Advanced Jeffcott Rotor Simulator

Author: Akshat Verma Stack: Python · NumPy · SciPy · PyWavelets · Matplotlib

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What is the Jeffcott Rotor?

The Jeffcott rotor (derived by H.H. Jeffcott in 1919) is the canonical model of a flexible rotating shaft. It consists of:

• A massless elastic shaft of stiffness $k$ (N/m), simply supported at both ends
• A rigid disk of mass $m$ (kg) mounted at the shaft midspan
• Viscous damping $c$ (N·s/m) representing bearing dissipation
• A mass eccentricity $e$ (m) — the offset of the centre of mass from the geometric centre

Despite its simplicity, the Jeffcott rotor captures the three most important phenomena in rotating machinery dynamics:

1. Critical speed resonance — the operating speed at which shaft flexibility causes catastrophic vibration amplification
2. Self-centring above critical speed — the counter-intuitive reduction in vibration at supercritical speeds
3. Phase reversal at resonance — the 90° → 180° phase shift that is the basis of industrial balancing procedures

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Equations of Motion

The governing equations in the inertial $(x, y)$ frame:

$$m\ddot{x} + c\dot{x} + kx = me\Omega^2\cos(\Omega t)$$

$$m\ddot{y} + c\dot{y} + ky = me\Omega^2\sin(\Omega t)$$

The right-hand side is the centrifugal unbalance force rotating at shaft speed $\Omega$. At resonance ($\Omega = \omega_n = \sqrt{k/m}$), the steady-state amplitude is:

$$X_{resonance} = \frac{e}{2\zeta}$$

where $\zeta = c / (2\sqrt{km})$ is the damping ratio. See the notebook for the complete derivation from first principles.

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Fault Configurations

The simulator implements four configurations, switchable via the CONFIG dictionary:

1. Healthy Rotor (fault='healthy')

Small residual eccentricity only. Produces a pure 1× spectrum and a circular orbit. Used as the diagnostic baseline.

2. Mass Unbalance (fault='unbalance')

Elevated eccentricity $e_u$ at a specified phase angle $\phi$. Produces an elevated 1× component with no other spectral changes — the orbit remains circular but larger.

Key diagnostic: 1× amplitude is elevated; 2× and higher are at the noise floor; orbit is round.

3. Breathing Crack — Mayes-Davies Model (fault='crack')

A transverse shaft crack opens and closes once per revolution under gravity bending. Modelled as:

$$k(\theta) = k_0\left(1 - \mu\cos\theta\right), \quad \theta = \Omega t$$

where $\mu \in [0,1]$ is the crack depth ratio (crack depth / shaft radius). This parametric stiffness variation generates super-harmonic response at 2×, even under purely 1× excitation.

Key diagnostic: Growing 2× component in the spectrum; inner loop in the orbit plot. 2×/1× amplitude ratio increases monotonically with crack depth $\mu$.

4. Bearing Clearance Nonlinearity (fault='bearing')

When shaft displacement exceeds radial bearing clearance $\delta$, the restoring force stiffens abruptly (journal contacts bearing shell):

$$F_{bearing}(x) = \begin{cases} -k_0 x & |x| \leq \delta \ -k_0 x - k_c(|x| - \delta),\text{sgn}(x) & |x| > \delta \end{cases}$$

This nonlinear switching generates super-harmonics (2×, 3×, ...) and potentially sub-harmonics (½×) depending on the severity.

Key diagnostic: Multiple harmonics (2×, 3×) in FFT; elevated kurtosis (> 4.0); clipped or flattened orbit.

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Signal Processing Outputs

For each fault configuration, the notebook produces:

Output	Tool	What it shows
Time domain $x(t)$, $y(t)$	Direct integration	Overall vibration level and waveform shape
Orbit plot	Parametric $x$ vs $y$	Shaft centreline trajectory — shape is fault-specific
FFT order spectrum	np.fft.rfft + Hann window	Frequency content normalised to running speed
STFT spectrogram	scipy.signal.stft	Time-varying frequency content
CWT scalogram	pywt.cwt (Morlet)	Multi-resolution time-frequency decomposition
Statistical features	RMS, kurtosis, crest factor	Scalar fault indicators for classification
Campbell diagram	Speed sweep	Amplitude vs speed — identifies critical speed

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Requirements

numpy
scipy
matplotlib
pywt          # pip install PyWavelets
pandas

Install all at once:

pip install numpy scipy matplotlib PyWavelets pandas jupyter

Running the Notebook

git clone https://github.com/akshatverma1602/Advanced-Jeffcott-Rotor-Dynamics-Simulator.git
jupyter notebook jeffcott_rotor_simulator.ipynb

Changing Parameters

**All parameters are in the CONFIG dictionary at the top of Section 1.

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Physical Context

In industrial practice, the Jeffcott rotor is the conceptual backbone of every rotating machinery vibration analysis. When a vibration analyst looks at data from a centrifugal pump, compressor, or steam turbine, they are mentally applying the Jeffcott model to interpret what they see:

• 1× dominant spectrum + round orbit → unbalance → ISO 1940 balancing procedure
• Elevated 2× appearing → crack or misalignment → shut down and inspect
• Multiple harmonics + high kurtosis → bearing clearance or rub → expedited maintenance
• Sub-synchronous component → oil whirl/whip → bearing redesign or speed change

This simulator generates vibration data that reproduces these exact signatures, providing a controlled environment to study fault evolution and develop/validate diagnostic algorithms.

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References

1. Jeffcott, H.H. (1919). The lateral vibration of loaded shafts in the neighbourhood of a whirling speed. Philosophical Magazine, Series 6, 37(219), 304–314.
2. Mayes, I.W. & Davies, W.G.R. (1984). Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 106(1), 139–145.
3. Vance, J.M., Zeidan, F. & Murphy, B. (2010). Machinery Vibration and Rotordynamics. Wiley.
4. Rao, J.S. (1996). Rotor Dynamics, 3rd ed. New Age International.
5. ISO 1940-1:2003 — Mechanical vibration — Balance quality requirements for rotors in a constant (rigid) state.
6. API Standard 610, 12th ed. — Centrifugal Pumps for Petroleum, Petrochemical and Natural Gas Industries.

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