FIELD OF THE INVENTION

The present invention relates to modal analysis, and more particularly to a computer- implemented method for modal analysis (such as a method for extracting vibration properties from vibration data and/or a computer implemented method for extracting structural properties from vibration measurements), a method for measuring vibration response and for modal analysis, a computer program, and vibration data processing apparatus (such as a digital apparatus for identifying modal properties from measured vibration data) and a system for obtaining free decay vibration response data and for modal analysis.

BACKGROUND OF THE INVENTION

Modal analysis, such as experimental modal testing, may be carried out to identify dynamic properties (hereinafter also referred to as modal properties) of tested physical structures from vibration data acquired by means of vibration sensors. These modal properties may then be used, among other purposes, to assess the structural performance of the tested physical structures when subjected to dynamic loads, such as wind, waves, machinery vibrations, traffic (e.g ., on a bridge), etc. In structural engineering, engineers are frequently confronted with the challenge of designing structures capable of withstanding the vibration loads induced during operation and/or environmentally induced vibration loads, and for this purpose rely on modal analysis methods. Therefore, a modal analysis method enabling more accurate estimations of the dynamic properties is highly relevant, e.g ., with a view to secure a safe performance and a proper serviceability over the lifetime of the physical structures.

Hence a method for modal analysis, which is more accurate and/or more robust would be advantageous.

SUMMARY OF THE INVENTION

It may be seen as an object of the present invention to provide a method for modal analysis which is more accurate and/or more robust. It may be a further object of the present invention to provide an alternative to the prior art.

Thus, the above-described object and several other objects are intended to be obtained in a first aspect of the invention by providing a computer-implemented method for modal analysis in the frequency domain of a physical structure, said computer-implemented method comprising :

• Obtaining free decay vibration response data expressed in the frequency domain for the structure,

• Formulating frequency domain equations, such as frequency domain analytical equations, based on the free decay vibration response data comparing two or more points in the frequency domain, such as separated by a single frequency step,

• Formulating a single or multiple order eigenvalue problem based on the formulated frequency domain equations, and

• Solving the single or multiple order eigenvalue problem thereby obtaining one or more modal parameters, such as one or more or all of a. one or more mode shapes, b. one or more natural frequencies, c. one or more modal participation vectors, and/or d. one or more damping ratios.

The invention may be advantageous for providing a computer-implemented method for modal analysis, which is accurate and/or robust. Another possible advantage may be that it enables dispensing with a need for (polynomial) fitting.

A possible advantage of this method may be that it enables performing an assessment of a physical modification of a physical structure, possibly enabling improving its structural dynamics and/or assessing the effect of the modification on the structural dynamics (and possibly verify an improvement), which may in turn be beneficial, e.g., for enabling (verifying) improving structural performance, safety and/or integrity of the physical structure.

It may be seen as an insight of the present inventors, that the present method yields accurate and/or robust results and/or enables dispensing with a need for a prior (polynomial) fitting. The invention and/or embodiments of the invention may be seen as corresponding to applying in the frequency domain a framework originally developed in the 1970'ies for the time domain and referred to in the art as the 'Ibrahim Time Domain (ITD) method', and it is surprising that adopting a method developed in another period of time and for another domain yields one or more of the above mentioned advantages. 'Modal analysis' is understood as is common in the art, such as the study of the dynamic properties of a system, such as in the frequency domain, such as a process of extracting modal parameters, such as natural frequencies, damping ratios (or loss or energy dissipation factors) and modal participation factor vectors (or operational factor vectors) from measured vibration data. 'Modal participation' is generally understood to be used interchangeably with 'modal participation factor'.

'Free decay vibration response data' is understood as is common in the art, such as wherein the term free vibration is used to indicate that, apart from optionally an external short duration perturbation, there is no external force causing/driving the motion, and that the motion is primarily or exclusively the result of initial conditions, such as an initial impulse or initial displacement of the mass element of the system from an equilibrium position and/or an initial velocity and/or acceleration. 'Data' is understood as is common in the art and may be understood to be digital data.

By 'obtaining free decay vibration response data' is understood any way of obtaining said data encompassing, but not limited to, measuring said data (either in displacement, velocity and acceleration), such as with the aid of vibration sensors. In embodiments, obtaining is given by measuring or receiving, such as receiving the data on a storage medium or at a data input port.

'Time domain' is understood as is common in the art, such as relating to data and/or functions arranged with respect to time, such as being temporally resolved.

'Frequency domain' is understood as is common in the art, such as relating to data and/or functions arranged with respect to frequency, such as being spectrally resolved.

'Frequency domain equations' or 'frequency domain analytical model' is understood as is common in the art, such as an equation describing displacement or velocities or accelerations as function of frequency. 'Formulating (an equation)' is understood as is common in the art, such as establishing or writing the equation and/or assigning numerical values to constants/invariants in the equation.

By 'comparing two or more points in the frequency domain on the free decay vibration response data expressed in the frequency domain' may be understood that two or more frequency domain equations are formulated, wherein each of said formulated frequency domain equations are formulated for a unique, such as a unique and discreet, frequency line (i.e., at least two frequency lines are different with respect to each other). 'Formulating a single or multiple order eigenvalue problem based on the formulated frequency domain equations' and 'solving the single or multiple order eigenvalue problem thereby obtaining one or more modal parameters' are each understood as is common in the art (and is exemplified elsewhere in the present application). It is understood that solving an eigenvalue problem might yield all of one or more mode shapes, one or more natural frequencies, one or more modal participation vectors, and one or more damping ratios.

Formulating an eigenvalue problem may be done solely from the frequency domain equation/function (e.g., the frequency response function or half spectrum) and optionally the Z-domain variable evaluated in all the spectral lines in the frequency range of interest.

According to an embodiment, there is presented the computer-implemented method, wherein the modal analysis in the frequency domain is based on a frequency domain analogue of an Ibrahim Time Domain method, such as comprising formulating equations in the frequency domain being analogues to Ibrahim equations in the time domain, such as Ibrahim equations formulated in the frequency domain, such as wherein different frequency lines are being used to formulate the eigenvalue problems. This formulation can be carried out for a single or multiple model order.

'Ibrahim Time Domain method' (ITD) is understood as is common in the art, such as described elsewhere in the present application and/or in chapter 9.4 of "Introduction to Operational Modal Analysis", Carlos Ventura, Rune Brincker, John Wiley & Sons, 2015, which is hereby incorporated by reference in entirety. A corresponding (analogue) method in the frequency domain may generally be referred to as Ibrahim Frequency Domain (IFD) method.

By 'a frequency domain analogue of an Ibrahim Time Domain method' reference is made to a method, which is similar to the Ibrahim Time Domain method except that it is expressed in the frequency domain (as is exemplified elsewhere in the present application).

According to an embodiment there is presented the computer-implemented method, where

• formulating the single or multiple order eigenvalue problem based on the formulated frequency domain equations comprises formulating equations in the frequency domain being analogues to Ibrahim equations in the time domain, such as Ibrahim equations formulated in the frequency domain, such as with the purpose of extracting the modal properties (e.g., natural frequencies, damping rations, mode shape vectors, modal participation factor vectors) from the measured frequency domain data, and/or where • formulating the single or multiple order eigenvalue problem based on the formulated frequency domain equations comprises using a plurality of, such as neighbouring, points in the frequency domain in an analogue way to the Ibrahim Time Domain method, such as comprising formulating equations in the frequency domain being analogues to Ibrahim equations in the time domain, such as Ibrahim equations formulated in the frequency domain, to formulate the single or multiple order eigenvalue problem, and optionally where

• solving the single or multiple order eigenvalue problem thereby obtaining one or more modal parameters is done with higher accuracy than obtainable with the Ibrahim time domain method.

'Ibrahim equations' are understood as is common in the art, such as two or more equations expressed in the time domain and describing free decay vibration as a function of time for two or more free decay time samples, such as Eq. (20). While 'Ibrahim equations' are strictly referring to equations formulated in the time domain, the use of 'Ibrahim equations' may in certain instances within the present application be understood (from the context, e.g., via the use of terminology identical similar to 'Ibrahim equation(s) in the frequency domain') that this reference (to Ibrahim equations) is being made to equations in the frequency domain being analogues to Ibrahim equations in the time domain.

By 'equations in the frequency domain being analogues to Ibrahim equations in the time domain' may be understood two or more equations expressed in the frequency domain, such as describing free decay vibration as a function of frequency, for two or more frequency lines, wherein the structure of the equation is similar to the Ibrahim equations, except that the reference to time has been exchanged with reference to frequency, such as in Eq. (29) and/or Eq. (36).

'Eigenvalue problems' are understood as is common in the art, such as in equations (31) and (38).

By 'higher accuracy' is understood that the one or more modal parameters is closer to the modal parameters true value. For example, a difference between an eigenfrequency determined according to this embodiment and the true eigenfrequency of the physical system is smaller than an eigenfrequency obtainable with the Ibrahim time domain method and the true eigenfrequency of the physical system.

According to an embodiment there is presented the computer-implemented method, further comprising providing graphical data relating to the modal analysis, such as • One or more illustrations of individual modal contribution, such as plots of the magnitude of the frequency domain modal decomposition (or modal coordinates),

• one or more phase plots, such as plots of the phase of the frequency domain modal decomposition (or modal coordinates), and/or

• one or more stabilization diagrams, such as to illustrate the robustness and accuracy of the modal properties' estimates obtained with models with increasing order.

A possible advantage of providing graphical data relating to the modal analysis may be that it enables a user (such as enables a user in an efficient, fast, simple, easy and/or convenient manner) to obtain information on the technical condition of the physical structure. The information may in turn enable the user to interact with the physical structure, for example to avoid technical malfunctions.

'Illustrations of individual modal contribution' may be understood as is common in the art, such as a graph showing modal contribution as a function of frequency. 'Modal contribution' may be understood as interchangeable with 'modal coordinates'.

'Phase plots' may be understood as is common in the art, such graphs showing phase as a function of frequency.

'Stabilization diagrams' may be understood as is common in the art, such as a diagram wherein poles are estimated with models with increasing order are plotted. It may be advantageous for identifying physical poles.

According to an embodiment there is presented the computer-implemented method, where the equations in the frequency domain being analogues to Ibrahim equations in the time domain, such as Ibrahim equations formulated in the frequency domain, are formulated using a Z-transform or a Laplace-transform. A possible advantage of the Z-transform may be that a formulation with the Z-transform is physically intuitive. The Laplace-transform may also be known as the S-transform.

According to an embodiment, there is presented the computer-implemented method, where the equations in the frequency domain being analogues to Ibrahim equations in the time domain in the frequency domain are formulated using Fourier transform.

According to an embodiment there is presented the computer-implemented method, where the equations in the frequency domain being analogues to Ibrahim equations in the time domain, such as Ibrahim equations formulated in the frequency domain, are formulated in the frequency domain using :

• one or more input-output time domain free decays analogues in the frequency domain, such as frequency response functions or transfer functions, and/or

• one or more output-only time domain free decays analogues in the frequency domain, such as power spectral density functions or half spectral density functions.

By 'input-output time domain free decays analogues in the frequency domain' may be understood analogues to functions for controlled loading in the time domain, wherein said analogues are applicable in the frequency domain.

By 'output-only time domain free decays analogues in the frequency domain' may be understood analogues to functions for random loading in the time domain, wherein said analogues are applicable in the frequency domain.

According to an embodiment (referred to as the Frequency Domain Eigen Realization Algorithm (FERA)) there is presented the computer-implemented method, where the equations in the frequency domain being analogues to Ibrahim equations in the time domain are formulated in the frequency domain using a formulation of the eigenvalue problem that is a frequency domain analogue to the Eigensystem Realization Algorithm (ERA) technique in the time domain.

The Frequency Domain Eigen Realization Algorithm (FERA) is the frequency domain analogue of the Time domain ERA. Similarly to the embodied pCF-MM/IFD the main advantage of the FERA with regard to its time domain counterpart is that the former provides clearer stabilization diagrams, which enables more accurate estimates for modal properties of tested structural system.

According to an embodiment there is presented the computer-implemented method comprising a modal model parameter estimation, which is carried out by estimating the modal parameters, such as the mode shapes <p _r and the continuous time poles, _r , (and, in turn, the natural frequencies and damping ratios), directly from the measured vibration data, such as directly from the free decay function Y _k , optionally via the solving of the Eigenvalue problem, such as in a single step and/or with no prior (polynomial curve) fitting. Fitting may be a problem, e.g., in the frequency domain since it might lead to the estimation of the so-called non-physical (or numerical) modal properties which may compromise the estimation's robustness and accuracy of the identified physical modal properties. The present embodiment may overcome said problem by relying on a modal model parameter estimation, which dispenses with a need for fitting. Another possible advantage may be that an additional step (i.e., the fitting step) is dispensed with, which implies that a more effective, robust and/or accurate approach may be provided.

According to an embodiment there is presented the computer-implemented method, wherein modal identification is automated, such as by a clustering-based algorithm.

According to an embodiment there is presented the computer-implemented method, wherein the method comprises:

• Obtaining free decay vibration response data expressed in the time domain for the structure, and

• Obtaining free decay vibration response data expressed in the frequency domain for the structure by transforming the free decay vibration response data expressed in the time domain to the frequency domain, such as using Z-transform or a Laplacetransform, and

• Optionally formulating an eigenvalue problem solely from the frequency domain function (e.g., the frequency response function or half spectrum) and the Z-domain variable evaluated in all the spectral lines in the frequency range of interest.

For example, there is obtained free decay vibration response data expressed in the time domain, which is then transformed into free decay vibration response data expressed in the frequency domain. This may be advantageous for avoiding having to provide free decay vibration response data expressed in the frequency domain prior to applying the embodiment, i.e., free decay vibration response data expressed in the time domain can be obtained, such as measured, and the method according to the present embodiment can in turn obtain those data and then transform them into the frequency domain.

According to a second aspect there is presented a method for measuring a vibration response and for modal analysis in the frequency domain of a physical structure, said method comprising : • Measuring a vibration response of the physical structure, such as measuring a vibration response with one or more sensors arranged for determining a movement of at least a part of the physical structure,

• Based on said vibration response, carrying out the modal analysis according to the computer-implemented method for modal analysis in the frequency domain of a structure according to the first aspect.

Measuring a vibration response can be carried out in any means yielding vibration response data, such as via one or more sensors (such as an accelerometer and/or a strain gauge) attached on or to the physical structure or measuring at a distance (such as a laser Doppler vibrometer), and capable of measuring displacement, velocity, or acceleration at points that cover a part or all of the physical structure.

According to an embodiment there is presented the computer-implemented method, where equations in the frequency domain being analogues to Ibrahim equations in the time domain, such as Ibrahim equations formulated in the frequency domain, are formulated in the frequency domain characterized by:

• Providing controlled input to the physical structure preceding or simultaneously with measuring, such as measuring a vibration response of the physical structure, and using free decays obtained in the time domain using the controlled input in order to estimate the free decays, such as impulse response functions, or

• Providing known random input to the physical structure simultaneously with measuring both the random input and the vibration response and using the free decays obtained by estimating the transfer functions of the physical structure, or

• Providing random input, such as unknown random input, to the physical structure preceding with measuring only the random responses and using free decays obtained in the time domain by estimating correlation functions of random responses of the physical structure.

By 'controlled input' may be understood controlled and/or known excitation, such as controlled force and/or know initial speed (or velocity) and/or displacement.

By 'known random input' may be understood controlled, known and measured excitations, such as random forces generated by mechanical exciters, also known by modal exciters.

By 'unknown random input' may be understood non-controlled and/or unknown excitation, such as excitation by wind and/or environmental vibrations. According to an embodiment there is presented the computer-implemented method, said method comprising :

(a) Carrying out a measurement of a vibration response and a modal analysis in the frequency domain of the physical structure according to the second aspect prior to a modification of the physical structure,

(b) Performing the physical modification of the physical structure,

(c) Carrying out a measurement of a vibration response and a modal analysis in the frequency domain of the physical structure according to the second aspect subsequent to the modification of the physical structure,

(d) Optionally comparing a result of the modal analysis of step (a) with a result of the modal analysis of step (c).

A possible advantage of this method may be that it enables performing a physical modification of the physical structure with a view to improve its structural dynamics and to assess the effect of the modification on the structural dynamics (and possibly verify an improvement), which may in turn be beneficial, e.g., for enabling (verifying) improving structural performance, safety and/or integrity of the physical structure.

According to a third aspect there is presented a computer-program and/or a computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out the computer-implemented method of the first aspect. Such a computer program and/or computer program product may be provided on any kind of computer readable medium or through a network.

According to a fourth aspect there is presented a data processing apparatus, such as a computer, such as a computer comprising or having access to the computer program and/or computer program product according to the third aspect, comprising a processor adapted to perform the computer-implemented method of the first aspect.

According to a fifth aspect there is presented a system for obtaining free decay vibration response data and for modal analysis, said system comprising :

• one or more sensors arranged for determining a displacement, velocity and/or acceleration of at least a part of the physical structure, and

• a data processing apparatus according to the fourth aspect, said system being arranged to execute the steps of the second aspect. BRIEF DESCRIPTION OF DRAWNGS

The first, second, third, fourth and fifth aspect according to the invention will now be described in more detail with regard to the accompanying figures. The figures show one way of implementing the present invention and is not to be construed as being limiting to other possible embodiments falling within the scope of the attached claim set.

Figure 1 shows North (left) and East (right) building facades of the Heritage Court Tower in Vancouver Downtown, Canada.

Figure 2 shows a stabilization diagram created with the embodiment designated pCF- MM/IFD by identifying models with orders, m, ranging from 1 to 40.

Figure 3 shows a stabilization diagram created with the pLSCF (also known as PolyMAX) technique by identifying models with orders, m, ranging from 1 to 40.

Figure 4 shows a stabilization diagram created with the embodiment designated FERA by identifying models with orders, r, ranging from 1 to 15.

Figure 5 shows a stabilization diagram created with the embodiment designated FERA technique by identifying models with orders, r, ranging from 1 to 10.

Figure 6 shows plots of the frequency-domain modal coordinates, H _m (s _a ), obtained with the embodiment pCF-MM (a) and with the with the classic FDD (b).

Figure 7 shows a photo of the T-Structure specimen (left), dimensions (center) and measured locations and directions (right).

Figure 8 shows plots of the frequency-domain modal coordinates, H _m (s _a ), obtained with the embodiment pCF-MM (a) and with the with the classic FDD (b).

Figure 9 shows the City Crest Tower, South-East (left) and South-West (right) facade.

Figure 10 shows measured directions and locations with 19 different datasets.

Figure 11 shows a stabilization diagram created with the embodiment designated pCF- MM/IFD by identifying models with orders, m, ranging from 1 to 50. Figure 12 shows vibration modes of the City Crest Tower identified with the embodiment designated pCF-MM/IFD.

DETAILED DISCLOSURE OF THE INVENTION

1. EXPERIMENTAL MODAL ANALYSIS THEORY

In experimental modal analysis, the main goal is to extract the modal properties from vibration measurements collected in vibration tests of structural systems. The modal properties extraction (or modal identification) can be carried out either in time domain or frequency domain. In the former, a free decay function is normally used as primary data. In this case, the modal properties are computed basically by fitting an analytical model to the measured free decay function. A time-domain free decay function containing the information of N _t inputs and N _o outputs can be modelled by the time domain modal model, as: (1) where is the mode shape matrix; A is a diagonal matrix containing the discrete-time poles; r is the modal participation factors matrix; the sub-index k denotes the discrete time (t = kAt) with At designating the sampling interval at which the free decay is recorded and (.) ^T denoting the matrix transpose operator. The matrices Φ , A and r have the following structure

(2) where (•)* designates the conjugate of a complex matrix, φi and y are, respectively, the mode shape vector and the modal participation vector corresponding to the i ^th vibration mode, and μi. = is the corresponding discrete time pole, with λi denoting the continuous-time poles which is related to the angular natural frequencies and the damping ratios , as

(3) where = 2πf , with f denoting the natural frequency in Hertz (cycles/sec). The free decay

Eq. (1) can be re-written in a partial form, i.e., as sum of the contributions of all vibration modes, as (4) where (.) ^H denotes the conjugate transpose (Hermitian) of a complex matrix. The free decay model as in Eq. (1) can be converted to the frequency domain by making use of the Laplace Transform, also known as S-Transform, or the Z-Transform. By doing so, and assuming zero initial condition for the free decay (Y _o = 0), the following frequency domain functions are obtained

(5)

(6) in Laplace domain and Z-domain, respectively, where denotes a diagonal matrix containing the continuous-time poles λ _ir s = jω and z = ejωΔ ^t are, respectively, the Laplace domain and the z-domain variables evaluated at the frequency ω. Eq. (5) and (6) can be re-written in a partial fraction form (i.e., as a sum of the modal contributions), as

(7)

(8)

It is worth mentioning that both frequency functions H(s) and H(z) denote both the so-called Frequency Response Function (FRF) and Half Spectrum (HS) in Z- and S-domain (Laplace domain). The poly-reference formulation of the classic Ibrahim Time Domain (ITD) described in the next section takes advantage of the fact that the free decay as in Eq. (1) can be shifted (positively or negatively) in the time axis. If a forward (or positive) discrete time shift r (which corresponds to t = rAt) is considered, Eq. (1) becomes (9) and the corresponding Z-transform assuming zero initial condition for the free decay (K _o = 0) (10)

Eq. (10) is central in the formulation of the embodiment of the invention designated as the poly-reference Complex Frequency-domain Modal Identification Technique Formulated in Modal Model pCF-MM (Ibrahim Frequency Domain (IFD)) described in detail in section 4. This equation follows from the property of the Z-transform in which a forward time shift of r in time domain corresponds to a multiplication by z ^r in the frequency domain, i.e.,

Another property of the free decay function that can be explored in the formulation of the pCF- MM (IFD) is related to its derivatives. The r ^th derivative of the free decay function as in Eq. (1) is given by (11) and the corresponding Laplace transform (12)

Similarly to the case of Eq. (10), Eq. (12) follows from the property of the Laplace Transform in which the r ^th derivative in the time domain corresponds to a multiplication by s ^r in the frequency domain, i.e.,

Actually, any transform can be used in the formulation of the pCF-MM/IFD. It is worth noting apart from forward time shifts and derivatives as in Eqs. (10) and (12), the backwards time shifts and integrations can also be used in the formulation of the embodied pCF-MM/IFD. Assuming backwards time shifts and integrations, Eqs. (10) and (12) become 2. THE TIME DOMAIN POLY REFERENCE (TDPR) ALSO KNOWN AS POLY-REFERENCE LEAST SQUARES COMPLEX EXPONENTIAL (pLSCE)

The TDPR/pLSCE is a time domain modal identification technique that was invented by Void et al. [1] in the eighties. The technique basically consists of estimating the modal properties from the measured free decay matrix in two steps. First, an Auto Regressive (AR) model is used to fit the measured free decay matrix in a linear least squares sense. Then the matrix coefficients of the fitted AR model are subsequently used to estimate the modal properties, i.e., the mode shape vectors, natural frequencies and damping rations. The AR model for the free decay can be formulated by writing down Eq. (1) for n discrete time shifts and by premultiplying each obtained equation by an AR matrix coefficient A _r (for r = 0,1, ••• , n + 1), yielding ( ₁₃₎

By summing up all the obtained equations, the following AR model is obtained (14)

Now, setting the left-hand side of Eq. (14) equal to zero and writing down the obtained equation for N free decay time samples (i.e., for k = 0,1, •••, N), yields

(15) by imposing A _n = I to force the system of equations to be determined. The system of equations (15) can be re-written in a compact form, as

(16) where

Once w ₀ and W ₁ are formed by using the measured free decays, Y _k , the AR coefficients can be determined in a linear least squares sense, by (17)

Once the AR coefficients A _r are determined by means of Eq. (17), the mode shape matrix and the discrete time poles can be found by setting the left right-hand side of Eq. (14) equal to a zero matrix. The non-trivial solution for the obtained equation is given by

(18)

The solution of Eq. (18) is found by computing the eigenvalue decomposition of the so-called Companion Matrix of the AR coefficients A _r . By forming such matrix, the following eigenvalue problem can be formulated or in a more compact form

(19) where

Is the Companion Matrix of the A _r coefficients and the corresponding eigenvectors. Finally, the mode shape matrix can be computed as 1 ^st block row of V, and the natural frequencies and damping ratios can be retrieved from discrete time eigenvalues A.

3. THE IBRAHIM TIME DOMAIN (ITD) IDENTIFICATION TECHNIQUE The ITD technique was invented by Samir Ibrahim in the 70's [2]. The underlying idea of the technique is to identify the modal properties, i.e., the mode shapes φ _r and the continuous time poles, λ _r , (and, in turn, the natural frequencies and damping ratios) directly from the free decay function Y _k , i.e., with no prior curve fitting as in the case of the TDPR technique discussed in the previous section. The derivation of the technique starts by writing down Eq. (1) for two different (and optionally consecutive) free decay time samples, e.g., for k and k + 1, and isolating r ^T in each equation

(20)

Combining Eqs. (20), the following equations are found

(21) and, finally, by combining Eqs. (21), and assuming a time lag k = 0 the following eigenvalue problem can be formulated

(22)

It is clear from Eq. (22) that θ> and A are determined by computing the eigenvalue decomposition of - (y _k+1 y _k ^r (y _k y _k ) ^_1 + y _k+1 y _k+1 ⁷ '(y _k y _k ) ^_1 ). Following a similar strategy used in the derivation of Eq. (22), a poly-reference implementation of the ITD technique can be formulated by writing down Eq. (1) for two sets of n discrete time shifts from the fc ^th and (fc + l) ^th time lag, i.e.,

The above equations can be written for a set of N time lags k (i.e., for k = 0, 1, 2, ...,N), yielding (23)

Isolating [r ^T Δir ^T Δ ^N r ^T ] in the equations above and comparing the resulting equations, the following expression is obtained

(24) or in a more compact form,

(25) where

Similarly to Eq. (21), an eigenvalue problem can formulated from Eq. (25), yielding (26)

Eq. (26) corresponds to the eigenvalue decomposition of |(M ₁ M _o ^r (M _o M _o ) ^-1 + M ₁ M ₁ ^r (M ₀ M ₁ ) ^-1 ), with M _o and M _} being Hankel matrices shifted by 1 time lag from each other, and >P and A designating the corresponding eigenvector and eigenvalue matrices. Once these matrices are determined, the mode shape matrix is retrieved as the 1 ^st block row of Ψ , and the continuous- time poles, A _r , and the corresponding natural frequencies and damping ratios are computed from the eigenvalue matrix A, as previously described. 4. THE EMBODIMENT OF THE INVENTION DESIGNATED PCF-MM ALSO CALLED IBRAHIM FREQUENCY DOMAIN (IFD) IDENTIFICATION TECHNIQUE

It should be highlighted that similarly to the ITD and pLSCE, the pCF-MM (or IFD) can be formulated either to extract the mode shape or the modal participation factor vectors from the measured frequency-domain function (e.g., FRF or HS). In the following, however, all the pCF- MM related approaches (i.e., the discreet- and continuous-time pCF-MM, the pCF-MM) are derived to extract the mode shape vectors. The key steps in formulating the pCF-MM/IFD are: (i) to use the frequency domain model in Z (or Laplace or Fourier domain), and (ii) to formulate the Ibrahim equations using this function. By using this strategy, a robust and totally new modal identification technique in the frequency domain can be derived, as hereinafter described in detail. Differently from the ITD, the underlying idea of pCF-MM or IFD herein derived is to estimate the modal properties from the measured frequency domain function, i.e., either from the Frequency Response Function (FRF) or the Half Spectrum (HS), which can be computed either by taking the Z-transform, S-transform (or Laplace Transform) or Fourier Transform of the shifted free decay function as in Eq. (9). In the following, however, the pCF-MM/IFD is formulated in Z-domain. For a zero shift (e.g., r = 0), Eq. (10) can be written down for two discrete consecutive frequency lines ω _a and ω _b (ω _b > ω _a ) spaced from each other by single discrete frequency step Aw, as ^{1 1} where, for simplicity, Similarly to the case of the ITD technique described in the previous section, Eqs. (29) can worked out so that the following equations are obtained

For simplicity, these equations can be re-written in more compact form, as ( _3Q) where Finally, Eqs. (30) can be combined to yield the following eigenvalue problem (31)

It is clear from Eq. (31) that and A are determined by computing the eigenvalue decomposition of . It should be highlighted that Eq. (31) is highly relevant in formulations of the present application, since it proves that the vibration properties (i.e., natural frequencies, damping ratios and mode shape vectors) can be extracted from the frequency domain functions estimated from the vibration measurements of the tested structural systems. Following a similar strategy used in the derivation of Eq. (26), a poly-reference implementation of the IFD technique can be formulated by firstly writing down Eqs. (10) for a set discrete forward time shifts, i.e., for r ranging from 0 to n, yielding the following set of equations

These equations can be condensed in the following matrix equation which, in turn, can be re-written as

(32)

Writing Eq. (32) for two frequency lines ω _a and a) _b (ω _b > ω _a ) spaced from each other by a single discrete frequency step Aw, yields the following equations

which is then combined into a single matrix equation as

(33)

Now, writing down Eq. (33) for all the N _f discrete frequency values available in the frequency band, i.e., for ω _a and ω _b ranging, respectively, from ω ₀ to ω _{N f} and from to ω _Nf , and combining the equations corresponding to each pair of evaluated frequency values in a single matrix equation, yields or in a more compact form,

(34) where Using a similar strategy as in the case of the poly-reference ITD, the following eigenvalue problem can be formulated from Eq. (33) (35) where ψ and A are determined by computing the eigenvalue decomposition of

5. ALTERNATIVE EMBODIMENT/FORMULATION FOR THE pCF-MM or IFD TECHNIQUE

As previously mentioned, the invented pCF-MM/IFD can also be formulated by applying the Laplace Transform to Eq. (11), which leads to FRF or HS formulated in Laplace domain. It is observed, however, that the formulation obtained with the Z-transform (described in section 4) is more robust and accurate in terms modal parameter estimates. An alternative formulation in S- (or Laplace) domain for the pCF-MM/IFD can be obtained by using the r ^th derivative of the free decay as in Eq. (12). This equation can be written down for two discrete consecutive frequency lines, ω _a and ω _b ( ω _b > ω _a ) shifted in the frequency axis from each other by single discrete frequency step Aw, as where, for simplicity, s ^r _a = and s _b = . Similarly to the case of the ITD technique described in section 3, Eqs. (36) can be worked out so that the following equations are obtained

These equations can be re-written in a more compact form, as where M _Oab = [H s _b ) - H(s _a )] and M _lab = [s _b H s _b ) - s _a H(s _a )]. Finally, Eqs. (37) can be combined to yield the following eigenvalue problem (38) It is clear from Eq. (38) that and A _c are determined by computing the eigenvalue decomposition of Following a similar strategy used in the derivation of Eq. (35), a poly-reference implementation of the alternative pCF- MM/IFD technique can be formulated by writing down Eqs. (12) for derivative orders r ranging from 0 to n, and subsequently combining the obtained equations. By follow this approach, the following set of equations is obtained

Similarly to the Eq. (32), this set of equations can be combined into single matrix equation, yielding

(39)

Now, writing down Eq. (39) for all the N _f discrete frequency values, i.e., for ω _a and ω _b ranging, respectively, from ω ₀ to ω _N and from to ω _Nf , and combining the equations corresponding to each pair of evaluated frequency values in a single matrix equation, yields or in a more compact form,

(39) where (40)

Similarly to the cases of the poly-reference ITD and the IFD described in sections 3 and 4, an eigenvalue problem can be formulated by using Eq. (39), yielding

(41) where ψ and A _c are determined by calculating the eigenvalue decomposition of

6. PRACTICAL IMPLEMENTATION OF THE pCF-MM (OR IFD) TECHNIQUE

The formulation of the pCF-MM/IFD synthesized by Eqs. (35) and (38) yields mode shape vectors <p _r and the corresponding continuous time poles _r not occurring in complex conjugate pairs as in Eq. (2), which makes it difficult to distinguish the physical from numerical modal properties. In order yield mode shape vectors and poles occurring in complex conjugate pairs, the eigenvalue problem of Eqs. (35) and (38) can be re-formulated as

(42) where Re(») denotes the real value of a complex quantity. In modal analysis, it may be advantageous to plot the so-called stabilization diagram to distinguish the physical properties from the numerical ones. An efficient way of constructing a stabilization diagram with proposed pCF-MM/IFD identification technique comprises the creation the system matrices /V _o and N ₁ (or O ₀ and O _x ) for the maximum model order, n, according to Eq. (35). Once these matrices are formed from the measured frequency domain function (e. g., FRF or HS), the eigenvalues, A _r , and the eigenvectors, Ψ _r , corresponding to the r ^th model order can be computed by evaluating expressions or which are expressed in MATLAB® notation.

7. FREQUENCY-DOMAIN DECOMPOSITION WITH pCF-MM/IFD

The pCF-M M can be used to decompose the frequency domain data into frequency-domain modal coordinates. If one varies ω _a in Eq. (31) within the frequency band of interest, say from

= 0 to a = while keeping ω _b = ω _Nf , then the eigenvalues, A(z _a ) , can be computed at each frequency line ω _a from (43)

Alternatively, Eq. (38) can be used to decompose the frequency domain function. In this case the modal decomposition is obtained by (44)

From this point onwards, the estimation of the natural frequencies and mode shape vectors is obtained by synthesizing the frequency-domain modal coordinates, H _m z _a ), from the eigenvalues, A _c (z _a ), computed at each the frequency line from (45) or (46) where the diag[»] operator stands for a vector containing the elements in the main diagonal of a matrix. Once the frequency-domain modal coordinates, HmZa) or H _m (s _a ), are synthesized from Eq. (45) or (46), the identification of the mode shape vectors and natural frequencies are estimated in a similar manner as with the classic FDD approach, i.e., by picking the peaks in the sorted synthesized plots of the frequency-domain modal coordinates.

8. FORMULATION OF THE FREQUENCY-DOMAIN EIGENSYSTEM REALIZATION ALGORITHM (FERA)

Differently from the pCF-MM/IFD described in the previous section, derivation of the FERA technique starts from the free decay expressed in terms of state space matrices. By making use of the state space formulation [3] it can be shown that the free decay, Y _k , can be also decomposed as (47) where C is the observation matrix and A the discrete-time state-space system matrix and G is a covariance matrix. When forward shifted by r time lags, Eq. (46) can be re-written in a general form, as (48)

By making use of the Z-transform and assuming zero initial condition for the free decay as in Eq. (47), i.e., K _o = 0, it is straightforward to stablish the two-way time-frequency domain transformation for Eq. (48), i.e., (49)

It worth highlighting that the embodied FERA can also be formulated for backward time shifts. In this case, Eq. (48) and (49) become

Now, assuming forward time shifts and writing the frequency-domain version of the free decay of equation with (49), both for different time shifts and frequency lines, i.e., for r = 0, and for z = z _Q ,z ...,z _Nf-1 and combining the resulting expressions in a single matrix equation, yields or in a more compact form

(50) where

An equation similar to (50) can be obtained by writing down the right-hand side of Eqs. (4) for z = z ₁ ,z _2t ... , z _Nf , yielding or in a more compact form

(51) where

Eqs. (50) and (51) can be combined in a single equation, as where N _o is given as in Eq. (34). Eq. (52) can be re-written in a compact form, as

No = or (53) where O and r are the so-called the observability and frequency-domain controllability matrices, respectively. An equation similar to (52) can be written for model orders r = 1, ... , n+1, yielding where N ₁ is given as in Eq. (34) and N (1,1) and N(1,0), respectively, as

Similarly to the case of Eq. (52), Eq. (54) can be decomposed as (55)

8.1. PRACTICAL IMPLEMENTATION

Alternatively, Eqs. (53) and (55) can be re-written to yield eigenvalues and eigenvectors occurring in complex conjugate pairs, as

(56) with Re(») and Im(») denoting the real and imaginary parts of a complex quantity. From this point onwards, it is straightforward to formulate the Frequency-domain ERA (FERA) modal identification method based on equations (56). Similarly to the time-domain ERA, this formulation starts out by taking the Singular Value Decomposition (SVD) of the block matrix N _0Re as in Eqs. (56) N _0Re = USV ^T (57)

By making use of Eqs. (56) and (57), the observability and controllability matrices can be estimated as

(58) respectively. Once these two matrices are estimated, the estimate of the discrete-time statespace system matrix, A, is obtained as

(59) where (») ⁺ denotes the pseudo inverse of a matrix. Similarly to the time-domain ERA, a convenient way of estimating A in Eq. (59) is by selecting only the first n singular values in S that correspond to n/2 vibration modes, and the corresponding singular vectors in U and V. If this approach is followed, a reduced discrete time state-space system matrix is obtained by evaluating the following expression

(60)

Once A is estimated from Eq. (60), the corresponding eigenvalues and eigenvectors are obtained by means of the eigenvalue decomposition (61)

The estimation of mode shape matrix from V is carried out in two steps. First, the observation matrix, C, is estimated from where W = [I _No 0], with I _N denoting an identity matrix with dimensions N _o x N _o . Finally, the mode shape matrix, 4>, can be estimated as (62)

9. APPLICATION EXAMPLES

Application example 1: Multiple model order identification from Heritage Court Tower (HTC) Half Spectral Matrix (Dataset 1) with pCF-MM/IFD

Figure 1 shows North (left) and East (right) building facades of the Heritage Court Tower in Vancouver Downtown, Canada. The HCT is a relatively regular 15 story reinforced concrete shear core building located at the corner of Hamilton and Robson in Vancouver, British Columbia, Canada. The HCT vibration measurements is a difficult multi-dataset application example because is based only on two reference sensors. The original publication of the ambient vibration test of the HCT is found in [32], and a full operational modal analysis was presented in the same year of this publication in [16].

Figure 2 shows a stabilization diagram created with the embodiment designated pCF- MM/IFD by identifying models with order, m, ranging from 1 to 40.

Figure 3 shows a stabilization diagram created with the pLSCF (also known as PolyMAX) technique by identifying models with order, m, ranging from 1 to 40.

Application example 2: Multiple order identification from Heritage Court Tower (HTC) Half Spectral Matrix (Dataset 1) with FERA

Figure 4 shows a stabilization diagram created with the embodiment designated FERA by identifying models with order, r, ranging from 1 to 120. Application example 3: Multiple order identification from Heritage Court Tower (HTC) Half Spectral Matrix (Dataset 2) with FERA

Figure 5 shows a stabilization diagram created with the embodiment designated FERA technique by identifying models with orders, r, ranging from 1 to 160.

Application example 4: Frequency-domain Modal Decomposition of Heritage Court Tower (HTC) Half Spectral Matrix (Dataset 2)

Figure 6 shows the magnitude (a) and phase angle (b) plots of the frequency-domain modal coordinates, H _m (s _a ), obtained with the embodiment pCF-MM and the Power Spectral Density (PSD) singular values (c) computed with the classic FDD [32].

Application example 5: Frequency-domain Modal Decomposition of the T-Structure Half Spectral Matrix

Figure 7 shows a photo of the T-Structure specimen (left), dimensions (center) and measured locations and directions (right), where both dimensions and locations are given in millimeters (mm).

Figure 8 shows plots of the magnitude (a) and phase angle (b) of frequency-domain modal coordinates, H _m (s _a ), obtained with the embodiment pCF-MM and the PSD singular values (c) computed the with the classic FDD [32].

Application example 6: Multi-dataset identification City Crest Tower in Vancouver, Canada with the invented pCF-MM/IFD

Figure 9 shows the City Crest Tower, South-East (left) and South-West (right) facade.

Figure 10 shows measured directions and locations with 19 different datasets (where the roving sensors are placed exclusively in the upper row, at levels 2-25, and the reference sensors are placed exclusively in the lower row, at level 29).

Figure 11 shows a stabilization diagram created with the embodiment designated pCF- MM/IFD by identifying models with order, m, ranging from 1 to 50. Figure 12 shows vibration modes of the City Crest Tower identified with the embodiment designated pCF-MM/IFD.

Although the present invention has been described in connection with the specified embodiments, it should not be construed as being in any way limited to the presented examples. The scope of the present invention is set out by the accompanying claim set. In the context of the claims, the terms "comprising" or "comprises" do not exclude other possible elements or steps. Also, the mentioning of references such as "a" or "an" etc. should not be construed as excluding a plurality. The use of reference signs in the claims with respect to elements indicated in the figures shall also not be construed as limiting the scope of the invention. Furthermore, individual features mentioned in different claims, may possibly be advantageously combined, and the mentioning of these features in different claims does not exclude that a combination of features is not possible and advantageous.

REFERENCES

[1] Void, H, Kundrat, J, Rocklin, GT, et al. A multi-input modal estimation algorithm for minicomputers. SAE paper 1982- 820194, 1982.

[2] Ibrahim, SR, and Milkulcik, EC. A method for direct identification of vibration parameters from the free response. Shock Vib Bull 1977; 47 : 183-196.

[3] Juang J.-N. Applied System Identification. Prentice Hall, Englewood Cliffs, NJ, USA, 1994.

[16] Brincker, R., Zhang, L., and Andersen, P., Modal Identification from Ambient Responses Using Frequency Domain Decomposition. In proceedings of the 18th International Modal Analysis Conference, San Antonio, TX., USA (2010)

[32] Ventura, C.E, and Horyna, T., Measured and calculated modal characteristics of the Heritage Court Tower in Vancouver, B.C. In Proceedings of the 18th International Modal Analysis Conference (IMAC), (pp. 1970-1974) (2000)