Model reduction-based Bayesian updating of non-classically damped systems using modal data from multiple setups

Abstract

This paper presents a Bayesian method for updating linear dynamical systems using complex modal data due to the effects of non-classical damping collected from multiple setups. The practical scenario of the availability of a limited number of sensors is avoided by considering measurements from multiple setups. Besides, the complex modal data is assumed to consist of the most probable values (MPVs) of the modal parameters and their posterior uncertainties, unlike the other approaches where only MPVs of the modal parameters are considered. System modal parameters are introduced as additional uncertain parameters to avoid the requirement of mode matching between the model-predicted and the measured modes. Since introducing the system mode shapes as additional parameters increases the problem dimensionality, the dynamic condensation technique is employed to reduce the finite element (FE) model to a smaller model with fewer degrees of freedom (DOFs) corresponding to the measured DOFs. Detailed formulation leading to the development of the posterior probability density function (PDF) is presented based on the current framework. Transitional Markov chain Monte Carlo (TMCMC) and Metropolis-within-Gibbs (MWG) sampling methodologies are used to approximate the posterior PDF. The effectiveness and efficiency of the proposed methodology are demonstrated using two numerical examples.

Appendices

Appendix A: development of the marginal PDF $p({{\varvec{\uptheta}}},{{\varvec{\upzeta}}},{{\varvec{\upomega}}}|D)$

The number of uncertain parameters in the posterior PDF $p({{\varvec{\uptheta}}},{{\varvec{\upzeta}}},{{\varvec{\upomega}}},{{\varvec{\Psi}}}|D)$ in Eq. (35) $n_{{\theta_{s} }} + 4N_{r} + \sum\nolimits_{s = 1}^{{N_{s} }} {\sum\nolimits_{s = 1}^{{N_{s} }} {n_{m}^{s} N_{r} } }$ is large. Hence, in this appendix section, an alternative approach is developed to estimate the marginal PDF $p({{\varvec{\uptheta}}},{{\varvec{\upzeta}}},{{\varvec{\upomega}}}|D)$ by integrating out ${{\varvec{\Psi}}}$ from the posterior PDF $p({{\varvec{\uptheta}}},{{\varvec{\upzeta}}},{{\varvec{\upomega}}},{{\varvec{\Psi}}}|D)$ in Eq. (35). This leads to a problem with the number of uncertain parameters equal to $n_{{\theta_{s} }} + 4N_{r}$ which is much lesser than $n_{{\theta_{s} }} + 4N_{r} + \sum\nolimits_{s = 1}^{{N_{s} }} {\sum\nolimits_{s = 1}^{{N_{s} }} {n_{m}^{s} N_{r} } }$. Applying the theorem of total probability to integrate out ${{\varvec{\Psi}}}$ in Eq. (35) gives:

$$p({{\varvec{\uptheta}}},{{\varvec{\upzeta}}},{{\varvec{\upomega}}}|D) = cp({{\varvec{\uptheta}}})p_{0} ({{\varvec{\upomega}}}|D)p_{0} ({{\varvec{\upzeta}}}|D)\prod\nolimits_{r} {I(\omega_{r} ,\zeta_{r} ,{{\varvec{\uptheta}}},D)}$$

(45)

where the integral $I_{r} \equiv I(\omega_{r} ,\zeta_{r} ,{{\varvec{\uptheta}}},D) = \prod\nolimits_{s} {\int {p_{0} ({{\varvec{\Psi}}}_{r,s} |D)p(\omega_{r} ,\zeta_{r} ,{{\varvec{\Psi}}}_{r,s} |{{\varvec{\uptheta}}})d{{\varvec{\Psi}}}_{r,s} } }$ is solved as follows:

$$\begin{aligned} I_{r} = \prod\limits_{s} \left| {2\pi {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}} } \right|^{{ - \frac{1}{2}}} \left| {2\pi {\mathbf{C}}_{e,r,s} } \right|^{{ - \frac{1}{2}}}\\ \int {\exp \left( { - \frac{1}{2}\left( {{{\varvec{\Psi}}}_{r,s}^{T} {\mathbf{Q}}_{r,s} {{\varvec{\Psi}}}_{r,s} - {\hat{\varvec{\Psi }}}_{r,s}^{T} {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}}^{ - 1} {{\varvec{\Psi}}}_{r,s} - {{\varvec{\Psi}}}_{r,s}^{T} {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}}^{ - 1} {\hat{\varvec{\Psi }}}_{r,s} + {\hat{\varvec{\Psi }}}_{r,s} {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}}^{ - 1} {\hat{\varvec{\Psi }}}_{r,s} } \right)} \right)d{{\varvec{\Psi}}}_{r,s} } \end{aligned}$$

(46)

and where ${\mathbf{Q}}_{r,s} = {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}}^{ - 1} + {\mathbf{X}}_{r,s}^{T} {\mathbf{C}}_{e,r,s}^{ - 1} {\mathbf{X}}_{r,s}$. By completing the squares for vector ${{\varvec{\Psi}}}_{r,s}$ and for symmetric invertible matrix ${\mathbf{Q}}_{r,s}$, we have the expression as:

$$I_{r} = \prod\limits_{s} {\left[ \begin{gathered} \left| {2\pi {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}} } \right|^{{ - \frac{1}{2}}} \left| {2\pi {\mathbf{C}}_{e,r,s} } \right|^{{ - \frac{1}{2}}} \exp \left( {\frac{1}{2}\left( {{\mathbf{a}}_{r,s}^{T} {\mathbf{Q}}_{r,s}^{ - 1} {\mathbf{a}}_{r,s} - {\hat{\boldsymbol{\Psi }}}_{r,s}^{T} {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}}^{ - 1} {\hat{\boldsymbol{\Psi }}}_{r,s} } \right)} \right) \hfill \\ \int {\exp \left( { - \frac{1}{2}({{\varvec{\Psi}}}_{r,s} - {\mathbf{Q}}_{r,s}^{ - 1} {\mathbf{a}}_{r,s} )^{T} {\mathbf{Q}}_{r,s} ({{\varvec{\Psi}}}_{r,s} - {\mathbf{Q}}_{r,s}^{ - 1} {\mathbf{a}}_{r,s} )} \right)d{{\varvec{\Psi}}}_{r,s} } \hfill \\ \end{gathered} \right]}$$

(47)

where ${\mathbf{a}}_{r,s} = {\hat{\mathbf{C}}}_{{{{\varvec{\upphi}}},r,s}}^{ - 1} {\hat{\mathbf{\Psi }}}_{r,s}$. The integral in the above equation is Aitken's integral that is solved to yield:

$$\,I_{r} \, = \prod\limits_{s} {\left| {2\pi {\hat{\mathbf{C}}}_{{{{\varvec{\upphi}}},r,s}} } \right|^{{ - \frac{1}{2}}} \left| {2\pi {\mathbf{C}}_{e,r,s} } \right|^{{ - \frac{1}{2}}} \exp \left( {\frac{1}{2}\left( {{\mathbf{a}}_{r,s}^{T} {\mathbf{Q}}_{r,s}^{ - 1} {\mathbf{a}}_{r,s} - {\hat{\mathbf{\Psi }}}_{r,s}^{T} {\hat{\mathbf{C}}}_{{{{\varvec{\upphi}}},r,s}}^{ - 1} {\hat{\mathbf{\Psi }}}_{r,s}^{T} } \right)} \right)\left| {2\pi {\mathbf{Q}}_{r,s}^{ - 1} } \right|^{\frac{1}{2}} } .$$

(48)

Further, using the identity $(A + UBV)^{ - 1} = A^{ - 1} - A^{ - 1} U(B^{ - 1} + VA^{ - 1} U)^{ - 1} VA^{ - 1}$ to solve for ${\mathbf{Q}}_{r}^{ - 1}$ gives:

$$I_{r} = \prod\limits_{s} {\left| {2\pi {{\varvec{\Xi}}}_{r,s} } \right|^{{ - \frac{1}{2}}} \exp \left( { - \frac{1}{2}{\hat{\mathbf{\Psi }}}_{r,s}^{T} {\mathbf{X}}_{r,s}^{T} {{\varvec{\Xi}}}_{r,s}^{ - 1} {\mathbf{X}}_{r,s} {\hat{\mathbf{\Psi }}}_{r,s} } \right)}$$

(49)

where ${{\varvec{\Xi}}}_{r,s} = {\mathbf{C}}_{e,r,s} + {\mathbf{X}}_{r,s} {\hat{\mathbf{C}}}_{{{{\varvec{\Phi}}},r,s}} {\mathbf{X}}_{r,s}^{T}$. By substituting I_r back into Eq. (49), it finally gives:

$$p({{\varvec{\uptheta}}},{{\varvec{\upomega}}},{{\varvec{\upzeta}}}|D) = cp({{\varvec{\uptheta}}})p_{0} ({{\varvec{\upomega}}}|D)p_{0} ({{\varvec{\upzeta}}}|D)\left( {\prod\limits_{r} {\prod\limits_{s} {\left| {2\pi {{\varvec{\Xi}}}_{r,s} } \right|^{{ - \frac{1}{2}}} } } } \right)\exp \left( { - \frac{1}{2}\sum\limits_{r} {\sum\limits_{s} {{\hat{\mathbf{\Psi }}}_{r,s}^{T} {\mathbf{X}}_{r,s}^{T} {{\varvec{\Xi}}}_{r,s}^{ - 1} {\mathbf{X}}_{r,s} {\hat{\mathbf{\Psi }}}_{r,s} } } } \right).$$

(50)

Appendix B: story stiffness and mass and TMD properties of Example 2

See Table 18.

Table 18 Story stiffness, mass and TMD properties of the 80 DOF system