mechanical-engineering-ai999x — offline, verified mechanical engineering solver with step-by-step answers

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License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Engineering · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

mechanical-engineering-ai999x is an offline, deterministic Python solver that answers mechanical engineering questions with verified results from 90 certified capabilities, gives 312 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

mechanical-engineering-ai999x is a compact, deterministic mechanical engineering engine for Python. It reads a question in plain English (or a typed call), picks a certified mechanical engineering capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified mechanical engineering capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 312 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: statics, beams and section properties, stress and strain, torsion, columns, fatigue, machine elements, vibrations and structural analysis.

Model overview

Property	Value
Package	mechanical-engineering-ai999x 1.0.0 (import mechanical_engineering_ai999x)
Type	Deterministic neuro-symbolic mechanical engineering engine (rule-based reading + certified symbolic capabilities)
Capabilities	90 certified procedures
Knowledge pack	version 1.0.1
Golden answers	312 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install mechanical-engineering-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/mechanical-engineering-ai999x/resolve/main/dist/mechanical_engineering_ai999x-1.0.0-py3-none-any.whl

import mechanical_engineering_ai999x as mechanical_engineering

mechanical_engineering.ask("Simply supported beam with point load: L = 6 m, P = 20 kN, a = 2 m, E = 200 GPa, I = 8e-5 m^4").show()
# -> SOLVED — RA = 13333.3, RB = 6666.67, M_max = 26666.7, V_max = 13333.3  (simply_supported_beam_point_load)
mechanical_engineering.ask("Principal stresses: sx = 50 MPa, sy = -10 MPa, txy = 40 MPa").show()
# -> SOLVED — s1_MPa = 70, s2_MPa = -30, tau_max_inplane_MPa = 50, theta_p_deg = 26.5651  (principal_stresses_2d)
mechanical_engineering.ask("Goodman safety factor: sa = 120 MPa, sm = 80 MPa, Se = 250 MPa, Sut = 700 MPa").show()
# -> SOLVED — n_fatigue = 1.68269, sigma_ar_MPa = 135.484  (goodman_fatigue_safety_factor)
mechanical_engineering.ask("Rainflow counting: history = [0, 5, -3, 4, -2, 6, -4, 1, 0]").show()
# -> SOLVED — ranges = [10.0, 9.0, 8.0, 6.0, 5.0, 1.0], counts = [0.5, 0.5, 0.5, 1.0, 1.0, 0.5], total_cycles = 4, max_range = 10  (rainflow_counting)
mechanical_engineering.solve("euler_buckling", E="200 GPa", I="8e-6 m^4", L="4 m", ends="fixed-pinned").show()
# -> SOLVED — P_cr = 2.01907e+06, P_cr_kN = 2019.07, K = 0.699156, Le = 2.79662  (euler_buckling)
mechanical_engineering.ask("Who created you?").message
# -> mechanical-engineering-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: mechanical-engineering-ai999x "<question>", mechanical-engineering-ai999x --capabilities. The outputs above were produced by running this exact code on mechanical-engineering-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

90 certified procedures. A few of them:

• Procedures: All critical (whirling) speeds of a shaft with several disks; Angle of twist of a stepped shaft; Bar fixed at both ends with an axial load at a step; Belt or rope friction on a drum (capstan equation); Bolt tightening torque and preload (T = K F d); Cantilever with a point load; Cantilever with a triangular (linearly varying) load; Cantilever with a uniformly distributed load; Combined axial force and biaxial bending of a rectangular section; Composite section of rectangles (parallel-axis theorem); Continuous beam under uniform loads (three-moment equation); Critical speed of a shaft carrying one disk; Critical stress of an intermediate column (Johnson parabola / Euler); Damping ratio, damped frequency and logarithmic decrement; Eccentrically loaded bolt group (elastic method); Eccentrically loaded column (secant formula); Endurance limit with the Marin factors; Euler buckling load of a column with given end conditions; Fatigue damage of a stress history (rainflow counting + Miner's rule); Fatigue damage on the trilinear (Eurocode 3) S-N curve; Fatigue life from the S-N line (10^3 to 10^6 cycles); Fatigue life with a mean-stress correction (Morrow, Goodman, SWT, Walker) and Basquin's law; Fatigue safety factor by the ASME-elliptic criterion; Fatigue safety factor by the Gerber criterion …

mechanical_engineering_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 90 formulas

Show every formula, by topic

Beams / cantilevers

Cantilever with a point load

$${\delta }_{tip}=\frac{P{a}^2(3L-a)}{6EI},{\theta }_{tip}=\frac{P{a}^2}{2EI}\backslash delta\_\{tip\}\; =\; \backslash frac\{Pa^2(3L-a)\}\{6EI\},\backslash quad\; \backslash theta\_\{tip\}\; =\; \backslash frac\{Pa^2\}\{2EI\}$$

Cantilever with a triangular (linearly varying) load

$${\delta }_{tip}=\frac{w_0{L}^4}{30EI},{\theta }_{tip}=\frac{w_0{L}^3}{24EI}\backslash delta\_\{tip\}\; =\; \backslash frac\{w\_0L^4\}\{30EI\},\backslash quad\; \backslash theta\_\{tip\}\; =\; \backslash frac\{w\_0L^3\}\{24EI\}$$

Cantilever with a uniformly distributed load

$${\delta }_{tip}=\frac{w{L}^4}{8EI},{\theta }_{tip}=\frac{w{L}^3}{6EI}\backslash delta\_\{tip\}\; =\; \backslash frac\{wL^4\}\{8EI\},\backslash quad\; \backslash theta\_\{tip\}\; =\; \backslash frac\{wL^3\}\{6EI\}$$

Beams / simply supported beams

Simply supported beam with a point load

$$M_{max}=\frac{Pab}{L},{\delta }_{max}=\frac{Pb(L^2-b^2{)}^{3\mathrm{/}2}}{9\sqrt{3}EIL}M\_\{max\}\; =\; \backslash frac\{Pab\}\{L\},\backslash quad\; \backslash delta\_\{max\}\; =\; \backslash frac\{Pb\backslash ,(L^2-b^2)^\{3/2\}\}\{9\backslash sqrt3\backslash ,EIL\}$$

Simply supported beam with a uniformly distributed load

$$M_{max}=\frac{w{L}^2}{8},{\delta }_{max}=\frac{5w{L}^4}{384EI}M\_\{max\}\; =\; \backslash frac\{wL^2\}\{8\},\backslash quad\; \backslash delta\_\{max\}\; =\; \backslash frac\{5wL^4\}\{384EI\}$$

Simply supported beam with several point loads

Beams / statically indeterminate beams

Fixed-fixed (built-in) beam with a point load

$$M_A=\frac{Pa{b}^2}{L^2},M_B=\frac{P{a}^{2}b}{L^2},{\delta }_P=\frac{P{a}^3{b}^3}{3EI{L}^3}M\_A\; =\; \backslash frac\{Pab^2\}\{L^2\},\backslash quad\; M\_B\; =\; \backslash frac\{Pa^2b\}\{L^2\},\backslash quad\; \backslash delta\_P\; =\; \backslash frac\{Pa^3b^3\}\{3EIL^3\}$$

Fixed-fixed (built-in) beam with a uniformly distributed load

$$M_{end}=\frac{w{L}^2}{12},M_{mid}=\frac{w{L}^2}{24},{\delta }_{max}=\frac{w{L}^4}{384EI}M\_\{end\}\; =\; \backslash frac\{wL^2\}\{12\},\backslash quad\; M\_\{mid\}\; =\; \backslash frac\{wL^2\}\{24\},\backslash quad\; \backslash delta\_\{max\}\; =\; \backslash frac\{wL^4\}\{384EI\}$$

Propped cantilever with a point load

$$R_B=\frac{P{a}^2(3L-a)}{2L^3},M_A=\frac{Pab(L+b)}{2L^2}R\_B\; =\; \backslash frac\{Pa^2(3L-a)\}\{2L^3\},\backslash quad\; M\_A\; =\; \backslash frac\{Pab(L+b)\}\{2L^2\}$$

Propped cantilever with a uniformly distributed load

$$R_B=\frac{3wL}{8},M_A=\frac{w{L}^2}{8},{\delta }_{max}\approx \frac{w{L}^4}{185EI}R\_B\; =\; \backslash frac\{3wL\}\{8\},\backslash quad\; M\_A\; =\; \backslash frac\{wL^2\}\{8\},\backslash quad\; \backslash delta\_\{max\}\; \backslash approx\; \backslash frac\{wL^4\}\{185EI\}$$

Columns / eccentric loading

Eccentrically loaded column (secant formula)

$${\sigma }_{max}=\frac{P}{A}[1+\frac{ec}{r^2}\sec (\frac{L}{2}\sqrt{\frac{P}{EI}}\left)\right]\backslash sigma\_\{max\}\; =\; \backslash frac\{P\}\{A\}\backslash Big[1\; +\; \backslash frac\{ec\}\{r^2\}\backslash sec\backslash Big(\backslash frac\{L\}\{2\}\backslash sqrt\{\backslash frac\{P\}\{EI\}\}\backslash Big)\backslash Big]$$

Columns / elastic buckling

Euler buckling load of a column with given end conditions

$$P_{cr}=\frac{{\pi }^{2}EI}{(KL{)}^2}P\_\{cr\}\; =\; \backslash frac\{\backslash pi^2EI\}\{(KL)^2\}$$

Columns / inelastic buckling

Critical stress of an intermediate column (Johnson parabola / Euler)

Failure theories / brittle materials

Safety factors of a brittle material (maximum normal stress, Coulomb-Mohr, modified Mohr)

$$\frac{{\sigma }_1}{S_{ut}}-\frac{{\sigma }_3}{S_{uc}}=\frac{1}{n},\frac{(S_{uc}-S_{ut}){\sigma }_A}{S_{uc}{S}_{ut}}-\frac{{\sigma }_B}{S_{uc}}=\frac{1}{n}\backslash frac\{\backslash sigma\_1\}\{S\_\{ut\}\}\; -\; \backslash frac\{\backslash sigma\_3\}\{S\_\{uc\}\}\; =\; \backslash frac1n,\backslash quad\; \backslash frac\{(S\_\{uc\}-S\_\{ut\})\backslash sigma\_A\}\{S\_\{uc\}S\_\{ut\}\}\; -\; \backslash frac\{\backslash sigma\_B\}\{S\_\{uc\}\}\; =\; \backslash frac1n$$

Failure theories / ductile materials

Von Mises and Tresca equivalent stresses and safety factors

$${\sigma }_{vm}=\sqrt{\frac{1}{2}[({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2]},n=\frac{S_y}{{\sigma }_{eq}}\backslash sigma\_\{vm\}\; =\; \backslash sqrt\{\backslash tfrac12\backslash big[(\backslash sigma\_1-\backslash sigma\_2)^2+(\backslash sigma\_2-\backslash sigma\_3)^2+(\backslash sigma\_3-\backslash sigma\_1)^2\backslash big]\},\backslash quad\; n\; =\; \backslash frac\{S\_y\}\{\backslash sigma\_\{eq\}\}$$

Fatigue / cumulative damage

Fatigue damage of a stress history (rainflow counting + Miner's rule)

$$D=\sum _i\frac{n_i}{N_c}(\frac{S_i}{S_c}{)}^{m}D\; =\; \backslash sum\_i\; \backslash frac\{n\_i\}\{N\_c\}\backslash Big(\backslash frac\{S\_i\}\{S\_c\}\backslash Big)^m$$

Fatigue damage on the trilinear (Eurocode 3) S-N curve

$$N=2\cdot {10}^6(\frac{\mathrm{\Delta }{\sigma }_C}{\mathrm{\Delta }\sigma }{)}^3(\mathrm{\Delta }\sigma \ge \mathrm{\Delta }{\sigma }_D),N=5\cdot {10}^6(\frac{\mathrm{\Delta }{\sigma }_D}{\mathrm{\Delta }\sigma }{)}^{5}N\; =\; 2\backslash cdot10^6\backslash Big(\backslash frac\{\backslash Delta\backslash sigma\_C\}\{\backslash Delta\backslash sigma\}\backslash Big)^3\backslash \; (\backslash Delta\backslash sigma\backslash ge\backslash Delta\backslash sigma\_D),\backslash quad\; N\; =\; 5\backslash cdot10^6\backslash Big(\backslash frac\{\backslash Delta\backslash sigma\_D\}\{\backslash Delta\backslash sigma\}\backslash Big)^5$$

Miner's rule damage with a single-slope S-N curve

$$D=\sum _i\frac{n_i}{N_i},N_i=N_c(\frac{S_c}{S_i}{)}^{m}D\; =\; \backslash sum\_i\backslash frac\{n\_i\}\{N\_i\},\backslash quad\; N\_i\; =\; N\_c\backslash Big(\backslash frac\{S\_c\}\{S\_i\}\backslash Big)^m$$

Fatigue / cycle counting

Rainflow cycle counting of a load history (ASTM E1049)

$$\mathrm{\mid }{x}_3-x_2\mathrm{\mid }\ge \mathrm{\mid }{x}_2-x_1\mathrm{\mid }\Rightarrow \text{count }\mathrm{\mid }{x}_2-x_1\mathrm{\mid }\backslash vert\; x\_3-x\_2\backslash vert\; \backslash ge\; \backslash vert\; x\_2-x\_1\backslash vert\; \backslash Rightarrow\; \backslash text\{count\; \}\; \backslash vert\; x\_2-x\_1\backslash vert$$

Fatigue / endurance limit

Endurance limit with the Marin factors

$$S_e=k_a{k}_b{k}_c{k}_d{k}_e{k}_f{S}_e^{\mathrm{\prime }},k_a=a{S}_{ut}^b,k_e=1-0.08z_{a}S\_e\; =\; k\_a\; k\_b\; k\_c\; k\_d\; k\_e\; k\_f\backslash ,S\_e\text{'},\backslash quad\; k\_a\; =\; aS\_\{ut\}^b,\backslash quad\; k\_e\; =\; 1\; -\; 0.08\backslash ,z\_a$$

Fatigue / mean-stress criteria

Fatigue safety factor by the ASME-elliptic criterion

$${\left(\frac{n{\sigma }_a}{S_e}\right)}^2+{\left(\frac{n{\sigma }_m}{S_y}\right)}^2=1\backslash left(\backslash frac\{n\backslash sigma\_a\}\{S\_e\}\backslash right)^2\; +\; \backslash left(\backslash frac\{n\backslash sigma\_m\}\{S\_y\}\backslash right)^2\; =\; 1$$

Fatigue safety factor by the Gerber criterion

$$\frac{n{\sigma }_a}{S_e}+{\left(\frac{n{\sigma }_m}{S_{ut}}\right)}^2=1\backslash frac\{n\backslash sigma\_a\}\{S\_e\}\; +\; \backslash left(\backslash frac\{n\backslash sigma\_m\}\{S\_\{ut\}\}\backslash right)^2\; =\; 1$$

Fatigue safety factor by the Soderberg criterion

$$\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_y}=\frac{1}{n}\backslash frac\{\backslash sigma\_a\}\{S\_e\}\; +\; \backslash frac\{\backslash sigma\_m\}\{S\_y\}\; =\; \backslash frac1n$$

Fatigue safety factor by the modified Goodman criterion

$$\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_{ut}}=\frac{1}{n}\backslash frac\{\backslash sigma\_a\}\{S\_e\}\; +\; \backslash frac\{\backslash sigma\_m\}\{S\_\{ut\}\}\; =\; \backslash frac1n$$

Fatigue / shaft design

Shaft diameter for a design factor (DE-Goodman)

$$d=\left(\frac{16n}{\pi }\right\{\frac{\sqrt{4(K_f{M}_a{)}^2+3(K_{fs}{T}_a{)}^2}}{S_e}+\frac{\sqrt{4(K_f{M}_m{)}^2+3(K_{fs}{T}_m{)}^2}}{S_{ut}}\}{)}^{1\mathrm{/}3}d\; =\; \backslash Big(\backslash frac\{16n\}\{\backslash pi\}\backslash Big\backslash \{\backslash frac\{\backslash sqrt\{4(K\_fM\_a)^2+3(K\_\{fs\}T\_a)^2\}\}\{S\_e\}\; +\; \backslash frac\{\backslash sqrt\{4(K\_fM\_m)^2+3(K\_\{fs\}T\_m)^2\}\}\{S\_\{ut\}\}\backslash Big\backslash \}\backslash Big)^\{1/3\}$$

Shaft fatigue safety factor (DE-Goodman)

$$\frac{1}{n}=\frac{16}{\pi d^3}\{\frac{\sqrt{4(K_f{M}_a{)}^2+3(K_{fs}{T}_a{)}^2}}{S_e}+\frac{\sqrt{4(K_f{M}_m{)}^2+3(K_{fs}{T}_m{)}^2}}{S_{ut}}\}\backslash frac1n\; =\; \backslash frac\{16\}\{\backslash pi\; d^3\}\backslash Big\backslash \{\backslash frac\{\backslash sqrt\{4(K\_fM\_a)^2+3(K\_\{fs\}T\_a)^2\}\}\{S\_e\}\; +\; \backslash frac\{\backslash sqrt\{4(K\_fM\_m)^2+3(K\_\{fs\}T\_m)^2\}\}\{S\_\{ut\}\}\backslash Big\backslash \}$$

Fatigue / stress-life

Fatigue life from the S-N line (10^3 to 10^6 cycles)

$$a=\frac{(f{S}_{ut}{)}^2}{S_e},b=-\frac{1}{3}{\log }_{10}\frac{f{S}_{ut}}{S_e},N=(\frac{{\sigma }_a}{a}{)}^{1\mathrm{/}b}a\; =\; \backslash frac\{(fS\_\{ut\})^2\}\{S\_e\},\backslash quad\; b\; =\; -\backslash frac13\backslash log\_\{10\}\backslash frac\{fS\_\{ut\}\}\{S\_e\},\backslash quad\; N\; =\; \backslash Big(\backslash frac\{\backslash sigma\_a\}\{a\}\backslash Big)^\{1/b\}$$

Fatigue life with a mean-stress correction (Morrow, Goodman, SWT, Walker) and Basquin's law

$${\sigma }_{ar}=\frac{{\sigma }_a}{1-{\sigma }_m\mathrm{/}{\sigma }_f^{\mathrm{\prime }}},{\sigma }_{ar}=\sqrt{{\sigma }_{max}{\sigma }_a},N_f=\frac{1}{2}(\frac{{\sigma }_{ar}}{{\sigma }_f^{\mathrm{\prime }}}{)}^{1\mathrm{/}b}\backslash sigma\_\{ar\}\; =\; \backslash frac\{\backslash sigma\_a\}\{1-\backslash sigma\_m/\backslash sigma\_f\text{'}\},\backslash quad\; \backslash sigma\_\{ar\}\; =\; \backslash sqrt\{\backslash sigma\_\{max\}\backslash sigma\_a\},\backslash quad\; N\_f\; =\; \backslash tfrac12\backslash Big(\backslash frac\{\backslash sigma\_\{ar\}\}\{\backslash sigma\_f\text{'}\}\backslash Big)^\{1/b\}$$

Fatigue strength at a given number of cycles (S-N line)

$$S_f=a{N}^{b}S\_f\; =\; aN^b$$

Machine elements / belts and band brakes

Belt or rope friction on a drum (capstan equation)

$$\frac{T_1}{T_2}=e^{\mu \theta }\left(e^{\mu \theta \mathrm{/}\sin (\beta \mathrm{/}2)}\text{ for a V-belt}\right)\backslash frac\{T\_1\}\{T\_2\}\; =\; e^\{\backslash mu\backslash theta\}\backslash quad\backslash Big(e^\{\backslash mu\backslash theta/\backslash sin(\backslash beta/2)\}\backslash \; \backslash text\{for\; a\; V-belt\}\backslash Big)$$

Power of a belt drive with centrifugal tension

$$\frac{T_1-m{v}^2}{T_2-m{v}^2}=e^{\mu \theta },P=(T_1-T_2)v\backslash frac\{T\_1\; -\; mv^2\}\{T\_2\; -\; mv^2\}\; =\; e^\{\backslash mu\backslash theta\},\backslash quad\; P\; =\; (T\_1-T\_2)v$$

Machine elements / bolted joints

Bolt tightening torque and preload (T = K F d)

$$T=K{F}_{i}dT\; =\; K\; F\_i\backslash ,d$$

Eccentrically loaded bolt group (elastic method)

$${\mathbf{F}}_i=-\frac{P}{n}\widehat{\mathbf{y}}+\frac{Pe}{\sum r_j^2}(y_i-\bar{y},-(x_i-\bar{x}))\backslash mathbf\; F\_i\; =\; -\backslash frac\{P\}\{n\}\backslash hat\{\backslash mathbf\; y\}\; +\; \backslash frac\{Pe\}\{\backslash sum\; r\_j^2\}\backslash ,(y\_i-\backslash bar\; y,\backslash ;\; -(x\_i-\backslash bar\; x))$$

Machine elements / clutches and brakes

Torque capacity of a disc clutch or brake

$$T_{wear}=\frac{Ff(D+d)}{4},T_{press}=\frac{Ff}{3}\frac{D^3-d^3}{D^2-d^2}T\_\{wear\}\; =\; \backslash frac\{Ff(D+d)\}\{4\},\backslash quad\; T\_\{press\}\; =\; \backslash frac\{Ff\}\{3\}\backslash ,\backslash frac\{D^3-d^3\}\{D^2-d^2\}$$

Machine elements / gear trains

Speeds of a simple planetary (epicyclic) gear train

$$\frac{{\omega }_s-{\omega }_c}{{\omega }_r-{\omega }_c}=-\frac{N_r}{N_s}\backslash frac\{\backslash omega\_s\; -\; \backslash omega\_c\}\{\backslash omega\_r\; -\; \backslash omega\_c\}\; =\; -\backslash frac\{N\_r\}\{N\_s\}$$

Machine elements / gears

Lewis bending stress of a gear tooth

$$\sigma =\frac{K_v{W}_t}{FmY}\backslash sigma\; =\; \backslash frac\{K\_v\; W\_t\}\{F\backslash ,m\backslash ,Y\}$$

Spur gear mesh geometry and contact ratio

$$Z=\sqrt{r_{a1}^2-r_{b1}^2}+\sqrt{r_{a2}^2-r_{b2}^2}-C\sin \varphi ,m_c=\frac{Z}{\pi m\cos \varphi }Z\; =\; \backslash sqrt\{r\_\{a1\}^2-r\_\{b1\}^2\}\; +\; \backslash sqrt\{r\_\{a2\}^2-r\_\{b2\}^2\}\; -\; C\backslash sin\backslash phi,\backslash quad\; m\_c\; =\; \backslash frac\{Z\}\{\backslash pi\; m\backslash cos\backslash phi\}$$

Machine elements / mechanisms

Four-bar linkage position analysis

$$K_1\cos {\theta }_4-K_2\cos {\theta }_2+K_3=\cos ({\theta }_2-{\theta }_4),{\theta }_4=2\arctan \frac{-B\mp \sqrt{B^2-4AC}}{2A}K\_1\backslash cos\backslash theta\_4\; -\; K\_2\backslash cos\backslash theta\_2\; +\; K\_3\; =\; \backslash cos(\backslash theta\_2-\backslash theta\_4),\backslash quad\; \backslash theta\_4\; =\; 2\backslash arctan\backslash frac\{-B\backslash mp\backslash sqrt\{B^2-4AC\}\}\{2A\}$$

Slider-crank mechanism: slider position, velocity and acceleration

$$x=r\cos \theta +\sqrt{l^2-r^2{\sin }^2\theta },v=\omega \frac{dx}{d\theta },a={\omega }^2\frac{d^{2}x}{d{\theta }^2}x\; =\; r\backslash cos\backslash theta\; +\; \backslash sqrt\{l^2\; -\; r^2\backslash sin^2\backslash theta\},\backslash quad\; v\; =\; \backslash omega\backslash frac\{dx\}\{d\backslash theta\},\backslash quad\; a\; =\; \backslash omega^2\backslash frac\{d^2x\}\{d\backslash theta^2\}$$

Machine elements / power screws

Power screw: raising and lowering torque, efficiency, self-locking

$$T_R=\frac{F{d}_m}{2}\frac{l+\pi f{d}_m\sec \alpha }{\pi d_m-fl\sec \alpha }+\frac{F{f}_c{d}_c}{2}T\_R\; =\; \backslash frac\{Fd\_m\}\{2\}\backslash ,\backslash frac\{l\; +\; \backslash pi\; fd\_m\backslash sec\backslash alpha\}\{\backslash pi\; d\_m\; -\; fl\backslash sec\backslash alpha\}\; +\; \backslash frac\{Ff\_cd\_c\}\{2\}$$

Machine elements / rolling bearings

Required catalogue rating of a rolling bearing

$$C_{10}=a_f{F}_D[\frac{x_D}{x_0+(\theta -x_0)(\ln 1\mathrm{/}{R}_D{)}^{1\mathrm{/}b}}{]}^{1\mathrm{/}a}C\_\{10\}\; =\; a\_fF\_D\backslash Big[\backslash frac\{x\_D\}\{x\_0\; +\; (\backslash theta-x\_0)(\backslash ln\; 1/R\_D)^\{1/b\}\}\backslash Big]^\{1/a\}$$

Rolling-bearing life at a given reliability

$$L_{10}=L_R(\frac{C}{F}{)}^a,x=x_0+(\theta -x_0)(\ln \frac{1}{R}{)}^{1\mathrm{/}b}L\_\{10\}\; =\; L\_R\backslash Big(\backslash frac\{C\}\{F\}\backslash Big)^a,\backslash quad\; x\; =\; x\_0\; +\; (\backslash theta-x\_0)\backslash Big(\backslash ln\backslash frac1R\backslash Big)^\{1/b\}$$

Machine elements / springs

Helical compression spring: rate, deflection and shear stress

$$k=\frac{G{d}^4}{8D^3{N}_a},\tau =K_s\frac{8FD}{\pi d^3},K_s=1+\frac{0.5}{C}k\; =\; \backslash frac\{Gd^4\}\{8D^3N\_a\},\backslash quad\; \backslash tau\; =\; K\_s\backslash frac\{8FD\}\{\backslash pi\; d^3\},\backslash quad\; K\_s\; =\; 1\; +\; \backslash frac\{0.5\}\{C\}$$

Wahl curvature factor of a helical spring

$$K_W=\frac{4C-1}{4C-4}+\frac{0.615}{C},\tau =K_W\frac{8FD}{\pi d^3}K\_W\; =\; \backslash frac\{4C-1\}\{4C-4\}\; +\; \backslash frac\{0.615\}\{C\},\backslash quad\; \backslash tau\; =\; K\_W\backslash frac\{8FD\}\{\backslash pi\; d^3\}$$

Pressure vessels / interference fits

Shrink (press) fit: interface pressure and hoop stresses

$$p=\frac{E\delta }{2R^3}\frac{(r_o^2-R^2)(R^2-r_i^2)}{r_o^2-r_i^2}p\; =\; \backslash frac\{E\backslash delta\}\{2R^3\}\backslash ,\backslash frac\{(r\_o^2-R^2)(R^2-r\_i^2)\}\{r\_o^2-r\_i^2\}$$

Pressure vessels / rotating disks

Stresses in a rotating disk (solid or with a central hole)

$${\sigma }_{\theta }(r_i)=\frac{3+\nu }{4}\rho {\omega }^2(r_o^2+\frac{1-\nu }{3+\nu }{r}_i^2),{\sigma }_{r,max}=\frac{3+\nu }{8}\rho {\omega }^2(r_o-r_i{)}^2\backslash sigma\_\backslash theta(r\_i)\; =\; \backslash frac\{3+\backslash nu\}\{4\}\backslash rho\backslash omega^2\backslash Big(r\_o^2\; +\; \backslash frac\{1-\backslash nu\}\{3+\backslash nu\}r\_i^2\backslash Big),\backslash quad\; \backslash sigma\_\{r,max\}\; =\; \backslash frac\{3+\backslash nu\}\{8\}\backslash rho\backslash omega^2(r\_o-r\_i)^2$$

Pressure vessels / thick-walled cylinders

Thick-walled cylinder under pressure (Lame equations)

$${\sigma }_{r,\theta }=\frac{p_i{a}^2-p_o{b}^2}{b^2-a^2}\mp \frac{(p_i-p_o)a^2{b}^2}{(b^2-a^2)r^2}\backslash sigma\_\{r,\backslash theta\}\; =\; \backslash frac\{p\_ia^2\; -\; p\_ob^2\}\{b^2-a^2\}\; \backslash mp\; \backslash frac\{(p\_i-p\_o)a^2b^2\}\{(b^2-a^2)r^2\}$$

Pressure vessels / thin-walled vessels

Thin-walled pressure vessel stresses (cylinder or sphere)

$${\sigma }_h=\frac{pr}{t},{\sigma }_l=\frac{pr}{2t}\backslash sigma\_h\; =\; \backslash frac\{pr\}\{t\},\backslash quad\; \backslash sigma\_l\; =\; \backslash frac\{pr\}\{2t\}$$

Section properties / arbitrary sections

Section properties of a polygonal section

$$A=\frac{1}{2}\sum c_i,I_x=\frac{1}{12}\sum c_i(y_i^2+y_i{y}_{i+1}+y_{i+1}^2)-A{\bar{y}}^{2}A\; =\; \backslash tfrac12\backslash sum\; c\_i,\backslash quad\; I\_x\; =\; \backslash tfrac1\{12\}\backslash sum\; c\_i\backslash ,(y\_i^2+y\_iy\_\{i+1\}+y\_\{i+1\}^2)\; -\; A\backslash bar\; y^2$$

Section properties / composite sections

Composite section of rectangles (parallel-axis theorem)

$$I_x=\sum _i(\frac{b_i{h}_i^3}{12}+A_i(y_i-\bar{y}{)}^2),I_{xy}=\sum _i{A}_i(x_i-\bar{x})(y_i-\bar{y})I\_x\; =\; \backslash sum\_i\backslash Big(\backslash frac\{b\_ih\_i^3\}\{12\}\; +\; A\_i(y\_i-\backslash bar\; y)^2\backslash Big),\backslash quad\; I\_\{xy\}\; =\; \backslash sum\_i\; A\_i(x\_i-\backslash bar\; x)(y\_i-\backslash bar\; y)$$

Section properties / standard shapes

Section properties of a T-section

$$\bar{y}=\frac{\sum A_i{y}_i}{\sum A_i},I_x=\sum (\frac{b_i{h}_i^3}{12}+A_i(y_i-\bar{y}{)}^2)\backslash bar\; y\; =\; \backslash frac\{\backslash sum\; A\_i\; y\_i\}\{\backslash sum\; A\_i\},\backslash quad\; I\_x\; =\; \backslash sum\backslash Big(\backslash frac\{b\_ih\_i^3\}\{12\}\; +\; A\_i(y\_i\; -\; \backslash bar\; y)^2\backslash Big)$$

Section properties of a channel (C) section

$$\bar{x}=\frac{\sum A_i{x}_i}{\sum A_i},I_y=\sum (\frac{h_i{b}_i^3}{12}+A_i(x_i-\bar{x}{)}^2)\backslash bar\; x\; =\; \backslash frac\{\backslash sum\; A\_i\; x\_i\}\{\backslash sum\; A\_i\},\backslash quad\; I\_y\; =\; \backslash sum\backslash Big(\backslash frac\{h\_ib\_i^3\}\{12\}\; +\; A\_i(x\_i-\backslash bar\; x)^2\backslash Big)$$

Section properties of a hollow circle (tube)

$$I=\frac{\pi (D^4-d^4)}{64},J=\frac{\pi (D^4-d^4)}{32}I\; =\; \backslash frac\{\backslash pi(D^4-d^4)\}\{64\},\backslash quad\; J\; =\; \backslash frac\{\backslash pi(D^4-d^4)\}\{32\}$$

Section properties of a hollow rectangle (box)

$$I_x=\frac{B{H}^3-b{h}^3}{12},Z_{p,x}=\frac{B{H}^2-b{h}^2}{4}I\_x\; =\; \backslash frac\{BH^3\; -\; bh^3\}\{12\},\backslash quad\; Z\_\{p,x\}\; =\; \backslash frac\{BH^2\; -\; bh^2\}\{4\}$$

Section properties of a solid circle

$$I=\frac{\pi d^4}{64},J=\frac{\pi d^4}{32},Z=\frac{\pi d^3}{32}I\; =\; \backslash frac\{\backslash pi\; d^4\}\{64\},\backslash quad\; J\; =\; \backslash frac\{\backslash pi\; d^4\}\{32\},\backslash quad\; Z\; =\; \backslash frac\{\backslash pi\; d^3\}\{32\}$$

Section properties of a solid rectangle

$$I_x=\frac{b{h}^3}{12},Z_x=\frac{b{h}^2}{6},r_x=\sqrt{I_x\mathrm{/}A}I\_x\; =\; \backslash frac\{bh^3\}\{12\},\backslash quad\; Z\_x\; =\; \backslash frac\{bh^2\}\{6\},\backslash quad\; r\_x\; =\; \backslash sqrt\{I\_x/A\}$$

Section properties of a symmetric I-section

$$I_x=\frac{b{h}^3-(b-t_w)(h-2t_f{)}^3}{12},Z_{p,x}=b{t}_f(h-t_f)+\frac{t_w(h-2t_f{)}^2}{4}I\_x\; =\; \backslash frac\{bh^3\; -\; (b-t\_w)(h-2t\_f)^3\}\{12\},\backslash quad\; Z\_\{p,x\}\; =\; bt\_f(h-t\_f)\; +\; \backslash frac\{t\_w(h-2t\_f)^2\}\{4\}$$

Section properties / unsymmetric sections

Section properties of an angle (L) section, with principal axes

$$I_{1,2}=\frac{I_x+I_y}{2}\pm \sqrt{(\frac{I_x-I_y}{2}{)}^2+I_{xy}^2},\tan 2\phi =\frac{-2I_{xy}}{I_x-I_y}I\_\{1,2\}\; =\; \backslash frac\{I\_x+I\_y\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash Big(\backslash frac\{I\_x-I\_y\}\{2\}\backslash Big)^2\; +\; I\_\{xy\}^2\},\backslash quad\; \backslash tan2\backslash varphi\; =\; \backslash frac\{-2I\_\{xy\}\}\{I\_x-I\_y\}$$

Statics / equilibrium of a particle

Tensions in two cables holding a weight

$$T_1=\frac{W\cos \beta }{\sin (\alpha +\beta )},T_2=\frac{W\cos \alpha }{\sin (\alpha +\beta )}T\_1\; =\; \backslash frac\{W\backslash cos\backslash beta\}\{\backslash sin(\backslash alpha+\backslash beta)\},\backslash quad\; T\_2\; =\; \backslash frac\{W\backslash cos\backslash alpha\}\{\backslash sin(\backslash alpha+\backslash beta)\}$$

Statics / force systems

Resultant of concurrent coplanar forces

$$R=\sqrt{(\sum F_i\cos {\theta }_i{)}^2+(\sum F_i\sin {\theta }_i{)}^2},{\theta }_R=\mathrm{atan2}(R_y,R_x)R\; =\; \backslash sqrt\{\backslash Big(\backslash sum\; F\_i\backslash cos\backslash theta\_i\backslash Big)^2\; +\; \backslash Big(\backslash sum\; F\_i\backslash sin\backslash theta\_i\backslash Big)^2\},\backslash quad\; \backslash theta\_R\; =\; \backslash operatorname\{atan2\}(R\_y,\; R\_x)$$

Statics / moments

Moment of a force about a point (2D)

$$M_O=(x-x_0)F_y-(y-y_0)F_x,d=\frac{\mathrm{\mid }{M}_O\mathrm{\mid }}{\mathrm{\mid }\mathbf{F}\mathrm{\mid }}M\_O\; =\; (x\; -\; x\_0)F\_y\; -\; (y\; -\; y\_0)F\_x,\backslash quad\; d\; =\; \backslash frac\{\backslash vert\; M\_O\backslash vert\; \}\{\backslash vert\; \backslash mathbf\; F\backslash vert\; \}$$

Stress analysis / combined loading

Combined axial force and biaxial bending of a rectangular section

$$\sigma =\frac{N}{A}+\frac{M_{x}y}{I_x}+\frac{M_{y}x}{I_y}\backslash sigma\; =\; \backslash frac\{N\}\{A\}\; +\; \backslash frac\{M\_xy\}\{I\_x\}\; +\; \backslash frac\{M\_yx\}\{I\_y\}$$

Stress analysis / principal stresses

Principal stresses and maximum shear (Mohr's circle, plane stress)

$${\sigma }_{1,2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{(\frac{{\sigma }_x-{\sigma }_y}{2}{)}^2+{\tau }_{xy}^2},\tan 2{\theta }_p=\frac{2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}\backslash sigma\_\{1,2\}\; =\; \backslash frac\{\backslash sigma\_x+\backslash sigma\_y\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash Big(\backslash frac\{\backslash sigma\_x-\backslash sigma\_y\}\{2\}\backslash Big)^2\; +\; \backslash tau\_\{xy\}^2\},\backslash quad\; \backslash tan2\backslash theta\_p\; =\; \backslash frac\{2\backslash tau\_\{xy\}\}\{\backslash sigma\_x-\backslash sigma\_y\}$$

Principal stresses of a three-dimensional stress state

$$\det (\mathit{\sigma }-\sigma \mathbf{I})=0,{\tau }_{max}=\frac{{\sigma }_1-{\sigma }_3}{2},{\tau }_{oct}=\frac{\sqrt{2}}{3}{\sigma }_{vm}\backslash det(\backslash boldsymbol\backslash sigma\; -\; \backslash sigma\backslash mathbf\; I)\; =\; 0,\backslash quad\; \backslash tau\_\{max\}\; =\; \backslash frac\{\backslash sigma\_1-\backslash sigma\_3\}\{2\},\backslash quad\; \backslash tau\_\{oct\}\; =\; \backslash frac\{\backslash sqrt2\}\{3\}\backslash sigma\_\{vm\}$$

Stress analysis / strain measurement

Strain rosette: strains, principal strains and principal stresses

$${\epsilon }_{1,2}=\frac{{\epsilon }_x+{\epsilon }_y}{2}\pm \sqrt{(\frac{{\epsilon }_x-{\epsilon }_y}{2}{)}^2+(\frac{{\gamma }_{xy}}{2}{)}^2},{\sigma }_1=\frac{E({\epsilon }_1+\nu {\epsilon }_2)}{1-{\nu }^2}\backslash varepsilon\_\{1,2\}\; =\; \backslash frac\{\backslash varepsilon\_x+\backslash varepsilon\_y\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash Big(\backslash frac\{\backslash varepsilon\_x-\backslash varepsilon\_y\}\{2\}\backslash Big)^2\; +\; \backslash Big(\backslash frac\{\backslash gamma\_\{xy\}\}\{2\}\backslash Big)^2\},\backslash quad\; \backslash sigma\_1\; =\; \backslash frac\{E(\backslash varepsilon\_1+\backslash nu\backslash varepsilon\_2)\}\{1-\backslash nu^2\}$$

Stress analysis / stress transformation

Plane stress transformation to rotated axes

$${\sigma }_{x^{\mathrm{\prime }}}=\frac{{\sigma }_x+{\sigma }_y}{2}+\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\theta +{\tau }_{xy}\sin 2\theta \backslash sigma\_\{x\text{'}\}\; =\; \backslash frac\{\backslash sigma\_x+\backslash sigma\_y\}\{2\}\; +\; \backslash frac\{\backslash sigma\_x-\backslash sigma\_y\}\{2\}\backslash cos2\backslash theta\; +\; \backslash tau\_\{xy\}\backslash sin2\backslash theta$$

Stress and strain / axial loading

Stepped bar under axial loads at its joints

$$\delta =\sum _i\frac{N_i{L}_i}{A_{i}E},{\sigma }_i=\frac{N_i}{A_i}\backslash delta\; =\; \backslash sum\_i\; \backslash frac\{N\_iL\_i\}\{A\_iE\},\backslash quad\; \backslash sigma\_i\; =\; \backslash frac\{N\_i\}\{A\_i\}$$

Stress and strain / constitutive law

Generalized Hooke's law: strains from a 3D stress state

$${\epsilon }_x=\frac{1}{E}[{\sigma }_x-\nu ({\sigma }_y+{\sigma }_z)],{\gamma }_{xy}=\frac{{\tau }_{xy}}{G},G=\frac{E}{2(1+\nu )}\backslash varepsilon\_x\; =\; \backslash frac\{1\}\{E\}\backslash big[\backslash sigma\_x\; -\; \backslash nu(\backslash sigma\_y+\backslash sigma\_z)\backslash big],\backslash quad\; \backslash gamma\_\{xy\}\; =\; \backslash frac\{\backslash tau\_\{xy\}\}\{G\},\backslash quad\; G\; =\; \backslash frac\{E\}\{2(1+\backslash nu)\}$$

Stress and strain / statically indeterminate members

Bar fixed at both ends with an axial load at a step

$$R_A=P\frac{k_1}{k_1+k_2},k_i=\frac{A_i{E}_i}{L_i},\delta =\frac{P}{k_1+k_2}R\_A\; =\; P\backslash frac\{k\_1\}\{k\_1+k\_2\},\backslash quad\; k\_i\; =\; \backslash frac\{A\_iE\_i\}\{L\_i\},\backslash quad\; \backslash delta\; =\; \backslash frac\{P\}\{k\_1+k\_2\}$$

Structural analysis / continuous beams

Continuous beam under uniform loads (three-moment equation)

$$M_{i-1}{L}_i+2M_i(L_i+L_{i+1})+M_{i+1}{L}_{i+1}=-\frac{w_i{L}_i^3+w_{i+1}{L}_{i+1}^3}{4}M\_\{i-1\}L\_i\; +\; 2M\_i(L\_i+L\_\{i+1\})\; +\; M\_\{i+1\}L\_\{i+1\}\; =\; -\backslash frac\{w\_iL\_i^3\; +\; w\_\{i+1\}L\_\{i+1\}^3\}\{4\}$$

Structural analysis / trusses

Member forces of a plane truss (method of joints)

$$\sum _{\text{members at }j}{F}_m{\widehat{\mathbf{e}}}_m+{\mathbf{R}}_j+{\mathbf{P}}_j=\mathbf{0}\backslash sum\_\{\backslash text\{members\; at\; \}\; j\}\; F\_m\backslash ,\backslash hat\{\backslash mathbf\; e\}\_\{m\}\; +\; \backslash mathbf\; R\_j\; +\; \backslash mathbf\; P\_j\; =\; \backslash mathbf\; 0$$

Torsion / circular shafts

Torsion of a hollow circular shaft

$${\tau }_{max}=\frac{16TD}{\pi (D^4-d^4)},\phi =\frac{TL}{GJ}\backslash tau\_\{max\}\; =\; \backslash frac\{16TD\}\{\backslash pi(D^4-d^4)\},\backslash quad\; \backslash varphi\; =\; \backslash frac\{TL\}\{GJ\}$$

Torsion of a solid circular shaft

$${\tau }_{max}=\frac{16T}{\pi d^3},\phi =\frac{TL}{GJ},J=\frac{\pi d^4}{32}\backslash tau\_\{max\}\; =\; \backslash frac\{16T\}\{\backslash pi\; d^3\},\backslash quad\; \backslash varphi\; =\; \backslash frac\{TL\}\{GJ\},\backslash quad\; J\; =\; \backslash frac\{\backslash pi\; d^4\}\{32\}$$

Torsion / compound shafts

Angle of twist of a stepped shaft

$$\phi =\sum _i\frac{T{L}_i}{G{J}_i},J_i=\frac{\pi d_i^4}{32}\backslash varphi\; =\; \backslash sum\_i\backslash frac\{TL\_i\}\{GJ\_i\},\backslash quad\; J\_i\; =\; \backslash frac\{\backslash pi\; d\_i^4\}\{32\}$$

Torsion / power transmission

Torque and minimum diameter of a power-transmission shaft

$$T=\frac{P}{\omega },d_{min}=(\frac{16T}{\pi {\tau }_{allow}}{)}^{1\mathrm{/}3}T\; =\; \backslash frac\{P\}\{\backslash omega\},\backslash quad\; d\_\{min\}\; =\; \backslash Big(\backslash frac\{16T\}\{\backslash pi\backslash tau\_\{allow\}\}\backslash Big)^\{1/3\}$$

Torsion / statically indeterminate shafts

Shaft fixed at both ends with a torque between them

$$T_A=T_0\frac{b\mathrm{/}{J}_2}{a\mathrm{/}{J}_1+b\mathrm{/}{J}_2},T_B=T_0-T_{A}T\_A\; =\; T\_0\backslash frac\{b/J\_2\}\{a/J\_1\; +\; b/J\_2\},\backslash quad\; T\_B\; =\; T\_0\; -\; T\_A$$

Vibrations / continuous systems

Natural frequencies of a uniform beam (Euler-Bernoulli)

$${\omega }_n=({\beta }_{n}L{)}^2\sqrt{\frac{EI}{\rho A{L}^4}},\cos \beta L\cosh \beta L=-1\text{ (cantilever)}\backslash omega\_n\; =\; (\backslash beta\_nL)^2\backslash sqrt\{\backslash frac\{EI\}\{\backslash rho\; AL^4\}\},\backslash quad\; \backslash cos\backslash beta\; L\backslash cosh\backslash beta\; L\; =\; -1\backslash \; \backslash text\{(cantilever)\}$$

Vibrations / critical speeds of shafts

All critical (whirling) speeds of a shaft with several disks

$$\det (\mathbf{A}\mathbf{M}-\frac{1}{{\omega }^2}\mathbf{I})=0\backslash det\backslash Big(\backslash mathbf\; A\backslash mathbf\; M\; -\; \backslash frac\{1\}\{\backslash omega^2\}\backslash mathbf\; I\backslash Big)\; =\; 0$$

Critical speed of a shaft carrying one disk

$$k=\frac{3EIL}{a^2{b}^2},{\omega }_{cr}=\sqrt{\frac{k}{m}}k\; =\; \backslash frac\{3EIL\}\{a^2b^2\},\backslash quad\; \backslash omega\_\{cr\}\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}$$

First critical speed of a shaft with several disks (Dunkerley)

$$\frac{1}{{\omega }_{cr}^2}=\sum _i{m}_i{a}_{ii},a_{ii}=\frac{a_i^2{b}_i^2}{3EIL}\backslash frac\{1\}\{\backslash omega\_\{cr\}^2\}\; =\; \backslash sum\_i\; m\_ia\_\{ii\},\backslash quad\; a\_\{ii\}\; =\; \backslash frac\{a\_i^2b\_i^2\}\{3EIL\}$$

First critical speed of a shaft with several disks (Rayleigh)

$${\omega }_{cr}^2=\frac{g\sum m_i{y}_i}{\sum m_i{y}_i^2}\backslash omega\_\{cr\}^2\; =\; \backslash frac\{g\backslash sum\; m\_iy\_i\}\{\backslash sum\; m\_iy\_i^2\}$$

Vibrations / damped vibration

Damping ratio, damped frequency and logarithmic decrement

$$\zeta =\frac{c}{2\sqrt{km}},{\omega }_d={\omega }_n\sqrt{1-{\zeta }^2},\delta =\frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}\backslash zeta\; =\; \backslash frac\{c\}\{2\backslash sqrt\{km\}\},\backslash quad\; \backslash omega\_d\; =\; \backslash omega\_n\backslash sqrt\{1-\backslash zeta^2\},\backslash quad\; \backslash delta\; =\; \backslash frac\{2\backslash pi\backslash zeta\}\{\backslash sqrt\{1-\backslash zeta^2\}\}$$

Free vibration response of a damped spring-mass system

$$x(t)=e^{-\zeta {\omega }_{n}t}(x_0\cos {\omega }_{d}t+\frac{v_0+\zeta {\omega }_n{x}_0}{{\omega }_d}\sin {\omega }_{d}t)x(t)\; =\; e^\{-\backslash zeta\backslash omega\_nt\}\backslash Big(x\_0\backslash cos\backslash omega\_dt\; +\; \backslash frac\{v\_0+\backslash zeta\backslash omega\_nx\_0\}\{\backslash omega\_d\}\backslash sin\backslash omega\_dt\backslash Big)$$

Vibrations / forced vibration

Steady-state response to a harmonic force

$$X=\frac{F_0\mathrm{/}k}{\sqrt{(1-r^2{)}^2+(2\zeta r{)}^2}},\tan \varphi =\frac{2\zeta r}{1-r^2}X\; =\; \backslash frac\{F\_0/k\}\{\backslash sqrt\{(1-r^2)^2\; +\; (2\backslash zeta\; r)^2\}\},\backslash quad\; \backslash tan\backslash phi\; =\; \backslash frac\{2\backslash zeta\; r\}\{1-r^2\}$$

Vibration due to rotating unbalance

$$X=\frac{m_{u}e{\omega }^2}{\sqrt{(k-M{\omega }^2{)}^2+(c\omega {)}^2}},F_T=X\sqrt{k^2+(c\omega {)}^2}X\; =\; \backslash frac\{m\_ue\backslash omega^2\}\{\backslash sqrt\{(k-M\backslash omega^2)^2\; +\; (c\backslash omega)^2\}\},\backslash quad\; F\_T\; =\; X\backslash sqrt\{k^2+(c\backslash omega)^2\}$$

Vibrations / free vibration

Natural frequency of a spring-mass system

$${\omega }_n=\sqrt{\frac{k}{m}},f_n=\frac{1}{2\pi }\sqrt{\frac{k}{m}}=\frac{1}{2\pi }\sqrt{\frac{g}{{\delta }_{st}}}\backslash omega\_n\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\},\backslash quad\; f\_n\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash sqrt\{\backslash frac\{k\}\{m\}\}\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash sqrt\{\backslash frac\{g\}\{\backslash delta\_\{st\}\}\}$$

Vibrations / multi-degree-of-freedom systems

Natural frequencies and first mode of a spring-mass chain

$$(\mathbf{K}-{\omega }^2\mathbf{M})\mathit{\varphi }=\mathbf{0}(\backslash mathbf\; K\; -\; \backslash omega^2\backslash mathbf\; M)\backslash boldsymbol\backslash phi\; =\; \backslash mathbf\; 0$$

Vibrations / torsional vibration

Torsional natural frequency of a two-rotor system

$$\omega =\sqrt{\frac{k_t(J_1+J_2)}{J_1{J}_2}},l_1=\frac{J_{2}L}{J_1+J_2}\backslash omega\; =\; \backslash sqrt\{\backslash frac\{k\_t(J\_1+J\_2)\}\{J\_1J\_2\}\},\backslash quad\; l\_1\; =\; \backslash frac\{J\_2L\}\{J\_1+J\_2\}$$

Vibrations / vibration isolation

Force (or base-motion) transmissibility of a vibration isolator

$$TR=\sqrt{\frac{1+(2\zeta r{)}^2}{(1-r^2{)}^2+(2\zeta r{)}^2}}TR\; =\; \backslash sqrt\{\backslash frac\{1\; +\; (2\backslash zeta\; r)^2\}\{(1-r^2)^2\; +\; (2\backslash zeta\; r)^2\}\}$$

Stiffness of an undamped isolator for a target transmissibility

$$TR=\frac{1}{r^2-1},k=m(\frac{\omega }{r}{)}^{2}TR\; =\; \backslash frac\{1\}\{r^2-1\},\backslash quad\; k\; =\; m\backslash Big(\backslash frac\{\backslash omega\}\{r\}\backslash Big)^2$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified mechanical engineering capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (statics, beams and section properties, stress and strain, torsion, columns, fatigue, machine elements, vibrations and structural analysis); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{mechanical_engineering_ai999x_1_0_0,
  title  = {mechanical-engineering-ai999x: an offline, verified mechanical engineering solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/mechanical-engineering-ai999x}
}

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Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com