From e77d7b5cf5693de8e99399ed8b8a356b98e372b4 Mon Sep 17 00:00:00 2001 From: Yaroslav Date: Wed, 15 Jul 2026 17:11:30 +0000 Subject: [PATCH] Synthetic generation model with impact gain G Wang-carrier impulse injection; G as fixed hyperparameter (not fault-calibrated). Co-Authored-By: Claude Opus 4.8 --- main.tex | 126 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 126 insertions(+) create mode 100644 main.tex diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..6a2ffc6 --- /dev/null +++ b/main.tex @@ -0,0 +1,126 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage{amsmath,amssymb} +\usepackage{geometry} +\geometry{a4paper,margin=25mm} +\usepackage{booktabs} +\usepackage{siunitx} + +\title{Synthetic Bearing-Fault Signal Generation\\ for Target-Normal-Only Synthetic-to-Real Diagnosis} +\author{} +\date{} + +\begin{document} +\maketitle + +\section{Setting} +We train a fault-type classifier using only (i) \emph{synthetic} fault signals and +(ii) \emph{real healthy} recordings from the target machine; no real fault labels are +used (target-normal-only). Synthetic faults are produced by injecting +physically-parameterized impulse trains into a real healthy carrier, following the +expert-knowledge model of Wang, Taal and Fink +(\texttt{arXiv:2107.01849}, Eqs.~5--6). + +\section{Generation model} +Let $n(t)$ be a measured healthy window from the target machine, normalized to unit +standard deviation, +\begin{equation} + n \leftarrow n / \big(\sigma(n)+\varepsilon_0\big). +\end{equation} +The synthetic signal of a given fault class is +\begin{equation} + \boxed{\;\varepsilon(t) \;=\; \sum_{i} A_i\, s\!\big(t - t_i\big) \;+\; \beta\, n(t)\;} + \label{eq:gen} +\end{equation} + +\paragraph{Impulse timing.} +Impacts recur at the fault period $T$ (in samples) with random slip: +\begin{equation} + t_i = i\,T + \delta_i,\qquad + T = \frac{f_s}{f_c},\qquad + f_c = o_{\text{fault}}\cdot f_r,\qquad + \delta_i \sim \mathcal{N}\!\big(0,(0.01\,T)^2\big), +\end{equation} +where $f_s$ is the sampling rate, $f_r$ the shaft rate (Hz) and +$o_{\text{fault}}$ the characteristic fault order (BPFO, BPFI, $2\times$BSF, \dots), +obtained from the bearing geometry. + +\paragraph{Single impact.} +Each impact is a short Hann window of duty $5\%$ of $T$, passed through a wide-band +band-pass filter (a proxy for the structural resonance), and scaled by the +\textbf{impact gain} $G$: +\begin{equation} + s(\tau) = G\cdot \mathrm{BP}_{[f_1,f_2]}\!\Big( w_{\text{Hann}}(\tau)\Big), + \qquad + \operatorname{len}(w) = \max\big(10,\ \lceil 0.05\,T\rceil\big). + \label{eq:impact} +\end{equation} + +\paragraph{Amplitude modulation (load-zone passage).} +\begin{equation} + A_i = \gamma_i \cdot \sum_{k=0}^{K}\alpha_k + \cos\!\Big(\tfrac{2\pi k\, t_i}{Q}\Big), + \qquad \gamma_i \sim \mathcal{N}(1,0.1^2), + \label{eq:mod} +\end{equation} +with $K=4$ and $\alpha=(1,\,0.76,\,0.38,\,0.11,\,0.05)$. The modulation period $Q$ +depends on the fault type: +\begin{center} +\begin{tabular}{lll} +\toprule +Fault & Modulation & $Q$ \\ +\midrule +Outer race (stationary) & none, $A_i=\gamma_i$ & $Q\to\infty$ \\ +Inner race & shaft rate & $Q = f_s/f_r$ \\ +Rolling element (ball) & cage rate & $Q = f_s/(o_{\text{FTF}}\,f_r)$ \\ +\bottomrule +\end{tabular} +\end{center} + +\paragraph{Background level.} +\begin{equation} + \beta \sim \mathcal{U}(0.25,\,1). +\end{equation} + +\paragraph{Normal class.} +The synthetic normal is the healthy carrier itself, $\varepsilon(t)=\beta\,n(t)$, so the +synthetic-normal distribution matches the real-normal distribution by construction. + +\section{The impact gain $G$} +$G$ in \eqref{eq:impact} sets the impact-to-carrier amplitude ratio, i.e.\ the +\emph{signal-to-noise ratio of the synthetic fault}. Because the carrier $n$ has unit +variance, $G$ is, in principle, machine-independent. + +\paragraph{Honest use (this work).} +$G$ is treated as a \textbf{fixed hyperparameter}, swept over $G\in\{15,40,100\}$. +It is \emph{not} calibrated against any real fault recording. Calibrating $G$ to match a +measured fault peak, +\begin{equation} + G^\star = \arg\min_{G}\Big|\, \operatorname{peak}_{o_{\text{fault}}} + \big(\text{synthetic}(G)\big) + - \operatorname{peak}_{o_{\text{fault}}}\big(\text{\emph{real fault}}\big)\Big|, + \label{eq:leak} +\end{equation} +would inject target-fault information into a target-normal-only protocol and is therefore +disallowed. + +\paragraph{Domain-randomization alternative.} +Equivalently, $G$ need not be chosen at all: applying data-agnostic augmentations to the +synthetic during training --- random gain $g\sim\mathcal{U}(0.5,1.5)$ and additive noise +at a random SNR $\sim\mathcal{U}(5,30)\,\text{dB}$, among others --- randomizes the fault +SNR over a range, removing $G$ as a tunable and keeping the pipeline leak-free. + +\section{Representation} +Each window (real or synthetic) is mapped to an \emph{envelope--order spectrum}: +\begin{equation} + x \;\xrightarrow{\ \mathrm{BP}_{[500,4000]\,\text{Hz}}\ }\; + \big|\mathcal{H}\{\cdot\}\big| \;\xrightarrow{\text{standardize}}\; + \big|\mathcal{F}\{\cdot\}\big| \;\xrightarrow{\ f\mapsto f/f_r\ }\; + \text{resample to } [0,30]\ \text{orders},\ 1000\ \text{bins}, +\end{equation} +where $\mathcal{H}$ is the Hilbert-envelope operator and the order axis normalizes out the +shaft speed. + +\end{document}