ME scope Application Note 19

Transcription

1 ME scope Application Note 19 Using the Stability Diagram to Estimate Modal Frequency & Damping The steps in this Application Note can be duplicated using any Package that includes the VES-4500 Advanced Modal Analysis option. Click here to download the ME scope Demo Project file for this App Note. INTRODUCTION In this Application Note a Stability diagram will be used to determine, The number of modes that are represented by resonance peaks in a set of spectral measurements A stable estimate of the modal frequency & damping of each mode In ME'scope, three different curve fitting methods can be used together with a Stability diagram to estimate frequency & damping. Two of the methods: the AF Polynomial and the Z-Polynomial method use the Rational Fraction Polynomial form of an FRF to curve fit the experimental data. The other method; the Complex Exponential method, curve fits a set of impulse response functions to estimate modal frequency & damping. DETERMINING THE MODEL SIZE The first and most important step in the use of any curve fitting method to estimate modal frequency & damping is to determine how many modes are represented in a set of measurements. In ME scope, this can be done in two ways, 1. Counting peaks on a Mode Indicator curve 2. Using the Stability diagram Counting Peaks on a Mode Indicator ME scope has two types of Mode Indicator functions; the Complex Mode Indicator Function (CMIF), and the Multivariate Mode Indicator Function (MMIF). Both will exhibit resonance peaks as indicators of resonances in a set of spectral data. During curve fitting, when the Count Peaks command is executed on the Mode Indicator tab, all of the peaks above the horizontal noise threshold are counted and displayed in the Peaks box on the tab. The same number in the Peaks box also appears in the Modes box on the Frequency and Damping tab. This number is used as the model size by the Polynomial curve fitting method. Each Mode Indicator can also be used to visually locate resonances so that more accuracy can be gained by curve fitting in smaller frequency cursor bands surrounding one or more peaks. The Local & Global Polynomial and the Quick Fit methods use a Mode Indicator and the number of Peaks counted on the Mode Indicator to determine the curve fitting model size. Stability Diagram When modes are heavily coupled together so that it is not clear how many resonances are represented in a particular frequency band, an incorrect number of peaks may be counted using a Mode Indicator. When resonance peaks cannot be accurately counted on a Mode Indicator curve, a Stability diagram method should be used instead. Using a Stability diagram method, curve fitting to applied to the measurement data using a number of different model sizes, ranging from 1 to a Maximum Model Size. The frequency & damping of a resonance is also called a pole. If the frequency & damping estimates of a pole are essentially the same for different model sizes, that pole is called a stable pole. The Stability diagram is a plot of pole estimates for many different model sizes. In the diagram below, pole estimates are displayed for model sizes up to 50 modes. The function in the background is a Modal Indicator which shows three resonance peaks. The poles are plotted on top of the Mode Indicator. Page 1 of 9

2 Each pole is displayed with, 1. A dot at the frequency of the pole Stability Diagram Showing Three Stable Poles. 2. A horizontal line equal in length to the half power point damping of the pole. Placing the mouse pointer near a pole will display its frequency & damping estimates, as shown above. Stable Pole Groups A pole estimate is said to be stable if its frequency & damping values are within user specified tolerances from one model size to the next on the Stability diagram. Three different tolerances are used to define a stable pole group, 1. Frequency tolerance (in Hz) 2. Damping tolerance (in Hz or %) 3. Minimum Number of poles the stable group Three stable pole groups are displayed in the figure above. All of the poles in a stable group are displayed with the same color. All poles with modal frequency dots in vertical alignment and horizontal damping lines of approximately the same length are defined as a stable group. At least 5 poles with frequency & damping estimates within 0.5 Hz of each other were required to define the three stable groups. These three tolerances are defined on the Stable Groups tab in the figure above. EXAMPLE #1 To learn how to use the Stability diagram, let s start with an example where the answer is already known. We will create a set of FRFs with known pole locations and random mode shapes in them. We will only use the Stability diagram to estimate the frequency & damping of the poles. The curve fitting step required to estimate the mode shapes will not be necessary. Execute File New Project in the ME scope window to start a new Project Execute File New Data Block in the ME scope window. A dialog box will open as shown below Page 2 of 9

3 Dialog for the New Data Block Command. You can also download the ME scope Demo Project file for this App Note by clicking on the link at the beginning of this App Note. Enter the parameters shown into the File New Data Block dialog box. These parameters will create a new Data Block with 100 time domain M#s in it. Each M# is a simulated impulse response containing three modes (at 100, 115, & 130 Hz with 5% damping) and a random residue (magnitude & phase). Each M# will have 2000 time domain samples, with seconds spacing between samples spanning a total time of 2 seconds. When all of the parameters have been entered into the dialog, click on OK Enter "100 M#s" into the next dialog box A Data Block window will open containing 100 Impulse Response Functions. The Fourier transform of an Impulse Response Function (IRF) is a Frequency Response Function (FRF). When this Data Block is transformed to the frequency domain using the FFT, it will contain 100 FRFs, which will then be used for curve fitting. Data Block Showing Impulse Response Functions. Page 3 of 9

4 The IRFs in your Data Block may look different than those shown above because the modal Residues were randomly generated. However this will not affect the frequency & damping estimates obtained by curve fitting the FRFs. To transform the Impulse Responses to FRFs, Execute Transform FFT in the BLK: 100 M#s window USING THE STABILITY DIAGRAM To begin curve fitting using the Stability diagram, Right click in the graphics area of BLK: 100 M#s and execute Curve Fitting from the menu During curve fitting the Curve Fitting panel is displayed on the right. It contains three tabs, one for each curve fitting step, 1. Mode Indicator 2. Frequency and Damping 3. Residues, Save Shapes The FRFs and Mode Indicator are displayed on the upper & lower left, as shown below. Click on Frequency and Damping tab to the right Click on Stability tab and then on the Stability button The AF Polynomial method (default) will be used to curve fit the 100 M#s. When it is finished, the poles estimates will displayed on top of the Mode Indicator as shown below. This is a Stability diagram. Data Block Showing a Stability Diagram. Notice that in addition to the three stable poles overlaid on the resonance peaks, there are also a number of stable poles groups near the edges of the Mode Indicator. These are called computational modes or poles. On the Stable Groups tab, slide the slider upward to increase the Min. Number of Stable Poles until the computational poles are no longer displayed on the Stability diagram Frequency & Damping Tolerances You can also decrease the frequency & damping tolerances to further limit the number of poles in each stable pole group. As soon as you change the tolerances, the Stability diagram will be redrawn to reflect your changes. Page 4 of 9

5 Saving the Average Frequency & Damping of a Stable Group There are two ways to save the average frequency & damping og the poles in a stable group. 1. Press the Save Groups button in the Stable Groups tab 2. Draw a selection box around some poles in a stable group on the Stability diagram Since there is usually so much information on a Stability diagram, it is useful to zoom in around some of the stable pole groups to display them more clearly. Increase the Min. Number of Stable Poles and zoom around the three stable pole groups displayed over the three resonance peaks, as shown below Press the Save Groups button The average frequency & damping of all of the stable groups will be displayed in the Modal Parameters spreadsheet on the right, as shown below. To use the section box to save the frequency & damping of a stable group, Click on the Stability diagram Hold down the Ctrl key and the left mouse button to display a "+" sign on the Stability diagram Drag to draw a selection box around some of the poles in a group Release the left mouse button to save the average frequency & damping in the Modal Parameters spreadsheet on the right For this ideal case, curve fitting yielded exactly the same frequency & damping values that were used to synthesize the 100 IRFs and hence the 100 FRFs that were curve fit using the AF Polynomial method. To try out the other two Stability diagram curve fitters Display the Stability tab, and select the Complex Exponential method from the Method list Press the Stability button When the curve fitter has finished the Stable Groups of pole estimates found by this method will be displayed on the Stability diagram. Page 5 of 9

6 Stability Diagram Showing Stable Poles From Complex Exponential Curve Fitter Scroll the bar to the right of the Stability diagram to display more of the poles in the stable groups, as shown above Press the Save Groups button to save the average frequency & damping values from this method into the Modal Parameters spreadsheet, as shown above EXAMPLE #2 In this example, we will use Stability diagram to identify the frequency & damping of the modes in a set of 99 experimental FRFs taken from the Jim Beam structure. This data is contained in BLK: Jim Beam FRFs which is part of the AppNote19.VTprj Project file. Click on the link at the beginning of this App Note to download the AppNote19.VTprj Project file Double click on BLK: Jim Beam FRFs to open its window Right click in the graphics area of BLK: Jim Beam FRFs and execute Curve Fitting from the menu Click on Stability tab and then on the Stability button Notice that there 10 resonance peaks are clearly displayed on the Mode Indicator, and that there are also stable groups of computational modes not near the 10 peaks. Turn on the Band cursor and surround the stable pole groups displayed on the 10 resonance peaks, as shown below Press the Save Groups button If frequency & damping estimates for more than 10 modes are saved in the Modal Parameters spreadsheet, one or more of the group tolerance parameters will have to be changed in order to reduce the number of stable groups Increase the Min. Number of Stable Poles to about 11, as shown below Press the Save Groups button When the frequency & damping tolerances are reduced, and the minimum number of stable poles is increased, fewer stable groups will be displayed on the Stability diagram Page 6 of 9

7 Estimating Modal Residues Stability Diagram Showing 10 Stable Groups within the Band Cursor Once the modal frequency & damping have been estimated, those estimates are used in a second curve fitting step to estimate modal residues. This curve fitting step is carried out on the Residues, Save Shapes tab. After the modal residue are estimates for each mode, a red fit function is synthesized and overlaid on the FRF data to indicate that the modal parameters estimates also represent an accurate model of the dynamics of the test article. The AF polynomial method estimates modal residues for each stable pole group by averaging together the residues of each pole in a stable group. This method does not require a second curve fitting step to estimate modal residues. Click on the Residues, Save Shapes tab and then on the Residues button on this tab A red fit function will be synthesized using the modal parameters and overlaid on each FRF. Use the scroll bar to the right of the FRFs to display the FRFs overlaid with their fit function, as shown below Red Fit Function Overlaid on an FRF Page 7 of 9

8 After some modes have been defined by adding their frequency & damping estimates to the Modal Parameters spreadsheet, any Residue curve fitting method can be used to estimate modal residues. Press the Save Shapes button to save the AF Polynomial estimates into a Shape Table Enter "AF Poly Modes" into the New File dialog box that opens Select Polynomial in the Method list on the Residues, Save Shapes tab Click on the Residues button The modal frequency & damping estimates in the Modal Parameters spreadsheet will be used by the Polynomial method to estimate the modal residues. Compare the figure below with the figure above, and notice that the red fit function is slightly different when residues are estimated using two different Residue curve fitters. Comparing Mode Shapes Red Fit Function from the Polynomial Residue Curve Fitter. The ME'scope approach to curve fitting is to use several different curve fitting methods to estimate modal parameters (frequency, damping, and mode shape), and then compare the results. If the modal parameter estimates from different curve fitting methods are essentially the same, the results did not depend on any particular curve fitting method and the estimates can be used as a valid representation of the structural dynamics. Press the Save Shapes button to save the Polynomial residue estimates into a separate Shape Table Press New File in the next dialog box that opens Enter "Polynomial Modes" into the New File dialog box The mode shapes from the two different curve fitting methods will be compared both numerically and in animation Close all windows except SHP: AF Poly Modes and SHP: Polynomial Modes" The Jim Beam structure model is also including in the AppNote19.VTprj Project file. To retrieve that file, Click on the link at the beginning of this App Note to download the AppNote19.VTprj Project file Double click on STR: Colored Jim Beam to open its window Right click in the graphics area of STR: Colored Jim Beam and execute Draw Compare Shapes from the menu Page 8 of 9

9 Animation of a pair of mode shapes, one from each of the two Shape Tables, should begin as shown below. Execute Windows Arrange Windows For Animation in the ME'scope window Click on a Select Shape button in either Shape Table to display the pair of mode shapes that have the Maximum MAC value among all mode shapes in the two Shape Tables. Two mode shapes that have a MAC value close to "1" are essentially the same mode shape. CONCLUSIONS In Example #1, we found that two different Stability diagram curve fitting methods precisely estimated the modal frequency & damping of three modes that were used to synthesize 100 Impulse Response Functions. The synthesized IRFs were transformed (using the FFT) to their equivalent FRFs. which were then curve fit to display stable pole estimates on a Stability diagram. In Example #2, we used the Stability diagram again to identify the frequency and damping of 10 modes of the Jim Beam structure from 99 experimental FRFs. Then two different residue curve fitting methods were used to obtain 3-dimensional mode shapes at 33 points on the Jim Beam structure. Finally, the mode shape pairs with maximum MAC values were displayed in animation. All 10 mode shape pairs from the two residue curve fitting methods had MAC values very close to "1", indicating that the two methods gave the same results. Although the Stability diagram was used in two examples where resonance peaks could be counted on a Mode Indicator curve, the Stability diagram can also be used to identify modes in data where counting resonance peaks is not possible. Page 9 of 9