Skip to main content
SpringerLink
Log in

The Direct Coupling of Modal and Impedance Based Components

  • Research paper
  • Published:
Experimental Techniques Aims and scope Submit manuscript

Abstract

Substructuring is a term used to describe the estimation of the dynamics of a coupled system assembly when only the dynamics of each uncoupled component is available. Existing approaches allow for the coupling of physical-to-physical models, physical-to-modal models, modal-to-modal models referred to as Component Mode Synthesis (CMS), and impedance-to-impedance models referred to as Frequency Based Substructuring (FBS). Often times, the component information may not be just modal data for both components or just FRF data for both components so that modal substructuring or FRF substructuring can be performed. In these cases, the component data needs to be converted from either modal data or FRF data to match the data of the other component. A method for directly coupling impedance- and modal-based components has not yet been addressed. A proposed Impedance to Modal Substructuring (IMS) approach addresses this situation by writing the equations in a form that allows the user to directly utilize modal data for one component and FRF data for the other component, offering more flexibility in coupling different component data sets. While intended to be used with experimental data, this approach may also implement analytical components. In this work, an approach was developed to allow for the direct coupling of impedance and modal models without the need for the user to convert component data type. The IMS approach derived in this work was validated using analytical and experimental data with various models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Data Availability

Data will be made available on request.

Abbreviations

K:

Stiffness matrix

k:

Singular or partitioned stiffness terms

M:

Mass matrix

m:

Singular or partitioned mass terms

C:

Damping matrix

I:

Identity matrix

U:

Mode shape matrix

u:

Individual mode shape/vector

F:

Applied force/forcing function

x:

Physical coordinate

p:

Modal coordinate

Λ:

Eigenvalue

H:

Matrix of FRFs

h:

Single FRF

ω:

Spectral frequency

λ:

Natural frequency (radians/second)

A:

Modal residue matrix

p:

System pole

σ:

Modal damping factor

j:

Denotes imaginary number

LR:

Lower residual term

UR:

Upper residual term

J:

Jacobian matrix for FRF calculation

[]:

Matrix

{}:

Column vector

\(\lfloor \ \rfloor\)  :

Row Vector

\(\bar{[\ ]}\)  :

Modal matrix

\(\dot{}\) :

First derivative

\(\ddot{}\) :

Second derivative

\(\odot\) :

Elementwise matrix multiplication

T:

Matrix transpose

*:

Complex conjugate

H:

Hermitian (complex conjugate transpose)

A:

Component ‘A’

B:

Component ‘B’

AB:

System model comprised of components ‘A’ and ‘B’

TIE:

Couple/tie matrix

N:

Arbitrary component number

Nm:

Number of modes

Ni:

Number of system outputs

Nf:

Number of spectral/frequency lines

Nc:

Number of connection DOF

c:

Coupling coordinate/DOF

i:

Output coordinate/DOF

j:

Input coordinate/DOF

q:

Non-coupling coordinate (beam A)

w:

Non-coupling coordinate (beam B)

References

  1. Avitabile P (1999) Why you can’t ignore those fixture resonances. J Sound Vib 33(3):20–26

    Google Scholar 

  2. Soine DE, Jones RJ, Harvie JM, Skousen TJ, Schoenherr TF (2018) Designing hardware for the boundary condition round robin challenge. In: Proceedings of the 36th International Modal Analysis Conference, Orlando

  3. Zwink B, Daniels B, Avitabile P (2020) Experimental application of boundary condition compensation map (from Field to Laboratory Response). In: Proceedings of the 38th International Modal Analysis Conference

  4. Schultz R, Schoenherr T, Owens B (2021) A proposed standard random vibration environment for BARC and the boundary condition challenge. In: Proceedings of the 39th International Modal Analysis Conference

  5. Schoenherr T (2020) Designing an optimized fixture for the BARC hardware using a parameterized model. In: Proceedings of the 38th International Modal Analysis Conference

  6. Hurty WC (1965) Dynamic analysis of structural systems using component modes. AIAA 3(4):678–685

    Article  Google Scholar 

  7. Craig RR Jr, Bampton MCC (1968) Coupling of substructures for dynamic analyses. AIAA 6(7):1313–1319

    Article  Google Scholar 

  8. MacNeal RH (1971) A hybrid method of component mode synthesis. Comput Struct 1(4): 581-601

  9. Rubin S (1975) Improved component-mode representation for structural dynamic analysis. AIAA 13(8):995–1006

    Article  Google Scholar 

  10. Craig RR Jr (1977) On the use of attachment modes in substructure coupling for dynamic analysis. In: 18th Structural Dynamics and Materials Conference, pp 89–99

  11. Martinez D, Carne T, Gregory D, Miller A (1984) Combined experimental/analytical modeling using component mode synthesis. In: 25th Structures, Structural Dynamics and Materials Conference, pp 140–152

  12. Baker M (1986) Component mode synthesis methods for test-based, rigidly connected flexible components. 25th Structures, Structural Dynamics and Materials Conference 23(3):316–322

  13. Duarte ML, Ewins DJ (1996) Improved experimental component mode synthesis (IECMS) with residual compensation based purely on experimental results. In: Proceedings of the 14th International Modal Analysis Conference, pp 641–647

  14. Morgan JA, Pierre C, Hulbert GM (1998) Calculation of component mode synthesis matrices from measured frequency response functions, part 1: theory. J Vibrations Acoust 120(2):503–508

    Article  Google Scholar 

  15. Weissenburger JT (1968) Effect of local modifications on the Eigenvalues and Eigenvectors of linear systems. J Appl Mech 35(2):327–332

  16. Pomazal RJ, Snyder VW (1969) The effect of local modifications on the eigenvalues and eigenvectors of damped linear systems. In: Proceedings of the 9th Aerospace Sciences Meeting, pp 215

  17. Avitabile P (2003) Twenty years of structural dynamic modification - A review. J Sound Vib 37(1):14–27

    Google Scholar 

  18. Klosterman AL (1971) On the experimental determination and use of modal representations of dynamic characteristics. University of Cincinnati

  19. Crowley JR, Klosterman AL, Rocklin GT, Vold H (1984) Direct structural modification using frequency response functions. In: Proceedings of the 2nd International Modal Analysis Conference

  20. Imregun M (1987) Structural modification and coupling dynamic analysis using measured FRF data. In: Proceedings of the 5th International Modal Analysis Conference, pp 1136–1141

  21. Otte D (1990) Coupling of structures using measured FRFs by means of SVD-based data reduction techniques. Proceedings of the 8th International Modal Analysis Conference

  22. Otte D, Leuridan J, Grangier H, Aquilina R (1991) Prediction of the dynamics of structural assemblies using measured FRF-data: some improved data enhancement techniques. In: Proceedings of the 9th International Modal Analysis Conference, pp 909–918

  23. Wyckaert K, Xu KQ, Mas P (1997) The virtues of static and dynamic compensations for FRF based substructuring. In: Proceedings of the 15th International Modal Analysis Conference, pp 1463–1468

  24. Cakar O, Sanliturk KY (2002) Elimination of noise and transducer effects from measured response data. In: Proceedings of the 6th Biennial Conference on Engineering Systems Design and Analysis

  25. Sanliturk K, Cakar O (2005) Noise elimination from measured frequency response functions. Mech Syst Signal Process 19(3):615–631

    Article  Google Scholar 

  26. Nicgorski D, Avitabile P (2010) Experimental issues related to frequency response function measurements for frequency-based substructuring. Proceedings of The 26th International Modal Analysis Conference

  27. Nicgorski D, Avitabile P (2010) Conditioning of FRF measurements for use with frequency based substructuring. Proceedings of the 27th International Modal Analysis Conference

  28. Butland A, Avitabil P (2010) A reduced orde, test verified component mode synthesis approach for system modeling applications. Mech Syst Signal Process 24(4):904–921

    Article  Google Scholar 

  29. Thibault L, Butland A, Avitabile P (2012) Variability improvement of Key Inaccurate Node Groups–VIKING. Topics in Modal Analysis II, vol 6. Springer, pp 603–624.

    Google Scholar 

  30. Avitabile P (2018) Proper projection vectors for the expansion of measured experimental modal data. J Eng Mech 144(5):04018024

  31. Allemang RJ (1982) A correlation coefficient for modal vector analysis. In: Proceedings of the 1st International Modal Analysis Conference, pp 110–116

  32. Heylen W, Lammens S (1996) FRAC: a consistent way of comparing frequency response functions. In: Proceedings of the Conference on Identification in Engineering Systems, pp 48–57

  33. Sandia National Laboratories, Round Robin Frame Structure [Online]. Available: https://wiki.sem.org/wiki/Round_Robin_Frame_Structure. Accessed 27 Feb 2022

  34. Seymour JA, Avitabile P (2023) Improved denoising methods for smoothing frequency response functions. Mech Syst Signal Process 204:110722. https://doi.org/10.1016/j.ymssp.2023.110722

Download references

Funding

No funding was received to assist with the preparation of this manuscript.

Author information

Authors and Affiliations

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

    J. A. Seymour & P. Avitabile

Authors
  1. J. A. Seymour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Avitabile
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. A. Seymour.

Ethics declarations

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

J. Seymour and P. Avitabile are members of SEM.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Seymour, J.A., Avitabile, P. The Direct Coupling of Modal and Impedance Based Components. Exp Tech (2024). https://doi.org/10.1007/s40799-023-00696-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40799-023-00696-4

Keywords

Search

Navigation