PRC Mechanical Engineer cheat sheet

Shaft diameter under combined steady bending M and torsion T (maximum shear theory): d^3 = 16/(pi*tau_allow) * sqrt(M^2 + T^2).

Belt tension ratio: flat belt T1/T2 = e^(mutheta); V-belt T1/T2 = e^(mutheta/sin beta), where beta is the half groove angle and theta the arc of contact in radians.

Helical spring: Wahl factor Kw = (4C - 1)/(4C - 4) + 0.615/C with spring index C = D/d; maximum shear stress tau = Kw8FD/(pid^3).

Bearing rating life L10 = (C/P)^p * 10^6 revolutions, with p = 3 for ball bearings and p = 10/3 for roller bearings.

Fatigue design lines: Soderberg sa/Se + sm/Sy = 1/n; Goodman sa/Se + sm/Su = 1/n; Gerber is parabolic and the least conservative of the three.

Clutch torque capacity: uniform wear T = muWn*(ro + ri)/2; uniform pressure T = (2/3)muWn(ro^3 - ri^3)/(ro^2 - ri^2).

Rectangular key: shear tau = 2T/(dwL) and bearing (crushing) stress sigma = 4T/(dhL), where h is key thickness and L key length.

Gear Lewis equation: bending stress sigma = Wt/(bpY), with p the circular pitch, Y the Lewis form factor, and p = pi*m for module m.

SAE/AISI 10xx designates plain carbon steel with the last two digits giving carbon points (1045 is about 0.45% C); 41xx is Cr-Mo and 43xx is Ni-Cr-Mo.

Taylor tool life: V*T^n = C, with n roughly 0.1-0.15 for high-speed steel and 0.2-0.4 for carbide; a larger n means tool life is less sensitive to speed.

Spindle speed N (rpm) = 1000V/(piD) for V in m/min and D in mm; in inch units N = 12V/(piD) with V in ft/min and D in inches.

Brinell hardness HB = 2P/[piD(D - sqrt(D^2 - d^2))], with P in kgf and D and d (ball and indentation diameters) in mm.

Heat-treatment sequence: annealing softens and relieves residual stress, normalizing refines grain, quenching forms martensite, tempering restores toughness.

E6011 electrode: E = electrode, 60 = 60 ksi minimum tensile strength, 1 = usable in all positions, 11 = high-cellulose coating usable on AC or DC either polarity.

Blanking shears out the desired slug as product; piercing cuts a hole whose slug is scrap; both are completed in a single press stroke.

Fits: clearance always yields positive allowance, interference always negative, transition may yield either; H7/g6 is a common close running fit.

Gray cast iron graphite flakes damp vibration and aid machinability; ductile (nodular) iron forms graphite nodules by magnesium or cerium treatment, raising strength and ductility.

Rankine cycle efficiency = [(h1 - h2) - (h4 - h3)]/(h1 - h4); reheating and regenerative feedwater heating both raise this efficiency.

One boiler horsepower = 33,475 Btu/h = 9.81 kW, defined as evaporating 34.5 lb of water per hour at 212 F.

Cooling tower approach = cold water temperature minus entering wet-bulb temperature; range = hot water temperature minus cold water temperature.

Two-stage compressor with perfect intercooling gives minimum work at P2 = sqrt(P1*P3), and the total work is half that of a single stage over the same overall pressure ratio.

Affinity laws for a fan or pump at constant impeller diameter and fluid density: Q proportional to N, head proportional to N^2, power proportional to N^3.

Air-standard Otto efficiency = 1 - 1/r^(k-1); Brayton efficiency = 1 - 1/rp^((k-1)/k); both increase with compression or pressure ratio.

Refrigeration COP = refrigeration effect / work input; Carnot refrigerator COP = TL/(TH - TL).

Boiler indirect (heat-loss) method efficiency = 100% minus the sum of stack, radiation, moisture and unburned-combustible losses.

Load factor = average load / peak load; demand factor = maximum demand / total connected load.

For ax^2 + bx + c = 0 the roots are x = [-b +/- sqrt(b^2 - 4ac)]/(2a); the discriminant b^2 - 4ac indicates real distinct, repeated, or complex roots.

Derivatives: d/dx(x^n) = nx^(n-1), d/dx(e^ax) = ae^ax, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(tan x) = sec^2 x.

Integrals: int(x^n dx) = x^(n+1)/(n+1) + C for n not equal to -1; int(dx/x) = ln|x| + C; int(e^ax dx) = e^ax/a + C.

Simpson's one-third rule: integral of f dx is approximately (h/3)[f0 + 4(sum of odd-index ordinates) + 2(sum of even-index ordinates) + fn], requiring an even number of intervals.

Mensuration: sphere V = 4pir^3/3 and S = 4pir^2; right cone V = pir^2h/3 with lateral area pirsqrt(r^2 + h^2).

Probability: binomial P = C(n,k)p^kq^(n-k); Poisson P = e^(-lambda)*lambda^k/k!; the normal curve encloses 68.3% of area within plus or minus one standard deviation.

Series sums: arithmetic Sn = n(a1 + an)/2; geometric Sn = a(1 - r^n)/(1 - r) for r not equal to 1, and an infinite geometric series converges to a/(1 - r) when |r| < 1.

L'Hopital's rule applies to 0/0 and infinity/infinity indeterminate forms: lim f/g = lim f'/g' provided the latter limit exists.

Effective annual rate i_eff = (1 + i/m)^m - 1 for nominal rate i compounded m times per year; continuous compounding gives i_eff = e^i - 1.

Capital recovery factor (A/P, i%, n) = i(1+i)^n/[(1+i)^n - 1]; sinking fund factor (A/F, i%, n) = i/[(1+i)^n - 1].

Perpetuity present worth P = A/i; capitalized cost = first cost + (annual operating cost)/i.

Depreciation: straight line d = (FC - SV)/n; sum-of-years-digits d_k = (n - k + 1)(FC - SV)/[n(n+1)/2]; double declining balance rate = 2/n applied to remaining book value.

Real interest rate adjusted for inflation: i_real = (1 + i_nominal)/(1 + f) - 1, where f is the inflation rate.

Break-even quantity Q = FC/(unit price - unit variable cost); simple payback period = initial investment / annual net cash flow.

Ohm's law V = IR with power P = VI = I^2*R = V^2/R; resistances add in series and conductances add in parallel.

First law for a closed system: Q - W = delta U; for steady-flow open systems: q - w = delta h + delta KE + delta PE.

Hooke's law sigma = E*epsilon within the elastic limit; Poisson's ratio nu = -(lateral strain)/(axial strain), about 0.3 for steel.