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Operational transfer path analysis based on deep neural network: Numerical validation

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Abstract

In this study, a novel transfer path analysis formulation using an emerging deep neural network model is presented and numerically validated for a multi-structural system. In the proposed formulation, only the operational responses of structures are utilized to identify the contributions of all paths to vibration responses at a point of interest in the frequency domain. To this end, the model parameters of a dense neural network model are initially determined using a training dataset to predict the responses precisely. Next, the contribution of each path is identified from the responses predicted by using the trained network model with the input associated with the selected path eliminated. To establish the correct model, the original operational datasets are augmented using the phase shift and the cross-spectrum. The path contributions estimated using the proposed formulation with another numerically generated operational dataset are compared with the known path contributions. The comparison results show that the proposed method based on a deep neural network model can successfully predict the path contributions using only operational responses.

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Abbreviations

B :

Constant vector added to nodes in the DNN

C nm :

Contribution of m-th input on the n-th ROI

c m :

Contribution of m-th input on the ROI

H :

Number of hidden layers + 1

J 0 :

Number of measured training datasets

J 1 :

Number of phases generated for phase augmentation

M :

Number of input

N :

Number of output

P :

Number of nodes in the DNN

R :

ROI in a system

R̃ :

Predicted ROI by the DNN

R̄ j :

Predicted ROI by putting the j-th input to zero

R̿ j :

Predicted ROI by putting all zeros except j-th input

R̆ :

Measured ROI

S :

Covariance matrix

T :

Transfer function matrix for a subsystem

T nm :

A transfer function for the m-th input and n-th ROI

U :

Input vector of the DNN

(U, Y):

Element of training dataset

W :

Weight matrix multiplied by the variables of nodes in the DNN

X :

Input response vector in a system

X̄ j :

Input vector with the j-th input to zero

X̿ j :

Input vector with all zeros except for the j-th input

X̆ :

Measured input response

Y :

Output vector of the DNN

Ỹ :

Predicted output vector with current model parameters

Z :

Hidden variable vector in the DNN

ℵ:

Training dataset

ℵ0 :

Initial datasets from measurements

ℵ1 :

Phase augmented training datasets

ℵ2 :

Cross-spectrum augmented training datasets by input

ℵ3 :

Cross-spectrum augmented training datasets by output

ϕ :

Phase of a response

θ̂ :

Trained model parameters

θ :

Model parameters for the DNN

θ̂ :

Trained model parameters

‖·‖:

Norm of a vector

x* :

Complex conjugate of a complex variable x

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Acknowledgments

This work was supported by a grant from the National Research Foundation of Korea (NRF) funded by the Korean government (MEST; Grant No. NRF-2018R1A2B2005391). This work was also partially supported by the Korea Basic Science Institute (KBSI) National Research Facilities & Equipment Center (NFEC) grant funded by the Korea government (Ministry of Education) (No. 2019R1A6C1010045).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Dongeui University, Busan, 47340, Korea

    Dooho Lee

  2. Department of Mechanical Engineering, Ajou University, Suwon-si, 16499, Korea

    Jin Woo Lee

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  1. Dooho Lee
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Correspondence to Dooho Lee.

Additional information

Recommended by Editor No-cheol Park

Dooho Lee received his B.S. degree from Seoul National University (Korea) in 1988, his M.S. from KAIST (Korea) in 1990, and his Ph.D. from KAIST (Korea) in 1994. He worked for Samsung Motors, Inc. (1995-1999) and is currently a Professor at Dongeui University. His research focuses on the design optimization of structural-acoustic systems, uncertainty propagation in dynamic problems, and sound transfer characteristics in human hearing systems.

Jin Woo Lee has been a Professor of Mechanical Engineering at Ajou University since 2009. His research interests are in the area of vibration, acoustics, acoustic and vibration metamaterials, topology optimization based designs and fluid-structure interactions of micro-cantilevers for RF-MEMS and AFM. His Ph.D. is from the School of Mechanical and Aerospace Engineering from Seoul National University in South Korea in 2003. He worked with Samsung Electronics Company from 2003 to 2006 and studied as a post-doctoral research associate at Seoul National University from 2006 to 2007. From 2007 to 2009, he was a postdoctoral research associate of Mechanical Engineering at Purdue University, West Lafayette, IN, USA.

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Lee, D., Lee, J.W. Operational transfer path analysis based on deep neural network: Numerical validation. J Mech Sci Technol 34, 1023–1033 (2020). https://doi.org/10.1007/s12206-020-0205-5

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  • DOI: https://doi.org/10.1007/s12206-020-0205-5

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