# Structural Condition Monitoring by Cumulative Harmonic Analysis of Random Vibration

Authors: Yoshinori Takahashi; Toru Taniguchi; Mikio Tohyama
Journal: Advances in Acoustics and Vibration
 (2008)

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## Abstract

Analysis of signals based on spectral accumulation has great potential for enabling the condition
of structures excited by natural forces to be monitored using random vibration records. This
article describes cumulative harmonic analysis (CHA) that was achieved by introducing a spectral
accumulation function into Berman and Fincham's  conventional cumulative analysis, thus enabling
potential new areas in cumulative analysis to be explored. CHA effectively enables system
damping and modal overlap conditions to be visualized without the need for transient-vibration
records. The damping and modal overlap conditions lead to a spectral distribution around dominant
spectral peaks due to structural resonance. This distribution can be revealed and emphasized
by CHA records of magnitude observed even within short intervals in stationary random vibration
samples.


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## Body

## 1. Introduction

This paper
describes spectral changes  visualized in a structural-vibration system using
cumulative harmonic analysis (CHA) [1] under random vibration conditions. Spectral changes in vibrations should be informative
for monitoring and diagnosing system health in structures. However, it is
difficult to track variations in the structural-transfer functions independent
of the source-signal characteristics. This is because the transfer-function
analysis of structural-vibration systems generally requires the specifications
of the excitation source [2, 3].Hirata [4] proposed a method of monitoring invisible changes in
structures based on the frequency distributions of the dominant spectral
components in every short interval period (SIP) under random and nonstationary
vibration conditions without any specific source-signal assumptions or
requirements. Changes in the modal resonance bandwidth, which represent the
damping conditions for modal vibration and must therefore be a significant
indicator of the health of a structure, can be evaluated from the frequency histogram
of the dominant components. As variance in the distribution increases, the
modal bandwidth widens, thus increasing damping. Hirata [5] also developed a method of
spectral analysis, called nonharmonic Fourier analysis, for extracting the
dominant low-frequency components of a vibration record in an SIP. However,
spectral analysis in SIPs is normally difficult for frequency resolution and
artifacts, because of time-windowing functions [6].Resonance can be interpreted as a
spectral-accumulation process of in-phase sinusoidal components with a
fixed-phase lag. Therefore, as the impulse response for a resonant system is
written as an infinite series from the signal-processing viewpoint, the effective
length of the sequence increases as system damping decreases but this is short
under heavy damping conditions. We examined the spectral-accumulation process
on a sample-by-sample basis to visualize the spectral changes in the dominant
modal-resonant components under random vibration.Berman and Fincham [7] previously formulated cumulative spectral analysis
(CSA) using a stepwise time-window function to evaluate the transient
characteristics of loudspeaker systems. We looked at the cumulative properties
of a spectral display of vibration records including transfer function changes.
In Section 2, we introduce a forgetting function into CSA instead of using a
stepwise time window. We called this cumulative harmonic analysis (CHA), and
its purpose was to emphasize the spectral-accumulation process, including the
increasing resonant spectral peak, on a sample-by-sample basis [8]. We describe CHA's
effectiveness in enabling changes in the transfer functions to be visualized
through numerical simulation experiments on a single vibrating system in
Section 3. In Section 4, we also describe CHA's effectiveness in a
two-degrees-of-freedom (2-DOF) vibrating system. Section 5 discusses CHA
visualization under conditions with damping changes.


## 2. Formulation of Cumulative Harmonic Analysis (CHA) from CSA

We formulate
CHA by introducing a forgetting function into CSA that corresponds to the
spectral-accumulation function. Assume that we have a signal sequence,x(n),
and define a spectral-accumulation function, w(n).
We define the cumulative harmonic analysis (CHA) of x(n) as(1)CHA(n,e−jΩ)≡∑m=0nX(m,e−jΩ),where(2)X(m,e−jΩ)≡w(m)x(m)e−jΩm,0≤m≤n.Substituting a forgetting
function such as(3)w(m)≡m+1,0≤m≤n,into spectral accumulation
function w(n),
we have(4)CHA(n,z−1)=∑m=0n(m+1)x(m)z−m=1x(0)z−0+2x(1)z−1+3x(2)z−2+⋯+nx(n)z−nin the z-plane including the unit circle where the
Fourier transform can be defined. Figure 1 has a schematic of CHA.
We use a triangular window as the spectral accumulation function, w(m)=m+1,
in this paper where the mathematical expression in the accumulation effect is
simple. However, we can freely define accumulation function w(n) as an exponential function, w(m)=eαm,
based on the degree of necessary spectral accumulation (or forgetting effect).Figure 1
Schematic for cumulative
harmonic analysis (CHA).The effect of the transfer-function pole on
frequency characteristics can be emphasized byCHA.
Assume that we have a simple decaying sequence(5)x(n)≡an(n=0,1,2,…),0<|a|<1.If we take the limit for CHA of the sequence above as(6)limn→∞CHA(n,z−1)≡limn→∞∑m=0n(m+1)amz−m=1(1−az−1)2,then we can see that CHA virtually increases the order of the pole
compared with regular discrete Fourier transform (DFT) (w(n)=1)(7)limn→∞DFT(n,z−1)=limn→∞∑m=0namz−m=11−az−1.Consequently, the  modal resonance is  visualized as a
way for the dominant-frequency components of vibration records to be narrowed
down to dominant  elements. Changes in the resonance of the structural transfer
functions can be expected to be  visualized even under stationary
random-vibration conditions. Figure2 shows what effect CHA has against regular
DFT using real and causal sequences including complex conjugate poles. In this
example, pole a in (5) has been expanded to a complex number
for a more general demonstration. The locations of the complex-conjugate poles
are shown in panel (d). The impulse response is illustrated in panel (a), the
magnitude record for CSA is in panel (c), and that for CHA is in (b). The
maximum magnitude is normalized to unity at every instant in both panels (b)
and (c). We can see the spectral-accumulation process is emphasized by CHA
where the resonant peak is increasing.Samples of CHA (panel b) and CSA (panel c) magnitude
displays for impulse response of single-degree-of-freedom resonance system
where poles of transfer function are located atπ/4  in z-plane. Panel (a) illustrates sequence
of impulse responses. Maximum magnitude is normalized to unity at every instant
in both panels (b) and (c) and we set N=800.
(a)(b)(c)(d)


## 3. CHA Example of Random Vibration Record in Single-Degree-of-Freedom (SDOF) System

Assume that
we have a random vibration record observed in a single-degree-of-freedom (SDOF)
system. These types of structural-vibration samples are normally easy to obtain
in SIPs due to the natural force of winds, ground movement, or both [4] without specifically having
to prepare source signals, which is impractical for large structures.Figure3 shows examples of CHA random-vibration
records, including an impulse response. In the examples, we have defined the
damping factor, h,
as(8)h≡−lnaΩp≅0.078.The axis representing time in
Figure 2 is normalized by the length of the impulse response record, which is
given by the reverberation time, TR,
which is estimated as(9)TR≡NTs≅6.9Ts−lna.Here, we define N as the length of an impulse response record,
and Ts denotes the sampling period of discrete
sequences.Examples of CHA and
spectrograms for random record of single-degree-of-freedom (SDOF) system. (a)
CHA magnitude and (b) spectrogram using STFT conditions are frame length, that
is, 0.18 with zero padding under conditions where total Fourier-transform
length is 91 and frame hop size is 1/10 frame length. Time intervals are
normalized by impulse response record length (1000 points). (c) Moving averaged
spectrogram using averaging STFT, (b) frame-by-frame, (d) random record to be
analyzed, and (e) impulse response record of vibration system where length is
defined by reverberation time. (f) Magnitude response and normalized frequency,
(g) close up of CHA magnitude in (a) at final observation instant accumulation
time of 1, and (h) pole plot of transfer function for impulse response in (e).Algorithms for (a) CHA, (b)
STFT, and (c) MA. DFT means anN-point DFT with zero padding.
(a)(b)(c)In Figure3, the CHA, STFT, and MA spectra with the
peaks plots were measured on a sample-by-sample or frame-by-frame basis and the
maximum magnitude records were normalized to unity in every instance of
observation. We can see that CHA in panel (a) visualizes the resonant
properties of the SDOF system better than the conventional spectrogram using
STFT in panel (b). The spectrograms using STFT conditions are  the frame length,
that is, 0.18 with zero padding under conditions where the total
Fourier-transform length is 91 and the frame hop size is 1/10 the frame length
(the time intervals are normalized by the impulse response record length (1000
points)). The MA spectrum in panel (c) also seems to enable stable analysis of
resonant properties. The MA spectrogram uses averaging STFT frame-by-frame
according to the frame progression. Panel (g) shows the CHA peak corresponding
to the impulse response spectrum peak in panel (f). This means that CHA can
promptly and effectively be used to determine the presumed modal frequency of
the transfer function included in the noise signal. The accumulation time
required to obtain a stable CHA-magnitude record seems to be around one-half
the length of the impulse-response record. The spectral components of CHA
obtained in every instance are interpolated by DFT with zero padding after the
vibration record. Accumulating  the interpolated spectra provides an accurate
estimate of the modal frequency and does not suffer from the artifacts of the
time-windowing function; consequently, the modal properties are emphasized by
virtually increasing the degree of the pole.


## 4. CHA Monitoring of Modal Overlap Conditions in Two-Degrees-of-Freedom (2-DOF) System

The damping
condition is a significant indicator of structural damage. This section
explains how CHA visualizes the effect of damping on the spectral distribution.
Structural-vibrating systems at low frequencies generally have low modal
overlap. However, if there is a pair of adjacent poles, the damping conditions
change the modal overlap, which is defined as [9, 10](10)M≡BΔω¯,B≅πδ=πhωp,where B is the modal bandwidth, Δω¯ is the average modal spacing, and ωp is the modal angular frequency of interest.
Figure 5 shows examples of the pole/zero plots for a 2-DOF system under low
modal-overlap conditions. Panel (b) is a plot with a single zero, and panel (a)
is that without zeros between the poles. A pair of same sign residues yields a
zero between the poles, and a pair of poles with different sign residues has no
zeros [11–13]. Here, we set Δω¯ as the distance between the two pole angular
frequencies in Figure 5. ωp is given by Ωp1.
We varied the damping conditions to control the modal overlap.Pole/zero plots for
two-degrees-of-freedom systems (2-DOF systems) and magnitude responses.
(a)(b)Figure6 shows the CHA magnitudes and MA spectrums for
the 2-DOF system illustrated in Figure 5. We can seen that CHA in both panels
(a) and (b) visualizes the resonant properties of the 2-DOF system better than
the conventional MA spectrograms. Panel (a) is a plot without zeros, and panel
(b) is one with a single zero. The modal overlap conditions in both panels were
set to M=0.5.
The top and middle panels illustrate the CHA and MA records. The bottom panels
in (a) or (b) show the results of spectral tracking [14] for prominent peaks of CHA
or MA spectrograms. Figure 7 shows histograms of all the peaks in Figure 6. We
can see that the CHA peaks correspond to the modal frequencies.CHA and MA displays and
tracking of prominent peaks under 2-DOF-system conditions. Panel (a): 2-DOF
system without zeros, panel (b): 2-DOF system with zero.
(a)(b)Figure 7
CHA and MA peak histogram
for 2-DOF system. Panel (a): 2-DOF system without zeros, panel (b): 2-DOF
system with zero.


## 5. CHA Monitoring under Conditions with Damping Changes

This section
discusses how CHA visualizes spectral changes due to damping conditions. Figure8 shows examples of modal-overlap conditions changing from M=0.25 to M=1 during CHA monitoring with or without a zero
between the poles. Panel (a) shows the case without zeros, and panel (b)
includes a zero between the poles. The bottom panels illustrate the
peak-tracking results of the CHA spectrogram. We can see that the variance in
the peak-tracking lines of the CHA peaks increases according to the change in
damping in both panels.Modal overlap changing
during CHA for 2-DOF system. Panel (a): 2-DOF system without zeros and panel (b): 2-DOF system with zero.
(a)(b)


## 6. Summary

We assessed
CHA and confirmed through numerical simulation experiments that it effectively
enabled system damping and modal-overlap conditions to be visualized without
having to use transient-response records. We developed a potential new area in
cumulative analysis by introducing a spectral-accumulation function into the
conventional method proposed by Berman and Fincham. Spectral properties of the
transfer function can be emphasized and visualized by using CHA-magnitude
records even for short intervals of stationary random-vibration records. For
simulated 2-DOF system vibration records differences and changes in the
modal-overlap conditions were observable in the contour plots of the CHA
magnitude as a function of time and frequency. Our computer simulation
confirmed that CHA can effectively be used to estimate the damped natural
frequency under SDOF conditions. The results indicate that CHA could be an
efficient method of monitoring and diagnosing structures without signal-source
requirements under stationary and nonstationary vibration conditions. However,
substantial field tests are required to develop a more practical monitoring
system.



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*Source: 261758-2008-08-03.xml*