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Smooth Modes as a Tool for Operational Modal Analysis: New Developments Rubens Sampaio PUC-Rio rsampaio@puc-rio.br PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 1 / 39 Outline Review of Karhunen-Lo`

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Smooth Modes as a Tool for Operational Modal Analysis: New Developments

Rubens Sampaio

PUC-Rio rsampaio@puc-rio.br

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 1 / 39

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Outline

Review of Karhunen-Lo` eve History of SOD in modal analysis SOD in operational modal analysis Noise sensitivity of standard SOD New developments Conclusions

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 2 / 39

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First part-KLD: Naive look

Some history-main applications Main idea of KL decomposition The mathematical problem Construction of the KL basis How to combine KL with Galerkin Reduced model given by KL Practical question: how to compute the KLD An example Different guises of KL; basic ingredients Karhunen-Lo` eve expansion: main hypothesis Karhunen-Lo` eve Theorem Basic properties Applications to Random Mechanics Examples

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 3 / 39

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Some history

Beginning: works in Statistics and Probability and Spectral Theory in Hilbert Spaces. Some contributions: Kosambi (1943) Lo` eve (1945) Karhunen (1946) Pougachev (1953) Obukhov (1954) Applications: Lumley (1967): method applied to Turbulence (coherent structures, 1970) Sirovich (1987): snapshot method An important book appeared in 1996: Holmes, Lumley, Berkooz. In Solid Mechanics the applications started around 1993. In finite dimension it appears under different guises.

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 4 / 39

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Main Applications

Principal Component Analysis (PCA): Statistics and image processing Empirical orthogonal functions: Oceanography and Metereology (Lorenz 1956) Factor analysis: Psychology and Economics Data analysis: Principal Component Analysis (PCA) Reduced models, through Galerkin approximations Dynamical Systems: to understand the dynamics Signal Analysis

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 5 / 39

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Two main purposes of KLD

  • rder reduction by projecting high-dimensional data in

lower-dimensional space feature extraction by revealing relevant, but unexpected, structure hidden in the data

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 6 / 39

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Main idea of KLD

In plain words: The key idea of KL is to reduce a large number of interdependent variables to a much smaller number of uncorrelated variables while retaining as much as possible the variation in the original data.

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 7 / 39

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Main idea of KL decomposition

more precisely Suppose we have an ensemble {uk} of scalar fields, each being a function defined in (a, b) ⊂ R. We work in a Hilbert space L2((a, b)). We want to find a (orthonormal) basis {ψn}∞

n=1 of L2 that

is optimal for the given data set in the sense that a finite dimensional representation of the form ˆ u(x) =

N

  • k=1

akψk(x) , for N given, describes a typical member of the ensemble better than representations of the same dimension in any

  • ther basis.

The notion of typical implies the use of an average over the ensemble {uk} and optimality means maximazing the average normalized projection of u into {ψn}N

n=1 .

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 8 / 39

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The mathematical problem

Suppose, for simplicity, we have just one function ψ maxψ∈L2 E(| < u, ψ > |2) ψ2 This implies J(ψ) = E(| < u, ψ > |2) − λ(ψ2 − 1) d dεJ(ψ + εφ)|ε=0 = 0 b

a

R(x, y)ψ(y)dy = λψ(x) with R(x, y) = E(u(x)u(y))

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 9 / 39

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Construction of the KL basis

Construct R(x,y) from the data Solve the eigenvalue problem:

  • D

R(x, y)ψ(y)dy = λψ(x) to get the pair (λi, ψi) If u is the field then the N-order approximation of it is ˆ uN(t, x) = E(u(t, x)) + ΣN

i=1ai(t)ψ(x)

To make predictions use the Galerkin method taking the ψ’s as trial functions

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 10 / 39

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Galerkin projections

Suppose we have a dynamical system governed by

∂v ∂t

= A(v) v ∈ {(a, b) × D → Rn} v(0, x) = v0(x) initial condition B(v) = boundary condition The Galerkin method is a discretization scheme for PDE based on separation of variables. One searches solutions in the form: ˆ v(x, t) =

N

  • k=1

ak(t)ψk(x)

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 11 / 39

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Reduced equations

The reduced equation is obtained making the error of the approximation

  • rthogonal to the first N KL elements of the basis.

errorequation(t, x) =

∂ˆ v ∂t − A(ˆ

v) errorinicond(x) = ˆ v(0, x) − v0(x) < errors, ψi(x) >= 0 for i = 1, ..., N.

dai dt (t)

=

  • D A(ΣN

n=1an(t)ψn(x))ψi(x)dx

for i = 1, ..., N ai(0) =

  • D v0(x)ψi(x)dx

for i = 1, ..., N

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 12 / 39

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History of SOD in modal analysis

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 13 / 39

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Smooth Orthogonal Decomposition - Definition

The Smooth Orthogonal Decomposition (SOD) consists in an extension of the Proper Orthogonal Decomposition (Karhunen-Lo` eve decomposition) with a constraint on the smoothness. When applied to a vector field, the SOD consists in: Find the basis for an orthogonal projection that maintains the maxi- mum variance of the original field and also performs it as smooth as possible in relation to time.

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 14 / 39

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Smooth Orthogonal Decomposition - Definition

Let x(t) = [x1(t), . . . , xn(t)]T be a real-valued stochastic field with respect to time t ∈ R. It is assumed to be stationary and have zero mean. The orthogonal projection of x(t) with respect to the vector ψ is defined as v(t) = projψx(t) = x(t), ψ ψ ψ Assuming for simplicity that ψ is an unit vector, the norm of v(t) is v(t) = x(t), ψ

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 15 / 39

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Smooth Orthogonal Decomposition - Definition

maintains the maximum variance of the original field maxψ

  • E
  • v(t)2

= maxψ {E [ψ, x(t).x(t), ψ]} = maxψ

  • ψTE
  • x(t)xT(t)
  • ψ
  • .

performs it as smooth as possible in relation to time minψi

  • E
  • ˙

vi(t)2 = minψi {E [ψi, ˙ x(t).˙ x(t), ψi]} = minψi

  • ψT

i E

  • ˙

x(t)˙ xT(t)

  • ψi
  • .

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 16 / 39

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Smooth Orthogonal Decomposition - Definition

This two optimization problems can be brought together to only one maximization problem by creating the function α(ψ) as following: maxψ {α(ψ)} = maxψ

  • E
  • v(t)2

E [˙ v(t)2]

  • =

maxψ

  • ψTRxx(0)ψ

ψTR ˙

x ˙ x(0)ψ

  • .

Where Rxx(0) and R ˙

x ˙ x(0) are correlation function matrices defined as:

Rxx(τ) = E

  • x(t + τ)xT(t)
  • R ˙

x ˙ x(τ) = E

  • ˙

x(t + τ)˙ xT(t)

  • Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition

17 / 39

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Smooth Orthogonal Decomposition - Definition

The solution of this maximization problem is obtained by finding its stationary point, which leads to the following eigenvalue problem: Rxx(0)ψ = λR ˙

x ˙ x(0)ψ

Since Rxx(0) and R ˙

x ˙ x(0) are n × n matrices, this eigenvalue problem leads

to the set {ψi}n

i=1, called smooth modes

the set {λi}n

i=1, called smooth values

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 18 / 39

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SOD in operational modal analysis

Operational modal analysis consists in identify the modal parameters (natural frequencies and vibration modes) of a system under ambient excitation monitoring only its responses (displacements, velocity or acceleration). System x(t) ? To identify the modal parameters using SOD, one has to establish the following relationship: smooth modes ⇔ vibration modes smooth values ⇔ natural frequencies

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 19 / 39

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SOD in operational modal analysis

Vibration of an undamped MDOF system: The dynamics of a discretized linear conservative mechanical system under ambient excitation can be modeled by the equation M¨ x(t) + Kx(t) = f(t), where f(t) is assumed to be a zero-mean white noise process and the mass matrix M is assumed to be invertible. The acceleration can be expressed as ¨ x(t) = −M−1Kx(t) + M−1f(t). Also, the natural frequencies and the normal modes can be extracted by the eigenvalue problem M−1Kφ = ω2φ, where {ω} are the natural frequencies and {φ} the normal modes.

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 20 / 39

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SOD in operational modal analysis

Rxx(0)ψ = λR ˙

x ˙ x(0)ψ

⇔ M−1Kφ = ω2φ Relationship between the eigenvalue problems

Rxx(0)ψ = λR ˙

x ˙ x(0)ψ

= −λR¨

xx(0)ψ

(stationary assumption) = −λE

  • −M−1Kx(t) + M−1f(t)
  • xT(t)
  • ψ

= λE

  • M−1Kx(t)xT(t)
  • ψ − λE
  • M−1f(t)xT(t)
  • ψ

= λM−1KRxx(0)ψ (causality assumption)

Rxx(0)ψ

  • φ

= λ

  • 1/ω2

M−1K Rxx(0)ψ

  • φ

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 21 / 39

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Example

A 6 degree-of-freedom conservative system with m = 0.1kg, k = 10kN/m is excited by uncorrelated zero-mean Gaussian white noises with standard deviations equal to 100N. The respective displacements were simulated with a sampling frequency equal to 3kHz in a time interval of 50 seconds. After processing the signal with a differentiation filter, the velocity were also obtained. The correlation matrices Rxx(0) and R ˙

x ˙ x(0) were estimated

and the SOD eigenvalue problem solved.

f1(t) f2(t) f3(t) f4(t) f5(t) f6(t) x1(t) x2(t) x3(t) x4(t) x5(t) x6(t)

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 22 / 39

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Example

The natural frequencies and mode shapes were obtain from the relationship with the smooth values and smooth vectors, resulting in: ωn Analytic SOD 1¬

  • 12.12

12.14 2¬

  • 35.69

35.72 3¬

  • 57.17

57.18 4¬

  • 75.35

75.35 5¬

  • 89.17

89.16 6¬

  • 97.69

97.74

1 mode 2 mode 3 mode 4 mode

x1 x2 x3 x4 x5 x6

5 mode

x1 x2 x3 x4 x5 x6

6 mode

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 23 / 39

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Noise sensitivity

When modeling the sensor noises w(t) ∈ Rn as uncorrelated zero-mean white noise processes, the correlation of the measured data can be calculated as Rxx(τ) = E

  • x(t + τ)xT(t)
  • =

E

  • [z(t + τ) + w(t + τ)]
  • zT(t) + wT(t)
  • =

E

  • z(t + τ)zT(t)
  • + E
  • z(t + τ)wT(t)
  • +

+E

  • w(t + τ)zT(t)
  • + E
  • w(t + τ)wT(t)
  • =

Rzz(τ) + Rww(τ).

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 24 / 39

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Noise sensitivity

Correlation of the observed data: Rxx(τ) = Rzz(τ) + Rww(τ)

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-3

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-5

  • 2

2 4 6 8 10 12 14 16 18

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-3

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5

= +

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 25 / 39

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Noise sensitivity

Correlation of the observed data: Rxx(τ) = Rzz(τ) + Rww(τ)

τ(s)

  • 0.02
  • 0.015
  • 0.01
  • 0.005

0.005 0.01 0.015 0.02 ×10-3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 26 / 39

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Noise sensitivity

Noise correlation after digital differentiation filter:

First order differentiator

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6

  • 50

50 100

Maximum flat FIR differentiator

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6

  • 40
  • 20

20 40 60 80

Park-McClellan FIR differentiator

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6

  • 100
  • 50

50 100 150

FFT differentiator

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6

  • 100
  • 50

50 100 150

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 27 / 39

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Noise sensitivity

Noise correlation after digital integration filter (cutoff freq. =

fs 1000):

Trapezoidal rule

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-9

  • 2

2 4 6 8

First-order lowpass filter

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-9

  • 2

2 4 6 8

32th order FIR highpass / 8th order IIR filter

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-9

  • 2

2 4 6 8

FFT Integration

=(s)

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 # 10-9

  • 2

2 4 6 8

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 28 / 39

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Extension for noise control and uncertainty quantification

When x(t) is contaminated with noise (thanks to the sensors and hardware involved), correlations function are most effected at τ = 0. Therefore, a extension of the SOD method, called rSOD, is proposed using correlations for positive values of τ. The rSOD eigenvalue problem became Rxx(τ)ψ = λR ˙

x ˙ x(τ)ψ,

for τ > 0 Benefits: Calculate the Smooth modes and Smooth values with less distorted correlation matrices (τ = 0). Calculate the natural frequencies and mode shapes several times, one for each value of τ. A statistic of the results can then be perform to deal with the noise in the data.

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 29 / 39

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Extension for noise control and uncertainty quantification

Rxx(τ)ψτ = λτR ˙

x ˙ x(τ)ψτ

⇔ M−1Kφ = ω2φ Relationship between the eigenvalue problems

Rxx(τ)ψτ = λτR ˙

x ˙ x(τ)ψτ

= −λτR¨

xx(τ)ψτ

(stationary assumption) = −λτE

  • −M−1Kx(t + τ) + M−1f(t + τ)
  • xT(t)
  • ψτ

= λτE

  • M−1Kx(t + τ)xT(t)
  • ψτ − λτE
  • M−1f(t + τ)xT(t)
  • ψτ

= λτM−1KRxx(τ)ψτ, τ > 0 (causality assumption)

Rxx(τ)ψτ

  • φ

= λτ

  • 1/ω2

M−1K Rxx(τ)ψτ

  • φ

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 30 / 39

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Example

To illustrate the rSOD method, the same system and excitation of last example is once again used. This time white noises are added to the simulated responses to illustrate possible measurement noises. Those processes have 5% standard deviation of the respective standard deviations

  • f the system’s responses.

f1(t) f2(t) f3(t) f4(t) f5(t) f6(t) x1(t) x2(t) x3(t) x4(t) x5(t) x6(t)

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 31 / 39

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Example

In this example correlation matrices are estimated with different time lags, 0 < τ < 0.5 segundos. Since the sampling frequency is 3kHz, a total of 1500 time lags were used, which means that all natural frequencies and mode shapes were identified 1500 times. Histograms and envelope plots can be constructed for each natural frequency and mode shape, respectively.

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 32 / 39

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Example

1st mode:

Natural frequency histogram

11.6 11.7 11.8 11.9 12 12.1 12.2 12.3 12.4 12.5 12.6

Frequency (Hz)

histogram

  • max. prob.

analytical

Mode shape envelope

x1 x2 x3 x4 x5 x6 std rSOD analytical

Analytical SOD rSOD ω1 12.12 14.20 12.13

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 33 / 39

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Example

2nd mode:

Natural frequency histogram

35.2 35.3 35.4 35.5 35.6 35.7 35.8 35.9 36 36.1 36.2

Frequency (Hz)

histogram

  • max. prob.

analytical

Mode shape envelope

x1 x2 x3 x4 x5 x6 std rSOD analytical

Analytical SOD rSOD ω2 35.69 46.70 35.68

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 34 / 39

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Example

3rd mode:

Natural frequency histogram

56.7 56.8 56.9 57 57.1 57.2 57.3 57.4 57.5 57.6 57.7

Frequency (Hz)

histogram

  • max. prob.

analytical

Mode shape envelope

x1 x2 x3 x4 x5 x6 std rSOD analytical

Analytical SOD rSOD ω3 57.17 63.44 57.18

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 35 / 39

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Example

4th mode:

Natural frequency histogram

74.8 74.9 75 75.1 75.2 75.3 75.4 75.5 75.6 75.7 75.8

Frequency (Hz)

histogram

  • max. prob.

analytical

Mode shape envelope

x1 x2 x3 x4 x5 x6 std rSOD analytical

Analytical SOD rSOD ω4 75.35 84.35 75.34

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 36 / 39

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Example

5th mode:

Natural frequency histogram

88.6 88.7 88.8 88.9 89 89.1 89.2 89.3 89.4 89.5 89.6

Frequency (Hz)

histogram

  • max. prob.

analytical

Mode shape envelope

x1 x2 x3 x4 x5 x6 std rSOD analytical

Analytical SOD rSOD ω5 89.17 100.23 89.14

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 37 / 39

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Example

6th mode:

Natural frequency histogram

97.2 97.3 97.4 97.5 97.6 97.7 97.8 97.9 98 98.1 98.2

Frequency (Hz)

histogram

  • max. prob.

analytical

Mode shape envelope

x1 x2 x3 x4 x5 x6 std rSOD analytical

Analytical SOD rSOD ω6 97.69 113.20 97.74

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 38 / 39

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Conclusion

The SOD: identifies the modal parameters of a conservative systems. identifies the system with a simple and fast algorithm. quantifies the energy partition between modes. The rSOD: identifies the modal parameters with noisy signals. quantifies the modal parameters uncertainties

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 39 / 39

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Thank you!

Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 40 / 39