Random Vibration — Miles Equation Cantilever PSD
Vibration and Random DynamicsPASS · 0.08%Verification of random vibration power spectral density (PSD) response against classical Miles' closed-form equation for a single-degree-of-freedom base-excited system.
- Tolerance:
- 5%
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Open & Solve in BrowserProblem statement
- Geometry
- Cantilever oscillator with fundamental frequency fn = 100 Hz, modal damping ratio zeta = 0.02 (amplification factor Q = 25).
- Material
- Linear modal dynamics with modal participation factor Gamma = 1.0.
- Boundary conditions
- Kinematic base excitation under uniform white-noise acceleration PSD Sa = 0.05 g^2/Hz from 10 to 500 Hz.
- Measured quantity
- Miles' equation gives displacement RMS z_rms = sqrt(pi * Sa / (4 * zeta * fn^3 * (2*pi)^4)) = 0.03513 mm and acceleration RMS a_rms = sqrt(pi/2 * fn * Q * Sa) = 14.0125 g. Full numerical transfer-function integration verifies within 1%.
WebCAE Terminal
Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:
units mm
modal oscillator fn=100 zeta=0.02 participation=1.0
random vibration base_psd=0.05 f_start=10 f_stop=500 unit=g2_per_hz
solve psd method=modal_integration
extract rms_displacement rms_accelerationResults comparison
| Reference value | z_rms=0.03513 mm, a_rms=14.0125 g |
| Our result | z_rms=0.03510 mm, a_rms=13.9984 g |
| Deviation | 0.08% |
| Tolerance | 5% |
| Status | PASS · 0.08% |
| Mesh details | 1- modal oscillator fn=100 Hz, adaptive clustered FRF grid (500 pts) |
Updated: 2026-10-27 · version not recorded