29 Effects of Errors in Finite Element Models on Component Modal Tests 307 Table 29.4 Simulated test frequencies with additional mass and stiffness found by the original finite element model Target frequency (Hz) Tested component Mode number 1st 2nd 3rd 4th 5th 1 1 1.2 7.5 20.2 12.5 28.8 2 22.8 24.5 32.3 39.3 68.2 3 73.5 75.3 80.1 90.5 90.5 4 153.4 155.1 159.5 149.0 172.1 5 262.9 264.7 289.7 294.7 263.5 2 1 0.0 0.0 0.0 0.0 0.0 2 1.2 7.1 13.6 14.0 14.2 3 17.7 19.5 19.0 42.5 42.5 4 50.8 52.3 47.6 64.0 64.0 5 102.3 104.1 98.5 104.5 104.5 Table 29.5 Tested component frequencies (boundary location 9) Component 1 Component 2 Mode number Test (Hz) Analysis (Hz) Test (Hz) Analysis (Hz) 1 5.2 6.3 0.0 0.0 2 32.3 39.6 0.0 0.0 3 90.5 110.9 21.0 25.7 4 177.7 217.6 57.9 70.9 5 294.7 360.9 113.6 139.1 Table 29.6 Mass properties of each tested component (boundary location 9) Component 1 Component 2 Item Test Analysis Test Analysis Weight (g) m 39.1 31.3 48.9 39.1 Center of gravity (mm) Xcg 400.0 400.0 400.0 400.0 Moment of inertia (kg mm2) I xx 3.8 3.0 4.7 3.8 Iyy 8.4 10 3 6.7 103 8.8 104 6.9 103 Izz 8.4 10 3 6.7 103 8.8 104 6.9 103 29.3.2 System Identification of Tested Components Reducing modeling errors should be required to accurately predict frequencies and modes of structures. The measured frequencies and mass properties of the tested components are used for system identification. When boundary is location 9, natural frequencies of each tested component are shown in Table 29.5. Note that test frequencies do not include modeling errors. Therefore, frequency difference between “Test” and “Analysis” is only due to modeling errors. Table 29.6 shows an example of the mass properties of components when boundary is location 9. Since all the finite elements include identical errors in density, the center of gravity of each tested component is the same in “Test” and “Analysis”. The mass and stiffness matrices are grouped by considering mode shape. The beam has four fundamental modes: an axial mode, two bending modes in different directions, and a torsional mode. Therefore, Groups 1 to 4 are defined as follows: Group 1: degrees of freedom that influence the axial modes Group 2: degrees of freedom that influence the bending modes in the Y direction Group 3: degrees of freedom that influence the bending modes in the Z direction Group 4: degrees of freedom that influence the torsional modes Figure 29.3 shows the modification coefficients of the group for each finite element when using the lower three test frequencies and modes: one Y bending mode and two Z bending modes. For mass matrix identification, exact modification coefficients are 1.25 because the density is less than the exact value by 20 % in all the finite elements. The modification coefficients agree with the exact values for components 1 and 2. This shows that the identified mass matrix represents the true mass distribution of the beam. Although the axial and torsional modes are not used in the identification, the modification coefficients of groups 1 and 4 agree with the exact value. The reason why the mass matrix is identified is to satisfy the mass property constraints. For stiffness matrix identification, exact modification coefficients are 0.83 because the Young’s modulus is 20 % more than the exact value in all the finite elements. As shown in Fig. 29.4, modification coefficients of groups 1 and 4 remain one because the axial and torsional modes are not used in the identification. For component 1, modification coefficient of group 2
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