Mass-Spring-Damper: Rotating Shaft Vibration Analysis

Why Vibration Analysis Matters

A pump shaft rotating at 3000 RPM develops a hairline crack. Months later, catastrophic failure. Total cost: $2.3M in downtime and damage.

This failure was predictable. The cracked shaft's natural frequency shifted into the operating range, and vibration amplitude increased measurably before failure. But nobody was watching the frequency domain.

This is why mechanical engineers simulate vibration behavior before building hardware — and why the mass-spring-damper model is one of the most important tools in structural dynamics.

The Mathematical Foundation

For a single-degree-of-freedom system (one mass, one spring, one damper), Newton's second law gives:

m·ẍ(t) + c·ẋ(t) + k·x(t) = F(t)

m = mass (kg)
c = damping coefficient (N·s/m)
k = spring stiffness (N/m)
F(t) = external force (N)
x(t) = displacement (m)

Rearranging into state-space or transfer function form (with s = Laplace variable):

X(s)              1/m
──── = ───────────────────────
F(s)    s² + (c/m)s + (k/m)

The system's key characteristics:

ωn = √(k/m)         Natural frequency (rad/s)
ζ  = c/(2√(km))     Damping ratio (dimensionless)
ωd = ωn√(1-ζ²)     Damped natural frequency

Building the Twinsys Model

Implementing this in Twinsys uses the Integrator chain approach: each block carries one physical quantity — acceleration, velocity, displacement.

          ┌─────────┐      ┌───────┐  a  ┌────────────┐  v     ┌────────────┐  x
F(t) ────►│   Sum   ├─────►│  1/m  ├────►│ Integrator ├────┬──►│ Integrator ├────┬──►
          │  + − −  │      └───────┘     └────────────┘    │   └────────────┘    │
          └──▲───▲──┘                                      │                     │
             │   │      ┌──────────┐                       │                     │
             │   └──────┤ Gain c   │◄──────────────────────┘                     │
             │          └──────────┘                                             │
             │          ┌──────────┐                                             │
             └──────────┤ Gain k   │◄────────────────────────────────────────────┘
                        └──────────┘

The feedback terms (spring force k·x and damper force c·v) are computed using Gain blocks and fed back to the Sum block as negative inputs.

Time step: this shaft rings at 22.5 Hz, so use a small fixed time step of 2·10⁻⁵ s. The Integrator chain needs about 2,000 steps per oscillation period to reproduce the resonance peak within about 5%; with a coarser step it overestimates the peak, and at 10⁻³ s the simulation diverges.

Case Study: Rotating Shaft on Bearings

A 50 kg rotating shaft is mounted on two bearings. Each bearing has an effective stiffness of 500,000 N/m. The damping from the oil film is approximately 2,000 N·s/m.

m = 50 kg
k = 2 × 500,000 = 1,000,000 N/m  (both bearings combined)
c = 2,000 N·s/m

ωn = √(1,000,000/50) = 141.4 rad/s = 22.5 Hz
ζ  = 2000 / (2√(1,000,000 × 50)) = 0.141  (lightly damped!)
ωd = 141.4 × √(1-0.141²) ≈ 140.0 rad/s

Critical observation: The natural frequency is 22.5 Hz. If the shaft has an imbalance (common with worn parts), it generates a sinusoidal force at the rotation frequency. At 3000 RPM = 50 Hz, we're well away from resonance. But at 1350 RPM = 22.5 Hz — resonance. During startup, the shaft passes through this resonance zone.

Resonance Simulation in Twinsys

Apply a sinusoidal force with F(t) = 100 · sin(ωt) N and sweep the frequency from 5 Hz to 30 Hz in simulation:

Amplitude Ratio (X/Xstatic)

  3.5 |                ***
  3.0 |               *   *
  2.5 |             **     *
  2.0 |           **        **
  1.5 |      *****            **
  1.0 |******                   *
  0.5 |
      └────────────────────────── Frequency (Hz)
       5    10   15   20   25   30
                        ↑
                  peak ≈ 22 Hz

Static deflection: X_static = F/k = 100/1,000,000 = 0.1 mm
Peak amplitude:    X_max ≈ F/(c·ωn) = 100/(2000×141.4) ≈ 0.35 mm
At resonance, deflection is about 3.5× the static deflection.

Adding the FFT Analysis

After running the Twinsys simulation, open the ChartPage and apply FFT to the displacement or acceleration signal. The FFT will show a sharp peak at 22.5 Hz — the natural frequency. As the shaft develops wear or imbalance, this peak grows.

This is exactly how predictive maintenance vibration monitoring works: track the FFT peak amplitude over time. When it starts growing, schedule maintenance before failure.

Twinsys's ChartPage supports:

• FFT: Magnitude spectrum (identify natural frequencies)
• STFT: Short-Time Fourier Transform (track frequency content over time)
• LP/HP/BP filters: Isolate specific frequency bands for bearing defect analysis

Two-DOF Extension: Coupled Shafts

Real machinery often has coupled subsystems. Two shafts connected by a flexible coupling become a 2-DOF system with two natural frequencies:

m1·ẍ1 + c1·ẋ1 + k1·x1 + kc·(x1-x2) = F1
m2·ẍ2 + c2·ẋ2 + k2·x2 + kc·(x2-x1) = F2

In Twinsys, each mass gets its own Integrator chain. The coupling force kc·(x1-x2) is computed with a Sum and Gain block and fed cross-wise into both chains. Suddenly you see two resonance peaks — the in-phase mode and the out-of-phase mode.

Summary Table: Key Parameters

Conclusion

The mass-spring-damper model is deceptively simple but reveals profound behavior: resonance, beating, mode shapes. Building it in Twinsys with an Integrator chain rather than a Transfer Function block gives you direct physical insight — each block represents a physical quantity (acceleration, velocity, displacement).

Combined with Twinsys's ChartPage FFT analysis, this approach provides the foundation for vibration-based predictive maintenance, structural health monitoring, and fatigue life estimation.

Try this: change c from 2000 to 1000 and observe how the resonance peak roughly doubles. Then set c=14142 (ζ=1.0) and observe critically damped behavior — no oscillation, fastest non-overshooting response.