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


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.


Dynamic load reconstruction Modal filters Input localization Multiple force identification 


General Terms


Degree of Freedom


Frequency Response Function


Modal Assurance Criterion


Single Degree of Freedom


Singular Value Decomposition



Indicates the kth mode


Indicates the rth DOF




Complex conjugate




Pseudo-inverse of transpose


Matrix truncated to a singular values or singular vectors


Conjugate Transpose



Correlation between reference and estimate


Modal viscous damping




Natural frequency


Total number of DOFs


Number of input DOFs




Number of modes


Number of observed DOFs


Number of spectral lines



(i × 1) Physical Input

\( \overline{F} \)

(m × 1) Modal Force


(N × 1) Primary Locator


(o × 1) Physical Response


(m × 1) Modal Response



(m × q) Estimate error


(m × q) Singular values from SVD


(m × m) Singular vectors from SVD


(q × q) Singular vectors from SVD


(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


(o × i) Residual terms from truncated modes


(N × m) Mode shape values for N DOF


(i × m) Mode shape values for input DOF


(o × m) Mode shape values for response DOF



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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