Experimental Mechanics

, Volume 55, Issue 8, pp 1551–1568 | Cite as

Dynamic Modeling of Stacked, Multi-Plate Flexible Systems

  • T. Lundstrom
  • C. Sidoti
  • N. JaliliEmail author


The vibration analysis and control of stacked-plate mechanical systems such as circuit board assemblies is an important technical problem that often requires a complete and accurate description of the open loop system dynamics prior to controller development. Often, a preliminary finite element model (FEM) of the assembly of interest is developed to estimate the dynamics of the system prior to the execution of a validation modal experiment. The results of this modal test must then be used to update the stiffness, mass and damping matrices to yield accurate transfer functions throughout the structure. This paper undertakes the mathematical development of a general, dynamic, multi-plate system. The work proceeds with the description of a three-plate system dynamic model and the physical test setup with test article used for validation and testing purposes. A model update is performed using differentiated velocity (in the frequency domain) data measured at discrete points on the test article with a laser Doppler vibrometer (LDV) and force measurements collected with an impedance head at a corner of the base plate. Using these data, accelerance frequency response functions (FRFs) were computed and the first eight flexible mode shapes were estimated and compared to the corresponding FEM shapes using both percent frequency difference and modal assurance criterion (MAC). A preliminary model update was performed with the addition/modification of discrete, rotary stiffness elements between plates and finished with a final update utilizing analytical model improvement (AMI) techniques employing a target modal matrix composed of both FEM modal vectors and expanded/smoothed experimental modal vectors. In addition, the experimental data was also used to estimate percent critical damping values for the first eight flexible modes and a Rayleigh damping model was developed for the system. The updated model was validated by comparing the magnitude/phase relationships of the base plate force input and several important response regions to that of the experimental results. The resonant frequencies of the first eight modes of the updated model differed from the physical system by no more than 0.3 % and the MAC values for the first eight modes all surpassed 0.97.


Multi-plate dynamic system modeling Modal analysis Structural dynamics Experimental modal analysis Dynamic model updating 


  1. 1.
    Den Hartog JP (1937) Vibration in Industry. J Appl Phys 8:76–83CrossRefGoogle Scholar
  2. 2.
    Frahm H (1911) Devices for damping vibrations of bodies. United States Patent Office, Patent Specification #989,958Google Scholar
  3. 3.
    Page C, Avitabile P, Niezrecki C (2012) Passive noise reduction using the modally enhanced dynamic absorber. Topics in Modal Analysis II, 6:637–638, Springer, New York.Google Scholar
  4. 4.
    Dayou J, Brennan M (2002) Global control of structural vibration using multiple-tuned tunable vibration neutralizers. J Sound Vib 258(2):347–357MathSciNetCrossRefGoogle Scholar
  5. 5.
    Vepirk AM (2003) Vibration protection of critical components of electronic equipment in harsh environmental conditions. J Sound Vib 259(1):161–175CrossRefGoogle Scholar
  6. 6.
    Ungar EE (2007) Chapter 59: introduction to principles of noise and vibration control. Handbook of noise and vibration control. Wiley, HobokenGoogle Scholar
  7. 7.
    Lin RM, Ewins DJ (1994) Analytical model improvement using frequency response functions. Mech Syst Signal Process 8(4):437–458CrossRefGoogle Scholar
  8. 8.
    Berman A, Najy EJ (1983) Improvement of a large analytical model using test data. AIAA J 21(8):1168–1173CrossRefGoogle Scholar
  9. 9.
    Dadfarnia M, Jalili N, Liu Z, Dawson DM (2004a) An observer-based piezoelectric control of flexible Cartesian robot arms: theory and experiment. Control Eng Pract 12:1041–1053Google Scholar
  10. 10.
    Kim S, Wang S, Brennan M (2011) Optimal and robust control of a flexible structure using an active dynamic vibration absorber. Smart Materials and Structures 20(4):5003Google Scholar
  11. 11.
    Song Z, Li F (2012) Active aeroelastic flutter analysis and vibration control of supersonic composite laminated plate. Compos Struct 94(2):702–713CrossRefGoogle Scholar
  12. 12.
    Paradies R, Ciresa P. (2009) Active wing design with integrated flight control using piezoelectric macro fiber composites. Smart Materials and Structures 18(3):5010Google Scholar
  13. 13.
    Hedrih K (2006) Transversal vibrations of double-plate systems. Acta Mech Sinica 22:487–501zbMATHCrossRefGoogle Scholar
  14. 14.
    Jeong K, Yoo G, Lee S (2004) Hydroelastic vibration of two identical rectangular plates. J Sound Vib 272:539–555CrossRefGoogle Scholar
  15. 15.
    Avitabile P, O’Callahan J, Tsuji H, DeClerck JP (2003) Reallocation of system mass and stiffness for achieving target specifications. Proceedings of the Twenty-First International Modal Analysis Conference, Orlando, FloridaGoogle Scholar
  16. 16.
    Avitabile P, O’Callahan J, Tsuji H, DeClerck JP (2004) Reallocation of system mass and stiffness for achieving target specifications using a superelement/substructuring methodology. Proceedings of the Twenty-Second International Modal Analysis Conference, Dearborn, MichiganGoogle Scholar
  17. 17.
    O’Callahan J, Avitabile P, Leung R (1984) Development Of mass and stiffness matrices for an analytical model using experimental modal data. Second International Modal Analysis Conference, Orlando, FLGoogle Scholar
  18. 18.
    O’Callahan J, Leung R (1985) Optimization of mass and stiffness matrices using a generalized inverse technique on the measured modes. Third International Modal Analysis Conference, Orlando, FloridaGoogle Scholar
  19. 19.
    O’Callahan J, Avitabile P, Riemer R (1989) System Equivalent Reduction Expansion Process (SEREP). Proceedings of the 7th International Modal Analysis Conference, Las Vegas, NVGoogle Scholar
  20. 20.
    Petyt M (1990) Introduction to finite element vibration analysis, vol 6. Cambridge University Press, New York, pp 229–293zbMATHCrossRefGoogle Scholar
  21. 21.
    Weaver W, Johnston P (1984) Finite elements for structural analysis, vol 6. Prentice-Hall, Inc, Englewood Cliffs, pp 200–231zbMATHGoogle Scholar
  22. 22.
    Song K (2000) Development of the velocity transformation function damped flat shell finite element for the experimental spatial dynamics modeling. Master’s Thesis, Virginia Polytechnic Institute and State UniversityGoogle Scholar
  23. 23.
    Welch P (1967) The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 15(2):70–73MathSciNetCrossRefGoogle Scholar
  24. 24.
    LMS Test.Lab 10A, LMS International, Leuven, BelgiumGoogle Scholar

Copyright information

© Society for Experimental Mechanics 2015

Authors and Affiliations

  1. 1.Piezoactive Systems Laboratory, Department of Mechanical and Industrial EngineeringNortheastern UniversityBostonUSA

Personalised recommendations