TRACKING CONTINUOUSLY SCANNING LASER DOPPLER VIBROMETER SYSTEMS AND METHODS

Abstract

A one-dimensional (1D) and two-dimensional (2D) scan scheme for a tracking continuously scanning laser Doppler vibrometer (CSLDV) system to scan the whole surface of a rotating structure excited by a random force. A tracking CSLDV system tracks a rotating structure and sweep its laser spot on its surface. The measured response of the structure using the scan scheme of the tracking CSLDV system is considered as the response of the whole surface of the structure subject to random excitation. The measured response can be processed by operational modal analysis (OMA) methods (e.g., an improved lifting method, an improved demodulation method, an improved 2D demodulation method). Damped natural frequencies of the rotating structure are estimated from the fast Fourier transform of the measured response. Undamped full-field mode shapes are estimated by multiplying the measured response using sinusoids whose frequencies are estimated damped natural frequencies.

Claims

1. A modal analysis method comprising: measuring a response of a rotating structure subject to random excitation with a tracking continuous scanning laser Doppler vibrometer system; determining a Fast Fourier Transform (FFT) of the response; applying a bandpass filter to the response with a passband that includes a damped natural frequency of the rotating structure to create a filtered response; determining a time interval between a minimum value and a maximum value of the filtered response; multiplying the filtered response in the time interval by sinusoidal signals to create a plurality of processed responses; and applying a lowpass filter to the plurality of processed responses to obtain an end-to-end undamped mode shape of the rotating structure.

2. The method of claim 1, wherein the sinusoidal signals include cos(ω.sub.d,it) and sin(ω.sub.d,it), where ω.sub.d,i is the damped natural frequency of the rotating structure.

3. The method of claim 1, wherein the tracking continuous scanning laser Doppler vibrometer system includes a camera, a scanner, and a single-point laser Doppler vibrometer.

4. The method of claim 1, wherein the rotating structure is rotating at a non-constant speed.

5. The method of claim 1, wherein the rotating structure is rotating at a constant speed.

6. The method of claim 1, wherein the time interval is measured by the system from a first end of a scan path to a second end.

7. The method of claim 1, further including determining end-to-end undamped mode shapes of the structure.

8. The method of claim 1, further including determining a first normalized end-to-end undamped mode shapes, a second normalized end-to-end undamped mode shape, and/or a third normalized end-to-end undamped mode shape of the rotating structure.

9. The method of claim 1, further including determining a first damped natural frequency, a second damped natural frequency, and/or a third damped natural frequency of the rotating structure.

10. The method of claim 1, wherein measuring the response of the rotating structure includes identifying a mark on the rotating structure.

11. The method of claim 10, wherein the mark is a black circular mark.

12. The method of claim 1, wherein measuring includes scanning along a two-dimensional path on the rotating structure.

13. The method of claim 1, wherein the rotating structure is a wind turbine blade.

14. A modal analysis method comprising: measuring a response of a rotating structure subject to random excitation with a tracking continuous scanning laser Doppler vibrometer system; interpolate positions of the response on a grid to generate a plurality of interpolated positions; rectifying the plurality of interpolated positions to create a plurality of rectified interpolated positions; identifying a plurality of zero-crossings from the plurality of rectified interpolated positions; determine a portion of the plurality of zero-crossings with a time increment; and interpolate and lift measurements at the portion of the plurality of zero-crossings.

15. The method of claim 14, wherein rectifying the plurality of interpolated positions includes determining negative absolute values of differences between the plurality of interpolated positions and a position of a virtual measurement point on a scan path.

16. The method of claim 14, wherein the time increment is equal to the inverse of a scan frequency.

17. The method of claim 14, wherein the tracking continuous scanning laser Doppler vibrometer system includes a camera, a scanner, and a single-point laser Doppler vibrometer.

18. The method of claim 17, wherein the method includes capturing images of the rotating structure.

19. The method of claim 14, further including determining a damped natural frequency, a damping ratio, and/or an undamped mode shape of the rotating structure.

20. The method of claim 14, wherein the rotating structure is a wind turbine blade.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The accompanying figures are provided by way of illustration and not by way of limitation.

[0042] FIG. 1 is a schematic of a rotating fan blade.

[0043] FIG. 2A is graph of simulate mirror signals for a TCSLDV system, specifically, the X-mirror signal for scanning a stationary structure.

[0044] FIG. 2B is X- and Y-mirror signals for scanning rotating structure.

[0045] FIG. 2C is the processed mirror signal by combining mirror signals in FIG. 2B with a simulated measured response.

[0046] FIG. 3 is a schematic of a rotating structure.

[0047] FIG. 4A is a graph of simulator mirror signals of a CSLDV system, specifically, the X-mirror signal for scanning a stationary structure.

[0048] FIG. 4B is X- and Y-mirror signals for scanning a rotating structure.

[0049] FIG. 4C is the processed mirror signal by combining mirror signals in FIG. 4B.

[0050] FIG. 5 is a schematic of a nonuniform rotating plate.

[0051] FIG. 6 is a schematic of a zigzag scan path on a rotating plate.

[0052] FIG. 7A is a picture of a TCSLDV system for rotating structure vibration measurement.

[0053] FIG. 7B is a schematic of a TCSLDV system.

[0054] FIG. 7C is a picture of a fan with a blade having a mark that is tracked by the camera.

[0055] FIG. 8A is a schematic of rotating structure vibration measurement using a TCSLDV system.

[0056] FIG. 8B illustrates an experimental setup for rotating structure vibration measurement using a TCSLDV system.

[0057] FIG. 9A is a graph of feedback signals from X- and Y-mirrors when R=15.41 rpm.

[0058] FIG. 9B is a graph of the measured response of the rotating fan blade under random excitation with R=15.41 rpm.

[0059] FIG. 9C is a graph of the processed mirror signals with R=15.41 rpm.

[0060] FIG. 10A is graph of the measured mark position when R=9.79 rpm.

[0061] FIG. 10B is a graph of the measured mark position when R=15.41 rpm.

[0062] FIG. 10C is a graph of the measured mark position when R=24.28 rpm.

[0063] FIG. 10D is a graph of the measured mark position with a non-constant rotation speed.

[0064] FIG. 10E is a graph of the measured rotation speeds of the fan blade with the constant and non-constant speeds based on the mark positions in FIGS. 10A-10D.

[0065] FIG. 11A is a graph of a FFT of the measured response of the rotating fan blade under random excitation when R=15.41 rpm.

[0066] FIG. 11B is a graph of the measured response in FIG. 11A that is filtered by a bandpass filter whose pass band is 27-28 Hz and processed mirror signal.

[0067] FIG. 11C is a graph of the FFT of the filtered measured response in FIG. 11A.

[0068] FIG. 11D is a graph of the second normalized mode shape obtained from the filtered measured response and processed mirror signal in FIG. 11B.

[0069] FIG. 12A is graph of the first normalized end-to-end undamped mode shape of the stationary fan blade.

[0070] FIG. 12B is a graph of the second normalized end-to-end undamped mode shape of the stationary fan blade.

[0071] FIG. 12C is a graph of the third normalized end-to-end undamped mode shape of the stationary fan blade.

[0072] FIG. 13A is a graph of the first normalized end-to-end undamped mode shape of a rotating fan blade with different rotation speeds.

[0073] FIG. 13B is a graph of the second normalized end-to-end undamped mode shape of a rotating fan blade with different rotation speeds.

[0074] FIG. 13C is a graph of the third normalized end-to-end undamped mode shape of a rotating fan blade with different rotation speeds.

[0075] FIG. 14 is a schematic of rotating structure vibration measurement using a TCSLDV system.

[0076] FIG. 15A is a graph of feedback signals for an X- and Y-mirror.

[0077] FIG. 15B is a graph of the measured response of the rotating fan blade under ambient excitation.

[0078] FIG. 15C is a graph of a processed mirror signal.

[0079] FIG. 15D is a graph of measured rotation speeds of the fan blade with a constant or non-constant rotation speed.

[0080] FIG. 15E is a graph of measured X and Y positions of the mark when the blade rotates.

[0081] FIG. 16A is a graph of laser spot position x/L from t=0 s to t=0.4 s with the horizontal dashed line indicating the position x/L=0.75.

[0082] FIG. 16B is a graph of interpolated laser spot positions with dots indicating instants when the laser spot arrived at x/L=0.75.

[0083] FIG. 16C is a graph of negatively rectified interpolated laser spot positions with dots indicating instants when the laser spot arrived at x/L=0.75.

[0084] FIG. 16D is a graph of measured response of the fan blade under ambient excitation with dots indicating lifted measurements at x/L=0.75.

[0085] FIG. 16E is a graph of lifted measurement at x/L=0.75 from t=0 s to t=25 s.

[0086] FIG. 17A is a graph of the correlation function between lifted measurement at x/L=0.5 and reference measurement at x/L=0.75 with non-negative time delays.

[0087] FIG. 17B is a graph of the amplitude of the power spectrum associated with the correlation function in FIG. 17A.

[0088] FIG. 17C is a graph of the phase of the power spectrum associated with the correlation function in FIG. 17A.

[0089] FIG. 18A is a graph of first normalized undamped mode shapes of the stationary fan blade.

[0090] FIG. 18B is a graph of first normalized undamped mode shapes of the stationary fan blade and the rotating fan blade with three different constant speeds.

[0091] FIG. 18C is a graph of operating deflection shapes (ODSs) of the rotating fan blade with three different constant speeds and a non-constant speed.

[0092] FIG. 19A is a picture of a tracking CSLDV system.

[0093] FIG. 19B is a picture of a fan.

[0094] FIG. 20A is a picture of an experimental setup for scanning a rotating fan blade.

[0095] FIG. 20B is a schematic of an experimental setup for scanning a rotating fan blade.

[0096] FIG. 20C is a schematic of scanning on a surface of the fan blade.

[0097] FIG. 21A is a graph of measured response of a rotating fan blade subject to random excitation with R=15.34 rpm.

[0098] FIG. 21B is a graph of X- and Y-mirror feedback signals with R=15.34 rpm.

[0099] FIG. 21C is a graph of processed mirror signals with R=15.34 rpm.

[0100] FIG. 22A is a graph of measured mark position when the fan blade is stationary.

[0101] FIG. 22B is a graph of measured mark position when R=9.71 rpm.

[0102] FIG. 22C is a graph of measured mark position when R=15.34 rpm.

[0103] FIG. 22D is a graph of measured mark position when R=22.05 rpm.

[0104] FIG. 22E is a graph of measured speeds of the rotating fan blade with different speeds based on mark positions in FIGS. 22A-22D.

[0105] FIG. 23A is a graph of estimated first normalized undamped full-field mode shape of the stationary fan blade using the 2D scan scheme and OMA method.

[0106] FIG. 23B is a graph of estimated second normalized undamped full-field mode shaped of the stationary fan blade using 2D scan scheme and OMA method.

[0107] FIG. 24A is a graph of estimated first normalized undamped full-field mode shape of the rotating fan blade with R=9.71 rpm.

[0108] FIG. 24B is a graph of estimated first normalized undamped full-field mode shape of the rotating fan blade with R=15.34 rpm.

[0109] FIG. 24C is a graph of estimated first normalized undamped full-field mode shape of the rotating fan blade with R=22.05 rpm.

[0110] FIG. 24D is a graph of estimated first normalized undamped mode shapes of the rotating fan blade on scan lines at the same position.

[0111] FIG. 24E is a schematic that illustrates positions of scan lines on which mode shapes in FIG. 24D and FIG. 25D are estimated.

[0112] FIG. 25A is a graph of estimated second normalized undamped full-field mode shape of the rotating fan blade with R=9.71 rpm.

[0113] FIG. 25B is a graph of estimated second normalized undamped full-field mode shape of the rotating fan blade with R=15.34 rpm.

[0114] FIG. 25C is a graph of estimated second normalized undamped full-field mode shape of the rotating fan blade with R=22.05 rpm.

[0115] FIG. 25D is a graph of estimated second normalized undamped mode shapes of the rotating fan blade on scan lines at the same position.

DETAILED DESCRIPTION

[0116] Section headings as used in this section and the entire disclosure herein are merely for organizational purposes and are not intended to be limiting.

[0117] All publications, patent applications, patents and other references mentioned herein are incorporated by reference in their entirety.

1. DEFINITIONS

[0118] The terms “comprise(s),” “include(s),” “having,” “has,” “can,” “contain(s),” and variants thereof, as used herein, are intended to be open-ended transitional phrases, terms, or words that do not preclude the possibility of additional acts or structures. The singular forms “a,” “and” and “the” include plural references unless the context clearly dictates otherwise. The present disclosure also contemplates other embodiments “comprising,” “consisting of” and “consisting essentially of,” the embodiments or elements presented herein, whether explicitly set forth or not.

[0119] For the recitation of numeric ranges herein, each intervening number there between with the same degree of precision is explicitly contemplated. For example, for the range of 6-9, the numbers 7 and 8 are contemplated in addition to 6 and 9, and for the range 6.0-7.0, the number 6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, and 7.0 are explicitly contemplated. Recitation of ranges of values herein are merely intended to serve as a shorthand method of referring individually to each separate value falling within the range, unless otherwise-Indicated herein, and each separate value is incorporated into the specification as if it were individually recited herein. For example, if a concentration range is stated as 1% to 50%, it is intended that values such as 2% to 40%, 10% to 30%, or 1% to 3%, etc., are expressly enumerated in this specification. These are only examples of what is specifically intended, and all possible combinations of numerical values between and including the lowest value and the highest value enumerated are to be considered to be expressly stated in this disclosure.

[0120] Unless otherwise defined herein, scientific and technical terms used in connection with the present disclosure shall have the meanings that are commonly understood by those of ordinary skill in the art. The meaning and scope of the terms should be clear; in the event, however of any latent ambiguity, definitions provided herein take precedent over any dictionary or extrinsic definition. Further, unless otherwise required by context, singular terms shall include pluralities and plural terms shall include the singular.

[0121] As used herein, the term “damped natural frequency” refers to the frequency at which a system with damping tends to oscillate in the absence of any driving forces.

2. DEMODULATION METHODOLOGY

[0122] Stanbridge and Ewins developed a demodulation method and a polynomial method for a CSLDV system to obtain operational deflection shapes (ODSs) of a structure under sinusoidal excitation. The demodulation method can be used to obtain ODSs of the structure by multiplying the response measured by the CSLDV system by sinusoid signals with the excitation frequency and applying a low-pass filter to the measured response multiplied by the sinusoidal signals. The polynomial method can be used to obtain ODSs of the structure by processing the discrete Fourier transform of the measured response. These two methods were applied to measurements of structures under impact and multi-sine excitations. These two methods are considered to be equivalent in measurement of the ODS if the structure is made of materials without mass and stiffness discontinuities and geometrically smooth. However, if the scan path of the laser spot passes over a damage on the structure, local anomaly caused by the damage can be shown in the ODS measured from the demodulation method while the local anomaly cannot be shown in the ODS measured from the polynomial method.

[0123] Chen et al. proposed a baseline-free method that combines the two methods to identify damage in beams under sinusoidal excitation, and the demodulation method was later used alone to identify damage in beams and plates. The demodulation method can only obtain ODSs of a structure under sinusoidal excitation. Based on the demodulation method, Xu et al. introduced a free response shape of a structure, which is a new type of vibration shapes. The free response shape corresponding to one mode of the structure can be used for baseline-free damage identification of the structure. A modal estimation method based on the concept of free response shape and demodulation method was developed to estimate modal parameters of the structure under impact excitation.

[0124] Nonuniform Rotating Euler-Bernoulli Beam Vibration Theory: With reference to FIG. 1, consider a fan blade that rotates about the z axis, which can be modeled as a nonuniform rotating Euler-Bernoulli beam with a length d attached to a rigid hub with a radius r. Two coordinate systems are considered: an inertial coordinate system O-XYZ, and a rotating coordinate system o-xyz whose origin o is at the center of the rigid hub, which is referred to as the rotation center; the z axis is parallel to the Z axis. The position of the origin O of the inertial coordinate system O-XYZ depends on the position of the camera of the TCSLDV system. Small transverse vibration of the beam along the z axis is considered and vibrations along x and y axes are neglected. The x axis passes through the rotation center and is tangent to the neutral axis of the beam at the point where the beam is attached to the hub.

[0125] The governing equation of the nonuniform rotating Euler-Bernoulli beam under a distributed random excitation force f(x, t) in the z direction and its associated boundary conditions are derived using the extended Hamilton's principle:

[00001]$\begin{array}{cc}\rho \left(x\right)u_{\mathrm{tt}}\left(x,t\right)+C\left[u_t\left(x,t\right)\right]+{\left[\mathrm{EI}\left(x\right)u_{xx}\left(x,t\right)\right]}_{xx}-u_{xx}\left(x,t\right){\dot{\theta }}^2\left(t\right){\int }_x^{r+d}\rho \left(\zeta \right)\zeta d\zeta +\rho \left(x\right)x{u}_x\left(x,t\right){\dot{\theta }}^2\left(t\right)=f\left(x,t\right),r\le x\le r+d,t>0& \left(1\right)\end{array}$$\begin{array}{cc}u\left(x,t\right)|x=r=u_x\left(x,t\right)|x=r=u_{xx}\left(x,t\right)|x=r+d={\left[\mathrm{EI}\left(x\right)u_{xx}\left(x,t\right)\right]}_x|x=r+d=0& \left(2\right)\end{array}$

where the subscripts denote partial differentiation, an overdot denotes a time derivative, x is the spatial position along the x axis, t is time, θ(t) is the angle between the x and X axes as shown FIG. 1, u is the transverse displacement of the beam at the position x and time t along the z axis, ρ(x) and EI(x) are the mass per unit length and flexural rigidity of the beam at x, respectively, and C(⋅) is the spatial damping operator. When {dot over (θ)}(t)=Ω is a constant, EQN. 1 can be written as

ρ(x)u.sub.tt(x,t)+C[u.sub.t(x,t)]+L[u(x,t)]=f(x,t),r≤x≤r+d,t>0  (3)

where L(⋅) is the spatial stiffness operator:

L[u]=[EI(x)u.sub.xx(x,t)].sub.xx−u.sub.xx(x,t)Ω.sup.2∫.sub.x.sup.r+dρ(ζ)ζdζ+ρ(x)xu.sub.x(x,t)Ω.sup.2,r≤x≤r+d,t>0  (4)

[0126] The term −u.sub.xx(x,t)Ω.sup.2∫.sub.x.sup.r+dρ(ζ)ζdζ+ρ(x)xu.sub.x(x,t)Ω.sup.2 in EQN. 4 is referred to as the centrifugal stiffening term due to rotation of the beam. The eigenvalue problem associated with the corresponding undamped rotating beam is

L[ϕ]=ω.sup.2ρ(x)ϕ  (5)

where ω is the undamped natural frequency of the beam, and ϕ is the corresponding mode shape or eigenfunction that satisfies boundary conditions corresponding to EQN. 2. Applying the boundary conditions for ϕ and integration by part, one has

[00002]$\begin{array}{cc}{\int }_r^{r+d}v\left(x\right)L\left[w\left(x\right)\right]\mathrm{dx}={\int }_r^{r+d}v\left(x\right)\left\{{\left[\mathrm{EI}\left(x\right)w_{xx}\left(x\right)\right]}_{xx}-w_{xx}\left(x\right){\Omega }^2{\int }_x^{r+d}\rho \left(\zeta \right)\zeta d\zeta +\rho \left(x\right)x{w}_x\left(x,t\right){\Omega }^2\right\}\mathrm{dx}={\int }_r^{r+d}\left[\mathrm{EI}\left(x\right)w_{xx}\left(x\right)v_{xx}\left(x\right)+{\Omega }^2{w}_x\left(x\right)v_x\left(x\right){\int }_x^{r+d}\rho \left(\zeta \right)\zeta d\zeta \right]\mathrm{dx}={\int }_r^{r+d}w\left(x\right)L\left[v\left(x\right)\right]\mathrm{dx},r\le x\le r+d,t>0& \left(6\right)\end{array}$

where v(x) and w(x) are two comparison functions that satisfy the boundary conditions for ϕ. Hence L(⋅) is self-adjoint. Let v(x)=w(x) in EQN. 6; one has

[00003]$\begin{array}{cc}{\int }_r^{r+d}w\left(x\right)L\left[w\left(x\right)\right]\mathrm{dx}={\int }_r^{r+d}\left[\mathrm{EI}\left(x\right)w_{xx}^2\left(x\right)+{\Omega }^2{w}_x^2\left(x\right){\int }_x^{r+d}\rho \left(\zeta \right)\zeta d\zeta \right]dx\ge 0,r\le x\le r+d,t>0& \left(7\right)\end{array}$

which means that L( ) is positive definite. One also has

∫.sub.r.sup.r+dv(x)ρ(x)w(x)dx=∫.sub.r.sup.r+dw(x)ρ(x)v(x)dx,r≤x≤r+d,t>0  (8)

∫.sub.r.sup.r+dw(x)ρ(x)w(x)dx=∫.sub.r.sup.r+dρ(x)w.sup.2(x)dx≥0,r≤x≤r+d,t>0  (9)

which mean that the spatial mass operator p(x) is self-adjoint and positive definite. By the expansion theorem, the solution to EQN. 3 can be expressed as

[00004]$\begin{array}{cc}u\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right)q_i\left(t\right)& \left(10\right)\end{array}$

where ϕ.sub.i(x) is the i-th eigenfunction of the corresponding undamped nonuniform rotating beam and q.sub.i(t) is the i-th generalized coordinate. Since the undamped nonuniform rotating beam is self-adjoint and L(⋅) and ρ(x) are positive definite, its eigenfunctions are real and can be normalized to satisfy the following orthonormality relations:

∫.sub.r.sup.r+dϕ.sub.i(x)ρ(x)ϕ.sub.j(x)dx=δ.sub.ij  (11)

∫.sub.r.sup.r+dϕ.sub.i(x)L[ϕ.sub.j(x)]dx=ω.sub.i.sup.2δ.sub.ij  (12)

where ω.sub.i is the i-th real undamped natural frequency of the rotating beam and δ.sub.ij is Kronecker delta that satisfies

[00005]$\begin{array}{cc}{\delta }_{ij}=\{\begin{array}{c}1,i=j\\ 0,i\ne j\end{array}& \left(13\right)\end{array}$

It is assumed that the spatial damping operator satisfies C(⋅)=k.sub.1L(⋅)+k.sub.2ρ(x), where k.sub.1 and k.sub.2 are two constants. Substituting EQN. 10 into EQN. 3, multiplying the resulting equation by ϕ.sub.j, integrating the resulting equation from x=r to x=r+d, and using EQN. 9 and EQN. 10 and the expression ∫.sub.r.sup.r+dϕ.sub.i(x)C[ϕ.sub.j(x)]dx=c.sub.iδ.sub.ij, where c.sub.i=k.sub.1ω.sub.i.sup.2+k.sub.2, yields

{umlaut over (q)}.sub.i(t)+c.sub.i{dot over (q)}.sub.i(t)+ω.sub.i.sup.2q.sub.i(t)=∫.sub.x.sup.r+dϕ.sub.i(x)f(x,t)dx  (14)

Considering an underdamped nonuniform rotating beam, one has c.sub.i=2ζ.sub.iω.sub.i with 0<ζ.sub.i<1

[00006]$\begin{array}{cc}{g}_i\left(t\right)=\frac{1}{{\omega }_{d,i}}{e}^{-{\zeta }_i{\omega }_{i}t}\sin \left({\omega }_{d,i}t\right)& \left(16\right)\end{array}$

being the i-th modal damping ratio of the beam. Since ω.sub.1≤ω.sub.i for i=1, 2, . . . , when 0<c.sub.i≤2ω.sub.1, all the modal damping ratios ζ.sub.i satisfy 0<ζ.sub.i<1. Assume that EQN. 1 has zero initial conditions; the solution to EQN. 14 can be expressed as

q.sub.i(t)=∫.sub.0.sup.t∫.sub.r.sup.r+dϕ.sub.i(x)f(x,t−τ)g.sub.i(τ)dxdτ  (15)

where
is the unit impulse response function of the beam corresponding to its i-th damped mode and ω.sub.d,i=ω.sub.i√{square root over (1−ζ.sub.i.sup.2)} is its i-th damped natural frequency. Substituting EQN. 15 into EQN. 10 yields

[00007]$\begin{array}{cc}u\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\int }_0^t{\int }_r^{r+d}{\varphi }_i\left(x\right)f\left(x,t-\tau \right)g_i\left(\tau \right)dxd\tau & \left(17\right)\end{array}$

If a concentrated force f.sub.a(t) is applied at a position x.sub.a on the beam, one has

f(x,t)=δ(x−x.sub.a)f.sub.a(t)  (18)

where δ is Dirac delta function. Substituting EQN. 18 into EQN. 17 yields

[00008]$\begin{array}{cc}u\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\varphi }_i\left(x_a\right){\int }_0^t{f}_a\left(t-\tau \right)g_i\left(\tau \right)d\tau & \left(19\right)\end{array}$

[0127] Improved Demodulation Method for TCSLDV Measurement of a Rotating Structure: virtual measurement points are assigned on the fan blade in FIG. 1 along a time-varying scan path D(t) when the TCSLDV system is used to periodically measure u by scanning its laser spot along the path D(t). The TCSLDV system registers discrete measurements of u with a finite sampling frequency F.sub.sa. The measured response of the TCSLDV system can be expressed by EQN. 19 when only one concentrated force acts on the blade. Substituting EQN. 16 into EQN. 19 and applying integration by part yield

[00009]$\begin{array}{cc}u\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\varphi }_i\left(x_a\right)\left\{-\frac{e^{-\zeta {\omega }_{i}t}{f}_a\left(0\right)}{{\omega }_{d,i}{\omega }_i^2}\left[\zeta {\omega }_i\sin \left({\omega }_{d,i}t\right)+{\omega }_{d,i}\cos \left({\omega }_{d,i}t\right)\right]+\frac{f_a\left(t\right)}{{\omega }_i^2}+{\int }_0^t\frac{e^{-\zeta {\omega }_{i}t}{f}_a^{\prime }\left(t-\tau \right)}{{\omega }_{d,i}{\omega }_i^2}\left[\zeta {\omega }_i\sin \left({\omega }_{d,i}\tau \right)+{\omega }_{d,i}\cos \left({\omega }_{d,i}\tau \right)\right]d\tau \right\}& \left(21\right)\end{array}$

EQN. 21 can be written as

[00010]$\begin{array}{cc}u\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\varphi }_i\left(x_a\right)\left[A_i\left(t\right)\cos \left({\omega }_{d,i}t\right)+B_i\left(t\right)\sin \left({\omega }_{d,i}t\right)+C_i\left(t\right)\right]& \left(22\right)\end{array}$

where A.sub.i(t), B.sub.i(t) and C.sub.i(t) are arbitrary functions of time related to the concentrated force f.sub.a(t) that is assumed to be a white-noise signal. Applying a bandpass filter with a passband that contains only one damped natural frequency of the rotating fan blade ω.sub.d,i to the measured response in EQN. 22, one has

[00011]$\begin{array}{cc}\begin{array}{c}{u}_i\left(x,t\right)={\Phi }_i\left(x\right)\cos \left({\omega }_{d,i}t-\alpha \right)\\ ={\Phi }_{I,i}\left(x\right)\cos \left({\omega }_{d,i}t\right)+{\Phi }_{Q,i}\left(x\right)\sin \left({\omega }_{d,i}t\right)\end{array}& \left(23\right)\end{array}$

where u.sub.i(x, t) is the signal obtained after u(x, t) is band-pass filtered, Φ.sub.i(x)=H.sub.iϕ.sub.i(x) in which H.sub.i is a scalar factor, Φ.sub.I,i(x)=Φ.sub.i(x)cos(ω.sub.d,it) and Φ.sub.Q,i(x)=Φ.sub.i(x)sin(ω.sub.d,it) are in-plane and quadrature components of Φ.sub.i(x), respectively, and α is a phase variable. EQN. 23 is multiplied by cos(ω.sub.d,it) and sin(ω.sub.d,it) to obtain Φ.sub.I,i(x) and Φ.sub.Q,i(x), respectively:

[00012]$\begin{array}{cc}\begin{array}{c}{u}_i\left(x,t\right)\cos \left({\omega }_{d,i}t\right)={\Phi }_{I,i}\left(x\right){\cos }^2\left({\omega }_{d,i}t\right)+{\Phi }_{Q,i}\left(x\right)\sin \left({\omega }_{d,i}t\right)\cos \left({\omega }_{d,i}t\right)\\ =\frac{1}{2}{\Phi }_{I,i}\left(x\right)+\frac{1}{2}{\Phi }_{I,i}\left(x\right)\cos \left(2{\omega }_{d,i}t\right)\\ +\frac{1}{2}{\Phi }_{Q,i}\left(x\right)\sin \left(2{\omega }_{d,i}t\right)\end{array}& \left(24\right)\end{array}$$\begin{array}{cc}\begin{array}{c}{u}_i\left(x,t\right)\sin \left({\omega }_{d,i}t\right)={\Phi }_{I,i}\left(x\right)\sin \left({\omega }_{d,i}t\right)\cos \left({\omega }_{d,i}t\right)+{\Phi }_{Q,i}\left(x\right){\sin }^2\left({\omega }_{d,i}t\right)\\ =\frac{1}{2}{\Phi }_{Q,i}\left(x\right)+\frac{1}{2}{\Phi }_{I,i}\left(x\right)\sin \left(2{\omega }_{d,i}t\right)\\ -\frac{1}{2}{\Phi }_{Q,i}\left(x\right)\cos \left(2{\omega }_{d,i}t\right)\end{array}& \left(25\right)\end{array}$

By applying a low-pass filter to u.sub.i(x, t)cos(ω.sub.d,it) and u.sub.i(x, t)sin(ω.sub.d,it),

[00013]$\frac{1}{2}{\Phi }_{I,i}\left(x\right)\cos \left(2{\omega }_{d,i}t\right),\frac{1}{2}{\Phi }_{Q,i}\left(x\right)\sin \left(2{\omega }_{d,i}t\right),\frac{1}{2}{\Phi }_{I,i}\left(x\right)\sin \left(2{\omega }_{d,i}t\right)$$\mathrm{and}-\frac{1}{2}{\Phi }_{Q,i}\left(x\right)\cos \left(2{\omega }_{d,i}t\right)$

in EQN. 24 and EQN. 25 can be eliminated so that Φ.sub.I,i(x) and Φ.sub.Q,i(x) can be obtained. When Φ.sub.I,i(x) and Φ.sub.Q,i(x) are obtained, Φ.sub.i(x) can be determined using the relation in EQN. 23 and the i-th normalized mode shape of the undamped rotating fan blade can be estimated by dividing Φ.sub.i(x) by its maximum value.

[0128] Method for Processing Mirror Signals of the TCSLDV System for Scanning a Rotating Structure: A scan path D(t) is usually a straight line on a rotating structure, which is the case considered here. If a CSLDV system is used to scan along a stationary structure and positions of the laser spot can be considered to be linearly related to the rotation angle of a mirror (e.g., the X-mirror) in the scanner in the CSLDV, only the X-mirror is needed to complete the scan and the mirror signal can be directly used as the position of the laser spot on the scan path D(t). However, both X- and Y-mirrors are needed to scan along a rotating structure such as the fan blade here. Thus use of only one mirror signal is not enough to describe the position of the laser spot on the scan path D(t). A simulated X-mirror signal for scanning a stationary structure is shown in FIG. 2A. Simulated X- and Y-mirror signals for scanning a rotating structure with a constant speed are shown in FIG. 2B. To obtain end-to-end mode shapes of the rotating structure with the constant speed, mirror signals that are similar to the one shown in FIG. 2A are needed to determine the position of the laser spot. A method is developed to combine X- and Y-mirror signals to describe the position of a laser spot on the scan path D(t) on the rotating structure.

[0129] In some embodiments, a mark is attached to a structure that rotates with a constant speed to identify positions of the structure and scan path at any time instant FIG. 1. Let the position of the rotation center relative to the inertial coordinate system O-XYZ be (a.sub.0, b.sub.0), the position of the mark be (a.sub.m(t), b.sub.m(t)), and the position of the laser spot be (a.sub.s(t), b.sub.s(t)); one has

M(t)=(a.sub.m(t)−a.sub.0,b.sub.m(t)−b.sub.0).sup.T=(d.sub.m cos(Ωt),d.sub.m sin(Ωt)).sup.T  (26)

P(t)=(a.sub.s(t)−a.sub.0,b.sub.s(t)−b.sub.0).sup.T=(d.sub.s cos(Ωt),d.sub.s sin(Ωt)).sup.T  (27)

where M(t) and P(t) are position vectors of the mark and laser spot, respectively, d.sub.m and d.sub.s(t) are distances between the rotation center and mark and between the rotation center and laser spot at time t that are shown in FIG. 1, respectively, and the superscript T denotes transpose of a vector. Note that d.sub.s(t) can be used to describe the position of the laser spot on the scan path D(t). Let M(t.sub.1) and M(t.sub.2) be position vectors of the mark at t=t.sub.1 and t=t.sub.2, respectively; the angle γ between M(t.sub.i) and M(t.sub.2) can be obtained by

[00014]$\begin{array}{cc}\gamma =a\cos \left(\frac{M\left(t_1\right).Math.M\left(t_2\right)}{.Math.M\left(t_1\right).Math..Math..Math.M\left(t_2\right).Math.}\right)& \left(28\right)\end{array}$

Therefore, the rotation speed of the structure R in revolutions per minute (rpm) can be written as

[00015]$\begin{array}{cc}R=\frac{30\gamma }{\pi \left(t_2-t_1\right)}& \left(29\right)\end{array}$

The adjusted position vector of the mark is

M.sup.a=((a.sub.m−a.sub.0)cos(β)+(b.sub.m−b.sub.0)sin(β),(a.sub.m−a.sub.0)sin(β)+(b.sub.m−b.sub.0)cos(β)).sup.T  (30)

where β is a phase variable that can be used to adjust the position of the mark in the inertial coordinate system O-XYZ. By changing the value of β, different positions of the mark can be obtained so that the TCSLDV system can sweep the laser spot along different scan paths. Two end points of the scan path can be obtained by

[00016]$\begin{array}{cc}\{\begin{array}{c}\left(a_1,b_1\right)=\left(a_0+e_1{M}_x,b_0+e_1{M}_y\right),C_1>1\\ \left(a_2,b_2\right)=\left(a_0+e_2{M}_x,b_0+e_2{M}_y\right),0<C_2<1\end{array}& \left(31\right)\end{array}$

where M.sub.x=(a.sub.m−a.sub.0)cos(β)+(b.sub.m−b.sub.0)sin(β), M.sub.y=(a.sub.m−a.sub.0)sin(β)+(b.sub.m−b.sub.0)cos(β), and e.sub.1 and e.sub.2 are two length factors that can be used to change positions of the two end points and the distance between them. The TCSLDV system can sweep its laser spot between the two end points while tracking the rotating structure. According to EQN. 27, the scan path lies on a straight line through the rotation center, and the distance between the rotation center and laser spot is

d.sub.s(t)=|P(t)|=√{square root over ((a.sub.s(t)−a.sub.0).sup.2+(b.sub.s(t)−b.sub.0).sup.2)}  (32)

[0130] Note that X- and Y-mirror signals obtained in experiment can be used as a.sub.s(t) and b.sub.s(t), respectively. The position of the laser spot on the scan path d.sub.s(t) can be obtained by combining X- and Y-mirror signals using EQN. 32. The processed mirror signal by this method is shown in FIG. 2C, where two dashed lines at t=t.sub.s and t=t.sub.e indicate time instants when the laser spot reaches the two end points of the scan path, respectively. Mirror signals and the measured response between t=t.sub.s and t=t.sub.e can be used to obtain an end-to-end undamped mode shape of the rotating structure. The scan frequency of the TCSLDV system is F.sub.sc=2/(t.sub.e−t.sub.s). Therefore, the duration of the time interval (t.sub.s, t.sub.e) is short when F.sub.sc is high.

[0131] Since the improved demodulation method can be applied to a measured response in a short time duration, the OMA method can also be applied to a structure that rotates with a prescribed time-varying speed to estimate its instantaneous damped natural frequencies and end-to-end undamped mode shapes in the short time duration since the rotation speed of the structure {dot over (θ)}(t) can be regarded as a constant in it and there are no other time-dependent coefficients in EQN. 1.

3. LIFTING METHODOLOGY

[0132] A lifting method that converts CSLDV measurement of a structure undergoing free vibration into measurements at multiple virtual measurement points as if there were transducers attached to it at these points in EMA. In this method, modal parameters of the structure can be estimated by obtaining and analyzing a set of frequency response functions between lifted measurements and an impact. A prior OMA method for a structure under ambient excitation was developed with use of a CSLDV system, where harmonic transfer functions and harmonic power spectra were employed. An OMA method that combines the lifting method and harmonic transfer functions was proposed, where processing and interpretation of CSLDV measurement become simpler. Since mode shapes of the structure estimated from the prior methods are represented by sums of spatially smooth harmonic functions, a large number of harmonic functions are required to describe the mode shapes if a scan path passes through a discontinuity on the structure. Xu et al. proposed a new OMA method for CSLDV measurement based on the lifting method to estimate modal parameters of a damaged structure and detect its local anomaly caused by damage. This OMA method requires the CSLDV system to scan the structure with a high scan frequency to obtain its modal parameters, which is difficult to achieve when its natural frequencies are high.

[0133] Rotating Euler-Bernoulli Beam Vibration Theory: As shown in FIG. 3, consider a fan blade that rotates about the z axis, which, without loss of generality, can be modeled as a uniform, rotating Euler-Bernoulli beam with a length 1 attached to a rigid hub with a radius b. Two coordinate systems are considered: an inertial coordinate system O-XYZ, and a rotating coordinate system o-xyz whose origin o is at the center of the rigid hub, which is referred to as the rotation center; the z axis is parallel to the Z axis. The position of the origin O of the inertial coordinate system O-XYZ depends on the position of the camera of the TCSLDV system. Small transverse vibration of the beam along the z axis is considered and vibrations along x and y axes are neglected. The x axis passes through the rotation center and is tangent to the neutral axis of the beam at the point where the beam is attached to the hub. Assume that the beam has linear viscous damping in the z direction with a damping coefficient c.

[0134] The governing equation of the uniform rotating Euler-Bernoulli beam under a distributed, external white-noise excitation force f(x, t) in the z direction and its associated boundary conditions are derived using the extended Hamilton's principle:

[00017]$\begin{array}{cc}\begin{array}{c}\rho w_{\mathrm{tt}}\left(x,t\right)+c{w}_t+{\mathrm{EIw}}_{xxxx}\left(x,t\right)\\ -\frac{1}{2}\rho {\dot{\theta }}^2\left(t\right)\left[\left({\left(b+l\right)}^2-x^2\right)w_{xx}\left(x,t\right)-2x{w}_x\left(x,t\right)\right]=f\left(x,t\right),\end{array}& \left(33\right)\end{array}$$\begin{array}{cc}\begin{array}{c}b\le x\le b+l,t>0\\ w\left(x,t\right).Math.x=b=w_x\left(x,t\right).Math.x=b=w_{xx}\left(x,t\right).Math.x=b+l=w_{xxx}\left(x,t\right).Math.x\\ =b+l=0\end{array}& \left(34\right)\end{array}$

where x is the spatial position along the x axis, t is time, θ(t) is the angle between the x- and X-axes, as shown FIG. 3, an overdot denotes a time derivative, w is the transverse displacement of the beam at the position x and time t along the z axis, and ρ and EI are the mass per unit length and flexural rigidity of the beam, respectively. Note that EQN. 33 and EQN. 34 are applicable to a rotating beam with a prescribed time-varying speed {dot over (θ)}(t). When {dot over (θ)}(t)=Ω is a constant, EQN. 33 can be written in the following form as

ρw.sub.tt(x,t)+cw.sub.t(x,t)+L[w(x,t)]=f(x,t),b≤x≤b+l,t>0  (35)

where L(⋅) is the stiffness operator:

[00018]$\begin{array}{cc}\begin{array}{c}L\left[w\right]={\mathrm{EIw}}_{xxxx}\left(x,t\right)-\frac{1}{2}\rho {\Omega }^2\left[\left({\left(b+l\right)}^2-x^2\right)w_{xx}\left(x,t\right)-2x{w}_x\left(x,t\right)\right],\\ b\le x\le b+l,t>0\end{array}& \left(36\right)\end{array}$

[0135] The term −½ρΩ.sup.2 [((b+l).sup.2−x.sup.2)w.sub.xx(x,t)−2xw.sub.x(x,t)] in EQN. 36 is referred to as the centrifugal stiffening term due to rotation of the beam. It can be shown that L(⋅) is self-adjoint and positive-definite with boundary conditions in EQN. 34. The eigenvalue problem associated with the corresponding undamped rotating beam is

L[ϕ]=ω.sup.2ρϕ  (37)

where ω is the undamped natural frequency of the beam, and ϕ is the corresponding mode shape or eigenfunction that satisfies boundary conditions corresponding to EQN. 34. By the expansion theorem, the solution to EQN. 35 can be expressed as

[00019]$\begin{array}{cc}w\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right)u_i\left(t\right)& \left(38\right)\end{array}$

where ϕ.sub.i(x) is the i-th eigenfunction of the undamped rotating beam and u.sub.i(t) is the i-th generalized coordinate. Since the undamped rotating beam is self-adjoint, its eigenfunctions can be normalized to satisfy the following orthonormality relations:

∫.sub.b.sup.b+lϕi(p)ρϕ.sub.j(p)dp=δ.sub.ij  (39)

∫.sub.b.sup.b+lϕi(p)L[ϕ.sub.j(p)]dp=ω.sub.n,i.sup.2δ.sub.ij  (40)

where ω.sub.n,i is the i-th undamped natural frequency of the rotating beam and δ.sub.ij is Kronecker delta that satisfies

[00020]$\begin{array}{cc}{\delta }_{ij}=\{\begin{array}{cc}1,& i=j\\ 0,& i\ne j\end{array}& \left(41\right)\end{array}$

Substituting EQN. 38 into EQN. 35, multiplying the resulting equation by ϕ.sub.j, integrating the resulting equation from x=b to x=b+l, and using EQN. 39 and EQN. 40 and the expression ∫.sub.b.sup.b+lϕ.sub.i(p)cϕ(p)dp=(c/ρ)δ.sub.ij=c.sub.iδ.sub.ij yield

ü.sub.i(t)+c.sub.i{dot over (u)}.sub.i(t)+ω.sub.n,i.sup.2u.sub.i(t)=∫.sub.b.sup.b+lϕ.sub.i(p)f(p,t)dp  (42)

[0136] Considering an underdamped rotating beam, one has c.sub.i=2ζω.sub.n,i, where 0<ζ.sub.i<1 is the i-th modal damping ratio of the beam. Since ω.sub.n,1≤ω.sub.n,i for i=1, 2, . . . , when 0<c<2ρω.sub.n,1, all the modal damping ratios ζ.sub.i satisfy 0<ζ.sub.i<1. Assume that EQN. 33 has zero initial conditions; the solution to EQN. 42 can be expressed by

u.sub.i(t)=∫.sub.0.sup.t∫.sub.b.sup.b+lϕ.sub.i(p)f(p,t)g.sub.i(t−τ)dpdτ  (43)

where g.sub.i(t) is the unit impulse response function corresponding to the i-th damped mode of the beam and ω.sub.d,i=ω.sub.n,i√{square root over (1−ζ.sub.i.sup.2)} is its i-th damped natural frequency. Substituting EQN. 43 into EQN. 38 yields

[00021]$\begin{array}{cc}w\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\int }_0^t{\int }_b^{b+l}{\varphi }_i\left(p\right)f\left(p,\tau \right)g_i\left(t-\tau \right)dpd\tau & \left(45\right)\end{array}$

If a concentrated force f.sub.a(t) is applied at a position x.sub.a on the beam, one has

f(x,t)=δ(x−x.sub.a)f.sub.a(t)  (46)

where δ is Dirac delta function. Substituting EQN. 46 into EQN. 45 yields

[00022]$\begin{array}{cc}w\left(x,t\right)=\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\varphi }_i\left(x_a\right){\int }_0^t{f}_a\left(\tau \right)g_i\left(t-\tau \right)d\tau & \left(47\right)\end{array}$

If m concentrated forces are applied on the structure, the response of the beam can be expressed as a superposition of those from all the forces by

[00023]$\begin{array}{cc}w\left(x,t\right)=\underset{j=1}{\overset{m}{.Math.}}\underset{i=1}{\overset{\infty }{.Math.}}{\varphi }_i\left(x\right){\varphi }_i\left(x_j\right){\int }_0^t{f}_j\left(\tau \right)g_i\left(t-\tau \right)d\tau & \left(48\right)\end{array}$$\begin{array}{cc}{g}_i\left(t\right)=\frac{1}{{\omega }_{d,i}}{e}^{-{\zeta }_i{\omega }_{n,i}t}\sin \left({\omega }_{d,i}t\right)& \left(44\right)\end{array}$

where x.sub.j is the position at which the j-th concentrated force f.sub.j is applied on the beam.

[0137] Lifting method for TCSLDV measurement of rotating structure: Virtual measurement points are assigned on the fan blade in FIG. 3 along a time-varying scan path r(t) when the TCSLDV system is used to periodically measure w by scanning its laser spot along the path r(t). The TCSLDV system registers discrete measurements of w with a finite sampling frequency F.sub.sa.

[0138] A scan path r(t) is usually a straight line on a rotating structure, which is the case considered here. If a CSLDV system is used to scan along a stationary structure and positions of the laser spot can be considered to be linearly related to the rotation angle of a mirror (e.g., X-mirror) in the scanner in the CSLDV, only the X-mirror is needed to complete the scan and the mirror signal can be directly used as the position of the laser spot on the scan path r(t). However, both X- and Y-mirrors are needed to scan along a rotating structure such as the fan blade here; thus use of only one mirror signal is not enough to describe the position of the laser spot on the scan path r(t). A simulated X-mirror signal for scanning a stationary structure is shown in FIG. 4A. Simulated X- and Y-mirror signals for scanning a rotating structure with a constant speed are shown in FIG. 4B. A new method is developed in this work to combine X- and Y-mirror signals to describe the position of a laser spot on the scan path r(t) on a rotating structure.

[0139] A mark is attached to a structure that rotates with a constant speed to identify positions of the structure and scan path at any time instant FIG. 3. Let the position of the rotation center relative to the inertial coordinate system XYZ be (u.sub.0, v.sub.0), the position of the mark be (u.sub.m(t), v.sub.m(t)), and the position of the laser spot be (u(t), v(t)); one has

[00024]$\begin{array}{cc}\{\begin{array}{c}{u}_m\left(t\right)-u_0=l_m\cos \left(\theta \left(t\right)\right)=l_m\cos \left(\Omega t\right)\\ u_m\left(t\right)-u_0=l_m\sin \left(\theta \left(t\right)\right)=l_m\sin \left(\Omega t\right)\end{array}& \left(49\right)\end{array}$$\begin{array}{cc}\{\begin{array}{c}u\left(t\right)-u_0=l\left(t\right)\cos \left(\theta \left(t\right)\right)=l\left(t\right)\cos \left(\Omega t\right)\\ v\left(t\right)-v_0=l\left(t\right)\sin \left(\theta \left(t\right)\right)=l\left(t\right)\sin \left(\Omega t\right)\end{array}& \left(50\right)\end{array}$

where l.sub.m is the distance between the rotation center and mark, and l(t) is the distance between the rotation center and laser spot at time t, which can be used to describe the position of the laser spot on the scan path r(t). The position vector of the mark is

V.sub.m(t)=((u.sub.m(t)−u.sub.s),(v.sub.m(t)−v.sub.0)).sup.T  (51)

where a superscript T denotes transpose of a vector. Let V.sub.m(t.sub.i) be the position vector at t=t.sub.1 and V.sub.m(t.sub.2) be that at t=t.sub.2; the angle θ.sub.12 between V.sub.m(t.sub.1) and V.sub.m(t.sub.2) can be obtained by

[00025]$\begin{array}{cc}{\theta }_{12}=a\cos \left(\frac{V_m\left(t_1\right).Math.V_m\left(t_2\right)}{.Math.V_m\left(t_1\right).Math..Math..Math.V_m\left(t_2\right).Math.}\right)& \left(52\right)\end{array}$

Therefore, revolutions per minute of the structure can be written as

[00026]$\begin{array}{cc}\mathrm{RPM}=\frac{30{\theta }_{12}}{\pi \left(t_2-t_1\right)}& \left(53\right)\end{array}$

The adjusted position vector of the mark is

V.sub.m.sup.α=((u.sub.m−u.sub.0)cos(α)+(v.sub.m−v.sub.0)sin(α),(u.sub.m−u.sub.0)sin(α)+(v.sub.m−v.sub.0)cos(α)).sup.T  (54)

where α is a phase variable that can be used to adjust the position of the mark in the inertial coordinate system O-XYZ. By changing the value of α, different positions of the mark can be obtained so that the TCSLDV system can sweep the laser spot along different scan paths. Two end points of the scan path can be obtained by

[00027]$\begin{array}{cc}\{\begin{array}{c}\left(u_1,v_1\right)=\left(u_0+C_1{V}_x,v_0+C_1{V}_y\right),C_1>1\\ \left(u_2,v_2\right)=\left(u_0+C_2{V}_x,v_0+C_2{V}_y\right),0<C_2<1\end{array}& \left(55\right)\end{array}$

where V.sub.x and V.sub.y are X and Y components of V.sub.m, respectively, and C.sub.1 and C.sub.2 are two length factors that can be used to change the distance between the two end points and positions of the two end points. The TCSLDV system can sweep its laser spot between the two end points while tracking the rotating structure. Since the scan path lies on a straight line through the rotation center, the distance between the rotation center and laser spot is

r(t)=√{square root over ((u−u.sub.0).sup.2+(v−v.sub.0).sup.2)}  (56)

where u and v are X- and Y-mirror signals, respectively. The position of the laser spot on the scan path r(t) can be obtained by combining X- and Y-mirror signals using EQN. 56. The processed mirror signal by this method is shown in FIG. 4C. The method can also be applied to a structure that rotates with a prescribed time-varying speed. The solution w in EQN. 47 can be written as functions of l(t) and t:

[00028]$\begin{array}{cc}w\left[l\left(t\right),t\right]=\underset{i=1}{\overset{n}{.Math.}}{\varphi }_i\left[l\left(t\right)\right]{\varphi }_i\left(x_a\right){\int }_0^t{f}_a\left(\tau \right)g_i\left(t-\tau \right)d\tau & \left(57\right)\end{array}$

where n is the number of modes that are measured in discrete measurements of w by the TCSLDV system, which is finite since F.sub.sa is finite. The number of virtual measurement points N on the scan path r(t) is determined by the sampling frequency F.sub.sa and scan frequency F.sub.sc as

[00029]$\begin{array}{cc}N=\frac{F_{sa}}{F_{sc}}& \left(58\right)\end{array}$

where F.sub.sc is equal to the number of complete scans in one second. Multiple series of discrete measurements of w in EQN. 57 are formed by lifting them in the lifting method. Each lifted w series corresponds to a virtual measurement point as if it were measured by a transducer attached to the rotating structure at that point. Note that N in EQN. 58 should be an integer since the laser spot needs to arrive at the same virtual points in each complete scan period; thus F.sub.sa should be an integer multiple of F.sub.sc. Therefore one has

l(t)=l(t+sT.sub.sc)  (59)

where s=0, 1, 2, . . . and T.sub.sc=1/F.sub.sc is the duration of a complete scan period. A complete scan means that the TCSLDV system sweeps its laser spot back and forth once on the scan path r(t). There is a constant sampling time difference T.sub.sa=1/F.sub.sa between two neighboring lifted w series, which means the lifted w series are not simultaneously registered by the TCSLDV system. Let measurement of w starts at t=0 in EQN. 57 when the laser spot arrives at an endpoint of the scan path r(t); the lifted w at the k-th virtual measurement point on the path r(t) can thus be written as a function of sT.sub.sc:

w.sub.k.sup.l(sT.sub.sc)=w[(k−1)T.sub.sa+(s−1)T.sub.sc]  (60)

where k=0, 1, . . . K and s=0, 1, . . . S, in which K and S are numbers of measurement points and complete scans, respectively. EQN. 60 shows that the laser spot arrives at the k-th virtual measurement point when t=(k−1)T.sub.sa+(s−1)T.sub.sc and the sampling frequency of w.sub.k.sup.l is equal to F.sub.sc. Measured w can be expressed by an S×K matrix W.sup.l that consists of lifted w at K virtual measurement points in S complete scans:

[00030]$\begin{array}{cc}{W}^l=\left[\begin{array}{cccc}w\left(0\right)& w\left(T_{\mathrm{sa}}\right)& .Math.& w\left(\left(K-1\right)T_{\mathrm{sa}}\right)\\ w\left(T_{\mathrm{sc}}\right)& w\left(T_{\mathrm{sa}}+T_{\mathrm{sc}}\right)& .Math.& w\left(\left(K-1\right)T_{\mathrm{sa}}+T_{\mathrm{sc}}\right)\\ .Math.& .Math.& \ddots & .Math.\\ w\left(\left(S-1\right)T_{\mathrm{sc}}\right)& w\left(T_{\mathrm{sa}}+\left(S-1\right)T_{\mathrm{sc}}\right)& .Math.& w\left(\left(K-1\right)T_{\mathrm{sa}}+\left(S-1\right)T_{\mathrm{sc}}\right)\end{array}\right]& \left(61\right)\end{array}$

where the s-th row consists of lifted w in the s-th scan period and the k-th column consists of lifted w at the k-th virtual measurement point.

[0140] Correlation between two TCSLDV measurements: Let w.sub.k.sub.1.sup.l and w.sub.k.sub.2.sup.l be lifted w at the k.sub.i-th and k.sub.2-th virtual measurement points on the scan path r(t); one has

[00031]$\begin{array}{cc}{w}_{k_1}^l\left(s_{k_1}{T}_{sc}\right)=\underset{i=1}{\overset{n}{.Math.}}{\varphi }_{i,k_1}{\int }_0^{\left(k_1-1\right)T_{sa}+\left(s_{k_1}-1\right)T_{sc}}{\varphi }_i\left(x_a\right)f_a\left(\tau \right)⨯g_i\left[\left(k_1-1\right)T_{sa}+\left(s_{k_1}-1\right)T_{sc}-\tau \right]d\tau & \left(62\right)\end{array}$$\begin{array}{cc}{w}_{k_2}^l\left(s_{k_2}{T}_{sc}\right)=\underset{i=1}{\overset{n}{.Math.}}{\varphi }_{i,k_2}{\int }_0^{\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}}{\varphi }_i\left(x_a\right)f_a\left(\tau \right)⨯g_i\left[\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}-\tau \right]d\tau & \left(63\right)\end{array}$

where ϕ.sub.i,k.sub.1(x) and ϕ.sub.i,k.sub.2(x) are values of the i-th normalized eigenfunction of the undamped rotating structure associated with the structure described by EQN. 33 at the k.sub.i-th and k.sub.2-th virtual measurement points on the scan path r(t), respectively, and S.sub.k.sub.1 and S.sub.k.sub.2 are numbers of complete scan periods of w.sub.k.sub.1.sup.l and w.sub.k.sub.2.sup.l, respectively. The correlation function between w.sub.k.sub.1.sup.l and w.sub.k.sub.2.sup.l can be expressed by the expected value of their product:

R.sub.k.sub.1.sub.k.sub.2[w.sub.k.sub.1.sup.l(m.sub.k.sub.1T.sub.sc),w.sub.k.sub.2.sup.l(m.sub.k.sub.2T.sub.sc)]=E[w.sub.k.sub.1.sup.l(m.sub.k.sub.1T.sub.sc)w.sub.k.sub.2.sup.l(m.sub.k.sub.2T.sub.sc)]  (64)

where E[⋅] is an expectation operator. Note that correlation functions can be calculated whether the structure rotates with a constant or time-varying speed, but modal parameters can be estimated by analyzing the correlation functions when the structure rotates with a constant speed. The laser spot of the TCSLDV system arrives at the k.sub.1-th and k.sub.2-th virtual measurement points on the scan path r(t) at t.sub.1=(k.sub.1−1)T.sub.sa+(s.sub.k.sub.1−1)T.sub.sc and t.sub.2=(k.sub.2−1)T.sub.sa+(s.sub.k.sub.2−1)T.sub.sc, respectively. Thus

(k.sub.2−1)T.sub.sa+(s.sub.k.sub.2−1)T.sub.sc=(k.sub.2−1)T.sub.sa+(s.sub.k.sub.1−1)T.sub.sc+T  (65)

where T=(s.sub.k.sub.2−s.sub.k.sub.1)T.sub.sc is the time-delay variable. If the structure rotates with a constant speed, the cross-correlation function in EQN. 63 becomes a function of s.sub.k.sub.1T.sub.sc and T by substituting EQN. 64 into EQN. 63 to obtain

R.sub.k.sub.1.sub.k.sub.2(m.sub.k.sub.1T.sub.sc,T)=E[w.sub.k.sub.1.sup.l(m.sub.k.sub.1T.sub.sc)w.sub.k.sub.2.sup.l(m.sub.k.sub.1T.sub.sc,T+T)]  (66)

which means that the reference point is the k.sub.1-th measurement point on the scan path r(t) and the measurement point is the k.sub.2-th virtual measurement point on the path r(t). Substituting EQNS. 62 and 63 into EQN. 66 yields

[00032]$$\begin{gathered} \begin{array}{cc} \\ {R}_{k_1{k}_2}\left(m_{k_1}{T}_{\mathrm{sc}},T\right)=E\left[\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{n}{.Math.}}{\varphi }_{i,k_1}{\varphi }_i\left(x_a\right){\varphi }_{j,k_2}{\varphi }_i\left(x_a\right){\int }_0^{\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}}⨯{\int }_0^{\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}}{g}_i\left[\left(k_1-1\right)T_{sa}+\left(s_{k_1}-1\right)T_{sc}-\sigma \right]⨯g_j\left[\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}-\tau \right]f_a\left(\sigma \right)f_a\left(\tau \right)d\tau d\sigma \right]& \left(67\right)\end{array} \end{gathered}$$

Since only f is a random variable in EQN. 67, EQN. 67 becomes

[00033]$\begin{array}{cc}{R}_{k_1{k}_2}\left(m_{k_1}{T}_{\mathrm{sc}},T\right)=\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{n}{.Math.}}{\varphi }_{i,k_1}{\varphi }_i\left(x_a\right){\varphi }_{j,k_2}{\varphi }_i\left(x_a\right){\int }_0^{\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}}⨯{\int }_0^{\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}}{g}_i\left[\left(k_1-1\right)T_{sa}+\left(s_{k_1}-1\right)T_{sc}-\sigma \right]⨯g_j\left[\left(k_2-1\right)T_{sa}+\left(s_{k_2}-1\right)T_{sc}+T-\tau \right]E\left[f_a\left(\sigma \right)f_a\left(\tau \right)\right]d\tau d\sigma & \left(68\right)\end{array}$

The expected value of f.sub.a(σ)f.sub.a(τ) is

E[f.sub.a(σ)f.sub.a(τ)]=α.sub.aδ(τ−σ)  (69)

where α.sub.a is a constant that depends on f.sub.a. Let λ=(k.sub.1−1)T.sub.sa+(s.sub.k.sub.i−1)T.sub.sc−σ; substituting EQN. 69 into EQN. 68 yields

[00034]$\begin{array}{cc}{\tilde{R}}_{k_1{k}_2}\left(T\right)=\underset{i=1}{\stackrel{n}{.Math.}}\underset{j=1}{\stackrel{n}{.Math.}}{\alpha }_a{\varphi }_{i,k_1}{\varphi }_i\left(x_a\right){\varphi }_{j,k_2}{\varphi }_i\left(x_a\right)⨯{\int }_0^{\infty }{g}_i\left(\lambda \right)g_j\left[T+\left(k_2-k_1\right)T_{sa}+\lambda \right]d\lambda & \left(70\right)\end{array}$

Substituting EQN. 35 into EQN. 70 yields

[00035]$\begin{array}{cc}{\tilde{R}}_{k_1{k}_2}\left(T\right)=\underset{j=1}{\overset{n}{.Math.}}{\varphi }_{j,k_2}\left[P_{k_1{k}_2,j}{e}^{-{\zeta }_j{\omega }_{n,j}\left(T+\left(k_2-k_1\right)T_{sa}\right)}⨯\cos \left({\omega }_{d,j}\left(T+\left(k_2-k_1\right)T_{sa}\right)\right)+Q_{k_1{k}_2,j}{e}^{-{\zeta }_j{\omega }_{n,j}\left(T+\left(k_2-k_1\right)T_{sa}\right)}⨯\sin \left({\omega }_{d,j}\left(T+\left(k_2-k_1\right)T_{sa}\right)\right)\right]& \left(71\right)\end{array}$$\mathrm{where}$$\begin{array}{cc}{P}_{k_1{k}_2,j}=\underset{i=1}{\overset{n}{.Math.}}\frac{{\alpha }_a{\varphi }_{i,k_1}{\varphi }_i\left(x_a\right){\varphi }_i\left(x_a\right)}{{\omega }_{d,i}{\omega }_{d,j}}{\int }_0^{\infty }{e}^{\left(-{\zeta }_i{\omega }_{n,i}-{\zeta }_j{\omega }_{n,j}\right)\lambda }⨯\sin \left({\omega }_{d,i}\lambda \right)\sin \left({\omega }_{d,j}\lambda \right)\mathrm{d\lambda }& \left(72\right)\end{array}$$\begin{array}{cc}{Q}_{k_1{k}_2,j}=\underset{i=1}{\overset{n}{.Math.}}\frac{{\alpha }_a{\varphi }_{i,k_1}{\varphi }_i\left(x_a\right){\varphi }_i\left(x_a\right)}{{\omega }_{d,i}{\omega }_{d,j}}{\int }_0^{\infty }{e}^{\left(-{\zeta }_i{\omega }_{n,i}-{\zeta }_j{\omega }_{n,j}\right)\lambda }⨯\sin \left({\omega }_{d,i}\lambda \right)\cos \left({\omega }_{d,j}\lambda \right)d\lambda & \left(73\right)\end{array}$

Let A.sub.j=√{square root over ((P.sub.k.sub.1.sub.k.sub.2.sub.,j).sup.2+(Q.sub.k.sub.1.sub.k.sub.2.sub.,j).sup.2)} be an amplitude constant and θ.sub.j=atan 2(Q.sub.k.sub.1.sub.k.sub.2.sub.,j, P.sub.k.sub.1.sub.k.sub.2.sub.,j) be a phase constant; EQN. 70 becomes

[00036]$\begin{array}{cc}{\tilde{R}}_{k_1{k}_2}\left(T\right)=R\left[\underset{j=1}{\overset{n}{.Math.}}{A}_j{\varphi }_{j,k_2}{e}^{-{\zeta }_j{\omega }_{n,j}\left(T+\left(k_2-k_1\right)T_{\mathrm{sa}}\right)+i{\omega }_{d,j}\left(T+\left(k_2-k_1\right)T_{\mathrm{sa}}-{\theta }_j\right)}\right]& \left(74\right)\end{array}$

where R[⋅] donates the real part of a complex variable and i=√{square root over (−1)}. Let Ã.sub.j=A.sub.je.sup.−iθ.sup.j be a complex factor corresponding to the j-th mode of the rotating structure with a constant speed; one has

[00037]$\begin{array}{cc}{\tilde{R}}_{k_1{k}_2}\left(T\right)=R\left[\underset{j=1}{\overset{n}{.Math.}}{Ã}_j{\varphi }_{j,k_2}{e}^{\left(-{\zeta }_j{\omega }_{n,j}+i{\omega }_{d,j}\right)\left(k_2-k_1\right)T_{\mathrm{sa}}+\left(-{\zeta }_j{\omega }_{n,j}+i{\omega }_{d,j}\right)T}\right]& \left(75\right)\end{array}$

The above provides complete derivation of a cross-correlation function between two lifted w measured on a rotating structure with a constant speed under white-noise excitation at a fixed position of the structure. By using EQN. 48 and following the above derivation, cross-correlation functions of the rotating structure under multiple white-noise excitations at multiple fixed positions of the structure can be obtained, which have the same form as that of EQN. 75.

[0141] By applying a standard OMA algorithm such as the PolyMAX algorithm to power spectra associated with cross-correlation functions, ω.sub.d,j, ζ.sub.j, and Ã.sub.jϕ.sub.j,k.sub.2e.sup.(−ζ.sup.j.sup.ω.sup.n,j.sup.+iω.sup.d,j.sup.)(k.sup.2.sup.−k.sup.1.sup.)T.sup.sa in EQN. 75 can be estimated, the first two of which are the j-th damped natural frequency and modal damping ratio of the rotating structure, respectively, and ϕ.sub.j,k.sub.2 in the third of which is the j-th undamped mode shape the rotating structure. The additional term e.sup.(−ζ.sup.j.sup.ω.sup.n,j.sup.+iω.sup.d,j.sup.)(k.sup.2.sup.−k.sup.1.sup.)T.sup.sa in the expression Ã.sub.jϕ.sub.j,k.sub.2e.sup.(−ζ.sup.j.sup.ω.sup.n,j.sup.+iω.sup.d,j.sup.)(k.sup.2.sup.−k.sup.1.sup.)T.sup.sa can be eliminated to obtain θ.sub.j,k.sub.2 with a scaling factor Ã.sub.j by multiplying the expression by e.sup.−(−ζ.sup.j.sup.ω.sup.n,j.sup.+iω.sup.d,j.sup.)(k.sup.2.sup.−k.sup.1.sup.)T.sup.sa since ω.sub.d,j and ζ.sub.j have been estimated and (k.sub.2−k.sub.1)T.sub.sa is known. Modal parameters of the rotating structure estimated are not affected by whether the structure is excited at one or multiple points, as long as the reference and measurement points and at least one excitation point are not nodal points of a mode of the structure of interest. ODSs of a rotating structure with a constant or time-varying speed can be estimated by analyzing correlation functions in EQN. 64 using, for example, the ODS module in the LMS Test.Lab software.

[0142] The present disclosure provides a modal analysis method comprising: measuring a response of a rotating structure subject to random excitation with a system; interpolate positions of the response on a grid to generate a plurality of interpolated positions; rectifying the plurality of interpolated positions to create a plurality of rectified interpolated positions; identifying a plurality of zero-crossings from the plurality of rectified interpolated positions; determine a portion of the plurality of zero-crossings with a time increment; and interpolate and lift measurements at the portion of the plurality of zero-crossings.

[0143] In some embodiments, rectifying the plurality of interpolated positions includes determining negative absolute values of differences between the plurality of interpolated positions and a position of a virtual measurement point on a scan path.

[0144] In some embodiments, the time increment is equal to the inverse of a scan frequency.

[0145] In some embodiments, the system is a TCSLDV system.

[0146] In some embodiments, the system includes a camera and the method includes capturing images of the rotating structure.

[0147] In some embodiments, the method further includes determining a damped natural frequency, a damping ratio, and/or an undamped mode shape of the rotating structure.

[0148] In some embodiments, the rotating structure is a wind turbine blade.

4. TWO-DIMENSIONAL SCAN SCHEME

[0149] Rotating Plate Model: A nonuniform rectangular rotating plate model is used to describe a rotating fan blade. The rotating plate that rotates about the z-axis has a length of 1 and width of b (FIG. 5). The plate is attached to a rigid hub. The inertial coordinate system O-XYZ whose origin O is at corners of pictures captured by the camera in the tracking CSLDV system and the rotating coordinate system o-xyz whose origin o is at the rotation center of the plate are used to describe the motion of the rotating plate. The distance between o and the fixed end of the rotating plate is r. Note that Z- and z-axes are parallel to each other. The angle between Y- and y-axes that is used to describe the angular position of the plate is θ(t), where t is time. By using the extended Hamilton's principle and only considering the vibration along the z-direction, the governing equation and associated boundary conditions of the nonuniform rotating plate in FIG. 5 subjected to a distributed random force f(x, y, t) along the z-direction can be derived as

ρ(x,y)u.sub.tt+∇.sup.2[D(x,y)∇.sup.2u]+C(u.sub.t)−{dot over (θ)}.sup.2(t)[u.sub.xx∫.sub.x.sup.r+lρ(p,y)pdp−ρ(x,y)xu.sub.x+u.sub.yy∫.sub.y.sup.±b/2ρ(x,p)pdp−ρ(x,y)yu.sub.y]=f(x,y,t),−b/2≤x≤b/2,r≤y≤r+l,t>0,  (76)

u|y=r=0,u.sub.x|y=r=0,u.sub.xx+vu.sub.yy|y=r+l=0,u.sub.xxx+u.sub.xyy|y=r+l=0,u.sub.xy|y=r+l=0,u.sub.yy+vu.sub.xx|x=±b/2=0,u.sub.yyy+u.sub.xxy|x=±b/2=0,u.sub.xy|x=±b/2=0,   (77)

where x and y are spatial positions along x- and y-directions, u is the plate displacement along the z-direction at the point (x, y) and time t, D(x, y) and v are the flexural rigidity and Poisson's ratio of the plate at (x, y), respectively, a subscript and an overdot denote partial differentiation with respect to x and t, respectively, ρ(x, y) is the mass per unit area of the rotating plate at (x, y), C is the spatial damping operator, and ∇ is the del operator.

[0150] The solution to the governing equation of the nonuniform rotating plate subject to a concentrated random force that is applied at the point (x.sub.a, y.sub.a) is derived in Example 4 as

u=Σ.sub.i=1.sup.∞ϕ.sub.i(x,y)ϕ.sub.i(x.sub.a,y.sub.a)∫.sub.0.sup.tf.sub.a(t−τ)(1/ω.sub.d,i)e.sup.−ζ.sup.i.sup.ω.sup.i.sup.τ sin(ω.sub.d,iτ)dτ,  (78)

where ϕ.sub.i (x, y) is the i-th full-field mode shape of the corresponding undamped rotating plate, and ω.sub.i, ζ.sub.i and ω.sub.d,i=ω.sub.i√{square root over (1−ζ.sub.i.sup.2)} are the i-th undamped natural frequency, modal damping ratio, and damped natural frequency of the plate, respectively.

[0151] Two-Dimensional (2D) Scan Scheme: A 2D zigzag scan path is generated on the surface of the rotating plate. Let coordinates of four corners of the plate be (X.sub.1, Y.sub.1), (X.sub.2, Y.sub.2), (X.sub.3, Y.sub.3) and (X.sub.4, Y.sub.4), and the scan path that is shown as dash lines in FIG. 6 starts from (X.sub.1, Y.sub.1) and ends at (X.sub.4, Y.sub.4). The scan path consists of multiple connected scan lines. The laser spot of the tracking CSLDV system is swept by an odd number of times on each scan line, and then start to be swept on the next scan line from the connection point of the scanned scan line and next scan line.

[0152] Since the plate rotates, coordinates of the four corners of the plate change with time. To generate a scan path on the rotating plate, coordinates of the corners need to be determined. A mark is attached somewhere outside the rotating plate to determine positions of the four corners. Let coordinates of the mark and rotation center be (X.sub.m, Y.sub.m) and (X.sub.0, Y.sub.0), respectively; the position of one corner of the rotating plate can be calculated by

[00038]$\begin{array}{cc}\{\begin{array}{c}{X}_1=X_0+s_1/s_m\left[\left(X_m-X_0\right)\cos \left(\alpha \right)-\left(Y_m-Y_0\right)\sin \left(\alpha \right)\right]\\ Y_1=Y_0+s_1/s_m\left[\left(Y_m-Y_0\right)\cos \left(\alpha \right)+\left(X_m-X_0\right)\sin \left(\alpha \right)\right]\end{array},& \left(79\right)\end{array}$

where α is the angle between the line passing through (X.sub.0, Y.sub.0) and (X.sub.m, Y.sub.m) and that passing through (X.sub.0, Y.sub.0) and (X.sub.1, Y.sub.1), and s.sub.1 and s.sub.m are constant distances between (X.sub.0, Y.sub.0) and (X.sub.1, Y.sub.1) and between (X.sub.0, Y.sub.0) and (X.sub.m, Y.sub.m), respectively. Note that α in EQN. 79 is a constant since the relative position between the mark and plate is fixed. Positions of other three corners can be similarly determined. Let the number of scan lines be N; positions of end points of the i-th scan line (X.sub.i.sup.1, Y.sub.i.sup.1) and (X.sub.i.sup.2, Y.sub.i.sup.2) can be calculated by

[00039]$\begin{array}{cc}\{\begin{array}{c}{X}_i^1=X_2+\frac{2\left(X_1-X_2\right)}{N-1}\left(\frac{N-i}{2}\right)\\ Y_i^1=Y_2+\frac{2\left(Y_1-Y_2\right)}{N-1}\left(\frac{N-i}{2}\right)\end{array},& \left(80\right)\end{array}$$\begin{array}{cc}\{\begin{array}{c}{X}_i^2=X_4+\frac{2\left(X_3-X_4\right)}{N-1}\left(\frac{N-i}{2}\right)\\ Y_i^2=Y_4+\frac{2\left(Y_3-Y_4\right)}{N-1}\left(\frac{N-i}{2}\right)\end{array},& \left(81\right)\end{array}$

when i is an odd number,

[00040]$\begin{array}{cc}\{\begin{array}{c}{X}_i^1=X_2+\frac{2\left(X_1-X_2\right)}{N-1}\left(\frac{N-i-1}{2}\right)\\ Y_i^{1_{=}}{Y}_2+\frac{2\left(Y_1-Y_2\right)}{N-1}\left(\frac{N-i-1}{2}\right)\end{array},& \left(82\right)\end{array}$$\begin{array}{cc}\{\begin{array}{c}{X}_i^2=X_4+\frac{2\left(X_3-X_4\right)}{N-1}\left(\frac{N-i+1}{2}\right)\\ Y_i^2=Y_4+\frac{2\left(Y_3-Y_4\right)}{N-1}\left(\frac{N-i+1}{2}\right)\end{array},& \left(83\right)\end{array}$

when i is an even number. Once positions of the two end points of the i-th scan line are determined, the laser spot can be swept between the two end points. This scheme can be applied to any quadrilateral rotating plate.

[0153] When the tracking CSLDV system is used to scan a rotating plate, feedback signals of X- and Y-mirrors of the scanner in the system can be used as X and Y positions of the laser spot. The method for processing mirror signals disclosed herein is used to process feedback signals of X and Y mirrors and processed mirror signals can be used to obtain mode shapes of the rotating plate. At any time instant, the distance between the laser spot and rotation center d.sub.s(t) is

d.sub.s(t)=√{square root over (((X.sub.s(t)−X.sub.0).sup.2+(Y.sub.s(t)−Y.sub.0).sup.2)},  (84)

where X.sub.s(t) and Y.sub.s(t) are X and Y positions of the laser spot at time t, respectively. One can just use d.sub.s(t) to describe the position of the laser spot on scan lines. The real time rotation speed of the plate R can also be obtained by

[00041]$\begin{array}{cc}R=\frac{30\gamma }{\pi \left(t_2-t_1\right)},& \left(85\right)\end{array}$

[0154] where t.sub.1 and t.sub.2 are two neighboring time instants when the tracking CSLDV system measures the surface velocity of the rotating plate, and

[00042]$\begin{array}{cc}\gamma =a\cos \left(\frac{\left(X_s\left(t_1\right)-X_0\right)\left(X_s\left(t_2\right)-X_0\right)+\left(Y_s\left(t_1\right)-Y_0\right)\left(Y_s\left(t_2\right)-Y_0\right)}{\sqrt{\left[{\left(X_s\left(t_1\right)-X_0\right)}^2+{\left(Y_s\left(t_1\right)-Y_0\right)}^2\right]\left[{\left(X_s\left(t_2\right)-X_0\right)}^2+{\left(Y_s\left(t_2\right)-Y_0\right)}^2\right]}}\right).& \left(86\right)\end{array}$

[0155] Improved 2D Demodulation Method: The time-varying 2D zigzag scan path is generated on the rotating plate based on the above scheme, and the tracking CSLDV system measures u by sweeping its laser spot along the zigzag scan path and registering discrete measurements of u. The solution to the governing equation of the rotating plate subject to a concentrated random force in EQN. 78 can be used as the measured response of the rotating plate using the 2D scan scheme. Note that when there are multiple random excitation forces applied on the rotating plate, the solution to its governing equation has the same form as that in EQN. 78. One can process the measured response from each scan line to estimate undamped mode shapes of the rotating plate on scan lines and combine mode shapes into an undamped full-field mode shape. Let x=x.sub.j be the x position of the j-th scan line; applying integration by part to EQN. 78 yields

[00043]$\begin{array}{cc}u\left(x_j,y,t\right)={.Math.}_{i=1}^{\infty }{\varphi }_i\left(x_j,y\right){\varphi }_i\left(x_a,y_a\right)\left[A_i\left(t\right)\cos \left({\omega }_{d,i}t\right)+B_i\left(t\right){\sin \left({\omega }_{d,i}t\right)}_{+C^i}\left(t\right)\right],& \left(87\right)\end{array}$$\mathrm{where}$$\begin{array}{cc}{A}_i\left(t\right)=-\frac{{\omega }_{d,i}{e}^{-{\zeta }_{i^{\omega }i}t}{f}_{a\left(0\right)}}{{\omega }_{d,i}{\omega }_i^2},& \left(88\right)\end{array}$$\begin{array}{cc}{B}_i\left(t\right)=-\frac{{\zeta }_i{\omega }_i{e}^{-{\zeta }_{i^{\omega }i}t}{f}_a\left(0\right)}{{\omega }_{d,i}{\omega }_i^2},& \left(89\right)\end{array}$$\begin{array}{cc}{C}_i\left(t\right)=\frac{f_a\left(t\right)}{{\omega }_i^2}+{\int }_0^t\frac{e^{-{\zeta }_i{\omega }_{i}t}{f}_a^{\prime }\left(t-\tau \right)}{{\omega }_{d,i}{\omega }_i^2}\left[{\zeta }_i{\omega }_i\sin \left({\omega }_{d,i}\tau \right)+{\omega }_{d,i}\cos \left({\omega }_{d,i}\tau \right)\right]d\tau .& \left(90\right)\end{array}$

[0156] Applying a bandpass filter that filters all damped natural frequencies of the rotating plate except the i-th damped natural frequency ω.sub.d,i to EQN. 87, the measured response becomes

u.sub.i(x.sub.j,y,t)=Φ.sub.i(x.sub.j,y)cos(ω.sub.d,it−β)=Φ.sub.I,i(x.sub.j,y)cos(ω.sub.d,it)+Φ.sub.Q,i(x.sub.j,y)sin(ω.sub.d,it),   (91)

where Φ.sub.i(x.sub.j, y)=H.sub.iΦ.sub.i(x.sub.j, y), in which H.sub.i is a scalar factor, β is a phase variable, Φ.sub.I,i(x.sub.j,y)=Φ.sub.i(x.sub.j,y)cos(β), and Φ.sub.Q,i(x.sub.j, y)=Φ.sub.i(x.sub.j, y)sin(β). EQN. 91 is multiplied by cos(ω.sub.d,it) and sin(ω.sub.d,it) to calculate Φ.sub.I,i(x.sub.j, y) and Φ.sub.Q,i(x.sub.j, y), respectively to yield

[00044]$\begin{array}{cc}{u}_i\left(x_j,y,t\right)\cos \left({\omega }_{d,i}t\right)={\Phi }_{I,i}\left(x_j,y\right){\cos }^2\left({\omega }_{d,i}t\right)+{\Phi }_{Q,i}\left(x_j,y\right)\sin \left({\omega }_{d,i}t\right)\cos \left({\omega }_{d,i}t\right)=\frac{1}{2}{\Phi }_{I,i}\left(x_j,y\right)+\frac{1}{2}{\Phi }_{I,i}\left(x_j,y\right)\cos \left(2{\omega }_{d,i}t\right)+\frac{1}{2}{\Phi }_{Q,i}\left(x_j,y\right)\sin \left(2{\omega }_{d,i}t\right),& \left(92\right)\end{array}$$\begin{array}{cc}{u}_i\left(x_j,y,t\right)\sin \left({\omega }_{d,i}t\right)={\Phi }_{I,i}\left(x_j,y\right)\sin \left({\omega }_{d,i}t\right)\cos \left({\omega }_{d,i}t\right)+{\Phi }_{Q,i}\left(x_j,y\right){\sin }^2\left({\omega }_{d,i}t\right)=\frac{1}{2}{\Phi }_{Q,i}\left(x_j,y\right)+\frac{1}{2}{\Phi }_{I,i}\left(x_j,y\right)\sin \left(2{\omega }_{d,i}t\right)-\frac{1}{2}{\Phi }_{Q,i}\left(x_j,y\right)\cos \left(2{\omega }_{d,i}t\right).& \left(93\right)\end{array}$