App Note #20
www.vibetech.com
2/25/2014
ME’scope Application Note #20 Multiple-Input Multiple-Output (MIMO) FRF Calculations INTRODUCTION Driving forces and response motions of a vibrating structure are related in a very straightforward manner in the frequency domain. The motions at N DOFs (points and directions) on the structure are related to forces applied to M DOFs by the following matrix of NM Frequency Response Functions (FRFs). Specifically:
X f Nx1 H f NxM F f Mx1
(1)
or
X 1 H 11 H 1m H nm X n H n1 X N H N 1 H Nm
H 1M F1 H nM Fm H NM FM
Each DOF is described by a point number and a direction. The excitation and response DOFs can be independent of one another, although it is common practice to measure the driving-point response motions. Driving-point FRFs are the diagonal elements of the FRF matrix, where the excitation and response DOF are the same. ME’scope contains commands for investigating all aspects of the Multiple-Input Multiple Output (MIMO) relationship of equation (1). You can: 1.
Calculate all the FRFs in the matrix from measured Forces and Responses.
2.
Calculate Multiple Responses, given a known matrix of FRFs and a vector of Forces. See Application Notes #21 and #22.
3.
Calculate Multiple Forces, given a known matrix of FRFs and a vector of Response motions. See Application Note #23.
Where: {X(f)}Nx1 is a vector containing N response motions.
In other words, provide any two elements of equation (1) and the third can be calculated using MIMO commands.
Xn(f) (abbreviated Xn) is the Fourier Spectrum of the response time-history, xn(t), at the nth response degree-of-freedom (DOF).
In this note we will calculate elements of the FRF matrix, given force inputs and response motions. Some realworld bridge test data will be used as our example.
{F(f)}Mx1 is a vector containing M forces.
In the process of calculating FRFs, you can specify averaging parameters, as well as calculate other functions such as Coherence and Auto Spectra. When hardware limitations require that you acquire data in a sequence of multiple Measurements Sets, ME’scope processes the Measurement Sets automatically, one at a time. All of these issues are discussed in this note.
{Fm(f)} (abbreviated Fm) is the Fourier Spectrum of the exciting force time-history, Fm(t), applied to the mth excitation DOF. [H(f)]NxM is a rectangular matrix of FRFs. Hnm(f) (abbreviated Hnm) is the FRF relating the motion at the nth response DOF to the force applied at the mth excitation DOF.
Steps in the application note can be duplicated using VT-550 Visual Modal Pro or any package that includes option VES-350 Advanced Signal Processing.
Z24 Bridge viewed from Bern-to-Zurich highway A1.
Page 1 of 7
App Note #20
www.vibetech.com
2/25/2014
BRIDGE DATA The example data is contained under the More Examples folder on the ME’scope installation CD. To access these files: Open ME’scope. Execute: File | Project | Open
Select Bridge 2 Shaker.PRJ from the More Examples subdirectory.
This will open the structure file, Z24 Bridge.STR, and the Data Block file, Z24 Bridge 2 Shaker Test Time Data.BLk. These files contain data from a random forced-vibration test that was conducted on the bridge using two hydraulic shakers.
Installing one of the two hydraulic shakers used in test. Deck of Z24 during tests (replacement bridge adjacent).
Close-up of highly sensitive seismic accelerometer. Installing a group of roving seismic accelerometers.
Page 2 of 7
App Note #20
www.vibetech.com
The two shakers were of different force capacities. The larger unit was mounted to drive DOF :1Z while the smaller shaker drove DOF :2Z on the center section, as shown below. The shaker positions remained fixed throughout the test and both shakers operated simultaneously.
2/25/2014
Bridge response motions were measured at 75 DOFs, using seismic accelerometers. The data was collected in nine sequentially acquired Measurement Sets. The accelerometers were moved to different DOFs for each Measurement Set. Every Measurement Set contained the two force signals at Reference DOFs :1Z and :2Z, plus acceleration at DOFs 1Z, 2Z and 2Y. Each Measurement Set also contained acceleration responses from additional unique Roving DOFs. Hence, the acceleration of 72 of the DOFs were each measured once, while the forces and accelerations at 3 DOFs were measured nine times. The repeated measurements allow us to examine the consistency among the 9 Measurement Sets. Since the repeated measurements contain both forces and motions at the same DOFs, each Measurement Set can also be tested for structural reciprocity. ď‚
Close the Z24 Bridge.STR window and open the Data Block window (shown below).
3D model showing shaker driving point locations. The shakers were driven with computer-generated white random noise spanning the 3 Hz to 30 Hz frequency range. The time domain signals and frequency spectra are shown below.
Z24 Bridge 2 Shaker Test Time Data.BLK Data Block. This Data Block contains 117 Time Waveforms. These include 72 unique accelerations plus 3 accelerations and 2 forces measured in each Measurement Set. Note that the force signals have the Units of N (Newtons), while the acceleration signals have Units of m/s^2 (meters/second2).
Typical Shaker Forces applied to DOFs :1Z and :2Z.
Each Trace has a DOF associated with it, which is listed in the Traces spreadsheet. The DOF consists of a sign, a Point number and a Direction letter followed by the Measurement Set number in braces. The colon (:) preceding each force DOF indicates it is a Reference (fixed) signal.
Page 3 of 7
App Note #20
www.vibetech.com
CALCULATING FRFs To calculate the FRFs between all response motions and the (2 in this case) Reference forces: Execute: Transform | MIMO | FRFs. The MIMO Analysis dialog will open.
2/25/2014
Each of the Time Waveforms in Z24 Bridge 2 Shaker Test Time Data.BLk has 65,536 samples of data in it. This permits a Spectrum Block Size of 32,768 spectral Lines, or the averaging of many spectra if a smaller Spectrum Block Size is chosen.
Enter 512 as the Spectrum Block Size.
Enter 128 as the Number of Averages.
Select Linear averaging.
Note that these selections will provide averaging with 50 Percent Overlap. This means that each spectrum will be calculated using 50% “new” data and 50% “old” data from the previous sampling window of time domain data.
In this case, both the Responses and Forces are contained in the same (Z24 Bridge 2 Shaker Test Time Data.BLk) Data Block. In general, these could be in two separate Data Blocks.
Select Hanning as the Time Domain Window. This helps reduce spectral leakage in the FRF calculations.
Refer to Application Note #1 for more information on leakage and time domain windowing functions.
Press the OK button. The MIMO Calculations dialog will open at the conclusion of the calculation.
Press OK. The Data Block Selection dialog will open.
Press the New File button. The New File dialog will open.
Notice that the Data Source contains only Time Waveforms. Alternatively, the forces and responses could be described using Auto & Cross Spectra. Since we are analyzing continuous random signals, the default Free Run Triggering is appropriate.
Check Coherence under Also Calculate.
To begin the FRF calculation process:
Press the Calculate button. The Spectrum Averaging dialog will open.
Page 4 of 7
App Note #20
www.vibetech.com
Enter Z24 FRFs & Cohs as the file name and press OK. The Z24 FRFs & Cohs window will open.
Close the Z24 Bridge 2 Shaker Test Time Data.BLk window.
2/25/2014
Coloring all 2Z referenced Traces red. All of the FRFs and Partial Coherences (that contain the Reference DOF :2Z) are now red Traces, as shown in the figure on the left. The Z24 FRFs & Cohs.BLK window. Driving Point FRFs Z24 FRFs & Cohs.BLK contains 495 Traces: 198 FRFs - 144 unique FRFs and 6 redundant FRFs measured 9 times each (once per Measurement Set).
We will first examine the driving point FRFs. We can bring these to the top of the Traces Spreadsheet by using the Sort Traces command.
99 Multiple Coherences – one for each Response.
Execute: Edit | Sort Traces | By.
198 Partial Coherences– two for each Response.
In the Sort Traces dialog that opens:
Hold down the Control key and click on 1Z, then click on 2Z in the Select From list. This selects both DOFs.
Select Ascending and press the Sort button.
EXAMINING THE RESULTS To simplify graphic comparisons, the color of all :2Zreferenced Traces will be changed to red:
All Traces containing the Roving response DOF 1Z are now at the top of the spreadsheet, followed by those containing the Roving response 2Z.
Execute: Edit | Select Traces | By.
In the Select Traces dialog that opens, choose Reference DOF, select 2Z and press Select.
All of the Traces that contain the Reference DOF :2Z are now highlighted in the Select column of the Data Block Spreadsheet.
Double-click on the Color column header. In the Trace Color dialog that opens, select Single Color and click on OK.
In the resulting Color dialog, click on red and press OK.
Page 5 of 7
App Note #20
www.vibetech.com
2/25/2014
Close the Sort Traces dialog. Execute: Format | Overlay Traces.
Select the nine 1Z:1Z and the nine 2Z:2Z driving point FRFs from Measurement Sets [1] through [9]. Execute: Format | Horizontal Axis. In Display Limits set Starting Value to 3 and Span to 27 to match the 3-30 Hz excitation frequency range of the tests. Execute: Display | Spreadsheet to hide the Traces Spreadsheet. Overlay plot of nine 2Z:1Z and nine 1Z:2Z FRFs The overlay plot (above) of the Z2:Z1 and Z1:Z2 cross FRFs illustrates excellent reciprocity between the two shaker locations. This symmetry indicates the bridge is behaving in a symmetric manner, which is consistent with another of the basic assumptions of a linear system that is made in modal analysis. Coherence Check We can also examine the Multiple and Partial Coherence Functions to verify the validity of our measurements. Execute: Format | Rows/Columns. Select the 3,1 format.
Comparison of nine estimates of two driving point FRFs.
Note that the Z2:Z2 FRFs (smaller shaker on the center span) are nearly identical. The differences in the nine Z1:Z1 FRFs are basically small gain changes. These driving-point FRFs verify that the nine Measurement Sets were made on a stationary linear system.
Raise the vertical scroll bar to its top-most position. Measurements M#1, M#2 and M#3 will be displayed as shown below.
Reciprocity Check Now let’s look at the structural reciprocity exhibited by the duplicate cross measurements in the nine Measurement Sets. Execute: Display | Spreadsheet to open the Spreadsheet.
Unselect the 18 driving-point FRFs just displayed.
Select the nine 2Z:1Z and the nine 1Z:2Z cross FRFs from Measurement Sets [1] through [9]. 2 FRFs and Multiple Coherence for Response DOF 1Z.
Page 6 of 7
App Note #20
www.vibetech.com
The Multiple Coherence Function (in the previous figure) answers the question: “How much of the response measured at DOF 1Z is linearly related to the two measured forces?” This is basically the same question answered by an ordinary Coherence function, when there is only one excitation force.
2/25/2014
sponse at 1Z is predominantly caused by the force applied at DOF :1Z. It is interesting to compare this result with the Response at DOF 2Z.
Like the ordinary Coherence, the Multiple Coherence Function is limited between 0 and 1. A value of 1 says the linearity is perfect; a value of 0 says the forces and the response are unrelated.
Move the vertical scroll bar down so that Traces M#48, M#49 and M#50 are displayed, as shown below.
Multiple & Partial Coherences for Response at 2Z. Multiple Coherence and :1Z & :2Z Partial Coherences.
Move the vertical scroll bar down so that Traces M#3, M#4, and M#5 are displayed together
The Partial Coherence Functions (the lower two Traces shown above), answer a different question: “How much of the Response measured at 1Z is due to the force at :1Z and how much is due to the force at :2Z?” Since this is a two-input case, there are two Partial Coherence functions for every Response. The Partial Coherence Function is also bounded between 0 and 1. At frequencies where its force is linearly related to the Response, the Partial Coherence approaches 1. At frequencies where the force and Response are not related, it approaches 0. The sum of all M Partial Coherence Functions for any single Response DOF should approach 1 or the value of the Multiple Coherence.
Note from the Partial Coherence Functions, that the dominant source of motion at DOF 2Z is not the shaker attached at the same location. It is the shaker located 20 meters away at :1Z.
SAVING YOUR PROJECT You will need the FRFs calculated in this Application Note for the calculations in Application Notes 21 and 22 and 23.
Execute: File | Project| Save As to save your work as a new project. The Save Project As dialog will open. Press Save All and enter My Z24 Bridge as the File name.
Hence, the preceding two figures tell us the two FRFs do a good job of “modeling” the Response at 1Z due to excitations at :1Z and :2Z except near the first resonance peak (3.9 Hz) where the Multiple Coherence dips to a value of 0.429. The two Partial Coherences show that the Re-
Page 7 of 7
Execute: File | Project | Close.