Dynamics: HHT-α numerical damping matches the amplification matrix of the scheme
Implicit transient dynamicsPASS · <0.01%Free vibration of a cantilever integrated with the HHT-α method (α = −0.1, β = (1−α)²/4, γ = (1−2α)/2) at a coarse step Δt = T₁/8, where the scheme dissipates visibly. The measured logarithmic decrement per period and the period are compared with the modulus and argument of the dominant complex eigenvalue of the one-step amplification matrix of an undamped oscillator evaluated at Ω = ω₁Δt. The same matrix gives ρ∞ = (1+α)/(1−α) for high-frequency modes.Free vibration of a cantilever integrated with the HHT-α method (α = −0.1, β = (1−α)²/4, γ = (1−2α)/2) at a coarse step Δt = T₁/8, where the scheme dissipates visibly. The measured logarithmic decrement per period and the period are compared with the modulus a… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Cantilever L = 1000 mm, section 40 × 60 mm, tet10 mesh, element size 50 mm.
- Material
- Steel: E = 200 GPa, ν = 0.3, ρ = 7850 kg/m³; equal-per-node mass (as the native modal analysis).
- Boundary conditions
- Root fixed; released from the static deflection under a 1000 N tip force. HHT-α, α = −0.1, Δt = T₁/8, 40 periods, no physical damping; ω₁ from the native modal solve of the same mesh.
- Measured quantity
- Logarithmic decrement per period and period of the tip displacement against δ = −2π·ln ρ/θ and T = 2πΔt/θ, where ρ·e^{±iθ} is the dominant complex eigenpair of the amplification matrix at Ω = ω₁Δt.
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
material E=2e11 nu=0.3 density=7850
primitive box dx=1000 dy=40 dz=60
mesh tet size=50 order=2
fix surface number 1
load surface number 2 force=1000 component=y
solve transient scheme=hht alpha=-0.1 dt=0.003809708907 tend=1.21910685 u0=static amplitude="0:0" mass=equal probes="1000,20,30" outputs=321 frames=0
probe transient_period_fit dir=y id="T"
probe transient_log_decrement_fit dir=y id="delta"
probe transient_log_decrement dir=y id="deltaPeaks"
probe transient_peak dir=y id="peak"Results comparison
| Reference value | δ = 0.023648T = 32.318 мс |
| Our result | δ = 0.023648T = 32.318 мс |
| Deviation | <0.01% |
| Tolerance | 5% |
| Status | PASS · <0.01% |
| Mesh details | тет10, размер 50 мм (126 тетр.); 320 шагов (принято 320, резок 0), 320 итераций Ньютона (макс 1/шаг), 2 факторизаций, линейный быстрый путь; ядро 76 мс, скрипт 2.0 с на нативе |
Updated: 2026-10-22 · WebCAE solver