Harmonic response: single-degree-of-freedom oscillator with Rayleigh damping
Harmonic analysisPASS · 0.06%A massless cantilever with a tip point mass is a single-degree-of-freedom oscillator. With Rayleigh damping C = αM + βK its modal damping ratio is ζ = (αm + βk)/(2mω₀) and the amplitude and phase of the forced response follow |H(ω)| = 1/√((k − mω²)² + (cω)²), φ = −atan2(cω, k − mω²) with c = αm + βk. The card checks the natural frequency, ζ, the amplitude and phase on nine frequencies, the position f₀√(1 − 2ζ²) and the height of the resonance peak.A massless cantilever with a tip point mass is a single-degree-of-freedom oscillator. With Rayleigh damping C = αM + βK its modal damping ratio is ζ = (αm + βk)/(2mω₀) and the amplitude and phase of the forced response follow |H(ω)| = 1/√((k − mω²)² + (cω)²),… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Massless cantilever L = 1.5 m, square section 25 × 25 mm (8 beam elements), point mass 50 kg at the free end.
- Material
- Steel: E = 200 GPa, ν = 0.3 (the beam itself carries no mass). Rayleigh coefficients α = 1 1/s, β = 0.012 s.
- Boundary conditions
- Fully fixed at the root. Harmonic tip force 10 N along y, frequency sweep 1-6 Hz. Natural frequency from the native modal solver, damping from the Rayleigh coefficients.
- Measured quantity
- Natural frequency f₀ = √(k/m)/2π with k = 3EI/L³; effective ζ; tip amplitude and phase at nine frequencies from 0.4 to 5 Hz (worst deviation); resonance peak position and height from a 6001-point sweep.
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:
# The card numbers come from cases/run_harmonic_verification.mjs (native modal + state/harmonicSolver.ts superposition with Rayleigh damping).
# This script reproduces the model and its native modal solution (f0); the harmonic sweep itself is the "solve harmonic ... rayleigh" command on solid bodies.
units mm
material E=2e11 nu=0.3 density=0
beam length=1500 section=rect w=25 h=25 n=8 scheme=cantilever P=1
mass node=1500,0,0 m=50
solve modal nmodes=3
probe frequency mode=1 id="f0"Results comparison
| Reference value | f₀ = 1.71223 Hz, ζ = 0.111026, f_peak = 1.691 Hz, u_peak = 7.83041 mm, phase(f₀) = −90° |
| Our result | f₀ = 1.71205 Hz, ζ = 0.111024, f_peak = 1.6908 Hz, u_peak = 7.83224 mm, phase(f₀) = -90.0559° |
| Deviation | 0.06% |
| Tolerance | 5% |
| Status | PASS · 0.06% |
| Element family | Beam element |
| Mesh details | 8 Euler–Bernoulli beam elements (massless) + point mass 50 kg at the tip; modal: webcae_solve (native, sidecar), superposition: state/harmonicSolver.ts |
Updated: 2026-10-21 · WebCAE solver