• We show here eigenvalue and modal frequency analyses of a cantilever using beam elements. The cross section of the cantilever is not a perfect circle since production tolerance is assumed, which means the difference between two geometric moments of inertia are given in the two directions of the cross section. In addition, non-structural weight is assumed, which is not taken into account in the pure beam analysis of model. The result is compared with that of MSC.Nastran.

  Fig. 1 Beam model

• The specification of the cantilever is as follows:
  
  • Cross section: circle
  • Length: 3m
  • Radius: 0.014m
  • Geometric moments of inertia: I_y = 2.9x10^-8m⁴, I_z = 3.1x10^-8m⁴
  • Torsional constant: 3.0�~10^-8m⁴
  • Mass density: 2.65�~10⁴N/m³ (=2.703��10³kg/m³)
  • non-structural weight: 2.414 N/m
  • Young's modulas: 7.1�~10¹⁰N/m²
  • Poisson ratio: 0.33

• The number of elements and nodes are 100 and 101, respectively.

• The following table shows the eigenvalues obtained from ADVENTURECluster and MSC.Nastran. The results coincide well enough.
  Table 1 Obtained eigenvalues

  Mode	ADVENTURECluster	Nastran
  1	2.04098	2.04097
  2	2.11018	2.11017
  3	12.7891	12.7891
  4	13.2227	13.2227
  5	35.8061	35.8061
  6	37.0202	37.0202
  7	70.1584	70.1583
  8	72.5373	72.5373
  9	115.965	115.965
  10	119.897	119.897
  11	173.213	173.213
  12	179.086	179.086
  13	241.900	241.900
  14	250.102	250.102
  15	322.022	322.022
  16	332.941	332.941
  17	398.623	398.623
  18	413.576	413.576
  19	427.599	427.599

• Then we perform modal frequency analysis. The excitation and evaluation points are are taken at the same point on the surface at the center of the cantilever and at the surface on the free end of the cantilever. The modal attenuation is assumed to be 2% of the critical damping under 10Hz region, and to be 5% of the critical damping over 10Hz region.

Fig. 2 Excitation points and evaluation points

• The displacement and phase responce in the x-direction at the point 1 and 2 in the figure are shown in the following figures. The results of MSC.Nastran and ADVENTURECluster coincide well enough.

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