Castilla y León PAU Technology and Engineering II cheat sheet

Crystal structures: BCC (α-Fe, ferrite), FCC (γ-Fe, austenite, Al, Cu), HCP (Mg, Zn, Ti). FCC has 4 atoms/cell and 12 slip systems; BCC has 2 atoms/cell and 48 slip systems; HCP has 3 atoms/cell and few slip systems, hence low ductility.

Carbon steel microstructure: ferrite (soft, BCC) + pearlite (lamellar ferrite/cementite) + cementite (Fe3C, hard and brittle). Increasing C content raises hardness and tensile strength but lowers ductility and weldability.

Martensite forms by diffusionless shear when austenite is quenched faster than the critical cooling rate; it is hard and brittle, so it must be tempered to relieve stresses and improve toughness.

Mechanical properties: Young's modulus E = σ/ε in the elastic region; yield strength σy at 0.2% offset; ultimate tensile strength is the maximum stress; ductility is measured as elongation at fracture or reduction of area.

Hardness tests: Brinell (10 mm ball, HBW), Vickers (diamond pyramid, HV, valid for thin layers), Rockwell (indenter + preload, HRC for hard steels). Hardness correlates roughly with tensile strength (σu ≈ 3.5·HB in MPa for steels).

Impact toughness: Charpy test measures absorbed energy; a ductile-to-brittle transition appears in BCC steels at low temperature, while FCC metals (austenitic stainless, Al) stay ductile.

Selection criteria (Ashby): translate requirements into function, constraints, objective and free variables; use material property charts (E vs ρ, strength vs density) and the performance index M = E^(1/2)/ρ for a stiff lightweight beam.

Corrosion: galvanic pair requires two different metals plus an electrolyte; the anodic metal corrodes. Avoid large cathode/anode area ratios; stainless steel passivates via a Cr2O3 layer.

Static equilibrium: ΣF = 0 and ΣM = 0 about any point. A simply supported beam with point load P at distance a from the left support (span L) gives reactions RA = P·b/L and RB = P·a/L.

Friction: Ff = μ·N. On an inclined plane at angle θ, the body slides when tan θ > μs; the angle at which sliding starts is the friction angle φ = arctan μs.

Kinematics: uniform circular motion has v = ω·r, a_n = ω²·r = v²/r, and T = 2π/ω. Belt drive ratio: n1·D1 = n2·D2 (ignoring slip).

Gear trains: i = ω_in/ω_out = product of driven teeth / product of driving teeth. A simple train with an idler does not change the ratio, only the direction.

Dynamics: Newton's second law ΣF = m·a; work W = F·d·cos θ; power P = W/t = F·v; kinetic energy Ec = ½mv²; potential energy Ep = mgh; efficiency η = useful output / input energy.

Torque and power in rotation: P = M·ω, with ω in rad/s. A motor delivering 10 N·m at 1500 rpm gives P = 10·(1500·2π/60) ≈ 1571 W.

Machine elements: rolling bearings are selected by dynamic load rating C and equivalent load P via L10 = (C/P)^p (p = 3 for ball, 10/3 for roller). Screw threads convert torque to axial force; leadscrew efficiency η = tan λ / tan(λ + φ).

Stress in elements: normal stress σ = F/A, shear τ = F/A, bending σ = M·y/I with I = b·h³/12 for a rectangular section; shafts transmit torque with τ = M_t·r/J, J = π·d⁴/32.

Ohm's law V = I·R; series resistances add (Req = ΣR); parallel: 1/Req = Σ(1/R). Voltage divider: Vout = Vin·R2/(R1+R2).

Kirchhoff's laws: KCL (sum of currents at a node = 0) and KVL (sum of voltages around a loop = 0). Use mesh or node analysis for multi-source circuits.

Power: P = V·I = I²·R = V²/R. A resistor rated ¼ W must satisfy I²·R ≤ 0.25 W to avoid overheating.

Thevenin equivalent: open-circuit voltage Vth in series with Rth measured with sources deactivated (voltage sources shorted, current sources opened). Maximum power transfer when RL = Rth.

Capacitors: C = Q/V; series 1/Ceq = Σ(1/C); parallel Ceq = ΣC. Time constant τ = R·C; charging v(t) = V(1 − e^(−t/τ)).

Inductors: v = L·di/dt; energy ½L·I²; τ = L/R. In DC steady state an inductor behaves as a short circuit and a capacitor as an open circuit.

Diodes: forward drop ≈ 0.7 V (Si) or 0.3 V (Ge); reverse-biased blocks. Half-wave rectifier gives Vdc = Vm/π; full-wave bridge gives Vdc = 2Vm/π.

Operational amplifier in linear mode: Vout = A·(V+ − V−) with virtual short at the inputs. Inverting gain = −Rf/Rin; non-inverting gain = 1 + Rf/R1.

Open loop: output does not affect the input; sensitive to disturbances. Closed loop: feedback compares the measured output with the setpoint, generating error e = r − y that the controller acts on.

Transfer function G(s) = Y(s)/X(s). Closed-loop transfer function with unity feedback is G/(1+G); the system is stable if all poles have negative real parts (Routh–Hurwitz).

PID controller: u(t) = Kp·e + Ki·∫e dt + Kd·de/dt. Proportional reduces error, integral removes steady-state error, derivative damps and anticipates but amplifies noise.

Sensors: NTC thermistor (resistance falls with temperature), PT100 (platinum, ≈100 Ω at 0 °C, ≈0.385 Ω/°C), LVDT (linear displacement), strain gauge (gauge factor GF = (ΔR/R)/(ΔL/L) ≈ 2), photodiode, inductive/capacitive proximity switch.

Actuators: DC motor speed proportional to voltage, torque to current; stepper motor moves in fixed steps (e.g. 1.8° = 200 steps/rev); servo motor with encoder closes the position loop; solenoid valve is on/off.

PLC scan cycle: read inputs → execute program → update outputs, repeated cyclically; scan time typically milliseconds. Ladder logic uses contacts (XIC, XIO) and coils (OTE).

Boolean logic: AND = series contacts, OR = parallel contacts, NOT = normally-closed contact. Karnaugh maps simplify functions; De Morgan: (A·B)' = A' + B', (A+B)' = A'·B'.

Sequential control: use timers (TON on-delay, TOF off-delay) and counters (CTU). Safety: emergency stop in normally-closed series, interlock prevents simultaneous conflicting outputs.

Pascal's principle: pressure transmitted undiminished; F1/A1 = F2/A2. A hydraulic press with area ratio 10 multiplies force by 10 while the piston travels 1/10 of the distance.

Flow and speed: Q = A·v, so cylinder extension speed v = Q/A. Force of a double-acting cylinder F = p·A (extend) and F = p·(A − A_rod) (retract).

Pressure units: 1 bar = 10⁵ Pa; 1 atm = 1.013 bar. Pneumatic networks typically work at 6–8 bar; hydraulics at 100–350 bar because oil is incompressible.

Air preparation unit (FRL): filter, regulator and lubricator in series at the inlet; a pressure regulator keeps working pressure constant, a non-return valve blocks reverse flow.

Directional valves: 5/2 valve (5 ports, 2 positions) drives a double-acting cylinder; 3/2 valve for single-acting. Designation 4/2 means 4 ways, 2 positions.

Speed control: meter-in (throttle before cylinder) for resistive loads, meter-out (throttle in exhaust) for overrunning loads to prevent runaway; flow control valve with bypass check.

Circuit symbols and logic: 5/2 bistable valve with two pilots; shuttle valve = OR function, two-pressure valve = AND function; quick exhaust valve increases return speed.

Accumulators store hydraulic energy and damp pressure peaks; relief valve limits maximum pressure; a hydraulic motor converts flow into rotation with torque M = p·V/2π (V = displacement per revolution).

Turning: cutting speed v = π·D·n/1000 (m/min), feed f (mm/rev), depth of cut a. Material removal rate Q ≈ v·f·a. Lathe operations: facing, cylindrical turning, threading, drilling with tailstock.

Milling and drilling: spindle speed n = 1000·v/(π·D). Drilling feed per revolution determines chip thickness; milling uses up-cut (conventional) or down-cut (climb) depending on feed direction vs cutter rotation.

Forming: hot rolling (above recrystallization temperature), forging, extrusion, drawing. Sheet bending: neutral axis length L = a + b + (π·α/180)·(R + k·e), with k ≈ 0.5 for R/e > 5.

Joining: arc welding (SMAW, MIG/MAG, TIG), resistance spot welding, soldering/brazing below parent metal melting. Rivets and bolts for demountable joints; adhesive bonding distributes stress but is sensitive to temperature.

Metrology: vernier caliper (0.02 mm resolution), micrometer (0.01 mm), dial gauge (0.01 mm), gauge blocks. Measurement uncertainty includes instrument resolution and repeatability.

Tolerances and fits: ISO system with holes (H) and shafts (h). Clearance fit (H7/g6), transition fit (H7/k6), interference fit (H7/p6). IT grades: IT6 fine, IT11 coarse.

Quality control: control charts (mean and range), Cp = (USL − LSL)/(6σ), Cpk accounts for centering. Six sigma target Cpk ≥ 1.33 for capable processes.

Production organisation: Gantt chart and PERT/CPM critical path; lean manufacturing eliminates the seven wastes (overproduction, waiting, transport, overprocessing, inventory, motion, defects).

Project phases: identification of need → feasibility study → preliminary design → detailed design → construction/manufacture → testing → exploitation → decommissioning. Each phase ends with a review gate.

Design methodology: define the problem and specifications, generate alternatives, evaluate and select (weighted criteria matrix), detail, verify. Specifications must be measurable (function, constraints, cost, safety).

Technical documentation: memory of the project, specifications, drawings (plan with scale, views, dimensions, tolerances, title block), bill of materials (BOM), operation manuals and maintenance plans.

Budget: total cost = direct costs (materials, labour, machinery) + indirect costs (overheads) + contingency. Break-even point Q = fixed costs / (unit price − unit variable cost).

Amortisation and payback: simple payback = initial investment / annual net cash flow. Net present value NPV = Σ CFt/(1+i)^t − I0; accept if NPV > 0.

Sustainability: life cycle assessment (LCA) covers extraction, manufacture, use and end-of-life. Ecodesign criteria: reduce material, use recycled/recyclable materials, reduce energy in use, design for disassembly.

Environmental impact: carbon footprint in kg CO2-eq; energy payback time of a system = embodied energy / annual energy saved; circular economy prioritises reuse, remanufacture and recycling over disposal.

Risk and safety: risk = probability × severity; hierarchy of controls (eliminate, substitute, engineering controls, administrative, PPE). CE marking and applicable standards for machinery safety.

CPU components: control unit, ALU, registers (PC, IR, accumulator). Performance = clock frequency × IPC; execution time = instruction count × CPI / frequency.

Memory hierarchy: registers → cache L1/L2/L3 → RAM → SSD/HDD. RAM is volatile (DRAM/SRAM), ROM/Flash non-volatile. Addressable memory = 2^n cells for n address bits.

Number systems: binary, octal, hexadecimal conversion; two's complement for signed integers; 1 byte = 8 bits; ASCII 7 bits, Unicode UTF-8 variable length.

Network models: OSI seven layers (physical, data link, network, transport, session, presentation, application) vs TCP/IP four layers. TCP is connection-oriented and reliable; UDP is connectionless.

Addressing: IPv4 is 32 bits, IPv6 is 128 bits. Subnet mask divides network and host parts; /24 means 24 network bits. MAC address 48 bits in data-link layer; DNS maps names to IP.

Topologies and devices: star, bus, ring, mesh; hub broadcasts, switch forwards by MAC, router forwards by IP. LAN vs WAN; Wi-Fi 802.11, Ethernet 802.3.

Programming basics: variables and types (int, float, char, boolean), operators, sequence/selection (if-else)/iteration (while, for), functions with parameters and return value, arrays and strings.

Algorithms: flowchart symbols (oval start/end, parallelogram I/O, rectangle process, diamond decision). Sorting (bubble O(n²), quicksort O(n log n)), linear vs binary search (binary requires sorted data, O(log n)).