ME scope Application Note 42 Multi-Input Multi-Output (MIMO) FRFs

ME scope Application Note 42 Multi-Input Multi-Output (MIMO) FRFs NOTE: This Application Note uses commands in the VES-3600 Advanced Signal Processing Check Help About to verify authorization of these options by your ME'scope license. Click here to download a Demo copy of the ME scope Project for this Application Note. What is a MIMO (Multi-Input Multi-Output) Model? Calculation of Transfer Functions, Outputs, and Inputs are all based upon use of a MIMO (Multi-Input Multi-Output) model of the dynamics of a structure. A MIMO model is a frequency domain model where Fourier spectra of multiple Inputs are multiplied by elements of a Transfer Function matrix to yield the Fourier Spectra of multiple Outputs. The MIMO model is expressed with the equation: where: {X(f)} = [H(f)] {F(f)} {F(f)} = Input Fourier spectra (m - vector). [H(f)] = Transfer Function matrix (n by m). {X(f)} = Output Fourier spectra (n - vector). m = number of Inputs. n = number of Outputs. f = frequency variable (radians per second). NOTE: Rows of the Transfer Function matrix correspond to Outputs, and columns correspond to Inputs. Each Input and Output corresponds to a measurement DOF (point & direction). Each Transfer Function is a Cross-channel function between two DOFs, an Input DOF and an Output DOF. Definitions Transfer Function A Transfer Function is defined as the Fourier spectrum of an Output divided by the Fourier spectrum of an Input. Frequency Response Function (FRF) An FRF is defined as the Fourier spectrum of a displacement, velocity, or acceleration response (Output) divided by the Fourier spectrum of the excitation force (Input) that caused the response. Transmissibility A Transmissibility is defined as the Fourier spectrum of an Output divided by the Fourier spectrum of an Input with the same units. Page 1 of 13

NOTE: An FRF and Transmissibility are special cases of a Transfer Function. MIMO Calculations Any one of the components of the MIMO model, (Inputs, Outputs, or Transfer Functions) can be calculated from the other two components using commands in the Transform MIMO menu. Inputs and Outputs can be either time or frequency domain M#s Transfer Functions can be either experimentally derived or synthesized from modal parameters The MIMO model is also used to calculate and display in animation a sinusoidal ODS due to multiple sinusoidal excitation forces at the same frequency. (See the Transform MIMO Sinusoidal ODS command description for details.) Frequency Response Function (FRF) Each FRF in a MIMO model defines the dynamic properties between a pair of DOFs of a structure, an Excitation DOF and a Response DOF. An FRF is defined as, FRF = (Fourier spectrum of a response / Fourier spectrum of an excitation force) The Excitation force is typically measured with a load cell. Response motion is typically measured with an acceleration, velocity, or displacement sensor. Experimental Modal Analysis In an EMA, FRF measurements are usually made under controlled conditions where the test article is artificially excited using a broadband excitation signal. In an impact test, the impactor applies a broad band impulsive force to the test article to excite many modes simultaneously In a shaker test, one or more shakers are attached to the test article and driven with uncorrelated broad band signals to excite many modes simultaneously Measuring Rows & Columns of the FRF Matrix NOTE: In an EMA, at least one row or column of the FRF matrix must be measured Columns of the FRF Matrix If a single excitation force is applied at the same fixed location and responses are measured at multiple points & directions, this corresponds to measuring elements from a single column of the FRF matrix. If multiple excitation forces are applied at multiple fixed locations and responses are measured at multiple points & directions, this corresponds to measuring elements from multiple columns of the FRF matrix. Rows of the FRF Matrix If a single response sensor is at a single fixed location and an excitation force is applied at multiple points & directions, this corresponds to measuring elements from a single row of the FRF matrix. If multiple response sensors at a multiple fixed locations and an excitation force is applied at multiple points & directions, this corresponds to measuring elements from multiple rows of the FRF matrix. 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. Page 2 of 13

NOTE: Each fixed sensor is called a Reference. Multiple Shaker Test Large structures with non-linear dynamic behavior are typically tested using multiple shakers, driven by pure or burst random excitation signals In a multiple shaker test, two or more (fixed) shakers are used to simultaneously excite the structure. 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. NOTE: 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 Illustrative Example FRFs can be calculated in two ways in ME'scope, 1. From the time waveforms of multiple excitation force and responses. 2. From the Auto spectra of the excitation forces, and the Cross spectra between the forces and the responses caused by them. Using Time Waveforms When FRFs are calculated from time waveforms, spectrum averaging is used to calculate estimates of Auto spectra of the excitation forces and the Cross spectra between all responses and excitation forces. Time domain windowing to reduce leakage, spectrum averaging to reduce extraneous noise, triggering, and overlap processing can be used during the calculation of these spectral estimates, (See Spectrum Averaging in the Signal Processing chapter for more details.) To illustrate the calculation of FRFs using time domain waveforms, Open the AppNote42.VTprj Demo Project file Open the BLK: Shaker Time Data file window. This file contains 117 time domain M#s. All of the data was not simultaneously acquired It was acquired in 9 Measurement Sets. All of the data in each Measurement Set was simultaneously acquired. Each Measurement Set contains 2 Force waveforms for the same 2 fixed Reference Inputs at DOFs 1Z & 2Z Each Measurement Set also contains the same 3 Roving response Outputs at DOFs 1Z, -2Y & 2Z There are 72 unique Roving response DOFs, distributed among the 9 Measurement Sets Page 3 of 13

A 2 Reference set of FRFs with 75 unique Roving response DOFs can be calculated from these waveforms. The FRFs are calculated using MIMO processing, as depicted in the diagram below. Editing the Force & Response M# Properties Block Diagram for Multiple Reference FRF Calculation. Notice that the Units column in the M#s spreadsheet in BLK: Shaker Time Data file is empty. NOTE: If sensor sensitivities and engineering units are not applied to force and response time domain M#s, the FRFs calculated from them will be un-calibrated, and therefore the modal parameters extracted from the FRFs are not a modal model, and cannot be used for SDM or MIMO calculations. Input Forces Execute M#s Sort Sort By in the BLK: Shaker Time Data window Select Input Output from the Sort By list in the dialog box that opens Select Input to place it in the Sort Using list on the right. Press Sort followed by the Close button Now, the M#s are ordered and measurements M#1 to M#18 are Inputs, followed by measurements M#19 to M#117.that are Outputs Select M#1 through M#18 Double click on the Units column header Select Lbf or N (whichever you prefer) in the Units dialog box, and click on OK Page 4 of 13

The 18 force M#s now have engineering units (Lbf or N), and are designated as Inputs to the MIMO model. Output Responses Right click on the Select column heading, and select Invert Selection from the menu Double click on the Units column header Select (g) in the Units dialog box,and click on OK Double click on the Select M# Column heading to un-select all M#s The 99 acceleration response M#s now have units of g's and are designated as Outputs of the MIMO model The forces and acceleration response time waveforms are now ready for MIMO processing. BLK: Shaker Time Data Window Showing Engineering Units & MIMO Inputs &Outputs. Calculating MIMO FRFs FRFs will be calculated using the multiple reference time domain waveforms in the BLK: Shaker Time Data window. The bridge was excited with 2 shakers simultaneously The shakers were driven by un-correlated broad band random signals with frequency spectra over the span of (3 to 30 Hz) The forces applied to the bridge and the acceleration responses due to the forces were simultaneously acquired in 9 Measurement Sets The Block Size of the Shaker Time Data.BLK file is 65536, (65536 samples per M#) The data was acquired at a 100 Hz sampling rate, (equivalent to a 0.01 second time increment between samples over a total time period of 655.35 seconds To verify these properties in the BLK: Shaker Time Data window, Right click in the graphics area and execute Data Block Properties from the menu FRFs will be calculated from this data with a Block Size of 2000 samples. This means that 4000 time domain samples are required to calculate each spectrum estimate. Spectrum averaging and overlap processing will be used to reduce extraneous noise in each final spectrum estimate Double click on the Select column in the M#s spreadsheet in the BLK: Shaker Time Data window until all M#s are un-selected. Execute Transform MIMO Transfer Functions. The following dialog box will open, Page 5 of 13

Make sure that BLK: Shaker Time Data is chosen in both the {Outputs} and {Inputs} source lists, as shown above. Check the Coherence box in the Include section. Click on the Calculate button. The Spectrum Averaging box will open. Make the following selections in the Spectrum Averaging dialog box, Spectrum Block Size: 2000 Number of Averages: 20 Linear Averaging Time Domain Window: Hanning Page 6 of 13

Notice that the Percent Overlap = 19%. This means that with each successive block of 4000 time domain samples used to calculate a spectral estimate contains 19% of the samples that were used in the previous block of samples. Hanning Window Because the Time Domain Window selection is Hanning, a Hanning window will be applied to each block of time domain samples before the FFT is applied to it. Use of the Hanning window is necessary to reduce the effects of leakage due to the non-periodicity of the random signals in each sampling window. (See Time Domain Windows in the Signal Processing Commands chapter for details.) Click on the OK button to calculate the FRFs & Coherences. When the calculations are completed, a dialog box will open allowing you to save the results. Click on the New File button Enter "MIMO FRFs" into the dialog box, and click on OK The BLK: MIMO FRFs window will open displaying the calculated FRFs and Coherences. A total of 495 M#s are contained in the Data Block, FRFs, Multiple Coherences and Partial Coherences. Removing Measurement Set Numbers MIMO FRFs NOTE: Measurement Set numbers are only required for performing FRF and Coherence calculations. After the calculations are completed, Measurement Sets are no longer required. To remove the Measurement Set numbers from the DOFs, Right click on the M#s spreadsheet and execute Edit Select M#s Select None to un-select all M#s Double click on the DOFs column header in the M#s spreadsheet In the DOF Generator dialog box (shown below), select Delete, check Measurement Set, and click on OK Notice that the [Measurement Set] has been removed from the DOFs of the M#s. Page 7 of 13

Sorting M#s to Group FRFs and Coherences Together Notice on the Title bar that the MIMO FRFs Data Block contains 495 M#s.of multiple Measurement Types. 198 FRFs (75 unique Roving DOFs & 24 duplicate Roving DOFs) times 2 reference DOFs 99 Multiple Coherences (75 unique Roving DOFs & 24 duplicate Roving DOFs 198 Partial Coherences (75 unique Roving DOFs & 24 duplicate Roving DOFs) times 2 reference DOFs The FRFs and Coherences are mixed together because each Measurement Set was calculated separately. They need to be sorted. Execute M#s Sort Sort By Choose Measurement Type from the Sort By list in the dialog box. Check Select All to place FRF, Multiple Coherence, Partial Coherence in the Sort Using list, as shown below Page 8 of 13

Press Sort to sort the M#s Choose Roving DOF from the Sort By drop down list. Check Select All to place all of the Roving DOFs in the Sort Using list on the right Press Sort followed by the Close button Deleting Duplicate FRFs & Coherences Notice that the first 18 M#s are nine duplicate copies of the same two FRFs with DOFs 1Z:1Z & 1Z:2Z. Select all 9 of the FRFs with DOFs 1Z:1Z Execute Format Overlay M#s to display the selected M#s. The are displayed below. From the overlaid FRFs (shown below) is clear that all nine FRFs for DOFs 1Z:1Z are duplicates on one another. Therefore, eight of the nine duplicate FRFs, as well as the Multiple & Partial Coherences for the duplicates, should be deleted. Multiple Coherence Overlaid FRFs for DOFs 1Z:1Z. Execute M#s Sort Sort By, and select Measurement Type from the Sort By list in the dialog box. Select Multiple Coherence to place it in the Sort Using list on the right. Check Renumber M#s Press Sort followed by the Close button Select M#1 through M#27, and un-select only three of them (with DOFs 1Z, -2Y, 2Z) Execute Edit Delete Selected M#s, and click on Yes in the next dialog box FRFs Execute M#s Sort Sort By, and select Measurement Type from the Sort By list in the dialog box. Select FRF to place it in the Sort Using list on the right. Press Sort followed by the Close button Page 9 of 13

Select M#1 through M#54, and un-select only six of them (with DOFs 1Z:2Z, 1Z:1Z, -2Y:2Z, - 2Y:1Z, 2Z:2Z, 2Z:1Z) Execute M#s Delete Selected, and click on Yes in the next dialog box Partial Coherence Repeat the steps above but Sort By Partial Coherence, to delete the duplicate Partial Coherences (with DOFs 1Z:2Z, 1Z:1Z, -2Y:2Z, -2Y:1Z, 2Z:2Z, 2Z:1Z) When all of the duplicate M#s have been deleted, the Data Block should contain 375 M#s, 5 M#s for each of 75 unique Roving DOFs. 2 FRFs, 1 Multiple Coherence, 2 Partial Coherences The M#s will be sorted again so that each group of 5 M#s consists of 2 FRFs, followed by 1 Multiple Coherence followed by 2 Partial Coherences Execute M#s Sort Sort By, and select Measurement Type from the Sort By list in the dialog box Select FRF, Multiple Coherence, Partial Coherence to place them in the Sort Using list Press Sort to sort the M#s Select Roving DOF from the Sort By list in the dialog box Check Select All to place all Roving DOFs in the Sort Using list, as shown below Press Sort followed by the Close button Multiple & Partial Coherence Partial Coherence shows the percentage contribution (0 to 1) of each shaker (Input) to each response (Output). Multiple Coherence shows the total contribution (0 to 1) of both Inputs to each Output Execute Format Strip Chart and choose 5 from the drop down list In the figure below, it is clear that the shaker at 1Z contributes more to the response at Roving DOF 1Z than the shaker at 2Z. Not only does FRF 1Z:2Z have more noise in it than FRF!Z:!Z, but the Partial Coherences clearly indicate this. Each pair of Partial Coherences shows how the two shakers contribute Page 10 of 13

to the overall response at each Roving DOF. The Multiple Coherence shows that the two shakers together caused most of the response at DOF 1Z in the frequency span (3 to 40Hz). The rest of the response is caused by extraneous noise. Animating Shapes from Multiple Reference FRFs MIMO FRF s, Multiple & Partial Coherences. A multiple reference set of FRFs is ideal for verifying that each resonance peak corresponds to a single mode of vibration. Since only the FRFs are needed for displaying shapes in animation, they will be copied to a separate Data Block. Execute M#s Select Select By in the BLK: MIMO FRFs window. Choose Measurement Type from the drop down list, and select FRF from the selection list. Press the Select button followed by the Close button Execute M#s Copy M#s to File, and click on No in the next dialog box Press the New File button in the dialog box that opens. Enter "MultiRef FRFs" into the dialog box and click on OK The BLK: Multi-Ref FRFs window will open containing 150 FRFs; 75 roving DOFs paired with 2 reference DOFs (1Z & 2Z). Page 11 of 13

Creating Measured M# Links To animate shapes directly from the FRFs, M# Links must be created for each Point & direction on the structure model where a response was actually measured. This is done by linking each M# in BLK: MultiRef FRFs to a DOF on the Bridge model. Execute Animate Create M# Links in the BLK: MultiRef FRFs window. Select STR: Z24 Bridge in the dialog box that opens. Click on OK in the dialog boxes that open. When the Measured M# Links have been created, the following dialog box will open. There are only 150 FRFs in the MultiRef FRFs Data Block so normally 150 M# Links would be created to link each FRF to a DOF of the structure. However, multiple Points on the railings of the bridge model where numbered with the same Point number. Therefore, a total of 218 M# Links were created. Creating Interpolated M# Links M# Links must also be created for all un-measured DOFs of the Bridge model. Otherwise, they won't move during shape animation. Execute M# Links Create Interpolated M# Links in the STR: Z24 Bridge window. Click on Yes and OK in the dialog boxes that open. When the Interpolated M# Links have been created, the following dialog box will open. Displaying ODS's Close all windows in ME'scope except BLK: MultiRef FRFs and STR: Z24 Bridge Execute Window Arrange Windows in the ME'scope window. Execute Draw Animate Shapes in the STR: Z24 Bridge window A dialog box will open saying that BLK: MultiRef FRFs has multiple references. Click on OK. Selecting a Reference Next, the M#s Select Select By dialog box will open where you must select M#s from one reference at a time. Page 12 of 13

Choose one of the references (1Z or 2Z) and click on the Select button Press the Imaginary part button in the BLK: MultiRef FRFs window. Execute Format Overlaid. Drag the (Line or Peak) cursor to a resonance peak to display the ODS at the resonance. Multiple Reference FRFs Verify a Single Resonance Choose reference 1Z in the M#s Select Select By dialog box, and click on the Select button NOTE: When the ODS at a resonance peak is displayed on a structure model from multi-reference FRFs, if the ODS does not change when a different reference is selected, this is evidence that the ODS is dominated by a single mode shape. Choose reference 2Z in the M#s Select Select By dialog box, and click on the Select button ODS from Reference 1Z Using the Peak Cursor. Page 13 of 13