Dynamics: Newmark trapezoid rule conserves energy over 100 periods
Implicit transient dynamicsPASS · <0.01%Undamped free vibration of a cantilever released from its static deflection, integrated with the Newmark average-acceleration rule (β = 1/4, γ = 1/2) over 100 periods of the first mode. For a linear system this rule conserves the discrete energy, so T + W_int − T₀ stays at rounding level while the unconditionally stable but non-dissipative scheme shows no amplitude drift.Undamped free vibration of a cantilever released from its static deflection, integrated with the Newmark average-acceleration rule (β = 1/4, γ = 1/2) over 100 periods of the first mode. For a linear system this rule conserves the discrete energy, so T + W_int… 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; consistent mass.
- Boundary conditions
- Root fixed; initial displacement = static solution under a 1000 N tip force, load amplitude 0:0 (released at t = 0), no damping. Δt = T₁/50, 100 periods.
- Measured quantity
- Largest |T + W_int + W_damp − W_ext − T₀| over all outputs divided by the energy scale (target ≤ 0.1 %).
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
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 dt=0.0006132104876 tend=3.066052438 u0=static amplitude="0:0" mass=consistent probes="1000,20,30" outputs=2000 frames=0
probe transient_energy_drift id="drift"
probe transient_energy_max kind=kinetic id="Tmax"
probe transient_energy_max kind=internal id="Wmax"
probe transient_period dir=y id="T"
probe transient_peak dir=y id="peak"Results comparison
| Reference value | дрейф энергии ≤ 0.1 % |
| Our result | дрейф энергии = 1.01e-9 (1.01e-7 %) |
| Deviation | <0.01% |
| Tolerance | 5% |
| Status | PASS · <0.01% |
| Mesh details | тет10, размер 50 мм (126 тетр.); 5000 шагов (принято 5000, резок 0), 5000 итераций Ньютона (макс 1/шаг), 1 факторизаций, линейный быстрый путь; ядро 1035 мс, скрипт 8.3 с на нативе |
Updated: 2026-10-22 · WebCAE solver