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Prediction of dynamic responses in a rectangular beam using the modal expansion method

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Abstract

Structures and equipment subject to dynamic loading are prone to a shortened life span due to excessive vibration levels, which can lead to fatigue failure of its components. Continuous monitoring of those systems can be a complicated and expensive task, due to the complexity and difficulty of accessibility to some locations, which makes it difficult to assess the structural integrity. One way to deal with this subject is to use finite element model, operational modal analysis, and the modal expansion method to predict dynamic responses in locations that have not been measured. In this work, a calibrated modal model of a rectangular aluminum beam suspended in air, in free–free boundary condition, was used to predict acceleration responses. The experimental modal matrix was obtained through impact tests and the system equivalent reduction expansion process (SEREP) technique was used to reduce the beam’s Finite Element model, thus ensuring the compatibility between numerical and experimental degrees of freedom. Model calibration was carried out using the local correspondence principle. Finally, by using the modal expansion method, the calibrated modal model was used to predict dynamic responses. Results showed high accuracy between the measured and the predicted acceleration signals.

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Abbreviations

[K]:

Stiffness matrix

[M]:

Mass matrix

[T]:

Reduction transformation matrix

\(\{\phi \}\) :

Experimentally obtained mode shapes (unscaled)

\(\small \{q(t)\}\) :

Modal coordinates

\(\tiny \small \{X(t)\},\{{\dot{X}}(t)\},\{{\ddot{X}}(t)\}\) :

Displacement, velocity and acceleration vectors

\(\{z\}\) :

Numeric mode adjustment coefficient vector

\(\small \{{\varPhi }\}\) :

Numerically obtained mode shapes

a, d:

Active and deleted degrees of freedom

i :

Iteration index

p :

Number of modes considered in modal truncation

[B]:

Numeric vibration modes cluster

[\(K_{\text {r}}\)], [\(M_{\text {r}}\)]:

Reduced stiffness and mass matrices

{F}:

Vector of loads acting on the system

COMAC:

Coordinate modal assurance criteria

EFDD:

Enhanced frequency-domain decomposition

FE:

Finite element

TRAC:

Time response assurance criteria

LC:

Local correspondence principle

MAC:

Modal assurance criteria

MAE:

Mean absolute error

RD:

Relative difference

RMSE:

Root mean squared error

SEREP:

System equivalent reduction expansion process

FRAC:

Frequency-response assurance criteria

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Authors and Affiliations

  1. Basic Engineering, Petrobras Av. Henrique Valadares, 28 - Centro, Rio de Janeiro, RJ, 20231-030, Brazil

    Claudio de O. Mendonça

  2. Laboratory of Dynamic Tests and Vibration Analysis (LEDAV), Ocean Engineering Program (PENO), Federal University of Rio de Janeiro (UFRJ), Av. Athos da Silveira Ramos - 149, Rio de Janeiro, RJ, 21941-909, Brazil

    Ricardo H. Gutiérrez & Ulisses A. Monteiro

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  1. Claudio de O. Mendonça
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  2. Ricardo H. Gutiérrez
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  3. Ulisses A. Monteiro
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Correspondence to Claudio de O. Mendonça.

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Mendonça, C.d.O., Gutiérrez, R.H. & Monteiro, U.A. Prediction of dynamic responses in a rectangular beam using the modal expansion method. Mar Syst Ocean Technol 16, 29–42 (2021). https://doi.org/10.1007/s40868-020-00091-3

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  • DOI: https://doi.org/10.1007/s40868-020-00091-3

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