Virtual Measurements in Experimental Structural Analysis

Transcription

1 in Experimental Structural Analysis Randall J. Allemang, PhD Structural Dynamics Research Lab University of Cincinnati Cincinnati, Ohio, USA COBEM97

2 - Introduction What are virtual Measurements? Virtual measurements are any measurements that are not directly derived from physical sensors Virtual measurements are developed from physical sensors via weighting and/or linear combinations (linear transformations) Virtual measurements are frequently used to reduce large physical data sets or to enhance characteristics within a physical data set Structural Dynamics Research Lab 2

3 Physical Measurements Physical measurements are generated as a function of placement of a physical sensor in a static/dynamic environment Physical measurements are difficult to interpret when multiple inputs/sources are generating the physical response of the sensor Structural Dynamics Research Lab 3

4 - Concept Virtual measurements utilize a linear transformation of physical sensors in order to preserve/enhance/clarify information in the original physical measurements The linear transformation is chosen depending upon the desired result, data reduction or data enhancement Structural Dynamics Research Lab 4

5 - Purpose Data Reduction - Reduce a large amount of redundant data to a manageable set (common in modal parameter estimation) Data Enhancement - Optimize the characteristic in the data relative to a specific mode of the structural system (modal filter) Structural Dynamics Research Lab 5

6 - History Virtual measurements have been commonly formulated to understand the underlying nature of multiple input, multiple output (MIMO) data acquisition/analysis situations Common examples are partial coherence, virtual coherence, principal force analysis Structural Dynamics Research Lab 6

7 - History Principal Force Example GFF = L NM GFF GFF... GFF GFF... GFF 11 1N N GFF i N N i i i O QP Structural Dynamics Research Lab 7

8 - History Principal Force Example Plot of eigenvalues GFF = V Λ V H Structural Dynamics Research Lab 8

9 - History Auto Power Spectrum of Forces H-Frame Example Structural Dynamics Research Lab 9

10 - History Principal Force Analysis H-Frame Example Structural Dynamics Research Lab 10

11 Linear Transformation: General Case - Reduce information in data set, commonly based upon eigenvalue decomposition (ED) or singular value decomposition (SVD). Special Case - Preserve information in data set relative to one or more modes of vibration, commonly based upon analytical or experimental modal vectors. Structural Dynamics Research Lab 11

12 General Linear Transformations Frequency Response Function Application H' = T H Structural Dynamics Research Lab 12

13 General Linear Transformations - FRF Eigenvalue Decomposition - Output DOF Space H H( ω) H( ω) = V Λ V N N N N N N N N o i s i s o N o N o o o N o N o H T v v... v... v Ne No lqlqlqn s = 1 2 k N e T Structural Dynamics Research Lab 13

14 General Linear Transformations - FRF Eigenvalue Decomposition - Input DOF Space H H( ω) H( ω) = V Λ V N N N N N N N N i o s o s i Ni Ni i i Ni Ni H T v v... v... v Ne Ni lqlqlqn s = 1 2 k N e T Structural Dynamics Research Lab 14

15 General Linear Transformations - FRF Singular Value Decomposition - Output DOF Space H = U Σ V N N N N N N N N N N o i s o o o o o i s H T u u... u... u Ne No lqlqlqn s = 1 2 k N e T Structural Dynamics Research Lab 15

16 General Linear Transformations - FRF Singular Value Decomposition - Input DOF Space H = U Σ V N N N N N N N N N N i o s i i i i i o s H T u u... u... u Ne Ni lqlqlqn s = 1 2 k N e T Structural Dynamics Research Lab 16

17 Typical Physical Measurements (280) Automotive Example Structural Dynamics Research Lab 17

18 Typical (20) Automotive Example Structural Dynamics Research Lab 18

19 Specialized Linear Transformations Reciprocal Modal Vector - Modal Filter Concept Enhance characteristics particular to each mode of vibration Requires an estimate of each mode of vibration Generates a modal coordinate Valid with time or frequency domain data Structural Dynamics Research Lab 19

20 Specialized Linear Transformations Reciprocal Modal Vector - Modal Filter Concept η r =l qlq T φ x r T lqlq= φ ψ r s R S 1, r T0, r = s s Structural Dynamics Research Lab 20

21 Specialized Linear Transformations Reciprocal Modal Vector - Method 1 Φ T Ψ = I Φ T = Ψ 1 Structural Dynamics Research Lab 21

22 Specialized Linear Transformations Reciprocal Modal Vector - Method 1 Ψ T M Ψ = I Φ T Ψ T M Structural Dynamics Research Lab 22

23 Specialized Linear Transformations Reciprocal Modal Vector - Method 2 l q N * * Qψ l q pr Q r rψ pr * H( ω) = ψ p r + mψ r * r ( jω λ ) ( jω λ ) r= 1 r lql q T Qψ lqm * * r pr T rψ pr φr H ω φ ψ r Q * ( ) = + p r r * ( jω λ ) ( jω λ ) r r r Structural Dynamics Research Lab 23

24 Modal Filter Estimates Reciprocal Modal Vector - Method 1 Reciprocal Modal Vector - Method 2 Analytical Modal Vectors and Mass Matrix Experimental Modal Vector Estimate SDOF - MDOF Parameter Estimation Complex Mode Indicator Function (CMIF) Structural Dynamics Research Lab 24

25 Complex Mode Indicator Function Plot of singular values H = U Σ V N N N N N N N N o i o i i i i i H Structural Dynamics Research Lab 25

26 Complex Mode Indicator Function Circular Plate Example Structural Dynamics Research Lab 26

27 Complex Mode Indicator Function Circular Plate Example Structural Dynamics Research Lab 27

28 Complex Mode Indicator Function Maxima in the primary and successive CMIF plots indicate the location of a modal frequency The singular vector associated with each maxima is a good estimate of the modal vector This estimate of the modal vector is used as a modal filter to isolate one modal coordinate in the enhanced frequency response function (efrf) Structural Dynamics Research Lab 28

29 Enhanced Frequency Response Function H pq ( ω) = 2 N r= 1 Q r ψ pr ψ qr ( jω λ ) r H pq ( ω) = 2 N r= 1 ψ pr Q r ψ qr ( jω λ ) r Structural Dynamics Research Lab 29

30 Enhanced Frequency Response Function efrf r ( ω) = Q r ψ qr ( jω λ ) r efrf r lql qlq 2 N T T lq r r s s= 1 ( ω) = φ H( ω) = φ ψ Q ψ s qs ( jω λ ) s Structural Dynamics Research Lab 30

31 Enhanced Frequency Response Function Circular Plate Example Structural Dynamics Research Lab 31

32 Enhanced Frequency Response Function Circular Plate Example Structural Dynamics Research Lab 32

33 Modal Filter, Time Domain - Example Space Truss Application Six (6) Input Actuators Thirty-Seven (37) Response Sensors Eight (8) Modal Filters Structural Dynamics Research Lab 33

34 Modal Filter, Time Domain - Example Physical Measurements Structural Dynamics Research Lab 34

35 Modal Filter, Time Domain - Example Physical Measurements Structural Dynamics Research Lab 35

36 Modal Filter, Time Domain - Example Structural Dynamics Research Lab 36

37 Modal Filter, Time Domain - Example Structural Dynamics Research Lab 37

38 Kalman Filtered Order Tracking - Example Structural Dynamics Research Lab 38

39 Kalman Filtered Order Tracking - Example Structural Dynamics Research Lab 39

40 Summary/Conclusions Virtual measurements are powerful tools which can be useful in understanding experimental structure analysis data Virtual measurements depend upon data set arrays with sufficient spatial information and linear, time invariant characteristics Virtual measurements yield reasonable solutions to practical problems Structural Dynamics Research Lab 40

41 Future Applications Since virtual measurements can be generated quickly as long as the linear transformations are known a priori, there are a number of interesting real time applications Flight flutter parameter estimation Multi-axis sensors (load, acceleration) Rigid and flexible body control Structural Dynamics Research Lab 41