System Theory- Based Iden2fica2on of Dynamical Models and Applica2ons

Transcription

1 System Theory- Based Iden2fica2on of Dynamical Models and Applica2ons K. C. Park Center for Aerospace Structures Department of Aerospace Engineering Sciences University of Colorado at Boulder, CO, USA and WCU Visiting Professor Division of Ocean Systems Engineering School of Mechanical Engineering KAIST, Daejeon, Korea

2 Road Map of Structural System Iden2fica2on ERA: Eigensystem Realization Algorithm CBSI: Common-Basis Structural Identification

3 Elements of Structural System Identification Experiment Design and/or Field Measurements Input signals or identification of excitation sources Sensor placement, signal/noise conditioning Determination of Frequency Response Functions (FRFs) and/or Impulse Response Functions (IRFs) from Input and Output (conditioned) Data Realization of State Space Model (A, B, C, D) from FRFS and/or IRFs: x(n+1) = A x(n) + B u(n) (1) y(n) = C x(n) + D u(n) (2) Derivation of Structural Modes, Mode Shapes and Damping Applications to Vibration, Noise and Damage Detection

4 Exciter (Input) Design and Sensor Placement Input must have a rich mix of frequency contents so that the output consists of an adequate range of response frequency components. The locations (positions) of sensors are critical to obtain the desired mode shapes. In addition, sensor types must correspond to the signal types. For example, on a cantilever beam, the strain sensors are placed near the clamp support, while the accelerometers are placed toward the tip. If higher mode shapes are needed, then more sensors need to be placed on or near the maximum modal amplitude locations. It is important to carry out FEM analysis to determine analytical modes and mode shapes!

5 Computation of Frequency Response Functions or Impulse Response Functions

6 Why do we need FRF or IRF? From the discrete state space model (1) and (2), by assuming x(n+1)= λ x(n) (3) the measured output y(n) can be expressed as y(n) = { C (λi A) -1 B + D } u(n) (4) Observe that if the input u(0) = δ(t), u(1) = = u(n) =0, then the output, which is the system impulse responses, is given by H = [h(0), h(1), h(2),, h(n)]= [D, CB, CAB,, CA (n-1) B] (5) which states that the left hand term is the measured quantities While the right hand term consists of model to be obtained: {Measured quantity} = {Model to be determined!}

7 Why do we need FRF or IRF? - Continued Each term in the analytical expression Y, H = {D, CB, CAB,, CA (n-1) B} is called the Markov parameters from which we wish to identify the model (A, B, C, D). In other words, system identification is a process, using the impulse response obtained from the measured signal H = [h(0), h(1), h(2),, h(n)] to determine the system model (A, B, C, D).

8 Why do we need FRF or IRF? - Continued In general, the input, U = [u(0), u(1), u(2),, u(n)], are not an impulse signal. Therefore, one must first obtain its impulse responses. The output, which is not the system impulse responses, is given by Y = [y(0), y(1), y(2),, y(n)] = {D, CB, CAB,, CA (n-1) B} U (6) where [u(0) u(1). n(n) ] [ 0 u(0). n(n-1) ] U = [ 0 0 u(0) n(n-2) ] (7) [ ] [ ] [ u(0) ]

9 Why do we need FRF or IRF? - Continued

10

11

12

13

14 Comments on FFT-based vs. Wavelet-based Determination of Impulse Response Functions Fourier bases span the en2re interval. Wavelets - basis func2ons give frequency info but are local in 2me.

15 Comparison of FFT vs. Wavelet for 1- DOF Example

16

17

18 System Realization Now that the impulse response (H) is obtained from the measured Data, how does one compute (A, B, C, D)? This is called Realization. In structural dynamics, among a plethora of algorithms, we will employ Eigensystem Realization Algorithm (ERA) due to Juang and Pappa. This is described next.

19 First, if there were no excita2on except the nonzero ini2al condi2on x(0), the corresponding output y for the successive discrete 2me steps, t = t, 2 t, 3 t,..., r t can be expressed as where matrix V p (mp n) is called an observability matrix.

20 Second, if we apply a train of excita2ons {u(0), u(1), u(2),..., u(s 1)}, the internal state variable vector xr can be wriyen as where W s is called a controllability matrix.

21 Third, One forms the following Hankel matrix

22 Details of Hankel Matrix:

23 Fourth, Carry Out Singular Decomposi2on of H ps (0):

24 Fi]h, Obtain V p and W s : Now we are ready to obtain (A, B, C) Construct H ps (1) = V p A W s Obtain generalized Inverse: Obtain A:

25

26 A Realization Example: Kalman s Problem

27 A single input/single output sequence given by Outut: y = {1,1,1,2,1,3,2,...} Input: u = {1,0,0,0,0,0,0,...}

28 Step 1: Construct Hankel matrix

29

30

31

32

33 Is a realization unique? The answer is no! There is a second way of realizing the above sequence. Instead of the singular decomposition employed, one can invoke the well-known LU-decomposition as H33(0) is symmetric:

34 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. One can verify that this second realization, although the sequences of x are completely different, gives the the same output y.

35 Singular Values and Similarity Transformation The singular values of the two A matrices, A 1 and A 2, are the same as they should be: Λ = diag(1.2056, j, j) Of course, their bases are different, which are related by a similarity transformation such that φ 1 = Tφ 2 so that the two realizations are related according to (A 1, B 1, C 1 ) = (T 1 A 2 T, T 1 B 2, C 2 T)

36

37 Discrete Case Con2nuum Case

38

39 Finally, we obtain the con2nuous, modal state- space realiza2on as Ques2on: How does one relate the modal state- space model to physical structural modes and mode shapes?

40 Derivation of Structural Modal Equation

41 The continuous state-space realization model can be expressed with displacement sensing, for each of statespace i-th mode (not structural mode!), as: The normal- mode form of the governing structural dynamics equa2ons expressed in a state- space form is expressed as:

42 It turns out that two transformations are needed to get the structural modal form from realization model. First, the Macmillan transformation given by

43 The resulting equations are, after this transformation, given by

44 The second transformation, known as the Alvin transformation is given by

45 The final form is thus given as

46 Reduced-Order Physical Structural Dynamics Equations

47 Full DOF Model: where the subscripts m and i denote the measured and unmeasured DOFs, respec2vely.

48 Model Reduction Scheme:

49 Model Reduction Scheme - Concluded

50 Applications: Damage Detection

51 Datotsu Earthquake Simulation Facility, Japan

52

53 Scale Model of Nuclear Containment Vessel

54

55

56 Simplified Model of the Test Article

57

58 Damage Indication based on global nodal stiffness changes Damage Indication based on localised flexibility changes

59

60 Localization Example for Engine Support Structure Problem

61 Damage indication based on global flexibility changes Damage indication based localized flexibility changes

62 Earthquake Resistant Concrete Column Damage Test (Courtesy: Prof. Pardoen of UC-Irvine)

63

64

65 Damage is at second element from the bottom in a shearing mode

66 I- 40 BRIDGE PROJECT ( ) Performed vibraaon tests on undamaged and damaged I- 40 Bridge over Rio Grande prior to demoliaon. Performed extensive finite element analyses Began to formally study vibraaon based damage detecaon algorithms.

67 I- 40 Bridge: Test/Analysis CorrelaAon Undamaged Mode 1 F=2.59 Hz Final Damage Mode 1 F=2.45 Hz

68