Ratio-Spectral Regime

Bearing Degradation Analysis

Independent validation of the 45° monotonic attractor using NASA IMS bearing dataset

Run-to-failure vibration data • Multiple test sets • Multiple bearings

42.91°

Mean Spectral Phase Across All Bearings

Deviation from 45°: only 2.09°

Spectral Regimes: Quadrature vs. Diagonal

〰️

Oscillatory Systems → 90°

Systems with bidirectional energy flow exhibit phase alignment dynamics.

• Wind turbines, Solar PV
• Nuclear shell structures
• Grid stability, Synchronization

Result: θ → 90° = maximum coherence

📉

Monotonic Systems → 45°

Systems that evolve in one direction only. Dynamics are ratio-based.

• Battery degradation ✓ Validated
• Bearing degradation ✓ Validated
• Equipment wear patterns

Result: θ = 45° = balanced ratio (damage ≈ life consumed)

NASA IMS Bearing Dataset

Bearings from NASA's Intelligent Maintenance Systems Center, run-to-failure tests with continuous vibration monitoring. Independent dataset from battery validation.

Bearings Analyzed

7,308

Total Samples

42.91°

Mean Phase

6/6

Confirmed

Individual Bearing Results

All bearings confirm 45° attractor (within 15° tolerance)

Test	Bearing	Status	Phase θ	Deviation	Result
Test 2	Bearing 1	Failed (outer race)	37.57°	7.43°	✓ Confirmed
Test 2	Bearing 2	Healthy	43.95°	1.05°	✓ Confirmed
Test 3/4	Bearing 1	Normal	44.72°	0.28°	✓ Confirmed
Test 3/4	Bearing 2	Normal	45.37°	0.37°	✓ Confirmed
Test 3/4	Bearing 3	Failed	42.63°	2.37°	✓ Confirmed
Test 3/4	Bearing 4	Normal	43.21°	1.79°	✓ Confirmed

Mean Phase

42.91°

Across all bearings

Std Deviation

2.55°

Remarkably tight clustering

Mean Deviation

2.22°

From theoretical 45°

Test Results Summary

✓ All Tests Confirmed

✓Test 2 Bearing 1 (failed): 37.57°
✓Test 2 Bearing 2 (healthy): 43.95°
✓Test 3/4 Bearings 1-4: All within 3°
✓Overall mean: 42.91° (2.09° from 45°)

Key Findings

45° Attractor: Confirmed across all bearings
Failed Bearings: Still follow the attractor
Healthy Bearings: Closest to exactly 45°
Independent Validation: Different dataset, same result
6/6 Success Rate: 100% confirmation

Methodology

Bearing health is tracked via RMS vibration amplitude. The spectral approach analyzes the geometric relationship between accumulated damage and operational lifetime.

RMS Vibration

Damage indicator

→

↓

Cumulative Damage

Integrated over time

→

↓

Life Consumed

Fraction of lifetime

→

↓

Spectral Phase θ

Phase geometry

Why 45° for Monotonic Systems?

For monotonic degradation, the spectral operator analyzes the geometric relationship between accumulated damage and life consumed. When damage accumulates proportionally with usage, these quantities maintain a balanced relationship.

The geometric balance point of equal contribution from both components is the diagonal — exactly 45°. Both battery and bearing degradation converge to this same attractor.

Linear degradation: balanced relationship → θ = 45°
Accelerating: damage-dominant → θ < 45°
Decelerating: time-dominant → θ > 45°