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Reconstruction of Nonlinear Contact Forces Beyond Limited Measurement Locations Using an SVD Modal Filtering Approach

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Abstract

Assessment of operating loads is often approached as an inverse process using vibration response data and some form of system model to reconstruct external forces. Any mismatch in dynamic behavior between the model and the real system may distort the estimated loads. However, if the changes in system behavior are of interest, then those changes may be characterized through force reconstruction as equivalent loads acting in conjunction with the original external inputs. While these system changes may be linear in nature, the concept may be extended to nonlinear behavior, including intermittent contact between components. A modal based methodology for localization and reconstruction of inputs at potentially unmeasured locations is applied to the characterization of local system changes through their equivalent loading. An analytical study of a simple M-C-K system subject to contact nonlinearity is presented for validation. An experimental study is then presented using a multi-plate structure subject to impact loading and an unmodeled contact condition.

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Abbreviations

DOF:

Degree of Freedom

FRAC:

Frequency Response Assurance Criterion

FRF:

Frequency Response Function

MAC:

Modal Assurance Criterion

SDOF:

Single Degree of Freedom

SVD:

Singular Value Decomposition

TRAC:

Time Response Assurance Criterion

k:

Indicates the kth mode

r:

Indicates the rth DOF

† :

Pseudo-inverse

* :

Complex conjugate

T :

Transpose

a :

Matrix truncated to a singular values or singular vectors

h :

Conjugate Transpose

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

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

[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

\( \left[\overline{H}\right] \) :

(m × m) Modal FRFs

[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

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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-2019-0265).

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Authors and Affiliations

  1. Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, 1 University Avenue, Lowell, MA, 01854, USA

    P. Logan, D. Fowler & P. Avitabile

  2. Air Force Research Laboratory, 306 W. Eglin Blvd, Eglin AFB, FL, 32542, USA

    J. Dodson

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  1. P. Logan
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  2. D. Fowler
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Correspondence to P. Logan.

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Logan, P., Fowler, D., Avitabile, P. et al. Reconstruction of Nonlinear Contact Forces Beyond Limited Measurement Locations Using an SVD Modal Filtering Approach. Exp Tech 44, 485–495 (2020). https://doi.org/10.1007/s40799-020-00371-y

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