FREE VIBRATION ANALYSIS OF A CANTILEVER
CRACKED BEAM WITH SUBSTRUCTURE ATTACHMENT

Free vibration analysis of a cracked cantilever beam with two types of additional substructure attachment is investigated using ANSYS program. The cantilever beam is used as a master structure with single substructure attachment in various locations (as 1-DOF mass attachment and 1-DOF mass-spring attachment) with influence of crack in different location and depths. The results for the changes of the natural frequencies of a cracked beam are compared with the results produced by Vahit et al. So the same geometrical properties have been studied. In additional work a cracked beam carrying two types of substructure attachment are compared with the results of the beam without a crack and with multi crack depth. In all calculations the beam has a uniform cross-section and the crack was modeled by reduction in the modulus of the beam. The reducing effects of the cracked beam on the natural frequencies had been more apparent with the substructure attached to the beam in different situations. The effect of mass-spring substructure is larger than the effect of the attachment when modeled as mass substructure for the same mass, with 17% for the first natural frequency and 2% for the second and third natural frequencies. The results can be used to identify cracks in simple beam structure; cracks have a clearer decreasing impact on
the natural frequencies.

THEORETICAL MODEL

The effect of the attachment models 1-Dof mass attachment model and 1-Dof mass-spring attachment model on the first three natural frequencies of a cantilever beam with a crack is given in form figures. Calculation in this study was carried out using ANSYS program via the following beam data: length (L=1m), height (h=0.01m), width
(b=0.01m), young modulus (E=211GPa), Poisson ratio (v=0.3), density (ρ=7860kg/m3) Three different crack location parameters Bc=Xc/L=0.2, 0.5 and 0.7 with a constant crack depth ratio a/h=0.4 are considered, as well as for Bc=0.5 different crack length are used as a/h=0.3,0.4 and 0.5. The mass parameter is kept constant as m/ρAL=0.3 with
corresponding mass location (Bm=Xm/L) along the beam length.
The analysis guide manuals in the ANSYS documentation set describe specific procedures for performing analyses for different engineering disciplines. A typical ANSYS analysis has three distinct steps:
- Build the model.

- Apply loads and obtain the solution.

- Review the results.

 Assumptions and Restrictions: These are:

1-      These structures have constant stiffness and mass effect.

2-      There is no damping.

3-      The structure has no time varying force, displacement, and temperatures applied (free vibration).

CONCLUSIONS
The main conclusions are summarized as:

1. Cracks have a clearer decreasing effect on the natural frequencies.

2. A crack near the free end will have smaller effect on the fundamental frequency than a crack closer to the fixed end.

3. The natural frequencies are almost unchanged when the crack is located away from the fixed end of the cantilever beam.
4. Natural frequencies of a cantilever beam depend on the location and depth of the crack.

5. The natural frequencies decrease when depth of the crack increases because of the bending moment along the beam.