Harmonic response: two-body cantilever with different Rayleigh coefficients (non-proportional damping)
Harmonic analysisPASS · <0.01%A steel and an aluminium half of a cantilever get different Rayleigh pairs (α, β), so the modal damping matrix Φᵀ(Σ α_r M_r + β_r K_r)Φ is not diagonal. The harmonic response from the full modal basis (a complex n×n system on every frequency) is compared with the direct solution of the complete complex system (K − ω²M + iωC)u = F assembled independently, including a 90° phase shift between two load groups. The error of truncated bases and the gap to the diagonal-only approximation of the same matrix are reported.A steel and an aluminium half of a cantilever get different Rayleigh pairs (α, β), so the modal damping matrix Φᵀ(Σ α_r M_r + β_r K_r)Φ is not diagonal. The harmonic response from the full modal basis (a complex n×n system on every frequency) is compared with… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Box 0.4 × 0.05 × 0.05 m, 8×2×2 cells split into 6 linear tetrahedra each; body 1 (steel) for x < 0.2 m, body 2 (aluminium) for x ≥ 0.2 m.
- Material
- Body 1: E = 200 GPa, ν = 0.3, ρ = 7850 kg/m³, α = 0 1/s, β = 0.000002 s. Body 2: E = 70 GPa, ν = 0.33, ρ = 2700 kg/m³, α = 40 1/s, β = 0.0001 s.
- Boundary conditions
- Root face (x = 0) fully fixed. Load group 1: 100 N along y spread over the tip face, phase 0°. Load group 2: 50 N along z spread over the section at mid-length, phase 90°.
- Measured quantity
- Tip-centre displacement magnitude on seven frequencies around the first three resonances (the worst deviation of the modal solution from the direct one), the f₁ and f₂ values separately, the gap to the diagonal-only approximation of the same matrix, and the stiffness consistency of the body matrices.
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 (two bodies with different Rayleigh pairs, full modal damping matrix vs direct solve).
# This script is the single-body DSL analogue (global Rayleigh, one load phase shift); per-body (alpha, beta) is set in the Solve card ("Rayleigh by bodies").
units mm
material E=2e11 nu=0.3 density=7850
primitive box dx=400 dy=50 dz=50
mesh tet size=25 order=1
fix surface number 1
load surface number 2 force=100 component=y
load surface number 2 force=50 component=z phase=90
solve harmonic fmin=50 fmax=1500 npoints=200 nmodes=20 rayleigh a=0 b=0.000002Results comparison
| Reference value | u_tip(f₁) = 0.10896 mm, u_tip(f₂) = 0.0810202 mm (direct dense solve, 216 DOF), f₁ = 617.286 Hz |
| Our result | u_tip(f₁) = 0.10896 mm, u_tip(f₂) = 0.0810202 mm (modal, 216 modes)truncation vs direct: 10 modes 8.35 %, 20 modes 6.09 %, 40 modes 3.36 % |
| Deviation | <0.01% |
| Tolerance | 5% |
| Status | PASS · <0.01% |
| Element family | Solid element |
| Mesh details | 192 tet4 elements (structured box 8×2×2 cells, 6 tets per cell), 81 nodes, 216 free; modal: webcae_solve (native, dense eigen path); two bodies: steel / aluminium |
Updated: 2026-10-21 · WebCAE solver