ME scope Application Note 24 Choosing Reference 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 excitation forces should be applied (or fixed response sensors be attached) to 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. How many fixed References (shakers, fixed impact points, or fixed response sensors) will be required to accurately identify the EMA modes of the structure? 2. Where should the fixed References be located? FEA Mode shapes of pinned-pinned FEA model. Page 1 of 11
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%. OPENING THE PROJECT Click on the link at the beginning of this App Note to download and then open AppNote24.VTprj This Project contains two data files; STR: Bridge Model and SHP: Bridge FEA Modes. MODE SHAPE ANIMATION To initiate mode shape animation, Execute: Draw Animate Shapes in the STR: Bridge Model window To select a mode for animation: Mode Shape of the Model Bridge. Click on the Select Shape button in the SHP: Bridge FEA Modes window Displaying Node Lines A Node Line is a line of points along which a mode shape has values of zero ( 0 ). To display the Node Lines of each mode shape, when the shape deformation reaches a maximum, Execute Animate Step Pause/Continue Execute Animate Contours Node Lines to display node lines on the animated shape display Uncheck Display } Display Objects Lines Show Lines to remove the lines from the bridge model Press the Select Shape button of each shape in the SHP: Bridge FEA Modes window to display its Node Lines Page 2 of 11
Node Lines of a Mode Shape Note that the four bending modes at 4.35, 11.97, 23.46 and 38.78 Hz all have parallel node lines that run across the width of the bridge (in the Y-direction). The 4.35 Hz mode has 2 node lines at the end-points of the bridge. The 11.97 Hz mode has 3 node lines, including one across the center span. The 23.46 Hz mode has 4 node lines including two near 1/3 and 2/3 of the span. The 38.78 Hz mode has 5 node lines, including one along the center of the bridge. The two torsion modes at 22.84 and 46.22 Hz have a node line along the center of the bridge (in the X-direction) and Y-direction node lines at the ends of the bridge. The 46.22 Hz mode has a third Y-direction node line running across the center of the bridge. Displaying Color Contours Color contours can also be displayed on a model to indicate the amount of deformation of each mode shape. The colors used for color contours are defined on the color palette on the Contour Colors tab in the SHP: Bridge FEA Modes window. To display color contours, Execute: File Shape Table Options in the SHP: Bridge FEA Modes window Select the Contour Colors tab Press the 8 Colors button Uncheck Scale Between Limits Check Interpolate Colors, and Press the OK button Uncheck Animate Contours Node Lines in the STR: Bridge Model window Check Animate Contours Contour Colors Contour Colors Setup in the Shape Table Options Box Page 3 of 11
The red areas are the anti-nodes of each mode shape. Anti-nodes are the locations of maximum response of each mode. These are highly desirable locations from which to either excite the structure or attach a reference response sensor. Color Key To display the color key thermometer that associates the magnitude of deformation with each contour color: Execute: Animate Contours Color Key The color key will show the mode shape values ranging from 0 (black) to the largest magnitude (red), as shown below Color Key Displayed with Color Contours. The color Contours and the Color Key can be also be displayed between limits. Execute File Shape Table Options, check the Scale Between Limits, and enter +1 for the High Limit and -1 for the Low Limit, as shown below SELECTING REFERENCE DOFs Color Contours Scaled Between Limits Most Experimental Modal Analysis (EMA), more popularly called modal testing, is done using a single reference testing method. In a single reference test, a single shaker is used to provide excitation at a single DOF, or a single DOF is impacted, or a single uni-axial accelerometer is attached to the structure to sense its motion at single DOF. Page 4 of 11
Single reference modal testing provides the best results when all of the modes of interest are highly responsive (at or near their anti-nodes) at the reference DOF We seek a Reference DOF on the model bridge that has a driving-point FRF with a large peak for each resonance. Finding such a point using trial FRF measurements can be very time consuming. Using FEA mode shapes to find a suitable Reference DOF is a much more efficient method. Some important facts about modes and FRFs are useful for finding a good Reference DOF The magnitude of each resonance peak in an FRF is proportional to the product of two mode shape components, one corresponding to the response DOF and the other corresponding to the reference DOF The magnitude of each resonance peak In a driving point FRF is proportional to the square of the mode shape component for the driving point DOF If a Reference DOF is chosen on a Node Line of a mode shape, a resonance peak for that mode will not appear in any FRF calculated from data acquired at the Reference DOF In a set of FRFs, the first evidence that a mode of vibration has been excited is the appearance of a resonance peak in at least some of the FRFs. Rather than guess at a good Reference DOF, using a set of FEA mode shapes is a far better way to locate a Reference DOF. SHAPE PRODUCT There is a command in the Shape Table that can assist you in locating a Reference DOF. The Shape Product command multiples two or more shapes together. By multiplying shapes together, all of the mode shape components that are zero 0 valued will cause the resulting Shape Product to be zero 0 valued for the same components, or DOFs. When the Shape Product is displayed in animation of a 3D model, all of its Node Lines (where the Shape Product is zero valued) will be clearly visible. RULE: A Reference DOF should be chosen at or near an anti-node of the Shape Product. In the Color Contour display of the Shape Product, all DOFs with non-zero deformation will be colored with different colors than those with little or no deformation. The colors corresponding to the largest Shape Product values will point out the best choices for a Reference DOF. To calculate and display the Shape Product, Execute: Tools Shape Product in the SHP: Bridge FEA Modes window The Shape Product dialog will open, indicating that 6 Shapes and 175 M#s (DOFs) will be used in the calculation. Press the Save Shape Product button In the dialog box that opens, press New File and enter Shape Product in the next dialog Execute Animate Animate Shapes in the SHP: Shape Product window Double click on the 3D View to display the Quad View Double click on the Top (Z-Axis) View to display it alone, as shown below Page 5 of 11
Color Contours of the Shape Product The regions where the Shape Product has the largest values are the best locations for the Reference DOF. Execute File Shape Options in the SHP: Shape Product window On the Contour Colors tab, choose a bright color for the small value regions, as shown above The color contour display of the Shape Product clearly shows that there six (6) equally attractive Reference DOFs on each side of the Model Bridge. Any one of these DOFs would be a good reference for exciting the bridge for a roving response test, or for locating a reference sensor for a roving impact test. To identify the DOF of a candidate reference point on the bridge model Execute Animate Step Pause/Continue to pause the animation Execute Animate Shapes Shape Values Click near any Point to select it. The M#s, DOFs, & shape values will be displayed next to the point Click near a selected Point again to un-select it and remove its displayed values Four DOFs (36Z, 40Z, 136Z & 140Z), shown in the figure below, are all good choices for the Reference DOF. Four Candidate Reference Points. Page 6 of 11
SYNTHESIZING DRIVING POINT FRFS We can verify the correctness of these Reference DOFs by synthesizing their driving-point FRFs and examining them for resonance peaks. Uncheck Animate Animate Shapes in the SHP: Shape Product window to terminate shape animation Execute: Tools Synthesize FRFs in the SHP: Bridge FEA Modes window. The Synthesize FRFs dialog shown below will open Enter Block Size = 2000, Ending Frequency = 75 Hz, and check Driving Points Only Hold down the Ctrl Key, select Roving DOFs 36Z, 40Z, 136Z, 140Z, and press the OK button In the next dialog box, press New File, enter Driving Point FRFs in the next dialog box, and press OK The BLK: Driving Point FRFs window will open with the four FRFs in it, as shown below Four Synthesized Driving Point FRFs Page 7 of 11
All six resonance peaks are clearly visible, but the two closely coupled modes (at 22.8 & 23.5 Hz) are just visible. Scroll through the four FRFs The four FRFs are virtually identical. Therefore, any one of these four DOFs can be used as a Reference DOF. Execute M#s CoQuad in the BLK: Driving Point FRFs window Place the mouse pointer near the two closely coupled modes and spin the mouse wheel to zoom in on the two resonance peaks Execute Display Maximize to expand the display, as shown below CoQuad Expanded Around Closely Coupled Modes In a real world test, measurement noise and distortion (non-linearity) would make these two resonant peaks hard to identify, and curve fitting the FRFs might not extract both modes. Random excitation together with spectrum averaging, and more frequency resolution in the FRFs are two ways to improve your curve fitting success. SINGLE REFERENCE FRFS In this case six noise-free FEA mode shapes will be used to synthesize a single reference set of FRFs using one of the previously identified Reference DOFs. Since the FRFs will be linear and noise -free, curve fitting them will yield a set of mode shapes the match perfectly with the original FEA mode shapes. Execute: Tools Synthesize FRFs in the SHP: Bridge FEA Modes window. The Synthesize FRFs dialog will open, as shown below Enter Block Size = 2000, Ending Frequency = 75 Hz Select Roving DOF 1Z Scroll to the bottom of the Roving DOF list, hold down the Shift Key, and click to select all of the Roving DOFs Select Reference DOF 36Z, and click on OK In the next dialog box, press New File, enter FRFs Ref 36Z in the next dialog box, and press OK The BLK: FRFs Ref 36Z window will open with 175 FRFs in it, as shown below Page 8 of 11
FRF-BASED CURVE FITTING 175 Single Reference FRFs Since the synthesized FRFs are linear, noise-free, and all six resonance peaks are visible and therefore can be counted, the Quick Fit curve fitting methods can be used on them without any further processing such as special windowing operations. Right click in the graphics area of the BLK: FRFs Ref 36Z window, and select Curve Fitting from the drop down menu On the Mode Indicator tab, select Multivariate Mode Indicator Function (MMIF) from the list Right click in the graphics area and select Curve Fit Quick Fit from the drop down menu The modal parameters for six modes will be displayed in the Modal Parameters spreadsheet on the right, and a red Fit Function will be overlaid on each FRF, as shown below. Scroll through the FRFs to examine the modal parameters and Fit Function for each FRF Page 9 of 11
To save the mode shapes into a Shape Table, Quick Fit of the FRFs. Right click in the graphics area and select Curve Fit Shapes Save shapes from the drop down menu In the dialog box that opens, press New File, enter Quick Fit Modes in the next dialog box, and press OK The SHP: Quick Fir Modes window will open with the Quick Fit modal parameters in it. MODE SHAPE COMPARISON Quick Fit Modal Parameters Notice that the Quick Fit mode shapes are called Residue Mode Shapes, whereas the FEA mode shapes were called UMM Mode Shapes. Application Note 05 explains the difference between these two types of mode shapes. Residue Mode Shapes can be Re-scaled into UMM Mode Shapes and vise versa. Regardless of how they are scaled, mode shape pairs can always be compared numerically using the Modal Assurance Criterian (MAC) Execute: Display MAC in the SHP: Bridge FEA Modes window. Page 10 of 11
Select SHP: Quick Fit Modes in the dialog box that opens, and click on OK The MAC plot below show that each Quick Fit mode shape is co-linear with the same numbered FEA mode shape. A visual comparison of the frequency & damping of each Quick Fit mode with the frequency & damping of each FEA mode demonstrate the accuracy of MDOF FRF-Based curve fitting, even when two modes are closely coupled. Comparison of Quick Fit & FEA Mode Shapes Page 11 of 11