Core Engineering Principles & Problem-Solving

Engineering fundamentals span MechE, CivE, and EE domains. Mastery requires understanding FBDs, material props, thermo laws, and systematic design methodology.

## Statics & Mechanics of Materials

Equilibrium conditions: Sum of forces = 0 (translational), sum of moments = 0 (rotational). For 2D problems: ΣFx=0, ΣFy=0, ΣM=0 gives 3 equations → can solve for 3 unknowns. Always start w/ FBD isolating the body of interest. Support reactions: pin (2 force components), roller (1 normal force), fixed (2 forces + 1 moment).

Stress and strain fundamentals: Normal stress σ = F/A (Pa or psi). Shear stress τ = V/A. Strain ε = ΔL/L (dimensionless). Hooke's Law: σ = Eε where E = Young's modulus (steel ~200 GPa, aluminum ~70 GPa, concrete ~30 GPa). Poisson's ratio ν = -εlateral/εaxial (steel ~0.3, rubber ~0.5). Shear modulus G = E/[2(1+ν)].

Beam analysis: Shear and moment diagrams essential for structural design. V(x) = -dM/dx, w(x) = -dV/dx where w = distributed load. Max bending stress σ = Mc/I where M = moment, c = distance from NA, I = second moment of area. For rectangular section: I = bh³/12. Deflection: EI(d⁴y/dx⁴) = w(x). Common cases memorized: cantilever w/ point load δmax = PL³/3EI, simply supported w/ center load δmax = PL³/48EI.

FOS: Factor of Safety = σyield/σactual. Typical FOS values: buildings 2-3, pressure vessels 3-4, aircraft 1.5-2 (weight-critical), consumer products 3-5. Never design below FOS=1.0. For fatigue loading, use endurance limit and SN curves; FOS against fatigue failure often higher.

## Thermodynamics

First Law (conservation of energy): Q - W = ΔU for closed systems. For open systems (SSSF): Q̇ - Ẇ = ṁ(Δh + ΔKE + ΔPE). Sign convention: Q positive into system, W positive out of system.

Second Law: Entropy always increases in isolated systems. Clausius: ∮(δQ/T) ≤ 0. Carnot efficiency ηmax = 1 - Tcold/Thot (absolute temps). No real engine exceeds Carnot. Entropy generation Sgen ≥ 0; = 0 only for REV processes.

Common cycles: Otto (gasoline engine) η = 1 - r^(1-k) where r = compression ratio, k = cp/cv. Diesel: higher compression ratio → higher η but heavier. Rankine (steam power): pump → boiler → turbine → condenser. Improve w/ reheat, regeneration, superheat. Refrigeration: reversed Rankine, COP = QL/(QH-QL) for cooling.

Heat transfer modes: Conduction q = -kA(dT/dx), Fourier's law. Convection q = hA(Ts-T∞), h depends on flow regime (natural vs forced, laminar vs turbulent). Radiation q = εσA(T⁴s-T⁴surr), Stefan-Boltzmann. Thermal resistance analogy: Rtotal = ΣRi in series, 1/Rtotal = Σ(1/Ri) in parallel. For composite walls: q = ΔT/Rtotal.

## Fluid Mechanics

Bernoulli's equation (steady, incompressible, inviscid along streamline): P₁/ρ + V₁²/2 + gz₁ = P₂/ρ + V₂²/2 + gz₂. Continuity: A₁V₁ = A₂V₂ for incompressible flow. Reynolds number Re = ρVD/μ determines flow regime: Re < 2300 laminar, Re > 4000 turbulent in pipes.

Pipe flow: Darcy-Weisbach hf = f(L/D)(V²/2g). Friction factor f from Moody chart or Colebrook equation. For laminar: f = 64/Re. Minor losses: hm = K(V²/2g) where K from tables (elbow ~0.9, valve varies). Pump selection: match system curve (static head + friction losses) to pump curve at BEP.

Dimensional analysis: Buckingham Pi theorem — if n variables involve m fundamental dimensions, then (n-m) dimensionless groups describe the system. Common groups: Re (inertia/viscous), Fr (inertia/gravity), Ma (inertia/compressibility), Nu (convection/conduction).

## Electrical Engineering Fundamentals

Ohm's Law: V = IR. Power: P = VI = I²R = V²/R. KVL: sum of voltages around loop = 0. KCL: sum of currents at node = 0. Series resistors: Rtotal = ΣRi. Parallel: 1/Rtotal = Σ(1/Ri). Voltage divider: Vout = Vin × R2/(R1+R2).

AC circuits: Impedance Z = R + jX where X = XL - XC. XL = ωL = 2πfL (inductor leads current by 90°). XC = 1/(ωC) (capacitor leads voltage by 90°). Power factor pf = cos(θ) where θ = angle between V and I. Real power P = VIcos(θ), reactive Q = VIsin(θ), apparent S = VI. Power factor correction: add capacitors to reduce reactive power, lower utility costs.

Three-phase systems: Balanced 3φ advantage — constant power delivery, smaller conductors, self-starting motors. Line vs phase: Wye VL = √3×Vφ, IL = Iφ. Delta VL = Vφ, IL = √3×Iφ. Total power P = √3×VL×IL×pf.

Motor selection: DC motors for precise speed control (armature voltage). AC induction most common industrial — rugged, low maintenance. Synchronous speed Ns = 120f/P (f=freq, P=poles). Slip s = (Ns-Nr)/Ns. VFDs enable variable speed on AC motors, improving efficiency vs throttling.

## Engineering Design Process

Systematic approach: 1) Define problem/requirements (functional specs, constraints, criteria), 2) Research/benchmarking (prior art, standards, codes), 3) Conceptual design (brainstorm, morphological chart), 4) Embodiment design (layout, preliminary calcs, material selection), 5) Detail design (full analysis, DFM/DFA, tolerancing), 6) Prototype and test, 7) Iterate.

Material selection: Ashby charts plot material properties (E vs ρ, σy vs ρ) to identify candidates. Performance indices guide selection: min weight rod in tension → maximize σy/ρ, min weight beam in bending → maximize σy^(2/3)/ρ. Consider: cost, availability, machinability, corrosion resistance, recyclability.

DFM principles: minimize part count, design for multiuse/symmetry, avoid tight tolerances where unnecessary (cost grows exponentially w/ tighter tolerance), design for ease of assembly (top-down, minimize reorientation), consider manufacturing method early (casting vs machining vs AM vs injection molding). GD&T per ASME Y14.5 for unambiguous tolerance specification.

Failure analysis: Common failure modes — yielding (exceeding σy), fracture (exceeding σult or fatigue), buckling (Euler: Pcr = π²EI/L² for slender columns), creep (high temp sustained load), corrosion (galvanic, pitting, stress-corrosion cracking). Root cause analysis: fishbone (Ishikawa) diagram, 5-Why method, fault tree analysis.

## Project Management for Engineers

Critical path method: identify all activities, dependencies, and durations. Forward/backward pass to find ES, EF, LS, LF. Float = LS-ES. Critical path = longest path (zero float). Crashing: add resources to critical activities to reduce project duration, but increases cost.

Risk management: identify → assess (probability × impact matrix) → mitigate → monitor. Technical risk mitigation: parallel development paths, early prototyping, TRL assessment. Schedule risk: add buffers on critical path, not every activity (Theory of Constraints).

Cost estimation: parametric (early stage, CER from historical data), bottom-up (detailed, WBS-based), analogous (similar past projects). Include contingency: 20-30% conceptual, 10-15% detailed design, 5-10% construction. Life cycle cost = development + production + operations + disposal.