Skip to main content
SpringerLink
Log in

Basics of Vibration Dynamics

  • Chapter
  • First Online:
Model Predictive Vibration Control

Abstract

This chapter intends to introduce the reader to the theoretical basics of vibration dynamics analysis. Every well designed active vibration control (AVC) system requires at least a fundamental understanding of the underlying vibration phenomenon. The simple point mass oscillator is used as an example to build more and more complicated systems gradually. After the free response of undamped and damped one degree of freedom systems is discussed, forced response from a harmonic source is considered as well. The basics of engineering vibration analysis of lumped mass multiple degree of freedom systems is investigated with a concise account on the eigenvalue problem and modal decomposition. The transversal vibration of a clamped-free cantilever beam is used as an example to show, how exact solutions for distributed parameter systems may be developed. The chapter is finished with a section discussing modeling techniques used in vibration control, such as first principle transfer function models, state-space models, FEM based models and experimental identification. The aim of this chapter is to introduce the mathematical description of vibration phenomena briefly, in order to characterize the nature of the mechanical systems to be controlled by the model predictive control (MPC) strategy presented in the upcoming chapters of this book.

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 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
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

Notes

  1. 1.

    Courtesy of the European Space Agency (ESA) and European Aeronautic Defence and Space Company (EADS)-Astrium.

  2. 2.

    Note that \(\zeta\) is not the same as \(\delta_d\).

  3. 3.

    It is customary to denote the mass matrix with M however in the upcoming chapter this symbol will be reserved for an entirely different concept.

  4. 4.

    Certain literature divides this constant to a \(c=c_0^2i^2,\) where \(c_0={{E}\over{\rho}}\) is the speed of the longitudinal wave and \(i\) is the radius of quadratic moment of inertia given by \(i={{J}\over{A}}.\)

  5. 5.

    In order to avoid confusion with Q used in later chapters as input penalty , the multiple DOF displacements transformed into the Laplace domain are denoted as \({{\fancyscript{Q}}}\) here.

  6. 6.

    Unless of course one uses an augmented system model with filters, observers etc.

  7. 7.

    A vibrating system without outside energy cannot be marginally stable, since that would create a system without energy dissipation and without damping.

  8. 8.

    Also known as The University of Newcastle Identification Toolbox (UNIT).

References

  1. Ansys Inc (2005) Release 10.0 Documentation for ANSYS. Ansys Inc., Canonsburg

    Google Scholar 

  2. Ansys Inc (2009) Release 12.1 Documentation for ANSYS. Ansys Inc. /SAS IP Inc., Canonsburg

    Google Scholar 

  3. Bandyopadhyay B, Manjunath T, Umapathy M (2007) Modeling, control and implementation of smart structures: a FEM-state space approach, 1st edn. Springer Verlag, Berlin

    Google Scholar 

  4. Beards CF (1996) Structural vibration: analysis and damping, 1st edn. Arnold, London

    Google Scholar 

  5. Benaroya H, Nagurka ML (2010) Mechanical vibration: analysis, uncertainities and control, 3rd edn. CRC Press, Taylor&Francis Group, Boca Raton

    Google Scholar 

  6. Carra S, Amabili M, Ohayon R, Hutin P (2008) Active vibration control of a thin rectangular plate in air or in contact with water in presence of tonal primary disturbance. Aerosp Sci Technol 12(1):54–61. doi: 10.1016/j.ast.2007.10.001, http://www.sciencedirect.com/science/article/B6VK2-4PXDM8C-1/2/db87a30acd2bfaefa3f97e3896bc9232, Aircraft Noise Reduction

  7. Cavallo A, De Maria G, Natale C, Pirozzi S (2007) Gray-box identification of continuous-time models of flexible structures. IEEE Trans Control Syst Technol 15(5):967–981. doi:10.1109/TCST.2006.890284

    Article  Google Scholar 

  8. Chiang RY, Safonov MG (1991) Design of \(\fancyscript{H}_{\infty}\) controller for a lightly damped system using a bilinear pole shifting transform. In: American control conference, pp 1927–1928

    Google Scholar 

  9. Clijmans L, Swevers J, Baerdemaeker JD, Ramon H (2000) Sprayer boom motion, Part 1: Derivation of the mathematical model using experimental system identification theory. J Agric Eng Res 76(1):61–69. doi:10.1006/jaer.2000.0530, http://www.sciencedirect.com/science/article/B6WH1-45F4RPT-2T/2/a326ca97e631682a5f8ae6e2120d30ed

    Google Scholar 

  10. de Silva CW (2006) Vibration: fundamentals and practice, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  11. Dong X, Meng G, Peng J (2006) Vibration control of piezoelectric smart structures based on system identification technique: numerical simulation and study. J Sound Vib 297:680–693

    Article  MathSciNet  Google Scholar 

  12. European space agency (ESA) and European aeronautic defence and space company (EADS)-Astrium (2004) Venus Express being prepared for vibration tests by Intespace facilities, Toulouse. Image ID: SEMW4L808BE

    Google Scholar 

  13. Fang B, Kelkar AG, Joshi SM, Pota HR (2004) Modelling, system identification, and control of acoustic-structure dynamics in 3-D enclosures. Control Eng Pract 12(8): 989–1004. doi:10.1016/j.conengprac.2003.10.003, http://www.sciencedirect.com/science/article/B6V2H-4BYR2KY-1/2/8a78e1d6cbd24a286de16096e8cf88e7, Special section on emerging technologies for active noise and vibration control systems

  14. Fraanje R, Verhaegen M, Doelman N, Berkhoff A (2004) Optimal and robust feedback controller estimation for a vibrating plate. Control Eng Pract 12(8):1017–1027. doi:10.1016/j.conengprac.2003.09.007, http://www.sciencedirect.com/science/article/B6V2H-4B0PNNN-1/2/2879914555f3304fea841d08ef5898de, Special section on emerging technologies for active noise and vibration control systems

  15. Fuller CR, Elliott SJ, Nelson PA (1996) Active control of vibration, 1st edn. Academic Press, San Francisco

    Google Scholar 

  16. Gibson S, Wills A, Ninness B (2005) Maximum-likelihood parameter estimation of bilinear systems. IEEE Trans Autom Control 50(10):1581–1596

    Article  MathSciNet  Google Scholar 

  17. Guzmán J, Rivera D, Dormido S, Berenguel M (2010) ITSIE: an interactive software tool for system identification education. Available http://aer.ual.es/ITSIE/

  18. Hatch MR (2000) Vibration simulation using MATLAB and ANSYS, 1st edn. Chapman and Hall / CRC press, Boca Raton

    Book  Google Scholar 

  19. Haverkamp B, Verhaegen M (1997) SMI Toolbox, 1st edn. Delft University of Technology

    Google Scholar 

  20. Hellerstein JL, Diao Y, Parekh S, Tilbury DM (2004) Feedback control of computing systems. Wiley/IEEE Press, Hoboken

    Book  Google Scholar 

  21. Inman DJ (2006) Vibration with control. Wiley, Chichester

    Book  Google Scholar 

  22. Inman DJ (2007) Engineering vibrations, 3rd edn. Pearson International Education (Prentice Hall), Upper Saddle River

    Google Scholar 

  23. Ji H, Qiu J, Zhu K, Badel A (2010) Two-mode vibration control of a beam using nonlinear synchronized switching damping based on the maximization of converted energy. J Sound Vib 329(14):2751–2767. doi:10.1016/j.jsv.2010.01.012, http://www.sciencedirect.com/science/article/B6WM3-4YG1R6R-2/2/88406f934e48ccfe56a6188409cc989e

  24. Kaiser OE, Pietrzko SJ, Morari M (2003) Feedback control of sound transmission through a double glazed window. J Sound Vib 263(4):775–795. doi:10.1016/S0022-460X(02)01259-2 , http://www.sciencedirect.com/science/article/B6WM3-48D38FJ-2/2/7f8ee2d801360f2532e76c9c53353872

    Google Scholar 

  25. Landau ID, Constantinescu A, Rey D (2005) Adaptive narrow band disturbance rejection applied to an active suspension–an internal model principle approach. Automatica 41(4):563–574. doi:10.1016/j.automatica.2004.08.022, http://www.sciencedirect.com/science/article/B6V21-4FB3X55-3/2/28887440b73dcde4fdbaefe4d507e857

    Google Scholar 

  26. Lara A, Bruch J, Sloss J, Sadek I, Adali S (2000) Vibration damping in beams via piezo actuation using optimal boundary control. Int J Solids Struct 37:6537–6554

    Article  MATH  Google Scholar 

  27. Li M, Lim TC, Lee JH (2008) Simulation study on active noise control for a 4-T MRI scanner. Magn Reson Imaging 26(3):393–400. doi:10.1016/j.mri.2007.08.003. http://www.sciencedirect.com/science/article/B6T9D-4R8KT3W-2/2/2797c565f329cf6cd1e567eefb69607e

  28. Liu J, Liu K (2006) A tunable electromagnetic vibration absorber: characterization and application. J Sound Vib 295(3–5):708–724. doi:10.1016/j.jsv.2006.01.033. http://www.sciencedirect.com/science/article/B6WM3-4JP9FXN-6/2/0b961839d0b922bbd94dcc5ce5c5f9e4

    Google Scholar 

  29. Ljung L (1999) System identification: theory for the user, 2nd edn. PTR Prentice Hall, Upper Saddle River

    Google Scholar 

  30. Luo Y, Xie S, Zhang X (2008) The actuated performance of multi-layer piezoelectric actuator in active vibration control of honeycomb sandwich panel. J Sound Vib 317(3–5):496–513. doi:10.1016/j.jsv.2008.03.047, http://www.sciencedirect.com/science/article/B6WM3-4SJR2GN-1/2/04c4aad317afe74e20e6f5810f689674

  31. Mahmoodi SN, Jalili N, Khadem SE (2008) An experimental investigation of nonlinear vibration and frequency response analysis of cantilever viscoelastic beams. J Sound Vib 311(3–5):1409–1419. doi:10.1016/j.jsv.2007.09.027, http://www.sciencedirect.com/science/article/B6WM3-4R113SP-1/2/4baf1df12aa15dbfcbdd0e4f13149b17

  32. Moshrefi-Torbati M, Keane A, Elliott S, Brennan M, Anthony D, Rogers E (2006) Active vibration control (AVC) of a satellite boom structure using optimally positioned stacked piezoelectric actuators. J Sound Vib 292(1–2):203–220. doi:10.1016/j.jsv.2005.07.040, http://www.sciencedirect.com/science/article/pii/S0022460X05005171

  33. Ninness B, Wills A (2006) An identification toolbox for profiling novel techniques. In: 14th IFAC symposium on system identification, pp 301–307

    Google Scholar 

  34. Ninness B, Wills A, Gibson S (2005) The University of Newcastle identification toolbox (UNIT). In: IFAC World Congress, pp 1–6

    Google Scholar 

  35. Ninness B, Wills A, Mills A (2009) System identification toolbox (ID Toolbox). Available http://sigpromu.org/idtoolbox/

  36. Pradhan S (2005) Vibration suppression of FGM shells using embedded magnetostrictive layers. Int J Solids Struct 42(9–10):2465–2488. doi:10.1016/j.ijsolstr.2004.09.049, http://www.sciencedirect.com/science/article/B6VJS-4F6SSGN-1/2/b6f9e2e6ffc65bfc0c4af5083e37df0b

    Google Scholar 

  37. Preumont A (2002) Vibration control of active structures, 2nd edn. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  38. Preumont A, Seto K (2008) Active control of structures, 3rd edn. Wiley, Chichester

    Book  Google Scholar 

  39. Rashid A, Nicolescu CM (2006) Active vibration control in palletised workholding system for milling. Int J Mach Tools Manuf 46(12–13):1626–1636. doi:10.1016/j.ijmachtools.2005.08.020, http://www.sciencedirect.com/science/article/B6V4B-4HGD76C-2/2/273540b1466f54bf47cc11a241007364

    Google Scholar 

  40. Rastgaar M, Ahmadian M, Southward S (2010) Experimental application of orthogonal eigenstructure control for structural vibration cancellation. J Sound Vib 329(19):3873–3887. doi:10.1016/j.jsv.2010.03.015, http://www.sciencedirect.com/science/article/B6WM3-502WG3V-1/2/a4d2874628bb1caebddd51cc6366b58c

    Google Scholar 

  41. Rossing TD, Moore RF, Wheeler PA (2001) The Science of Sound, 3rd edn. Addison Wesley, San Francisco

    Google Scholar 

  42. Rossiter JA (2003) Model-based predictive control: a practical approach, 1st edn. CRC Press, Boca Raton

    Google Scholar 

  43. Schon T, Wills A, Ninness B (2011) System identification of nonlinear state-space models. Automatica 37(1):39–49

    Google Scholar 

  44. Seba B, Nedeljkovic N, Paschedag J, Lohmann B (2005) \(\fancyscript{H}_{\infty}\) feedback control and Fx-LMS feedforward control for car engine vibration attenuation. Appl Acoust 66(3):277–296. doi:10.1016/j.apacoust.2004.07.015, http://www.sciencedirect.com/science/article/B6V1S-4DTKD2W-1/2/d413b004e2a2e14e9df7fcf75f2df02f

  45. Sima V, Simal D, Van Huffel S (2002) SLICOT system identification software and applications. In: 2002 IEEE international symposium computer aided control system design proceedings, pp 45–50. doi:10.1109/CACSD.2002.1036927

  46. Sima V, Sima DM, Huffel SV (2004) High-performance numerical algorithms and software for subspace-based linear multivariable system identification. J Comput Appl Math 170(2):371–397. doi:10.1016/j.cam.2003.12.046, http://www.sciencedirect.com/science/article/B6TYH-4BYR6W6-B/2/e0433bca6148658f54c9a683d281c020

    Google Scholar 

  47. Skullestad A, Hallingstad O (1998) Vibration parameters identification in a spacecraft subjected to active vibration damping. Mechatronics 8(6):691–705. doi:10.1016/S0957-4158(97)00051-2, http://www.sciencedirect.com/science/article/B6V43-3W18XD5-4/2/d6e2d2a478a77ff09e9f6f7dfe7fd503

  48. Sloss J, Bruch J, Sadek I, Adali S (2003) Piezo patch sensor/actuator control of the vibrations of a cantilever under axial load. Compos Struct 62:423–428

    Article  Google Scholar 

  49. Šolek P (2009) Techincká Mechanika II, 1st edn. Slovenská technická univerzita v Bratislave, Nakladatel’stvo STU, Bratislava (Technical Mechanics II) in Slovak language.

    Google Scholar 

  50. Starek L (1985) Vyššia dynamika, 2nd edn. SVŠT, Bratislava (Advanced Dynamics) in Slovak language.

    Google Scholar 

  51. Starek L (2006) Kmitanie mechanických sústav. Slovenská technická univerzita v Bratislave, Nakladatel’stvo STU, Bratislava (Vibration of Mechanical Systems) in Slovak language.

    Google Scholar 

  52. Starek L (2009) Kmitanie s riadením, 1st edn. Slovenská technická univerzita v Bratislave, Nakladatel’stvo STU, Bratislava (Vibration with Control) in Slovak language.

    Google Scholar 

  53. Sumali H, Meissner K, Cudney HH (2001) A piezoelectric array for sensing vibration modal coordinates. Sens Actuators A 93(2):123–131. doi: 10.1016/S0924-4247(01)00644-6 http://www.sciencedirect.com/science/article/B6THG-43N2J67-5/2/b7e5396a71e7ee7fae9ef93cb254b7b7

    Google Scholar 

  54. Takács G, Rohal’-Ilkiv B (2008) Experimental identification of a structure with active vibration cancelling. Acta Mech Slovaca 12(3-B):795–803

    Google Scholar 

  55. Takács G, Rohal’-Ilkiv B (2009) MPC with guaranteed stability and constraint feasibility on flexible vibrating active structures: a comparative study. In: Hu H (ed) Proceedings of the eleventh IASTED international conference on control and applications, Cambridge

    Google Scholar 

  56. Takács G, Rohal’-Ilkiv B (2009) Newton-Raphson based efficient model predictive control applied on active vibrating structures. In: Proceedings of the European Control Control Conference, Budapest

    Google Scholar 

  57. Takács G, Rohal’-Ilkiv B (2009) Newton-Raphson MPC controlled active vibration attenuation. In: Hangos KM (ed) Proceedings of the 28. IASTED international conference on modeling, identification and control, Innsbruck

    Google Scholar 

  58. Takács G, Rohal’-Ilkiv B (2010) Capacitive proximity sensor position feedback in active vibration control of lightly damped cantilevers. In: Shokin YI, Bychkov I, Potaturkin O (eds) Proceedings of the 3rd IASTED international multiconference ACIT-CDA, Novosibirsk, pp 692–700

    Google Scholar 

  59. The Mathworks (2011) Matlab system identification toolbox v7.4.2 (R2011a), Natick. Software. Available http://www.mathworks.com/help/toolbox/ident/

  60. Weber B, Feltrin G (2010) Assessment of long-term behavior of tuned mass dampers by system identification. Eng Struct 32(11):3670–3682. doi: 10.1016/j.engstruct.2010.08.011 http://www.sciencedirect.com/science/article/B6V2Y-5119FWV-2/2/20320a260725a111ed30767718b9b50b

    Google Scholar 

  61. Wills A, Ninness B (2008) On gradient-based search for multivariable system estimates. IEEE Trans Autom Control 53(1):298–306

    Article  MathSciNet  Google Scholar 

  62. Wills A, Ninness B, Gibson S (2009) Maximum likelihood estimation of state space models from frequency domain data. IEEE Trans Autom Control 54(1):19–33

    Article  MathSciNet  Google Scholar 

  63. Wills AG, Bates D, Fleming AJ, Ninness B, Moheimani SOR (2008) Model predictive control applied to constraint handling in active noise and vibration control. IEEE Trans Control Syst Technol 16(1):3–12

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Institute of Automation, Measurement and Applied Informatics, Slovak University of Technology in Bratislava, Námestie slobody 17, 812 31, Bratislava 1, Slovakia

    Gergely Takács & Boris Rohal’-Ilkiv

Authors
  1. Gergely Takács
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Rohal’-Ilkiv
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag London Limited

About this chapter

Cite this chapter

Takács, G., Rohal’-Ilkiv, B. (2012). Basics of Vibration Dynamics. In: Model Predictive Vibration Control. Springer, London. https://doi.org/10.1007/978-1-4471-2333-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-2333-0_2

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-2332-3

  • Online ISBN: 978-1-4471-2333-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation