ME scope Application Note 29 FEA Model Updating of an Aluminum Plate NOTE: You must have a package with the VES-4500 Multi-Reference Modal Analysis and VES-8000 FEA Model Updating options enabled to reproduce these results. Click here to download the ME scope Project ZIP file for carrying out this example. Structural Dynamics Modification (SDM) has become a practical tool for improving the engineering designs of mechanical systems. It provides a very quick and inexpensive approach for investigating the effects of design modifications to a structure, thus eliminating the need for costly prototype fabrication and testing. Modal Models SDM is unique in that it works directly with a modal model of the structure, either an Experimental Modal Analysis (EMA) modal model, a Finite Element Analysis (FEA) modal model, or a hybrid modal model consisting of both EMA and FEA modal parameters. EMA mode shapes are extracted from experimental data, and are referred to as EMA modes. FEA mode shapes are extracted from a finite element computer model, and are referred to as FEA modes. A modal model is a set of properly scaled mode shapes. SDM assumes that the mode shapes are scaled to Unit Modal Masses. They are referred to an UMM mode shapes. A modal model preserves the mass and elastic properties of the structure, and therefore represents its dynamic properties. Design Modifications Once the dynamic properties of an unmodified structure are defined in the form of its modal model, SDM can be used to predict the dynamic effects of certain kinds of mechanical design modifications to the structure. These modifications can be as simple as point mass, linear spring, or linear damper additions to or removals from the structure, or more complex modifications that are modeled using FEA elements such as plates (membranes) and solid elements. SDM is computationally very efficient. SDM solves an eigenvalue problem in modal space, whereas FEA mode shapes are obtained by solving an eigenvalue problem in physical space. Another advantage of SDM is that the modal model of the unmodified structure only has to contain data for the DOFs (points and directions) where the modification elements are attached to the structure. SDM then provides a new modal model of the modified structure, as depicted in Figure 1. Figure 1. SDM Input-Output Diagram Page 1 of 7
FEA Modal Updating Because of its computational speed, SDM can be used to quickly evaluate thousands of modifications to the physical properties of an FEA model. Although the mode shapes of an FEA model may correlate well with the EMA mode shapes of a structure, the FEA and EMA modal frequencies often do not correlate well. Since modal frequencies are very sensitive to the physical properties of a structure, the difference between FEA and EMA modal frequencies can result from choosing inaccurate physical properties for the FEA model. In this example, two physical properties of the aluminum plate shown in Figure 1 will be updated in order to make its FEA frequencies match its EMA frequencies more closely. The two properties are its thickness and its material density. SDM will be used to explore the influence of 100 s of different values of these two properties on the FEA modal frequencies. The FEA model will then be updated with the property values that yield FEA modal frequencies that correlate closest with the EMA frequencies. The plate and its RIB stiffener are shown in Figure 1. The RIB will not be considered in this example. The dimensions of the plate are 20 inches (508 mm) by 25 inches (635 mm) by 3/8 inches (9.525 mm) thick. Roving Impact Modal Test A roving impact modal test was conducted on the plate without the RIB stiffener attached to it. The plate was tested while resting on bubble pack, which approximated free-free boundary conditions. A grid of 30 points were impacted, as shown in Figure 2. Thirty FRFs were calculated, each one from the spectrum of an impact force and the spectrum of its resulting (fixed) reference acceleration response. The plate was impacted in the Z-direction, normal to its surface. Therefore, the FRFs provided Z-direction motion at each of the 30 points. The EMA modal frequencies, damping and mode shapes of 14 modes were extracted from the FRFs by curve fitting them. Figure 1. Aluminum Plate Figure 2. Impact Test Points to Obtain EMA Plate Modes Page 2 of 7
To curve fit the FRFs and display the EMA mode shapes in animation, Press the EMA Plate Modes Hotkey The EMA modal parameters are estimated by curve fitting the 30 FRFs calculated from the roving impact test. Mode shapes for 14 modes were obtained by curve fitting the FRFs. Each mode shape has 30 DOFs (1Z through 30Z). A curve fit on one of the FRFs is shown in Figure 3. FEA Modes of the Plate Figure 3. Curve Fit of an Experimental FRF An FEA model of the plate was constructed using 80 FEA plate (membrane) elements. The following properties of the aluminum material in the plate were, 1) Young s modulus of elasticity: 1E7 lbf/in^2 (6.895E4 N/mm^2) 2) Density: 0.101 lbm/in^3 (2.796E-6 kg/mm^3) 3) Poisson s Ratio: 0.33. 4) Plate thickness: 0.375 in (9.525 mm) To solve for the first 20 modes of the FEA model and compare them in animation with the EMA modes, Press the FEA Plate Modes Hotkey The FEA model shown in Figure 4 has 99 points (or nodes). The eigen-solution for the first 20 FEA modes includes 6 rigid body modes and 14 flexible body modes. Each FEA mode shape has 593 DOFs (three translational and 3 rotational DOFS at each point). The points (nodes) of the FEA model that coincide with the EMA test points have been given the same point numbers as the test points. This is required in order to calculate Modal Assurance Criterion (MAC) values using common DOFs between pairs of the FEA and EMA mode shapes. NOTE: The FEA mode shapes are scaled to UMM, so they constitute an FEA modal model of the plate, which is required for FEA Model Updating. Page 3 of 7
Mode Shape Comparison Figure 4. FEA model using 80 FEA Quad Plate Elements The Modal Assurance Criterion (MAC) values between the EMA mode shapes and the first 14 flexible body FEA mode shapes are displayed in the bar chart in Figure 5. The diagonal bars are all greater than 0.90, indicating that the 14 flexible EMA mode shapes correlated one-for-one with the first flexible 14 FEA mode shapes. The worstcase pair of mode shapes has a MAC value of 0.98. These MAC values indicate a very good correlation between the EMA and FEA mode shapes using their matching shape components (DOFs 1Z through 30Z). Modal Frequency Comparison Figure 5. MAC Values of FEA & EMA Mode Shapes-Plate without RIB The modal frequencies of the matching pairs of FEA and EMA mode shapes are listed in Table 1. Each EMA modal frequency is higher than the frequency of its matching FEA mode. The highest frequency pair is different by 100 Hz. Nevertheless, the FEA and EMA mode shapes are closely correlated. These differences indicate that the stiffness of the actual aluminum plate is greater than the stiffness of the FEA model. These frequency differences could be reduced by increasing the modulus of elasticity or reducing the density of the aluminum property of the FEA plates, or by increasing the thickness of the FEA plates. Page 4 of 7
FEA Model Updating Shape Number FEA Frequency (Hz) EMA Frequency (Hz) EMA Damping (Hz) MAC 1 91.38 101.5 0.04487 0.98 2 115.5 129.1 0.264 0.99 3 190.1 208.1 0.4977 0.99 4 217.3 242.0 0.1089 0.99 5 251.1 284.0 0.1444 0.99 6 332.3 367.5 0.6455 0.98 7 412.0 468.7 0.1659 0.98 8 424.3 477.0 0.3509 0.98 9 495.7 567.1 2.979 0.99 10 563.6 643.2 0.9498 0.99 11 625.9 713.6 3.583 0.98 12 653.6 741.9 0.9449 0.98 13 688.7 802.0 0.4814 0.98 14 756.6 858.6 3.087 0.98 Table 1. FEA vs. EMA Modes-Plate without RIB The physical properties used for the FEA plate elements were, 1) Young s modulus of elasticity: 1E07 lbf/in^2 (6.895E4 N/mm^2) 2) Density: 0.101 lbm/in^3 (2.796E-6 kg/mm^3) 3) Poisson s Ratio: 0.33. 4) Plate thickness: 0.375 in (9.525 mm) The Plate is made from 6061-T651 aluminum. The correct density for that type of aluminum is 0.0975 lbm/in^3 (2.966E-6 kg/mm^3. This is less than the density used in the FEA model. In addition, the Plate elements were assigned a nominal thickness of 0.375 in (9.525 mm). Errors in either or both of these parameters could cause the FEA modal frequencies to be less than their corresponding EMA frequencies. In ME scope, SDM is used in a manner similar to Modal Sensitivity analysis. It will be used to solve and rank modal solutions where the FEA density and the FEA plate thickness are each given multiple values between minimum and maximum limits. Difference between Modal Sensitivity and FEA Model Updating In order to calculate the new modes of a modified structure, SDM only requires a modal model of the unmodified structure together with the FEA modification elements, as shown in Figure 1. To create the solution equations, the properties of the modification elements are converted into mass, stiffness, and damping modification matrices, which are then transformed into modal coordinates using the mode shapes of the unmodified structure. These matrices in modal coordinates are added to the modal matrices of the unmodified structure, and the new equations are solved for the new modes. In order to update the properties of an FEA model, SDM is used in a different way. First, the mass and stiffness properties of the unmodified FEA model are subtracted from the mass and stiffness properties of the modified FEA model. Then the differences are added to the modal matrices of the unmodified model. In order to do this, the FEA properties of the unmodified FEA model are required. The Difference: FEA Model Updating requires the element properties of the unmodified FEA model whereas Modal Sensitivity analysis does not. Page 5 of 7
To perform FEA Model Updating in ME scope, Press the FEA Model Updating Hotkey The FEA Model Updating window will open, as shown in Figure 6. Figure 6 shows the FEA Model Updating window prior to performing a Model Updating. The frequencies of the FEA modes are listed in the first column of the upper spreadsheet. The modal frequency of each corresponding EMA mode (with the highest MAC value) is listed in the next column. Select mode pair 7 in the upper spreadsheet Select the plate thickness and density properties in the lower spreadsheet The solution spaces for the FEA properties are listed in the lower spreadsheet. The solution space for each property is defined by three parameters, (minimum, maximum. and number of steps). The solution space for the plate thickness is defined with 10 evenly spaced steps between a minimum value of 0.375 and a maximum value of 0.50. The solution space for the density is defined with 10 evenly spaced steps between a minimum value of 0.90 and a maximum value of 0.11. SDM will calculate 100 solutions using all combinations of values in the solution spaces of these two properties. Press the Calculate button on the bottom of the FEA Model Updating window. After all solutions have ben calculated, the solutions are ranked. The best solution is the one with mode 7 closest in frequency to the target frequency of 101.5 Hz. Use the scroll bar on the right to scroll through the solutions Figure 5. FEA Model Updating Prior to Calculation Page 6 of 7
Figure 6. FEA Model Updating After Calculation The best solution is displayed when the scroll bar is as the top of its slider. The updated density = 0.944 more closely matches the handbook density for 6061-T651 aluminum. The updated thickness= 0.403 in. is greater than the nominal thickness originally used. Press the Save Mode Shapes button followed by the Close button in the FEA Model Updating window Press the Continue button on the progress bar in the lower left corner of the ME scope window As the comparison display sweeps through the updated FEA mode shapes (calculated by SDM) and the EMA mode shapes, notice that each pair of shapes with Maximum MAC is also much closer. Updated FEA Modes So far, SDM was used to find the best parameters for updating the FEA model. When the Save Mode Shapes button was pressed, SDM was used to calculate new mode shapes using the updated parameters. The FEA model updating process will be repeated, but this time the properties of the FEA model itself will be updated and a new eigen-solution will be calculated for the updated FEA model. Then, the new FEA mode shapes will be compared in animation with the EMA mode shapes. Press the Updated FEA Modes Hotkey Repeat the steps above, but press the Update Properties button instead of the Save Mode Shapes button on the FEA Model Updating window This time, the comparison display will sweep through the Updated FEA and EMA mode shape pairs. Notice that each pair of shapes with Maximum MAC is also much closer in frequency with one another. Page 7 of 7