Vibration in mechanical systems (motors, bearings, robots) is typically treated as a stochastic process due to noise, varying loads, and sensor error.

Statistics are used to characterize, detect, and make decisions from vibration data.

RMS (Root Mean Square) represents the average energy level of the vibration signal:

\[ x_{\text{RMS}} = \sqrt{\frac{1}{N}\sum_{i=1}^N x_i^2} \]

Where:

• \(N\) = total number of samples
• \(x_i\) = vibration amplitude at sample \(i\)

A higher RMS value indicates stronger vibration.

Fourier transform moves signal to frequency domain. Power Spectral Density (PSD) is a statistical estimator of energy distribution:

\[ S_{xx}(f) = \lim_{T\to\infty}\frac{1}{T} |X_T(f)|^2 \]

Where:

• \(S_{xx}(f)\) = Power Spectral Density at frequency \(f\)
• \(T\) = observation time window
• \(X_T(f)\) = Fourier transform of the signal

PSD estimates are used for diagnosing resonances and faults.

Estimate natural frequency and damping. Logarithmic decrement:

\[ \delta = \ln\left(\frac{x_1}{x_2}\right), \quad \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} \]

Where:

• \(\delta\) = logarithmic decrement
• \(x_1, x_2\) = successive peak amplitudes
• \(\zeta\) = damping ratio

Point estimates and confidence intervals quantify uncertainty.

Typical tests:

• t-test: compare RMS before/after maintenance
• ANOVA: multiple operating conditions

Example null hypothesis: \[ H_0: \mu_{\text{before}} = \mu_{\text{after}} \]

Applications: fault detection, predictive maintenance.

Data Collection:

Accelerometers mounted on machines collect vibration signals Sampling rates: 1,000 - 50,000 Hz Stored as time-series (CSV files)

Example data:

Step-by-step process:

1. Descriptive Statistics: Calculate RMS, mean, variance, kurtosis from raw signal
2. Parameter Estimation: Estimate natural frequency \(f_n\) and damping ratio \(\zeta\)
3. Hypothesis Testing: Compare baseline (healthy) vs. current condition
   • \(H_0\): Machine is good (no significant change in RMS)
   • \(H_1\): Machine needs maintenance (RMS increased)
4. Decision Making: Use p-value with significance level \(\alpha = 0.05\)

Example: If p-value < 0.05, we reject \(H_0\) and schedule maintenance.

We model a simple harmonic vibration as:

\[ x(t) = A \sin(2\pi f t + \phi) \]

Where:

• \(A\) is the amplitude
  
• \(f\) is the frequency (Hz)
  
• \(\phi\) is the phase (radians)
  
• \(t\) is time

This represents the fundamental form of a periodic vibration signal.

From that we have the graph:

A <- 1        # amplitude
f <- 50       # frequency in Hz
phi <- 0      # phase

# Time vector (1 second duration)
time <- seq(0, 1, length.out = 1000)

# Harmonic signal x(t) = A * sin(2*pi*f*t + phi)
signal <- A * sin(2 * pi * f * time + phi)

# Build data frame
df <- data.frame(time, signal)

# Plot
ggplot(df, aes(time, signal)) +
  geom_line() +
  labs(title = "Simple Harmonic Motion",
       x = "Time (s)",
       y = "Amplitude")