Lectures on. Engineering System Identification ( Lund University, Spring 2015)

Transcription

1 Lectures on Engineering System Identification ( Lund University, Spring 2015) Lecture 2: Elements of System Identification, Identification Process and Examples!!! Instructor: K. C. Park, University of Colorado at Boulder!

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3 Why do we need a good analytical model? A good analytical model can provide the experimentalist with instrumentation needs, e.g., how many sensors and where to mount them, the location(s) and levels of excitations, anticipated response records( frequency ranges, and response magnitudes), among others. No one model can serve for different applications: Is it for vibration/acoustic control? For damage detection? Or design optimization? Etc.. Eventually, it is meaningful to obtain experimentally verified models for the engineers to utilize them for design improvements, performance evaluation, cost reduction, and certification for codes and standards.

4 Preliminary test runs (from a single to more than dozens runs) 1. Check each output to examine the sampling rates set from the analytical model is adequate; 2. Readjust the filter bands (roll-off ranges, etc.) if necessary; 3. Readjust the signal conditioners for each output line if necessary; 4. Check all the lines from the sensors and exciters to the A/D convertor lines that are fed into the computer; 5. Obtain the auto and cross correlation plots and examine carefully whether the frequency contents as well as the anti-resonances are adequately detected; 6. Now, make sure to minimize the temperature variations and environmental vibrations during experiments; 7. Run test, but if at all possible, repeat several runs and obtain preliminary (input-output ratio) transfer functions or preliminary impulse response functions; 8. If necessary, adjust sensor locations and repeat Steps 1-7.

5 Common overlook by novice experimentalists (including myself) 1. Inadequate pre-testing modeling; 2. Wrong choices of excitations; 3. Wrong choices of sensors and their locations; 4. Mal-attachments of sensors and actuators; 5. Assuming textbook boundary conditions; 6. Inconsistencies of units employed; 7. Ignoring noises and other uncertainties;

6 Of a plethora of system identification techniques, which one(s) should I choose? 1. Know the system characteristics that you are trying to identify (This is where a good analytical model and subsequent analyses play key roles!) 2. Begin with a technique that you are most familiar. If you find that the one you are familiar with may not be adequate, then search for other techniques 3. Be mindful of the limitations of the chosen identification technique; no technique is free from any deficiencies. 4. Do not stretch too much to stick with a technique just because you are familiar with. A good engineer remains a good one by adapting himself/herself to new techniques available.

7 The following examples are provided to give you a glimpse of the levels of your learning experience as to what kind of problems you may be able to tackle after your exposure to the present lectures. We may undertake similar problems as part of our small assignments, notably for control problems.

8 Example: Damage Identification utilizing transmission zeros (G. Reich s Ph.D Thesis) Poles vs. Zeros poles zeros Poles used extensively for: System Identification Modal Analysis Control Design Input Tailoring of Resonators Zeros ignored except for: Control Applications: Dead-band avoidance Closed-loop Performance Indication Sensor/Actuator Pair Selection and Placement

9 Motivation Most model-based DD methodologies require a large number of sensors. Closely tied to modern modal testing practices. Analysis uses measured modes/mode shapes as starting point. Large matrix size, large computational cost. Present method may require fewer sensors at select locations. Localized identification, separate processing and observation of select locations Smaller matrix size, smaller computational cost.

10 DD/HM Related Research Global Local Flexibility/Modeshape Mayes [1992], Pandey [1991,1994], Toksoy [1994] Peterson, Doebling, Alvin [1998], Park, Reich, Alvin [1998] Updating Zimmerman & Kaouk [1994, 1995] Doebling [1996], Denoyer [1996], Hemez [1993] Control/Filter Lim [1995, 1996], Cobb & Leibst [1997], Slater & Shelley [1993] Kranock [1998] Transmission Zero Afolabi [1987] This thesis Flexibility/mode shapes - direct comparison of undamaged/ damaged values related to sensor/dof location Updating - correlations of analytical DOF to model changes Control/filter - minimize modal force errors though fictitious controller/filter Transmission zeros - comparison of FRF elements correlated to sensor/dof locations

11 Motivation, cont d. Global vs. Local, cont d. In the global case stiffness and mass properties are associated with nodes. globally coupled K g = " k 1 + k 2 k 2 0 % $ ' $ k 2 k 2 + k 3 k 3 ' # $ 0 k 3 k 3 & ' A node may have contributions from many elements. In the local case stiffness and mass properties are associated with elements. uncoupled K = " k % $ ' $ 0 k 2 k ' $ 0 k 2 k ' $ ' $ k 3 k 3 ' $ ' # k 3 k 3 & A local node has contributions only from the corresp. element.

12 Definition of Transmission Zeros y g ( s) = H g ( s) f g ( s) H g ( s) = ( K g + iωd g ω 2 M g ) 1 1 =K g y g ( s) : displacement output f g ( s) : force input h gi, j ( s) = cof K g i, j det K g i, cof K j g = ( 1) i+j j,i det K g K g j,i is the submatrix of K g without rows j and columns i i, zeros of h gi, j ( s) roots of cof K j g = 0

13 Transmission Zeros, cont d. Transmission zeros of the MIMO transfer function Antiresonances of the SISO element of the transfer function matrix

14 Partitioned vs. global flexibility $ ( K φ + iωd φ ω 2 M φ ) B ' φ + - & ) T, %& B φ 0 ().- φ ˆ /- 0 ˆ 1 - = + - f, ˆ φ.- 0 λ b / (4) ˆ φ = H φ ˆ f φ H φ = Z φ Z φ B φ ( B T φ Z φ B φ ) 1 B T φ Z φ Z φ = ( K φ + iωd φ ω 2 M φ ) 1 φ = Φ φ 1 Lu g L T Φ φ T f φ = f g H φ = Φ φ 1 LH g L T Φ φ T H g = ( K g + iωd g ω 2 M g ) 1

15 Roadmap

16 A Simple Example Global TZ [Afolabi, 1987] K g = " k 1 + k 2 k 2 0 % $ k 2 k 2 + k 3 k ' $ 3 ' # $ 0 k 3 k 3 &' M g = " m % $ 0 m 2 0 ' $ ' # $ 0 0 m 3 &' zeros of node 1: z g1,1 = k 2 k 3 + ( k 3 m 2 + k 2 m 3 + k 3 m 3 )s 2 + m 2 m 3 s 4 zeros of node 2: z g2,2 = k 1 k 3 + k 2 k 3 + ( k 3 m 1 + k 1 m 3 + k 2 m 3 )s 2 + m 1 m 3 s 4 Independent of k1 f(k1, k2, k3) f(k1, k2, k3) zeros of node 3: z g3,3 = k 1 k 2 + k 1 k 3 + k 2 k 3 + ( k 2 m 1 + k 3 m 1 + k 1 m 2 + k 2 m 2 )s 2 + m 1 m 2 s 4

17 Simple Example, cont d. Element-by-element (Localized) TZ SL = " 1 0 0% $ ' $ ' # $ 0 1 1& ' so that the localized TF matrix elements are: Independent of k1 Independent of k3 zeros of element 1: z ε1,1 = k 2 k 3 + ( k 3 m 2 + k 2 m 3 + k 3 m 3 )s 2 + m 2 m 3 s 4 Independent of k2 zeros of element 2: z ε2,2 = k 1 k 3 + ( k 3 m 1 + k 3 m 2 + k 1 m 3 + k 3 m 3 )s 2 + ( m 1 m 3 + m 2 m 3 )s 4 zeros of element 3: z ε3,3 = k 1 k 2 + ( k 2 m 1 + k 1 m 2 + k 2 m 2 + k 1 m 3 + k 2 m 3 )s 2 + ( m 1 m 2 + m 1 m 3 )s 4

18 Simple Example, cont d. Comparison of Global FRFs Collocated input/output sets Damage in element 2 h g1,1 h g2,2 h g3,3

19 Simple Example, cont d. Comparison of Localized FRFs Both zeros are invariant - damage occurs here. h ε1,1 h ε2,2 h ε3,3

20 Indeterminate Plate 12 elements, 3 DOF per node (1 displ. & 2 rotations) Strains can be either Mindlinor Kirchhoff-type Strain-displacement matrix limitations: n e e ε = n e n R where n ε e # strain DOF / element n e # elemental DOF / element n R e # RBM / element Also, S e must have full row rank.

21 Indeterminate Plate DD Results Global TZ Strain-basis TZ

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23 Ladder Limited Mode Problem 16 elements, 3 DOF per node (2 displ. & 1 rotation) Lowest 12 modes plus residual flexibility term identified due to burst-random input Cumulative TZ variation computed based on state-space form of modal values

24 Ladder Limited Mode Case, cont d. On the left, the full analytical spectrum is used to compute the TZ variation. On the right, only the 12 identified modes plus the residual is used to compute the TZ variation. Full Spectrum Limited Modes

25 NUPEC RCCV Experimental Test Scale model of a Reinforced Concrete Containment Vessel (RCCV) Vibration test histories due to base motion recorded before and after an internal pressure test (used to create damage typical for this application)

26 RCCV Experimental Test, cont d. 22 accelerometer locations chosen for system identification 8 modes identified below 120 Hz for both tests After realization, modes transformed to beam model for damage detection analysis

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28 RCCV Experimental Test, cont d.

29 References" " 1. Alvin, K. F., Robertson, A. N., Reich, G. W. and Park, K. C., Structural system identification: from reality to models, Computers & Structures, 81(2003), ! 2. Reich, G. W. and Park, K. C., A Theory for Strain-Based Structural System Identification, Journal of Applied Mechanics, 68(4), ! 3. Alvin, K. F. and Park, K. C., Extraction of Substructural Flexibilities from Global Frequencies and Mode Shapes, AIAA Journal, vol. 37, no.11, 1999, p ! 4. Park, K. C. and Reich, G. W., A Theory for Strain-Based Structural System Identification, in: Adaptive Structures and Technologies, ed. by N. W. Hagood and M. J. Attala, Technomic Pub., 1999, ! 5. Park, K. C. and Reich, G. W., A Procedure to Determine Accurate Rotations from Measured Strains and Displacements for system Identification, Proc. 17th International Modal Analysis Conference, 8-11 February 1999, Kissimmee, FL.! 6. Robertson, A. N. and Park, K. C., An Investigation of Time Efficiency in Wavelet-Based Markov Parameter Extraction Methods, Proc AIAA SDM Conference, Paper No. AIAA , April , Long Beach, CA.! 7. Robertson, A. N., Park, K. C. and Alvin, K. F., Extraction of Impulse Response Data via Wavelet Transform for Structural System Identification, ASME Journal of Vibrations and Acoustics, 120, No.1, January 1998, ! 8. Robertson, A. N., Park, K. C. and Alvin, K. F., Identification of Structural Dynamics Models Using Wavelet-Generated Impulse Response Data, ASME Journal of Vibrations and Acoustics, 120, No.1, January 1998, ! 9. Alvin, K. F., Park, K. C. and Peterson, L. D., Extraction of Undamped Normal Modes and Full Modal Damping Matrix from Complex Modal Parameters, AIAA Journal, Volume 35, Number 7, 1997, ! 10. G. W. Reich and K. C. Park, Localized system identification and structural health monitoring from vibration test data, Proc AIAA SDM Conference, Paper No. AIAA , April , Kissimmee, FL.! 11. Robertson, A. N., Park, K. C. and Alvin, K. F., MIMO System Structural System Identification: A Revisit via Wavelet Transform, Proc. the 14th Intnl. Modal Analysis Conf., Dearborn, MI, Feb ! 12. Alvin, K. F. and Park, K. C., A Second-Order Structural Identification Procedure via System Theory-Based Realization, AIAA Journal, 32(2), February 1994, ! 13. Alvin, K. F., Park, K. C. and Peterson, L. D., A Minimal-Order Experimental Component Mode Synthesis: New Results and Challenges, AIAA Journal, 33(8), August 1995, ! 14. Alvin, K. F., Peterson, L. D. and Park, K. C., Experimental Identification of Normal Modes and Damping in an Actively Controlled Structure, Proc AIAA SDM Conference, Paper No. AIAA , April , Hilton Head, SC.! 15. Alvin, K. F., Park, K. C. and Peterson, L. D., Consistent Model Reduction of Modal Parameters for Reduced-Order Active Control, AIAA Journal of Guidance, Control and Dynamics, 18(4), July-August 1995, ! 16. Alvin, K. F., Peterson, L. D. and Park, K. C., A Method for Determining Minimum-Order Mass and Stiffness Matrices from Modal Test Data, AIAA Journal, 33(1), January 1995, !!