Single Mode Problem

Abstract

This chapter develops the computational algorithm for determining the most probable value (MPV) and covariance matrix of modal parameters in Bayesian operational modal analysis with data in a single setup and for well-separated modes. The MPV of the mode shape is analytically derived in terms of other parameters, which effectively reduces the numerical optimization problem for MPV to involve only four parameters independent of the number of measured degrees of freedom. The asymptotic behavior of MPV for modes with high signal-to-noise ratio is analyzed. Analytical formulas are derived for systematically computing the covariance matrix of parameters given the measured data. Examples with synthetic data, laboratory data and field data are presented to illustrate the algorithms and their applications.

In this chapter we discuss efficient methods for calculating the posterior most probable values (MPV) and covariance matrix of modal parameters in Bayesian operational modal analysis (OMA) with data collected in a single setup. The mode of interest is assumed to be well-separated from others so that it dominates the selected frequency band. The general case of multiple modes is considered in Chap. 13. Similar techniques in this chapter are used for developing algorithms for well-separated modes from multi-setup data in Chap. 14.

Mathematically, the posterior MPV minimizes the negative log-likelihood function (NLLF) and so its determination involves an optimization problem. The algorithm for MPV makes use of the fact that the NLLF can be written as a quadratic function of mode shape. This allows the most probable mode shape to be determined analytically in terms of the remaining parameters. The dimension of the optimization problem is then reduced to four only, regardless of the number of measured DOFs. A good initial guess can be obtained by considering the asymptotic behavior of the MPV when the signal-to-noise ratio of data is high.

The algorithm presented in this chapter is based on Au (2011); see also Zhang and Au (2013). The presentation has been simplified and reorganized. Computer programing of the posterior covariance matrix using the analytical formulas in Sect. 12.4 is simpler as it has taken advantage of a more systematic procedure based on Lagrange multiplier concepts (Sect. 11.2.3).

Recall the problem context from Sect. 10.1 where the measured ambient vibration history is denoted by $\{ {\hat{\mathbf{y}}}_{j} \}_{j = 0}^{N - 1}$ $\left( {n \times 1} \right);$ $n$ is the number of measured DOFs and N is the number of samples per data channel. The scaled Fast Fourier Transform (FFT) of $\{ {\hat{\mathbf{y}}}_{j} \}_{j = 0}^{N - 1}$ at frequency ${\text{f}}_{k} = k/N\Delta t$ (Hz) is

$$\hat{{\mathcal{F}}}_{k} = \sqrt {\frac{\Delta t}{N}} \sum\limits_{j = 0}^{N - 1} {{\hat{\mathbf{y}}}_{j} e^{{ - 2\pi {\mathbf{i}}jk/N}} }$$

(12.1)

where $\Delta t$ (s) is the sampling interval. The set of modal parameters ${\varvec{\uptheta }}$ comprises

$$\mathop f\limits_{\substack{\displaystyle {\text{Natural}} \, \\ {\displaystyle \text{frequency (Hz)}} } } \,\mathop \zeta \limits_{\substack{\displaystyle {\text{Damping}} \, \\ {\displaystyle \text{ratio}} } } \,\mathop S\limits_{\substack{\displaystyle {\text{Modal force}} \\ \, {\displaystyle \text{PSD}} } } \,\mathop {S_{e} }\limits_{\substack{\displaystyle {\text{Prediction}} \, \\ {\displaystyle \text{error PSD}} } } \,\mathop {\varvec{\upvarphi }}\limits_{\substack{\displaystyle n \times 1 \\ {\displaystyle \text{Mode shape}} } }$$

The mode shape is subjected to unit norm constraint, ${\varvec{\upvarphi }}^{T} {\varvec{\upvarphi }} = 1.$ The modal force PSD has the same unit as the PSD of acceleration. The prediction error PSD has the same unit as the PSD of data.

1 Alternative Form of NLLF

The NLLF can be rewritten in a form that facilitates the determination of the most probable mode shape. Recall the NLLF from Sect. 10.1:

$$L({ \varvec{\uptheta}}) = nN_{f} \ln \pi + \sum\limits_{k} {\ln |{\mathbf{E}}_{k} ({\varvec{\uptheta}})|} + \sum\limits_{k} {\hat{{\mathcal{F}}}_{k}^{*} {\mathbf{E}}_{k} ({\varvec{\uptheta}})^{ - 1} \hat{{\mathcal{F}}}_{k} }$$

(12.2)

where ${\mathbf{E}}_{k} ({\varvec{\uptheta }})$ is the theoretical PSD matrix of data for given ${\varvec{\uptheta }}.$ Assuming a single mode in the selected frequency band,

$${\mathbf{E}}_{k} = SD_{k} {\varvec{\upvarphi \upvarphi }}^{T} + S_{e} {\mathbf{I}}_{n}$$

(12.3)

$$D_{k} = \frac{{(2\pi {\text{f}}_{k} )^{{ - 2{\text{q}}}} }}{{(1 - \beta_{k}^{2} )^{2} + (2\zeta \beta_{k} )^{2} }}\quad \beta_{k} = \frac{f}{{{\text{f}}_{k} }}\quad {\text{q}} = \left\{ {\begin{array}{*{20}c} 0 & {\text{acceleration data}} \\ 1 & {\text{velocity data}} \\ 2 & {\text{displacement data}} \\ \end{array} } \right.$$

(12.4)

where ${\mathbf{I}}_{n}$ denotes the $n \times n$ identity matrix. The key is to express $|{\mathbf{E}}_{k} |$ and ${\mathbf{E}}_{k}^{ - 1}$ in explicit form using the eigenvector representation of ${\mathbf{E}}_{k}$. Let $\{ {\mathbf{b}}_{i} \}_{i = 1}^{n}$ ($n \times 1)$ be an orthonormal basis with ${\mathbf{b}}_{1} = {\varvec{\upvarphi }}$. Substituting ${\mathbf{I}}_{n} = \sum\nolimits_{i = 1}^{n} {{\mathbf{b}}_{i} {\mathbf{b}}_{i}^{T} }$ into (12.3) and collecting terms,

$${\mathbf{E}}_{k} = \underbrace {{(SD_{k} + S_{e} )}}_{\text{eigenvalue}}{\varvec{\upvarphi \upvarphi }}^{T} + \sum\limits_{i = 2}^{n} {\underbrace {{S_{e} }}_{\text{eigenvalue}}{\mathbf{b}}_{i} {\mathbf{b}}_{i}^{T} }$$

(12.5)

This shows that ${\mathbf{E}}_{k}$ has eigenvalues $\{ SD_{k} + S_{e} , S_{e} , \ldots , S_{e} \}$ and eigenvectors $\{ {\varvec{\upvarphi }},{\mathbf{b}}_{2} , \ldots ,{\mathbf{b}}_{n} \}$. The determinant of ${\mathbf{E}}_{k}$ is equal to the product of its eigenvalues:

$$|{\mathbf{E}}_{k}| = (SD_{k} + S_{e} )S_{e}^{n - 1}$$

(12.6)

Replacing the eigenvalues in (12.5) by their reciprocals gives ${\mathbf{E}}_{k}^{ - 1}$:

$${\mathbf{E}}_{k}^{ - 1} = (SD_{k} + S_{e} )^{ - 1} {\varvec{\upvarphi \upvarphi }}^{T} + \sum\limits_{i = 2}^{n} {S_{e}^{ - 1} {\mathbf{b}}_{i} {\mathbf{b}}_{i}^{T} }$$

(12.7)

Substituting $\sum\nolimits_{i = 2}^{n} {{\mathbf{b}}_{i} {\mathbf{b}}_{i}^{T} } = {\mathbf{I}}_{n} - {\varvec{\upvarphi \upvarphi }}^{T}$ and rearranging gives

$${\mathbf{E}}_{k}^{ - 1} = S_{e}^{ - 1} \left[ {{\mathbf{I}}_{n} - \left( {1 + \frac{{S_{e} }}{{SD_{k} }}} \right)^{ - 1} {\varvec{\upvarphi \upvarphi }}^{T} } \right]$$

(12.8)

Substituting (12.6) and (12.8) into (12.2) and noting

$$\underbrace {{\mathop {\hat{\mathcal{F}}_{k}^{*} }\limits_{ 1\times n} \mathop {\varvec{\upvarphi }}\limits_{n \times 1} }}_{\text{scalar}}\underbrace {{\mathop {{\varvec{\upvarphi }}^{T} }\limits_{1 \times n} \mathop {\hat{\mathcal{F}}_{k} }\limits_{n \times 1} }}_{\text{scalar}} = \underbrace {{\mathop {({\varvec{\upvarphi }}^{T} {\hat{\mathcal{F}}}_{k} )}\limits_{\text{scalar}} \mathop {({\hat{\mathcal{F}}}_{k}^{*} {\varvec{\upvarphi }})}\limits_{\text{scalar}} }}_{\text{real scalar}} = \mathop {{\varvec{\upvarphi }}^{T} }\limits_{1 \times n} \underbrace {{\text{Re} ({\hat{\mathcal{F}}}_{k}{\hat {\mathcal{F}}}_{k}^{*} )}}_{\substack{ n \times n \\ {\text{real sym}} .} }\mathop {\varvec{\upvarphi }}\limits_{n \times 1}$$

(12.9)

gives

$$L = nN_{f} \ln \pi + \sum\limits_{k} {\ln (SD_{k} + S_{e} )} + (n - 1)N_{f} \ln S_{e} + S_{e}^{ - 1} d - S_{e}^{ - 1} {\varvec{\upvarphi }}^{T} {\mathbf{A}}{\varvec{\upvarphi }}$$

(12.10)

where

$${\mathbf{A}} = \sum\limits_{k} {(1 + \frac{{S_{e} }}{{SD_{k} }})^{ - 1} {\hat{\mathbf{D}}}_{k} }$$

(12.11)

$${\hat{\mathbf{D}}}_{k} = \text{Re} (\hat{{\mathcal{F}}}_{k} \hat{{ \mathcal{F}}}_{k}^{*} )\qquad d = \sum\limits_{k} {\hat{{ \mathcal{F}}}_{k}^{*} \hat{{ \mathcal{F}}}_{k} }$$

(12.12)

In deriving (12.10), we have assumed ${\varvec{\upvarphi }}^{T} {\varvec{\upvarphi }} = 1$. When using this form of the NLLF for determining the MPV by optimization, the mode shape substituted in the expression should always have unit norm.

2 Algorithm for MPV

In (12.10), only the term ${\varvec{\upvarphi }}^{T} {\mathbf{A}}{\varvec{\upvarphi }}$ depends on ${\varvec{\upvarphi }}$. The most probable mode shape, denoted by $\hat{{\varvec{\upvarphi }}}$, should therefore maximize the term subjected to ${\varvec{\upvarphi }}^{T} {\varvec{\upvarphi }} = 1$. As ${\mathbf{A}}$ is positive definite, it follows that $\hat{{\varvec{\upvarphi}}}$ is the eigenvector of ${\mathbf{A}}$ with the largest eigenvalue (Section C.4). Evaluating L at ${\varvec{\upvarphi }} = \hat{{\varvec{\upvarphi }}}$, the MPV of $\{ f, \zeta , S, S_{e} \}$ can be obtained by minimizing

$$\tilde{L}(f,\zeta , S, S_{e} ) = nN_{f} \ln \pi + \sum\limits_{k} {\ln (SD_{k} + S_{e} )} + (n - 1)N_{f} \ln S_{e} + S_{e}^{ - 1} (d - \hat{\lambda })$$

(12.13)

where $\hat{\lambda } = \hat{{\varvec{\upvarphi }}}^{T} {\mathbf{A}}\hat{{\varvec{\upvarphi }}}$ is the largest eigenvalue of ${\mathbf{A}}.$ Note that $\hat{\lambda }$ depends on $\{ f,\zeta , S, S_{e} \}$ but the dependence has been omitted for notational simplicity.

Below are two algorithms for calculating the MPV of ${\varvec{\uptheta}} = \{ f,\zeta , S, S_{e} ,{\varvec{\upvarphi }}\}$. The first one follows directly from the considerations above. The second one is iterative. It is a special case of the multi-mode algorithm in Chap. 13.

Algorithm 1

• 0. Set initial guess for $\{ f,\zeta ,S,S_{e} \}$. See Sect. 12.3.1.

• 1. Determine the MPV of $\{ f,\zeta ,S,S_{e} \}$ by minimizing $\tilde{L}$ in (12.13).

• 2. Determine the MPV of ${\varvec{\upvarphi }}$ as the eigenvector (largest eigenvalue) of ${\mathbf{A}}$ in (12.11) at the MPV of $\{ f,\zeta ,S,S_{e} \}$.

Algorithm 2

The following is an iterative algorithm. At any one step, only the mentioned parameters are updated; the remaining ones are kept at their current value.

• 0. Set initial guess for $\{ f,\zeta ,S,S_{e} ,{\varvec{\upvarphi }}\}$. See Sect. 12.3.1.

• 1. Update $\{ f,\zeta ,S,S_{e} \}$ by minimizing L in (12.10) w.r.t. $\{ f,\zeta ,S,S_{e} \}$.

• 2. Update ${\varvec{\upvarphi }}$ as the eigenvector (largest eigenvalue) of ${\mathbf{A}}$ in (12.11).

Repeat Steps 1 and 2 until convergence.

3 High s/n Asymptotics of MPV

The MPV of modal parameters exhibits characteristic behavior when the signal-to-noise (s/n) ratio of data is high, in the sense that

$$\gamma_{k} = \frac{{SD_{k} }}{{S_{e} }} \gg 1$$

(12.14)

In this case $(1 + S_{e} /SD_{k} )\sim 1$. Substituting into (12.11),

$${\mathbf{A}}\sim {\mathbf{A}}_{0} = \sum\limits_{k} {{\hat{\mathbf{D}}}_{k} } = \text{Re} \sum\limits_{k} {\hat{\mathcal{F}}_{k} {\hat{\mathcal{F}}}_{k}^{*} }$$

(12.15)

The most probable mode shape is therefore asymptotically given by the eigenvector (largest eigenvalue) of ${\mathbf{A}}_{0}$, which can be directly computed from data without iterations.

Replacing $\hat{\lambda }$ in the expression of $\tilde{L}$ in (12.13) by the largest eigenvalue of ${\mathbf{A}}_{0}$ results in an expression that does not yield a reasonable value for the MPV of $S$ or $S_{e}$, suggesting that the leading order w.r.t. these parameters has been lost in the approximation. To obtain the correct asymptotic form, the first order of ${\mathbf{A}}$ should be retained. Substituting $(1 + S_{e} /SD_{k} )^{ - 1} \sim 1 - S_{e} /SD_{k}$ into (12.11) gives

$${\mathbf{A}}\sim \sum\limits_{k} {{\hat{\mathbf{D}}}_{k} } - \frac{{S_{e} }}{S}\sum\limits_{k} {D_{k}^{ - 1} {\hat{\mathbf{D}}}_{k} }$$

(12.16)

For the summand in (12.13),

$$\begin{aligned} \ln (SD_{k} + S_{e} ) & = \ln \left[ {SD_{k} (1 + \frac{{S_{e} }}{{SD_{k} }})} \right] \\ & = \ln SD_{k} + \ln (1 + \frac{{S_{e} }}{{SD_{k} }})\sim \ln S + \ln D_{k} + \frac{{S_{e} }}{{SD_{k} }} \\ \end{aligned}$$

(12.17)

Substituting (12.16) and (12.17) into (12.13) gives

$$\begin{aligned} & \tilde{L}(f,\zeta ,S,S_{e} )\sim nN_{f} \ln \pi + \sum\limits_{k} {\ln D_{k} } + \frac{{S_{e} }}{S}\sum\limits_{k} {D_{k}^{ - 1} } \\ & \quad + \left[ {(n - 1)N_{f} \ln S_{e} + S_{e}^{ - 1} (d - \hat{\lambda }_{0} )} \right] + \left[ {N_{f} \ln S + S^{ - 1} \sum\limits_{k} {\frac{{\hat{d}_{k} }}{{D_{k} }}} } \right] \\ \end{aligned}$$

(12.18)

$$\hat{\lambda }_{0} = \hat{{\varvec{\upvarphi }}}^{T} {\mathbf{A}}_{0} \hat{{\varvec{\upvarphi }}}\qquad \hat{d}_{k} = \hat{{\varvec{\upvarphi }}}^{T} {\hat{\mathbf{D}}}_{k} \hat{{\varvec{\upvarphi }}}$$

(12.19)

The variation of $S_{e} S^{ - 1} \sum\nolimits_{k} {D_{k}^{ - 1} }$ w.r.t. $S_{e}$ or $S$ is asymptotically small compared to the bracketed terms and so $\tilde{L}$ depends on $S_{e}$ dominantly through the first bracket and on $S$ through the second bracket. Minimizing the first and second bracket gives the MPV of $S_{e}$ and $S$, respectively:

$$\hat{S}_{e} \sim \frac{{d - \hat{\lambda }_{0} }}{{(n - 1)N_{f} }}\quad \hat{S}\sim \frac{1}{{N_{f} }}\sum\limits_{k} {\frac{{\hat{d}_{k} }}{{D_{k} }}}$$

(12.20)

3.1 Initial Guess of MPV

The initial guess of MPV for numerical optimization can be set as follow. The initial guess for f can be picked from the singular value (SV) spectrum of data (Sect. 7.3). The initial guess for $\zeta$ can be set as 1%. The initial guess for $S$ and $S_{e}$ can be calculated using (12.20). The initial guess for ${\varvec{\upvarphi }}$ can be taken as the eigenvector (largest eigenvalue) of ${\mathbf{A}}_{0}$ in (12.15).

4 Posterior Covariance Matrix

Under a Gaussian approximation of the posterior PDF, the posterior covariance matrix of the set of modal parameters ${\varvec{\uptheta }}$ $\left( {n_{\theta } \times 1} \right)$ satisfying constraint (on mode shape norm) is given by (Sect. 11.2)

$$\mathop {{\hat{\mathbf{C}}}}\limits_{{n_{\theta } \times n_{\theta } }} = \mathop {(\nabla {\hat{\mathbf{v}}}_{c} )}\limits_{{n_{\theta } \times p}} \underbrace {{(\nabla^{2} \hat{L}_{c} )^{ + } }}_{p \times p}{\mathop {(\nabla {\hat{\mathbf{v}}}_{c} )}\limits_{{p \times n_{\theta } } }}^{T}$$

(12.21)

where ‘+’ denotes the pseudo-inverse, i.e., ignoring the zero eigenvalues from constraint singularity; $\nabla^{2} \hat{L}_{c}$ is the Hessian of $L_{c} ({\mathbf{u}}) = L({\mathbf{v}}_{c} ({\mathbf{u}}))$ w.r.t. a set of free parameters ${\mathbf{u}}$ $\left( {p \times 1} \right)$ at the MPV; ${\mathbf{v}}_{c} ({\mathbf{u}})$ maps ${\mathbf{u}}$ to ${{\varvec{\uptheta }}}$ that always satisfies the constraint $G({\mathbf{v}}_{c} ({\mathbf{u}})) = 0$; $G({\varvec{\uptheta}})$ is the norm constraint equation on mode shape; $\nabla {\hat{\mathbf{v}}}_{c}$ is the gradient of ${\mathbf{v}}_{c} ({\mathbf{u}})$ at the MPV. A hat ‘^’ on quantity denotes that it is evaluated at the MPV. Using the formula based on Lagrange multiplier (Sect. 11.2.3),

$$\mathop {\nabla^{2} \hat{L}_{c} }\limits_{p \times p} = \mathop {(\nabla {\hat{\mathbf{v}}}_{c} )^{T} }\limits_{{p \times n_{\theta } }} (\mathop {\nabla^{2} \hat{L}}\limits_{{n_{\theta } \times n_{\theta } }} + \hat{\lambda }_{1} \mathop {\nabla^{2} \hat{G}}\limits_{{n_{\theta } \times n_{\theta } }} )\mathop {(\nabla {\hat{\mathbf{v}}}_{c} )}\limits_{{n_{\theta } \times p}}$$

(12.22)

where $\nabla^{2} \hat{L}$ is the Hessian of the NLLF in (12.10) (ignoring mode shape norm constraint) and

$$\hat{\lambda }_{1} = - \frac{{(\nabla \hat{G})(\nabla \hat{L})^{T} }}{{(\nabla \hat{G})(\nabla \hat{G})^{T} }}$$

(12.23)

is the value of the Lagrange multiplier at the MPV.

The terms in (12.22) can be determined as follow. Let $\varpi = [f,\zeta ,S,S_{e} ]^{T}$ comprise the modal parameters other than mode shape. One can take

$${\mathbf{u}} = \left[ {\begin{array}{*{20}c} \varpi \\ {\varvec{\upvarphi }} \\ \end{array} } \right]\quad {\varvec{\uptheta }} = \left[ {\begin{array}{*{20}c} \varpi \\ {\varvec{\upvarphi }} \\ \end{array} } \right]\quad {\mathbf{v}}_{c} ({\mathbf{u}}) = \left[ {\begin{array}{*{20}c} \varpi \\ {{\varvec{\upvarphi }}/||{\varvec{\upvarphi }}||} \\ \end{array} } \right]\quad G({\varvec{\uptheta }}) = 1 - {\varvec{\upvarphi }}^{T} {\varvec{\upvarphi }}$$

(12.24)

so that $G({\mathbf{v}}_{c} ({\mathbf{u}})) \equiv 0$ for any ${\mathbf{u}}$. Direct differentiation and evaluating at the MPV gives

$$\nabla \hat{G} = [\begin{array}{*{20}c} {{\mathbf{0}}_{1 \times 4} } & { - 2\hat{{\varvec{\upvarphi }}}^{T} } \\ \end{array} ]\qquad \nabla^{2} \hat{G} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{4 \times 4} } & {} \\ {} & { - 2{\mathbf{I}}_{n} } \\ \end{array} } \right]$$

(12.25)

$$\nabla \hat{L} = [\begin{array}{*{20}c} {\nabla_{\varpi } \hat{L}} & { - 2S_{e}^{ - 1} \hat{{\varvec{\upvarphi }}}^{T} {\hat{\mathbf{A}}}} \\ \end{array} ]\qquad \nabla {\hat{\mathbf{v}}}_{c} = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{4 \times 4} } & {} \\ {} & {{\mathbf{I}}_{n} - \hat{{\varvec{\upvarphi }}}\hat{{\varvec{\upvarphi }}}^{T} } \\ \end{array} } \right]$$

(12.26)

where $\nabla_{\varpi }$ denotes the gradient w.r.t. $\varpi$; ${\hat{\mathbf{A}}}$ denotes the value of ${\mathbf{A}}$ at the MPV. Substituting into (12.23) gives

$$\hat{\lambda }_{1} = - S_{e}^{ - 1} \hat{\lambda }$$

(12.27)

where $\hat{\lambda }$ is the largest eigenvalue of ${\hat{\mathbf{A}}}.$ Due to mode shape norm constraint, $\nabla \hat{L}_{c}$ has one zero eigenvalue with eigenvector $[{\mathbf{0}}_{4 \times 1} ;\hat{{\varvec{\upvarphi }}}].$

It remains to determine $\nabla^{2} \hat{L},$ which can be obtained via finite difference approximation or analytical expression. The latter is discussed in the next two sections. Section 12.4.1 expresses the derivatives of $L$ in terms of those of ${\mathbf{E}}_{k}$. It has the advantage of unifying derivation and computer coding effort with other formulations. Section 12.4.2 presents alternative expressions based on a condensed form of the NLLF. It can speed up computation especially when $n$ is large, but requires additional coding effort specific to the present context.

4.1 General Expressions

Let a superscripted variable in parenthesis denote a derivative w.r.t. the variable. For any parameters x and y,

$$\begin{aligned} L^{(x)} & = \sum\limits_{k} {(\ln } |{\mathbf{E}}_{k} |)^{(x)} + \sum\limits_{k} {{\hat{\mathcal{F}}}_{k}^{*} ({\mathbf{E}}_{k}^{ - 1} )^{(x)} {\hat{\mathcal{F}}}_{k} } \\ L^{(xy)} & = \sum\limits_{k} {(\ln } |{\mathbf{E}}_{k} |)^{(xy)} + \sum\limits_{k} {{\hat{\mathcal{F}}}_{k}^{*} ({\mathbf{E}}_{k}^{ - 1} )^{(xy)} {\hat{\mathcal{F}}}_{k} } \\ \end{aligned}$$

(12.28)

The derivatives of $\ln |{\mathbf{E}}_{k} |$ and ${\mathbf{E}}_{k}^{ - 1}$ can be expressed in terms of the derivatives of ${\mathbf{E}}_{k}$. For any $x$ and $y,$

$$\begin{aligned} (\ln |{\mathbf{E}}_{k} |)^{(x)} & = tr[{\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(x)} ] \, \quad ({\mathbf{E}}_{k}^{ - 1} )^{(x)} = - {\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(x)} {\mathbf{E}}_{k}^{ - 1} \\ (\ln |{\mathbf{E}}_{k} |)^{(xy)} & = tr[{\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(xy)} - {\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(x)} {\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(y)} ] \\ ({\mathbf{E}}_{k}^{ - 1} )^{(xy)} & = {\mathbf{E}}_{k}^{ - 1} [{\mathbf{E}}_{k}^{(x)} {\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(y)} + {\mathbf{E}}_{k}^{(y)} {\mathbf{E}}_{k}^{ - 1} {\mathbf{E}}_{k}^{(x)} - {\mathbf{E}}_{k}^{(xy)} ]{\mathbf{E}}_{k}^{ - 1} \\ \end{aligned}$$

(12.29)

The derivatives of ${\mathbf{E}}_{k}$ are given in Table 12.1. The expressions involve the derivatives of $D_{k}$, which are given in Table 12.2. For better conditioning, ${\mathbf{E}}_{k}^{ - 1}$ should be calculated using the formula below (recalled from (12.8)), where the ill-conditioning due to small $S_{e}$ has been segregated out in the factor $S_{e}^{ - 1}$ :

Table 12.1 Derivatives of ${\mathbf{E}}_{k} = SD_{k} {\varvec{\upvarphi \upvarphi }}^{T} + S_{e} {\mathbf{I}}_{n}$

Table 12.2 Derivatives of $D_{k}^{ - 1} = [(1 - \beta_{k}^{2} )^{2} + (2\zeta \beta_{k} )^{2} ](2\pi {\text{f}}_{k} )^{{2{\text{q}}}}$ and $D_{k}$

$${\mathbf{E}}_{k}^{ - 1} = S_{e}^{ - 1} \left[ {{\mathbf{I}}_{n} - \left( {1 + \frac{{S_{e} }}{{SD_{k} }}} \right)^{ - 1} {\varvec{\upvarphi \upvarphi }}^{T} } \right]$$

(12.30)

4.2 Condensed Expressions

Computing the derivatives of the NLLF via those of ${\mathbf{E}}_{k}$ involves matrix computations of size $n$ (the number of DOFs) which can be large in applications. It is possible to obtain the derivatives in a condensed form using (12.10) to speed up computations. Let $\varpi = [f,\zeta ,S,S_{e} ]^{T}$ and $\nabla^{2} L$ be partitioned as

$$\nabla^{2} L = \left[ {\begin{array}{*{20}c} {\mathop {L^{(\varpi \varpi )} }\limits_{4 \times 4} } & {\mathop {L^{{(\varpi {\varvec{\upvarphi }})}} }\limits_{4 \times n} } \\ {{\text{sym}} .} & {\mathop {L^{{({\varvec{\upvarphi}\varvec{\upvarphi} })}} }\limits_{n \times n} } \\ \end{array} } \right]$$

(12.31)

The partition $L^{{(\varpi {\varvec{\upvarphi }})}}$ involves $L^{{(x{\varvec{\upvarphi }})}}$ for $x = f,\zeta ,S,S_{e}$. These are given in Table 12.3. The partition $L^{(\varpi \varpi )}$ involves $L^{(xy)}$ for $x,y = f,\zeta ,S,S_{e}$. To facilitate presentation, the NLLF in (12.10) is separated into two parts:

$$L = L_{D} + L_{Q}$$

(12.32)

$$L_{D} = nN_{f} \ln \pi + \sum\limits_{k} {\ln (SD_{k} + S_{e} )} + (n - 1)N_{f} \ln S_{e}$$

(12.33)

$$L_{Q} = S_{e}^{ - 1} (d - {\varvec{\upvarphi }}^{T} {\mathbf{A}}{\varvec{\upvarphi }}) = S_{e}^{ - 1} (d - \sum\limits_{k} {B_{k} q_{k} } )$$

(12.34)

$$B_{k} = (1 + \frac{{S_{e} }}{{SD_{k} }})^{ - 1} \quad q_{k} = {\varvec{\upvarphi }}^{T} {\hat{\mathbf{D}}}_{k} {\varvec{\upvarphi }}$$

(12.35)

Table 12.3 Derivatives $L^{{(x{\varvec{\upvarphi }})}}$ for $x = f,\zeta ,S,S_{e}$

Note that $B_{k}$ only depends on $\{ f,\zeta ,S,S_{e} \}$ and $q_{k}$ only on ${\varvec{\upvarphi }}.$ The second derivatives of $L_{D}$ and $L_{Q}$ are given in Tables 12.4 and 12.5, respectively. They involve the derivatives of $D_{k}$ and $B_{k}$, which are given in Tables 12.2 and 12.6, respectively. Figure 12.1 shows the information flow. The quantities $D_{k}$, $B_{k}$, $\gamma_{k} = SD_{k} /S_{e}$ and $q_{k} = {\varvec{\upvarphi }}^{T} {\hat{\mathbf{D}}}_{k} {\varvec{\upvarphi }}$ appear repeatedly and so they may be computed and stored upfront. The expressions in Tables 12.3, 12.4, 12.5 and 12.6 are applicable for general parameter values, not necessarily at the MPV.

Table 12.4 Derivatives of $L_{D} = nN_{f} \ln \pi + \sum\nolimits_{k} {\ln (SD_{k} + S_{e} )} + (n - 1)N_{f} \ln S_{e}$

Table 12.5 Derivatives of $L_{Q} = S_{e}^{ - 1} (d - {\varvec{\upvarphi}}^{T} {\mathbf{A}}{\varvec{\upvarphi}})$

Table 12.6 Derivatives of $B_{k}^{ - 1} = (1 + S_{e} /SD_{k} )$ and $B_{k}$

Fig. 12.1

Information flow of derivative calculations

5 Synthetic Data Examples

In this section, we present examples on Bayesian OMA with well-separated modes based on synthetic data. Example 12.1 provides a basic illustration of Bayesian modal identification results and their interpretation. Examples 12.2 and 12.3 illustrate the effect of modeling error and the choice of bandwidth, respectively.

Example 12.1

(Ten-storied building) Consider the horizontal vibration of a ten-storied shear building with uniform mass 1000 tons per floor, interstory stiffness 1767 kN/mm and damping ratio 1% in all modes. The first three modes have natural frequencies 1, 2.98 and 4.89 Hz. The structure is subjected to i.i.d. white noise excitation at all floors, each with PSD $S_{w} = 96.2{\text{N}}^{2} /{\text{Hz}}.$ This structure was considered in Example 10.1. The acceleration on 5/F and the roof are measured for 600 s at 100 Hz. They are contaminated by i.i.d. data noise of PSD $S_{e} = 1(\upmu{\text{g}})^{2} /{\text{Hz}}.$ Figure 12.2 shows the root PSD and SV spectrum of data. The frequency band used for modal identification is indicated by the horizontal bar below each peak.

Fig. 12.2

a Root PSD and b root SV spectrum using data on 5/F and the roof, ten-storied building (synthetic data). Horizontal bar indicates selected frequency band

Posterior Statistics

Table 12.7 summarizes the modal identification results. The exact value of modal force PSD of the first mode was calculated in Example 10.1; the values for other modes can be calculated similarly. The MPVs are close to the exact values in a manner consistent with the posterior c.o.v.s (coefficient of variation = standard deviation/MPV). The exact values are generally within two standard deviations from the MPVs. The MPV of the prediction error PSD appears to be biased high as the mode number increases. This is because the frequency band of the higher modes contains unmodeled pseudo-static contribution from the lower modes.

Table 12.7 Modal identification results, ten-storied building (synthetic data)

The most probable mode shapes (omitted here) are very close to the exact mode shapes. In Table 12.7, the mode shape c.o.v. is equal to the square root sum of eigenvalues of the mode shape covariance matrix (Sect. 11.3.3). It is of the same order of magnitude as the hyper-angle between the exact and most probable mode shape.

Table 12.8 shows the posterior correlation coefficients among $f$, $\zeta$, $S$ and $S_{e}$. Correlation is generally small, with the only exception between $\zeta$ and $S$, of about 66%. This is no coincidence and can be explained based on uncertainty laws (Chap. 15).

Table 12.8 Posterior correlation coefficient (%), ten-storied building (synthetic data)

Posterior marginal PDF

Figure 12.3 shows the posterior marginal PDFs of modal parameters. The exact marginal PDF (solid line) was obtained by numerically integrating the likelihood function w.r.t. the remaining parameters and normalizing so that its area is 1. The approximate PDF (dashed line) is a Gaussian PDF centered at the MPV and with a standard deviation obtained from the posterior covariance matrix. Obtaining the exact marginal PDF is computationally expensive. It is only done here for comparison. The figure shows that the Gaussian approximation is generally good.

Fig. 12.3

Posterior marginal PDF of modal parameters. Solid line exact by numerical integration; dashed line Gaussian approximation

Effect of Data Duration

Modal identification results depend on data, and clearly its duration. Figure 12.4 shows the results for data durations of 600, 1200, 2400 and 4800 s. The MPV (dot) need not coincide with the exact value but the latter is generally within the ± two standard deviation error bar. As data duration increases, the MPV converges (in a random manner) to the exact value and the error bar shortens. In real applications the exact value (dashed line) is unknown (or does not even exist) and so identification precision is assessed primarily based on the error bars.

Fig. 12.4

Modal identification results for different data durations. Dot MPV; error bar ±two standard deviations. Dashed line exact value

Effect of Signal-to-Noise Ratio

Identification uncertainty depends on data noise intensity. Figure 12.5 shows the posterior c.o.v. of modal parameters of the first mode when the data is contaminated with measurement noise of PSD $S_{e}$, ranging from 100 $(\upmu{\text{g}})^{2} /{\text{Hz}}$ to 0.01 $(\upmu{\text{g}})^{2} /{\text{Hz}}.$ The results are plotted w.r.t. the ‘modal s/n ratio’ (Sect. 7.7.3)

$$\gamma = \frac{S}{{4S_{e} \zeta^{2} }}$$

(12.36)

calculated using the MPVs determined in each case. For reference, in the nominal case, $S_{e} =$ 1 $(\upmu{\text{g}})^{2} /{\text{Hz}}$ and $\gamma \approx 700.$ The figure shows that the posterior c.o.v. decreases significantly with $\gamma$ when $\gamma$ is small (say, <100). When $\gamma$ is large the posterior c.o.v. settles at a non-zero value. Intuitively, even when there is no measurement noise, there is still uncertainty in the modal parameters because the input excitation is unknown.■

Fig. 12.5

Posterior c.o.v. of modal parameters (first mode) for different $S_{e}$

Example 12.2

(Modeling error from unmodeled modes) Consider synthetic data generated at 100 Hz for 1200 s according to

$${\hat{\mathbf{y}}}_{j} = {\varvec{\upvarphi }}_{1} \ddot{\eta }_{1} (t_{j} ) + {\varvec{\upvarphi }}_{2} \ddot{\eta }_{2} (t_{j} ) + {\varvec{\upvarepsilon}}(t_{j} )$$

(12.37)

where $\ddot{\eta }_{i} (t)$ ($i = 1,2)$ is the modal acceleration satisfying

$$\ddot{\eta }_{i} (t) + 2\zeta_{i} \omega_{i} \dot{\eta }_{i} (t) + \omega_{i}^{2} \eta_{i} (t) = p_{i} (t)$$

(12.38)

with $\omega_{i} = 2\pi f_{i}$ (rad/s); $f_{1} =$ 1 Hz, $f_{2} =$ 1.5 Hz, $\zeta_{1} = \zeta_{2} =$ 1%;

$${\varvec{\upvarphi }}_{1} = \left[ {\begin{array}{*{20}c} 1 & 2 & 2 \\ \end{array} } \right]^{T} /3\quad {\varvec{\upvarphi }}_{2} = \left[ {\begin{array}{*{20}c} { - 2} & 1 & 2 \\ \end{array} } \right]^{T} /3$$

(12.39)

and $p_{i} (t)$ is the modal excitation, stationary Gaussian with PSD $S_{i}$; $S_{1} = 1(\upmu{\text{g}})^{2} /{\text{Hz}}$; $S_{2}$ will be specified later. The prediction error ${\varvec{\upvarepsilon}}(t)$ is i.i.d. Gaussian white noise with PSD $S_{e} = 1(\upmu{\text{g}})^{2} /{\text{Hz}}$. The mode at 1 Hz is identified using FFT in the band [0.95, 1.00] Hz and assuming single mode. Here we investigate the effect of modeling error due to the unmodeled mode at 1.5 Hz.

When $S_{2} = 0$ the second mode is absent and there is no modeling error. The PSD and SV spectrum of data in this case are shown in Fig. 12.6. The lowest two lines in (b) reflect data noise PSD. When $S_{2} > 0$, the contribution of the second mode in the selected band induces modeling error. Figure 12.7 shows the root PSD and SV spectrum when $S_{2} = 300(\upmu{\text{g}})^{2} /{\text{Hz}}$. As seen in (b), the presence of the second mode increases the second largest singular value. This appears like an increase in the noise PSD in the selected band.

Fig. 12.6

a Root PSD and b root SV spectrum when $S_{2} = 0$

Fig. 12.7

a Root PSD and b root SV spectrum when $S_{2} = 300(\upmu{\text{g}})^{2} /{\text{Hz}}$

Figure 12.8 shows the modal identification results for different values of $S_{2}$ up to $300({\upmu g})^{2} /{\text{Hz}}$. The results are plotted w.r.t. the ‘modeling error ratio’, defined as the PSD ratio of the second to the first mode at 1 Hz. When the ratio is $10^{ - 5}$ the results essentially correspond to $S_{2} = 0$. Even in this case the MPV need not be equal to the exact value because the data duration is finite. Departure of identification results from this nominal (‘model error free’) case reflects modeling error. For $\{ f_{1} ,\zeta_{1} ,S_{1} \}$, there is a slight departure in MPV from the nominal value but it is still small compared to identification uncertainty (error bar). For $S_{e}$, when the modeling error ratio is small, (say $< 10^{ - 3}$), the identification result reflects the PSD of data noise $( {1(\upmu{\text{g}})^{2} /{\text{Hz}}} )$. For larger ratios it reflects the PSD contribution of the second mode. In all cases the identified mode shapes (not shown) are close to the exact ones, in the worst case with a c.o.v. of about 10%.■

Fig. 12.8

Modal identification results for different values of $S_{2}$, plotted versus modeling error ratio, defined as the PSD ratio of the second to the first mode at 1 Hz. Dot MPV; error bar ±two standard deviations

Example 12.3

(Bandwidth trade-off) Consider Example 12.2 again with the largest modeling error ratio, i.e., $S_{2} = 300(\upmu{\text{g}})^{2} /{\text{Hz}}$. It was demonstrated that the effect of the unmodeled second mode on the identification results of $\{ f_{1} ,\zeta_{1} ,S_{1} \}$ was insignificant. This example further illustrates that this in fact depends on the selected bandwidth, which in general should play a balance between identification precision (the wider the better) and modeling error (the narrower the better).

Figure 12.9 shows the modal identification results for different selected bands, parameterized as $(1 \pm 0.01\kappa )$ where $\kappa$ is a ‘bandwidth factor’. For reference, the selected band in the last example was [0.95, 1.05], i.e., $\kappa = 5$. As seen in the figure, as $\kappa$ increases from small values, identification uncertainty (error bar) decreases, reflecting the positive effect of increasing FFT data size. Identification uncertainty is insensitive to $\kappa$ when $5 < \kappa < 10$, where it is trading off with modeling error. Bias is not significant until $\kappa > 15$, as reflected by the error bar not covering the exact value (dashed line). Thus, when the selected band is narrow, modeling error is small but identification uncertainty could be unacceptably large. When the band is too wide, results can be significantly biased due to modeling error. The band should trade off these two factors. Here a value between 5 and 10 is appropriate.■

Fig. 12.9

Modal identification results with different selected bands $(1 \pm 0.01\kappa)$

6 Laboratory/Field Data Examples

In this section we present examples on Bayesian OMA with well-separated modes based on experimental data measured under laboratory or field environment. They illustrate the typical results one may obtain in reality and how to interpret them, taking into account potential modeling errors.

Example 12.4

(Laboratory shear frame) Consider the three-storied laboratory shear frame in Fig. 12.10. The four corners of each floor are instrumented with two piezoelectric accelerometers along the horizontal x- and y-directions. This gives a total of $3 \times 4 \times 2 = 24$ measured DOFs. Ambient data was recorded for 600 s at 2048 Hz. It was later decimated to 512 Hz for analysis.

Fig. 12.10

Laboratory shear frame; a photo; b measured DOFs

Data and spectra

Figure 12.11 shows the measured time histories at DOFs 1 and 2. Figure 12.12 shows the root PSD and SV spectrum. The spectra indicate a noise PSD in the order of $100 \, \upmu{\text{g}}/\sqrt {\text{Hz}}$ to 20 $\upmu{\text{g}}/\sqrt {\text{Hz}}$ from low to high frequencies. The peaks of the top line in (b) indicate potential modes. Ten well-separated modes can be easily recognized.

Fig. 12.11

Measured time histories at DOFs 1 and 2, laboratory frame (detrended)

Fig. 12.12

One-sided Root PSD a and root SV b spectrum of data at DOFs 1 and 2 of laboratory frame. Horizontal bar indicates selected frequency band

Modal Identification Using DOFs 1 and 2

The measured data at DOFs 1 and 2 are already sufficient for reliable modal identification. The horizontal bars below the peaks in Fig. 12.12b indicate the bands selected for modal identification. They are hand-picked by zooming in the individual bands. The identification results are summarized in Table 12.9. The c.o.v. of frequencies are all below 0.1%. The c.o.v. of damping ratio ranges from a few to fifty percent. The c.o.v. of modal force PSD is of a similar order as the c.o.v. of damping ratio. The MPV of prediction error PSD decreases with the mode number, which is consistent with the spectra in Fig. 12.12. The modal s/n ratio $\gamma = S/4S_{e} \zeta^{2}$ is defined as before in (12.36), calculated using the MPVs in each case. It is generally quite high, ranging from a few hundreds to a few thousands.

Table 12.9 Modal identification results using DOFs 1 and 2, laboratory frame

Modal Identification Using All Measured DOFs

Identifying modes using more DOFs allows one to examine their nature through the mode shapes. The results using all the 24 measured DOFs are summarized in Table 12.10. The increase in the number of measured DOFs directly increases the modal force PSD $S$ and modal s/n ratio $\gamma = S/4S_{e} \zeta^{2}$. Despite the substantial increase in $\gamma$, there is little difference in the MPV and c.o.v. of the natural frequency and damping ratio.

Table 12.10 Modal identification results using all 24 measured DOFs, laboratory frame

Figure 12.13 shows the identified mode shapes (MPV) with their nature indicated, e.g., TX2 for the second translational mode along the x direction; R1 for the first torsional mode. Based on lumped mass structural dynamics, the frame theoretically has nine modes, comprising three translational modes in the x and the y direction, and three rotational modes. Figure 12.13 contains an additional mode S1 whose frequency and mode shape lie between R2 and R3.■

Fig. 12.13

Mode shapes (MPV) identified using all 24 measured DOFs. Title above each plot shows the mode number, nature, frequency and damping ratio (MPV)

Example 12.5

(Tall building) Consider a tall building measuring 50 m × 50 m in plan and over 300 m in height. Accelerations along two horizontal (x and y) directions at four corners on the roof were measured for 1800 s under normal wind situation in the late afternoon. Figure 12.14 shows the measured time histories at one corner. The root PSD and SV spectrum are shown in Fig. 12.15. The SV spectrum suggests six potential modes in the bands shown. The nature of each mode is determined from the mode shape found later.

Fig. 12.14

Biaxial (x and y) measured acceleration time histories (detrended) at one corner on the roof of a tall building

Fig. 12.15

One-sided root PSD (a) and root SV spectrum (b) of biaxial (x and y) data measured at one corner on the roof of a tall building. Horizontal bar indicates selected band

Modal Identification

The modal identification results using the biaxial data at one corner on the roof are shown in Table 12.11. The MPV of $\sqrt {S_{e} }$ is a few $\upmu{\text{g}}/\sqrt {\text{Hz}}$ and tends to decrease with the mode number. This is consistent with the spectra in Fig. 12.15. The modal s/n ratio ranges from a few hundreds to a thousand. The modal force has a root PSD of about 1 $\upmu{\text{g}}/\sqrt {\text{Hz}}$. The c.o.v. of damping ratio tends to decrease with the mode number. This is partly because the effective data length as a multiple of natural period is longer for higher modes with higher frequencies.

Table 12.11 Modal identification results using biaxial data at one corner on the roof of tall building

Table 12.12 shows the modal identification results using data at all four corners (8 DOFs). Comparing with Table 12.11, comments similar to Example 12.4 can be made. Using all measured DOFs directly increases the modal force PSD and modal s/n ratio but it makes little difference in the MPV and posterior c.o.v.s. The root modal force PSD roughly doubles because the number of DOFs with similar mode shape values is quadrupled. Figure 12.16 shows the identified mode shapes (MPV).■

Table 12.12 Modal identification results using data at four corners on the roof of tall building

Fig. 12.16

Identified mode shapes (MPV) using data at four corners on the roof of tall building. Title above each plot shows the mode number, nature, frequency and damping ratio (MPV)

Example 12.6

(Smart phone data, footbridge) This example demonstrates ambient modal identification using triaxial acceleration data recorded on a smart phone. As shown in Fig. 12.17, the phone was placed on one side of the bridge deck at 1/4 span of the Queen’s Park Bridge in Chester, UK. Figure 12.18 shows the measured time history collected for 300 s at 50 Hz. Figure 12.19 shows the root PSD and SV spectrum of data. The spectra suggest a noise PSD of a few hundreds of $\upmu{\text{g}}/\sqrt {\text{Hz}}$, consistent with Fig. 6.2 in Sect. 6.3. The peaks suggest potential modes between 0.5 and 5 Hz. The band around 0.8 Hz is likely to contain a single mode. The situation is less clear with other bands, which can possibly contain close modes. This can usually be resolved using sensors from multiple locations.

Fig. 12.17

a Queen’s Park Bridge and b location of sensor (smart phone)

Fig. 12.18

Triaxial data (detrended) at 1/4 span of Queen’s Park Bridge. The x, y and z direction correspond to the transverse, longitudinal and vertical direction of the bridge

Fig. 12.19

One-sided root PSD (a) and SV (b) spectrum of triaxial data, Queen’s Park Bridge. Horizontal bar indicates selected band

Modal Identification

Table 12.13 shows the modal identification results. The most probable mode shapes are shown in Table 12.14, where their likely nature are also indicated. These results should be interpreted with a level of confidence consistent with the quality of data, which is much lower than the previous two examples. For reference, the quantization resolution is about 0.001 g (Example 6.1) and its contribution to the root PSD of noise is about 60 ${\upmu \text{g}/}\sqrt {\text{Hz}}$ (Sect. 6.6.2). The overall noise PSD is a few hundreds of $\upmu{\text{g}}/\sqrt {\text{Hz}}$; see Fig. 12.19 or Table 12.13 $\left( {\sqrt {S_{e} } } \right).$ The identification results assume a single mode dominating the selected band, which is debatable in the current case. The Bayesian algorithm always returns results as along as a converged solution in the MPV can be obtained. The posterior c.o.v. need not account for modeling error.■

Table 12.13 Modal identification results, triaxial smart phone data, Queen’s Park Bridge

Table 12.14 Most probable mode shapes, triaxial smart phone data, Queen’s Park Bridge