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Experimental Techniques

, Volume 44, Issue 1, pp 113–125 | Cite as

Reconstruction of External Forces Beyond Measured Points Using a Modal Filtering Decomposition Approach

  • P. LoganEmail author
  • P. Avitabile
  • J. Dodson
Research paper
  • 53 Downloads

Abstract

A methodology for the reconstruction of dynamic loads from vibration response data is developed which employs a well-correlated finite element model to permit load identification at locations where no response measurements are available; this is extended from a force reconstruction approach based upon modal filtering. Force reconstruction is approached from a modal perspective, where modal responses are estimated from physical response data, followed by independent reconstruction of modal forces for each modal oscillator. Modal force estimates are analyzed in aggregate via singular value decomposition to permit identification of input locations which are compatible with the estimated modal force patterns. Modal forces are then returned to the physical domain via transformation by modal matrices. Experimental validation is performed with a free-free beam subject to multiple impact.

Keywords

Dynamic load reconstruction Modal filters Input localization Multiple force identification 

Nomenclature

General Terms

DOF

Degree of Freedom

FRF

Frequency Response Function

MAC

Modal Assurance Criterion

SDOF

Single Degree of Freedom

SVD

Singular Value Decomposition

Subscripts

k

Indicates the kth mode

r

Indicates the rth DOF

Superscripts

Pseudo-inverse

*

Complex conjugate

T

Transpose

-T

Pseudo-inverse of transpose

a

Matrix truncated to a singular values or singular vectors

h

Conjugate Transpose

Scalars

α

Correlation between reference and estimate

ζ

Modal viscous damping

ω

Frequency

ωn

Natural frequency

N

Total number of DOFs

i

Number of input DOFs

j

√-1

m

Number of modes

o

Number of observed DOFs

q

Number of spectral lines

Vectors

F

(i × 1) Physical Input

\( \overline{F} \)

(m × 1) Modal Force

PL

(N × 1) Primary Locator

X

(o × 1) Physical Response

p

(m × 1) Modal Response

Matrices

[ε]

(m × q) Estimate error

[Σ]

(m × q) Singular values from SVD

[ϒ]

(m × m) Singular vectors from SVD

[V]

(q × q) Singular vectors from SVD

[H]

(o × i) Physical FRFs

[\( \overline{H} \)]

(m × m) Modal FRFs

[\( \overline{P} \)]

(m × q) Reference modal responses over multiple spectral lines

[\( \hat{P} \)]

(m × q) Estimated modal responses over multiple spectral lines

[R]

(o × i) Residual terms from truncated modes

[U]

(N × m) Mode shape values for N DOF

[UINPUT]

(i × m) Mode shape values for input DOF

[URESPONSE]

(o × m) Mode shape values for response DOF

Notes

Acknowledgements

Some of the work presented herein was partially funded by Air Force Research Laboratory Award FA8651-16-2-0006 “Nonstationary System State Identification Using Complex Polynomial Representations” as well as by NSF Civil, Mechanical and Manufacturing Innovation (CMMI) Grant No. 1266019 entitled “Collaborative Research: Enabling Non-contact Structural Dynamic Identification with Focused Ultrasound Radiation Force”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency. The authors are grateful for the support obtained.

Distribution A. Approved for public release; distribution unlimited. (96TW-2018-0386).

Compliance with Ethical Standards

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© The Society for Experimental Mechanics, Inc 2019

Authors and Affiliations

  1. 1.Structural Dynamics and Acoustic Systems LaboratoryUniversity of Massachusetts LowellLowellUSA
  2. 2.Air Force Research LaboratoryEglin AFBUSA

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