Erratum for “Framework for Probabilistic Assessment of Maximum Nonlinear Structural Response Based on Sensor Measurements: Discretization and Estimation” by Ajay Saini and Iris Tien

Saini, Ajay; Tien, Iris

doi:10.1061/(ASCE)EM.1943-7889.0001632

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May 25, 2019

Erratum for “Framework for Probabilistic Assessment of Maximum Nonlinear Structural Response Based on Sensor Measurements: Discretization and Estimation” by Ajay Saini and Iris Tien

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Authors: Ajay Saini, S.M.ASCE https://orcid.org/0000-0002-7882-278X [email protected], and Iris Tien, Ph.D., A.M.ASCE [email protected]Author Affiliations

Publication: Journal of Engineering Mechanics

Volume 145, Issue 8

https://doi.org/10.1061/(ASCE)EM.1943-7889.0001632

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The following corrections should be made to the original paper:

With bold variables indicating vectors and matrices and italic variables indicating scalars, the equations in the discretization section of the methodology should be as follows:

The structural equation of motion is

M {\ddot{u}}_{s} + C {\dot{u}}_{s} + F (u_{s}) = - B_{2} M 1 a_{g} + f

(1)

In first-order form, Eq. (1) becomes

{\dot{z}}_{s} = [\begin{matrix} 0 & I \\ 0 & - M^{- 1} C \end{matrix}] z_{s} + [\begin{matrix} 0 \\ - M^{- 1} F (z_{s}) \end{matrix}] + [\begin{matrix} 0 \\ - 1 \end{matrix}] B_{2} a_{g} + [\begin{matrix} 0 \\ M^{- 1} \end{matrix}] f

(2)

The state space representation of the full dynamical system is

\dot{z} = [\begin{matrix} 0 & 1 & 0 & 0 \\ - ω_{g}^{2} & - 2 ξ_{g} ω_{g} & 0 & 0 \\ 0 & 0 & 0 & I \\ 1 B_{2} ω_{g}^{2} & 12 B_{2} ξ_{g} ω_{g} & 0 & - M^{- 1} C \end{matrix}] z + [\begin{matrix} 0 \\ 0 \\ 0 \\ - M^{- 1} F (z_{s}) \end{matrix}] + [\begin{matrix} 0 \\ - 1 \\ 0 \\ 0 \end{matrix}] w + [\begin{matrix} 0 \\ 0 \\ 0 \\ M^{- 1} \end{matrix}] f

(8)

In the exact discretization solution, Eq. (10) is

\int_{z_{k}}^{z_{k + 1}} \frac{dz}{A_{c} (z) z + b_{c} w + B_{c} f} = \int_{t}^{t + Δ t} d t

(10)

with the given system expressed as

\dot{z} = A_{c} (z) + b_{c} w + B_{c} f

. The transformation to the differential form uses

y = \exp (V (z)) 1

. The governing equation for

z_{4}

is

{\dot{z}}_{4} = f_{4} (z, t) = 1 B_{2} ω_{g}^{2} z_{1} + 12 B_{2} ξ_{g} ω_{g} z_{2} - M^{- 1} F (z_{3}) - M^{- 1} C z_{4} + M^{- 1} f

(15)

{[(V (z) 1)]}_{k}^{k + 1} = I \int d t

(17)

with the elements of

V

being

{[V_{22}]}_{k}^{k + 1} = \frac{\ln (- ω_{g}^{2} z_{1_{k}} - 2 ξ_{g} ω_{g} z_{2_{k + 1}} - w_{k}) - \ln (- ω_{g}^{2} z_{1_{k}} - 2 ξ_{g} ω_{g} z_{2_{k}} - w_{k})}{- 2 ξ_{g} ω_{g}}

(19)

{[V_{33}]}_{k}^{k + 1} \times z_{4_{k}} = z_{3_{k + 1}} - z_{3_{k}}

(20)

{[V_{44}]}_{k}^{k + 1} = i n v (- M^{- 1} C) (\ln (1 B_{2} ω_{g}^{2} z_{1 k} + 12 B_{2} ξ_{g} ω_{g} z_{2 k} - M^{- 1} F (z_{3_{k}}) - M^{- 1} C z_{4_{k + 1}} - M^{- 1} f_{k}) - \ln (1 B_{2} ω_{g}^{2} z_{1 k} + 12 B_{2} ξ_{g} ω_{g} z_{2 k} - M^{- 1} F (z_{3_{k}}) - M^{- 1} C z_{4_{k}} - M^{- 1} f_{k}))

(21)

The system evolution equations for

z_{3}

and

z_{4}

are

z_{3_{k + 1}} = z_{3_{k}} + z_{4_{k}} Δ t

(24)

z_{4_{k + 1}} = (- {MC}^{- 1}) [{(1 B_{2} ω_{g}^{2} z_{1} + 12 B_{2} ξ_{g} ω_{g} z_{2} - M^{- 1} F (z_{3_{k}}) - M^{- 1} C z_{4_{k}} - M^{- 1} f_{k}) e^{- M^{- 1} C Δ t}} - (- 1 B_{2} ω_{g}^{2} z_{1} - 12 B_{2} ξ_{g} ω_{g} z_{2} + M^{- 1} F (z_{3_{k}}) + M^{- 1} f_{k})]

(25)

In matrix form, this is written as

z_{k + 1} = A_{1} z_{k} + F_{1} + b w_{k} + B f_{k}

(26)

where

A_{1} = [\begin{matrix} 1 & Δ t & 0 & 0 \\ \frac{- ω_{g}^{2} (e^{- 2 ξ_{g} ω_{g} Δ t} - 1)}{- 2 ξ_{g} ω_{g}} & e^{- 2 ξ_{g} ω_{g} Δ t} & 0 & 0 \\ 0 & 0 & I & I Δ t \\ B_{2} ω_{g}^{2} (- {MC}^{- 1}) (e^{- M^{- 1} C Δ t} - I) & 2 B_{2} ξ_{g} ω_{g} (- {MC}^{- 1}) (e^{- M^{- 1} C Δ t} - I) & 0 & e^{- M^{- 1} C Δ t} \end{matrix}]

(27)

F_{1} = [\begin{matrix} 0 \\ 0 \\ 0 \\ C^{- 1} (e^{- M^{- 1} C Δ t} - 1) F (z_{s}) \end{matrix}]

(28)

B = (- M C^{- 1}) (e^{- M^{- 1} C Δ t} - I) B_{c}

(30)

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Journal of Engineering Mechanics

Volume 145 • Issue 8 • August 2019

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Received: Nov 29, 2018

Accepted: Dec 12, 2018

Published online: May 25, 2019

Published in print: Aug 1, 2019

Discussion open until: Oct 25, 2019

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Ajay Saini, S.M.ASCE https://orcid.org/0000-0002-7882-278X [email protected]

Ph.D. Candidate, School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Dr., Atlanta, GA 30332-0355 (corresponding author). ORCID: https://orcid.org/0000-0002-7882-278X. Email: [email protected]

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Iris Tien, Ph.D., A.M.ASCE [email protected]

Assistant Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Dr., Atlanta, GA 30332-0355. Email: [email protected]

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