ME scope Application Note 25 Choosing Response DOFs for a Modal Test

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1 ME scope Application Note 25 Choosing Response DOFs for a Modal Test The steps in this Application Note can be duplicated using any ME'scope Package that includes the VES-3600 Advanced Signal Processing option. Check Help About to verify authorization of this option. Click here to download the ME scope Demo Project file for this App Note. INTRODUCTION Finite Element Analysis (FEA) can provide excellent guidance for conducting an experimental modal analysis (an EMA). A well thought out FEA-based test plan reduces testing time and leads to better experimental results. In this note we will use FEA mode shapes to plan where data should be acquired from a real-world structure in order to adequately define its experimental mode shapes. Experimentally derived mode shapes are called EMA mode shapes, and mode shapes derived from an FEA model are called FEA mode shapes. The FEA mode shapes used in this App Note were calculated using the VES-8000 Finite Element Modeling option in ME'scope, but that option is not required to carry out the steps in this App Note. We will address two important questions, 1. What is the minimum number of test DOFs (degrees-of-freedom or points & directions) required to distinguish one EMA mode shape from another? 2. How many test DOFs are needed to make a satisfactory graphic display of the EMA mode shapes of interest? HOW MANY TEST DOFs ARE REQUIRED? Most FEA models contain far more DOFs than can be acquired experimentally. During a Roving Response modal test, a much smaller number of structural responses will typically be acquired. During a Roving Impact modal test, a much smaller number of DOFs will typically be impacted. How many test DOFs are required to compare EMA mode shapes with FEA mode shapes, and how should the test DOFs be selected? Three factors are important, 1. The test DOFs should be a subset of the FEA DOFs. This is required so that comparisons can be directly made between EMA mode shapes and FEA mode shapes 2. Sufficient test DOFs must be acquired to allow unambiguous verification of every EMA mode shape. (This can be a smaller number of points than is required to display a detailed picture of each EMA mode shape.) 3. If a properly scaled EMA modal model of the structure is desired for Structural Dynamics Modification (SDM) or MIMO Modeling & Simulation studies, then at least one driving point measurement (where the excitation DOF and response DOF are the same) must be made. Modal frequency & damping can be verified using only a single test DOF. However, more test DOFs are required not only to distinguish one mode shape from another but also to numerically compare EMA mode shapes with FEA mode shapes. Rule of Thumb: A commonly accepted rule of thumb is to acquire at least 10 DOFs per cycle of each mode shape, assuming that the mode shape exhibits a sinusoidal waveform over the span of the structure. We will find that using a set of FEA mode shapes is a much better way to determine how many test DOFs are required, and more importantly where they should be located on the test article. Page 1 of 10

2 THE FEA MODEL The FEA model of the test article is a small-scale model of a bridge with a span of 27 feet and a width of 3 feet. Both ends of the bridge are pinned to rigid supports. The FEA modal model contains six modes below 50 Hz. Each FEA mode shape has 175 DOFs (degrees-of-freedom or points & directions). All DOFs are in the vertical Z-direction. All of the modes have modal damping of less than 3%. Six FEA mode shapes containing 175 Z-Direction DOFs. This FEA bridge model was introduced in Application Note 24. In that note the optimum excitation (or Reference) DOFs were determined for a single reference test and a multiple reference test. Opening the Project Click on the link at the beginning of this App Note to download and then open AppNote25.VTprj This Project contains two data files; STR: Bridge Model and SHP: Bridge FEA Modes. SELECTING DOFs FOR MODAL TEST Clearly, the 175 DOFs of the FEA mode shapes shown above are more than adequate to distinguish between the six mode shapes. However, acquiring data at 175 DOFs is far more than necessary for a modal test. We desire to perform a modal test using the smallest sub-set of these DOFs that will still uniquely define each mode shape. The Modal Assurance Criterion (MAC) provides a means of numerically identifying the uniqueness of each mode shape. Right click in the upper spreadsheet area of SHP: Bridge FEA Modes and execute: Display MAC Select SHP: Bridge FEA Modes in the dialog box that opens, and press OK Page 2 of 10

3 The MAC bar chart will be displayed as shown below. MAC is a correlation coefficient between a pair of mode shapes. It is a measure of the "co-linearity" of two shapes. All MAC values are between "1" and "0". If their MAC value is "1", the shapes lie on the same straight line. If their MAC value is less than "1", they do not lie on the same straight line. If MAC =1, the two mode shapes are identical to one another If MAC > 0.9 the two mode shapes are similar to one another If MAC < 0.9 the two mode shapes are different from one another Execute Display Spreadsheet in the Display MAC window to display the MAC values in a spreadsheet as shown below Each of the "diagonals" of this spreadsheet has a value of "1" and each of the "off-diagonals" has a very small value. Another way of interpreting this is that the six mode shapes are linearly independent of one another. Since these mode shapes are linearly independent of one another, they are also said to be orthogonal (at right angles) to one another. Now we will investigate using a much smaller number of DOFs than the entire 175 DOFs. Right click again in the upper spreadsheet area of SHP: Bridge FEA Modes and execute: Display MAC to display the MAC bar chart again Page 3 of 10

4 Click on M#40 (40Z) in the Select M# column in the lower spreadsheet of SHP: Bridge FEA Modes to display the bar chart as shown below MAC values when only DOF 40Z is selected. With any single DOF selected, the MAC values are all "1's", indicating that all mode shapes are co-linear and cannot be distinguished from one another using only 1 DOF. In App Note 24, DOF 40Z was found to be an optimum Reference DOF for a single reference test. Also, DOFs 65Z & 88Z were found to be optimum Reference DOFs for a multiple reference test. Select M#65 (65Z) and M#88 (88Z) in the lower spreadsheet of SHP: Bridge FEA Modes to display the MAC bar chart as shown below MAC values when DOFs 40Z, 65Z & 88Z are selected. With these three DOFs selected, the off-diagonal elements of the bar chart still have values greater 0.9, indicating that this set of three DOFs cannot discriminate between mode shapes 2, 3 & 6. Additional shape DOFs must be selected to improve the linear independence of the mode shapes. Linearly Independent Shapes: If all off-diagonal MAC bars have values of 0.8 or less, the mode shapes will be considered to be independent of one another and therefore are uniquely defined. The objective of modal testing is to select as few test DOFs as possible to achieve a set of linearly independent EMA mode shapes. Select additional DOFs 11Z, 36Z, 61Z, 115 Z & 140Z in the lower spreadsheet of SHP: Bridge FEA Modes With these DOFs selected, the criterion above is easily met. As shown below, the worst-case off-diagonal bar has a value Page 4 of 10

5 MAC bars with eight DOFs selected. These MAC bars indicate that these eight test DOFs are sufficient to uniquely distinguish the six mode shapes from one another. Two different modal tests could be performed using these eight test DOFs, 1. A single reference modal test acquiring responses at the eight DOFs with DOF 40Z as the reference 2. A multiple reference modal test acquiring responses at the eight DOFs with DOFs 65Z & 88Z as the references SPATIAL RESOLUTION To display the test points on the bridge model Right click in the graphics are of STR: Bridge Model and execute Edit Current Objects Points Right click in the graphics are of STR: Bridge Model and execute Display Display Objects Points Point Labels Select the eight points in the Points spreadsheet as shown below These eight test points would provide EMA mode shapes with sufficient linear independence of one another to be correlated with the FEA mode shapes using MAC. However, this small number of shape DOFs is not sufficient for displaying the EMA mode shapes in animation on the bridge model. To display the FEA shapes in animation as shown below, Right click in the graphics are of STR: Bridge Model and execute Display Display Objects Points Point Labels again to turn off the Point labels display Right click in the graphics are of STR: Bridge Model and execute Draw Animate Shapes Right click in the graphics are of STR: Bridge Model and execute Animate Deformation Arrows Page 5 of 10

6 Spatial Resolution Using the Rule-of-Thumb Eight Test DOFs in animation using Arrows Inspection of the FEA mode shapes discloses that they all have sine-like deformation along the X-axis of the bridge. The highest spatial sine wave of these six mode shapes is two sine cycles along the length of the bridge. The fifth (38.78 Hz) mode shape exhibits two cycles of this sine -like deformation. Using the previously stated Rule-of-Thumb, there should be about 20 equally-spaced test points in the X-direction of the bridge. The deformation in the Y-direction of all six FEA mode shapes appears to be a straight line. Three equally-spaced points in the Y-direction will be sufficient to identify shapes in this direction. 57 Point Modal Test Right click in the graphics are of STR: Bridge Model and execute Animate Draw Structure Hold down the Ctrl key and click near every other point in the grid as shown below Right click in the graphics are of STR: Bridge Model and execute Display Display Objects Points Point Labels Bridge Test Model Showing 58 Test Points As shown above, selecting every other point will result in a grid of 19 points along the X-axis and 3 points along the Y-axis giving a total of 57 test points. The six points at each end of the bridge are fixed boundary points which do not move and therefore do not require measurements. Page 6 of 10

7 In addition, the test points must contain DOFs 40Z, 65Z & 88Z to be used in either a single reference test (using 40Z as the reference DOF) or a multiple reference test (using DOFs 65Z & 88Z as references). Points 40 & 88 are already part of the grid of 57 test points shown above. Select point 65 also to give the bridge modal test model a total of 58 points Because the six end points of the bridge are fixed, they don't need to be tested. Hence, the total number of test points is reduced to 52 points with measurement only in the vertical (Z-axis) direction. This is a realistic size for a modal test. To verify that this test model provides sufficient spatial resolution to uniquely identify all six EMA mode shapes, Make sure that the 58 points are still selected on the bridge model as shown above Right click in the graphics are of STR: Bridge Model and execute M# Links Select M#s The 58 M#s corresponding to the modal test points are now selected in SHP: Bridge FEA Modes as shown below. MAC will now be used to verify that EMA mode shapes with these selected DOFs will provide sufficient spatial resolution to uniquely define each mode shape. Right click in the upper spreadsheet area of SHP: Bridge FEA Modes and execute Display MAC Select SHP: Bridge FEA Modes in the dialog box that opens, and press OK As shown below, the MAC bar chart shows linear independence of the mode shapes with the 58 selected DOFs. MAC Values of Mode Shapes from a 58 Point Modal Test. Page 7 of 10

8 27 Point Modal Test Even though the spatial sampling provided by the 58 test point model will provide six unique mode shapes, what if a smaller test model was used? Would it still provide six unique mode shapes? The figure below shows a 27 point test model. Again, since the bridge endpoints are fixed, only 21 points need to be measured. This model provides 9 points along the X-axis, less than half the number required by the Rule of Thumb. To verify that this test model provides sufficient spatial resolution to uniquely identify all six EMA mode shapes, Make sure that the 27 points are still selected on the bridge model as shown above Right click in the graphics area of STR: Bridge Model and execute M# Links Select M#s The 27 M#s corresponding to the modal test points are now selected in SHP: Bridge FEA Modes as shown below. Page 8 of 10

9 MAC will now be used to verify that EMA mode shapes with these selected points will provide sufficient spatial resolution to uniquely define each mode shape. Right click in the upper spreadsheet area of SHP: Bridge FEA Modes and execute: Display MAC Select SHP: Bridge FEA Modes in the dialog box that opens, and press OK As shown below, the MAC bar chart still shows linear independence of the mode shapes, even when only 27 test points are used. SUMMARY MAC Values of Mode Shapes from a 27 Point Modal Test. The MAC bar charts showed that all six mode shapes were orthogonal to one another for both a 57 point and 27 point modal tests. Apparently we made a "lucky choice" of test points when only 27 points were selected on the bridge model. The six EMA mode shapes with those DOFs would still orthogonal to one another. For both the 57 point and 27 point modal tests, the test points were carefully chosen so that the off-diagonal MAC values were "0". For most structures the off-diagonal MAC values would not be "0". Using FEA mode shapes, if modal test DOFs are chosen so that the off-diagonal MAC values of the FEA mode shapes are less that 0.8, that should be sufficient to uniquely identify each EMA mode shape. As illustrated by the figure below, the animated mode shapes of the 58 point Bridge Model are easily identified as the same mode shapes of the original 175 point Bridge Model. On the other hand, the mode shapes of the 27 point Bridge Model are not as easily identified as the mode shapes of the original 175 point Model. In particular, mode shapes 4 & 5 exhibit insufficient spatial resolution. In this App Note, two important tools were used to determine how many test modal points to use and where to locate them, 1. Assuming that the mode shapes are sinusoidal in nature, a simple 10 point-per-cycle Rule-of-Thumb was used to determine the minimum number of test points required 2. The Modal Assurance Criterion (MAC) was used to choose FEA mode shape DOFs so that the EMA modes with those same DOFs would be linearly independent of one another, and therefore uniquely identified Page 9 of 10

10 Mode shapes using 58 Point (left) and 27 Point (right) test models. Page 10 of 10