Skip to main content
SpringerLink
Log in

Experimental Modal Parameter Evaluation Methods

  • Reference work entry
  • First Online:
Handbook of Experimental Structural Dynamics

  • 1335 Accesses

Abstract

Modern experimental modal analysis (EMA) methods provide a number of modal parameter solutions based upon different models, different model orders and different numerical processing of the redundant data and/or results. Evaluation of the modal parameter solutions provides a way of obtaining a single unique set of modal parameters that best represents the measured experimental data. The early portion of this chapter is a review of some of the experimental modal analysis (EMA) methods covered in detail in Chap. 10, “Experimental Modal Analysis Methods” in this handbook. This is followed by presenting a number of numerical tools that are used in connection with the EMA methods to evaluate and validate the number of modal parameters that can be estimated from a multiple input, multiple output (MIMO) set of measured data. Some tools like complex and multivariate mode indication functions (CMIF and MvMIF) can be used to determine the model order and/or number of modal frequencies that can be estimated from the experimental data. These tools can be applied independent of the EMA method that is used and are particularly useful when close or repeated modal frequencies are present in the experimental data. Additionally, various consistency diagrams, pole surface plots and modal parameter clustering methods are defined that become part of, and enhance, the EMA method used to estimate the modal parameters. Finally, the last portion of this chapter overviews methods that are primarily post processing tools to evaluate and validate the modal parameters that have been estimated. Methods include techniques for normalizing, conditioning and presenting the modal vectors, like the modal vector complexity plot (MVCP) along with techniques for using the estimated modal vectors to estimate other functions like the enhanced frequency response function (eFRF) which can be used to validate the physical validity of the estimated modal vectors. Orthogonality of modal vectors along with consistency of modal vectors, as measured by the modal assurance criterion (MAC), also falls into this category of evaluation and validation tools that are applied after the modal parameters have been estimated. The chapter finishes with a brief example of how several of the evaluation and validation tools can be combined into an autonomous modal parameter estimation method.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 799.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 899.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Leuridan J, Brown D, Allemang R (1985) Time domain parameter identification methods for linear modal analysis: a unifying approach. ASME Paper Number 85-DET-90

    Google Scholar 

  2. Allemang RJ, Brown, D (1987) Volume III: modal parameter estimation. USAF Report: AFWAL-TR-87-3069. Experimental modal analysis and dynamic component synthesis, no 3, pp 130

    Google Scholar 

  3. Allemang R, Brown, D, Fladung W (1994) Modal parameter estimation: a unified matrix polynomial approach. In: Proceedings, International Modal Analysis Conference (IMAC), pp 501–514

    Google Scholar 

  4. Allemang R, Brown D (1996) A unified matrix polynomial approach to modal identification, Allied Publishers Limited pp 379–390

    Google Scholar 

  5. Allemang R, Brown D (1998) A unified matrix polynomial approach to modal identification. J Sound Vib 211(3):301–322

    Article  Google Scholar 

  6. Allemang, R, Phillips A (2004) The unified matrix polynomial approach to understanding modal parameter estimation: an update. In: Proceedings, International Seminar on Modal Analysis (ISMA)

    Google Scholar 

  7. Spitznogle F (1971) Improvements in the complex exponential signal analysis computational algorithm. Report Number U1-829401-5, Representation and Analysis of Sonar Signals, vol 1, pp 37

    Google Scholar 

  8. Brown D, Allemang R, Zimmerman, R, Mergeay M (1979) Parameter estimation techniques for modal analysis. SAE Paper Number 790221, SAE Transactions, vol 88, pp 828–846

    Google Scholar 

  9. Vold, H, Rocklin T (1982) The numerical implementation of a multi-input modal estimation algorithm for mini-computers. In: Proceedings, International Modal Analysis Conference (IMAC), pp 542–548

    Google Scholar 

  10. Vold H, Kundrat J, Rocklin, T, Russell R (1982) A multi-input modal estimation algorithm for mini-computers. SAE Trans 91(1):815–821

    Google Scholar 

  11. Ibrahim, S, Mikulcik E (1977) A method for the direct identification of vibration parameters from the free response. Shock Vib Bull 47:183–198

    Google Scholar 

  12. Pappa R (1982) Some statistical performance characteristics of the ITD modal identification algorithm. AIAA Paper Number 82-0768, p 19

    Google Scholar 

  13. Fukuzono K (1986) Investigation of multiple-reference ibrahim time domain modal parameter estimation technique. M. S. Thesis, p 220

    Google Scholar 

  14. Juang, J, Pappa R (1985) An eigensystem realization algorithm for modal parameter identification and model reduction. AIAA J Guid Control Dyn 8(4):620–627

    Article  Google Scholar 

  15. Juang J (1987) Mathematical correlation of modal parameter identification methods via system realization theory. J Anal Exp Modal Anal 2(1):1–18

    Google Scholar 

  16. Longman RW, Juang J-N (1989) Recursive form of the eigensystem realization algorithm for system identification. AIAA J Guid Control Dyn 12(5):647–652

    Article  MathSciNet  Google Scholar 

  17. Zhang L, Kanda H, Brown, D, Allemang R (1985) A polyreference frequency domain method for modal parameter identification. ASME Paper No. 85-DET-106, vol 8, pp 8+

    Google Scholar 

  18. Lembregts F, Leuridan J, Zhang, L, Kanda H (1986) Multiple input modal analysis of frequency response functions based on direct parameter identification. In: Proceedings, International Modal Analysis Conference (IMAC)

    Google Scholar 

  19. Lembregts F (1988) Frequency domain identification techniques for experimental multiple input modal analysis. Doctoral Dissertation, Katholieke University of Leuven, Belgium, vol 213

    Google Scholar 

  20. Lembregts F, Leuridan, J, Van Brussel H (1989) Frequency domain direct parameter identification for modal analysis: state space formulation. Mech Syst Signal Process 4(1):65–76

    Article  Google Scholar 

  21. Coppolino R (1981) A simultaneous frequency domain technique for estimation of modal parameters from measured data. SAE Paper No. 811046, vol 12

    Google Scholar 

  22. Craig R, Kurdila A, Kim HM (1990) State-space formulation of multi-shaker modal analysis. J Anal Exp Modal Anal 5(3):169–183

    Google Scholar 

  23. Leuridan J (1984) Some direct parameter model identification methods applicable for multiple input modal analysis. Doctoral Dissertation, vol 384, pp 384+

    Google Scholar 

  24. Natke H (1988) Updating computational models in the frequency domain based on measured data: a survey. Probab Eng Mech 3(1):28–35

    Article  Google Scholar 

  25. Richardson, M, Formenti D (1982) Parameter estimation from frequency response measurements using rational fraction polynomials. In: Proceedings, International Modal Analysis Conference (IMAC), pp 167–182

    Google Scholar 

  26. Shih C, Tsuei Y, Allemang, R, Brown D (1988) A frequency domain global parameter estimation method for multiple reference frequency response measurements. Mech Syst Signal Process (MSSP) 2(4):349–365

    Article  Google Scholar 

  27. Shih C (1989) Investigation of numerical conditioning in the frequency domain modal parameter estimation methods, Doctoral Dissertation, vol 127, pp 127+

    Google Scholar 

  28. Van der Auweraer, H, Leuridan J (1987) Multiple input orthogonal polynomial parameter estimation. Mech Syst Signal Process (MSSP) 1(3):259–272

    Article  Google Scholar 

  29. Van der Auweraer H, Snoeys R, Leuridan JM (1986) A Global frequency domain modal parameter estimation technique for mini-computers. ASME J Vib Acoust Stress Reliab Des 10, vol 11

    Google Scholar 

  30. Vold H (1986) Orthogonal polynomials in the polyreference method. In: Proceedings, International Seminar on Modal Analysis (ISMA-11)

    Google Scholar 

  31. Vold, H, Shih CY (1988) Numerical sensitivity of the characteristic polynomial. In: Proceedings, International Seminar on Modal Analysis (ISMA-13)

    Google Scholar 

  32. Vold H (1990) Numerically robust frequency domain modal parameter estimation. Sound Vib Mag (SVM) 24(1):38–40

    Google Scholar 

  33. Vold H (1990) Statistics of the characteristic polynomial in modal analysis. In: Proceedings, International Seminar on Modal Analysis (ISMA), pp 53–57

    Google Scholar 

  34. Vold H, Napolitano K, Hensley, D, Richardson M (2008) Aliasing in modal parameter estimation, an historical look and new innovations. In: Proceedings, International Modal Analysis Conference (IMAC), vol 16, pp 12–17

    Google Scholar 

  35. Fladung, W, Vold H (2016) An improved implementation of the orthogonal polynomial modal parameter estimation algorithm using the orthogonal complement. In: Proceedings, International Modal Analysis Conference (IMAC)

    Google Scholar 

  36. Fladung, W, Vold H (2016) An Orthogonal view of the polyreference least-squares complex frequency modal parameter estimation algorithm. In: Proceedings, International Modal Analysis Conference (IMAC)

    Google Scholar 

  37. Van der Auweraer H, Guillaume P, Verboven, P, Vanlanduit S (2001) Application of a fast-stabilizing frequency domain parameter estimation method. ASME J Dyn Syst Meas Control 123(4):651–658

    Article  Google Scholar 

  38. Guillaume P, Verboven P, Vanlanduit S, der Auweraer V H and Peeters B (2003) A polyreference implementation of the least-squares complex frequency domain estimator. In: Proceedings, International Modal Analysis Conference (IMAC), p 12

    Google Scholar 

  39. Verboven P, Guillaume P, Cauberghe B, Parloo, E, Vanlanduit S (2003) Stabilization charts and uncertainty bounds for frequency domain linear least squares estimators. In: Proceedings, International Modal Analysis Conference (IMAC), pp 1–10

    Google Scholar 

  40. Verboven P, Cauberghe B, Vanlanduit S, Parloo, E, Guillaume P (2004) The secret behind clear stabilization diagrams: the influence of the parameter constraint on the stability of the poles. In: Proceedings, Society of Experimental Mechanics (SEM) Annual Conference, vol 17, pp 1–17

    Google Scholar 

  41. Verboven P (2002) Frequency domain system identification for modal analysis. Doctoral Dissertation

    MATH  Google Scholar 

  42. Cauberghe B (2004) Application of frequency domain system identification for experimental and operational modal analysis. Doctoral Dissertation, p 259

    Google Scholar 

  43. Shih C, Tsuei Y, Allemang, R, Brown D (1988) Complex mode indication function and its application to spatial domain parameter estimation. Mech Syst Signal Process (MSSP) 2(4):367–377

    Article  Google Scholar 

  44. Williams R, Crowley, J, Vold H (1985) The multivariable mode indicator function in modal analysis. In: Proceedings, International Modal Analysis Conference (IMAC), pp 66–70

    Google Scholar 

  45. Phillips AW, Allemang RJ, Pickrel CR (1997) Clustering of modal frequency estimates from different solution sets. In: Proceedings, International Modal Analysis Conference (IMAC), pp 1053–1063

    Google Scholar 

  46. Phillips, A, Allemang R (2014) Normalization of experimental modal vectors to remove modal vector contamination. In: Proceedings, International Modal Analysis Conference (IMAC), p 12

    Google Scholar 

  47. Phillips A, Allemang, R, Brown D (2011) Autonomous modal parameter estimation: methodology. In: Proceedings, International Modal Analysis Conference (IMAC), p 22

    Google Scholar 

  48. Allemang R (1980) Investigation of some multiple input/output frequency response function experimental modal analysis techniques. Doctoral Dissertation, pp 141–214

    Google Scholar 

  49. Phillips, A, Allemang R (1998) The enhanced frequency response function (eFRF): scaling and other issues. In: Proceedings, International Seminar on Modal Analysis (ISMA)

    Google Scholar 

  50. Guyan R (1965) Reduction of stiffness and mass matrices. AIAA J 3(2):380

    Article  Google Scholar 

  51. Irons B (1965) Structural eigenvalue problems: elimination of unwanted variables. AIAA J 3(5):961–962

    Google Scholar 

  52. Downs B (1979) Accurate reduction of stiffness and mass matrices for vibration analysis and a rationale for selecting master degrees of freedom. ASME Paper Number 79-DET-18, p 5

    Google Scholar 

  53. Sowers J (1978) Condensation of free body mass matrices using flexibility coefficients. AIAA J 16(3):272–273

    Article  Google Scholar 

  54. O’Callahan J, Avitabile, P, Riemer R (1989) System equivalent reduction expansion process (SEREP). In: Proceedings, International Modal Analysis Conference (IMAC), pp 29–37

    Google Scholar 

  55. Kammer D (1987) Test analysis model development using an exact modal reduction. J Anal Exp Modal Anal 2(4):174–179

    Google Scholar 

  56. Kammer D (1990) Sensor placement for on-orbit modal identification and correlation of large space structures. In: Proceedings, American Control Conference, pp 2984–2990

    Google Scholar 

  57. Allemang, R, Brown D (1982) A correlation coefficient for modal vector analysis. In: Proceedings, International Modal Analysis Conference (IMAC), pp 110–116

    Google Scholar 

  58. Allemang R (2002) The modal assurance criterion (MAC): twenty years of use and abuse. In: Proceedings, International Modal Analysis Conference (IMAC), pp 397–405

    Google Scholar 

  59. Heylen W (1990) Extensions of the modal assurance criterion. J Vib Acoust (JVA) 112:468–472

    Article  Google Scholar 

  60. O’Callahan J (1998) Correlation considerations – Part 4 (Modal vector correlation techniques). In: Proceedings, International Modal Analysis Conference (IMAC), pp 197–206

    Google Scholar 

  61. Brechlin E, Bendel, K, Keiper W (1998) A New Scaled Modal Assurance Criterion for Eigenmodes Containing Rotational Degrees of Freedom. In: Proceedings, International Seminar on Modal Analysis (ISMA), vol ISMA23, pp 7

    Google Scholar 

  62. Wei J, Wang, W, Allemang R (1990) Model correlation and orthogonality criteria based on reciprocal modal vectors. In: Proceedings, SAE Noise and Vibration Conference, pp 607–616

    Google Scholar 

  63. Fotsch, D, Ewins D (2000) Applications of MAC in the frequency domain. In: Proceedings, International Modal Analysis Conference (IMAC), pp 1225–1231

    Google Scholar 

  64. Fotsch, D, Ewins D (2001) Further Applicatios of the FMAC. In: Proceedings, International Modal Analysis Conference (IMAC), pp 635–639

    Google Scholar 

  65. Lieven, N, Ewins D (1988) Spatial correlation of mode shapes, the coordinate modal assurance criterion (COMAC). In: Proceedings, International Modal Analysis Conference (IMAC), pp 690–695

    Google Scholar 

  66. Hunt D (1992) Application of an enhanced coordinate modal assurance criterion (ECOMAC). In: Proceedings, International Modal Analysis Conference (IMAC), pp 66–71

    Google Scholar 

  67. Milecek S (1994) The use of modal assurance criterion extended. In: Proceedings, International Modal Analysis Conference (IMAC), pp 363–369

    Google Scholar 

  68. Samman M (1996) A modal correlation coefficient (MCC) for detection of kinks in mode shapes. ASME J Vib Acoust 118(2):271–272

    Article  Google Scholar 

  69. Samman M (1997) Structural damage detection using the modal correlation coefficient (MCC). In: Proceedings, International Modal Analysis Conference (IMAC), pp 627–630

    Google Scholar 

  70. Mitchell L (2001) Increasing the sensitivity of the modal assurance criteria to small mode shape changes: the IMAC. In: Proceedings, International Modal Analysis Conference (IMAC), pp 64–69

    Google Scholar 

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

    Google Scholar 

  72. Allemang R (1990) Vibrations: experimental modal analysis, UC-SDRL-CN-20-263-663/664, pp 7-1–7-2

    Google Scholar 

  73. Van der Auweraer H, Iadevaia M, Emborg U, Gustavsson M, Tengzelius, U, Horlin N (1998) Linking test and analysis results in the medium frequency range using principal field shapes. In: Proceedings, International Seminar on Modal Analysis (ISMA), p 8

    Google Scholar 

  74. Pascual R, Golinval, J, Razeto M (1997) A Frequency domain correlation technique for model correlation and updating. In: Proceedings, International Modal Analysis Conference (IMAC), pp 587–592

    Google Scholar 

  75. Avitabile, P, Pechinsky F (1994) Coordinate orthogonality check (CORTHOG). In: Proceedings, International Modal Analysis Conference (IMAC), pp 753–760

    Google Scholar 

  76. Hawkins F (1965) An automatic resonance testing technique for exciting normal modes of comples structures. In: Symposium IUTAM, Recent Progress in the Mechanics of Linear Vibrations, pp 37–41

    Google Scholar 

  77. Hawkins F J. (1967) GRAMPA – an automatic technique for exciting the principal modes of vibration of complex structures. Royal Aircraft Establishment, vol RAE-TR-67-211

    Google Scholar 

  78. Taylor GA, Gaukroger, D, Skingle C (1967) MAMA – a semi-automatic technique for exciting the principal modes of vibration of complex structures. Aeronautical Research Council, vol ARC-R/M-3590, pp 20

    Google Scholar 

  79. Takahashi, K, Furusawa M (1987) Development of automatic modal analysis. In: Proceedings, International Modal Analysis Conference (IMAC), pp 686–692

    Google Scholar 

  80. Yam Y, Bayard D, Hadaegh F, Mettler E, Milman M and Scheid R (1989) Autonomous frequency domain identification: theory and experiment. NASA JPL Report JPL Publication 89-8, p 204

    Google Scholar 

  81. Lim T, Cabell, R, Silcox R (1996) On-line identification of modal parameters using artificial neural networks. J Vib Acoust (JVA) 118(4):649–656

    Article  Google Scholar 

  82. Pappa R, Woodard S E, Juang J (1997) A benchmark problem for development of autonomous structural modal identification. In: Proceedings, International Modal Analysis Conference (IMAC), pp 1071–1077

    Google Scholar 

  83. Pappa R, James, G, Zimmerman D (1998) Autonomous modal identification of the space shuttle tail rudder. J Spacecr Rocket 35(2):163–169

    Article  Google Scholar 

  84. Chhipwadia K, Zimmerman, D, James III G (1999) Evolving autonomous modal parameter estimation. In: Proceedings, International Modal Analysis Conference (IMAC), pp 819–825

    Google Scholar 

  85. James III G, Zimmerman, D, Chhipwadia K (1999) Application of autonomous modal identification to traditional and ambient data sets. In: Proceedings, International Modal Analysis Conference (IMAC), pp 840–845

    Google Scholar 

  86. Vanlanduit S, Verboven P, Schoukens, J, Guillaume P (2001) An automatic frequency domain modal parameter estimation algorithm. In: Proceedings, International Conference on Structural System Identification, pp 637–646

    Google Scholar 

  87. Verboven P, Parloo E, Guillame, P, Overmeire M (2001) Autonomous modal parameter estimation based on a statistical frequency domain maximum likelihood approach. In: Proceedings, International Modal Analysis Conference (IMAC), pp 1511–1517

    Google Scholar 

  88. Verboven P, Parloo E, Guillaume, P, Van Overmeire M (2002) Autonomous structural health monitoring, part i: modal parameter estimation and tracking. Mech Syst Signal Process (MSSP) 16(4):637–657

    Article  Google Scholar 

  89. Parloo E, Verboven P, Guillaume, P, Van Overmeire M (2002) Autonomous structural health monitoring – Part II: vibration-based in-operation damage assessment. Mech Syst Signal Process (MSSP) 16(4):659–675

    Article  Google Scholar 

  90. Vanlanduit S, Verboven P, Guillaume, P, Schoukens J (2003) An automatic frequency domain modal parameter estimation algorithm. J Sound Vib (JSV) 265:647–661

    Article  Google Scholar 

  91. Lanslots J, Rodiers, B, Peeters B (2004) Automated Pole-Selection: Proof-of-Concept and Validation. In: Proceedings, International Seminar on Modal Analysis (ISMA)

    Google Scholar 

  92. Mevel L, Sam, A, Goursat M (2004) Blind modal identification for large aircrafts. In: Proceedings, International Modal Analysis Conference (IMAC), p 8

    Google Scholar 

  93. Lau J, Lanslots J, Peeters, B, der Auweraer V (2007) Automatic modal analysis: reality or myth?. In: Proceedings, International Modal Analysis Conference (IMAC), p 10

    Google Scholar 

  94. Liu J, Ying H, Shen, S, Dong S (2007) The function of modal important index in autonomous modal analysis. In: Proceedings, International Modal Analysis Conference (IMAC), p 6

    Google Scholar 

  95. Mohanty P, Reynolds, P, Pavic A (2007) Automatic interpretation of stability plots for analysis of a non-stationary structure. In: Proceedings, International Modal Analysis Conference (IMAC), p 7

    Google Scholar 

  96. Poncelet F, Kerschen, G, Golinval J (2008) Operational modal analysis using second-order blind identification. In: Proceedings, International Modal Analysis Conference (IMAC), p 7

    Google Scholar 

  97. Rainieri C, Fabbrocino, G, Cosenza E (2009) Fully automated OMA: an opportunity for smart SHM systems. In: Proceedings, International Modal Analysis Conference (IMAC), p 9

    Google Scholar 

  98. Allemang R, Phillips, A, Brown D (2011) Autonomous modal parameter estimation: statistical considerations. In: Proceedings, International Modal Analysis Conference (IMAC), p 17

    Google Scholar 

  99. Brown D, Allemang, R, Phillips A (2011) Autonomous modal parameter estimation: application examples. In: Proceedings, International Modal Analysis Conference (IMAC), p 17

    Google Scholar 

  100. Phillips A, Allemang, R, Pickrel C (1998) Estimating modal parameters from different solution sets. In: Proceedings, International Modal Analysis Conference (IMAC), p 10

    Google Scholar 

  101. Allemang, R, Phillips A (2014) Spatial Information in Autonomous Modal Parameter Estimation, Shock and Vibration Journal: Special ICEDYN Issue, pp 1–18

    Google Scholar 

  102. Allemang R, Phillips, A, Brown D (2011) Combined state order and model order formulations in the unified matrix polynomial method (UMPA). In: Proceedings, International Modal Analysis Conference (IMAC)

    Google Scholar 

  103. DeClerck, J, Avitabile P (1996) Development of several new tools for modal pre-test evaluation. In: Proceedings, International Modal Analysis Conference (IMAC), pp 1272–1277

    Google Scholar 

  104. Cafeo J, Lust, R, Meireis U (2000) Uncertainty in Mode Shape Data and its Influence on the Comparison of Test and Analysis Models. In: Proceedings, International Modal Analysis Conference (IMAC), pp 349–355

    Google Scholar 

  105. Paez, T, Hunter N (1998) Fundamental concepts of the bootstrap for statistical analysis of mechanical systems. Exp Tech 21(3):35–38

    Article  Google Scholar 

  106. Hunter, N, Paez T (1998) Applications of the bootstrap for mechanical system analysis. Exp Tech 21(4):34–37

    Article  Google Scholar 

  107. Efron, B, Gong G (1983) A leisurely look at the bootstrap, the jackknife and cross-validation. Am Stat 37(1):36–48

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structural Dynamics Research Laboratory, Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA

    R. J. Allemang & A. W. Phillips

Authors
  1. R. J. Allemang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. W. Phillips
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. J. Allemang .

Editor information

Editors and Affiliations

  1. Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA

    Randall Allemang

  2. Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, Lowell, MA, USA

    Peter Avitabile

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Society for Experimental Mechanics

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Allemang, R.J., Phillips, A.W. (2022). Experimental Modal Parameter Evaluation Methods. In: Allemang, R., Avitabile, P. (eds) Handbook of Experimental Structural Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4547-0_12

Download citation

Publish with us

Policies and ethics

Search

Navigation