Castilla-La Mancha PAU Technology and Engineering II (Tecnología e Ingeniería II - UCLM 2026) cheat sheet
Castilla-La Mancha PAU Technology and Engineering II (Tecnología e Ingeniería II - UCLM 2026) · Keep the key ideas within reach.
Materials Science and Engineering: Properties, Structure, Testing and Selection
Mechanical stress is σ = F/A (N/m² = Pa); normal strain is ε = ΔL/L₀ (dimensionless); elastic modulus E = σ/ε in the linear (Hooke) region.
Hooke's law σ = E·ε applies only below the proportional limit; beyond the yield strength (σy, 0.2 % offset for metals) deformation becomes plastic and permanent.
Tensile test outputs: ultimate tensile strength (maximum σ), yield strength, elongation at break A% = (Lf−L₀)/L₀·100, and reduction of area Z%.
Hardness scales: Brinell (HB, ball indenter, loads 500–3000 kgf), Vickers (HV, 136° diamond pyramid, load-independent), Rockwell (HRC for hard steels). Empirical relation: σu(MPa) ≈ 3.45·HB.
Impact toughness is measured by Charpy Izod-type tests (J) and falls sharply at the ductile–brittle transition temperature; FCC metals (Cu, Al) stay ductile at low T, BCC steels do not.
Crystal structures: BCC (Fe-α, W), FCC (Fe-γ, Cu, Al), HCP (Mg, Zn). Atomic packing factor: FCC/HCP 0.74, BCC 0.68.
Binary phase diagrams: lever rule gives phase fractions; eutectic point = lowest melting composition; Fe-C eutectoid at 0.77 % C, 727 °C gives pearlite; martensite forms by fast quenching.
Selection criteria: specific strength σ/ρ and specific stiffness E/ρ; corrosion resistance via galvanic series (avoid coupling dissimilar metals), and cost per unit of function.
Mechanical Systems and Machine Design: Mechanisms, Transmission, Structures and Resistance of Materials
Velocity ratio of a gear pair: i = ω₁/ω₂ = z₂/z₁ = d₂/d₁; for a train i_total = product of driven teeth / product of driver teeth.
Belt drives: i = n₁/n₂ = D₂/D₁; slip reduces real ratio; crossed belts reverse rotation direction. Chain drives keep a fixed ratio (no slip).
Four-bar linkage Grashof condition: shortest + longest ≤ sum of the other two links ⇒ at least one link can fully rotate.
Screw mechanism: axial advance per turn = pitch p; torque–load relation T ≈ F·p/(2π·η) with efficiency η.
Power transmitted by a rotating shaft: P = T·ω = 2π·n·T/60 (T in N·m, n in rpm, P in W).
Axial stress σ = N/A; shear stress τ = V/A; bending stress σ = M·y/I with I = b·h³/12 for a rectangle; torsion τ = T·r/J with J = π·d⁴/32 for a solid circle.
Beam deflection and moment depend on support type: simply supported (Mmax = qL²/8 for uniform load), cantilever (Mmax = qL²/2 at the fixed end).
Buckling (Euler): Pcr = π²·E·I/(k·L)², with k = 1 pinned-pinned, 0.5 fixed-fixed, 2 fixed-free; slenderness λ = L/i governs validity.
Fatigue failure occurs below static strength after many cycles; stress concentration at notches (Kt) and surface finish are critical design factors.
Electrical and Electronic Systems: Circuits, Machines, Power Electronics and Control
Ohm's law V = I·R; series R_eq = ΣR; parallel 1/R_eq = Σ(1/R); power P = V·I = I²R = V²/R.
Kirchhoff: ΣI = 0 at a node; ΣV = 0 around a loop. Thévenin equivalent: Vth = open-circuit voltage, Rth = resistance seen with sources killed.
AC: X_L = 2πfL, X_C = 1/(2πfC), Z = √(R² + (X_L−X_C)²); power factor cos φ = R/Z; three-phase P = √3·V_L·I_L·cos φ.
Transformer: V₁/V₂ = N₁/N₂ = I₂/I₁; ideal transformer conserves apparent power S = V·I.
DC machine: back-emf E = k·Φ·n; torque T = k·Φ·I. Induction motor slip s = (ns − n)/ns, synchronous speed ns = 60f/p.
Diodes: forward bias ≈ 0.7 V (Si), 0.3 V (Ge); half-wave rectifier output mean = Vmax/π; full-wave/bridge = 2·Vmax/π; ripple reduced by filter capacitors.
Transistor as switch: cutoff (no base current), saturation (VCE ≈ 0.2 V); as amplifier, IC = β·IB in the active region.
Operational amplifier: ideal A → ∞, input impedance → ∞; inverting gain = −R₂/R₁, non-inverting gain = 1 + R₂/R₁.
Control systems: open-loop vs closed-loop; PID controller output u = Kp·e + Ki∫e dt + Kd·de/dt; stability assessed by gain/phase margin on the Bode plot.
Energy Resources, Generation, Conversion and Sustainability
Energy units: 1 kWh = 3.6 MJ; 1 tep = 41.87 GJ; 1 cal = 4.18 J; power = energy/time (1 W = 1 J/s).
Thermal power plant efficiency η = W_net/Q_in = 1 − T_f/Q_in; Carnot limit η = 1 − Tc/Th (absolute temperatures).
Renewables: wind power P = ½·ρ·A·v³ (Betz limit 16/27 ≈ 59 %); hydro P = ρ·g·Q·H·η; PV panel output depends on irradiance and temperature (derating ≈ −0.4 %/°C).
Spain's generation mix and the 2030/2050 decarbonisation targets drive integration of variable renewables, requiring storage (pumped hydro, batteries) and grid flexibility.
Environmental impact indicators: CO₂ emissions per kWh (coal ≈ 900 g, gas ≈ 400 g, nuclear/wind/PV life-cycle < 50 g).
Energy conversion chains: chemical → thermal → mechanical → electrical; each stage has losses, so overall η = product of stage efficiencies.
Nuclear fission releases ≈ 200 MeV per U-235 nucleus; a reactor is critical when k = 1 (multiplication factor).
Sustainability criteria: energy payback time, life-cycle analysis (LCA), and the waste hierarchy (reduce, reuse, recycle, recover).
Manufacturing, Production Systems and Automation
Casting: sand mould with pattern, sprue, riser; shrinkage allowance compensates solidification contraction; die casting suits high-volume non-ferrous parts.
Forming: hot rolling (recrystallisation above T_recrystallisation), forging, extrusion, deep drawing; cold working increases strength but causes work hardening and needs annealing.
Machining parameters: cutting speed v = π·D·n/1000 (m/min), feed f (mm/rev), depth of cut a; material removal rate MRR ≈ v·f·a.
Welding: arc (electrode), MIG/MAG (continuous wire with gas shield), TIG (non-consumable tungsten electrode), resistance spot welding for sheet metal.
Joining and finishing: riveting, bolting, adhesive bonding; surface treatments (galvanising, anodising, painting) for corrosion protection.
CNC programming with G-codes (G00 rapid, G01 linear interpolation, G02/G03 circular) and M-codes (M03 spindle on, M08 coolant); coordinates relative (G91) or absolute (G90).
Automation: PLC ladder logic, sensors/actuators, open-loop vs closed-loop control, and industrial robots classified by kinematics (Cartesian, SCARA, articulated, delta).
Production metrics: takt time = available time/demand; OEE = availability × performance × quality; lean tools (JIT, kanban, 5S).
Engineering Project Methodology, Technical Drawing, Standards and ICT Tools
Project phases: idea/needs analysis → feasibility study → preliminary and detailed design → execution → testing → documentation and closure.
Project documents: memory (justification, calculations, annexes), plans/drawings, specification (pliego de condiciones) and budget (presupuesto with measurements and unit prices).
Gantt chart shows scheduling over time; PERT/CPM network computes critical path (longest path, zero slack) and earliest/latest times.
Technical drawing standards: ISO/UNE line types (continuous thick for visible edges, dashed for hidden), first-angle (European) vs third-angle projection, scale = drawing/real.
Dimensioning rules: dimension lines outside the figure, no redundant dimensions, tolerances and fits per ISO 286 (H7/g6), geometric tolerancing (GD&T symbols).
Views: front, top and side elevation; auxiliary views for inclined faces; sections and hatching at 45° to show internal features.
ICT tools: CAD (2D/3D parametric modelling), CAM, CAE (FEM simulation), spreadsheets for budgets, and collaborative project management software.
Budget calculation: total = Σ(unit price × quantity) plus indirect costs and VAT; amortisation and payback period evaluate economic viability.
Materials Science and Engineering: Properties, Structure, Testing and Selection
Mechanical stress is σ = F/A (N/m² = Pa); normal strain is ε = ΔL/L₀ (dimensionless); elastic modulus E = σ/ε in the linear (Hooke) region.
Hooke's law σ = E·ε applies only below the proportional limit; beyond the yield strength (σy, 0.2 % offset for metals) deformation becomes plastic and permanent.
Tensile test outputs: ultimate tensile strength (maximum σ), yield strength, elongation at break A% = (Lf−L₀)/L₀·100 and reduction of area Z%.
Hardness scales: Brinell (HB, ball indenter, loads 500–3000 kgf), Vickers (HV, 136° diamond pyramid, load-independent), Rockwell (HRC for hard steels). Empirical relation: σu(MPa) ≈ 3.45·HB.
Impact toughness is measured by Charpy Izod-type tests (J) and falls sharply at the ductile–brittle transition temperature; FCC metals (Cu, Al) stay ductile at low T, BCC steels do not.
Crystal structures: BCC (Fe-α, W), FCC (Fe-γ, Cu, Al), HCP (Mg, Zn). Atomic packing factor: FCC/HCP 0.74, BCC 0.68.
Binary phase diagrams: lever rule gives phase fractions; eutectic point = lowest melting composition; Fe-C eutectoid at 0.77 % C, 727 °C gives pearlite; martensite forms by fast quenching.
Selection criteria: specific strength σ/ρ and specific stiffness E/ρ; corrosion resistance via galvanic series (avoid coupling dissimilar metals), and cost per unit of function.
Mechanical Systems and Machine Design: Mechanisms, Transmission, Structures and Resistance of Materials
Velocity ratio of a gear pair: i = ω₁/ω₂ = z₂/z₁ = d₂/d₁; for a train i_total = product of driven teeth / product of driver teeth.
Belt drives: i = n₁/n₂ = D₂/D₁; slip reduces real ratio; crossed belts reverse rotation direction. Chain drives keep a fixed ratio (no slip).
Four-bar linkage Grashof condition: shortest + longest ≤ sum of the other two links ⇒ at least one link can fully rotate.
Screw mechanism: axial advance per turn = pitch p; torque–load relation T ≈ F·p/(2π·η) with efficiency η.
Power transmitted by a rotating shaft: P = T·ω = 2π·n·T/60 (T in N·m, n in rpm, P in W).
Axial stress σ = N/A; shear stress τ = V/A; bending stress σ = M·y/I with I = b·h³/12 for a rectangle; torsion τ = T·r/J with J = π·d⁴/32 for a solid circle.
Beam deflection and moment depend on support type: simply supported (Mmax = qL²/8 for uniform load), cantilever (Mmax = qL²/2 at the fixed end).
Buckling (Euler): Pcr = π²·E·I/(k·L)², with k = 1 pinned-pinned, 0.5 fixed-fixed, 2 fixed-free; slenderness λ = L/i governs validity.
Fatigue failure occurs below static strength after many cycles; stress concentration at notches (Kt) and surface finish are critical design factors.
Electrical and Electronic Systems: Circuits, Machines, Power Electronics and Control
Ohm's law V = I·R; series R_eq = ΣR; parallel 1/R_eq = Σ(1/R); power P = V·I = I²R = V²/R.
Kirchhoff: ΣI = 0 at a node; ΣV = 0 around a loop. Thévenin equivalent: Vth = open-circuit voltage, Rth = resistance seen with sources killed.
AC: X_L = 2πfL, X_C = 1/(2πfC), Z = √(R² + (X_L−X_C)²); power factor cos φ = R/Z; three-phase P = √3·V_L·I_L·cos φ.
Transformer: V₁/V₂ = N₁/N₂ = I₂/I₁; ideal transformer conserves apparent power S = V·I.
DC machine: back-emf E = k·Φ·n; torque T = k·Φ·I. Induction motor slip s = (ns − n)/ns, synchronous speed ns = 60f/p.
Diodes: forward bias ≈ 0.7 V (Si), 0.3 V (Ge); half-wave rectifier output mean = Vmax/π; full-wave/bridge = 2·Vmax/π; ripple reduced by filter capacitors.
Transistor as switch: cutoff (no base current), saturation (VCE ≈ 0.2 V); as amplifier, IC = β·IB in the active region.
Operational amplifier: ideal A → ∞, input impedance → ∞; inverting gain = −R₂/R₁, non-inverting gain = 1 + R₂/R₁.
Control systems: open-loop vs closed-loop; PID controller output u = Kp·e + Ki∫e dt + Kd·de/dt; stability assessed by gain/phase margin on the Bode plot.
Energy Resources, Generation, Conversion and Sustainability
Energy units: 1 kWh = 3.6 MJ; 1 tep = 41.87 GJ; 1 cal = 4.18 J; power = energy/time (1 W = 1 J/s).
Thermal power plant efficiency η = W_net/Q_in; Carnot limit η = 1 − Tc/Th (absolute temperatures).
Renewables: wind power P = ½·ρ·A·v³ (Betz limit 16/27 ≈ 59 %); hydro P = ρ·g·Q·H·η; PV panel output depends on irradiance and temperature (derating ≈ −0.4 %/°C).
Spain's generation mix and the 2030/2050 decarbonisation targets drive integration of variable renewables, requiring storage (pumped hydro, batteries) and grid flexibility.
Environmental impact indicators: CO₂ emissions per kWh (coal ≈ 900 g, gas ≈ 400 g, nuclear/wind/PV life-cycle < 50 g).
Energy conversion chains: chemical → thermal → mechanical → electrical; each stage has losses, so overall η = product of stage efficiencies.
Nuclear fission releases ≈ 200 MeV per U-235 nucleus; a reactor is critical when k = 1 (multiplication factor).
Sustainability criteria: energy payback time, life-cycle analysis (LCA), and the waste hierarchy (reduce, reuse, recycle, recover).
Manufacturing, Production Systems and Automation
Casting: sand mould with pattern, sprue, riser; shrinkage allowance compensates solidification contraction; die casting suits high-volume non-ferrous parts.
Forming: hot rolling (recrystallisation above T_recrystallisation), forging, extrusion, deep drawing; cold working increases strength but causes work hardening and needs annealing.
Machining parameters: cutting speed v = π·D·n/1000 (m/min), feed f (mm/rev), depth of cut a; material removal rate MRR ≈ v·f·a.
Welding: arc (electrode), MIG/MAG (continuous wire with gas shield), TIG (non-consumable tungsten electrode), resistance spot welding for sheet metal.
Joining and finishing: riveting, bolting, adhesive bonding; surface treatments (galvanising, anodising, painting) for corrosion protection.
CNC programming with G-codes (G00 rapid, G01 linear interpolation, G02/G03 circular) and M-codes (M03 spindle on, M08 coolant); coordinates relative (G91) or absolute (G90).
Automation: PLC ladder logic, sensors/actuators, open-loop vs closed-loop control, and industrial robots classified by kinematics (Cartesian, SCARA, articulated, delta).
Production metrics: takt time = available time/demand; OEE = availability × performance × quality; lean tools (JIT, kanban, 5S).
Engineering Project Methodology, Technical Drawing, Standards and ICT Tools
Project phases: idea/needs analysis → feasibility study → preliminary and detailed design → execution → testing → documentation and closure.
Project documents: memory (justification, calculations, annexes), plans/drawings, specification (pliego de condiciones) and budget (presupuesto with measurements and unit prices).
Gantt chart shows scheduling over time; PERT/CPM network computes critical path (longest path, zero slack) and earliest/latest times.
Technical drawing standards: ISO/UNE line types (continuous thick for visible edges, dashed for hidden), first-angle (European) vs third-angle projection, scale = drawing/real.
Dimensioning rules: dimension lines outside the figure, no redundant dimensions, tolerances and fits per ISO 286 (H7/g6), geometric tolerancing (GD&T symbols).
Views: front, top and side elevation; auxiliary views for inclined faces; sections and hatching at 45° to show internal features.
ICT tools: CAD (2D/3D parametric modelling), CAM, CAE (FEM simulation), spreadsheets for budgets, and collaborative project management software.
Budget calculation: total = Σ(unit price × quantity) plus indirect costs and VAT; amortisation and payback period evaluate economic viability.