GATE ME flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Euler's theorem (homogeneous functions)

If f(x,y) is homogeneous of degree n, then x∂f/∂x + y∂f/∂y = nf. Its partial derivatives are homogeneous of degree n−1, which is used to simplify Euler-type problems.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic equation. It allows powers of a matrix to be reduced and gives A⁻¹ as a polynomial in A for a nonsingular matrix.

Laplace transform of derivative

L{f′(t)} = sF(s) − f(0), and L{f″(t)} = s²F(s) − sf(0) − f′(0). These turn linear ODEs with initial conditions into algebraic equations in s.

Newton–Raphson convergence

The iteration x(n+1) = x(n) − f(x(n))/f′(x(n)) converges quadratically near a simple root. It fails if f′ is zero or nearly zero near the root.

Cauchy–Riemann equations

A complex function f = u + iv is analytic where ux = vy and uy = −vx, with continuous partial derivatives. Satisfying them lets u be the harmonic conjugate of v.

Simpson's 1/3 rule

Approximates ∫f dx by (h/3)[f0 + 4(f1 + f3 + …) + 2(f2 + f4 + …) + fn]. It requires an even number of intervals and has error of order h⁴.

Principal stresses by Mohr's circle

σ1,2 = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]. The maximum shear stress equals the circle radius and acts on planes at 45° to the principal planes.

Euler buckling load

Pcr = π²EI/(Leff)², where Leff depends on end conditions: L pinned–pinned, 0.5L fixed–fixed, 0.7L fixed–pinned and 2L fixed–free. It applies to long, slender columns.

Torsion formula for circular shafts

Shear stress τ = Tr/J and twist angle θ = TL/GJ. For a solid shaft J = πd⁴/32; for a hollow shaft J = π(do⁴ − di⁴)/32.

Goodman fatigue criterion

For fluctuating stress with alternating component σa and mean σm, failure is predicted by σa/σe + σm/σu = 1, where σe is the endurance limit and σu the ultimate strength.

Logarithmic decrement

For a damped free vibration, δ = ln(x1/x2) = 2πζ/√(1 − ζ²). For small damping, ζ ≈ δ/(2π), giving the damping ratio from successive amplitudes.

Grashof condition

A four-bar linkage has at least one fully rotating link when the sum of the shortest and longest links is less than or equal to the sum of the other two links.

Bernoulli equation

For steady, inviscid, incompressible flow along a streamline, p/ρg + V²/2g + z is constant. Each term is a head: pressure, velocity and elevation.

Reynolds number in pipes

Re = ρVD/μ = VD/ν. Pipe flow is laminar below about 2000, transitional between 2000 and 4000, and turbulent above 4000.

Hagen–Poiseuille flow

For fully developed laminar pipe flow, the pressure drop is Δp = 128μLQ/(πD⁴) and the friction factor is f = 64/Re. The velocity profile is parabolic.

Fin efficiency

For a rectangular fin with an adiabatic tip, η = tanh(mL)/(mL), where m = √(hP/kA). A longer fin or a higher conductivity increases efficiency.

Dittus–Boelter equation

For turbulent flow inside a smooth pipe, Nu = 0.023Re^0.8Pr^n, with n = 0.4 for heating and 0.3 for cooling of the fluid.

Stefan–Boltzmann law

The net radiation from a grey surface is Q = εσA(T⁴ − Tsurr⁴), where σ = 5.67×10⁻⁸ W/m²K⁴ and ε is the emissivity.

LMTD of a heat exchanger

LMTD = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2), using the terminal temperature differences. Counter-flow gives a higher LMTD than parallel flow for the same end temperatures.

Carnot efficiency

The maximum efficiency between reservoirs at Th and Tc is 1 − Tc/Th, with absolute temperatures. No real heat engine operating between the same reservoirs can exceed it.

Hall–Petch relationship

Yield strength σy = σ0 + kd^(−1/2), where d is the mean grain diameter. Refining grains raises strength and improves toughness by impeding dislocation motion.

Fick's first law

The diffusion flux is J = −D dC/dx, where D = D0 exp(−Q/RT). The negative sign shows that diffusion proceeds down the concentration gradient.

Chvorinov's rule

Solidification time t = B(V/A)², where V is volume and A is the heat-transfer surface area. A riser must have a larger V/A ratio than the casting it feeds.

Taylor's tool life equation

VTⁿ = C relates cutting speed V and tool life T. The exponent n is about 0.1–0.15 for HSS and 0.4–0.6 for carbide tools.

Merchant's shear angle relation

2φ + β − α = 90°, where φ is the shear angle, β the friction angle and α the rake angle. A larger rake angle raises the shear angle and lowers cutting force.

Economic order quantity

EOQ = √(2DS/H), with D annual demand, S ordering cost and H holding cost per unit per year. At the EOQ, ordering cost equals holding cost.

Critical path in CPM

The critical path is the longest path through the network and has zero total float. Any delay on it delays the project completion time.

Process capability index Cpk

Cpk = min[(USL − μ)/(3σ), (μ − LSL)/(3σ)]. It penalises off-centre processes, unlike Cp, which uses only the specification width and σ.

Rolling maximum draft

The maximum reduction per pass is Δhmax = μ²R, where μ is the coefficient of friction and R the roll radius. Larger rolls or higher friction allow a larger draft.

Successive percentage change

Changes of x% and y% give a net change of x + y + xy/100 percent. A rise then an equal fall of x% leaves a net loss of x²/100 percent.

Alligation rule

To mix components of value c and d to obtain a mean value m, the required quantities are in the ratio (m − c):(d − m), with c < m < d.

Difference of CI and SI

For two years at rate R, the difference between compound and simple interest is P(R/100)². For three years it is P(R/100)²(3 + R/100).