articleepubs.surrey.ac.uk/849950/1/pddr final submission.docx · web viewthe logarithmic decrement...

Probability Distribution of Decay Rate: a statistical time-domain damping parameter for structural damage

identificationAli M. Aya, Suiyang Khooa, Ying Wangb*

a Deakin University, 75 Pigdons Road, Waurn Ponds, VIC, 3216, Australiab University of Surrey, Guildford, Surrey, GU2 7XH, the United Kingdom

Abstract

This paper proposes a novel vibration-based damage identification method, named the probability distribution of decay-rate (PDDR). By introducing a statistical framework, the PDDR method estimates damage-induced changes in overall damping behaviour of a free-vibration dynamic system. Utilising free-vibration impulse-response (IR) data, a one-dimensional dataset of local maxima-minima points is constructed. A statistical analysis of this dataset is then performed to derive damage-sensitive parameters. It is demonstrated that through the use of a statistical analysis framework, a number of enhancements are attained in terms of both robustness and leniency in estimating the significantly nonlinear property of overall damping. An impact hammer test is conducted in the laboratory to verify the efficacy of the proposed PDDR method. The test was performed on a scale-model steel Warren truss bridge structure, subjected to bolt-connection failures. The comparison results between the PDDR method and the standard experimental modal analysis (EMA) method confirm that the former is effective for damage identification of complex structures, particularly with real experimental data and steel-frame structure assemblies.

Keywords: PDDR; probability distribution; variance; impact hammer; damping; decay-rate; structural damage; Statistical analysis; Warren-truss;

1. Introduction

Any modern society is inevitably reliant on the performance of civil infrastructure systems. This reliance combined with a demand in the enhancement of the load-carrying capacity and the service-life leads to the increasing attention for structural health monitoring (SHM). Popular for its efficiency and ease of use, the impact hammer test method in civil SHM has taken a significant role, particularly for small to medium-sized structures. An impact hammer test involves an impulse excitation delivered to a test structure, and then its impulse-responses (IRs) are recorded. After processing measured IRs, a transfer function is applied, allowing for the extraction of modal parameters. Generally, by assuming a linear time-invariant (LTI) system model, extracted modal parameters are precisely analysed for an accurate correlation with discernible physical changes in the test structure. This procedure is fundamentally based upon the link between the physical state of a structure to its resultant dynamic behaviour. Specifically, structural and physical damages cause a reduction in overall stiffness and an increase in structural flexibility; both of which are observable via dynamic properties [1].

However, despite numerous studies in literature employing this foundational relationship for vibration-based damage identification (VBDI), the modal damping property is seldom utilised as a damage-sensitive property [2, 3], compared to its alternatives of natural frequencies and

* * Corresponding author. Tel.: +44-1483684090E-mail address: [email protected] .

mailto:[email protected]

2

mode shapes. Yet a number of studies have endeavoured or at least noted its potential. These researchers were able to provide plausible results to support the scheme that changes in modal damping ratios serve as an effective and significantly sensitive damage indicator.

One of the earliest studies, Savage and Hewlett [4] presented an experimental validation of a non-destructive testing method on reinforced concrete beams subjected to crack damages. Changes in damping ratios of up to 80% were reported before and after the subjected damage of shear failures near the support of the elongated concrete beams, outperforming the feature of changes in natural frequencies in regards to sensitivity. Salane and Baldwin Jr [5] explored the potential of modal damping, plus two other modal parameters, for damage identification on both scale-model and full-scale bridges. Discernible changes in modal damping ratios were reported on both tests. However, the scale-model bridge’s damping ratio estimates exhibited a consistently positive trend against damage severity, while those of the full-scale bridge did not. Peroni et al. [6] examined the changes in modal damping of sandwich panels subjected to debonding-type damages. The experimentally derived modal damping values from the first few modes indicated a slight increase in damping coefficients with increased damages. Montalvão et al. [7] developed a method to localise damage on CFRP plates using the damping parameter. It was concluded that a number of uncertainties are involved in accurate damping estimation. Mustafa et al. [8] proposed an energy-based damping evaluation method. The results demonstrated it can largely enhance the accuracy and reliability of structural damage identification, even on a complex steel truss bridge. It is also noted that real structures often do not exhibit viscous damping behaviour. Li et al. [9] used the Rayleigh damping model in a sensitivity-based model updating method for damping identification. The results showed the proposed approach can identify the damping ratios of both the first and second modes simultaneously. A more detailed review can be found in Cao et al. [10].

Collectively, the reviewed studies have highlighted a number of limitations which hinder the utilisation of modal damping parameters as a damage-sensitive feature. Firstly, it is often difficult to determine active damping forces in analytical models, as opposed to the physical properties of mass and stiffness. Secondly, the estimation of modal damping parameters suffers issues such as individual mode identification with mode coupling and crossovers, mode validation, consistency in presence of resonant peaks, etc. Acknowledging these limitations, an alternative class of methods, known as time-domain analysis has been proposed.

The logarithmic decrement method is the simplest method to estimate the damping ratio or decay-rate of an IR waveform. However, it is only applicable to single-degree-of-freedom (SDOF) systems, essentially resulting in a single damping ratio expression of the exponential decay-rate [11]. Liao and Wells [12] and Ge and Sutherland [13] are noted as incremental steps forward, where previously identified resonant frequencies were isolated in frequency sub-bands, using such band-pass filters as Butterworth filtering. The logarithmic decrement was then applied individually to resultant mode-isolated time-domain waveforms. Although notably accurate in certain experimental studies, an underlying limitation is a requirement for the prior knowledge regarding locations of confirmed resonant frequencies. Further, the Butterworth filtering requires widely spaced modes with no mode coupling, which limits its application to damage detection in complex structures.

An advancement to the direct signal filtering is to fit the Impulse Response Functions (IRFs) directly onto the measured transient acceleration responses, which leads to the identified resonant frequencies as best fits across a specified bandwidth. The methods in this category include Smith Least Squares (SLS) algorithm [14], Limit Envelopes technique [15], Linear Prediction Singular Value Decomposition (LPSVD), and the Least Squares Complex

3

Exponential (LSCE) method [16].Two other methods that implement a certain type of transformation technique to extract

modal parameters are the Hilbert-Huang Transform (HHT) [17], and Wavelet Transform method [18]. For example, Curadelli et al. [18] proposed a structural damage identification method using the instantaneous damping coefficient, acquired via Wavelet transform technique applied to transient acceleration responses. It presented through both numerical and experimental results that damage resulted in substantial variations in coefficients, characterising damping of experimentally measured systems.

In summary, time-domain methods for damping estimation are implemented in one of three ways: (1) filtering techniques; (2) direct-fitting approaches to identify resonant frequencies; and (3) decomposition techniques via time-frequency analysis. The core objective in all three categories of analysis is to first deconstruct a mode-superposed IR signal, and then to estimate the identified damping ratio parameters per identified mode. The discernible changes in identified damping parameters are then correlated to structural damages in the dynamic system.

In this study, a novel statistical analysis method is explored for damping estimation and is further developed as a damage identification methodology denoted as the probability distribution of decay-rate (PDDR). It aims to circumvent the deconstruction step of a mode-superposed IR measurement, which is usually challenging for complex structures. Assuming that a structure is a multi-degree-of-freedom (MDOF) linear viscously-damped system, the proposed PDDR method represents the system’s overall damping characteristic as a single statistical variance parameter of established sample datasetU , constructed using time-domain IR measurements. This paper firstly introduces the theoretical background and implementation of the PDDR method, and subsequently, verifies the method through an experimental study on a scale Warren-truss bridge model.

2. Theoretical Framework

2.1. Background

For an underdamped, LTI MDOF free-vibration system, the general displacement function can be stated as a discretised Q number of constituent modes of vibration, as follows:

x (t )=∑i=1

Q

A i e−ζi ωni

t sin (√1−ζ i2 ωni

t+ψ i ) (1)

where x (t) is the displacement at time t ; Ai represents the response amplitude per vibration mode i; ζ i is the damping ratio of mode i; ωni

is the circular frequency of mode i; and ψ i is the phase of mode i. Similarly, the corresponding velocity x (t ), and acceleration transient responses x (t ), can be provided as a summation of modes of vibration, derived from Eq. (1).

Regardless of specific response function x (t ) ; x (t ); x (t), the decay-rate element of such a system can be expressed using the envelope function Aenv ( t ):

AT .env (t )=∑i=1

Q

Aenv ie−ζi ωni

t (2)

where AT .env ( t ) denotes a summation of all exponential decay envelopes Aenv i. To obtain

accurate estimations of the damping ratiosζ i, per identified mode, the surveyed literature of time-domain methods provide either filtering, direct fitting, or time-frequency analysis

4

approaches. The simplest one is the logarithmic decrement method, but it is only applicable under the condition of a linear SDOF system (Q=1 ¿. The damping ratio estimateζ is determined as follows:

ζ = δ√4 π2+δ2 (3)

where the logarithmic decrement is:

δ= 1m

ln( xn

xn+2 m) (4)

m∈ N+¿¿, and xn, xn+2 m are two successive local maxima or minima acceleration data points identified on an exponentially decaying sinusoid waveform, spaced by m number of periods.

Recalling Eq. (1), with normalised amplitude and SDOF condition (Q=1), the following general IR function is defined (assume ψ=0):

x (t )=e−ζ ωn t sin (√1−ζ 2ωnt ) (5)

Furthermore, let us apply the conditions for a finite length of0<t ≤tmax, where tmax is a given constant. The resultant function Eq. (5) is plotted in Figure 1 (unitised), where local maxima-minima points are also illustrated.

Figure 1. Amplitude-normalised SDOF plot (0<ζ <1 ) illustrating notation for local maxima-minima points

Based on Eqs. (3) and (4), the damping ratio for an underdamped system (0<ζ <1 ) can be estimated. For example, by utilising two successive local maxima (x¿¿ r ; xr+2)¿, i.e. m=1, the damping ratio is estimated as follows:

ζ est .r=[1+(2 π )2(ln xr

xr+2)−2 ]

−12

(6)

where ζ est .r is the estimated damping ratio using the rth pair.Since the damping ratio estimated in Eq. (6) utilises only two local maxima, an averaging

method to improve its accuracy would be used:

5

ζ est=ζ est .1+ζ est .2+…+ζ est . R

R(7)

where R denotes the number of successive pairs taken.R=N−2, where N is the total number of local maxima-minima. For a theoretical SDOF model with zero noise or disturbances, thenζ est .1=ζ est .2=…=ζ est .R.

2.2. The PDDR method

The PDDR method aims to overcome the fundamental shortcomings of the logarithmic decrement method, by following a statistical probabilistic framework. It utilises the identical data points, but constructs a one-dimensional datasetU , collating the normalised amplitudes of both local maxima and minima points, within a specified range of 0<t ≤tmax, instead of taking local maxima-minima and then averaging. The dataset U takes the form (also applicable for x (t ) ; x( t)):

U={x1; x2;…; xn ;…;x N } ,where xn (t∨t ≤ tmax ) (8)

The parameter of interest is the variance of dataset U , i.e., dispersion parameter, defined as:

σ 2=∑n=1

N

( xn−μ )2

N(9)

where μ and N denote the mean value and the total number of data points of dataset U , respectively.

If Eqs. (7) and (9) are expressed as two multivariate functions, they share an identical set of function-variables. From Eq. (7):

ζ est≔g ( x1 ; x2 ;…; xn;…; xN ) ,where xn (t∨t ≤ tmax ) (10)

and from Eq. (9):

σ 2≔h ( x1; x2 ;…;xn ;…; xN ) , where xn ( t∨ t ≤t max ) (11)

In view of LTI damped SDOF system, with the displacement function (Eq. (5)), it is proposed that function g ( x1; x2;…; x N ) is strictly negatively monotonically related to functionh ( x1; x2 ;…;x N ), provided that t max>0 andx (t¿¿max)>0¿. The proof is given in the Appendix.

Based on this, the following points are reached:1) For LTI SDOF systems, the damping ratio (determined via logarithmic decrement method)

is strictly negatively monotonically related to the variance parameter of datasetU , defined in Eq. (8).

2) However, the monotonic relation (see Eq.(A-1)) begins to break down as tmax → ∞.3) The logarithmic decrement method in time domain is validated to work in LTI SDOF

systems, however, once Q>1 in Eqs. (1)-(2) for MDOF systems, then Axioms 2 to 5 in Appendix become inapplicable by definition.

4) Since the structural damage inherently has an effect on the natural frequencyωi, of an SDOF system, the condition for a constant set ωi in Eq. (5) is violated, and this leads to the assumed constant tmax to also become a dependent variable.

6

The motivation of the PDDR method is that while the logarithmic decrement method is inherently inapplicable for MDOF systems, multivariate functionh ( x1; x2 ;…;x N ), retains a significant level of accuracy in estimation of overall damping property. This robustness is primarily achieved with the following two features:1) The PDDR method shifts from a standard single, or multiple, sets of estimates of damping

ratios to a statistical parameter of variance (σ 2). This framework allows a controlled level of uncertainty while maintaining the accuracy of damage identification, though a decrease in precision occurs by coupling all damping ratio estimates in an MDOF system as a single overall damping estimate. While the logarithmic decrement allows zero leniencies in Axioms 2, 3, and 5 (Appendix), functionh ( x1; x2 ;…;x N ) allows flexibility following a statistical analysis perspective.

2) The damage-induced changes in the natural frequencies of an MDOF system (∆ ωni ) is mitigated by employing a magnitude weighted frequency average (MWFA) parameter, where a frequency average of the underlying IR waveform is computed and set to govern the variabletmax. This function will be explained in the next section.

The following section presents the steps for the implementation of the PDDR method.

3. Structural damage identification using the PDDR Method

The PDDR method is structured into two phases. Phase 1 is comprised of data acquisition (IR measurement), signal processing, and formatting. DatasetsU are constructed from measured IR waveforms. Phase 2 collates all datasetsU , and then conducts nonparametric statistical analysis on the variance property of datasetsU , resulting in damage detection via hypothesis testing. Lastly, damage identification is performed by the correlation analysis of variance property versus damage severity. Figure 2 provides an outline to the PDDR method.

3.1. Phase 1: Establishing Datasets U3.1.1. Signal Processing and MWFA

Phase 1 begins with the IR signal acquisition, where a standard impact hammer test procedure is performed, based on a Single-Input-Multiple-Output (SIMO) test setup. After data acquisition of input-force and output-acceleration time histories, signal processing and formatting are performed. In this step, the measured IR waveforms are first normalised, in both measurement duration and magnitude for the correct signal averaging and damage scenario comparison. Two denoising filters are then applied for the effective implementation of the peak-picking process algorithm. The details of the signal processing steps can be found in detail in study Ay [19].

Following this step, a magnitude weighted frequency average (MWFA) estimate of a given processed IR measurement is determined. The MWFA parameter is a single quantitative estimate, providing a basic measure of dominant resonant frequencies within a specified frequency band. It is acquired from the raw magnitude spectrum of computed Power Spectral Density (PSD).

In this study, the nonparametric Welch method is employed [20] for the estimation of MWFA. This method mitigates the large fluctuations and leakages in measured power spectra by splitting a discrete record X ( γ ) with a length of Γ into Lsegments with Θ samples, such that L=Γ /Θ. The following process provides critical PSD settings required to avoid any loss in spectrum fidelity (mainly power reduction) while addressing statistical estimator variance.

7

Consider an output acceleration measurement x (t ) as a transient and finite sampled signal. The measurement x (t ) produces a one-sided Power Spectral Estimate (PSE) G xx ( f ), through a one-sided Discrete Fourier transform (DFT). The DFT can be defined using the autocorrelation functionR xx ( τ ), as per the Wiener-Khinchin theorem [20], such that:

G xx ( f )=∑τ=1

∞

R xx (τ ) e−i 2 π f τ t max , f ≥ 0 (12)

where i is the imaginary unit, tmax is the total discrete sample size of the transient measurement x (t ), and the autocorrelation function R xx ( τ ) takes on the measurement x (t ) as a random process in form of the sampled data ¨X ( γ ) with Γ discrete points.

Figure 2. Process diagram of the PDDR method divided into two phases

8

In Eq. (12), the features of Welch’s method can be specified, including the overlapping settings, the applied spectral window, and the frequency resolution within a set bandwidth. After splitting a discrete PSD spectrum X ( γ ) into Lsegments, the Welch’s method applies a window W (γ ) to each segment, so that a sequence of X1 (γ )W ( γ ) ,…, X Γ (γ )W (γ ) is established. Accordingly, Eq. (12) transfers the applied windows onto a series of periodograms as follows [21]:

G xx❑ ( f )= 1

H Θ [∑γ=0

Θ−1

X τ (γ )W (γ ) e−i 2 π f γ ]2

, τ=0,1 , …, (L−1 ) (13)

where H is the normalisation factor for the power spectra. The Welch’s estimate of PSD is the average of periodograms in Eq. (13), such that:

G xxw ( f )= 1

L∑τ=0

L−1

Gxx ( f ) (14)

From this averaging procedure, the Welch’s estimate provides the quantity for the raw PSD in the following manner:

E [G xxw ( f ) ]=G xx ( f ) W (f ) (15)

where, G xx ( f ) is the raw PSD and W ( f ) is the power normalised DFT of window W (γ ), defined respectively as follows [22]:

G xx ( f )= 1Γ

[ X ( f )]2 (16)

and

W ( f )= 1H Θ [∑γ=0

Θ−1

W (γ ) e−i 2π f γ] (17)

The above process is utilised to extract the MWFA parameter from measurements x (t ). The following settings are applied: (1) the rectangular window on transient IR x (t ), due to an energy versus time distribution that is considerably uneven; (2) a significantly long sample (measurement lengths, T ) to attenuate the window leakages to a negligible level; and (3) a greater attention given to data fidelity over statistical stability to retain resonant peaks magnitudes. The MWFA parameter is defined as follows:

MWFA=∑i=1

N

f i|G xx❑ ( f )|i

∑i=1

N

f i

(18)

where within the frequency band 0≤ f ≤ f max, N is the total number of the identified peaks, with a linear amplitude |G xx

w ( f )|i and corresponding frequency f i. The peak detection is completed using Sequential Quadratic Polynomials (SQP algorithm), detailed in studies [23, 24].

3.1.2. Dataset U

After computing the MWFA quantities for every output x (t ) measurement, the subsequent step is to extract local maxima-minima points by adjusting for changes in natural frequencies and

9

amplitude discrepancies.For each damage scenario (DS), or a new measurement, a processed time-domain IR is

recorded, which establishes the following set of IRs per accelerometer:

Az : { xDS 0 ( t ) , xDS 1 (t ) , …, xDSJ (t ) } , where , { xDS ji( t )|−1.0 ≤ x (t ) ≤ 1.0}. (19)

where Az denotes accelerometer z, and DSji represents the ith record for the jth damage scenario (from 0 to J). Then, an end-truncation point is established based on the baseline measurement (DS0), by utilising a sliding root mean square (RMS) criterion, implemented as follows:

Ψ s(t)=√ 1T2−T 1

∫T1

T2

[ x DS 0 ( t )]2 dt , (20)

where xDS 0 (t ) is the normalised baseline IR measurement, and T 2−T 1 is the sample time width in seconds, which is set in this study to a value governed by the sampling frequency f s. The truncation process is detailed in Ay [19].

The resultant end-truncation time tmax : DS 0 for the baseline condition serves as a reference quantity for end-truncations of the remaining, or the newly added IR measurements { x DS1 (t ) , …, xDSJ ( t ) }. These IR measurements taken after baseline DS0 are end-truncated based on a multiple of tmax : DS 0, using previously computed MWFA quantity per IR measurement xDS ji

( t ). They are determined as follows:

tmax : DS ji=( MWFA DS 0

MWFADS ji)tmax : DS 0 (21)

The last step of phase 1 for PDDR method is to determine the statistical variance σ DS ji2 of

each corresponding datasetU DS ji.

3.1.3. Statistical Hypothesis testing

Phase 2 of the PDDR method encompasses the statistical analyses, involving both damage detection through statistical hypothesis testing and damage identification by correlation analysis. In this study, damage identification is limited in scope to a supervised learning process, whereby the identification is based on a training dataset with the correlated damage severity/location and the change in overall damping behaviour.

The first objective, damage detection, is achieved by assessing the homogeneity of damage-sensitive parameters, variance σ DS ji

2 . Precisely, a hypothesis testing is performed, which provide a binary result, where the null hypothesis denotes no damage detected, and the rejection of the null hypothesis denotes damage detected.

The nonparametric Fligner-Killeen (F-K) test [25] is specifically chosen for this study, as it: (1) does not assume the normality of the measurement data sets; (2) is robust against substantial departures from normality; (3) is not sensitive to outliers as it uses ranked absolute values computed for centred samples and weights; and (4) does not require group sample sizes to be equal.

The F-K test is a Type-I error hypothesis test. The homogeneity of variances (null hypothesis) H 0 and alternate conditionH A are defined respectively as:

10

H 0 :σ DS 02 =σDS 1

2 =…=σDSJ2 (22)

and

H A :σ DS ji

2 ≠ σ DS j ' i

2 , for some DS ji≠ DS j 'i (23)

In practice, damage detection is performed based on initial baseline measurements, followed by subsequent updated (new) measurements.

The nonparametric F-K test statistic is computed, assuming χ2-distribution with J degrees of freedom, as follows [25, 26]:

QFK=( N ¿−1

∑l=1

N¿

(a¿ (l ) )2 )∑ni=1

N¿

{ ∑DS ji=DS 0

DSJ

a¿ (R ( DS ji) ni )}2

(24)

where N ¿ is total dataset size and a¿ ( l ) denotes score quantities computed in an increasing order against ranking function ( R ), such that R( DS ji) ni

=R|U ( DS ji ) ni

¿ |. The more detailed procedure can be found in Ay [19].

After computing the F-K statistic, QFK, an approximate level α s-test is then conducted, where the null hypothesis H 0 (Eq. (22)) is rejected under the condition ofQFK ≥ χα

2 (J ). In this study, α s=0.05 (95% confidence level).

Furthermore, a pairwise test is conducted that elaborates the initial binary result to determine the pair (or group of pairs) that resulted in group heteroscedasticity. From the perspective of SHM, this serves two purposes. First, the pairwise p-value coefficients (or statistical significance) indicate the level of sensitivity damage-sensitive feature σ DS ji

2 has among tabulated p-values. Second, it allows an estimate of damage severity by one-to-one comparison. For the pairwise comparisons of damage scenario datasetsU DSQ, the multiple significance results are computed. Consequently, the condition of multiplicity arises, risking Type I statistical error. To address this issue, a Bonferroni correction is implemented [27].

3.1.4. Dataset Outliers

To prepare the dataset for damage identification, outliers need to be eliminated, where the truncation is applied based on the computation of the Interquartile Range (IQR) quantities. To ensure the data retention across all datasets of IQR ranges, the maximum quantity of the datasets U is selected. Furthermore, an optional IQR multiple is applied, allowing for a wider range, and only discarding extreme outliers. Figure 3 presents this multiple equal to 1.5. Upon truncation, the datasets U are processed as truncated datasets, defined as follows:

U =¿ (25)

where ε low∗¿; εup∗¿¿¿ denote the lower and upper outlier limits, computed using maximum IQR quantity. Note, U ⊆U , and U variance is σ❑

2 .

11

Figure 3. Example boxplot representation (left) of dataset U constructed from a typical exponentially decaying sinusoid (transient response)

3.1.5. Confidence Intervals

Before assessing the correlation between the variance propertyσ 2 and the structural damage scenario, it is important to first quantify a confidence margin for propertyσ 2.

A suitable method to determine confidence margins is to assume a nonparametric bootstrap methodology, which is inherently an estimate from empirical data and avoids any assumptions made on the distributions of compiled datasetsU DSj. The bootstrap-based confidence margins or percentiles essentially works on the empirical data, where the independent and identically distributed sample sets S¿ of dataset U are constructed, following a reiterative process to estimate a bootstrap statisticθ¿. The estimated statistic for confidence margins is the variance parameterσ 2. Hence, a bootstrapping resampling method is implemented to obtain 95% confidence intervals of variance parameters in this study. Accordingly, the lower and upper confidence limits are computed as follows:

q . low=Φ [ z0+z0+z(α bt /2)

1− a( z0+z(α bt/2)) ] (26)

and,

q . up=Φ[ z0+z0+ z(1−αbt /2)

1−a( z0+z(1−α bt /2)) ] (27)

where Φ [ ∙ ] is the cumulative distribution function for Gaussian distribution; z0 is the bias-correction; zα bt

is the α bt quantile of Gaussian distribution; and a is the acceleration parameter calculated in this study using the jackknife estimation [28]. Subsequently, using Eqs. (6)-(7), the required 95% confidence level limits to original parameter σ DSn

2 , are defined as follows:

12

θq .low¿ = θ( B+1 )( α bt/2)

¿ (28)

and,

θq .up¿ =θ( B+1) (1−α bt /2)

¿ (29)

3.1.6. Correlation Analysis

The final step of damage identification involves the analysis of the relationship between the variance parameter and structural damage scenarioDS ji. By this point of analysis, a set of J+1 measurements { DS 0 ,DS 1 , … DSJ } have been acquired, and among the group of measurements, the damage has been detected with the rejection of the null hypothesisH 0. The datasets have been truncated to eliminate outliers and are now ready to be examined quantitatively using the updated damage-sensitive parameterσ DS ji

2 . The damage identification is realised by either a direct-match to a previously labelled or identified measurement, or a total shift assessment from the baseline or the closest labelled measurement.

To apply the PDDR method to SHM, the group assessment via the hypothesis testing for heteroscedasticity would be completed with every new measurement taken from the test structure. Thus, if the null hypothesis were to be rejected, one would immediately deduce that the final measurement reflects a statistically significant change (damage) in the structural state. Under this type of application, this step would serve as an indicative correlation between all measurements taken over time. Thus, one would be able to assess if the change in damping property, as reflected in the parameterσ❑

2 , depicted a gradual change over time, or a sudden shift between the last two measurements.

The damage severities or scenarios, though denoted as an ordinal list of numbers for a case study, may not imply a linear change in the structural damping of the system from a physics-based perspective. Although the subjected damages may be incrementally increasing, the resultant change in damping property of the system does not guarantee linearity. Thus, this study adopts the Spearman’s rank correlation coefficient as a means to determine a coefficient quantity between increased severities of damage and proposed a damage-sensitive parameterσ❑

2 .

4. Experimental Study: Warren-Truss Bridge Model

4.1. Experimental Setup and Procedure

A non-destructive impact hammer test was performed on a scale-model steel Warren-truss bridge, with a SIMO configuration. The independent variable of the experiment was set as an incrementally increasing number of connection failures, simulated by loosening test bolts at four different splice-brackets.

The test structure was erected as a single span bridge with eight equilateral triangular sections, creating a total span of 5.5m, with 0.65m width. All the structural members had an identical square-box cross-section, with the dimensions of 30×30×3mm (AS/NZS1163:2009). The lengths of the structural members were in three sets, i.e. the floor deck beams with 500mm, the equilateral truss members with 600mm, and lateral floor deck bracing with 800mm. All the splice brackets were identical in dimensions with a thickness of 5mm, and two M10 bolt connections at both ends of the member. All the bolts were high tensile ISO Grade 8.8, featuring lock-nuts and serrated washers to prevent the possible rattling or loosening due to vibration testing. All the bolts within the test structure were fully tightened to a torque setting of 25N·m.

13

Figure 4 presents an isometric view of the test bridge. The structure was assembled in the laboratory and was supported by two steel sawhorses bolted onto the concrete floor. The boundary conditions for the impact hammer testing were set as fixed-fixed. The impact hammer strike point was set as the midpoint of the second lower chord, and the strikes were in the −Z direction. A total of three uniaxial piezoelectric accelerometers were attached at midpoints of the fourth, fifth and sixth floor-beams, respectively.

Figure 4. Experiment schematic of steel Warren truss structure: (i) impact hammer Endevco®2304 and accelerometer Endevco®61C13, (ii) data acquisition module NI PXIe-4492, (iii) PC NI® PXIe-1078

The signal acquisition was configured as one input force f (t), and three output accelerations x (t ) per strike. For all the four channels, the sampling frequency was set to f s=20 kHz, with a total sampling timeT=5.0 s, programmed to start recording based on rising slope condition of instantaneous input force by impact hammer. A total of four strikes were performed per damage scenario, which allows for noise minimization.

The experiment consisted of seven damage scenarios (see Table 1). The designed damage scenarios were aimed to replicate the structural conditions of the connection failures, which were achieved by loosening four diagonal test bolts per splice bracket. A test bolt was deemed loosened after two complete counter-clockwise revolutions made by a wrench.

The first damage scenario (DS 0) was recorded as the baseline (intact-state), with all bolts tightened to a 25N·m torque setting. The following damage scenarios were set as the loosening of test bolts in sets of four, in an accumulating manner. All the loosened test bolts were part of the bottom deck splice brackets, indicated as SB in Figure 4.

14

Table 1. Damage scenario list of the Warren-truss bridge experiment

4.2. PDDR Phase 1 Implementation

The PDDR Phase 1 results were presented in Tables Table 2-Table 4. In all three accelerometers, the MWFA parameter increases with each increase in damage severity, however not strictly monotonically. In connection to this, end-truncation parameterstmax, in all the three accelerometers, changed marginally, indicating a relatively small offset adjustment made by the MWFA parameter. Overall, the average MWFA shift of the three datasets was 8.886%.

Table 2. PDDR phase 1 results for accelerometer A1Damage scenario dataset

MWFA (Hz ) tmax (seconds) Dataset Size IQR σ 2

(× 10−3 )U DS 0 261.4909 1.0062 526 0.11829 30.44851

U DS1 264.8003 0.9936 521 0.08417 42.49838U DS 2 264.1875 0.9959 579 0.05688 30.12321U DS 3 276.9856 0.9499 563 0.02828 25.32197U DS 4 277.2275 0.9490 583 0.02566 18.08111U DS 5 283.3387 0.9286 547 0.01346 22.74035U DS 6 282.6902 0.9307 553 0.01153 25.37206

Table 3. PDDR phase 1 results for accelerometer A2Damage scenario dataset

MWFA (Hz ) tmax (seconds) Dataset Size IQR σ 2

(× 10−3 )U DS 0 267.9446 1.1671 550 0.15599 41.82031U DS 1 276.3606 1.1315 548 0.10525 48.68298U DS2 284.0890 1.1007 575 0.04861 19.78541U DS 3 324.1007 0.9648 548 0.03592 24.54251U DS4 284.2695 1.1000 601 0.02781 23.19632U DS 5 306.5091 1.0202 533 0.01985 20.16279U DS 6 305.4687 1.0237 564 0.01807 22.16162

Table 4. PDDR phase 1 results for accelerometer A3Damage MWFA (Hz ) tmax (seconds) Dataset Size IQR σ 2

Damage scenario dataset Description of damage Damage locationU DS 0 Intact structure N/A

U DS 1 4 test bolts loosened SB1U DS 2 8 test bolts loosened SB1-2U DS 3 12 test bolts loosened SB1-3U DS 4 16 test bolts loosened SB1-4U DS 5 20 test bolts loosened SB1-5U DS 6 24 test bolts loosened SB1-6

15

scenario dataset (× 10−3 )

U DS 0 262.7978 1.2641 525 0.21156 78.03002U DS1 284.1713 1.1690 579 0.09744 38.81942U DS 2 273.0904 1.2164 560 0.08438 40.61835U DS 3 311.1321 1.0677 576 0.06608 34.22367U DS 4 300.4944 1.1055 598 0.05083 28.61497U DS 5 276.4933 1.2014 554 0.02480 26.72626U DS 6 301.3006 1.1025 550 0.01912 24.94467

Two other significant changes observed in Tables Table 2-Table 4 were noted as IQR values and variance σ 2. In all three accelerometers, the damage-sensitive variance propertyσ 2, was observed to have a weak negative correlation to damage severity, not monotonic with deviations between damage scenarios DS1-DS3. However, this was not the case for the IQR values, which presented a strong negative correlation to damage severities and were considered perfectly monotonic. This observation suggests a requirement to consider the theoretical premise that the outer data points of datasets U correlate to an erratic segment of transient responses, and hence the PDDR method requires the cleaning of outliers.

The IQR quantities in all three dataset groups were significantly narrow with respect to the absolute limits of−1.0 ≤ x ≤+1.0; where the largest IQR from each accelerometer dataset group was noted as 11.89%, 15.59 %, and 21.11% of the fixed maximum range, respectively. This indicates the changes in the overall exponential decay-rate of the transient responses, where a narrower IQR suggests a stiffer structure with a higher decay-rate.

4.3. PDDR Phase 2 Implementation

By Phase 2, seven datasets U have been established for all the signal acquisition channels. The unprocessed estimates of variance σ 2 have been computed. The subsequent steps show statistical analyses conducted on the variance property of datasets U for the damage identification.

4.3.1. F-K Test: group assessment

An effective means to visually assess changes in the IQR parameters and the related group heteroscedasticity is to use boxplots. Figure 5 presents the initial qualitative assessment boxplots, divided per dataset group (accelerometer channel).

The visual assessment of Figure 5 indicates: (1) the characteristic similarity in boxplots among the three datasets, mainly in the IQR and whisker sizes for each corresponding damage scenario and a significant narrowing between damage scenarios DS 0−DS 1; (2) a very large number of outliers present across all damage scenarios for all the datasets, suggesting a very rapid decay rate of the transient responses; and (3) in terms of the visible trend, the box and whisker sizes notably decreased in size incrementally with each increase in damage severity or scenario; which provided strong indication that the IQR values of the dataset and damage severity are negatively correlated.

Subsequently, a nonparametric F-K test was performed based on Eqs. (22) and (23). The computation results for each accelerometer were presented in Table 5. Given an identical number of datasets recorded by each accelerometer, all the three computations have a between-group degree of freedom of 6 [29]. By observing the resultant F-statistic values with the

16

associated p-values, it was noted that the extreme values obtained in both parameters were due to the combination of a substantially large squared deviation and relatively small between-group degrees of freedom.

The significantly large F-statistic values observed in all three-accelerometer data strongly supported heteroscedasticity between groups{U DS 0 , …,U DS 6 }, which was further confirmed in boxplots in Figure 5.

All the obtained p-values are noted less than the significance value of0.05, resulting in the rejection of the null hypothesis in all cases. This result, from a damage detection perspective, means that the damage can be detected from all three accelerometers.

Figure 5. Boxplots of datasetsU DSn per accelerometer group for steel Warren-truss bridge structure experiment

Table 5. Analysis of variance between group datasets U DSn for Warren truss bridge experiment using nonparametric F-K test

4.3.2. F-K Test: pairwise assessment

Subsequently, the F-K test was reapplied on the pairwise comparison basis, which served to break down the group assessment results and to extract key pairs that triggered a statistically

Accelerometer Between-groups Squared Deviation F-statistic p-value

A1 30515881.510 115.055 0.0000A2 27305481.600 97.928 0.0000A3 28570763.000 101.766 0.0000

17

significant result. The results for accelerometers A1-A3 are provided in following Tables 6-8, respectively. A pairwise correction factor is applied, utilizing Bonferroni’s correction method.

Table 6. F-K test pairwise comparison for accelerometer A1 test results for Warren truss bridge structure experiment


p−valueDamage Scenario

DS0 DS1 DS2 DS3 DS4 DS5 DS6DS0 - 0.041 0.000 0.000 0.000 0.000 0.000DS1 0.041 - 0.000 0.000 0.000 0.000 0.000DS2 0.000 0.000 - 0.199 0.003 0.000 0.000DS3 0.000 0.000 0.199 - 0.113 0.000 0.000DS4 0.000 0.000 0.003 0.113 - 0.032 0.009DS5 0.000 0.000 0.000 0.000 0.032 - 0.549DS6 0.000 0.000 0.000 0.000 0.009 0.549 -


From the pairwise p-values, an ordinal pattern was observed between the adjacent damage scenario pairs. For the data from accelerometers A1 and A2, the adjacent damage scenario pairsDS 0−DS 1, DS 3−DS 4, DS 4−DS 5, and DS 5−DS 6 resulted in a p-value greater than the significance level ofα BC=0.0071. Thus, for these adjacent pairs, the null hypothesis was not rejected, and thus a statistically significant difference in damage-sensitive variance was not





18

recognized. These results were consistent with the theoretical premise of the PDDR method and the

independent variable of the experiment, where the damage scenarios were designed with incrementally increasing damage severity. Furthermore, it indicated that for a majority of the adjacent damage scenario pairs, a statistically insignificant change occurred, i.e., no detectable or distinguishable change in structural state. Once the difference of the damage severities was greater than one increment (number of bolt connection failures), all the pairwise comparisons’ p-values resulted in a statistically significant difference, which leads to the rejection of the null hypothesis in Eq. (22).

It is also noted that the pairDS 4−DS 6, from accelerometer A2, was the only non-adjacent pair, which registered a p-value greater than the adjusted significance levelα BC. However, this did not violate the ordinal pattern noted earlier, as the closer pairs within this segment ( i.e., DS 4−DS 5 andDS 5−DS 6) correctly resulted in a much larger level p-value.

4.3.3. Variance correlation analysis

The experimental datasets U were further assessed for identifiable correlation between the damage-sensitive parameter versus increased damage severity. Before this, some appropriate modifications were made in means of minimizing the effects of outliers and narrowing the focus down to a desired region.

Following the PDDR method outlined in Figure 2, all the established datasets U were first truncated to only retain data points which fell within the area of interest, defined based on the IQR parameter. This parameter was specifically defined based on the largest IQR quantity per accelerometer group of datasets. The IQR sizes of each dataset were computed for evaluation, presented in Table 9, in addition to group maximum IQR values.

Table 9. Summary table of IQR for constructed datasets U with group maximums denoted *.To

maintain the uniformity across accelerometer groups, and given the relative similarity in group maximum IQR sizes, an identical truncation window was applied for all 21 datasets. The applied window size was defined using groups’ maximum IQR parameter, i.e., 0.2104, which was then rounded up to the amplitude range of0.25, and translated to the standardised limits of−0.125 ≤ x≤ 0.125. The truncated datasets in all three accelerometers, denotedU , were defined for this experiment results as follows:

U DS ji= { x∈U DS ji∨−0.125< x<0.125 } (30)

Subsequently, a bootstrapping method was applied to the resultant truncated datasets U for the improved accuracy with the confidence level set at the 95% limit. The lower and upper confidence limits, θq .low

¿ ; θq . up¿ , were presented in Table 10, in conjunction with the

corresponding dataset U variance parameters σ 2.

The interquartile range ( IQR)Damage scenario dataset U DSj

Accelerometer DS 0 DS 1 DS 2 DS 3 DS 4 DS 5 DS 6A1 0.1176* 0.0838 0.0566 0.0280 0.0254 0.0132 0.0115A2 0.1554* 0.1052 0.0483 0.0353 0.0278 0.0196 0.0179A3 0.2104* 0.0969 0.0837 0.0656 0.0505 0.0248 0.0189

19

Table 10. Truncated dataset variance and lower-upper confidence limits (95%) for accelerometers A1-A3: Warren-truss bridge experiment

AccelerometerTruncated

damage scenario dataset

σ 2 (× 10−3 ) θq .low¿ (×10−3 ) θq .up

¿ (× 10−3 )

A1

U DS0 3.45219 3.11367 3.79304

U DS 1 2.43723 2.08271 2.79491

U DS2 1.80166 1.53255 2.09985

U DS 3 1.47239 1.21201 1.75224

U DS4 0.99542 0.81291 1.20268

U DS 5 1.02840 0.80768 1.26345

U DS6 0.78491 0.61809 0.95323

A2

U DS 0 3.61447 3.18851 3.99234

U DS 1 2.55324 2.22057 2.88585

U DS 2 1.70876 1.43907 1.99067

U DS3 1.54470 1.26517 1.83138

U DS 4 1.07823 0.87178 1.30732

U DS5 1.13971 0.91463 1.37929

U DS 6 0.88672 0.68132 1.10029

A3

U DS0 3.98372 3.50494 4.46095

U DS 1 2.41553 2.11341 2.71347

U DS2 2.13361 1.82096 2.39444

U DS 3 1.95379 1.68462 2.23080

U DS4 1.60891 1.37106 1.87688

U DS 5 1.15536 0.90335 1.40006

U DS6 1.01821 0.85135 1.18319

20

Figure 6. Warren-truss bridge experiment: Truncated variance σ 2 vs. damage scenario for accelerometer A1



Figure 9. Warren-truss bridge experiment: Percentage shift of truncated variance σ 2 from the intact state (DS0) vs. damage scenario for accelerometers A1-A3 and Accelerometer Averaged

The tabulated results were shown graphically in Figures 6-8, where the limits θq .low¿ ; θq . up

¿ are the error bars on a scatter plot of the variance moment σ 2 versus the incrementally increasing damage severity. By observing these plots, a visually distinguishable negative correlation was observed in all the three accelerometer results, where the increase in damage severity ( i.e., number of bolt failures) resulted in a significant decrease in variance, σ 2. Collectively, all the three accelerometers’ data is presented in Figure 9, which further indicated that all these data presented a very similar trend.

The final part of the PDDR method is to define and compute a correlation coefficient, quantitatively assessing the monotonic relationship between the damage scenarios and damage-sensitive parameters. As this study also serves to validate the PDDR method’s efficacy on real vibration measurements, the computed correlation values would directly support or contradict the proposed method.

21

The Spearman’s correlation coefficientr S, defined in this experiment’s, is as follows (computed per accelerometer):

ρ s=1−6∑

i=0

J

d i2

(J +1 ) ( (J +1 )2−1 )(31)

where d i2 is the square of the difference between two ranked ( R ( ∙ ) ) test variables, such that:

d i2=R ( ji )−R ( σ DS ji

2 ) . (32)

The Spearman’s correlation statistic assesses the monotonicity of two input variables. Rather than a certain form of regression, this experiment produces the following results shown in Table 11. It is important to note that any type of regression analysis would theoretically be invalid in this experiment; as the PDDR method’s final damage-sensitive parameter σ 2 does not directly correlate to structural damages of a dynamic system (in this case failed bolt connections), while instead provides an estimate of the overall physical damping of the system.

Table 11. Warren-truss bridge experiment: Spearman’s rank correlation coefficients for comparison of computed PDDR truncated variance parameters σ 2 vs. damage scenarios Spearman’s rank correlation

Accelerometer (sensor channel)A1 A2 A3 Averaged

Spearman’s Rho (r¿¿ s)¿ or Correlation coefficient

-0.9642 -0.9642 -1.0000 -1.0000

Significance (2-tailed) (α=0.01 ) 0.000454 0.000454 0.000000 0.000000Number of entries

(J+1) 7 7 7 7

From the computed Spearman’s coefficients (r¿¿ s)¿ in Table 11, the data from A1 and A2 presented the identical monotonic relationship between the increase in bolt-connection failures (damage scenario) and damage-sensitive parameter σ 2. A perfectly negative correlation is not met, and specifically, the monotonicity condition is violated in damage scenarios DS4 and DS5. However, the data from A3 showed a fully monotonic relationship, which was validated by the computed correlation coefficient equal to−1.0.

4.4. PDDR versus standard EMA

The results based on the proposed PDDR method, the truncated variance estimates σ 2, were then compared against the standard modal damping ratio estimates, through a standard EMA procedure.

4.4.1. EMA Procedure

A suitable EMA and curve-fitting procedure, the LSCE method, was implemented in this study. This method essentially works by computing system poles λk in time-domain directly utilising time-domain Impulse Response Functions (IRF) via auto-regressive moving average (ARMA) model in time-domain [30]. The inverse Fourier transforms (IFFT ) are taken from the precomputed FRFs. The estimation of natural frequencies and modal damping ratios from

22

identified resonances was completed using the following LSCE formulation:hrs ( t )=IFFT [ H rs (ω ) ] (33)

and in terms of the summation of the directly fitted exponential functions, the Prony’s method was utilised:

hrs ( t )=∑k=1

n

( A rs ( k ) eλ k t+ Ars (k )e

λk t ) (34)

where Ars (k ) ; A rs ( k ) are the kth residue functions, defining the displacement of the rth DOF in the sth mode, and the bar denotes the complex conjugate.

Collectively, five vibration modes were identified within the frequency band 0≤ f ≤ 500 Hz. Table 12 presents the extracted natural frequencies with the corresponding modal damping ratios. A graphical representation of the extracted modal parameters is presented in Figure 10, in the form of a resynthesized FRF summation curve. Figures Figure 11-Figure 12 present the changes in modal damping ratios with damage and relative percentage changes from baseline (intact state) of denoted DS0, respectively. For the specific modes, the damping ratios fluctuate with the increase of damage severity with no clear pattern. However, for mode average results, a rough increasing trend can be identified.

Table 12. Natural frequencies and damping ratios of first five identified modes vs. damage scenariosNatural Frequency, Hz (Damping Ratio, %)

Damage Scenario Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

DS0 144.713 (3.195) 198.287 (1.261) 269.012 (1.673) 283.712 (1.145) 433.137 (0.779)DS1 143.613 (3.484) 197.012 (2.094) 268.088 (1.539) 282.113 (3.177) 433.512 (0.894)DS2 147.788 (2.030) 196.087 (3.571) n/a 283.287 (1.280) 435.063 (1.207)DS3 n/a 195.938 (1.658) n/a 277.238 (1.578) 441.062 (2.154)DS4 n/a 190.362 (2.497) 253.337 (1.678) 275.327 (2.634) n/aDS5 n/a n/a 251.887 (1.736) 275.013 (2.591) n/aDS6 n/a n/a 250.988 (1.942) 274.012 (2.920) n/a

23

Figure 10. Resynthesized FRF summation curve using LSCE method extracted modal parameters

Figure 11. Modal damping ratios of first five identified modes vs. damage scenario

Figure 12. Percentage changes in modal damping ratios from baseline (DS0) state vs. damage scenarios

The following observations were made in regards to the extracted natural frequencies and damping ratios:

(1) Except for mode 4, all the other modes were affected by the resonant peak disappearances at certain intervals for the subjected damage scenarios, recognized by the unstable poles in the LSCE curve-fitting process. Mode 1 was noted as most affected by the missing data points between DS3-DS6, more than half of the subjected damage scenarios. To investigate this, an alternative curve-fitting method, Rational Fraction Polynomial method was used. It can identify some resonant peaks within frequency proximity of missing modes. However, the direct FRF plots indicated that most of the identified peaks were either entirely wiped out or were far too wide and flat for accurate curve fitting.

(2) For the natural frequency changes, the results from modes 2, 3, and 4 indicated a weak negative correlation against damage severity, while those form modes 1 and 5 presented no distinguishable correlation. The largest percentage shift was observed in mode 3, among DS0-DS6, with a shift of -6.70% equating to a decrease of 18.02Hz. The shift in mode 2 was -3.99% equating to a decrease of 7.92Hz.

(3) For the modal damping ratios, mode 5 was the only one which presented a perfect positive correlation with the damage severity, while the mode disappears from DS4 to DS6. The other modes presented no identifiable correlation. The largest change from the initial baseline measurement (DS0) was observed in mode 4 between DS0 and DS1, with an increase in modal damping ratio of 183%.

4.4.2. Comparison of Results

In addition to the modal damping ratios, the equally weighted mode-averaged modal damping ratio quantities are presented in Table 13. Similar to the PDDR method, quantitative correlation coefficients were determined using Spearman’s coefficient.

24

Table 13. Warren-truss bridge experiment: Spearman’s rank correlation coefficients for comparison of estimated modal damping ratios (5 modes) vs. damage scenarios

Figure 13. Percentage changes from baseline DS0 condition of PDDR truncated variance σ 2 (negated values) and mode-averaged modal damping ratios

From the derivations provided in Appendix, it is theoretically shown that the PDDR method’s damage-sensitive parameter variance(σ ¿¿2)¿, has a strictly negative correlation against a dynamic system’s damping property (for SDOF underdamped systems). The SHM literature generally indicates the expectation for modal damping ratios to increase with the increases of structural damage. Thus, for a direct comparison of the variance parameter (σ ¿¿2)¿ against FRF-extracted modal damping ratios(ζ ), Figure 13 presents an inverted plot of variance quantities σ 2 (multiplied by−1) for parameterσ 2 results.

The identified set of modal damping ratios for the first five modes using the LSCE method individually presented inconclusive results to serve as a damage-sensitive indicator. Precisely, only one of the five identified modes presented that the modal damping ratios had a statistically significant correlation to the subjected bolt-connection damages. A key hindrance was the disappearance of the identifiable modes within the frequency band, leading to the gaps in the data points for accurate damage identification procedures. It is noted that once the modal damping ratios were averaged (equally weighted), a full set of data points covering all damage scenarios was observed, with a visually distinguishable correlation. However, this correlation was deemed questionable with a Spearman’s coefficient 0.6785.

The direct comparison plot in Figure 13 presented that both parameters are similarly sensitive to the changes in the subjected damage. This is supported by a computed Pearson correlation between accelerometer averaged truncated variance σ 2 versus mode-averaged modal

Identified Mode

1 2 3 4 5Mode

AveragedSpearman’s Rho (r¿¿ s)¿ or

Correlation coefficient-0.5000 0.5000 0.9000 0.4285 1.0000 0.6785

Significance (2-tailed) (α=0.01 ) 0.66666 0.391002 0.037386 0.337368 0.0000 0.093750

Number of entries 3 5 5 7 4 7

25

damping ratios ζ , yielding a value of−0.734. Although the averaged modal damping ratios presented even higher sensitivity to the existence of damage, they showed a strong diversion from a monotonical relationship with the damage severities, for example, DS3-DS6. In contrast, the PDDR truncated variance parameters indicate a near-perfect monotonic relationship, in all three-accelerometers, and the averaged results (see Table 11).

5. Discussions & Conclusions

This paper presented the PDDR method, a novel statistical structural damage identification framework, which employs overall damping estimates as a damage-sensitive feature. It is formulated to be applicable for both damage detection through statistical hypothesis testing, and damage identification through correlation analysis. This method was first introduced with the relevant background and its basis of the logarithmic decrement method. It was then detailed in the form of an algorithmic process. To verify its effectiveness, a Warren truss bridge model under different bolt connection damage severities was tested. The results indicated that the truncated variance obtained from the PDDR method has a perfect negative monotonic relationship, while the modal damping ratios obtained from the standard EMA method have a moderate to weak correlation. The results demonstrate that the efficacy of the PDDR method has been experimentally validated.

5.1. Advantages

The merits of the PDDR method are identified as follows: (1) It can be applied autonomously as an algorithm reiterated with each new IRF measurement,

following the process as shown in Figure 2;(2) Since the damping properties are estimated directly in time-domain using IRF

measurements, it can avoid the issues of the standard EMA method, e.g., spectral leakages, curve-fitting and modal extraction considerations;

(3) The noise contamination issue can be mitigated through the use of a statistical analysis approach, where the damage-sensitive parameter is the statistical variance moment of compiled datasetU . Additionally, the dataset integrates all the identified local maxima-minima data points which can reduce the effects of random measurement errors (in contrast with the logarithmic decrement method) and enhance its robustness;

(4) While it is acknowledged that the variance moment estimates provide no physics-based interpretation of the test structures, e.g. damping at separate vibration modes (MDOF), it serves as an excellent indicator of changes in the overall damping estimated at various points (accelerometer location) through a robust statistical analysis method. It performs well on a notably complex test structure (the Warren-truss bridge).

5.2. Limitations

The limitations of the PDDR method are also noted as follows: (1) To accurately model the operational/environmental changes, it needs a number of

measurements under different conditions, even if the damage is definitely not present. Thus, a routine and frequent ‘monitoring’ strategy must take place for the approach to be adequately ‘trained’ with multiple baseline data.

(2) The proposed method does not necessarily rely on the impact hammer testing, because it only uses the output data. However, the input force needs to be an impulse excitation. For large-scale civil infrastructure systems, it is not easy but still achievable, e.g. using jumping

26

test.

5.3. Future Research

A number of potential research directions are identified: (1) A shift from nonparametric statistical analyses to parametric distribution estimation and

analysis, whereby the truncation of outliers from the original datasets U is avoided.(2) The optimal placement of sensors and impact point can enhance the performance of the

PDDR method. This will need the construction of finite element models (FEM) and modal analysis.

(3) The proposed method is essentially a data-driven approach to structural damage identification. Its performance is expected to improve through the integration of physics-based method, e.g. FEM updating approaches.

(4) The practical application of the PDDR method needs to consider the operational and environmental condition changes, which requires the appropriate processing of the long-term monitoring data. A Bayesian framework is noted for this potential enhancement.

Acknowledgement

The last author would like to thank EPSRC for its financial support to him through the First Grant Scheme (EP/R021090/1).

References

[1] A. Morassi and F. Vestroni, Dynamic Methods for Damage Detection in Structures. Springer Vienna, 2008.

[2] S. Adhikari, Structural Dynamic Analysis with Generalized Damping Models: Analysis. John Wiley & Sons, 2013.

[3] S. W. Doebling, C. R. Farrar, M. B. Prime, and D. W. Shevitz, "Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review," Los Alamos National Lab., NM (United States)1996.

[4] R. Savage and P. Hewlett, "A new NDT method for structural integrity assessment," NDT international, vol. 11, no. 2, pp. 61-67, 1978.

[5] H. Salane and J. Baldwin Jr, "Identification of modal properties of bridges," Journal of Structural Engineering, vol. 116, no. 7, pp. 2008-2021, 1990.

[6] I. Peroni, A. Paolozzi, and A. Bramante, "Effect of debonding damage on the modal damping of a sandwich panel," in 9th Conference International Modal Analysis Conference (IMAC), 1991, vol. 1, pp. 1617-1622.

[7] D. Montalvão, A. Ribeiro, and J. Duarte-Silva, "A method for the localization of damage in a CFRP plate using damping," Mechanical Systems and Signal Processing, vol. 23, no. 6, pp. 1846-1854, 2009.

[8] S. Mustafa, Y. Matsumoto, and H. Yamaguchi, "Vibration-Based Health Monitoring of an Existing Truss Bridge Using Energy-Based Damping Evaluation," Journal of Bridge Engineering, vol. 23, no. 1, p. 04017114, 2017.

[9] J. Li, H. Hao, G. Fan, P. Ni, X. Wang, C. Wu, J.M. Lee and K.H. Jung, "Numerical and experimental verifications on damping identification with model updating and vibration monitoring data". Smart Structures and Systems, 20(2), pp.127-137, 2017.

[10] M.S. Cao, G.G. Sha, Y.F. Gao and W. Ostachowicz, "Structural damage identification using damping: a compendium of uses and features". Smart Materials and Structures, 26(4), p.043001, 2017

[11] C. W. de Silva, Vibration Damping, Control, and Design. CRC Press, 2007.

27

[12] Y. Liao and V. Wells, "Modal parameter identification using the log decrement method and band-pass filters," Journal of Sound and Vibration, vol. 330, no. 21, pp. 5014-5023, 2011.

[13] C. Ge and S. Sutherland, "Application of experimental modal analysis to determine damping properties for stacked corrugated boxes," Mathematical Problems in Engineering, vol. 2013, 2013.

[14] J. Cooper, "Parameter estimation methods for flight flutter testing," Advanced Aeroservoelastic Testing and Data Analysis, vol. 10, p. 2, 1995.

[15] A. Tondl, "The application of skeleton curves and limit envelopes to analysis of nonlinear vibration," Shock and Vibration Inform. Center The Shock and Vibration Digest, vol. 7, no. 7, pp. 3-20, 1975.

[16] D. Brown, R. Allemang, R. Zimmerman, and M. Mergeay, "Parameter estimation techniques for modal analysis," SAE Technical paper0148-7191, 1979.

[17] J. N. Yang, Y. Lei, S. Lin, and N. Huang, "Hilbert-Huang based approach for structural damage detection," Journal of engineering mechanics, vol. 130, no. 1, pp. 85-95, 2004.

[18] R. Curadelli, J. Riera, D. Ambrosini, and M. Amani, "Damage detection by means of structural damping identification," Engineering Structures, vol. 30, no. 12, pp. 3497-3504, 2008.

[19] A. M. Ay, "Impulse vibration based structural damage identification methods in time domain," Deakin University2017.

[20] P. Welch, "The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms," IEEE Transactions on audio and electroacoustics, vol. 15, no. 2, pp. 70-73, 1967.

[21] G. Kitagawa, Introduction to Time Series Modeling. CRC Press, 2010.[22] J. K. Sinha, Vibration Analysis, Instruments, and Signal Processing. CRC Press, 2014.[23] G. Palshikar, "Simple algorithms for peak detection in time-series," in Proc. 1st Int. Conf. Advanced

Data Analysis, Business Analytics and Intelligence, 2009, pp. 1-13.[24] A. M. Ay, Y. Wang, and S. Khoo, "Signal Processing for Time Domain Analysis of Impact Hammer

Test Data," presented at the 8th European Workshop On Structural Health Monitoring (EWSHM 2016), Spain, Bilbao, 5-8 July 2016, 2016.

[25] M. A. Fligner and T. J. Killeen, "Distribution-free two-sample tests for scale," Journal of the American Statistical Association, vol. 71, no. 353, pp. 210-213, 1976.

[26] T. P. Hettmansperger and J. W. McKean, Robust Nonparametric Statistical Methods, Second Edition. CRC Press, 2010.

[27] T. V. Perneger, "What's wrong with Bonferroni adjustments," British Medical Journal, vol. 316, no. 7139, p. 1236, 1998.

[28] B. Efron and R. J. Tibshirani, The jackknife. Springer, 1993.[29] G. A. Milliken and D. E. Johnson, Analysis of Messy Data Volume 1: Designed Experiments, Second

Edition. CRC Press, 2009.[30] Z. F. Fu and J. He, Modal Analysis. Elsevier Science, 2001.[31] M. D. Mesarovic, D. Macko, and Y. Takahara, Theory of Hierarchical, Multilevel, Systems. Elsevier

Science, 2000.[32] I. H. Bernstein, C. P. Garbin, and G. K. Teng, Applied Multivariate Analysis. Springer New York,

2012.[33] T. Świątkowski, "On the conditions of monotonicity of functions," Fundamenta Mathematicae, vol.

59, no. 2, pp. 189-201, 1966.

28

Appendix

Suppose two mappings (multivariate functions) of g and h have a common domain and take on values in the same partially ordered set. Functions { g;h } are monotonically related if the values of one function is related in a monotonic fashion to the corresponding values of the other [31]; where, g is said to be (negatively) monotonically related toh, if and only if, g ( x1; x2;…; x N ) ≤ g (x1

' ; x2' ;…; xN

' ) wheneverh ( x1; x2 ;…;x N ) ≥ h( x1' ; x2

' ;…; xN' ). Further,

the two functions are strictly monotonically related if, and only if [31, 32]:g ( x1; x2;…; x N )<g (x1

' ; x2' …; xN

' ) ,whenever h ( x1; x2 ;…;x N )>h ( x1

' ; x2' ;…; xN

' )(A-1)

where, xn≠ xn' , and datasets ( x1; x2 ;…;x N ) and (x1

' ; x2' …; xN

' ) adhere to Eq. (8).Two significant points arise from the monotonic relation of function g to h:

i. Functions g ( x1; x2;…; x N ) andh ( x1; x2 ;…;x N ), are only mutually monotonically related, if and only if, functions g;h are strictly monotonically related, i.e., a non-strict monotonic relationship is not symmetrical, but reflexive.

ii. An efficient approach to prove a monotonic relation between two functions is to determine a corresponding general monotone function, say Λ. Hence, function g ( x1; x2;…; x N ) is shown to be monotonically related toh ( x1; x2 ;…;x N ), if there exists a corresponding monotone function Λ, from R (h ) onto R ( g ), such that g ( x1; x2;…; x N )=Λ (h ( x1; x2; …; x N )), for all x j in their domains, where R ( g ) and R (h ), are the range of the functions g and h. However, given that functions g ( x1; x2;…; x N ) and h ( x1; x2 ;…;x N ) are multivariate, a monotonic function Λ, is analytically intractable, since no general definition of monotonicity exists, i.e., no total order ofR2 exists [31-33]. This condition is further complicated by the fact that set parameterstmax and ωi are variables, which directly influences Eq. (6).

In view of the second point, let us confine our theoretical framework with a number of appropriate restrictions. Thus, we are to establish the following conditions, where parameters tmax and ωn are given constants, such that:

0<tmax ≪∞ , (A-2)

ωn

2π>tmax , (A-3)

x j (t|t <tmax ¿≠0 , (A-4)

Assuming conditions in Eqs. (A-2)-(A-4) is true, the following axioms are reached:Axiom 1. From Eq. (5), the amplitude of the waveform is normalised, and thus:

−1≤ x ≤1. (A-5)

Axiom 2. Since in Eq. (5), Q=1, and parameters {ζ , ω } are single constants (SDOF), then:

|x1|≥|x2|…≥|xn|≥ …≥|xN|, where|xn|≠ 0 (A-6)

Axiom 3. From Eqs. (5)-(7), and with the assumptions of no external forces and disturbances

29

acting on the SDOF system:

ln ( xn

xn+2)=ln( xm

xm+2) , where{n ,m }∈(1 , …, N ). (A-7)

Axiom 4. From Eq. (6), the period of waveform remains constant, and thus:x j (t∨t=(t 1+mT )) , wherem=0 ,0.5 ,1 ,1.5 , … (A-8)

with T denotes the period of the waveform of IRF in Eq. (5).Axiom 5. From Eq. (6), we can define the local maxima-minima amplitudes’ constraints:

{x1 , x3 , x5 , …∨0<xn≤ 1 }{x2 , x4 , x6 , …∨−1≤ xn<0 }}given ψ=0 , (A-9)

where ψ is the phase angle of IRF.Let us now recall the first half of Eq.(A-1), which defines the strict negative monotonic

relation of functions { g ;h }:g ( x1; x2;…; x N )<g (x1

' ; x2' ;…; xN

' ) . (A-10)

Using the definition of multivariate function g ( x1; x2;…; x N ) in Eq. (10), combined with Axioms 2-5, it is evident that in order for the condition in Eq. (A-10) to be true, the input variable sets {x1 ;x2 ;…; xN } and {x1

' ;x2' ;…; xN

' }, must meet the following condition:

{x1 ≥ x1' }∧{x2>x2

' ;…; xN >xN' }, (A-11)

where, x1≥ x1' is the first local maxima (ψ=0), both normalised in amplitude. Using condition

in Eq.(A-11), we now apply this to the second half of Eq.(A-1), and thus we reach the following:

{x1 ≥ x1' }∧{x2>x2

' ;…; xN >xN' }→ h ( x1; x2; …; x N )>h (x1

' ; x2' ;…; xN

' ) . (A-12)

From this deduction, we conclude that functions g ( x1; x2;…; x N ) and h ( x1; x2 ;…;x N ) are strictly (negatively) monotonically related.

articleepubs.surrey.ac.uk/849950/1/pddr final submission.docx · web viewthe logarithmic decrement...

Documents