40 Nonlinear Reduced Order Modeling of a Curved Axi-Symmetric Perforated Plate: Comparison with Experiments 439 reduced density was not updated because it can be computed from the geometric properties of the perforations (ex. size of hole and count) and hence should be quite accurate. On the other hand, the effective modulus is dependent on any residual stresses from the addition of perforations, or imperfections of the perforation location geometry. The fundamental nonlinear normal mode of the plate was computed from this model using the procedure discussed in [3, 4]. Specifically, a reduced order model was created using the Implicit Condensation method [12–14] and the nonlinear reduced order model (NLROM) was integrated in the NNMCont Matlab routine provided by Peeters et al. [6] to compute the nonlinear modes. In [3, 4] this approach was found to provide an excellent approximation for the backbone of each NNM of a geometrically nonlinear beam. In the NLROMs presented here, only modes 1 and 6 are used as a modal basis since they are the first two symmetric modes with a nodal radius and hence both important to the response near the first natural frequency. 40.2.2 Experimental Structure The article under investigation is a circular perforated plate with rolled ends which is shown in Fig. 40.2. A mechanical punch was used to create the circular perforations in a flat 16 gauge (1.52 mm thick) 409 stainless steel plate in an array of equilateral triangles with 10.16 mm long edges. Once this process was completed, the plate was formed around a 317.5 mm diameter mold with the excess trimmed so a lip of 24.5 mm remained. The plate was then welded to an 89 mm high cylinder made from a 14 gauge (1.9 mm thick) 409 stainless steel plate that was cold rolled to the 317.5 mm diameter as shown in Fig. 40.2b. The welded plate assembly was then bolted to a fixture with twelve 6.4 mm evenly spaced holes. An 8000 N shaker was used to provide base excitation at a single harmonic to trace the nonlinear normal modes of the fundamental frequency of vibration. It is important to note that all stated dimensions are nominal and subject to variation as is observed in the final geometry of the two plates investigated here. The two plates considered for investigation are labeled perforated plate 1 (PP01) and perforated plate 2 (PP02). The initial geometry of both perforated plates were measured with 3D digital image correlation providing initial coordinates of the entire surface of the plates. The two plates presented here had an imperfect initial curvature induced by the method of formation as seen in Fig. 40.3. PP01 has an induced curvature with a peak deformation of 3.73 mm when compared with the nominal model. This is shown in Fig. 40.3a, b, where the colormap segments the coordinate location of the measured surface. PP02 has an induced curvature with a peak deformation of 4.84 mm when compared with the nominal model as seen in Fig. 40.3c, d. Fig. 40.2 Experimental setup of the perforated plates. (a) Perforated plate before welding into test configuration, (b) perforated plate welded into the supporting cylinder
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