Benchmark UP6

Longitudinal Vibrations of a Mass on a Rubber Spring


Let us consider a cantilever of length l and width and depth z0, fully fixed at x=0. The cantilever is made of the Neo-Hookean material of density ρ (considered zero). At position x=l an additional mass m is added, equally distributed to all four nodes located at x=l. The cantilever, initally at rest, is subjected to acceleration g in -x direction. The numerical solution is tested on motion of the point at position x=l.

Image depicts cantilever fixed on one end and with mass on other one (dimensions, orientaion of gravitational loading).

m = 0.1 kg g = 1 m · s-2 l = 1 m z0 = 0.01 m κ = 108 Pa μ = 3 .33¯ · 105 Pa ρ = 0 kg · m-3


The analytical solution of the problem, considering small strain limit, yields
d2λ dt2 + 3μz02ml λ + gl =0
which gives us frequency f=z0 3μml=5.033Hz.

FE Model

The model consists of twenty-five BRICK24 elements.

Theory ANSYS (HYPER58) femCalc (BRICK24)
f [Hz] f [Hz] Ratio [-] f [Hz] Ratio [-]
5.033 5.038 1.001 5.045 1.002