Vibration Simulator
Simulate forced mechanical vibrations, evaluate dynamic magnification factors, and design vibration isolation mounts with real-time resonance curve and phase lag tracking.
What is Forced Harmonic Vibration and Resonance?
Forced vibration occurs when a mechanical assembly is continuously driven by an external periodic excitation force (such as an unbalanced rotating shaft, reciprocating engine piston, or acoustic sound field). Unlike free vibration which decays to zero, forced vibration persists at the exact driving frequency of the excitation source.
When the driving frequency coincides with the mechanical system's natural resonance frequency (\(r = \omega / \omega_n \approx 1\)), the Magnification Factor surges to extreme levels, causing massive dynamic deflections and destructive structural stresses if adequate viscous damping is not provided.
How to Use the Vibration Simulator
Model mechanical resonance and test vibration isolation in four easy steps.
Set Mass & Stiffness
Adjust mass (m) and spring stiffness (k) to determine the baseline natural frequency fn = √(k/m) / (2π).
Sweep Driving Frequency
Vary driving frequency from sub-resonant (r < 1) through resonance (r = 1) up to the isolation zone (r > 1.414).
Tune Viscous Damping
Increase damping ratio ζ to suppress violent resonance peak spikes during machine start-up.
Check Transmissibility
Inspect the transmissibility ratio (TR) to ensure effective vibration isolation below 1.0.
Key Dynamic Analysis Features
Engineered for rotating machinery technicians, aerospace engineers, and university physics.
Frequency Response (FRF) Plot
Plots dynamic magnification factor M versus frequency ratio r with an active operating coordinate marker.
Vibration Isolation Engine
Evaluates transmissibility ratio TR to identify the safe operational isolation region (r > √2).
Phase Lag Computation
Calculates exact phase lag angle ϕ between applied excitation force and resultant displacement response.
Governing Forced Vibration Equations
The equation of motion for a single-degree-of-freedom system driven by a harmonic force \(F(t) = F_0 \cos(\omega t)\) is:
The steady-state vibration response amplitude \(X\) and phase lag angle \(\phi\) are:
Where \(r = \omega / \omega_n\) is the frequency ratio. The phase lag \(\phi\) is:
The force Transmissibility Ratio \(TR\) defining the fraction of dynamic force transmitted to the foundation is:
Practical Engineering Applications
Industrial Rotating Machinery Isolation
Large diesel generators, centrifugal fans, and air compressors are mounted on elastomeric rubber or steel spring mounts designed so that \(r > 2.5\). This isolates over 90% of dynamic operating vibration from shaking building floors and foundations.
Aerospace Satellite Launch Shaker Testing
Before spaceflight, satellites undergo qualification on high-tonnage electrodynamic shake tables. Engineers sweep through sinusoidal and random vibration profiles to verify that internal avionics do not encounter destructive resonance during rocket staging.
Frequently Asked Questions
Mechanical resonance occurs when a periodic external driving force excites a structure at or near its natural frequency (frequency ratio r = ω / ωn ≈ 1). At this frequency, energy transfer is maximized, causing vibration amplitudes to surge dramatically, which can lead to catastrophic structural failure if damping is low.
The Magnification Factor (M = X / Xstatic) is the ratio of dynamic steady-state vibration amplitude to the static deflection that would occur under a steady constant force F0. At low frequencies r << 1, M ≈ 1; at resonance r = 1, M reaches 1 / (2ζ).
Vibration isolation requires operating in the isolation zone where the frequency ratio r > √2 ≈ 1.414. Only in this region does the transmissibility ratio drop below 1.0 (TR < 1), meaning the force transmitted to the foundation is less than the applied excitation force.
In the isolation region (r > √2), increasing damping reduces the resonance peak during run-up, but slightly increases high-frequency transmissibility because the viscous damper transmits forces directly across the isolation mount.