PE Industrial cheat sheet

Normal stress sigma = F/A; engineering strain epsilon = deltaL/L0; Young's modulus E = sigma/epsilon only within the linear-elastic region.

Thermal expansion: deltaL = alpha * L * deltaT, where alpha is the coefficient of thermal expansion of the material.

Worst-case tolerance stack-up = sum of individual tolerances; statistical (root-sum-square) stack-up = sqrt(sum of squared tolerances), valid for independent, roughly normal variations.

Factor of safety = failure strength / allowable stress; design stress = strength / FS.

A clearance fit always has the hole limits larger than the shaft limits; an interference fit always has the shaft larger than the hole.

Common conversions: 1 hp = 0.7457 kW; 1 Btu = 1.055 kJ; 1 psi = 6.895 kPa; 1 in = 25.4 mm; 1 kgf = 9.807 N.

Ohm's law V = IR and electrical power P = VI = I^2R; three-phase power = sqrt(3)VIpf.

A dimensionally homogeneous equation has identical units on both sides; unit checking catches algebraic errors before computation.

Capital recovery factor: (A/P, i, n) = i(1+i)^n / [(1+i)^n - 1]; EUAC = PW * (A/P, i, n).

Present worth of a uniform series: (P/A, i, n) = [(1+i)^n - 1] / [i(1+i)^n]; single-payment (P/F, i, n) = (1+i)^-n.

Net present worth = sum of discounted cash flows; accept an independent project when NPW >= 0 at the MARR.

IRR is the discount rate making NPW = 0; accept when IRR >= MARR. Multiple sign changes in the cash flow can create multiple IRRs.

Straight-line depreciation: D = (first cost - salvage) / n. MACRS uses declining balance switching to straight line, half-year convention, and ignores salvage.

Simple payback period = first cost / annual net savings; it ignores the time value of money and cash flows after payback.

Break-even quantity Q = fixed cost / (unit price - unit variable cost); contribution margin per unit is price minus variable cost.

Inflation relation: (1 + i_market) = (1 + i_real)(1 + f); to deflate, divide the market rate by (1 + f).

Benefit-cost ratio = PW(benefits) / PW(costs); a ratio >= 1.0 corresponds to NPW >= 0 at the same discount rate.

Addition rule: P(A or B) = P(A) + P(B) - P(A and B); if independent, P(A and B) = P(A)*P(B); if mutually exclusive, P(A and B) = 0.

Bayes: P(A|B) = P(B|A)P(A) / P(B), where the denominator is the total probability of B over all causes.

Binomial: mean = np, variance = np(1-p). Poisson: mean = variance = lambda*t, used for counts of rare events per unit time or area.

Exponential distribution: mean = 1/lambda, standard deviation = 1/lambda, memoryless; it models interarrival and time-to-failure data.

Normal distribution: about 68.3% of the area lies within +/-1 sigma, 95.4% within +/-2 sigma, 99.7% within +/-3 sigma.

Confidence interval for a mean with known sigma: xbar +/- z(alpha/2) * sigma / sqrt(n); use t when sigma is estimated and n is small.

Type I error = rejecting a true null hypothesis (probability alpha); Type II error = failing to reject a false null (probability beta); power = 1 - beta.

Central limit theorem: sample means approach normality with mean mu and standard error sigma/sqrt(n) as n increases, regardless of population shape.

R-squared is the fraction of total variation in the response explained by the fitted regression model; it does not by itself prove causation.

A linear program has a linear objective and linear constraints; if an optimum exists it occurs at a vertex (extreme point) of the feasible region.

The simplex method moves between adjacent basic feasible solutions, improving the objective at each step until no improving pivot exists.

Shadow price = dual variable of a constraint = marginal change in the objective per unit increase in the right-hand side, valid only inside the sensitivity range.

Binding constraints have nonzero shadow prices; nonbinding (slack) constraints have zero shadow price.

Integer programs are solved by LP relaxation plus branch and bound; the LP relaxation objective is a bound on the integer optimum.

M/M/1 queue: utilization rho = lambda/mu must be < 1; L = rho/(1-rho), Lq = rho^2/(1-rho), W = 1/(mu - lambda).

Little's Law: L = lambda*W (items in system = arrival rate times time in system); it holds for any stable steady-state system.

Transportation problem: total supply must equal total demand; Vogel's approximation gives a good initial basic feasible solution.

Monte Carlo simulation uses random sampling of input distributions; increasing replications narrows the confidence interval on the output by 1/sqrt(n).

Chvorinov's rule: solidification time t = B*(V/A)^2, where V/A is the volume-to-surface-area ratio; risers must have a larger V/A than the casting they feed.

Taylor tool life: V*T^n = C; increasing cutting speed reduces tool life, with n typically 0.1-0.15 for HSS and 0.2-0.4 for carbide.

Turning spindle speed: N = 12V/(pi*D) with V in ft/min and D in inches; cutting time depends on length, feed rate, and rpm.

Welding heat input per unit length = (voltage * current) / travel speed; higher heat input widens the heat-affected zone.

Blanking cuts out the desired part; piercing cuts a hole and the slug is scrap. Shear clearance is roughly 5-10% of sheet thickness.

Cold working increases strength and hardness while reducing ductility; annealing restores ductility by recrystallization.

Powder metallurgy: powder blending, compaction, and sintering produce near-net shapes with controlled porosity and minimal machining.

Additive manufacturing build time scales with part height divided by layer thickness, so layer thickness trades resolution against throughput.

Brazing and soldering join metals without melting the base metal; brazing filler melts above 450 C, soldering below that.

EOQ = sqrt(2DS/H), where D is annual demand, S is order/setup cost per order, and H is annual holding cost per unit.

Total annual inventory cost at EOQ = ordering cost + holding cost, and these two components are equal at the EOQ quantity.

Reorder point = expected demand during lead time + safety stock; safety stock = z * sigma_demand * sqrt(lead time).

MRP net requirement = gross requirement - on-hand inventory - scheduled receipts + safety stock, then lot-sized by the lot-sizing rule.

The master production schedule disaggregates the aggregate production plan into end-item quantities by time period; MRP then explodes the bill of materials.

Johnson's rule for a two-machine flow shop: take the shortest processing time; if it is on machine 1 schedule first, if on machine 2 schedule last.

SPT sequencing minimizes average flow time, average waiting time, and average work-in-process on a single machine.

EDD (earliest due date) sequencing minimizes the maximum lateness on a single machine.

Kanban cards = (demand rate * replenishment lead time * (1 + safety factor)) / container quantity.

Load-distance score = sum over all department pairs of (material flow * rectilinear or Euclidean distance); lower scores indicate better block layouts.

Systematic Layout Planning uses a relationship chart with closeness ratings A, E, I, O, U, X (absolutely necessary through undesirable).

CRAFT improves an existing layout by pairwise exchanges of departments, accepting a swap only if the load-distance cost decreases.

Cycle time = available time per period / required output; minimum number of workstations = ceil(sum of task times / cycle time).

Line balance efficiency = sum of task times / (number of workstations * cycle time); balance delay = 1 - efficiency.

Flow patterns: straight line suits simple sequential flow; U-shape shortens travel and allows one operator to cover entry and exit.

Material handling equipment selection hinges on unit load, travel distance, throughput rate, and path flexibility (AGV vs conveyor vs forklift).

Cube utilization improves with higher rack heights and narrower aisles, but narrow aisles cut lift-truck speed and raise safety risk.

Control chart limits are set at the process mean +/- 3 sigma; Xbar-R charts monitor variables data, p and c charts monitor attribute data.

Cp = (USL - LSL) / (6sigma) measures spread only; Cpk = min(USL - mu, mu - LSL) / (3sigma) accounts for both spread and centering.

Cp high but Cpk low indicates the process is capable but off-center; Cp low indicates excessive variation that must be reduced.

Taguchi loss: L = k*(x - T)^2, where k = loss at specification limit / (tolerance half-width)^2; loss grows quadratically from target.

Six Sigma quality corresponds to 3.4 defects per million opportunities when the 1.5 sigma long-term shift is included.

Operating characteristic curve plots probability of lot acceptance against lot fraction defective for a given sampling plan; AQL and LTPD anchor the plan.

The seven basic quality tools: check sheet, histogram, Pareto chart, cause-and-effect diagram, scatter diagram, control chart, and flowchart.

Gage R&R separates repeatability (same appraiser, same part) from reproducibility (different appraisers) to quantify measurement system variation.

Normal time = observed time * performance rating; standard time = normal time * (1 + allowance factor) for personal, fatigue, and delay allowances.

Work sampling estimates the proportion of time spent in an activity from random observations; required n = z^2p(1-p)/e^2.

Learning curve: T_n = T_1 * n^b, where b = log(learning rate)/log(2); an 80% curve gives b = -0.322.

The NIOSH lifting equation computes a recommended weight limit using load constant, horizontal and vertical location, asymmetry, coupling, frequency, and duration multipliers.

Anthropometric design rules: size for the 5th percentile for reach and clearance of movement, the 95th percentile for clearance and strength-limited tasks.

RULA and REBA are observational posture-scoring tools that rank tasks by musculoskeletal risk and flag jobs needing redesign.

Principles of motion economy: use gravity-fed bins, keep both hands busy, use symmetric and natural curved motions, and locate tools within the normal work area.

Metabolic rate drives heat stress allowances; heavy work in hot environments requires longer rest allowances than light work.

Bullwhip effect: demand variability amplifies upstream, driven by batch ordering, price promotions, forecast updating, and shortage gaming/rationing.

Safety stock = z * sigma_demand * sqrt(lead time); z is set by the target cycle service level (1.28 for 90%, 1.65 for 95%, 2.33 for 99%).

Newsvendor critical ratio = underage cost / (underage cost + overage cost); the optimal order quantity sets the in-stock probability equal to this ratio.

ABC classification ranks SKUs by annual dollar usage; A items are few but carry most value and warrant tight control.

Incoterms allocate cost, risk, and responsibility: EXW places least obligation on the seller, FOB transfers risk at the ship's rail, DDP places most on the seller.

Total logistics cost tradeoff: faster transport cuts inventory and stockout cost but raises freight; the optimum minimizes the sum, not freight alone.

Cross-docking and milk-run routes reduce handling and inventory by consolidating flows; hub-and-spoke networks trade route density against distance.

Fill rate measures units shipped versus units ordered, while cycle service level measures the probability of no stockout per replenishment cycle.

Hierarchy of controls, most to least effective: elimination, substitution, engineering controls, administrative controls, personal protective equipment.

OSHA incidence rate = (number of recordable cases * 200,000) / total hours worked; 200,000 = 100 workers * 40 hours * 50 weeks.

Chemical exposure limits: TLV-TWA is an 8-hour time-weighted average, STEL is a 15-minute short-term limit, and ceiling is never to be exceeded.

Noise: an 85 dBA 8-hour TWA triggers a hearing conservation program, and exposure limits use a 5 dB exchange rate (90 dBA for 4 hours).

Fire requires fuel, oxygen, and heat (ignition); removal of any leg of the fire triangle prevents or extinguishes combustion.

Fire classes: A ordinary combustibles, B flammable liquids, C energized electrical, D combustible metals, K cooking oils; the extinguishing agent must match the class.

Permit-required confined spaces require atmospheric testing, ventilation, an attendant, and rescue provisions before entry.

Pollution prevention hierarchy: source reduction first, then reuse, recycling, treatment, and disposal as the last resort; ISO 14001 structures an environmental management system.