increase total entropy of the universe.

Continuity: ρ₁A₁V₁ = ρ₂A₂V₂; for incompressible flow, Q = AV is constant along a stream tube.

Bernoulli along a streamline (incompressible, inviscid, steady): P/γ + V²/2g + z = constant; add head loss hL for real fluids.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below about 2300 and turbulent above about 4000 in circular pipes.

Darcy-Weisbach head loss hf = f(L/D)(V²/2g); in laminar pipe flow f = 64/Re, independent of roughness.

Fourier's law: q = −kA dT/dx; for a composite plane wall, resistances R = L/(kA) add in series, like electrical resistances.

Newton's law of cooling: q = hA(Ts − T∞); the Biot number Bi = hL/k compares internal conduction resistance with surface convection resistance.

Pump affinity laws: Q ∝ N, H ∝ N², P ∝ N³ for the same impeller diameter; trimming diameter gives Q ∝ D, H ∝ D², P ∝ D³.

Specific speed Ns = N√Q/H^(3/4) classifies impellers: low Ns favors radial (centrifugal), high Ns favors axial flow.

NPSH available must exceed NPSH required to avoid cavitation; NPSHA = (Patm − Pv)/(ρg) + z − hf, with z positive when the pump is below the source.

Best efficiency point (BEP) is where a pump should operate; running far left or right of BEP increases vibration, radial thrust, and recirculation damage.

System curve H = Hstatic + kQ² intersects the pump curve at the operating point; throttling or speed change moves the operating point along the curve.

Compressor work for isentropic compression of an ideal gas: w = cpT₁[(P₂/P₁)^((k−1)/k) − 1]; intercooling between stages reduces total work.

For multi-stage compression, equal pressure ratio per stage minimizes work: r = (Pfinal/Pinitial)^(1/n) for n stages.

Reciprocating compressor volumetric efficiency drops with clearance: ηv = 1 + C − C(P₂/P₁)^(1/n), where C is clearance fraction.

For turbines, stage efficiency and reheat factor matter; the Rankine cycle improves with reheat and regenerative feedwater heating.

Cutting speed V = πDN/1000 (m/min with D in mm); increasing speed raises tool temperature and tool wear, following the Taylor tool-life equation VT^n = C.

In orthogonal machining, shear angle φ relates to rake α and friction angle β by 2φ + β − α = 90° (Merchant's relation).

Casting: risers must freeze after the casting; Chvorinov's rule t = B(V/A)² shows that a high volume-to-surface ratio feeds a low one.

Welding symbol: the arrow side is below the reference line and the other side above; the tail carries process or specification references.

For a fillet weld, throat thickness = 0.707 × leg size; weld capacity depends on throat area, not leg area.

Preheat reduces hydrogen-assisted cold cracking in steels; carbon equivalent CE = C + Mn/6 + (Cr+Mo+V)/5 + (Ni+Cu)/15 above about 0.4 raises cracking risk.

Arc welding current and heat input: HI = (V·I·60)/(1000·v) kJ/mm, where v is travel speed in mm/min; excessive heat input causes distortion and HAZ softening.

GMAW uses a consumable wire electrode with shielding gas; GTAW uses a non-consumable tungsten electrode and requires separate filler, giving cleaner welds.

Powder metallurgy and forging change grain flow; forging aligns grain flow with stress direction, improving fatigue strength over machining from bar stock.

Machine guarding must prevent access to the point of operation; guards are fixed, interlocked, adjustable, or self-adjusting, and must not create new hazards.

Safety distance for presence-sensing devices: D = K·T + C, where K ≈ 1.6 m/s (hand speed) and T is total stopping time including response delay.

Lockout/tagout requires isolating all energy sources, verifying zero energy, and applying a personal lock; each worker applies their own lock.

ASME BPVC Section VIII governs pressure vessel design; Section IX covers welding procedure and performance qualification (WPS/PQR/WPQ).

ASME B31.3 covers process piping; B31.1 covers power piping; the applicable code is chosen by service, not by convenience.

Non-destructive testing methods: PT finds surface-breaking defects, MT finds surface and near-surface in ferromagnetic parts, UT and RT find internal defects.

Radiographic sensitivity is measured by IQI (penetrameter) wire or hole size; ultrasonic testing uses calibration blocks and detects planar flaws better than RT.

Time value of money: F = P(1 + i)^n; present worth P = F/(1 + i)^n; for annuities use the capital recovery factor A = P·i(1+i)^n/[(1+i)^n − 1].

Net present value decision rule: accept independent projects with NPV > 0; for mutually exclusive alternatives choose the highest NPV at a common analysis period.

Internal rate of return is the i that makes NPV = 0; IRR can give multiple roots when cash flows change sign more than once, so use NPV instead.

Depreciation: straight-line D = (B − S)/n; MACRS uses declining-balance with a half-year convention and ignores salvage value.

Critical path method: the critical path has zero total float; crashing shortens the project at the least cost per unit time along the critical path.

Earned value: CV = EV − AC, SV = EV − PV, CPI = EV/AC, SPI = EV/PV; CPI below 1 indicates cost overrun.

Professional ethics: engineers must hold paramount public safety, health, and welfare; they sign and seal only work they are competent in and disclose conflicts of interest.

Bidding ethics: the engineer may not accept compensation from both owner and contractor for the same project without full disclosure and consent of all parties.