ME scopeves. VES-4600 Advanced Modal Analysis. (February 8, 2019)

Transcription

1 ME scopeves VES-4600 Advanced Modal Analysis (February 8, 2019)

2 Notice Information in this document is subject to change without notice and does not represent a commitment on the part of Vibrant Technology. Except as otherwise noted, names, companies, and data used in examples, sample outputs, or screen shots, are fictitious and are used solely to illustrate potential applications of the software. Warranty Vibrant Technology, Inc. warrants that (a) the software in this product will perform substantially in accordance with the accompanying documentation, for a period of one (1) year from the date of delivery, and that (b) any hardware accompanying the software will be free from defects in materials and workmanship for a period of one (1) year from the date of delivery. During this period, if a defect is reported to Vibrant Technology, replacement software or hardware will be provided to the customer at no cost, excluding delivery charges. Any replacement software will be warranted for the remainder of the original warranty period or thirty (30) days, whichever is longer. This warranty shall not apply to defects resulting from improper or inadequate maintenance by the customer, customer supplied software or interfacing, unauthorized modification or misuse, operation outside of the environmental specifications for the product, or improper site preparation or maintenance. In the event that the software does not materially operate as warranted above, the sole remedy of the customer (and the entire liability of Vibrant Technology) shall be the correction or detour of programming errors attributable to Vibrant Technology. The software should not be relied on as the sole basis to solve a problem whose incorrect solution could result in injury to a person or property. If the software is employed in such a manner, it is at the entire risk of the customer, and Vibrant Technology disclaims all liability for such misuse. NO OTHER WARRANTY IS EXPRESSED OR IMPLIED. VIBRANT TECHNOLOGY SPECIFICALLY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MATERIAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANT ABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE REMEDIES PROVIDED HEREIN ARE THE CUSTOMER'S SOLE AND EXCLUSIVE REMEDIES. VIBRANT TECHNOLOGY SHALL NOT BE LIABLE FOR ANY DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES IN CONNECTION WITH THE FURNISHING, PERFORMANCE, OR USE OF THIS PRODUCT, WHETHER BASED ON CONTRACT, TORT, OR ANY OTHER LEGAL THEORY. Copyright The software described in this document is copyrighted by Vibrant Technology, Inc. or its suppliers and is protected by United States copyright laws and international treaty provisions. Unauthorized reproduction or distribution of this program, or any portion of it, may result in severe civil and criminal penalties, and will be prosecuted to the maximum extent possible under the law. You may make copies of the software only for backup or archival purposes. No part of this manual may be reproduced or transmitted in any form or by any means for any purpose without the express written permission of Vibrant Technology. Copyright by Vibrant Technology, Inc. All rights reserved. Printed in the United States of America. Vibrant Technology, Inc E. Arapahoe Rd. Tower II, Suite 600 Centennial, CO USA phone: (831) fax: (831) support@vibetech.com

3 Table of Contents Advanced Modal Analysis... 3 Additional Data Block Commands... 3 Multi-Reference Methods... 3 Stability Diagram... 3 Animate Menu... 3 Display Menu... 3 Tools menu... 3 Modes menu... 3 Additional Shape Table Commands... 3 Animate Menu... 3 Display Menu... 3 Tools menu... 3 Special Mouse & Keyboard Commands... 3 Mode Indicator Graph... 3 Stability Diagram... 4 Selecting Modes... 4 When Is Multi-Reference Modal Analysis Necessary?... 4 Single Reference versus Multi-Reference FRFs... 4 Multi-Reference Modal Testing... 4 Multiple Shaker Test... 4 Multiple Reference Roving Impact Test... 5 Multi-Reference Mode Indicators... 5 Multi-Reference CMIF... 5 Multi-Reference MMIF... 6 Modal Participation Factors... 6 Multi-Reference Parameter Estimation... 6 Multi-Reference Polynomial... 6 Stability Methods... 6 AF Polynomial (Alias Free Polynomial)... 6 Complex Exponential... 6 Z Polynomial... 6 Methods Column... 6 Stability Diagram... 7 Why Use a Stability Diagram?... 7 What is a Stable Pole Group?... 7 i

4 Table of Contents Poles Diagram... 8 Changing the Stable Group Tolerances... 8 Percentage of Critical Damping or 3 db Bandwidth Damping... 9 Displaying Pole Value Balloons... 9 Save Groups Button... 9 Stability Pole Selection Box Stability Clear Stability Diagram Animate Normalize Shapes Display Complexity Plot Normal Shape Complex Shape Shape Table Complexity Plot Data Block Complexity Plot Normalized Shapes on the Complexity Plot MPC (Modal Phase Co-linearity) Flipping the Phases of the Right-Hand Shape Display Magnitude Ranking Data Block Magnitude Ranking Which Magnitudes Are Ranked? Magnitude Value Shape Table Magnitude Ranking Which Magnitudes Are Ranked? Magnitude Value Tools Measurement Expansion Modes Modal Decomposition Display Shape M#s M, C, K Tools Participation Matrix Least-Squared-Error Solution Tools Shape Expansion Script Menu Curve Fit Stability Diagram Parameters Curve Fit Stability Reset Curve Fit Save Stable Groups ii

5 VES-4600 Advanced Modal Analysis If the VES-4600 Advanced Modal Analysis option is authorized by your ME'scope license, the following commands are enabled in the Data Block and Shape Table windows. Check Help About to verify authorization of this option. Additional Data Block Commands The following commands are added to a Data Block window. Multi-Reference Methods Multi-Reference curve fitting methods are enabled on the Mode Indicator, Frequency and Damping, and Residues tabs. Stability Diagram Two Stability tabs are added to the Frequency & Damping tab. The Stability tab contains several Multi-Reference curve fitting methods. The Stability or Poles diagrams are also enabled on the Mode Indicator display. Animate Menu Animate Normalize Shapes Display Menu Display Complexity Plot Display Magnitude Ranking Tools menu Tools Measurement Expansion Modes menu Modes Modal Decomposition Additional Shape Table Commands The following commands are added to a Shape Table window. Animate Menu Animate Normalize Shapes Display Menu Display Complexity Plot Display Magnitude Ranking Display Shape M#s M, C, K Display Poles Tools menu Tools Participation Matrix Tools Shape Expansion Special Mouse & Keyboard Commands Mode Indicator Graph Hold down the Alt key and place the mouse pointer near a frequency estimate (vertical line) on the Mode Indicator graph to display its Frequency & Damping balloon. 3

6 When Is Multi-Reference Modal Analysis Necessary? Click the left mouse button to display its Frequency & Damping box permanently on the Mode Indicator graph. Click & drag to move the nearest Frequency & Damping box on the Mode Indicator graph. Click the right mouse button to erase all Frequency & Damping boxes on the Mode Indicator graph. Stability Diagram Click & drag to draw a selection box enclosing a group of stable poles on the Stability or Poles diagram. Selecting Modes Right click near a frequency estimate (vertical line) on the Mode Indicator to select the mode in the Modal Parameters spreadsheet. Display the Band cursor and enclose one of more vertical lines on the Mode Indicator to select the modes in the Modal Parameters spreadsheet When Is Multi-Reference Modal Analysis Necessary? The term Multi-Reference Modal Analysis means that FRFs are acquired using two or more references (fixed input or output sensors) and are curve fit using Multi-Reference curve fitting methods. Multi-Reference Modal Analysis is required when a structure has resonances that occur under one of the following conditions; 1. Closely Coupled Modes: One resonance peak represents two or more modes. 2. Repeated Roots: Two or more modes having the same natural frequency but different mode shapes. 3. Local Modes: Different resonance peaks occur in FRFs from different references. In each of the above cases, multiple reference curve fitting is required in order to properly extract all modal parameters from a set of FRFs. Multiple reference FRFs correspond to multiple rows or columns of the FRF matrix in the MIMO model. Single Reference versus Multi-Reference FRFs A single reference set of FRFs is, The minimum requirement for extracting experimental modal parameters using FRF-based curve fitting Obtained by exciting the structure with a single (fixed) exciter or using a single (fixed) response sensor Is a single row or column of elements in the FRF matrix of a MIMO model of the structure Not enough for extracting closely coupled modes, repeated roots, or local modes of a structure A multi-reference set of FRFs is, Required for extracting closely coupled modes, repeated roots, or local modes of a structure Obtained by exciting the structure with multiple (fixed) exciters or using multiple (fixed) response sensors Consists of multiple rows or columns of elements in the FRF matrix of a MIMO model of the structure Useful for extracting modes when a structure has high modal density (many resonance peaks in small frequency bands) Multi-Reference Modal Testing A multiple reference modal test is done using either multiple (fixed) exciters with sensors to measure the forces, or multiple (fixed) response sensors. Each fixed sensor is called a Reference. Multiple Shaker Test In a multiple shaker test, two or more (fixed) shakers are used to simultaneously excite the structure. 4

7 Advanced Modal Analysis The multiple shakers must be driven with uncorrelated broad band signals. In a multiple reference test, an FRF between each response and each reference force is calculated Also, Multiple & Partial Coherences are calculated The FRFs are elements of two or more columns of the FRF matrix in a MIMO model of the structure. Large structures with non-linear dynamic behavior are typically tested using multiple shakers, driven by pure or burst random excitation signals. Random excitation together with spectrum averaging is used to "average out" the non-linear dynamic behavior of the structure from the spectra and hence the FRFs. Multiple Reference Roving Impact Test In a multiple reference roving impact test, two or more (fixed) response sensors are used, and the structure is excited one DOF at a time with a roving impactor. This test is the same as performing two or more single Reference modal tests, but takes no more time to complete than a single reference roving impact test The FRFs are elements of two or more rows of the FRF matrix in a MIMO model of the structure Multi-Reference Mode Indicators As part of the Multi-Reference Modal Analysis option, both Multi-Reference CMIFs (Complex Mode Indicator Functions) and Multi-Reference MMIFs (Multivariate Mode Indicator Functions) are added to the Mode Indicator methods list on the Mode Indicator tab. A peak at or near the same frequency in two or more Multi-Reference Indicator curves indicates closely coupled modes or repeated roots. Multi-Reference CMIF CMIFs Indicating Two Closely Coupled Modes Near 200 Hz. The Multi-Reference CMIF curves are calculated by performing a singular value decomposition of multireference FRF data, resulting in a separate CMIF curve for each reference. Each peak on each CMIF curve is an indication of a resonance. 5

8 Multi-Reference Parameter Estimation Multi-Reference MMIF The Multi-Reference MMIF curves are calculated by performing an eigen-solution of multi-reference FRF data, resulting in a separate MMIF curve for each reference. Each peak (or valley) on each MMIF curve is an indication of a resonance. Modal Participation Factors The Multi-Reference CMIF & MMIF calculations also provide Modal Participation Factor curves for each reference of FRF data. Modal participation factors are used to weight each reference of data during Multi-Reference curve fitting. Multi-Reference Parameter Estimation ME'scope contains several different Multi-Reference Parameter Estimation (curve fitting) methods. Each method uses the modal participation of each resonance in each reference to weight the data during curve fitting Multi-Reference Polynomial This method uses a multi-reference version of the Rational Fraction Orthogonal Polynomial method, together with modal participations from a multiple reference Mode Indicator function The curve fitting model size in the Modes box on the Polynomial tab is used for estimating modal Frequency & Damping Stability Methods These methods utilize a Stability diagram which doesn't require peak counting on a Mode Indicator function. When the Multi-Reference Modal Analysis is enabled, Stability and Stable Groups tabs are added to the Frequency & Damping tab in a Data Block window. The Stability tab contains curve fitting methods for estimating modal frequency & damping using a progression of curve fitting model sizes, from one mode up to the Max. Model Size listed on the tab. AF Polynomial (Alias Free Polynomial) An extension of the Rational Fraction Orthogonal Polynomial method. The term "alias free" refers to its characteristic of placing computational modes toward the edges of the curve fitting band, instead of aliasing them throughout the band. Complex Exponential A time domain method that estimates poles by curve fitting Impulse Response Functions (IRFs). During curve fitting, the Inverse FFT is applied to each FRF to obtain its corresponding IRF. Z Polynomial An extension of the Rational Fraction Orthogonal Polynomial method. Uses the Z transform to transform frequency to a unit circle, resulting in numerically stable solution equations. Methods Column The curve fitting methods used to estimate the parameters of each mode are listed in the Frequency & Damping Method column, and the Residues Method column. The following abbreviations are used for the curve fitting methods in the Multi-Reference Modal Analysis option, "AF Poly" Alias Free Polynomial "Comp Exp" Complex Exponential "Z Poly" Z Polynomial 6

9 Advanced Modal Analysis "M-Poly" Multi-Reference Polynomial "M-AF Poly" Multi-Reference AF Polynomial "M-Comp Exp" Multi-Reference Complex Exponential "M-Z Poly" Multi-Reference Z Polynomial Stability Diagram When the Multi-Reference Modal Analysis option is enabled in ME'scope, a Stability tab and a Stable Groups tab are added to the Frequency and Damping tab during curve fitting. Why Use a Stability Diagram? The Stability methods do not require that resonance peaks be counted on the Mode Indicator. A Stability diagram is a plot of pole (modal frequency & damping) estimates obtaining from multiple curve fitting model sizes. Stability curve fitting starts with a model size for one (1) mode and ends with a model size in the Max Model Size box on the Stability tab. Each modal frequency estimate is displayed as a vertical line on the Stability diagram. Each modal damping estimate is displayed as a horizontal line on the Stability diagram. What is a Stable Pole Group? Stability Diagram Showing Computational Modes. The Stability diagram is displayed on top of a Mode Indicator graph. A Stable Pole Group must meet all of the following criteria; All poles with frequency estimates within the Frequency Tolerance listed on the Stable Groups tab are candidates for a Stable Pole Group. All poles with damping estimates within the Damping Tolerance listed on the Stable Groups tab are candidates for a Stable Pole Group. 7

10 Stability Diagram If the number of poles satisfying the Frequency & Damping Tolerances is greater than or equal to the Min. Number of Stable Poles listed on the Stable Groups tab, they are added to a Stable Group. All poles in a Stable Group are displayed using the same color on the Stability or Poles diagram.\ Stable Group Colors alternate between the top two Contour Colors in the File Data Block Options box. Poles Diagram Stability Diagram Showing Stable Pole Groups. Poles estimates can also be displayed on a Poles diagram, as shown below. Check the Poles box on the Stable Groups tab to display the Poles diagram Modal frequency estimates are plotted along the horizontal axis and modal damping estimates along the vertical axis on the Poles diagram. Changing the Stable Group Tolerances Poles Diagram. The Stability diagram is updated whenever the Tolerances or the Min. Number of Stable Poles on the Stable Groups tab is changed. 8

11 Advanced Modal Analysis To change a Stability Tolerance, Click on its radio button, and scroll the slider bar on the right side of the Stable Groups tab. Or type a number into the Tolerance or Min. Number of Stable Poles box on the Stable Groups tab Press the Reset button to reset all of the Stability Tolerances to default values. Percentage of Critical Damping or 3 db Bandwidth Damping Damping can be chosen either as a percentage of critical damping (%) or as the 3 db or half power point damping (Hz). The current type of damping is indicated by its selected radio button. See the FRFs in Terms of Modal Parameters section for definitions of damping terms. Displaying Pole Value Balloons To display a pole value balloon on the Stability or Poles diagram, Hold down the Alt key Place the mouse pointer near a pole to display its pole value balloon Left click near a pole to permanently display its pole value balloon To move a pole value balloon, Release the Alt key Place the mouse pointer on the balloon Click & drag the balloon To remove a balloon from the Stability or Poles diagram, Right click near the pole on the Stability or Poles diagram Save Groups Button When the Save Groups button is pressed, Stable Group Showing Pole Value Balloons. The average value of all poles in each visible Stable Pole Group is added to the Modal Parameters spreadsheet. 9

12 Animate Normalize Shapes If the Band cursor is displayed, the average value of the poles in each Stable Pole Group within the band is added to the Modal Parameters spreadsheet. Stability Pole Selection Box Enables the drawing of a selection box to select one or more poles from the Stability or Poles diagram, and save their average values into the Modal Parameters spreadsheet Hold down the Ctrl key to display the selection box mouse pointer ("+") on the Stability or Poles diagram. Click & drag to draw a selection box to enclose one or more desired poles. All pole values within the selection box are averaged together and added to the Modal Parameters spreadsheet. Stability Clear Stability Diagram Clears the Stability Diagram in the Mode Indicator graph area. Animate Normalize Shapes Displays shapes as either complex of normalized. When enabled, normalized shapes are displayed during shape animation from an Animation or Comparison Source. This command can be executed from a Structure, Shape Table or Acquisition window. See Display Complexity Plot for details Display Complexity Plot Opens the Complexity Plot window. This command can be executed from either a Data Block or a Shape Table. Normal Shape Each shape component of a normal shape has a phase of 0 or 180 degrees During shape animation, a normal shape exhibits a standing wave motion, and its Node Lines will not move The shape components of a normal shape lie on a straight line in a Complexity Plot An FEA model with no damping yields normal mode shapes. All phases of a normal mode shape are 0 & 180 degrees. Complex Shape Each shape component of a complex shape can have an arbitrary phase During shape animation, a complex shape can exhibit a traveling wave motion, and its Node Lines can move The shape components of a complex shape do not lie on a straight line in a Complexity Plot Experimental mode shapes can be complex shapes for several reasons; 1. Real structures with heavy damping in them can have complex mode shapes 2. Measurement errors can introduce arbitrary phases into the shape component estimates 3. Curve fitting errors can introduce arbitrary phases into the shape component estimates 10

13 Advanced Modal Analysis Shape Table Complexity Plot A Shape Table Complexity Plot displays the magnitudes & phases for all (or selected) shapes for all (or selected) M#s, Data Block Complexity Plot Shape Table Complexity Plot. A Data Block Complexity Plot displays the magnitudes & phase all (or selected) M# values at the current Cursor position. Normalized Shapes on the Complexity Plot Data Block Complexity Plot. When complex shapes are "normalized", they behave like normal shapes. When Display Normalize Shapes is checked, complex shape components are displayed on the left, and normalized shape components are displayed on the right of the Complexity Plot, as shown below The normalization line (dashed line) on a Complexity Plot is used to normalize each complex shape. When a shape is normalized, the magnitude of each shape component is retained but the phase is changed to either 0 or 180 degrees. When a shape is normalized, the red (+) shape components are given 0 degrees phase, and the blue (-) shape components are given 180 degrees phase. 11

14 Display Magnitude Ranking To rotate the normalization line to a different position, 1. Click & drag near the normalization line on the Complexity Plot 2. Or execute Display Set Normalization Angle and enter an angle into the dialog box. MPC (Modal Phase Co-linearity) Complexity Plot with Shape Normalization Turned ON. If only one shape is displayed in the Complexity Plot, its MPC (Modal Phase Co-linearity) value is also displayed. MPC values range between 0 &1. If MPC = 1, all components the shape lie on a straight line. If MPC < 1, some shape components do not lie on a straight line. If MPC is "close to 1", this indicates that the structure is lightly damped, or that the mode shape is a normal mode shape. Flipping the Phases of the Right-Hand Shape During Comparison Animation, if two similar shapes appear to be animating 180 degrees out of phase with one another, the phases of right-hand shape can be changed by 180 degrees so that the two shapes animate more closely together. To flip the sign of the right-hand shape; 1. Execute Animate Compare Shapes Flip Sign in the Structure window to multiply the right-hand shape by "-1". 2. Or rotate the normalization line on the Complexity Plot by dragging it to flip the phase of the shape components. Display Magnitude Ranking Displays a Magnitude Ranking bar chart in a separate window This command can be executed from a Data Block or Shape Table. Data Block Magnitude Ranking This bar chart displays the magnitudes of the measurement values at the current cursor position, or of all measurement values if no cursors are displayed. M# magnitudes are plotted on the vertical axis versus M#s (and non-zero DOFs) on the horizontal axis. 12

15 Advanced Modal Analysis The magnitudes are ranked from the largest on the left to the smallest on the right. The Magnitude Ranking chart is updated when the cursor is moved, or when different M#s are selected. The Contour Colors defined in the File Data Block Options box are used for the Bar colors Which Magnitudes Are Ranked? Magnitude Ranking Chart. If the Real part of the measurements is displayed, the Real parts are ranked. If the Imaginary part of the measurements is displayed, the Imaginary parts are ranked. Otherwise, the magnitudes of the measurements are ranked. Magnitude Value Hover the mouse pointer over a magnitude bar to display its value at the bottom of the window. Shape Table Magnitude Ranking This bar chart displays the magnitude of all (or selected) shape components (M#s), ordered from the largest to the smallest. Shape component magnitudes are plotted on the vertical axis versus M#s on the horizontal axis. The magnitudes are ranked from the largest on the left to the smallest on the right. The Contour Colors defined in the File Shape Table Options box are used for the Bar colors 13

16 Tools Measurement Expansion Which Magnitudes Are Ranked? Shape Magnitude Ranking Chart. If the Real parts of the M#s are displayed, the Real parts are ranked. If the Imaginary parts of the M#s are displayed, the Imaginary parts are ranked. Otherwise, the magnitudes of the M#s are ranked. Magnitude Value Hover the mouse pointer over a magnitude bar to display its value at the bottom of the window. Tools Measurement Expansion Expands the measurements in a Data Block using a set of shapes. This command uses the same equation for linearly relating one set of shapes to another as the Tools Shape Expansion command in a Shape Table, and the Modes Modal Decomposition command in a Data Block This command is useful for; 1. Expanding a Data Block with a few DOFs using shapes with more DOFs in them 2. Expanding a Data Block of time or frequency measurements using FEA shapes with more DOFs in them The Measurement Expansion algorithm performs a least-squared-error curve fit of the shapes to each sample of the Data Block data at the common DOFs between the shapes and the Data Block. A participation matrix of the shapes is calculated for each sample of Data Block data Then the participation matrix is multiplied by the shapes to obtain the expanded values of the measurements at each sample. Only mode shapes are used for Measurement Expansion. Modal Frequency & Damping are not used. The shapes must meet the following conditions, 1. The shapes are valid for the structure regardless of boundary conditions 2. The shapes are linearly independent of one another for the common DOFs between the shapes and the Data Block If these two conditions are met, the shapes provide a valid expanded set of measurements for all of the unmeasured DOFs of the structure represented by the DOFs in the shapes. 1. Execute this command in the Data Block containing the measurements to be expanded 2. A file dialog box will open from which the Shape Table containing shapes to be used for measurement expansion can be selected 14

17 Advanced Modal Analysis 3. After the calculation is completed, another file dialog box will open allowing you to save the Data Block with the expanded measurements in it Measurement Expansion can be controlled by selecting shapes and/or M#s in either the Data Block or Shape Table before executing this command. Modes Modal Decomposition Decomposes time or frequency measurements into "resonance curves" that represent the contribution of each mode shape or ODS to the measurement data. Solves the following equation for each M#, at each sample of data in a Data Block, where: [ Shape Table] = matrix of mode shapes or operating deflection shapes (ODS's). Each matrix column is a mode shape or ODS. {Weights} = vector of weights (also called a participation vector), for each sample of time or frequency data in the Data Block {ODS} = the ODS of data at each sample for each sample of time or frequency data in the Data Block Each resonance curve is saved as a new M# in a new Data Block. For example, if 10 mode shapes are used to decompose the time or frequency data in a Data Block, 10 M#s, each one a resonance curve for each mode, are calculated. Only mode shapes are used for Modal Decomposition. Modal Frequency & Damping are not used. Data Block Showing Modal Decomposition Resonance Curves Overlaid. Each modal decomposition resonance curve can be curve fit to extract the experimental modal frequency & damping associated with that resonance. Display Shape M#s M, C, K Displays the effective mass, damping & stiffness (also called the generalized mass, damping & stiffness) of each mode shape in the M#s spreadsheet. Effective mass, damping & stiffness are the values each mode would have if it were a single Mass-Spring-Damper system located at a DOF of the mode shape. This command can only be used with UMM mode shapes. The effective mass, damping & stiffness are calculated for each mode with the formulas, 15

18 Tools Participation Matrix where: Effective Mass = 1 / (Freq x Real Part (DP Residue) + Damp x Imaginary Part (DP Residue) Effective Damping = 2 x Damp x Effective Mass Effective Stiffness = (Freq 2 + Damp 2 ) x Effective Mass Freq - damped natural frequency of the mode. Damp - half power point damping of the mode. DP Residue - driving point Residue for each component of the mode shape. The driving point Residue is calculated from each component of the UMM mode shape. Tools Participation Matrix Calculates and displays the shape Participation Matrix. The Participation Matrix equates the shapes in one Shape Table with the shapes of another Shape Table. Each Shape Table can contain ODS's, FEA Mode Shapes, EMA Mode Shapes, or Engineering Data Shapes. Two sets of complex valued shapes can be assembled into two matrices [U] & [V], where each column of each matrix contains a shape. The two shape matrices are equated to each other with the following matrix equation, Least-Squared-Error Solution The Participation Matrix is computed as a least-squared-error solution to the above equation using the formula below. The two shape matrices [U] & [V] must have at least some matching DOFs (shape components) If the shapes [U] are linearly independent of one another for the matching DOFs, the Participation Matrix can be computed. Linear independence can be validated by displaying the MAC values of the mode shapes. Only mode shapes are used to calculate the Participation Matrix. Modal Frequency & Damping are not used. 16

19 Advanced Modal Analysis Tools Shape Expansion Expands the number of DOFs in a Shape Table using shapes with more DOFs in them from another Shape Table. This command is useful for; 1. Expanding ODS's with a few DOFs using mode shapes with many DOFs in them. 2. Expanding EMA mode shapes with a few DOFs using FEA mode shapes with many DOFs in them. The shape expansion algorithm performs a least-squared-error curve fit of the shapes with many M#s to the shapes with few M#s using the common DOFs between the two Shape Tables. The Participation Matrix of the shapes with many M#s in the shapes with few M#s is calculated. Then the participation matrix is multiplied by the shapes with many M#s to obtain the expanded shapes. The frequency & damping of each shape with few M#s is retained with each expanded shape. After a Participation Matrix [Part] has been calculated between the matching components of two shape matrices [U] & [V], an expanded set of shapes [Ve] is calculated with the following equation, Only mode shapes are used for Shape Expansion. Modal Frequency & Damping are not used. The shapes must meet the following conditions, 1. The shapes are valid for the structure regardless of boundary conditions. 2. The shapes are linearly independent of one another for the common DOFs between the shapes and those in the Shape Table If these conditions are met, the shapes provide a valid expanded set of measurements for all of the unmeasured DOFs of the structure represented by the DOFs in the shapes. If a set of FEA mode shapes with many M#s meets the above conditions, then expanding an ODS or a set of EMA mode shapes using the FEA mode shapes provides a valid Expanded ODS or set of EMA mode shapes for all DOFs of the structure represented by the DOFs in the FEA mode shapes. 17

20 Script Menu 1. Execute this command in the Shape Table containing the shapes with few M#s 2. A file dialog box will open from which the Shape Table containing shapes with many M#s can be selected 3. After the calculation is completed, another file dialog box will open allowing you to save the new Shape Table with the expanded shapes in it. Shape expansion can be controlled by selecting shapes and/or M#s in either Shape Table before executing this command. Script Menu The commands in this menu can be added to a Script window to automate the execution of these and other ME'scope commands. Curve Fit Stability Diagram Creates a stability diagram by curve fitting all or selected M#s in a Data Block. Parameters 1. Stability Method (AF Polynomial, Complex Exponential, Z Polynomial) 2. Damping in Percent (Yes) or Hertz (No) 3. Frequency Tolerance 4. Damping Tolerance 5. Min No. of Stable Poles 6. Show All (yes or no) 7. Poles (yes or no) Curve Fit Stability Reset Resets the stability diagram parameters to default values. Curve Fit Save Stable Groups Saves the average value of all of the Stable Pole Groups from the Stability diagram into the Modal Parameters spreadsheet. The frequency & damping of all of the poles in a Stable Group are averaged together, and the average value of the poles in each Group is saved in the Modal Parameters spreadsheet. 18