AMIE Section B Mechanical cheat sheet

Euler buckling load: P_cr = π²EI/(L_e)², with effective length L_e = KL (K = 0.5 fixed-fixed, 0.7 pinned-fixed, 1.0 pinned-pinned, 2.0 fixed-free).

Second-order linear ODE with constant coefficients: solve the characteristic equation; distinct real roots give exponentials, repeated roots give (A + Bx)e^{rx}, complex roots give e^{αx}(A cos βx + B sin βx).

Laplace transforms: L{f'} = sF(s) − f(0); L{e^{at}} = 1/(s − a); L{sin ωt} = ω/(s² + ω²); use partial fractions for inverse transforms.

For a function continuous on [a, b], the mean value theorem gives f'(c) = (f(b) − f(a))/(b − a) for some c in (a, b).

Matrix solution of linear systems: AX = B is consistent only if rank(A) = rank([A|B]); a unique solution requires rank = number of unknowns.

Normal distribution: about 68% of values lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

Fourier series of period 2π: a₀/2 + Σ(aₙ cos nx + bₙ sin nx); coefficients are obtained by integrating f(x) times the corresponding sine or cosine over one period.

For small oscillations of a simple pendulum, the period is T = 2π√(L/g), independent of mass and, for small amplitudes, of amplitude.

First-angle projection places the object between observer and plane; the view from above is drawn below the front view, and the view from the left is drawn on the right.

Third-angle projection places the plane between observer and object; the top view is drawn above the front view and the left-side view on the left.

A section view uses uniformly spaced 45° hatching lines to indicate cut material; adjacent parts in an assembly get hatching in opposite directions.

Dimension lines are drawn thin and terminate in arrowheads; extension lines start about 1 mm from the object outline and extend about 2 mm beyond the dimension line.

An isometric drawing uses axes at 30° to the horizontal on both sides of the vertical; true lengths are drawn along the axes, and circles appear as ellipses.

Tolerance is the permitted variation (upper limit minus lower limit); allowance is the intentional difference between mating part sizes (clearance positive, interference negative).

Geometric tolerancing symbols: flatness and straightness need no datum; perpendicularity, parallelism, and position require a datum reference.

A thread is designated by nominal diameter and pitch, e.g. M20 × 2.5; the pitch is the axial distance between corresponding points on adjacent threads.

First law for a closed system: Q − W = ΔU; for a steady-flow open system: q − w = Δh + ΔKE + ΔPE.

For an ideal gas, ΔU = m c_v ΔT and ΔH = m c_p ΔT; c_p − c_v = R and γ = c_p/c_v.

Carnot efficiency of a heat engine: η = 1 − T_L/T_H (absolute temperatures); COP of a refrigerator = T_L/(T_H − T_L).

Entropy change for an ideal gas: Δs = c_p ln(T₂/T₁) − R ln(P₂/P₁); for a reversible adiabatic process, Δs = 0.

Fourier's law of conduction: q = −kA dT/dx; for a plane wall, q = kA(T₁ − T₂)/L.

Newton's law of cooling: q = hA(T_s − T_∞); the Biot number Bi = hL_c/k indicates whether internal or surface resistance dominates.

Stefan–Boltzmann law: q = εσA(T_s⁴ − T_surr⁴), with σ = 5.67 × 10⁻⁸ W/(m²·K⁴); a black body has ε = 1.

For a heat exchanger, LMTD = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂); the effectiveness–NTU method applies when outlet temperatures are unknown.

Continuity for incompressible flow: A₁V₁ = A₂V₂; mass conservation in general: ρ₁A₁V₁ = ρ₂A₂V₂.

Bernoulli's equation along a streamline: P/ρg + V²/2g + z = constant, valid for steady, inviscid, incompressible flow.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below about 2000 and turbulent above about 4000.

Darcy–Weisbach head loss: h_f = f(L/D)(V²/2g); the friction factor f = 64/Re for laminar flow.

Hydraulic turbine efficiency: η_o = shaft power/(ρgQH); specific speed N_s = N√P/H^{5/4} classifies turbine type (Pelton low, Francis medium, Kaplan high).

Centrifugal pump: head H = (P₂ − P₁)/ρg + (V₂² − V₁²)/2g + (z₂ − z₁); cavitation occurs when local pressure falls below vapour pressure.

Pitot tube measures velocity from stagnation pressure: V = √(2(P_stag − P_static)/ρ).

Hydraulic jump converts supercritical to subcritical flow; Froude number Fr = V/√(gD) distinguishes the regimes (Fr < 1 subcritical).

Axial stress σ = P/A; elastic strain ε = σ/E; Poisson's ratio ν = −lateral strain/longitudinal strain.

Bending stress σ = My/I, with maximum stress σ_max = M/Z; for a rectangular section Z = bd²/6 and I = bd³/12.

Torsion of a circular shaft: τ = Tr/J, with J = πd⁴/32 for a solid shaft; angle of twist θ = TL/GJ.

Principal stresses: σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ²]; maximum shear stress equals half the difference of principal stresses.

Euler column formula applies when the slenderness ratio exceeds the critical value; below it, use the Rankine–Gordon formula.

Gear train: velocity ratio = N_driven/N_driver = product of driven teeth/product of driver teeth; simple gear trains with idlers change direction only.

Four-bar linkage Grashof condition: s + l ≤ p + q gives at least one full rotation of the shortest link relative to its neighbours.

Critical speed of a shaft: ω_c = √(g/δ), where δ is the static deflection; operation near critical speed causes large vibration.

Allowable stress design: σ_allow = S_y/FOS for ductile materials and S_u/FOS for brittle materials.

Soderberg, Goodman, and Gerber criteria combine mean and alternating stresses; Goodman: σ_a/S_e + σ_m/S_u = 1.

For a bolt under initial preload F_i and external load P, the added load on the bolt is K_b P/(K_b + K_c), where K_b and K_c are bolt and clamped-member stiffnesses.

Power transmitted by a belt: P = (T₁ − T₂)V; the ratio T₁/T₂ ≤ e^{μθ} limits the torque before slip.

Flat key width is typically d/4 and height d/6 of the shaft diameter; the key is designed for shear and crushing.

Journal bearing Sommerfeld number S = (μN/P)(r/c)²; minimum film thickness must exceed the surface finish to avoid metal contact.

S-N curve: fatigue strength falls with cycles and levels off at the endurance limit S_e (about 0.5 S_u for steels) beyond about 10⁶ cycles.

Spring rate of a helical compression spring: k = Gd⁴/(8D³n), where d is wire diameter, D mean coil diameter, and n active coils.

Cutting tool life follows Taylor's equation: VTⁿ = C; n is about 0.1–0.15 for HSS and 0.2–0.4 for carbide.

Merchant's shear angle relation: 2φ + β − α = 90°, where β is friction angle and α is rake angle.

Metal cutting: chip thickness ratio r = t₁/t₂ = sin φ/cos(φ − α); cutting force components are measured with a dynamometer.

Casting: riser volume must exceed shrinkage volume; Chvorinov's rule gives solidification time t = C(V/A)².

Welding: heat input per unit length = VI/v; arc welding efficiency converts electrical power to heat in the joint.

Gauge (sheet thickness) decreases as gauge number increases; the drawing operation reduces diameter while thickness stays nearly constant.

CNC interpolation: G01 is linear, G02 clockwise arc, G03 counter-clockwise arc; feed rate is set by the F word.

Metrology: least count of a vernier caliper = 1 main scale division − 1 vernier scale division; micrometer least count is usually 0.01 mm.

Economic order quantity: EOQ = √(2DS/H), where D is annual demand, S ordering cost, and H holding cost per unit per year.

Break-even quantity = fixed cost/(selling price − variable cost per unit).

PERT expected time: t_e = (a + 4m + b)/6; variance = ((b − a)/6)².

Critical path is the longest path through the network; its total float is zero, and it sets the project duration.

Work measurement: standard time = observed time × rating factor × (1 + allowance).

Learning curve: time for the nth unit T_n = T₁ n^{log r/log 2}, where r is the learning rate (commonly 80%).

CPM crashing trades extra cost for reduced duration; the crash slope = (crash cost − normal cost)/(normal time − crash time).

Inventory reorder point = demand during lead time + safety stock; ABC analysis classifies items by annual consumption value.

Ohm's law V = IR; series resistances add, parallel conductances add: 1/R_p = Σ1/R_i.

Power in DC circuits: P = VI = I²R = V²/R; for three-phase systems P = √3 V_L I_L cos φ.

For an RLC series circuit, resonant frequency f₀ = 1/(2π√(LC)); at resonance impedance is purely resistive and current is maximum.

Transformer turns ratio: V₂/V₁ = N₂/N₁ = I₁/I₂; ideal transformer conserves apparent power.

Three-phase star connection: line voltage = √3 × phase voltage; delta connection: line current = √3 × phase current.

Diode forward drop is about 0.7 V for silicon and 0.3 V for germanium; a half-wave rectifier produces ripple frequency equal to supply frequency.

Transistor current gain β = I_C/I_B; for a common-emitter amplifier, voltage gain is negative, indicating phase inversion.

Induction motor synchronous speed N_s = 120f/P; slip s = (N_s − N_r)/N_s and rotor frequency = s × supply frequency.

Program control structures: sequence, selection (if/else, switch), and iteration (for, while, do-while); each loop needs a terminating condition.

Array indexing in C starts at 0; an array of n elements has valid indices 0 to n − 1, and overflow is undefined behaviour.

Pointer holds an address; dereferencing accesses the value stored there. Passing a pointer to a function allows the caller's variable to be modified.

Call by value copies the argument; call by reference passes the address, so changes inside the function affect the original variable.

A stack is LIFO and a queue is FIFO; both support insertion and deletion, but at different ends.

Time complexity: linear search is O(n), binary search O(log n), bubble sort O(n²), and merge sort O(n log n).

A database relation in third normal form has no transitive dependency of a non-key attribute on the primary key.

File handling in C: fopen returns a FILE pointer, fclose releases it; fscanf/fprintf handle formatted I/O, and EOF signals end of file.

A report structure: title, abstract, objectives, methodology, results, discussion, conclusion, and references; each section serves a distinct purpose.

Calibration compares instrument readings against a standard of known value; the correction is standard reading minus indicated reading.

Accuracy is closeness to the true value; precision is repeatability of readings; a measurement can be precise but inaccurate.

Propagation of uncertainty: for a sum or difference, absolute uncertainties add; for a product or quotient, relative uncertainties add.

Systematic errors shift all readings in one direction; random errors scatter about a mean and are reduced by averaging repeated readings.

A control experiment holds all variables constant except the one being tested, isolating the cause of an observed effect.

Plot a calibration curve with the independent variable on the x-axis and fit a straight line; the slope gives sensitivity.

Results must be reproducible: record raw data, instrument model, and conditions so another worker can repeat the test.