engineering pack

46 nodes, in noodlelab[engineering] and above.

Engineering/Control Analysis

Bode Plot

engineering.bode_plot

Magnitude (dB) and phase (degrees) against frequency (rad/s), with the gain and phase margins read as for an open loop (NaN where the curve does not cross 0 dB or −180°). Link the open loop L(s), the controller and plant in series, to see how far the closed loop is from instability.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
min_frequency	Quantity[rad/s]	0: automatic Default '0.0 rad / s'.
max_frequency	Quantity[rad/s]	0: automatic Default '0.0 rad / s'.
points	int	Default 500.
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
frequency	NDArray[float64]
magnitude_db	NDArray[float64]
phase_deg	NDArray[float64]
plot	Figure
gain_margin_db	float
phase_margin_deg	float
summary	dict[str, float]

Controllability & Observability

engineering.controllability

Whether the inputs can steer every state (controllable) and the outputs reveal every state (observable): the ranks of the controllability matrix [B AB A²B …] and the observability matrix [C; CA; CA²; …] against the number of states. Pole placement and LQR need a controllable system.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Outputs

Name	Type	Description
controllable	bool
observable	bool
controllable_rank	int
observable_rank	int
states	int
controllability_matrix	NDArray[float64]
observability_matrix	NDArray[float64]
summary	dict[str, float]

DC Gain

engineering.dc_gain

The steady-state gain: the output for a constant unit input, once everything has settled, G(0) (or G(1) when discrete). Infinite with an integrator.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
result	float

Impulse Response

engineering.impulse_response

The response to a unit impulse (a hammer blow): the system’s own motion, which rings at its natural frequencies. peak is the value largest in size, with its sign.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
duration	Quantity[s]	0 s: automatic, from the slowest pole Default '0.0 s'.
points	int	Default 1000.
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
t	Quantity[s]
y	NDArray[float64]
plot	Figure
peak	float
final_value	float
summary	dict[str, float]

Initial Response

engineering.initial_response

The free motion from an initial state with no input: a structure released from a deflected shape. The states are those of the state-space model (Mass-Spring-Damper: positions, then velocities, in SI units).

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
initial_state	str	One value per state, comma separated: 0.01, 0, 0, 0 Default '1, 0'.
duration	Quantity[s]	0 s: automatic, from the slowest pole Default '0.0 s'.
points	int	Default 1000.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
t	Quantity[s]
y	NDArray[float64]
plot	Figure
peak	float
final_value	float
summary	dict[str, float]

Nyquist Plot

engineering.nyquist_plot

The open loop L(jω) drawn in the complex plane for all frequencies. The closed loop (unity negative feedback) is stable when the curve encircles −1 anticlockwise once for each unstable open-loop pole: Z = N + P unstable closed-loop poles, with N the clockwise encirclements.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
plot	Figure
encirclements	int
open_loop_unstable_poles	int
closed_loop_stable	bool
summary	dict[str, float]

Pole-Zero Map

engineering.pole_zero_map

The poles (×) and zeros (○) in the complex plane, and a table with each pole’s natural frequency ωn, damping ratio ζ and time constant τ. Poles in the right half plane (outside the unit circle, when discrete) make the system unstable; lightly damped ones make it ring.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
poles	NDArray[complex128]
zeros	NDArray[complex128]
table	DataFrame
plot	Figure
stable	bool
min_damping	float
dc_gain	float
summary	dict[str, float]

Root Locus

engineering.root_locus

Where the closed-loop poles go as the gain K of the loop K L(s) rises from 0: they start at the open-loop poles (×) and end at its zeros (○) or run off to infinity. critical_gain is the smallest gain at which the closed loop turns unstable (infinite if it never does, NaN if it is never stable).

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
gain	float	Mark the poles at this gain K Default 1.0.
max_gain	float	0: automatic Default 0.0.
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
plot	Figure
poles_at_gain	NDArray[complex128]
stable_at_gain	bool
critical_gain	float
summary	dict[str, float]

Simulate

engineering.simulate

The response to any input signal u(t), from rest: a measured road profile, a set-point schedule, a sine sweep. t in seconds (a plain array counts as seconds), one value of u per time.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
t	Quantity | NDArray[floating]	Times, evenly spaced
u	NDArray[floating]
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
t	Quantity[s]
y	NDArray[float64]
plot	Figure
peak	float
final_value	float
summary	dict[str, float]

Stability Margins

engineering.stability_margins

How far an open loop L(s) is from instability once the loop is closed with unity negative feedback. The gain margin (dB) is how much the gain can rise, at the phase crossover (phase −180°); the phase margin (°) how much phase lag can be added, at the gain crossover (|L| = 1). Infinite when the curve never crosses. stability_margin is the closest the Nyquist curve comes to −1 (1 is far, 0 is unstable).

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.

Outputs

Name	Type	Description
gain_margin_db	float
phase_margin_deg	float
gain_crossover	Quantity[rad/s]
phase_crossover	Quantity[rad/s]
stability_margin	float
closed_loop_stable	bool
summary	dict[str, float]

Step Response

engineering.step_response

The response to a unit step: overshoot (%), 10–90 % rise time, settling time (within settling % of the final value for good), time of the peak and final value. An unstable system has no final value, overshoot or settling time (NaN). The system comes from Transfer Function, Feedback and the other builders (its coefficients used to be typed here).

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
duration	Quantity[s]	0 s: automatic, from the slowest pole Default '0.0 s'.
points	int	Default 1000.
input	str	Its name; empty: the first Default ''.
output	str	Its name; empty: the first Default ''.
settling	float	The settling band, % of the final value Default 2.0.

Outputs

Name	Type	Description
t	Quantity[s]
y	NDArray[float64]
plot	Figure
final_value	float
overshoot	float
rise_time	Quantity[s]
settling_time	Quantity[s]
peak_time	Quantity[s]
stable	bool
summary	dict[str, float]

Engineering/Control Design

LQR

engineering.lqr

The linear-quadratic regulator: the state feedback u = −K x that minimises ∫ (xᵀQx + uᵀRu) dt. Larger Q entries hold those states closer; larger R entries spend less input. closed_loop is the system with the feedback in place, driven by a reference r added to the input.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
Q	Quantity | NDArray[floating] | None	State weights; empty: the identity Optional.
R	Quantity | NDArray[floating] | None	Input weights; empty: the identity Optional.

Outputs

Name	Type	Description
gain	NDArray[float64]
closed_loop	LTI	A linear system: from Transfer Function, State Space, PID Controller…
poles	NDArray[complex128]
summary	dict[str, float]

Pole Placement

engineering.pole_placement

The state feedback u = −K x that puts the closed-loop poles where you say: further left is faster, and a pair a ± bj has damping ratio −a/√(a² + b²). The system must be controllable.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
poles	str	One per state, comma separated: -2, -3 ± 1j Default '-2, -3'.

Outputs

Name	Type	Description
gain	NDArray[float64]
closed_loop	LTI	A linear system: from Transfer Function, State Space, PID Controller…
poles	NDArray[complex128]
summary	dict[str, float]

Engineering/Controls

Convert System

engineering.convert_system

The same system as a transfer function or in state space. Without the optional slycot package, only single-input, single-output systems convert to a transfer function.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
to	Literal['transfer function', 'state space']	Default 'state space'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Discretize

engineering.discretize

A continuous system as a discrete one sampled every sample_time, for a digital controller: G(s) becomes G(z).

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
sample_time	Quantity[s]	Default '0.01 s'.
method	Literal['zoh', 'foh', 'tustin', 'matched', 'euler', 'backward_diff']	zoh: a sample held until the next; tustin: the bilinear transform Default 'zoh'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Feedback

engineering.feedback

Close the loop around system: G / (1 + G H) with negative feedback, the usual kind, where the output is compared with the set point. With unity feedback (no H) and the open loop L = C G linked, this is the closed loop T = L / (1 + L).

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
feedback	Any	H(s) in the feedback path; empty: 1 (unity) Optional.
sign	Literal['negative', 'positive']	Default 'negative'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

First-Order System

engineering.first_order_system

K / (τ s + 1): a lag such as a heater, a tank or an actuator, reaching 63 % of its final value after one time constant. A delay is approximated by a Padé filter of pade_order.

Inputs

Name	Type	Description
gain	float	Default 1.0.
time_constant	Quantity[s]	Default '1.0 s'.
delay	Quantity[s]	A dead time, as a Padé approximation Default '0.0 s'.
pade_order	int	Default 3.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Mass-Spring-Damper

engineering.mass_spring_damper

The state-space model of masses, springs and dampers, M x’’ + C x’ + K x = F, for Step Response, Feedback and the rest. The states are the positions x1… and velocities v1…, the inputs the forces F1… at force_at, and the outputs as measure says. M in kg, K in N/m and C in N·s/m (other units are converted; plain numbers are SI).

Inputs

Name	Type	Description
mass	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array, or plain numbers in SI units
stiffness	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array, or plain numbers in SI units
damping	Quantity | NDArray[floating] | None	A matrix: a quantity holding a 2-D array, or plain numbers in SI units Optional.
force_at	str	Where forces act, by degree of freedom: 1, or 1, 2 Default '1'.
measure	str	The outputs: x1 (position), v1 (velocity), a1 (acceleration) Default 'x1'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Minimal Realisation

engineering.minimal_realisation

The system without its cancelling poles and zeros: (s + 1)/((s + 1)(s + 2)) is 1/(s + 2). Without the optional slycot package, a state-space system must have one input and one output.

Inputs

Name	Type	Description
system	LTI	A linear system: from Transfer Function, State Space, PID Controller…
tolerance	float	How close a pole and zero cancel Default 1e-06.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

PID Controller

engineering.pid_controller

A PID controller, C(s) = Kp + Ki/s + Kd s/(Tf s + 1). The derivative is filtered, as any real one is: without the filter it would be improper, amplifying noise without limit. Gains are in consistent SI units.

Inputs

Name	Type	Description
kp	float	Proportional gain Default 1.0.
ki	float	Integral gain, per second Default 0.0.
kd	float	Derivative gain, in seconds Default 0.0.
derivative_filter	Quantity[s]	Tf: the derivative acts through 1/(Tf s + 1) Default '0.01 s'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Parallel

engineering.parallel

Two systems side by side on the same input, their outputs added (or b subtracted from a).

Inputs

Name	Type	Description
a	LTI	A linear system: from Transfer Function, State Space, PID Controller…
b	LTI	A linear system: from Transfer Function, State Space, PID Controller…
sign	Literal['add', 'subtract']	Default 'add'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Second-Order System

engineering.second_order_system

The standard second-order system, K ωn² / (s² + 2ζωn s + ωn²): a mass on a spring and damper, an RLC circuit. Below ζ = 1 it overshoots.

Inputs

Name	Type	Description
natural_frequency	Quantity[rad/s]	Default '1.0 rad / s'.
damping_ratio	float	ζ: 1 is critical Default 0.5.
gain	float	Default 1.0.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Series

engineering.series

Systems one after another: the signal goes through a, then b (then c). A controller, an actuator and a plant in series make the open loop L(s).

Inputs

Name	Type	Description
a	LTI	A linear system: from Transfer Function, State Space, PID Controller…
b	LTI	A linear system: from Transfer Function, State Space, PID Controller…
c	LTI | None	A linear system: from Transfer Function, State Space, PID Controller… Optional.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

State Space

engineering.state_space

A system in state space: x’ = A x + B u, y = C x + D u. A is n×n, B n×m, C p×n and D p×m (zero when not linked). A in 1/time is taken in 1/s; the others in consistent SI units.

Inputs

Name	Type	Description
A	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array, or plain numbers in SI units
B	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array, or plain numbers in SI units
C	Quantity | NDArray[floating]	A matrix: a quantity holding a 2-D array, or plain numbers in SI units
D	Quantity | NDArray[floating] | None	A matrix: a quantity holding a 2-D array, or plain numbers in SI units Optional.
sample_time	Quantity[s]	0 s: continuous; else discrete Default '0.0 s'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Transfer Function

engineering.transfer_function

A transfer function G(s) = num(s) / den(s), typed as polynomial coefficients in descending powers of s: 1, 2, 1 is s² + 2s + 1. With a sample time it is a discrete G(z) instead.

Inputs

Name	Type	Description
numerator	str	Coefficients in descending powers of s, e.g. 1, 2, 1 Default '1'.
denominator	str	Coefficients in descending powers of s, e.g. 1, 2, 1 Default '1, 0.8, 1'.
sample_time	Quantity[s]	0 s: continuous (s); else discrete (z) Default '0.0 s'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Transfer Function (Expression)

engineering.transfer_function_expression

A transfer function typed as an expression in s, with other symbols taken from values (as plain numbers in SI units), such as K/(tau*s + 1) with K and tau from a Values node.

Inputs

Name	Type	Description
text	str	A function of s: 10/(s*(s + 2)), K/(tau*s + 1) Default '10/(s*(s + 2))'.
values	SymbolValues | None	Optional.
variable	str	Default 's'.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Zero-Pole-Gain

engineering.zero_pole_gain

A transfer function from its zeros, poles and gain: G(s) = k (s − z₁)(s − z₂)… / ((s − p₁)(s − p₂)…). a ± bj is a complex pair.

Inputs

Name	Type	Description
zeros	str	Comma separated; empty: none Default ''.
poles	str	Comma separated: -1, -2 ± 3j Default '-1, -2'.
gain	float	Default 1.0.

Outputs

Name	Type	Description
result	LTI	A linear system: from Transfer Function, State Space, PID Controller…

Engineering/Decibels

From dB

engineering.from_db

The ratio a value in decibels stands for.

Inputs

Name	Type	Description
db	float
kind	Literal['power', 'amplitude']	power: 10 log10; amplitude (voltage, pressure): 20 log10 Default 'power'.

Outputs

Name	Type	Description
result	float

To dB

engineering.to_db

A ratio in decibels: 10 log10 of a power ratio, 20 log10 of an amplitude ratio. Use a ratio to 1 mW for dBm, to 1 W for dBW.

Inputs

Name	Type	Description
ratio	float
kind	Literal['power', 'amplitude']	power: 10 log10; amplitude (voltage, pressure): 20 log10 Default 'power'.

Outputs

Name	Type	Description
result	float

dB Budget

engineering.db_budget

Add up gains and losses in dB, one per line (# starts a comment). The table has the running total after each item, for a report.

Inputs

Name	Type	Description
items	str	One per line: name and value in dB (losses negative) Default 'Transmit power 30 dBm\nCable loss -2\nAntenna gain 12\nPath loss -120'.
extra	float	Added to the total, e.g. from another node Default 0.0.

Outputs

Name	Type	Description
total	float
table	DataFrame

Engineering/Fluids

Pipe Pressure Drop

engineering.pipe_pressure_drop

The friction pressure drop along a straight, full, circular pipe, Δp = f (L / D) ρ v² / 2. The Darcy friction factor f is 64 / Re in laminar flow and from the Haaland equation otherwise. Roughness: about 0.0015 mm drawn tubing and plastic, 0.045 mm commercial steel, 0.26 mm cast iron.

Inputs

Name	Type	Description
flow_rate	Quantity[L/s]	Default '2.0 l / s'.
diameter	Quantity[mm]	Default '50.0 mm'.
length	Quantity[m]	Default '100.0 m'.
roughness	Quantity[mm]	Default '0.045 mm'.
density	Quantity[kg/m^3]	Default '998.0 kg / m ** 3'.
viscosity	Quantity[Pa*s]	Default '0.001 Pa * s'.

Outputs

Name	Type	Description
pressure_drop	Quantity[kPa]
velocity	Quantity[m/s]
reynolds	float
friction_factor	float
regime	str

Reynolds Number

engineering.reynolds_number

Re = ρ v L / μ, with length the pipe diameter (or a body’s characteristic length), and the pipe-flow regime: laminar below 2300, turbulent above 4000. The defaults are water at 20 °C.

Inputs

Name	Type	Description
velocity	Quantity[m/s]
length	Quantity[m]	Default '0.05 m'.
density	Quantity[kg/m^3]	Default '998.0 kg / m ** 3'.
viscosity	Quantity[Pa*s]	Default '0.001 Pa * s'.

Outputs

Name	Type	Description
reynolds	float
regime	str

Engineering/Materials

Material

engineering.material

Typical properties of a common material: stiffness, density, strength, thermal expansion and conductivity. For preliminary design: check a datasheet before relying on the strengths.

Inputs

Name	Type	Description
name	str	Default 'Aluminium 6061-T6'.

Outputs

Name	Type	Description
youngs_modulus	Quantity[GPa]
poisson_ratio	float
density	Quantity[kg/m^3]
yield_strength	Quantity[MPa]
ultimate_strength	Quantity[MPa]
thermal_expansion	Quantity[1/K]
thermal_conductivity	Quantity[W/(m*K)]
properties	dict[str, Any]

Material Table

engineering.material_table

Every built-in material as a table, with specific stiffness and strength (per unit density), for filtering, plotting or a Decision Matrix.

Outputs

Name	Type	Description
result	DataFrame

Engineering/Sections

Section Properties

engineering.section_properties

Area, second moments of area, elastic section modulus (about x) and least radius of gyration.

Inputs

Name	Type	Description
shape	Literal['rectangle', 'hollow rectangle', 'circle', 'tube', 'I-beam']	Default 'rectangle'.
width	Quantity[mm]	Default '50.0 mm'.
height	Quantity[mm]	Default '100.0 mm'.
wall	Quantity[mm]	Default '5.0 mm'.
web	Quantity[mm]	Default '5.0 mm'.

Outputs

Name	Type	Description
area	Quantity[mm^2]
ixx	Quantity[mm^4]
iyy	Quantity[mm^4]
section_modulus	Quantity[mm^3]
radius_of_gyration	Quantity[mm]
summary	dict[str, float]

Engineering/Structures

Axial Stress

engineering.axial_stress

Direct stress under an axial force, σ = F / A (tension positive).

Inputs

Name	Type	Description
force	Quantity[N]
area	Quantity[mm^2]

Outputs

Name	Type	Description
result	Quantity[MPa]

Beam

engineering.beam

Maximum deflection, bending moment and shear of a prismatic beam, and its deflected shape. load is the total load: a point load at midspan (at the free end of a cantilever), or spread uniformly over the span. Fixed-fixed moments are the largest, at the supports.

Inputs

Name	Type	Description
length	Quantity[m]	Default '2.0 m'.
load	Quantity[kN]	Default '5.0 kN'.
youngs_modulus	Quantity[GPa]	Default '200.0 GPa'.
second_moment	Quantity[mm^4]	Default '4000000.0 mm ** 4'.
support	Literal['simply supported', 'cantilever', 'fixed-fixed']	Default 'simply supported'.
load_type	Literal['point', 'uniform']	uniform: the load spread over the span Default 'point'.
points	int	Default 101.

Outputs

Name	Type	Description
max_deflection	Quantity[mm]
max_moment	Quantity[N*m]
max_shear	Quantity[N]
x	NDArray[float64]
deflection	NDArray[float64]
plot	Figure
summary	dict[str, float]

Bending Stress

engineering.bending_stress

The largest bending stress in a section, σ = M / Z.

Inputs

Name	Type	Description
moment	Quantity[N*m]
section_modulus	Quantity[mm^3]

Outputs

Name	Type	Description
result	Quantity[MPa]

Euler Buckling

engineering.euler_buckling

The elastic buckling load of a slender column, P = π² E I / (K L)², with the effective length factor K for the end conditions. Give the area for the critical stress and slenderness ratio K L / r; stocky columns (slenderness below about 100 in steel) yield before they buckle.

Inputs

Name	Type	Description
youngs_modulus	Quantity[GPa]
second_moment	Quantity[mm^4]
length	Quantity[m]
area	Quantity[mm^2] | None	Optional.
ends	Literal['pinned-pinned', 'fixed-free', 'fixed-pinned', 'fixed-fixed']	Default 'pinned-pinned'.

Outputs

Name	Type	Description
critical_load	Quantity[kN]
critical_stress	Quantity[MPa]
slenderness	float

Safety Factor

engineering.safety_factor

How far a design is from failing: the factor of safety capacity / demand (e.g. yield strength / stress), and the margin of safety capacity / (required × demand) − 1, which passes when it is ≥ 0.

Inputs

Name	Type	Description
capacity	Quantity
demand	Quantity
required	float	The factor the design needs Default 1.5.

Outputs

Name	Type	Description
factor	float
margin	float
passes	bool

Von Mises Stress

engineering.von_mises

The equivalent (von Mises) stress of a plane stress state, to compare with the yield strength: √(σx² − σx σy + σy² + 3 τxy²).

Inputs

Name	Type	Description
sigma_x	Quantity[MPa]
sigma_y	Quantity[MPa]	Default '0.0 MPa'.
tau_xy	Quantity[MPa]	Default '0.0 MPa'.

Outputs

Name	Type	Description
result	Quantity[MPa]

Engineering/Thermal

Convection

engineering.convection

Heat flow from a surface to a fluid, Q = h A ΔT (Newton’s law of cooling). Typical h: 5–25 W/(m²·K) still air, 10–200 forced air, 500–10 000 forced water.

Inputs

Name	Type	Description
coefficient	Quantity[W/(m^2*K)]	Default '10.0 W / K / m ** 2'.
area	Quantity[m^2]	Default '1.0 m ** 2'.
temperature_difference	Quantity[delta_degC]	Default '20.0 Δ°C'.

Outputs

Name	Type	Description
heat_flow	Quantity[W]
resistance	Quantity[K/W]

Pipe Conduction

engineering.pipe_conduction

Radial heat flow through a pipe wall or insulation layer, Q = 2π k L ΔT / ln(r₂ / r₁).

Inputs

Name	Type	Description
conductivity	Quantity[W/(m*K)]
inner_radius	Quantity[mm]	Default '25.0 mm'.
outer_radius	Quantity[mm]	Default '50.0 mm'.
length	Quantity[m]	Default '1.0 m'.
temperature_difference	Quantity[delta_degC]	Default '20.0 Δ°C'.

Outputs

Name	Type	Description
heat_flow	Quantity[W]
resistance	Quantity[K/W]

Thermal Expansion

engineering.thermal_expansion

The change in length of a free bar, ΔL = α L ΔT.

Inputs

Name	Type	Description
length	Quantity[m]
expansion	Quantity[1/K]	Default '1.2e-05 / K'.
temperature_change	Quantity[delta_degC]	Default '50.0 Δ°C'.

Outputs

Name	Type	Description
result	Quantity[mm]

Wall Conduction

engineering.wall_conduction

Heat flow through a plane wall, Q = k A ΔT / t, and its thermal resistance t / (k A) (add resistances in series for layered walls).

Inputs

Name	Type	Description
conductivity	Quantity[W/(m*K)]
thickness	Quantity[mm]	Default '100.0 mm'.
area	Quantity[m^2]	Default '1.0 m ** 2'.
temperature_difference	Quantity[delta_degC]	Default '20.0 Δ°C'.

Outputs

Name	Type	Description
heat_flow	Quantity[W]
resistance	Quantity[K/W]

Engineering/Trade

Decision Matrix

engineering.decision_matrix

Rank options (one per row) by a weighted sum of criteria. Each criterion column is scaled from 0 (worst option) to 1 (best), so units do not matter, then weighted by the magnitude of its weight. The score is out of 100.

Inputs

Name	Type	Description
table	DataFrame
option	str	Default ''.
criteria	str	column:weight, comma separated; a negative weight: lower is better Default ''.

Outputs

Name	Type	Description
ranking	DataFrame
best	str
plot	Figure