Dynamics: free damped vibration of a cantilever with Rayleigh damping
Implicit transient dynamicsPASS · 0.07%A prismatic steel cantilever is pulled by a tip force and released (initial displacement = static solution, load amplitude 0:0). Rayleigh damping C = a·M + b·K₀ is tuned for ζ = 2 % on the first mode. The logarithmic decrement of the tip displacement and the damped period are compared with δ = 2πζ/√(1−ζ²) and with the first natural period of the native modal analysis of the same mesh.A prismatic steel cantilever is pulled by a tip force and released (initial displacement = static solution, load amplitude 0:0). Rayleigh damping C = a·M + b·K₀ is tuned for ζ = 2 % on the first mode. The logarithmic decrement of the tip displacement and the d… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Cantilever L = 1000 mm, section 40 × 60 mm, bending about the weak axis (y); quadratic tetrahedra (tet10), element size 25 mm.
- Material
- Steel: E = 200 GPa, ν = 0.3, ρ = 7850 kg/m³; consistent-with-modal (equal-per-node) mass.
- Boundary conditions
- Root face fixed. Tip force 1000 N along y applied statically, released at t = 0. Newmark β = 1/4, γ = 1/2, Δt = T₁/100, 12 periods, Rayleigh a = ζω₁, b = ζ/ω₁ with ζ = 0.02.
- Measured quantity
- Logarithmic decrement from the positive tip-displacement peaks and the mean damped period against 2πζ/√(1−ζ²) and T₁/√(1−ζ²), T₁ = 1/f₁ from the native modal solve of the same mesh.
WebCAE Terminal
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units mm
material E=2e11 nu=0.3 density=7850
primitive box dx=1000 dy=40 dz=60
mesh tet size=25 order=2
fix surface number 1
load surface number 2 force=1000 component=y
solve transient dt=0.0003056426759 tend=0.3667712111 u0=static amplitude="0:0" rayleigh a=4.111458119 b=0.00009728908538 mass=equal probes="1000,20,30" outputs=1200 frames=0
probe transient_period dir=y id="T"
probe transient_log_decrement dir=y id="delta"
probe transient_peak dir=y id="peak"
probe transient_energy_drift id="drift"Results comparison
| Reference value | δ = 0.12569T_d = 30.57 мс |
| Our result | δ = 0.12561T_d = 30.586 мс |
| Deviation | 0.07% |
| Tolerance | 5% |
| Status | PASS · 0.07% |
| Mesh details | тет10, размер 25 мм (974 тетр.); 1201 шагов (принято 1201, резок 0), 1201 итераций Ньютона (макс 1/шаг), 2 факторизаций, линейный быстрый путь; ядро 2993 мс, скрипт 19.6 с на нативе |
Updated: 2026-10-22 · WebCAE solver