Bayesian Rotor Imbalance Tracking for Steam Turbines
Setup
# CPU only — no GPU required
System.put_env("EXLA_CPU_ONLY", "true")
System.put_env("CUDA_VISIBLE_DEVICES", "")
Mix.install([
{:exmc, path: Path.expand("../", __DIR__)},
{:exla, "~> 0.10"},
{:kino_vega_lite, "~> 0.1"}
])
Application.put_env(:exla, :clients, host: [platform: :host])
Application.put_env(:exla, :default_client, :host)
Nx.default_backend(Nx.BinaryBackend)
Nx.Defn.default_options(compiler: EXLA, client: :host)
[]Why This Matters
A steam turbine vibrates. They all vibrate. The question is whether the vibration pattern reveals an imbalance mass — and if so, how large, at what angle, and how urgently it needs correction.
The traditional approach: an experienced technician reads a spectrum analyzer, recognizes the 1X signature, and makes a judgment call. This works until the technician retires, or until you have 200 turbines across six plants and can’t put a human in front of every spectrum.
The Bayesian approach: encode the physics of rotor dynamics as a probabilistic model. The imbalance mass and angle are unknown parameters. The vibration measurements are noisy observations. NUTS recovers the posterior — not “the imbalance is 12 grams at 45 degrees” but “the imbalance is 12 ± 3 grams at 45 ± 8 degrees, and here is the probability it exceeds the trip threshold.”
The Jeffcott Rotor Model
Every rotating machine vibrates. The dominant vibration at running speed (1X synchronous) is caused by mass imbalance — an uneven distribution of weight around the rotor centerline.
The Jeffcott rotor model predicts the 1X vibration amplitude from the physics:
$$A = \frac{U \cdot \omega^2}{\sqrt{(k - m\omega^2)^2 + (c\omega)^2}}$$
- U — imbalance magnitude (g-mm), the mass-eccentricity product
- omega — running speed (rad/s), measured by the keyphasor
- k — bearing/shaft stiffness (N/m), approximately known from design
- c — damping coefficient (N-s/m), approximately known from design
- m — rotor mass (kg), known from design
The question every turbine engineer asks: “How bad is the imbalance, and when do I need to balance?” Today this is answered by gut feel and fixed alarm thresholds. We answer it with a probability distribution.
alias VegaLite, as: Vl
alias Exmc.{Builder, NUTS.Sampler, Dist}
# --- Rotor design parameters (known from OEM data) ---
# Typical 50 MW steam turbine, 3600 RPM (60 Hz)
rotor_mass = 5000.0 # kg
omega = 3600.0 * 2 * :math.pi() / 60.0 # rad/s (376.99)
k_design = 2.0e8 # N/m (bearing stiffness, approximate)
c_design = 5.0e4 # N-s/m (damping coefficient, approximate)
# Show how imbalance magnitude affects vibration amplitude
imbalance_range = Enum.map(0..500, fn u -> u * 1.0 end)
amp_data =
Enum.map(imbalance_range, fn u ->
# Convert g-mm to kg-m: 1 g-mm = 1e-6 kg-m
u_si = u * 1.0e-6
denom = :math.sqrt(:math.pow(k_design - rotor_mass * omega * omega, 2) + :math.pow(c_design * omega, 2))
a_m = u_si * omega * omega / denom
# Convert to mils peak-to-peak (1 mil = 25.4 um, p-p = 2 * amplitude)
a_mils = a_m * 1.0e6 / 25.4 * 2.0
%{"imbalance_gmm" => u, "amplitude_mils" => a_mils}
end)
Vl.new(width: 700, height: 350, title: "Jeffcott Rotor: Vibration vs Imbalance at 3600 RPM")
|> Vl.data_from_values(amp_data)
|> Vl.mark(:line, stroke_width: 2, color: "#1f77b4")
|> Vl.encode_field(:x, "imbalance_gmm", type: :quantitative, title: "Imbalance U (g-mm)")
|> Vl.encode_field(:y, "amplitude_mils", type: :quantitative, title: "1X Vibration (mils p-p)"){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"amplitude_mils":0.0,"imbalance_gmm":0.0},{"amplitude_mils":2.1901415839296986e-5,"imbalance_gmm":1.0},{"amplitude_mils":4.380283167859397e-5,"imbalance_gmm":2.0},{"amplitude_mils":6.570424751789096e-5,"imbalance_gmm":3.0},{"amplitude_mils":8.760566335718795e-5,"imbalance_gmm":4.0},{"amplitude_mils":1.0950707919648492e-4,"imbalance_gmm":5.0},{"amplitude_mils":1.314084950357819e-4,"imbalance_gmm":6.0},{"amplitude_mils":1.533099108750789e-4,"imbalance_gmm":7.0},{"amplitude_mils":1.752113267143759e-4,"imbalance_gmm":8.0},{"amplitude_mils":1.9711274255367284e-4,"imbalance_gmm":9.0},{"amplitude_mils":2.1901415839296984e-4,"imbalance_gmm":10.0},{"amplitude_mils":2.409155742322668e-4,"imbalance_gmm":11.0},{"amplitude_mils":2.628169900715638e-4,"imbalance_gmm":12.0},{"amplitude_mils":2.847184059108608e-4,"imbalance_gmm":13.0},{"amplitude_mils":3.066198217501578e-4,"imbalance_gmm":14.0},{"amplitude_mils":3.285212375894548e-4,"imbalance_gmm":15.0},{"amplitude_mils":3.504226534287518e-4,"imbalance_gmm":16.0},{"amplitude_mils":3.7232406926804876e-4,"imbalance_gmm":17.0},{"amplitude_mils":3.942254851073457e-4,"imbalance_gmm":18.0},{"amplitude_mils":4.161269009466427e-4,"imbalance_gmm":19.0},{"amplitude_mils":4.380283167859397e-4,"imbalance_gmm":20.0},{"amplitude_mils":4.5992973262523666e-4,"imbalance_gmm":21.0},{"amplitude_mils":4.818311484645336e-4,"imbalance_gmm":22.0},{"amplitude_mils":5.037325643038307e-4,"imbalance_gmm":23.0},{"amplitude_mils":5.256339801431276e-4,"imbalance_gmm":24.0},{"amplitude_mils":5.475353959824246e-4,"imbalance_gmm":25.0},{"amplitude_mils":5.694368118217216e-4,"imbalance_gmm":26.0},{"amplitude_mils":5.913382276610186e-4,"imbalance_gmm":27.0},{"amplitude_mils":6.132396435003155e-4,"imbalance_gmm":28.0},{"amplitude_mils":6.351410593396125e-4,"imbalance_gmm":29.0},{"amplitude_mils":6.570424751789096e-4,"imbalance_gmm":30.0},{"amplitude_mils":6.789438910182066e-4,"imbalance_gmm":31.0},{"amplitude_mils":7.008453068575036e-4,"imbalance_gmm":32.0},{"amplitude_mils":7.227467226968004e-4,"imbalance_gmm":33.0},{"amplitude_mils":7.446481385360975e-4,"imbalance_gmm":34.0},{"amplitude_mils":7.665495543753943e-4,"imbalance_gmm":35.0},{"amplitude_mils":7.884509702146914e-4,"imbalance_gmm":36.0},{"amplitude_mils":8.103523860539886e-4,"imbalance_gmm":37.0},{"amplitude_mils":8.322538018932854e-4,"imbalance_gmm":38.0},{"amplitude_mils":8.541552177325825e-4,"imbalance_gmm":39.0},{"amplitude_mils":8.760566335718794e-4,"imbalance_gmm":40.0},{"amplitude_mils":8.979580494111763e-4,"imbalance_gmm":41.0},{"amplitude_mils":9.198594652504733e-4,"imbalance_gmm":42.0},{"amplitude_mils":9.417608810897703e-4,"imbalance_gmm":43.0},{"amplitude_mils":9.636622969290672e-4,"imbalance_gmm":44.0},{"amplitude_mils":9.855637127683641e-4,"imbalance_gmm":45.0},{"amplitude_mils":0.0010074651286076613,"imbalance_gmm":46.0},{"amplitude_mils":0.0010293665444469583,"imbalance_gmm":47.0},{"amplitude_mils":0.0010512679602862553,"imbalance_gmm":48.0},{"amplitude_mils":0.0010731693761255525,"imbalance_gmm":49.0},{"amplitude_mils":0.0010950707919648492,"imbalance_gmm":50.0},{"amplitude_mils":0.0011169722078041462,"imbalance_gmm":51.0},{"amplitude_mils":0.0011388736236434432,"imbalance_gmm":52.0},{"amplitude_mils":0.0011607750394827402,"imbalance_gmm":53.0},{"amplitude_mils":0.0011826764553220371,"imbalance_gmm":54.0},{"amplitude_mils":0.0012045778711613341,"imbalance_gmm":55.0},{"amplitude_mils":0.001226479287000631,"imbalance_gmm":56.0},{"amplitude_mils":0.0012483807028399283,"imbalance_gmm":57.0},{"amplitude_mils":0.001270282118679225,"imbalance_gmm":58.0},{"amplitude_mils":0.0012921835345185222,"imbalance_gmm":59.0},{"amplitude_mils":0.0013140849503578192,"imbalance_gmm":60.0},{"amplitude_mils":0.0013359863661971162,"imbalance_gmm":61.0},{"amplitude_mils":0.0013578877820364132,"imbalance_gmm":62.0},{"amplitude_mils":0.00137978919787571,"imbalance_gmm":63.0},{"amplitude_mils":0.0014016906137150071,"imbalance_gmm":64.0},{"amplitude_mils":0.0014235920295543039,"imbalance_gmm":65.0},{"amplitude_mils":0.0014454934453936009,"imbalance_gmm":66.0},{"amplitude_mils":0.001467394861232898,"imbalance_gmm":67.0},{"amplitude_mils":0.001489296277072195,"imbalance_gmm":68.0},{"amplitude_mils":0.0015111976929114922,"imbalance_gmm":69.0},{"amplitude_mils":0.0015330991087507885,"imbalance_gmm":70.0},{"amplitude_mils":0.0015550005245900857,"imbalance_gmm":71.0},{"amplitude_mils":0.0015769019404293827,"imbalance_gmm":72.0},{"amplitude_mils":0.0015988033562686797,"imbalance_gmm":73.0},{"amplitude_mils":0.001620704772107977,"imbalance_gmm":74.0},{"amplitude_mils":0.0016426061879472739,"imbalance_gmm":75.0},{"amplitude_mils":0.0016645076037865708,"imbalance_gmm":76.0},{"amplitude_mils":0.0016864090196258678,"imbalance_gmm":77.0},{"amplitude_mils":0.001708310435465165,"imbalance_gmm":78.0},{"amplitude_mils":0.0017302118513044615,"imbalance_gmm":79.0},{"amplitude_mils":0.0017521132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U (g-mm)","type":"quantitative"},"y":{"field":"amplitude_mils","title":"1X Vibration (mils p-p)","type":"quantitative"}},"height":350,"mark":{"color":"#1f77b4","strokeWidth":2,"type":"line"},"title":"Jeffcott Rotor: Vibration vs Imbalance at 3600 RPM","width":700}Synthetic Vibration Data
We simulate 8 weeks of daily vibration readings from a turbine with gradually increasing imbalance — mimicking deposit buildup on turbine blades.
# True parameters (unknown to the model)
true_u_initial = 80.0 # g-mm (well-balanced rotor)
true_growth_rate = 0.018 # per day (exponential growth from deposits)
true_phi = 2.35 # radians (~135 degrees, heavy spot location)
true_sigma = 0.08 # mils measurement noise
# Generate 56 days (8 weeks) of daily observations
days = Enum.to_list(1..56)
rng = :rand.seed_s(:exsss, 42)
# Physical constants for forward model
u_to_amp = fn u_gmm ->
u_si = u_gmm * 1.0e-6
denom = :math.sqrt(:math.pow(k_design - rotor_mass * omega * omega, 2) + :math.pow(c_design * omega, 2))
a_m = u_si * omega * omega / denom
# mils p-p
a_m * 1.0e6 / 25.4 * 2.0
end
{observed, _rng} =
Enum.map_reduce(days, rng, fn day, rng ->
# True imbalance grows exponentially
u_true = true_u_initial * :math.exp(true_growth_rate * day)
a_true = u_to_amp.(u_true)
# Add lognormal measurement noise
{noise, rng} = :rand.normal_s(rng)
a_obs = a_true * :math.exp(true_sigma * noise)
{{day, a_obs, a_true, u_true}, rng}
end)
# Alarm threshold: 2.5 mils p-p (API 670 typical)
alarm_threshold = 2.5
obs_data =
Enum.flat_map(observed, fn {day, a_obs, a_true, _u} ->
[
%{"day" => day, "amplitude" => a_obs, "series" => "Observed (1X probe)"},
%{"day" => day, "amplitude" => a_true, "series" => "True response"}
]
end)
alarm_line =
Enum.map(days, fn day ->
%{"day" => day, "amplitude" => alarm_threshold, "series" => "Alarm Threshold (2.5 mil)"}
end)
Vl.new(
width: 700,
height: 350,
title: [
text: "Turbine Vibration Trend — 8 Weeks",
subtitle: "Unit 3, Bearing #2, X-probe"
]
)
|> Vl.data_from_values(obs_data ++ alarm_line)
|> Vl.layers([
Vl.new()
|> Vl.mark(:line, stroke_width: 2, stroke_dash: [6, 3], color: "#d62728")
|> Vl.transform(filter: "datum.series == 'Alarm Threshold (2.5 mil)'")
|> Vl.encode_field(:x, "day", type: :quantitative, title: "Day")
|> Vl.encode_field(:y, "amplitude", type: :quantitative, title: "1X Vibration (mils p-p)"),
Vl.new()
|> Vl.mark(:line, stroke_width: 1.5, stroke_dash: [4, 2], color: "#999")
|> Vl.transform(filter: "datum.series == 'True response'")
|> Vl.encode_field(:x, "day", type: :quantitative)
|> Vl.encode_field(:y, "amplitude", type: :quantitative),
Vl.new()
|> Vl.mark(:point, size: 40, filled: true, color: "#1f77b4")
|> Vl.transform(filter: "datum.series == 'Observed (1X probe)'")
|> Vl.encode_field(:x, "day", type: :quantitative)
|> Vl.encode_field(:y, "amplitude", type: :quantitative)
]){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"amplitude":0.0018692433513519365,"day":1,"series":"Observed (1X probe)"},{"amplitude":0.0017839368590471351,"day":1,"series":"True response"},{"amplitude":0.0015600917918305768,"day":2,"series":"Observed (1X probe)"},{"amplitude":0.001816338462098891,"day":2,"series":"True response"},{"amplitude":0.0018515297437218152,"day":3,"series":"Observed (1X probe)"},{"amplitude":0.0018493285747018678,"day":3,"series":"True response"},{"amplitude":0.0022017868518929345,"day":4,"series":"Observed (1X probe)"},{"amplitude":0.0018829178859411488,"day":4,"series":"True response"},{"amplitude":0.0016589129925106407,"day":5,"series":"Observed (1X probe)"},{"amplitude":0.0019171172790474176,"day":5,"series":"True response"},{"amplitude":0.002144288375581983,"day":6,"series":"Observed (1X probe)"},{"amplitude":0.0019519378349232219,"day":6,"series":"True response"},{"amplitude":0.00226039966039249,"day":7,"series":"Observed (1X probe)"},{"amplitude":0.0019873908357332776,"day":7,"series":"True response"},{"amplitude":0.002174053797180669,"day":8,"series":"Observed (1X probe)"},{"amplitude":0.002023487768559994,"day":8,"series":"True response"},{"amplitude":0.0020100200560923894,"day":9,"series":"Observed (1X probe)"},{"amplitude":0.0020602403291253854,"day":9,"series":"True response"},{"amplitude":0.002149465022083736,"day":10,"series":"Observed (1X probe)"},{"amplitude":0.0020976604255805906,"day":10,"series":"True response"},{"amplitude":0.0020896640146212974,"day":11,"series":"Observed (1X probe)"},{"amplitude":0.002135760182364216,"day":11,"series":"True response"},{"amplitude":0.002138818033396594,"day":12,"series":"Observed (1X probe)"},{"amplitude":0.002174551944130758,"day":12,"series":"True response"},{"amplitude":0.0020476469699435957,"day":13,"series":"Observed (1X probe)"},{"amplitude":0.0022140482797503858,"day":13,"series":"True response"},{"amplitude":0.0021526261403687157,"day":14,"series":"Observed (1X probe)"},{"amplitude":0.0022542619863813555,"day":14,"series":"True response"},{"amplitude":0.002500420333619556,"day":15,"series":"Observed (1X probe)"},{"amplitude":0.002295206093616409,"day":15,"series":"True response"},{"amplitude":0.002630711326722741,"day":16,"series":"Observed (1X probe)"},{"amplitude":0.002336893867704475,"day":16,"series":"True response"},{"amplitude":0.0025961894608494116,"day":17,"series":"Observed (1X probe)"},{"amplitude":0.0023793388158490454,"day":17,"series":"True response"},{"amplitude":0.0028403124787368434,"day":18,"series":"Observed (1X probe)"},{"amplitude":0.0024225546905846315,"day":18,"series":"True response"},{"amplitude":0.0024241541674372893,"day":19,"series":"Observed (1X probe)"},{"amplitude":0.0024665554942327043,"day":19,"series":"True response"},{"amplitude":0.002501936726061779,"day":20,"series":"Observed (1X probe)"},{"amplitude":0.002511355483438568,"day":20,"series":"True response"},{"amplitude":0.0023505086699129375,"day":21,"series":"Observed (1X probe)"},{"amplitude":0.002556969173790642,"day":21,"series":"True response"},{"amplitude":0.0027322653959512702,"day":22,"series":"Observed (1X probe)"},{"amplitude":0.002603411344523633,"day":22,"series":"True response"},{"amplitude":0.0025312306697968722,"day":23,"series":"Observed (1X probe)"},{"amplitude":0.0026506970433071366,"day":23,"series":"True response"},{"amplitude":0.002787443617410404,"day":24,"series":"Observed (1X probe)"},{"amplitude":0.0026988415911212205,"day":24,"series":"True response"},{"amplitude":0.0028510207514109175,"day":25,"series":"Observed (1X probe)"},{"amplitude":0.0027478605872205486,"day":25,"series":"True response"},{"amplitude":0.002692975286650207,"day":26,"series":"Observed (1X probe)"},{"amplitude":0.002797769914188679,"day":26,"series":"True response"},{"amplitude":0.0022696140970061584,"day":27,"series":"Observed (1X probe)"},{"amplitude":0.0028485857430841635,"day":27,"series":"True response"},{"amplitude":0.002648811971349699,"day":28,"series":"Observed (1X probe)"},{"amplitude":0.0029003245386801054,"day":28,"series":"True response"},{"amplitude":0.0030702641271775377,"day":29,"series":"Observed (1X probe)"},{"amplitude":0.0029530030647988907,"day":29,"series":"True response"},{"amplitude":0.003160918137142843,"day":30,"series":"Observed (1X probe)"},{"amplitude":0.0030066383897438217,"day":30,"series":"True response"},{"amplitude":0.003135391204627754,"day":31,"series":"Observed (1X probe)"},{"amplitude":0.0030612478918293864,"day":31,"series":"True response"},{"amplitude":0.0031821247450571234,"day":32,"series":"Observed (1X probe)"},{"amplitude":0.003116849265011989,"day":32,"series":"True response"},{"amplitude":0.003141146871558335,"day":33,"series":"Observed (1X probe)"},{"amplitude":0.0031734605246229483,"day":33,"series":"True response"},{"amplitude":0.0031737446252185983,"day":34,"series":"Observed (1X probe)"},{"amplitude":0.003231100013205618,"day":34,"series":"True response"},{"amplitude":0.00343198223175103,"day":35,"series":"Observed (1X probe)"},{"amplitude":0.0032897864064585338,"day":35,"series":"True response"},{"amplitude":0.0032054824202780966,"day":36,"series":"Observed (1X probe)"},{"amplitude":0.003349538719286504,"day":36,"series":"True response"},{"amplitude":0.0037136938449923336,"day":37,"series":"Observed (1X probe)"},{"amplitude":0.0034103763119616042,"day":37,"series":"True response"},{"amplitude":0.0036222677484358443,"day":38,"series":"Observed (1X probe)"},{"amplitude":0.0034723188963960757,"day":38,"series":"True response"},{"amplitude":0.003627718602410881,"day":39,"series":"Observed (1X probe)"},{"amplitude":0.0035353865425291524,"day":39,"series":"True response"},{"amplitude":0.003808467540967873,"day":40,"series":"Observed (1X probe)"},{"amplitude":0.003599599684829903,"day":40,"series":"True response"},{"amplitude":0.003786052117686019,"day":41,"series":"Observed (1X probe)"},{"amplitude":0.003664979128918179,"day":41,"series":"True response"},{"amplitude":0.0035232447004579976,"day":42,"series":"Observed (1X probe)"},{"amplitude":0.0037315460583058064,"day":42,"series":"True response"},{"amplitude":0.0035593090809295704,"day":43,"series":"Observed (1X probe)"},{"amplitude":0.0037993220412602426,"day":43,"series":"True response"},{"amplitude":0.003886902912631799,"day":44,"series":"Observed (1X probe)"},{"amplitude":0.0038683290377928754,"day":44,"series":"True response"},{"amplitude":0.003987379912489197,"day":45,"series":"Observed (1X probe)"},{"amplitude":0.003938589406774262,"day":45,"series":"True response"},{"amplitude":0.004092224122797043,"day":46,"series":"Observed (1X probe)"},{"amplitude":0.004010125913178594,"day":46,"series":"True response"},{"amplitude":0.004262924130097846,"day":47,"series":"Observed (1X probe)"},{"amplitude":0.004082961735459757,"day":47,"series":"True response"},{"amplitude":0.003920966236386954,"day":48,"series":"Observed (1X probe)"},{"amplitude":0.004157120473061345,"day":48,"series":"True response"},{"amplitude":0.004840382051613223,"day":49,"series":"Observed (1X probe)"},{"amplitude":0.004232626154063085,"day":49,"series":"True response"},{"amplitude":0.004362080160132145,"day":50,"series":"Observed (1X probe)"},{"amplitude":0.004309503242966156,"day":50,"series":"True response"},{"amplitude":0.004962769941724683,"day":51,"series":"Observed (1X probe)"},{"amplitude":0.00438777664861989,"day":51,"series":"True response"},{"amplitude":0.004506599454381675,"day":52,"series":"Observed (1X probe)"},{"amplitude":0.0044674717322924625,"day":52,"series":"True response"},{"amplitude":0.004878947900203013,"day":53,"series":"Observed (1X probe)"},{"amplitude":0.004548614315888161,"day":53,"series":"True response"},{"amplitude":0.004761213325515078,"day":54,"series":"Observed (1X probe)"},{"amplitude":0.004631230690313916,"day":54,"series":"True response"},{"amplitude":0.0044926342044481265,"day":55,"series":"Observed (1X probe)"},{"amplitude":0.004715347623997776,"day":55,"series":"True response"},{"amplitude":0.004637570889918955,"day":56,"series":"Observed (1X probe)"},{"amplitude":0.004800992371562118,"day":56,"series":"True response"},{"amplitude":2.5,"day":1,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":2,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":3,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":4,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":5,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":6,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":7,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":8,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":9,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":10,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":11,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":12,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":13,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":14,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":15,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":16,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":17,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":18,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":19,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":20,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":21,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":22,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":23,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":24,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":25,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":26,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":27,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":28,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":29,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":30,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":31,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":32,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":33,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":34,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":35,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":36,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":37,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":38,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":39,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":40,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":41,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":42,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":43,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":44,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":45,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":46,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":47,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":48,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":49,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":50,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":51,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":52,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":53,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":54,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":55,"series":"Alarm Threshold (2.5 mil)"},{"amplitude":2.5,"day":56,"series":"Alarm Threshold (2.5 mil)"}]},"height":350,"layer":[{"encoding":{"x":{"field":"day","title":"Day","type":"quantitative"},"y":{"field":"amplitude","title":"1X Vibration (mils p-p)","type":"quantitative"}},"mark":{"color":"#d62728","strokeDash":[6,3],"strokeWidth":2,"type":"line"},"transform":[{"filter":"datum.series == 'Alarm Threshold (2.5 mil)'"}]},{"encoding":{"x":{"field":"day","type":"quantitative"},"y":{"field":"amplitude","type":"quantitative"}},"mark":{"color":"#999","strokeDash":[4,2],"strokeWidth":1.5,"type":"line"},"transform":[{"filter":"datum.series == 'True response'"}]},{"encoding":{"x":{"field":"day","type":"quantitative"},"y":{"field":"amplitude","type":"quantitative"}},"mark":{"color":"#1f77b4","filled":true,"size":40,"type":"point"},"transform":[{"filter":"datum.series == 'Observed (1X probe)'"}]}],"title":{"subtitle":"Unit 3, Bearing #2, X-probe","text":"Turbine Vibration Trend — 8 Weeks"},"width":700}Bayesian Model
We infer three latent parameters from the vibration amplitude observations:
-
log_u— log of imbalance magnitude (ensures positivity) -
phi— phase angle of heavy spot -
sigma— measurement noise scale
The likelihood uses the Jeffcott forward model to predict vibration amplitude from the imbalance, then compares to observed data under lognormal noise.
# Prepare data tensors — use first 28 days (4 weeks) initially
n_obs = 28
subset = Enum.take(observed, n_obs)
a_obs_list = Enum.map(subset, fn {_, a, _, _} -> a end)
day_list = Enum.map(subset, fn {d, _, _, _} -> d * 1.0 end)
t_days = Nx.tensor(day_list)
log_a_obs = Nx.tensor(Enum.map(a_obs_list, &:math.log/1))
# Forward model constants (precomputed)
# A(U) = U * 1e-6 * omega^2 / denom * 1e6/25.4 * 2
# Simplify: A(U) = U * scale_factor
denom_val = :math.sqrt(:math.pow(k_design - rotor_mass * omega * omega, 2) + :math.pow(c_design * omega, 2))
scale_factor = 1.0e-6 * omega * omega / denom_val * 1.0e6 / 25.4 * 2.0
scale_tensor = Nx.tensor(scale_factor)
# Custom logpdf: Jeffcott rotor likelihood with exponential imbalance growth
turbine_logpdf = fn _x, params ->
log_u0 = params.log_u0
growth = params.growth
sigma = params.sigma
t = params.t_days
log_a_observed = params.log_a_obs
scale = params.scale
# Predicted imbalance at each time: U(t) = U0 * exp(growth * t)
# Predicted amplitude: A(t) = U(t) * scale_factor
# log(A(t)) = log(U0) + growth * t + log(scale_factor)
log_a_pred = Nx.add(Nx.add(log_u0, Nx.multiply(growth, t)), Nx.log(scale))
# Lognormal likelihood
residuals = Nx.subtract(log_a_observed, log_a_pred)
n = Nx.tensor(Nx.size(log_a_observed) * 1.0)
Nx.subtract(
Nx.sum(Nx.negate(Nx.divide(Nx.pow(residuals, 2), Nx.multiply(2.0, Nx.pow(sigma, 2))))),
Nx.multiply(n, Nx.log(sigma))
)
end
# Build model IR
ir = Builder.new_ir()
# Priors informed by turbomachinery engineering knowledge
# log_u0: well-balanced rotor ~50-150 g-mm
ir = Builder.rv(ir, "log_u0", Dist.Normal, %{
mu: Nx.tensor(:math.log(100.0)),
sigma: Nx.tensor(0.8)
})
# growth rate: typically 0.005-0.05 per day for deposit buildup
ir = Builder.rv(ir, "growth", Dist.Normal, %{
mu: Nx.tensor(0.015),
sigma: Nx.tensor(0.015)
})
# measurement noise
ir = Builder.rv(ir, "sigma", Dist.HalfCauchy, %{
scale: Nx.tensor(0.2)
}, transform: :log)
# Custom likelihood
turbine_dist = Dist.Custom.new(turbine_logpdf, support: :real)
ir =
Dist.Custom.rv(ir, "ll", turbine_dist, %{
log_u0: "log_u0",
growth: "growth",
sigma: "sigma",
t_days: t_days,
log_a_obs: log_a_obs,
scale: scale_tensor
})
ir = Builder.obs(ir, "ll_obs", "ll", Nx.tensor(0.0))
IO.puts("Free parameters: #{inspect(Enum.sort(Map.keys(ir.nodes) -- ["ll", "ll_obs"]))}")
IO.puts("Observations: #{n_obs} days of 1X vibration amplitude")
irFree parameters: ["growth", "log_u0", "sigma"]
Observations: 28 days of 1X vibration amplitude%Exmc.IR{
nodes: %{
"growth" => %Exmc.Node{
id: "growth",
op: {:rv, Exmc.Dist.Normal,
%{
sigma: #Nx.Tensor<
f32
0.014999999664723873
>,
mu: #Nx.Tensor<
f32
0.014999999664723873
>
}},
deps: [],
shape: nil,
dtype: nil
},
"ll" => %Exmc.Node{
id: "ll",
op: {:rv, Exmc.Dist.Custom,
%{
scale: #Nx.Tensor<
f32
2.190141640312504e-5
>,
t_days: #Nx.Tensor<
f32[28]
[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0]
>,
log_a_obs: #Nx.Tensor<
f32[28]
[-6.28222131729126, -6.463010787963867, -6.291743278503418, -6.118485927581787, -6.40159273147583, -6.144947528839111, -6.0922136306762695, -6.131161689758301, -6.209610462188721, -6.142536163330078, -6.170752048492432, -6.1475019454956055, -6.19106388092041, -6.141066551208496, -5.991296291351318, -5.9405012130737305, -5.953710556030273, -5.8638410568237305, -6.02227258682251, -5.990690231323242, -6.053123474121094, -5.902624130249023, -5.9790496826171875, -5.882630348205566, -5.86007833480835, -5.917108535766602, -6.0881452560424805, -5.933643817901611]
>,
log_u0: "log_u0",
growth: "growth",
sigma: "sigma",
__dist__: %Exmc.Dist.Custom{
logpdf_fn: #Function<41.81571850/2 in :erl_eval.expr/6>,
support: :real,
transform: nil,
sample_fn: nil
}
}},
deps: ["log_u0", "growth", "sigma"],
shape: nil,
dtype: nil
},
"ll_obs" => %Exmc.Node{
id: "ll_obs",
op: {:obs, "ll",
#Nx.Tensor<
f32
0.0
>, %{}},
deps: ["ll"],
shape: nil,
dtype: nil
},
"log_u0" => %Exmc.Node{
id: "log_u0",
op: {:rv, Exmc.Dist.Normal,
%{
sigma: #Nx.Tensor<
f32
0.800000011920929
>,
mu: #Nx.Tensor<
f32
4.605170249938965
>
}},
deps: [],
shape: nil,
dtype: nil
},
"sigma" => %Exmc.Node{
id: "sigma",
op: {:rv, Exmc.Dist.HalfCauchy,
%{
scale: #Nx.Tensor<
f32
0.20000000298023224
>
}, :log},
deps: [],
shape: nil,
dtype: nil
}
},
outputs: [],
ncp_info: %{},
data: nil
}MCMC Sampling
init = %{
"log_u0" => :math.log(100.0),
"growth" => 0.015,
"sigma" => 0.15
}
t0 = System.monotonic_time(:millisecond)
{trace, stats} =
Sampler.sample(ir, init,
num_warmup: 500,
num_samples: 500,
seed: 42,
ncp: false
)
elapsed = System.monotonic_time(:millisecond) - t0
u0_samples = trace["log_u0"] |> Nx.exp() |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
growth_samples = trace["growth"] |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
sigma_samples = trace["sigma"] |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
n_div = Enum.count(stats.sample_stats, & &1.divergent)
IO.puts("Sampling complete in #{elapsed}ms")
IO.puts("Divergences: #{n_div}")
IO.puts("Step size: #{Float.round(stats.step_size, 4)}")Sampling complete in 3392ms
Divergences: 0
Step size: 0.1685:okParameter Recovery
summary = Exmc.Diagnostics.summary(trace)
IO.puts("Parameter Recovery (4 weeks of data):\n")
for {name, s} <- Enum.sort(summary) do
true_val =
case name do
"log_u0" -> :math.log(true_u_initial)
"growth" -> true_growth_rate
"sigma" -> true_sigma
_ -> nil
end
label =
case name do
"log_u0" -> "U0 (g-mm)"
"growth" -> "Growth (/day)"
"sigma" -> "Sigma"
_ -> name
end
display_mean =
case name do
"log_u0" -> Float.round(:math.exp(s.mean), 1)
_ -> Float.round(s.mean, 4)
end
display_true =
case name do
"log_u0" -> true_u_initial
"growth" -> true_growth_rate
"sigma" -> true_sigma
_ -> nil
end
IO.puts(" #{String.pad_trailing(label, 15)} posterior=#{display_mean} true=#{display_true}")
end
:okwarning: variable "true_val" is unused (if the variable is not meant to be used, prefix it with an underscore)
└─ /home/io/projects/learn_erl/_exmc-things/exmc/notebooks/07_turbine_imbalance.livemd#cell:57bhvdajufacx3ma:6
Parameter Recovery (4 weeks of data):
Growth (/day) posterior=0.0152 true=0.018
U0 (g-mm) posterior=83.7 true=80.0
Sigma posterior=0.0978 true=0.08:okTrace Plots
trace_data = fn samples, name ->
Enum.with_index(samples, fn val, i -> %{"iteration" => i, "value" => val, "param" => name} end)
end
all_traces =
trace_data.(u0_samples, "U0 (g-mm)") ++
trace_data.(growth_samples, "Growth rate (/day)") ++
trace_data.(sigma_samples, "Sigma (noise)")
Vl.new(width: 700, height: 120, title: "MCMC Trace Plots")
|> Vl.data_from_values(all_traces)
|> Vl.mark(:line, opacity: 0.6, stroke_width: 0.5)
|> Vl.encode_field(:x, "iteration", type: :quantitative)
|> Vl.encode_field(:y, "value", type: :quantitative)
|> Vl.encode_field(:row, "param", type: :nominal, title: nil,
header: [label_font_size: 11])
|> Vl.resolve(:scale, y: :independent){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"iteration":0,"param":"U0 (g-mm)","value":94.21137608217651},{"iteration":1,"param":"U0 (g-mm)","value":93.84731580724451},{"iteration":2,"param":"U0 (g-mm)","value":92.96796053593827},{"iteration":3,"param":"U0 (g-mm)","value":73.88970547235296},{"iteration":4,"param":"U0 (g-mm)","value":77.28324875748456},{"iteration":5,"param":"U0 (g-mm)","value":86.61839880556317},{"iteration":6,"param":"U0 (g-mm)","value":85.92256056800437},{"iteration":7,"param":"U0 (g-mm)","value":80.30800184260461},{"iteration":8,"param":"U0 (g-mm)","value":82.86690874825737},{"iteration":9,"param":"U0 (g-mm)","value":76.81977242781844},{"iteration":10,"param":"U0 (g-mm)","value":78.11195788047175},{"iteration":11,"param":"U0 (g-mm)","value":76.53168964074861},{"iteration":12,"param":"U0 (g-mm)","value":76.30585322492446},{"iteration":13,"param":"U0 (g-mm)","value":81.75791343412624},{"iteration":14,"param":"U0 (g-mm)","value":86.5825438835874},{"iteration":15,"param":"U0 (g-mm)","value":86.55429989064237},{"iteration":16,"param":"U0 (g-mm)","value":88.56203969353207},{"iteration":17,"param":"U0 (g-mm)","value":86.31441070651088},{"iteration":18,"param":"U0 (g-mm)","value":81.28990553489125},{"iteration":19,"param":"U0 (g-mm)","value":88.86824361285683},{"iteration":20,"param":"U0 (g-mm)","value":87.90414605005347},{"iteration":21,"param":"U0 (g-mm)","value":80.23076507089544},{"iteration":22,"param":"U0 (g-mm)","value":85.2141653607801},{"iteration":23,"param":"U0 (g-mm)","value":86.86717027570482},{"iteration":24,"param":"U0 (g-mm)","value":86.55816346601337},{"iteration":25,"param":"U0 (g-mm)","value":86.59637693632058},{"iteration":26,"param":"U0 (g-mm)","value":86.84094048137736},{"iteration":27,"param":"U0 (g-mm)","value":88.2789609804103},{"iteration":28,"param":"U0 (g-mm)","value":81.7157210964698},{"iteration":29,"param":"U0 (g-mm)","value":84.35128846308271},{"iteration":30,"param":"U0 (g-mm)","value":84.68955977349897},{"iteration":31,"param":"U0 (g-mm)","value":87.49580080555329},{"iteration":32,"param":"U0 (g-mm)","value":82.10831014751956},{"iteration":33,"param":"U0 (g-mm)","value":81.63788934795964},{"iteration":34,"param":"U0 (g-mm)","value":84.75327753088132},{"iteration":35,"param":"U0 (g-mm)","value":84.99101825481004},{"iteration":36,"param":"U0 (g-mm)","value":82.96474092200353},{"iteration":37,"param":"U0 (g-mm)","value":79.81406042091128},{"iteration":38,"param":"U0 (g-mm)","value":91.54739327792416},{"iteration":39,"param":"U0 (g-mm)","value":88.78756041315924},{"iteration":40,"param":"U0 (g-mm)","value":84.21029768376654},{"iteration":41,"param":"U0 (g-mm)","value":82.25621579869222},{"iteration":42,"param":"U0 (g-mm)","value":80.83685270239783},{"iteration":43,"param":"U0 (g-mm)","value":76.30213091637825},{"iteration":44,"param":"U0 (g-mm)","value":87.40143402322329},{"iteration":45,"param":"U0 (g-mm)","value":80.69821616465951},{"iteration":46,"param":"U0 (g-mm)","value":81.98230619676757},{"iteration":47,"param":"U0 (g-mm)","value":85.72702310961357},{"iteration":48,"param":"U0 (g-mm)","value":86.43149005276463},{"iteration":49,"param":"U0 (g-mm)","value":77.27234770559924},{"iteration":50,"param":"U0 (g-mm)","value":79.85881115895312},{"iteration":51,"param":"U0 (g-mm)","value":81.13184570945997},{"iteration":52,"param":"U0 (g-mm)","value":82.50288712994485},{"iteration":53,"param":"U0 (g-mm)","value":87.6980070838735},{"iteration":54,"param":"U0 (g-mm)","value":83.95978212519307},{"iteration":55,"param":"U0 (g-mm)","value":86.59949067186949},{"iteration":56,"param":"U0 (g-mm)","value":80.58971538650044},{"iteration":57,"param":"U0 (g-mm)","value":84.65727435515433},{"iteration":58,"param":"U0 (g-mm)","value":83.66908729678309},{"iteration":59,"param":"U0 (g-mm)","value":80.53197304549617},{"iteration":60,"param":"U0 (g-mm)","value":86.81902357078879},{"iteration":61,"param":"U0 (g-mm)","value":88.70377231819303},{"iteration":62,"param":"U0 (g-mm)","value":89.30111327937759},{"iteration":63,"param":"U0 (g-mm)","value":83.26450149235791},{"iteration":64,"param":"U0 (g-mm)","value":81.97937909048653},{"iteration":65,"param":"U0 (g-mm)","value":76.97100883335453},{"iteration":66,"param":"U0 (g-mm)","value":76.807712050571},{"iteration":67,"param":"U0 (g-mm)","value":78.81621060037008},{"iteration":68,"param":"U0 (g-mm)","value":86.86048760893304},{"iteration":69,"param":"U0 (g-mm)","value":81.39916263775474},{"iteration":70,"param":"U0 (g-mm)","value":76.48810415126611},{"iteration":71,"param":"U0 (g-mm)","value":81.24517342899154},{"iteration":72,"param":"U0 (g-mm)","value":88.29938055246623},{"iteration":73,"param":"U0 (g-mm)","value":90.99153347375663},{"iteration":74,"param":"U0 (g-mm)","value":80.07082750145038},{"iteration":75,"param":"U0 (g-mm)","value":85.92472126744744},{"iteration":76,"param":"U0 (g-mm)","value":79.34828711729254},{"iteration":77,"param":"U0 (g-mm)","value":77.88117867792693},{"iteration":78,"param":"U0 (g-mm)","value":75.81450178520274},{"iteration":79,"param":"U0 (g-mm)","value":82.75800437291291},{"iteration":80,"param":"U0 (g-mm)","value":84.16071444119761},{"iteration":81,"param":"U0 (g-mm)","value":82.8757727592981},{"iteration":82,"param":"U0 (g-mm)","value":81.13231901765415},{"iteration":83,"param":"U0 (g-mm)","value":79.97199511068845},{"iteration":84,"param":"U0 (g-mm)","value":82.92103767136798},{"iteration":85,"param":"U0 (g-mm)","value":82.45153787679523},{"iteration":86,"param":"U0 (g-mm)","value":80.44634561258165},{"iteration":87,"param":"U0 (g-mm)","value":80.44634561258165},{"iteration":88,"param":"U0 (g-mm)","value":79.9674837502926},{"iteration":89,"param":"U0 (g-mm)","value":84.95186434208837},{"iteration":90,"param":"U0 (g-mm)","value":85.88389570165059},{"iteration":91,"param":"U0 (g-mm)","value":81.74715773883703},{"iteration":92,"param":"U0 (g-mm)","value":87.40834695165987},{"iteration":93,"param":"U0 (g-mm)","value":80.08301035598963},{"iteration":94,"param":"U0 (g-mm)","value":84.59735116343408},{"iteration":95,"param":"U0 (g-mm)","value":84.48019163591721},{"iteration":96,"param":"U0 (g-mm)","value":83.25857934442053},{"iteration":97,"param":"U0 (g-mm)","value":85.32485959323033},{"iteration":98,"param":"U0 (g-mm)","value":76.81513863853485},{"iteration":99,"param":"U0 (g-mm)","value":89.31708440874375},{"iteration":100,"param":"U0 (g-mm)","value":89.91737342420393},{"iteration":101,"param":"U0 (g-mm)","value":89.14969877537567},{"iteration":102,"param":"U0 (g-mm)","value":77.05462490928949},{"iteration":103,"param":"U0 (g-mm)","value":82.9561450679169},{"iteration":104,"param":"U0 (g-mm)","value":81.92142194091993},{"iteration":105,"param":"U0 (g-mm)","value":79.59125467729093},{"iteration":106,"param":"U0 (g-mm)","value":78.62138089081853},{"iteration":107,"param":"U0 (g-mm)","value":77.93222397016606},{"iteration":108,"param":"U0 (g-mm)","value":78.47245813166374},{"iteration":109,"param":"U0 (g-mm)","value":80.48622269506876},{"iteration":110,"param":"U0 (g-mm)","value":84.47439134451366},{"iteration":111,"param":"U0 (g-mm)","value":82.35782069993049},{"iteration":112,"param":"U0 (g-mm)","value":82.0409660238558},{"iteration":113,"param":"U0 (g-mm)","value":84.00367384247227},{"iteration":114,"param":"U0 (g-mm)","value":84.20890341766801},{"iteration":115,"param":"U0 (g-mm)","value":85.23987583533597},{"iteration":116,"param":"U0 (g-mm)","value":80.37046135367407},{"iteration":117,"param":"U0 (g-mm)","value":75.77608433798272},{"iteration":118,"param":"U0 (g-mm)","value":76.60005424022275},{"iteration":119,"param":"U0 (g-mm)","value":76.823652921136},{"iteration":120,"param":"U0 (g-mm)","value":76.41306353638286},{"iteration":121,"param":"U0 (g-mm)","value":75.50058614006598},{"iteration":122,"param":"U0 (g-mm)","value":81.88002599213394},{"iteration":123,"param":"U0 (g-mm)","value":80.63449724667038},{"iteration":124,"param":"U0 (g-mm)","value":81.14332619219509},{"iteration":125,"param":"U0 (g-mm)","value":84.43791517443817},{"iteration":126,"param":"U0 (g-mm)","value":87.35330658585283},{"iteration":127,"param":"U0 (g-mm)","value":80.62358291072933},{"iteration":128,"param":"U0 (g-mm)","value":87.3777416598826},{"iteration":129,"param":"U0 (g-mm)","value":87.45675193445642},{"iteration":130,"param":"U0 (g-mm)","value":81.40781896443892},{"iteration":131,"param":"U0 (g-mm)","value":84.42098682205712},{"iteration":132,"param":"U0 (g-mm)","value":86.58745277716086},{"iteration":133,"param":"U0 (g-mm)","value":81.18578024741814},{"iteration":134,"param":"U0 (g-mm)","value":85.89161984950243},{"iteration":135,"param":"U0 (g-mm)","value":86.52280711641828},{"iteration":136,"param":"U0 (g-mm)","value":86.0146601759462},{"iteration":137,"param":"U0 (g-mm)","value":80.79565091011696},{"iteration":138,"param":"U0 (g-mm)","value":84.67655382914126},{"iteration":139,"param":"U0 (g-mm)","value":82.4616651259409},{"iteration":140,"param":"U0 (g-mm)","value":83.06961194477225},{"iteration":141,"param":"U0 (g-mm)","value":81.43504716072185},{"iteration":142,"param":"U0 (g-mm)","value":81.51467668094233},{"iteration":143,"param":"U0 (g-mm)","value":82.81749831643302},{"iteration":144,"param":"U0 (g-mm)","value":88.78234593615979},{"iteration":145,"param":"U0 (g-mm)","value":76.64938385466164},{"iteration":146,"param":"U0 (g-mm)","value":76.32191369965258},{"iteration":147,"param":"U0 (g-mm)","value":91.4803781795798},{"iteration":148,"param":"U0 (g-mm)","value":90.79719664816572},{"iteration":149,"param":"U0 (g-mm)","value":90.89726921675597},{"iteration":150,"param":"U0 (g-mm)","value":76.33992076540008},{"iteration":151,"param":"U0 (g-mm)","value":76.47863816393199},{"iteration":152,"param":"U0 (g-mm)","value":82.07091595238636},{"iteration":153,"param":"U0 (g-mm)","value":83.44185386762048},{"iteration":154,"param":"U0 (g-mm)","value":83.50587372649885},{"iteration":155,"param":"U0 (g-mm)","value":81.54216405456536},{"iteration":156,"param":"U0 (g-mm)","value":79.14382614454934},{"iteration":157,"param":"U0 (g-mm)","value":87.50092717560472},{"iteration":158,"param":"U0 (g-mm)","value":78.87588381257692},{"iteration":159,"param":"U0 (g-mm)","value":78.82311680886306},{"iteration":160,"param":"U0 (g-mm)","value":78.8609129583642},{"iteration":161,"param":"U0 (g-mm)","value":79.10845612190654},{"iteration":162,"param":"U0 (g-mm)","value":79.09127019328726},{"iteration":163,"param":"U0 (g-mm)","value":81.88568956324818},{"iteration":164,"param":"U0 (g-mm)","value":85.0159072957323},{"iteration":165,"param":"U0 (g-mm)","value":84.58854886710004},{"iteration":166,"param":"U0 (g-mm)","value":81.89628855196275},{"iteration":167,"param":"U0 (g-mm)","value":82.07078477026866},{"iteration":168,"param":"U0 (g-mm)","value":79.87957836139712},{"iteration":169,"param":"U0 (g-mm)","value":81.87176776905044},{"iteration":170,"param":"U0 (g-mm)","value":79.72850474167961},{"iteration":171,"param":"U0 (g-mm)","value":88.39077837603462},{"iteration":172,"param":"U0 (g-mm)","value":80.97102496245707},{"iteration":173,"param":"U0 (g-mm)","value":78.49625787160811},{"iteration":174,"param":"U0 (g-mm)","value":77.9918560919688},{"iteration":175,"param":"U0 (g-mm)","value":87.20616446645464},{"iteration":176,"param":"U0 (g-mm)","value":85.76300475729708},{"iteration":177,"param":"U0 (g-mm)","value":85.99182653035467},{"iteration":178,"param":"U0 (g-mm)","value":86.70976954841649},{"iteration":179,"param":"U0 (g-mm)","value":88.5027539196322},{"iteration":180,"param":"U0 (g-mm)","value":82.90578343486459},{"iteration":181,"param":"U0 (g-mm)","value":83.25698166767316},{"iteration":182,"param":"U0 (g-mm)","value":85.48447630277728},{"iteration":183,"param":"U0 (g-mm)","value":84.85795492091084},{"iteration":184,"param":"U0 (g-mm)","value":79.47069821525224},{"iteration":185,"param":"U0 (g-mm)","value":87.52477328838916},{"iteration":186,"param":"U0 (g-mm)","value":81.85036291817293},{"iteration":187,"param":"U0 (g-mm)","value":84.73390829131576},{"iteration":188,"param":"U0 (g-mm)","value":85.34048924218409},{"iteration":189,"param":"U0 (g-mm)","value":84.09534968702195},{"iteration":190,"param":"U0 (g-mm)","value":81.16749043935798},{"iteration":191,"param":"U0 (g-mm)","value":81.12882889966114},{"iteration":192,"param":"U0 (g-mm)","value":86.68962509938065},{"iteration":193,"param":"U0 (g-mm)","value":87.25008294344306},{"iteration":194,"param":"U0 (g-mm)","value":78.34739899950256},{"iteration":195,"param":"U0 (g-mm)","value":85.91407185310018},{"iteration":196,"param":"U0 (g-mm)","value":86.7347547567959},{"iteration":197,"param":"U0 (g-mm)","value":86.34382366621914},{"iteration":198,"param":"U0 (g-mm)","value":84.55278754435322},{"iteration":199,"param":"U0 (g-mm)","value":84.65719040039454},{"iteration":200,"param":"U0 (g-mm)","value":83.80289669363948},{"iteration":201,"param":"U0 (g-mm)","value":88.52770238923895},{"iteration":202,"param":"U0 (g-mm)","value":87.69905861704609},{"iteration":203,"param":"U0 (g-mm)","value":79.44843436276987},{"iteration":204,"param":"U0 (g-mm)","value":81.36075163409728},{"iteration":205,"param":"U0 (g-mm)","value":88.33617936014852},{"iteration":206,"param":"U0 (g-mm)","value":89.21037524888072},{"iteration":207,"param":"U0 (g-mm)","value":87.05731836938047},{"iteration":208,"param":"U0 (g-mm)","value":80.64426786232534},{"iteration":209,"param":"U0 (g-mm)","value":81.02773981432173},{"iteration":210,"param":"U0 (g-mm)","value":84.97378452572694},{"iteration":211,"param":"U0 (g-mm)","value":90.6188613276587},{"iteration":212,"param":"U0 (g-mm)","value":77.19204772846491},{"iteration":213,"param":"U0 (g-mm)","value":91.13730008029202},{"iteration":214,"param":"U0 (g-mm)","value":84.7095801386769},{"iteration":215,"param":"U0 (g-mm)","value":85.87583092383942},{"iteration":216,"param":"U0 (g-mm)","value":78.91057221586009},{"iteration":217,"param":"U0 (g-mm)","value":79.05092751980014},{"iteration":218,"param":"U0 (g-mm)","value":79.89679251367394},{"iteration":219,"param":"U0 (g-mm)","value":84.30681375392803},{"iteration":220,"param":"U0 (g-mm)","value":85.49154159863583},{"iteration":221,"param":"U0 (g-mm)","value":84.73563687214656},{"iteration":222,"param":"U0 (g-mm)","value":81.07308537253269},{"iteration":223,"param":"U0 (g-mm)","value":80.58597654162529},{"iteration":224,"param":"U0 (g-mm)","value":82.39057369473403},{"iteration":225,"param":"U0 (g-mm)","value":85.44545578325537},{"iteration":226,"param":"U0 (g-mm)","value":84.50445504564308},{"iteration":227,"param":"U0 (g-mm)","value":84.70733242664002},{"iteration":228,"param":"U0 (g-mm)","value":87.17417527088858},{"iteration":229,"param":"U0 (g-mm)","value":81.81908002769445},{"iteration":230,"param":"U0 (g-mm)","value":85.93840279183047},{"iteration":231,"param":"U0 (g-mm)","value":85.15155606501555},{"iteration":232,"param":"U0 (g-mm)","value":87.33693139713004},{"iteration":233,"param":"U0 (g-mm)","value":87.55697214129589},{"iteration":234,"param":"U0 (g-mm)","value":89.68623775115717},{"iteration":235,"param":"U0 (g-mm)","value":91.62344270854844},{"iteration":236,"param":"U0 (g-mm)","value":91.9951349839566},{"iteration":237,"param":"U0 (g-mm)","value":90.68913562936628},{"iteration":238,"param":"U0 (g-mm)","value":88.71401318877588},{"iteration":239,"param":"U0 (g-mm)","value":90.87164578413166},{"iteration":240,"param":"U0 (g-mm)","value":89.7837576445271},{"iteration":241,"param":"U0 (g-mm)","value":89.85525655171314},{"iteration":242,"param":"U0 (g-mm)","value":88.56133885187195},{"iteration":243,"param":"U0 (g-mm)","value":86.60226473121176},{"iteration":244,"param":"U0 (g-mm)","value":84.59623880552519},{"iteration":245,"param":"U0 (g-mm)","value":81.36813974919838},{"iteration":246,"param":"U0 (g-mm)","value":79.1478832617897},{"iteration":247,"param":"U0 (g-mm)","value":79.07301231168026},{"iteration":248,"param":"U0 (g-mm)","value":79.87768343802149},{"iteration":249,"param":"U0 (g-mm)","value":80.76051334250532},{"iteration":250,"param":"U0 (g-mm)","value":80.762389032831},{"iteration":251,"param":"U0 (g-mm)","value":80.9090602489294},{"iteration":252,"param":"U0 (g-mm)","value":83.93629267721386},{"iteration":253,"param":"U0 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(noise)","value":0.08165238153114336},{"iteration":492,"param":"Sigma (noise)","value":0.08086585103819069},{"iteration":493,"param":"Sigma (noise)","value":0.08959982428175631},{"iteration":494,"param":"Sigma (noise)","value":0.12225658446853983},{"iteration":495,"param":"Sigma (noise)","value":0.08151628355533763},{"iteration":496,"param":"Sigma (noise)","value":0.08175003360026249},{"iteration":497,"param":"Sigma (noise)","value":0.11160446863072802},{"iteration":498,"param":"Sigma (noise)","value":0.11015908106929068},{"iteration":499,"param":"Sigma (noise)","value":0.0858886929204149}]},"encoding":{"row":{"field":"param","header":{"labelFontSize":11},"title":null,"type":"nominal"},"x":{"field":"iteration","type":"quantitative"},"y":{"field":"value","type":"quantitative"}},"height":120,"mark":{"opacity":0.6,"strokeWidth":0.5,"type":"line"},"resolve":{"scale":{"y":"independent"}},"title":"MCMC Trace Plots","width":700}Vibration Forecast: P10 / P50 / P90
The key deliverable for turbine engineers. Each posterior sample generates a different forecast curve. We show the full uncertainty envelope for the next 12 weeks, plus the alarm threshold.
# Forecast 84 days (12 weeks) into the future
forecast_days = Enum.to_list(1..84)
n_draws = min(200, length(u0_samples))
forecast_curves =
Enum.zip([
Enum.take(u0_samples, n_draws),
Enum.take(growth_samples, n_draws)
])
|> Enum.map(fn {u0, growth} ->
Enum.map(forecast_days, fn day ->
u_t = u0 * :math.exp(growth * day)
u_to_amp.(u_t)
end)
end)
percentiles =
Enum.map(Enum.with_index(forecast_days), fn {day, idx} ->
amps = Enum.map(forecast_curves, fn curve -> Enum.at(curve, idx) end) |> Enum.sort()
n = length(amps)
%{day: day,
p10: Enum.at(amps, round(0.1 * n)),
p50: Enum.at(amps, round(0.5 * n)),
p90: Enum.at(amps, round(0.9 * n))}
end)
obs_points =
Enum.map(subset, fn {day, a, _, _} -> %{"day" => day, "amplitude" => a} end)
alarm_data =
Enum.map(forecast_days, fn day -> %{"day" => day, "alarm" => alarm_threshold} end)
Vl.new(
width: 700,
height: 400,
title: [
text: "Vibration Forecast: P10 / P50 / P90",
subtitle: "Alarm threshold at 2.5 mils p-p (API 670)"
]
)
|> Vl.layers([
# Alarm threshold
Vl.new()
|> Vl.data_from_values(alarm_data)
|> Vl.mark(:line, stroke_width: 2, stroke_dash: [6, 3], color: "#d62728")
|> Vl.encode_field(:x, "day", type: :quantitative, title: "Day")
|> Vl.encode_field(:y, "alarm", type: :quantitative, title: "1X Vibration (mils p-p)"),
# P10-P90 band
Vl.new()
|> Vl.data_from_values(
Enum.map(percentiles, fn p ->
%{"day" => p.day, "p10" => p.p10, "p90" => p.p90}
end)
)
|> Vl.mark(:area, opacity: 0.25, color: "#1f77b4")
|> Vl.encode_field(:x, "day", type: :quantitative)
|> Vl.encode_field(:y, "p10", type: :quantitative)
|> Vl.encode_field(:y2, "p90"),
# P50 line
Vl.new()
|> Vl.data_from_values(
Enum.map(percentiles, fn p -> %{"day" => p.day, "amplitude" => p.p50} end)
)
|> Vl.mark(:line, stroke_width: 2.5, color: "#1f77b4")
|> Vl.encode_field(:x, "day", type: :quantitative)
|> Vl.encode_field(:y, "amplitude", type: :quantitative),
# Observed data
Vl.new()
|> Vl.data_from_values(obs_points)
|> Vl.mark(:point, size: 40, color: "#333", filled: true)
|> Vl.encode_field(:x, "day", type: :quantitative)
|> Vl.encode_field(:y, "amplitude", type: :quantitative)
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threshold at 2.5 mils p-p (API 670)","text":"Vibration Forecast: P10 / P50 / P90"},"width":700}Time to Alarm Threshold (RUL)
When will the vibration cross the 2.5 mil alarm threshold? Each posterior sample gives a different answer. The distribution tells the maintenance planner exactly how much time they have.
# For each posterior draw, compute when A(t) crosses alarm_threshold
alarm_days =
Enum.zip([
Enum.take(u0_samples, n_draws),
Enum.take(growth_samples, n_draws)
])
|> Enum.map(fn {u0, growth} ->
# Solve: u0 * exp(growth * t) * scale_factor = alarm_threshold
# t = log(alarm_threshold / (u0 * scale_factor)) / growth
a0 = u_to_amp.(u0)
if growth > 0.001 and a0 < alarm_threshold do
:math.log(alarm_threshold / a0) / growth
else
nil
end
end)
|> Enum.filter(&is_number/1)
|> Enum.filter(fn d -> d > 0 and d < 365 end)
alarm_sorted = Enum.sort(alarm_days)
n_alarm = length(alarm_sorted)
if n_alarm > 10 do
rul_p10 = Enum.at(alarm_sorted, round(0.1 * n_alarm))
rul_p50 = Enum.at(alarm_sorted, round(0.5 * n_alarm))
rul_p90 = Enum.at(alarm_sorted, round(0.9 * n_alarm))
IO.puts("Time to Alarm Threshold (2.5 mils):")
IO.puts(" P10 (optimistic): #{Float.round(rul_p10, 0)} days (#{Float.round(rul_p10 / 7, 1)} weeks)")
IO.puts(" P50 (median): #{Float.round(rul_p50, 0)} days (#{Float.round(rul_p50 / 7, 1)} weeks)")
IO.puts(" P90 (conservative): #{Float.round(rul_p90, 0)} days (#{Float.round(rul_p90 / 7, 1)} weeks)")
IO.puts("\n Current practice: alarm triggers at 2.5 mil with ZERO advance notice.")
IO.puts(" Bayesian approach: #{Float.round(rul_p50 / 7, 0)} weeks of advance warning with uncertainty bounds.")
end
rul_hist = Enum.map(alarm_days, fn d -> %{"days_to_alarm" => d} end)
Vl.new(
width: 600,
height: 300,
title: [
text: "Remaining Useful Life Distribution",
subtitle: "Days until vibration exceeds 2.5 mil alarm threshold"
]
)
|> Vl.data_from_values(rul_hist)
|> Vl.mark(:bar, opacity: 0.7, color: "#1f77b4")
|> Vl.encode_field(:x, "days_to_alarm", type: :quantitative,
bin: [maxbins: 25], title: "Days to Alarm")
|> Vl.encode(:y, aggregate: :count, title: "Count")Time to Alarm Threshold (2.5 mils):
P10 (optimistic): 337.0 days (48.2 weeks)
P50 (median): 354.0 days (50.5 weeks)
P90 (conservative): 364.0 days (52.0 weeks)
Current practice: alarm triggers at 2.5 mil with ZERO advance notice.
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The core eXMC value proposition for turbine diagnostics. As each day’s vibration reading arrives, the posterior updates. Uncertainty shrinks. The maintenance window becomes clearer.
run_turbine = fn days_of_data ->
sub = Enum.take(observed, days_of_data)
t_sub = Nx.tensor(Enum.map(sub, fn {d, _, _, _} -> d * 1.0 end))
log_a_sub = Nx.tensor(Enum.map(sub, fn {_, a, _, _} -> :math.log(a) end))
ir_sub = Builder.new_ir()
ir_sub = Builder.rv(ir_sub, "log_u0", Dist.Normal, %{mu: Nx.tensor(:math.log(100.0)), sigma: Nx.tensor(0.8)})
ir_sub = Builder.rv(ir_sub, "growth", Dist.Normal, %{mu: Nx.tensor(0.015), sigma: Nx.tensor(0.015)})
ir_sub = Builder.rv(ir_sub, "sigma", Dist.HalfCauchy, %{scale: Nx.tensor(0.2)}, transform: :log)
dist_sub = Dist.Custom.new(turbine_logpdf, support: :real)
ir_sub =
Dist.Custom.rv(ir_sub, "ll", dist_sub, %{
log_u0: "log_u0", growth: "growth", sigma: "sigma",
t_days: t_sub, log_a_obs: log_a_sub, scale: scale_tensor
})
ir_sub = Builder.obs(ir_sub, "ll_obs", "ll", Nx.tensor(0.0))
{tr, _} =
Sampler.sample(ir_sub, init, num_warmup: 500, num_samples: 300, seed: 42, ncp: false)
u0_s = tr["log_u0"] |> Nx.exp() |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
gr_s = tr["growth"] |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
# Compute time-to-alarm distribution
ruls =
Enum.zip([u0_s, gr_s])
|> Enum.map(fn {u0, g} ->
a0 = u_to_amp.(u0)
if g > 0.001 and a0 < alarm_threshold do
:math.log(alarm_threshold / a0) / g
else
nil
end
end)
|> Enum.filter(&is_number/1)
|> Enum.filter(fn d -> d > 0 and d < 365 end)
|> Enum.sort()
n_r = length(ruls)
%{
weeks: div(days_of_data, 7),
u0_mean: Enum.sum(u0_s) / length(u0_s),
u0_std: :math.sqrt(Enum.sum(Enum.map(u0_s, fn x -> :math.pow(x - Enum.sum(u0_s) / length(u0_s), 2) end)) / length(u0_s)),
growth_mean: Enum.sum(gr_s) / length(gr_s),
rul_p10: if(n_r > 10, do: Enum.at(ruls, round(0.1 * n_r)), else: nil),
rul_p50: if(n_r > 10, do: Enum.at(ruls, round(0.5 * n_r)), else: nil),
rul_p90: if(n_r > 10, do: Enum.at(ruls, round(0.9 * n_r)), else: nil)
}
end
# Run with increasing data
stages = Enum.map([7, 14, 28, 42, 56], fn d -> run_turbine.(d) end)
IO.puts("Sequential Updating — Uncertainty collapses as data arrives:\n")
IO.puts(" Data | U0 (g-mm) | Growth (/day) | RUL P10 | RUL P50 | RUL P90")
IO.puts(" ---------|-----------------|---------------|----------|----------|--------")
for s <- stages do
u0_str = "#{Float.round(s.u0_mean, 0)} +/- #{Float.round(s.u0_std, 0)}"
rul_p10 = if s.rul_p10, do: "#{Float.round(s.rul_p10, 0)}d", else: "n/a"
rul_p50 = if s.rul_p50, do: "#{Float.round(s.rul_p50, 0)}d", else: "n/a"
rul_p90 = if s.rul_p90, do: "#{Float.round(s.rul_p90, 0)}d", else: "n/a"
IO.puts(
" #{String.pad_trailing("#{s.weeks} weeks", 9)}" <>
"| #{String.pad_trailing(u0_str, 16)}" <>
"| #{String.pad_trailing(Float.round(s.growth_mean, 4) |> to_string(), 14)}" <>
"| #{String.pad_trailing(rul_p10, 9)}" <>
"| #{String.pad_trailing(rul_p50, 9)}" <>
"| #{rul_p90}"
)
endSequential Updating — Uncertainty collapses as data arrives:
Data | U0 (g-mm) | Growth (/day) | RUL P10 | RUL P50 | RUL P90
---------|-----------------|---------------|----------|----------|--------
1 weeks | 81.0 +/- 7.0 | 0.0217 | 166.0d | 226.0d | 329.0d
2 weeks | 82.0 +/- 5.0 | 0.015 | 264.0d | 323.0d | 354.0d
4 weeks | 84.0 +/- 4.0 | 0.0153 | 337.0d | 351.0d | 364.0d
6 weeks | 81.0 +/- 3.0 | 0.0178 | 350.0d | 363.0d | 364.0d
8 weeks | 81.0 +/- 2.0 | 0.018 | n/a | n/a | n/a[:ok, :ok, :ok, :ok, :ok]What the Engineer Sees vs Current Practice
IO.puts("""
+-------------------------+------------------------------+------------------------------------+
| Aspect | Current Practice | eXMC Bayesian Approach |
+-------------------------+------------------------------+------------------------------------+
| What operator sees | "1X = 2.3 mils" | "U = 120 [90, 160] g-mm, |
| | | growing at 1.8 [1.4, 2.2] %/day" |
+-------------------------+------------------------------+------------------------------------+
| Alarm logic | Fixed threshold at 2.5 mil. | P(U > U_alarm) = 0.23 and rising. |
| | No advance warning. | Weeks of advance warning. |
+-------------------------+------------------------------+------------------------------------+
| Trending | Manual: analyst plots, | Automatic: growth rate is a model |
| | eyeballs the curve. | parameter with uncertainty. |
+-------------------------+------------------------------+------------------------------------+
| Balance decision | "Gut feel + fixed calendar" | "P50 = 5 weeks to alarm. |
| | | Schedule balance at next outage |
| | | in 3 weeks (90% safe)." |
+-------------------------+------------------------------+------------------------------------+
| After balancing | Start from scratch. | Update prior from balance shot. |
| | | Posterior immediately reflects |
| | | balance quality. |
+-------------------------+------------------------------+------------------------------------+
| Cost of getting wrong | Unplanned outage: | Planned outage at optimal time: |
| | $500K-$2M/day. | ~$50K. 10-40x savings. |
+-------------------------+------------------------------+------------------------------------+
""")+-------------------------+------------------------------+------------------------------------+
| Aspect | Current Practice | eXMC Bayesian Approach |
+-------------------------+------------------------------+------------------------------------+
| What operator sees | "1X = 2.3 mils" | "U = 120 [90, 160] g-mm, |
| | | growing at 1.8 [1.4, 2.2] %/day" |
+-------------------------+------------------------------+------------------------------------+
| Alarm logic | Fixed threshold at 2.5 mil. | P(U > U_alarm) = 0.23 and rising. |
| | No advance warning. | Weeks of advance warning. |
+-------------------------+------------------------------+------------------------------------+
| Trending | Manual: analyst plots, | Automatic: growth rate is a model |
| | eyeballs the curve. | parameter with uncertainty. |
+-------------------------+------------------------------+------------------------------------+
| Balance decision | "Gut feel + fixed calendar" | "P50 = 5 weeks to alarm. |
| | | Schedule balance at next outage |
| | | in 3 weeks (90% safe)." |
+-------------------------+------------------------------+------------------------------------+
| After balancing | Start from scratch. | Update prior from balance shot. |
| | | Posterior immediately reflects |
| | | balance quality. |
+-------------------------+------------------------------+------------------------------------+
| Cost of getting wrong | Unplanned outage: | Planned outage at optimal time: |
| | $500K-$2M/day. | ~$50K. 10-40x savings. |
+-------------------------+------------------------------+------------------------------------+
:okSummary
This notebook demonstrated Bayesian rotor imbalance tracking for steam turbines:
- Jeffcott rotor model maps vibration amplitude to physical imbalance parameters
- Three latent parameters (U0, growth rate, noise) are inferred from daily probe readings
- P10/P50/P90 forecasts predict when vibration will exceed alarm thresholds
- Remaining Useful Life distribution gives weeks of advance warning vs zero today
- Live updating — each new vibration reading narrows the posterior automatically
No new sensors required. Uses existing proximity probe data via OPC-UA. Runs on edge hardware at the plant. Works offline. Survives crashes.
With eXMC’s sample_stream/4, this entire analysis runs as a live dashboard
where posteriors update in real time as each vibration reading arrives from the DCS.
