La Rioja PAU Tech & Engineering II (UR) cheat sheet

Tensile test gives the stress-strain curve: engineering stress σ = F/A₀, engineering strain ε = ΔL/L₀; the elastic zone obeys Hooke's law σ = E·ε, and the slope of the linear zone is Young's modulus E.

Ductility is measured as elongation at break A% = ((L_f − L₀)/L₀)·100 and reduction of area Z% = ((A₀ − A_f)/A₀)·100; brittle materials show A% below roughly 5 %.

Resilience (elastic energy per volume) equals the area under the elastic part, σ_y²/(2E); toughness equals the total area under the curve to fracture and is high only when strength and ductility are both high.

Hardness scales are not interchangeable: Brinell (HB, hard ball, coarse), Vickers (HV, diamond pyramid, universal), Rockwell (HRC cone for hard steels, HRB ball for soft metals); conversion tables are empirical.

The tensile strength of a metal is roughly proportional to its hardness times a constant (σ_u ≈ k·HB with k ≈ 3.3–3.5 MPa per HB unit for steels); this is a rule of thumb, not a law.

Crystal defects govern properties: point defects (vacancies, interstitials), line defects (dislocations explain plastic slip and work hardening), planar defects (grain boundaries: smaller grains give higher yield stress by the Hall-Petch relation σ_y = σ₀ + k·d^(−1/2)).

Cooling rate controls microstructure in steels: fast cooling gives martensite (hard, brittle), moderate gives bainite, slow gives pearlite/ferrite; the TTT and CCT diagrams map these paths, and the eutectoid composition is 0.77 % C at 727 °C.

Ashby material selection uses performance indices derived from the objective: for a light stiff tie-rod, maximize E/ρ; for a light stiff beam in bending, maximize E^(1/2)/ρ; for a light strong beam, maximize σ_y^(2/3)/ρ.

Corrosion protection is by barrier (coatings), sacrificial anode (zinc on steel, galvanizing), or cathodic protection; galvanic corrosion accelerates when two dissimilar metals in contact have a large potential difference and a small anode-to-cathode area ratio.

Static equilibrium of a rigid body requires ΣF_x = 0, ΣF_y = 0, ΣM = 0 (three equations in 2D); a two-force member carries force only along the line joining its pins, and a three-force member in equilibrium has concurrent lines of action.

A pin support gives two reaction components, a roller gives one perpendicular to the surface, and a fixed (built-in) support gives two forces plus a moment; the number of unknowns must not exceed the equilibrium equations for a statically determinate problem.

Normal stress σ = N/A and axial deformation δ = N·L/(E·A); the sign convention is tension positive, and a statically indeterminate bar is solved by combining compatibility of deformations with equilibrium.

Bending: the elastic flexure formula is σ = M·y/I, maximum at y = c; the section modulus is W = I/c, so σ_max = M/W. The neutral axis passes through the centroid of the section.

Deflection of a simply supported beam with a central point load P is δ = P·L³/(48·E·I); for a uniformly distributed load q it is δ = 5·q·L⁴/(384·E·I); a cantilever with an end load P deflects δ = P·L³/(3·E·I).

Shear stress in a beam is τ = V·Q/(I·b), maximum at the neutral axis for a rectangular section (τ_max = 1.5·V/A); torsion of a circular shaft gives τ = T·r/J with J = π·d⁴/32 for a solid shaft.

Buckling of a slender column follows Euler's formula P_cr = π²·E·I/(L_e)², where the effective length L_e depends on end conditions (pinned-pinned L, fixed-free 2L, fixed-pinned 0.7L, fixed-fixed 0.5L); validity requires slenderness above the limit λ = √(π²·E/σ_p).

Mechanisms: the degree of freedom of a planar linkage is M = 3·(n − 1) − 2·j₁ − j₂ (Grübler/Kutzbach), where n is the number of links, j₁ lower pairs and j₂ higher pairs; a four-bar linkage is a crank-rocker, double-crank or double-rocker depending on Grashof's condition s + l ≤ p + q.

Gear trains: the velocity ratio is i = ω_in/ω_out = z_out/z_in for a single pair; for a compound train multiply the ratios of each stage, and the sign depends on the number of external meshes (each external mesh reverses direction).

Power transmission relations: P = M·ω = F·v, ω = 2π·n/60 with n in rpm; belt drives give i = d₂/d₁ with slip reducing the real ratio, and a screw thread converts torque to axial force by F = 2π·M/(p·η) with p the lead.

DC circuits: Ohm's law V = I·R, series resistances add, parallel conductances add (R_p = R₁R₂/(R₁+R₂)); Kirchhoff's current law (ΣI at a node = 0) and voltage law (ΣV around a loop = 0) solve any linear network.

The superposition theorem applies only to linear circuits; Thévenin's theorem replaces any two-terminal linear network by V_th in series with R_th, where V_th is the open-circuit voltage and R_th the resistance seen with sources killed (voltage sources shorted, current sources opened).

Maximum power transfer to a load occurs when R_L = R_th, giving P_max = V_th²/(4·R_th) and an efficiency of 50 % at that point.

AC: for sinusoidal quantities the RMS value is V_p/√2; impedance of a resistor is R, of an inductor jωL, of a capacitor 1/(jωC); resonance occurs at f₀ = 1/(2π√(LC)) where the impedance is purely resistive and the power factor is 1.

Three-phase systems: in a star (Y) connection the line voltage is √3 times the phase voltage and the line current equals the phase current; in delta the line current is √3 times the phase current; total balanced power P = √3·V_L·I_L·cos φ.

Power factor correction: the capacitor bank needed to raise cos φ from cos φ₁ to cos φ₂ is Q_C = P·(tan φ₁ − tan φ₂), connected in parallel with the load; this reduces line current and losses without changing the active power consumed.

Transformers: the voltage ratio equals the turns ratio V₁/V₂ = N₁/N₂ = I₂/I₁ (ideal); real transformers are characterized by short-circuit (copper) and open-circuit (iron) tests, and the equivalent circuit refers impedances to one side by the square of the turns ratio.

DC machines: the induced EMF is E = k·Φ·ω (or E = p·Z·Φ·n/(60·a)) and torque T = k·Φ·I_a; a shunt motor has a nearly constant speed, a series motor has high starting torque and must never run unloaded.

Power electronics: a diode rectifies uncontrolled, a thyristor (SCR) conducts after a gate pulse and turns off at zero current, a MOSFET suits high-frequency low/medium power, an IGBT suits medium/high power; a buck converter steps voltage down with duty cycle D (V_o = D·V_i) and a boost converter steps up (V_o = V_i/(1 − D)).

Operational amplifiers: the ideal op-amp has infinite input impedance, zero output impedance and infinite open-loop gain; the inverting configuration gives V_o = −(R_f/R_in)·V_i and the non-inverting gives V_o = (1 + R_f/R₁)·V_i.

Energy is conserved but exergy is destroyed: the first law gives the energy balance, the second law limits conversion efficiency to the Carnot value η = 1 − T_cold/T_hot (absolute temperatures), which no real cycle can exceed.

Thermal power plants: Rankine cycle efficiency improves by raising steam pressure and temperature and by lowering condenser pressure; the condenser rejects the large latent heat, which is why real efficiencies are around 35–45 % for fossil plants.

Renewable capacity factor is the ratio of actual annual energy output to the energy the plant would produce at rated power all year; solar PV reaches roughly 15–25 % in Spain and wind roughly 25–40 % depending on site.

Wind power scales with the cube of wind speed: P = ½·ρ·A·v³·C_p, where the Betz limit caps C_p at 16/27 ≈ 0.593; the swept area A = π·R², so doubling the rotor radius quadruples the power.

Hydropower: P = ρ·g·Q·H·η, with Q the flow in m³/s and H the net head in metres; pumped storage is currently the dominant grid-scale electricity storage technology.

Solar PV: a cell's I-V curve has a short-circuit current I_sc, an open-circuit voltage V_oc and a maximum power point where the fill factor FF = P_max/(V_oc·I_sc) is highest; irradiance sets the current and cell temperature mainly reduces the voltage.

Transmission losses in a line are P_loss = I²·R = (P/V)²·R for a given transmitted power, which is why high voltage is used for long-distance transport; a transformer changes voltage with negligible power loss in the ideal case.

Efficiency chain: overall efficiency is the product of the stage efficiencies; a motor-pump set with 90 % motor, 95 % coupling and 70 % pump efficiency delivers 0.90·0.95·0.70 ≈ 0.60 of the input power.

Sustainability metrics: the energy payback time is the energy invested divided by the annual energy produced; CO₂ intensity is expressed in g CO₂/kWh and life-cycle values are roughly 10–50 for nuclear and wind, 400–1000 for coal.

A closed-loop control system compares the reference r(t) with the measured output y(t) to form the error e = r − y; the controller acts on the error, and feedback reduces sensitivity to disturbances and parameter variation at the cost of possible instability.

PID control: u = K_p·e + K_i·∫e dt + K_d·de/dt; K_p reduces rise time, K_i eliminates steady-state error but adds overshoot and can destabilize, K_d damps and improves settling but amplifies noise.

A first-order system G(s) = K/(τs + 1) reaches 63.2 % of its final value in one time constant τ and is essentially settled at 4τ (2 % band); a second-order system is characterized by damping ratio ζ and natural frequency ω_n, with ζ < 1 giving oscillatory response.

Stability in the s-plane requires all closed-loop poles to have negative real parts; the Routh-Hurwitz criterion detects sign changes in the first column and hence right-half-plane poles without solving the polynomial.

Sensors: a potentiometer measures angular position, an LVDT measures linear displacement, a thermocouple generates a Seebeck EMF proportional to temperature difference, a Pt100 changes resistance with temperature, an incremental encoder gives pulses proportional to rotation and an absolute encoder gives a coded position.

Actuators: a DC motor's speed is set by armature voltage, a stepper motor moves in fixed steps (typical 1.8° per full step, 200 steps per revolution) and holds position without feedback, a servo motor closes the loop internally, and a solenoid gives two-position action.

A PLC executes cyclically: read inputs into the image table, solve the program, update outputs; the scan time is the sum of these phases, and a watchdog timer faults the CPU if a scan exceeds the limit.

PLC programming per IEC 61131-3 uses ladder diagram (LD), function block diagram (FBD), structured text (ST), instruction list (IL) and sequential function chart (SFC); ladder contacts in series implement AND, in parallel OR, and a normally closed contact negates a signal.

Robot geometry: a Cartesian robot has three prismatic axes, a SCARA has two revolute horizontal axes plus a vertical prismatic axis, and a six-axis anthropomorphic arm has six revolute joints; the workspace is the reachable set of end-effector positions.

Degrees of freedom and mobility follow the same Grübler formula as linkages; forward kinematics computes the end-effector pose from joint values and inverse kinematics the reverse, which may have multiple solutions.

First-angle projection (ISO, used in Europe) places the view on the far side of the object: the top view goes below the front view and the left-side view goes to the right; third-angle projection (ASME, US) places views on the near side.

The three principal views align: the front and top views share the same width, the front and side views share the same height, and the top and side views share the same depth - the rule of correspondence of projections.

Line types: continuous thick for visible edges, continuous thin for dimension and projection lines, dashed for hidden edges, dash-dot thin for centre and symmetry axes, and continuous thin wavy for break lines.

Scale is the drawing-to-reality ratio; an enlargement has a scale greater than 1 (e.g. 2:1), a reduction less than 1 (e.g. 1:5); standard ISO 5455 series include 1:1, 1:2, 1:5, 1:10, 2:1, 5:1.

Dimensioning rules per ISO 129: dimensions are given in true size regardless of scale, the unit (mm) is stated once in the title block and not repeated, dimension lines are not crossed by other lines, and each feature is dimensioned once only - no duplicates.

Tolerances and fits: the upper and lower deviations define the tolerance zone; a hole basis system uses H for the hole (e.g. H7) and letters a–h for clearance, j–n for transition and p–zc for interference shaft fits; ISO tolerance grades IT01 to IT18 decrease in precision as the number rises.

Surface roughness is specified by Ra in micrometres with a symbol; smaller Ra means a smoother surface and higher cost, so roughness is specified only where functionally needed.

Sections and cuts: a full section removes half the object along a cutting plane, a half section shows one quarter removed for symmetric parts, and hatching at 45° thin lines indicates cut material; adjacent parts are hatched in opposite directions.

CAD practice: parametric modelling drives geometry from dimensions and constraints, so editing a parameter regenerates the model; layers organize line types and colours, blocks/assemblies reuse components, and drawings are derived from the 3D model so changes propagate.

Project phases in engineering: identify the need, feasibility study, preliminary design, detailed design, construction/manufacturing, testing and commissioning, and closure with as-built documentation; each phase ends in a review gate.

The project documents typically comprise memory (objectives, calculations, annexes), plans (drawings), specifications (materials and tests), budget and schedule; the memory justifies the technical decisions and the budget quantifies them.

Budget structure: unit price times quantity gives the execution material cost; add labour, indirect costs (typically a percentage of direct cost) and industrial profit, then apply VAT to obtain the total contract price.

Technical documentation standards: drawings follow ISO/UNE-EN ISO, electrical installations the Low Voltage Electrotechnical Regulation (REBT), and products the CE marking directives; the declaration of conformity and technical file are legally required for CE.

Design methodology: the VDI 2221/2222 approach separates clarifying the task, conceptual design (functional structures, working principles), embodiment and detail; a functional analysis uses a black box with inputs, outputs and a function structure.

Design for X: DFM simplifies manufacturing, DFA reduces part count and assembly time, DFE minimizes environmental impact; a bill of materials (BOM) lists every component with reference, description and quantity.

Quality tools: a Pareto chart ranks causes by frequency (roughly 80 % of problems from 20 % of causes), a cause-effect (Ishikawa) diagram groups causes into categories, and FMEA scores severity, occurrence and detection to give a risk priority number.

ICT tools in projects: spreadsheets for budgets and calculations, CAD for geometry, CAE/FEA for simulation, PDM/PLM for version and lifecycle control, and Gantt or critical-path scheduling for time planning; the critical path is the longest sequence of dependent tasks and sets the minimum project duration.

Tensile test gives the stress-strain curve: engineering stress σ = F/A₀, engineering strain ε = ΔL/L₀; the elastic zone obeys Hooke's law σ = E·ε, and the slope of the linear zone is Young's modulus E.

Ductility is measured as elongation at break A% = ((L_f − L₀)/L₀)·100 and reduction of area Z% = ((A₀ − A_f)/A₀)·100; brittle materials show A% below roughly 5 %.

Resilience (elastic energy per volume) equals the area under the elastic part, σ_y²/(2E); toughness equals the total area under the curve to fracture and is high only when strength and ductility are both high.

Hardness scales are not interchangeable: Brinell (HB, hard ball, coarse), Vickers (HV, diamond pyramid, universal), Rockwell (HRC cone for hard steels, HRB ball for soft metals); conversion tables are empirical.

The tensile strength of a metal is roughly proportional to its hardness times a constant (σ_u ≈ k·HB with k ≈ 3.3–3.5 MPa per HB unit for steels); this is a rule of thumb, not a law.

Crystal defects govern properties: point defects (vacancies, interstitials), line defects (dislocations explain plastic slip and work hardening), planar defects (grain boundaries: smaller grains give higher yield stress by the Hall-Petch relation σ_y = σ₀ + k·d^(−1/2)).

Cooling rate controls microstructure in steels: fast cooling gives martensite (hard, brittle), moderate gives bainite, slow gives pearlite/ferrite; the TTT and CCT diagrams map these paths, and the eutectoid composition is 0.77 % C at 727 °C.

Ashby material selection uses performance indices derived from the objective: for a light stiff tie-rod, maximize E/ρ; for a light stiff beam in bending, maximize E^(1/2)/ρ; for a light strong beam, maximize σ_y^(2/3)/ρ.

Corrosion protection is by barrier (coatings), sacrificial anode (zinc on steel, galvanizing), or cathodic protection; galvanic corrosion accelerates when two dissimilar metals in contact have a large potential difference and a small anode-to-cathode area ratio.

Static equilibrium of a rigid body requires ΣF_x = 0, ΣF_y = 0, ΣM = 0 (three equations in 2D); a two-force member carries force only along the line joining its pins, and a three-force member in equilibrium has concurrent lines of action.

A pin support gives two reaction components, a roller gives one perpendicular to the surface, and a fixed (built-in) support gives two forces plus a moment; the number of unknowns must not exceed the equilibrium equations for a statically determinate problem.

Normal stress σ = N/A and axial deformation δ = N·L/(E·A); the sign convention is tension positive, and a statically indeterminate bar is solved by combining compatibility of deformations with equilibrium.

Bending: the elastic flexure formula is σ = M·y/I, maximum at y = c; the section modulus is W = I/c, so σ_max = M/W. The neutral axis passes through the centroid of the section.

Deflection of a simply supported beam with a central point load P is δ = P·L³/(48·E·I); for a uniformly distributed load q it is δ = 5·q·L⁴/(384·E·I); a cantilever with an end load P deflects δ = P·L³/(3·E·I).

Shear stress in a beam is τ = V·Q/(I·b), maximum at the neutral axis for a rectangular section (τ_max = 1.5·V/A); torsion of a circular shaft gives τ = T·r/J with J = π·d⁴/32 for a solid shaft.

Buckling of a slender column follows Euler's formula P_cr = π²·E·I/(L_e)², where the effective length L_e depends on end conditions (pinned-pinned L, fixed-free 2L, fixed-pinned 0.7L, fixed-fixed 0.5L); validity requires slenderness above the limit λ = √(π²·E/σ_p).

Mechanisms: the degree of freedom of a planar linkage is M = 3·(n − 1) − 2·j₁ − j₂ (Grübler/Kutzbach), where n is the number of links, j₁ lower pairs and j₂ higher pairs; a four-bar linkage is a crank-rocker, double-crank or double-rocker depending on Grashof's condition s + l ≤ p + q.

Gear trains: the velocity ratio is i = ω_in/ω_out = z_out/z_in for a single pair; for a compound train multiply the ratios of each stage, and the sign depends on the number of external meshes (each external mesh reverses direction).

Power transmission relations: P = M·ω = F·v, ω = 2π·n/60 with n in rpm; belt drives give i = d₂/d₁ with slip reducing the real ratio, and a screw thread converts torque to axial force by F = 2π·M/(p·η) with p the lead.

DC circuits: Ohm's law V = I·R, series resistances add, parallel conductances add (R_p = R₁R₂/(R₁+R₂)); Kirchhoff's current law (ΣI at a node = 0) and voltage law (ΣV around a loop = 0) solve any linear network.

The superposition theorem applies only to linear circuits; Thévenin's theorem replaces any two-terminal linear network by V_th in series with R_th, where V_th is the open-circuit voltage and R_th the resistance seen with sources killed (voltage sources shorted, current sources opened).

Maximum power transfer to a load occurs when R_L = R_th, giving P_max = V_th²/(4·R_th) and an efficiency of 50 % at that point.

AC: for sinusoidal quantities the RMS value is V_p/√2; impedance of a resistor is R, of an inductor jωL, of a capacitor 1/(jωC); resonance occurs at f₀ = 1/(2π√(LC)) where the impedance is purely resistive and the power factor is 1.

Three-phase systems: in a star (Y) connection the line voltage is √3 times the phase voltage and the line current equals the phase current; in delta the line current is √3 times the phase current; total balanced power P = √3·V_L·I_L·cos φ.

Power factor correction: the capacitor bank needed to raise cos φ from cos φ₁ to cos φ₂ is Q_C = P·(tan φ₁ − tan φ₂), connected in parallel with the load; this reduces line current and losses without changing the active power consumed.

Transformers: the voltage ratio equals the turns ratio V₁/V₂ = N₁/N₂ = I₂/I₁ (ideal); real transformers are characterized by short-circuit (copper) and open-circuit (iron) tests, and the equivalent circuit refers impedances to one side by the square of the turns ratio.

DC machines: the induced EMF is E = k·Φ·ω (or E = p·Z·Φ·n/(60·a)) and torque T = k·Φ·I_a; a shunt motor has a nearly constant speed, a series motor has high starting torque and must never run unloaded.

Power electronics: a diode rectifies uncontrolled, a thyristor (SCR) conducts after a gate pulse and turns off at zero current, a MOSFET suits high-frequency low/medium power, an IGBT suits medium/high power; a buck converter steps voltage down with duty cycle D (V_o = D·V_i) and a boost converter steps up (V_o = V_i/(1 − D)).

Operational amplifiers: the ideal op-amp has infinite input impedance, zero output impedance and infinite open-loop gain; the inverting configuration gives V_o = −(R_f/R_in)·V_i and the non-inverting gives V_o = (1 + R_f/R₁)·V_i.

Energy is conserved but exergy is destroyed: the first law gives the energy balance, the second law limits conversion efficiency to the Carnot value η = 1 − T_cold/T_hot (absolute temperatures), which no real cycle can exceed.

Thermal power plants: Rankine cycle efficiency improves by raising steam pressure and temperature and by lowering condenser pressure; the condenser rejects the large latent heat, which is why real efficiencies are around 35–45 % for fossil plants.

Renewable capacity factor is the ratio of actual annual energy output to the energy the plant would produce at rated power all year; solar PV reaches roughly 15–25 % in Spain and wind roughly 25–40 % depending on site.

Wind power scales with the cube of wind speed: P = ½·ρ·A·v³·C_p, where the Betz limit caps C_p at 16/27 ≈ 0.593; the swept area A = π·R², so doubling the rotor radius quadruples the power.

Hydropower: P = ρ·g·Q·H·η, with Q the flow in m³/s and H the net head in metres; pumped storage is currently the dominant grid-scale electricity storage technology.

Solar PV: a cell's I-V curve has a short-circuit current I_sc, an open-circuit voltage V_oc and a maximum power point where the fill factor FF = P_max/(V_oc·I_sc) is highest; irradiance sets the current and cell temperature mainly reduces the voltage.

Transmission losses in a line are P_loss = I²·R = (P/V)²·R for a given transmitted power, which is why high voltage is used for long-distance transport; a transformer changes voltage with negligible power loss in the ideal case.

Efficiency chain: overall efficiency is the product of the stage efficiencies; a motor-pump set with 90 % motor, 95 % coupling and 70 % pump efficiency delivers 0.90·0.95·0.70 ≈ 0.60 of the input power.

Sustainability metrics: the energy payback time is the energy invested divided by the annual energy produced; CO₂ intensity is expressed in g CO₂/kWh and life-cycle values are roughly 10–50 for nuclear and wind, 400–1000 for coal.

A closed-loop control system compares the reference r(t) with the measured output y(t) to form the error e = r − y; the controller acts on the error, and feedback reduces sensitivity to disturbances and parameter variation at the cost of possible instability.

PID control: u = K_p·e + K_i·∫e dt + K_d·de/dt; K_p reduces rise time, K_i eliminates steady-state error but adds overshoot and can destabilize, K_d damps and improves settling but amplifies noise.

A first-order system G(s) = K/(τs + 1) reaches 63.2 % of its final value in one time constant τ and is essentially settled at 4τ (2 % band); a second-order system is characterized by damping ratio ζ and natural frequency ω_n, with ζ < 1 giving oscillatory response.

Stability in the s-plane requires all closed-loop poles to have negative real parts; the Routh-Hurwitz criterion detects sign changes in the first column and hence right-half-plane poles without solving the polynomial.

Sensors: a potentiometer measures angular position, an LVDT measures linear displacement, a thermocouple generates a Seebeck EMF proportional to temperature difference, a Pt100 changes resistance with temperature, an incremental encoder gives pulses proportional to rotation and an absolute encoder gives a coded position.

Actuators: a DC motor's speed is set by armature voltage, a stepper motor moves in fixed steps (typical 1.8° per full step, 200 steps per revolution) and holds position without feedback, a servo motor closes the loop internally, and a solenoid gives two-position action.

A PLC executes cyclically: read inputs into the image table, solve the program, update outputs; the scan time is the sum of these phases, and a watchdog timer faults the CPU if a scan exceeds the limit.

PLC programming per IEC 61131-3 uses ladder diagram (LD), function block diagram (FBD), structured text (ST), instruction list (IL) and sequential function chart (SFC); ladder contacts in series implement AND, in parallel OR, and a normally closed contact negates a signal.

Robot geometry: a Cartesian robot has three prismatic axes, a SCARA has two revolute horizontal axes plus a vertical prismatic axis, and a six-axis anthropomorphic arm has six revolute joints; the workspace is the reachable set of end-effector positions.

Degrees of freedom and mobility follow the same Grübler formula as linkages; forward kinematics computes the end-effector pose from joint values and inverse kinematics the reverse, which may have multiple solutions.

First-angle projection (ISO, used in Europe) places the view on the far side of the object: the top view goes below the front view and the left-side view goes to the right; third-angle projection (ASME, US) places views on the near side.

The three principal views align: the front and top views share the same width, the front and side views share the same height, and the top and side views share the same depth - the rule of correspondence of projections.

Line types: continuous thick for visible edges, continuous thin for dimension and projection lines, dashed for hidden edges, dash-dot thin for centre and symmetry axes, and continuous thin wavy for break lines.

Scale is the drawing-to-reality ratio; an enlargement has a scale greater than 1 (e.g. 2:1), a reduction less than 1 (e.g. 1:5); standard ISO 5455 series include 1:1, 1:2, 1:5, 1:10, 2:1, 5:1.

Dimensioning rules per ISO 129: dimensions are given in true size regardless of scale, the unit (mm) is stated once in the title block and not repeated, dimension lines are not crossed by other lines, and each feature is dimensioned once only - no duplicates.

Tolerances and fits: the upper and lower deviations define the tolerance zone; a hole basis system uses H for the hole (e.g. H7) and letters a–h for clearance, j–n for transition and p–zc for interference shaft fits; ISO tolerance grades IT01 to IT18 decrease in precision as the number rises.

Surface roughness is specified by Ra in micrometres with a symbol; smaller Ra means a smoother surface and higher cost, so roughness is specified only where functionally needed.

Sections and cuts: a full section removes half the object along a cutting plane, a half section shows one quarter removed for symmetric parts, and hatching at 45° thin lines indicates cut material; adjacent parts are hatched in opposite directions.

CAD practice: parametric modelling drives geometry from dimensions and constraints, so editing a parameter regenerates the model; layers organize line types and colours, blocks/assemblies reuse components, and drawings are derived from the 3D model so changes propagate.

Project phases in engineering: identify the need, feasibility study, preliminary design, detailed design, construction/manufacturing, testing and commissioning, and closure with as-built documentation; each phase ends in a review gate.

The project documents typically comprise memory (objectives, calculations, annexes), plans (drawings), specifications (materials and tests), budget and schedule; the memory justifies the technical decisions and the budget quantifies them.

Budget structure: unit price times quantity gives the execution material cost; add labour, indirect costs (typically a percentage of direct cost) and industrial profit, then apply VAT to obtain the total contract price.

Technical documentation standards: drawings follow ISO/UNE-EN ISO, electrical installations the Low Voltage Electrotechnical Regulation (REBT), and products the CE marking directives; the declaration of conformity and technical file are legally required for CE.

Design methodology: the VDI 2221/2222 approach separates clarifying the task, conceptual design (functional structures, working principles), embodiment and detail; a functional analysis uses a black box with inputs, outputs and a function structure.

Design for X: DFM simplifies manufacturing, DFA reduces part count and assembly time, DFE minimizes environmental impact; a bill of materials (BOM) lists every component with reference, description and quantity.

Quality tools: a Pareto chart ranks causes by frequency (roughly 80 % of problems from 20 % of causes), a cause-effect (Ishikawa) diagram groups causes into categories, and FMEA scores severity, occurrence and detection to give a risk priority number.

ICT tools in projects: spreadsheets for budgets and calculations, CAD for geometry, CAE/FEA for simulation, PDM/PLM for version and lifecycle control, and Gantt or critical-path scheduling for time planning; the critical path is the longest sequence of dependent tasks and sets the minimum project duration.