GATE PI flashcards

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

Chip thickness ratio

The ratio r = t1/t2 of uncut chip thickness to chip thickness after cutting. It relates to shear angle by tan(phi) = r*cos(alpha)/(1 - r*sin(alpha)), where alpha is the rake angle.

Taylor tool life equation

Empirical relation V*T^n = C linking cutting speed V and tool life T for a given tool-work combination. Exponent n is about 0.1-0.15 for HSS and 0.2-0.4 for carbide.

Merchant shear angle relation

At minimum cutting force, 2*phi + beta - alpha = 90 degrees, where phi is shear angle, beta the friction angle at the rake face, and alpha the rake angle. It predicts the shear plane orientation.

Chvorinov's rule

Solidification time t = B*(V/A)^2, where V is casting volume, A is heat-transfer surface area, and B is a mould constant. A riser must have a greater modulus V/A than the casting.

Heat input in welding

Heat input per unit length equals eta*V*I/v, where V is arc voltage, I is current, v is travel speed, and eta is arc efficiency. Higher heat input widens the heat-affected zone.

True strain in forming

True strain epsilon = ln(L/L0) = ln(A0/A), based on instantaneous dimensions. It is additive across passes, unlike engineering strain, and is used with flow stress curves.

Maximum draft in rolling

The largest reduction possible in a rolling pass is delta_max = mu^2*R, where mu is the friction coefficient and R the roll radius. Beyond this the rolls cannot bite the strip.

Shear strain in orthogonal cutting

Shear strain on the shear plane is gamma = cos(alpha)/[sin(phi)*cos(phi - alpha)]. It increases as shear angle decreases, indicating more severe deformation and higher cutting energy.

Hollomon strain-hardening equation

Flow stress sigma = K*epsilon^n, where K is the strength coefficient and n the strain-hardening exponent. A larger n means the metal hardens more before necking.

Turning material removal rate

For a turning operation MRR = pi*D*N*f*d, with D the workpiece diameter, N rotational speed, f feed per revolution, and d depth of cut. Cutting speed is V = pi*D*N.

Resistance spot welding heat

Heat generated at the weld nugget is H = I^2*R*t, where I is current, R the contact resistance, and t the weld time. Current has the strongest effect because it enters squared.

Economic order quantity

EOQ = sqrt(2*D*S/H), where D is annual demand, S ordering cost per order, and H holding cost per unit per year. At EOQ, ordering cost equals holding cost.

Exponential smoothing forecast

F(t+1) = alpha*D(t) + (1 - alpha)*F(t), where D(t) is actual demand and alpha is the smoothing constant. Larger alpha gives more weight to recent demand.

PERT expected time

Expected activity time t_e = (a + 4m + b)/6, with a optimistic, m most likely, and b pessimistic estimates. Variance is ((b - a)/6)^2 and sums along the critical path.

Line balancing cycle time

Cycle time = available production time per period divided by required output per period. Minimum workstations is the ceiling of total task time divided by cycle time.

Process capability indices

Cp = (USL - LSL)/(6*sigma) measures potential capability. Cpk = min[(USL - mean), (mean - LSL)]/(3*sigma) measures actual capability and accounts for centring.

Standard time calculation

Standard time = observed time * performance rating * (1 + allowance fraction). Normal time is observed time times rating; allowances cover personal needs, fatigue, and unavoidable delays.

M/M/1 queue measures

With arrival rate lambda and service rate mu, utilisation rho = lambda/mu. Average number in system L = rho/(1 - rho) and average queue wait W_q = lambda/[mu*(mu - lambda)].

Reorder point

The inventory level at which a new order is placed, equal to demand rate multiplied by lead time. It may be raised by a safety stock to cover demand variability.

Crash cost per unit time

Computed as (crash cost - normal cost)/(normal time - crash time) for an activity. To shorten a project, crash the cheapest critical activity and recheck the critical path.

X-bar chart control limits

Control limits for the sample mean are grand mean +/- A2*R_bar, where A2 depends on subgroup size and R_bar is the average range. The R chart monitors within-sample variability.

Balance delay

Balance delay = (n*C - sum of task times)/(n*C), where n is the number of workstations and C the cycle time. It measures idle time fraction on an assembly line.

Successive percentage change

Two successive changes of a% and b% are equivalent to a single change of a + b + ab/100 percent. Two successive discounts therefore multiply, not add, the remaining price.

Average speed for equal distances

If a journey covers equal distances at speeds u and v, average speed is 2uv/(u + v), the harmonic mean. It is always less than the arithmetic mean of the two speeds.

Combined work rate

If A alone takes a days and B alone takes b days, their combined one-day work is 1/a + 1/b. Together they finish the job in ab/(a + b) days.

Combination formula

The number of ways to choose r items from n is nCr = n!/[r!(n - r)!]. Use combinations when order does not matter and permutations nPr = n!/(n - r)! when it does.

Bayes' rule

P(A|B) = P(B|A)*P(A)/P(B). It updates a prior probability P(A) with evidence B, and is used for conditional probability problems involving sequential events.

Profit and loss percentage

Profit percent = (SP - CP)/CP * 100, where SP is selling price and CP cost price. Two items sold at the same price with equal gain and loss percentages always give a net loss of (x/10)^2 percent.

Geometric series sum

Sum of n terms of a geometric progression is a*(r^n - 1)/(r - 1) for r not equal to 1, where a is the first term and r the common ratio. Infinite sum is a/(1 - r) when |r| < 1.

Weighted average

Weighted average = sum(w_i*x_i)/sum(w_i), where w_i are the weights. For grouped data, the mean uses class frequencies as weights and class midpoints as values.

Train crossing time

Time for a train to cross another train or a platform equals the sum of their lengths divided by the relative speed. Opposite directions add speeds; same direction subtracts them.

Arrangements with repeated letters

The number of distinct arrangements of a word with n letters, where letters repeat p, q, ... times, is n!/(p!*q!*...). Divide by the factorial of each repetition count.