PEC EPE Mechanical cheat sheet

Euler identity: e^(iθ) = cos θ + i sin θ; a sinusoid A cos(ωt + φ) can be represented as Re[A e^(i(ωt+φ))].

Laplace derivative: L{f′(t)} = sF(s) − f(0⁺); L{f″(t)} = s²F(s) − s f(0⁺) − f′(0⁺).

Linear system Ax = b: no solution if rank(A) < rank([A|b]); unique if rank(A) = rank([A|b]) = n; infinitely many if equal but less than n.

Newton-Raphson: x_(k+1) = x_k − f(x_k)/f′(x_k); quadratic convergence near a simple root.

Taylor expansion about a: f(x) = Σ f^(n)(a)(x−a)^n/n!, with remainder f^(n+1)(ξ)(x−a)^(n+1)/(n+1)!.

Mean value theorem: if f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) satisfies f′(c) = [f(b)−f(a)]/(b−a).

Normal standardization: Z = (X−μ)/σ; a two-sided 95% confidence interval for a large-sample mean is x̄ ± 1.96 s/√n.

Unit conversions: 1 N = 1 kg·m/s²; 1 Pa = 1 N/m²; 1 J = 1 N·m = 1 W·s; 1 bar = 100 kPa.

Axial stress and deformation: σ = P/A and δ = PL/(AE) for a prismatic linearly elastic bar.

Bending: σ = My/I, with maximum bending stress at the farthest fiber; for a rectangle, I = bh³/12 about the centroidal axis.

Torsion: τ = Tρ/J, maximum at the outer surface; for a solid circular shaft, J = πd⁴/32 and τ_max = 16T/(πd³).

Euler column buckling: P_cr = π²EI/(KL)²; common K values are 0.5 fixed-fixed, 0.7 fixed-pinned, 1.0 pinned-pinned, 2.0 fixed-free.

Thin-walled pressure vessel: hoop stress σ_h = pr/t; longitudinal stress σ_l = pr/(2t); hoop stress is twice longitudinal for a closed cylinder.

Von Mises equivalent stress in plane stress: σ_vm = √(σ_x² − σ_xσ_y + σ_y² + 3τ_xy²).

Fatigue criteria: Goodman uses σ_a/S_e + σ_m/S_ut = 1/n; Soderberg uses σ_a/S_e + σ_m/S_y = 1/n and is more conservative.

Stress concentration: K_t = σ_max/σ_nom; fatigue notch factor K_f = 1 + q(K_t − 1), where q is notch sensitivity.

First law closed system: ΔU = Q − W; steady-flow energy equation per unit mass: q − w = Δh + Δ(V²/2) + gΔz.

Carnot efficiency: η_Carnot = 1 − T_C/T_H, with absolute temperatures; no engine between fixed reservoirs can exceed it.

Ideal-gas entropy change: Δs = c_p ln(T2/T1) − R ln(P2/P1) = c_v ln(T2/T1) + R ln(v2/v1).

Fourier conduction: q″ = −k dT/dx; for a plane wall, q = kA(T_1 − T_2)/L; thermal resistance R_cond = L/(kA).

Newton cooling: q = hA(T_s − T_∞); convection resistance R_conv = 1/(hA).

Radiation: q = εσA(T_s⁴ − T_sur⁴), σ = 5.67 × 10⁻⁸ W/(m²·K⁴); radiation resistance is nonlinear because of T⁴.

Heat exchanger LMTD: q = U A F ΔT_lm; for counterflow, ΔT_lm = (ΔT_1 − ΔT_2)/ln(ΔT_1/ΔT_2).

COP definitions: COP_R = Q_L/W_in; COP_HP = Q_H/W_in; for a reversible cycle, COP_HP = COP_R + 1.

Continuity: ρ_1 A_1 V_1 = ρ_2 A_2 V_2; for incompressible flow, A_1 V_1 = A_2 V_2.

Bernoulli along a streamline: p/(ρg) + V²/(2g) + z = constant for steady, incompressible, inviscid flow without shaft work.

Reynolds number: Re = ρVD/μ = VD/ν; pipe flow is laminar below Re ≈ 2300 and turbulent above Re ≈ 4000.

Darcy-Weisbach head loss: h_f = f(L/D)(V²/(2g)); minor losses h_m = K V²/(2g).

Pump affinity laws at constant impeller diameter: Q ∝ N, H ∝ N², P ∝ N³.

NPSH available: NPSH_a = p_atm/(ρg) + z_s − h_f − p_v/(ρg); cavitation risk rises when NPSH_a < NPSH_r.

Hydraulic power: P_hyd = ρgQH; pump overall efficiency η = P_hyd/P_shaft.

Flow measurement: for an orifice or venturi, Q = C_d A_2 √[2Δp/(ρ(1 − (A_2/A_1)²))].

Iron-carbon eutectoid: at 0.76 wt% C and about 727 °C, austenite transforms to pearlite; hypoeutectoid steels contain proeutectoid ferrite, hypereutectoid contain cementite.

Heat treatment: annealing softens and relieves stress; normalizing refines grain; quenching forms martensite; tempering restores toughness.

Chvorinov's rule: solidification time t = C(V/A)²; a riser must remain liquid longer than the casting it feeds.

Welding: the heat-affected zone is unmelted base metal altered by thermal cycles; preheating reduces cracking risk in high-carbon and alloy steels.

Taylor tool life: V T^n = C; increasing cutting speed V reduces tool life T for given n and C.

Sheet-metal bending: elastic recovery causes springback; bend allowance accounts for neutral-axis arc length.

Thermoplastics remelt on heating; thermosets are cross-linked and char or degrade rather than flow.

Casting defect rules: shrinkage porosity is reduced by directional solidification; gas porosity is reduced by degassing and controlled moisture.

Little's law: WIP = throughput × flow time; applies to stable systems in steady state.

EOQ: Q* = √(2DS/H); reorder point = demand during lead time + safety stock.

OEE = availability × performance × quality; availability = run time/planned production time.

Process capability: C_p = (USL − LSL)/(6σ); C_pk = min(USL − μ, μ − LSL)/(3σ).

PERT: t_e = (a + 4m + b)/6; variance = ((b − a)/6)².

Learning curve: time for unit n = T_1 n^b, where b = ln(learning rate)/ln 2; an 80% curve has b ≈ −0.322.

Queue utilization: ρ = λ/μ; a single-server queue is stable only if ρ < 1.

Inventory turnover = cost of goods sold/average inventory; higher turnover means less capital tied up, but stockout risk must be managed.

Critical path: longest path through the network; its total float is zero, so any delay delays project completion.

EVM: CV = EV − AC; SV = EV − PV; CPI = EV/AC; SPI = EV/PV.

EAC = BAC/CPI when current cost performance is expected to continue; ETC = EAC − AC.

Total float = LS − ES = LF − EF; free float = ES_successor − EF_activity.

Threat responses: avoid, mitigate, transfer, accept; opportunity responses: exploit, enhance, share, accept.

RACI: each task has one accountable person; responsible does work, consulted gives input, informed is notified.

Contract risk: fixed-price contracts place cost risk mainly on the seller; cost-reimbursable contracts place it mainly on the buyer.

Communication channels = n(n − 1)/2 for n stakeholders; scope baseline = scope statement + WBS + WBS dictionary.

Paramount obligation: engineers shall hold public health, safety, and welfare paramount in professional practice.

Competence rule: perform services only in areas of competence established by education, training, or experience; disclose limits when entering new areas.

Conflict of interest: disclose financial or personal interests that could influence judgment and avoid or manage the conflict.

Confidentiality: client or employer information must be protected unless disclosure is required by law or necessary to protect public safety.

Bribery and gifts: no improper payments or gifts to influence decisions; comply with applicable anti-corruption laws.

Whistleblowing: report serious public-safety dangers to an appropriate authority after internal channels fail, using proportionate disclosure.

Intellectual property: give credit for others' work, avoid plagiarism, and respect patents, trade secrets, and confidentiality agreements.

Sustainable practice: consider environmental impact, resource efficiency, and lifecycle consequences in design and recommendations.