Calculations

Calculations

• Common-cause split

  The independent and the shared part of a failure rate.

  independent = λ (1 − β); common = λ β

  Fault trees and reliability block diagrams logic.js Every app
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• Gamma function

  Γ(x), used for the mean life.

  Lanczos approximation, g = 7, 9 coefficients; reflection below 0.5

  Weibull life data weibull.js Every app
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• Weibull distribution functions

  Failure probability, reliability, density, hazard, quantile, mean and B-life.

  F = 1 − e^(−(t ÷ η)^β); quantile = η (−ln(1 − F))^(1 ÷ β); mean = η Γ(1 + 1 ÷ β)

  Weibull life data weibull.js Every app
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• Log-likelihood

  How well a shape and scale explain the failure and suspension times.

  Σ failures ln f(t) + Σ suspended ln R(t)

  Weibull life data weibull.js Every app
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• Maximum-likelihood fit

  Shape, scale and their confidence bounds from failure times, with suspensions.

  g(β) = Σ tᵝ ln t ÷ Σ tᵝ − 1 ÷ β − mean ln t = 0; η = (Σ tᵝ ÷ r)^(1 ÷ β)

  Weibull life data weibull.js Every app
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• Median ranks

  The plotting position of each failure, adjusted for suspended units.

  F = (rank − 0.3) ÷ (n + 0.4); rank increment = (n + 1 − previous rank) ÷ (1 + units at or after this one)

  Weibull life data weibull.js Every app
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• Rank regression fit

  Shape and scale from a straight line on the Weibull plot.

  β = S_xy ÷ S_xx on (ln t, ln(−ln(1 − F))); η = e^(−a ÷ β)

  Weibull life data weibull.js Every app
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• Duration drawn for a task

  One trial's duration of a task given as minimum, most likely and maximum.

  u < f: min + √(u (max − min)(mode − min)); else max − √((1 − u)(max − min)(max − mode)); duration = round(v)

  Schedule risk (Monte Carlo) simulate.js Every app
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• Finish percentiles, mean and spread

  The finish day at each confidence level over all trials.

  P = sorted[⌈p × n⌉ − 1]; mean = Σ ÷ n; sd = √(Σ (x − mean)² ÷ n)

  Schedule risk (Monte Carlo) simulate.js Every app
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• Criticality and sensitivity

  The share of trials in which a task was critical, and the correlation of its duration with the finish.

  criticality = times critical ÷ n; sensitivity = Σ (d − d̄)(F − F̄) ÷ √(Σ (d − d̄)² × n × sd²)

  Schedule risk (Monte Carlo) simulate.js Every app
  64-bit

• Source signals

  Step, ramp, sine and pulse sources as functions of time.

  sine: offset + amp × sin(2π f t + phase); ramp: v₀ + (v₁ − v₀)(t − t₀) ÷ (t₁ − t₀)

  Lumped system models model.js Every app
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• Resistive element laws

  The flow through a resistor, damper, orifice, diode or user law for a given difference across it.

  resistor: i = v ÷ R; orifice: i = v ÷ √(K √(v² + ε²)); diode: i = Is (e^(v ÷ nVt) − 1), a straight line above 40 nVt

  Lumped system models model.js Every app
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• Storage elements

  Capacitors, masses, tanks, inductors and springs: what they store and how it enters the equations.

  capacitive: i + d(C v)/dt = 0; inductive: v + d(L i)/dt = 0; spring: L = 1 ÷ k

  Lumped system models model.js Every app
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• Reported flow through an element

  The current, heat flow, flow or force reported for an element.

  the element's law at the solved state; diode: Is (e^(min(v, 40 nVt) ÷ nVt) − 1)

  Lumped system models model.js Every app
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• Dense linear solve

  Solves the Newton step.

  Gaussian elimination with partial pivoting

  Lumped system models solve.js Every app
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• Newton iteration

  Finds the state that satisfies the equations at one time.

  J_ij = (F_i(x + h e_j) − F_i(x − h e_j)) ÷ 2h, h = 1e-7 × max(|x_j|, 1); stop when |Δxᵢ| ≤ rtol |xᵢ| + atol + 1e-11 × max(1, largest |x|)

  Lumped system models solve.js Every app
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• Operating point

  The steady state, or the state set by the given initial values.

  one Newton solve from zero

  Lumped system models solve.js Every app
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• One time step

  The state one step later.

  trapezoid: (2 ÷ h)(S − S₀) − Ṡ₀ + g = 0; backward Euler: (1 ÷ h)(S − S₀) + g = 0

  Lumped system models solve.js Every app
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• Step-size control

  Chooses the time step from an estimate of the error, when adaptive stepping is on.

  err = max |x(two half steps) − x(one step)| ÷ (rtol |x| + atol), ÷ 3 for the trapezoid; h ← h × clamp(0.9 err^(−1/3), 0.2, 5)

  Lumped system models solve.js Every app
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• Value between stored times

  A node value at a time between two computed points.

  y = y₀ + (y₁ − y₀)(t − t₀) ÷ (t₁ − t₀)

  Lumped system models solve.js Every app
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• Parameter expressions

  The value of a parameter written as an expression, such as 2*3.3e3.

  numbers, parameters, + − × ÷ ^, brackets, pi

  Lumped system models modelica.js Every app
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• Thermal conductance to resistance

  A thermal conductor from a model file as the engine's thermal resistance.

  R = 1 ÷ G

  Lumped system models modelica.js Every app
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