Madrid PAU Tech & Engineering II cheat sheet

Density ρ = m/V; specific weight γ = ρg. Typical values: steel ≈ 7850 kg/m³, aluminium ≈ 2700 kg/m³, oak ≈ 700 kg/m³.

Hooke's law in tension: σ = E·ε, valid only below the proportional limit. E: steel ≈ 210 GPa, aluminium ≈ 70 GPa, copper ≈ 110 GPa.

Ductile materials (low-carbon steel) show a yield plateau and large plastic strain before fracture; brittle materials (cast iron, glass, ceramics) fracture with little deformation and are designed by maximum tensile stress.

Hardness scales are not interchangeable: Brinell (HB, hard indenters, coarse materials), Vickers (HV, microhardness and thin layers), Rockwell (HRC for hardened steels).

Selection rule: match property to function - stiffness (E) governs deflection, strength (σy, σu) governs failure, toughness (area under σ–ε curve) governs impact, thermal conductivity governs heat paths.

Ferrous alloys: carbon content raises hardness and lowers ductility; alloying with Cr/Ni gives stainless steel (passive Cr₂O₃ layer). Non-ferrous: Al alloys (light, corrosion-resistant), Cu alloys (conductivity, bearings), Ti alloys (high strength-to-weight, biocompatible).

Polymers: thermoplastics soften reversibly on heating (PE, PVC, PMMA); thermosets cross-link irreversibly (epoxy, polyester); elastomers recover large strains. Composites combine matrix and reinforcement to give anisotropic, high specific stiffness.

Environmental criteria: embodied energy, recyclability, toxicity, and life-cycle assessment. Recycling aluminium saves roughly 95 % of the energy needed for primary production; steel is highly recyclable but melting is energy-intensive.

Corrosion prevention options: galvanising (sacrificial Zn), cathodic protection, protective coatings, and selecting alloys with a passive oxide layer.

Static equilibrium of a planar rigid body: ΣFx = 0, ΣFy = 0, ΣM about any point = 0. Three independent equations give at most three unknown reactions.

Support types and reactions: roller (1 vertical reaction), pinned/hinge (2 reactions, H and V), fixed/clamped (2 forces + 1 moment). A structure with more unknowns than equilibrium equations is statically indeterminate.

Truss analysis by the method of joints: each joint is in equilibrium under concurrent forces; members carry only axial load (tension or compression) if joints are pin-connected and loads act at joints.

Internal forces in a beam are found by the method of sections: axial force N, shear force V, and bending moment M. Shear equals the slope of the moment diagram; a point load produces a kink in V and a slope change in M.

Bending stress: σ = M·y/I, maximum at the extreme fibre where y = c. Section modulus W = I/c, so σmax = M/W. Beam deflection scales with L³ for a point load (δ = FL³/48EI for a simply supported centre load).

Stress and strain: normal stress σ = F/A₀, engineering strain ε = ΔL/L₀, shear stress τ = F/A. Safety factor n = σfailure/σworking; design requires n > 1 with a margin chosen by code.

Euler buckling of a pinned-end column: Pcr = π²EI/L². Slenderness ratio λ = L/i (i = √(I/A)) determines whether buckling or yielding governs.

Second moment of area for a rectangle about its centroid: I = b·h³/12. Increasing depth h is far more efficient than increasing width b for bending stiffness.

Velocity ratio of a gear pair: i = ωdriver/ωdriven = Zdriven/Zdriver (inverse of the tooth ratio). For a train, multiply the stage ratios; idler gears change direction only.

Worm-and-wheel: i = number of teeth on wheel / number of worm starts; a single-start worm gives a large reduction and is self-locking at small lead angles.

Belt and pulley: v = ω·r is the same on both pulleys (no slip), so ω₁r₁ = ω₂r₂. Crossed belts reverse rotation; open belts keep the same direction.

Four-bar linkage Grashof condition: shortest + longest ≤ sum of the other two links guarantees at least one full rotation (crank-rocker).

Power and torque: P = T·ω (ω in rad/s), P = F·v for linear motion. Efficiency η = Pout/Pin < 1; losses arise from friction, slip, and deformation.

Screw jack / lead screw: torque to raise a load T = F·p/(2πη) approximately, where p is the lead; the mechanical advantage is large but efficiency is low (self-locking screws η < 0.5).

Cam-follower: the displacement diagram relates cam angle to follower lift; a knife-edge follower traces the cam profile, a roller follower reduces friction, and spring force must exceed follower inertia at high speed.

Transmission ratio of a chain drive is set by sprocket tooth counts; chains need lubrication and slack adjustment, unlike timing belts which run dry and slip-free.

DC circuits: Ohm's law V = I·R; series resistances add (R = R₁ + R₂), parallel resistances add reciprocally (1/R = 1/R₁ + 1/R₂). Kirchhoff: ΣI at a node = 0, ΣV around a loop = 0.

Power in a resistor: P = V·I = I²R = V²/R. Voltage divider: Vout = Vin·R₂/(R₁ + R₂). Thevenin equivalent: Vth = open-circuit voltage, Rth = resistance seen with sources deactivated.

Magnetic force on a conductor: F = B·I·L·sinθ; on a moving charge F = qvB·sinθ. Faraday's law: ε = −dΦ/dt; Lenz's law fixes the direction so the induced current opposes the change.

Transformer (AC): Vs/Vp = Ns/Np = Ip/Is; ideal power is conserved, but real efficiency is below 100 % (copper and iron losses).

Diode: forward-biased above ≈ 0.7 V (Si) or 0.3 V (Ge), reverse-biased blocks. Half-wave rectifier conducts one half-cycle; a bridge rectifier conducts both, giving ripple frequency 2f.

Transistor as a switch: cut-off (no base current, VCE ≈ VCC) and saturation (VCE ≈ 0.2 V). In active mode IC = β·IB.

Logic gates: AND (Y = A·B), OR (Y = A + B), NOT (Y = Ā), NAND and NOR are universal gates. Boolean identities: A + A·B = A, A + Ā·B = A + B.

Combining resistances with real sources: internal resistance r reduces terminal voltage to V = ε − I·r; maximum power transfer to a load occurs when RL = r.

Open-loop control applies an action without measuring the output; closed-loop (feedback) control compares the measured output with the setpoint and acts on the error e = setpoint − measured value.

PID controller output: u(t) = Kp·e + Ki·∫e dt + Kd·de/dt. Proportional gain reduces steady-state error but can cause oscillation; integral action eliminates offset; derivative action damps overshoot.

Sensor selection: thermocouple or RTD for temperature, strain gauge for force, LVDT for displacement, photoelectric or inductive proximity for presence, encoder for angular position and speed.

Actuators: DC/stepper/servo motors, solenoids, pneumatic and hydraulic cylinders. Stepper motors move in discrete steps (step angle = 360°/steps per revolution) and need no feedback for simple positioning.

PLC scan cycle: read inputs, execute the program, update outputs, repeat. Ladder logic uses contacts (X) and coils (Y); latching (self-holding) circuits retain state after the start button is released.

Robot degrees of freedom equal the number of independent joints; a 6-DOF arm can position and orient its end effector in 3D. Workspace is the reachable volume; repeatability is tighter than absolute accuracy.

Robot configurations: Cartesian (x-y-z), cylindrical, spherical, SCARA (horizontal assembly), and articulated (anthropomorphic). Pneumatic grippers are fast and cheap; vacuum grippers suit flat, non-porous parts.

Safety and reliability: emergency-stop circuits are hard-wired and normally closed so a broken wire stops the machine; sensors and actuators should fail to a safe state.

First-angle projection (ISO/DIN): the object sits between the observer and the plane, so the view from the left is drawn on the right. Third-angle (ANSI) places each view on the same side as the observer.

Line types: continuous thick for visible edges, dashed for hidden edges, chain thin for centre lines and axes, and thin continuous for dimension and projection lines.

Scale: drawing length / real length. Enlargement (2:1), full size (1:1), and reduction (1:2, 1:5, 1:10, 1:50) are chosen so the drawing fits the sheet while preserving readability.

Dimensioning rules: dimensions give true size, never scale-measured; avoid duplication; place dimensions outside the outline; use a leader for diameters (Ø) and radii (R); tolerance is the permitted variation between upper and lower limits.

Standard sheet sizes follow ISO 216 (A0 = 1 m², each A(n+1) is half of A(n)); title blocks carry scale, material, author, date, and drawing number.

Project method: define the need, gather constraints, generate alternatives, evaluate against criteria (cost, safety, environment, feasibility), select, prototype, test, and document. A Gantt chart schedules tasks and dependencies.

Technical report structure: objective, background, specifications, calculations with units, results, conclusions, and references. All magnitudes must carry SI units and consistent significant figures.

Evaluation of solutions uses weighted decision matrices; criteria weights must sum to 1 and each option is scored against the same scale for a fair comparison.