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A structural response reconstruction method based on a continuous-discrete state space model

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Abstract

Structural response reconstruction is an important technique for structural health monitoring. However, aiming at the problem of discretization error in the discrete state space model used in structural response reconstruction, a novel structural response reconstruction method based on a continuous-discrete state space model is proposed, which can avoid the discretization error and improve the reconstruction performance. Firstly, the structure is modelled with a continuous-discrete state space model and the process noise covariance matrix is transformed into the process noise gain matrix. Secondly, square root cubature Kalman filter is applied for response reconstruction and the Gaussian nested implicit Runge-Kutta method is used for state recursion. Then the ill-conditioned matrix inverse problem is solved in the posterior estimation with the Tikhonov regularization method. Finally, the proposed method is validated through numerical simulation of a two-dimension truss and response test of an overhanging beam. The results show that the proposed method can achieve high reconstruction accuracy even when the noise level is 30 %, demonstrating its effectiveness and robustness in practical engineering applications.

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Abbreviations

q :

Modal coordinate

ξ :

Damping matrix

ω :

Modal frequency matrix

Φ :

Displacement mode shape matrix

L :

Dapping matrix of excitation load

f(•):

Continuous state transition function

h(•):

System observation equation

u :

External incentives

t :

Time

x :

System state vector

P :

Covariance matrix of the system state vector

w(t ) :

Standard brownian motion process

Q :

Process noise gain matrix

y :

Observation vector

v k :

Gaussian white noise

R k :

Covariance matrix of the measured value

Δt :

Time interval

x 0 :

Initial value of the system state vector

P 0 :

Initial covariance matrix of the system state vector

y u :

Observed value of external excitation

X :

Process noise covariance

e i :

Unit vector

n :

Dimension of the system state vector

\({\boldsymbol{\hat x}}\) :

Cubature sample vector

S :

Square root of the covariance matrix

X :

Combine of the cubature sample vector

Ψ :

Transient matrix

J :

Jacobian matrix

le :

Iterative residual

ge :

Evaluation of the cumulative residual

ε l :

Upper bound of iterative residual

ε g :

Upper bound of evaluation of the cumulative residual

T :

Step size

K :

Kalman gain

y r :

Reconstruction response

y c :

Real response

y noise :

Measurement noise

m :

Noise level

N :

Normal distribution vector

δ :

Sampling points

α RMSE :

Root mean squared error

E :

The mathematical expectation

Tria :

Triangulation of matrix

std :

Standard deviation

RMSE :

Root mean squared error

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (No: 62161018) and Outstanding Graduate Student “Innovation Star” Project of Gansu Provincial Department of Education (2023CXZX-567).

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Authors and Affiliations

  1. School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Fanghua Chen & Zhenrui Peng

Authors
  1. Fanghua Chen
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Corresponding author

Correspondence to Zhenrui Peng.

Additional information

Fanghua Chen was born in Guangdong Province in China. He received the bachelor’s degree in vehicle engineering from Lanzhou Jiaotong University, Lanzhou, China, in 2020, where he is also currently pursuing the master’s degree in vehicle engineering. His main research interests include modal analysis, load identification, and structural response reconstruction.

Zhenrui Peng received the bachelor’s degree in mechanical engineering from Lanzhou Jiaotong University, Lanzhou, China, in 1995, and the Ph.D. degree in control science and engineering from Zhejiang University, Hangzhou, China, in 2007. He presided over three projects of the National Natural Science Foundation of China and published over 90 papers up to now, and over 30 papers have been indexed by SCI, EI, and ISTP. His major research fields include fault diagnosis of mechanical equipment, finite element model updating, and structural response reconstruction.

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Chen, F., Peng, Z. A structural response reconstruction method based on a continuous-discrete state space model. J Mech Sci Technol 37, 5713–5723 (2023). https://doi.org/10.1007/s12206-023-1011-7

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  • DOI: https://doi.org/10.1007/s12206-023-1011-7

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