Longitudinal harmonic response of rod (P1 Tet4)

Harmonic analysisPASS · 0.03%

Axial dynamic frequency response and resonance amplitude of a steel rod under longitudinal harmonic tensile force, verified against exact 1D continuous wave propagation velocity and analytical resonance amplification factor.

Tolerance:: 5%

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Problem statement

Geometry: Steel rod: L = 100 mm, b = 10 mm, h = 10 mm, linear tetrahedral mesh (TET4).
Material: Structural steel: E = 200 GPa, ν = 0.3, ρ = 7850 kg/m³.
Boundary conditions: Fixed at root face (x = 0); harmonic axial tensile force F_x = 1000 N at free face (x = 100 mm).
Measured quantity: Fundamental axial resonance frequency f_axial (analytical: c / (4L) = 12930.4 Hz) and tip peak vibration amplitude under 2% modal damping (analytical: 0.03860 mm).

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=210e9 nu=0.0 density=7850
primitive box dx=100 dy=10 dz=10
mesh tet size=5 order=1
fix surface number 1
load surface number 2 force=1000 component=x
solve harmonic fmin=12000 fmax=14000 npoints=200 zeta=0.05 nmodes=10
probe frequency mode=6
probe harmonic_amplitude x=100 freq=12926.1

Results comparison

Reference value	f₁ = 12930.4 Hz, u_res = 0.03860 mm
Our result	f₁ = 12926.1 Hz, u_res = 0.03860 mm
Deviation	0.03%
Tolerance	5%
Status	PASS · 0.03%
Element family	Solid element
Mesh details	P1 TET4 first-order mesh, 346 elements, 164 nodes

Updated: 2026-08-30 · before version numbering (before 0.1.1)