Modern Analytical Methods Applied to Mechanical Engineering Systems

  • P. Hagedorn
  • W. Seemann
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 387)


In the following lectures modern analytical methods are applied to several industrial systems. The first application is the use of singular perturbation analysis in the eigenvalue problem of a vibrating string with a small bending stiffness. This models the behavior of overhead transmission lines. A more complicated system is the eigenvalue problem of a cylinder vibrating in an cylindrical duct, which is filled with a viscous fluid.

The next lecture shows the use of analytical methods in the modelling of ultrasonic motors. Of special interest is the coupling between the electric and the mechanical field in the piezoelectric patches. Here Hamilton’s principle for electromechanical systems is of great importance. It allows to find approximate solutions fulfilling additional constraint equations.

In overhead transmission lines also wind excited vibrations are important as they may lead to fatigue. Models for the corresponding mechanism are given as well as the analysis of special vibration absorbers designed by modern methods of vibration theory.

The last lecture deals with three problems of nonholonomic systems and of stability and instability theorems. The first problem shows that the augmented Lagrangian is not stationary for nonholonomic systems. The second gives a simple exercise in Liapunov stability. The last deals with the Lagrange-Dirichlet theorem and its inverses.


Nonholonomic System Bonding Layer Outer Solution Ultrasonic Motor Singular Perturbation Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Anderson, K.; Hagedorn, P.: On the Energy Dissipation in Spacer Dampers in Bundled Conductors of Overhead Transmission Lines, Journal of Sound and Vibration (1995), 180 (4), 539–556.CrossRefGoogle Scholar
  2. [2]
    Chen, S.S.; Wambsganss, M.W.; Jendrzejczyk, J.A.: Added Mass and Damping of a Vibrating Rod in Confined Viscous Fluids, Journal of Applied Mechanics, Vol. 43, 1976, 325–329.CrossRefGoogle Scholar
  3. [3]
    Claren, R.; Diana, G.: Vibrazioni dei conduttori, L’Energia Elettrica, No. 10, 1966.Google Scholar
  4. [4]
    Cole, J.D.: Perturbation Methods in Applied Mathematics, Blaisdell Publ. Comp., 1968.MATHGoogle Scholar
  5. [5]
    Cremer, L.; Heckl, M.: Structure-Borne Sound, Springer-Verlag, Berlin, Heidelberg, New York, London, 1988.Google Scholar
  6. [6]
    Cremer, L.: Vorlesungen über Technische Akustik, Springer-Verlag, Berlin, Heidelberg, New York, 1971.Google Scholar
  7. [7]
    Diana, G.; Falco, M.: On the forces transmitted to a vibrating cylinder by a blowing fluid, Meccanica, Vol. 6, 1971, 9–22.CrossRefGoogle Scholar
  8. [8]
    Doocy, E.S.; Hard, A.R.; Rawlins, C.B.; Ikegami, R.: Transmission line reference book, Wind induced Conductor Motion, Electric Power Research Institute, Palo Alto, Cal. 1979.Google Scholar
  9. [9]
    Van Dyke, M.: Perturbation Methods in Fluid Mechanics, Parabolic Press, Palo Alto, California, 1975.Google Scholar
  10. [10]
    Eckhaus, W.: Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979.MATHGoogle Scholar
  11. [11]
    Eckhaus, W.: Matched Asymptotic Expansions and Singular Perturbations, North-Holland, Amsterdam, 1973.MATHGoogle Scholar
  12. [12]
    Farquharson, F.; Mehugh, R.E.: Wind tunnel investigation of conductor vibration with use of rigid models, Trans. AIEE, Vol. 75, 1956, Part III, 871–878.Google Scholar
  13. [13]
    Hagedorn, P.: Nonlinear Oscillations, Clarendon Press, Oxford, 1988.Google Scholar
  14. [14]
    Hagedorn, P.: Ein einfaches Rechenmodell zur Berechnung winderregter Schwingungen an Hochspannungsleitungen mit Dämpfern, Ing.-Arch., Vol. 49, 1980, 161–177.CrossRefMATHGoogle Scholar
  15. [15]
    Hagedorn, P.: On the computation of damped wind-excited vibrations of overhead transmission lines, Journal of Sound and Vibration, Vol. 83, 1982, 253–271.MathSciNetGoogle Scholar
  16. [16]
    Hagedorn, P.; Wallaschek, J.: Traveling Wave Ultrasonic Motors, Part I: Working Principle and Mathematical Modelling of the Stator, Journal of Sound and Vibration, Vol. 155 (1): 31–46, 1993.Google Scholar
  17. [17]
    Hagedorn, P.: Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Archive for Rational Mechanics and Analysis, Vol. 42, 1971, 4, 281–316.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Hagedorn, P. and Mahwin, J.: A Simple Variational Approach to an Inversation of the Lagrange-Dirichlet Theorem, Archive for Rational Mechanics and Analysis, 120 (1992), 327–335.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Horacec, J.; Zolotarev, I.: Acoustic-Structural coupling of Vibrating Cylindrical Shells with Flowing Fluid, Journal of Fluids and Structures, Vol. 5, 1991, 487–501.Google Scholar
  20. [20]
    Johnson, K.L.: Contact Mechanics, Cambridge University Press, Cambridge, 1985.CrossRefMATHGoogle Scholar
  21. [21]
    Kaplun, S.: Fluid Mechanics and Singular Perturbations, Ed. by Lagerstrom, P.A., Howard, L.N. and Liu, C.S., Academic Press, New York, 1967.Google Scholar
  22. [22]
    Kevorkian, J.; Cole, J.D.: Perturbation Methods in Applied Mechanics, Applied Mathematical Sciences, Vol. 34, Springer-Verlag, New York, 1980.Google Scholar
  23. [23]
    Nayfeh, A.; Mook, D.T.: Nonlinear Oscillations, John Wiley and Sons, New York, 1979.MATHGoogle Scholar
  24. [24]
    Morse, P.M.; Ingard, K.U.: Theoretical Acoustics, 1968, New York: MCGraw-Hill.Google Scholar
  25. [25]
    Nascimento, N.; Hagedorn, P.: Stochastic field processes in the mathematical modelling of damped transmission line vibrations, Paper presented at the Fifth International Conference on Mathematical Modelling, Berkeley, 1985.Google Scholar
  26. [26]
    Nascimento, N.: Stochastische Schwingungen eindimensionaler, kontinuierlicher mechanischer Systeme, Doctoral Thesis, TH Darmstadt, 1984.Google Scholar
  27. [27]
    Nayfeh, A.H.: Perturbation Methods, John Wiley and Sons, New York, 1973.MATHGoogle Scholar
  28. [28]
    Neimark, J.F. and Fufaev, N.A.: Dynamics of Nonholonomic Systems, American National Society, Rhodes Island, 1972.MATHGoogle Scholar
  29. [29]
    O’Malley, R.E.: Introduction to Singular Perturbations, Academic Press, New York, 1974.MATHGoogle Scholar
  30. [30]
    Palauiadov, V. P.: On stability of an Equlibrium in a Potential Field, Functional Analysis and its Application, Vol. 11 (1977), No. 4, 42–55 (in Russian).Google Scholar
  31. [31]
    Pars, B.A.: Analytical Dynamics, Heinemann, London, 1968.Google Scholar
  32. [32]
    Paîdoussis, M.P.; Nguyen, V.B.; Misra, A.K.: A Theoretical Study of the Stability of Cantilevered Coaxial Cylindrical Shells Conveying Fluid, Journal of Fluids and Structures, Vol. 5, 1991, 127–164.Google Scholar
  33. [33]
    Rawlins, C.B.: Power imparted by wind to a model of a vibrating conductor, Electrical Products Division, ALCOA Labs., Massena, NY, 1982.Google Scholar
  34. [34]
    Schäfer, B.: Dynamical modelling of wind-induced vibrations of overhead lines, International Journal of Non-Linear Mechanics, Vol. 19, 1984, 455–467.CrossRefMATHGoogle Scholar
  35. [35]
    Seemann, W.: Stresses in a Thin Piezoelectric Element Bonded to a Half-space, Applied Mechanics Reviews, to appear November 1997.Google Scholar
  36. [36]
    Seemann, W.; Hagedorn, P.: On the Realizability of a Traveling Wave Linear Motor, 13th International Modal Analysis Conference,Conference Proceedings, Nashville 1995, USA, pp. 1778–1784.Google Scholar
  37. [37]
    Seemann, W.; Wolf, K.; Straub, A.; Hagedorn, P.; Chang, F.-K.: Bonding Stresses between Piezoelectric Actuators and elastic Beams, Proceedings of the SPIE Conference on Smart Materials and Structures, San Diego, 1997.Google Scholar
  38. [38]
    Seemann, W.; Wauer, J.: Vibrating Cylinder in a Cylindrical Duct Filled with an Incompressible Fluid of Low Viscosity, Acta Mechanica, Vol. 113, pp. 93–107, 1995.CrossRefMATHGoogle Scholar
  39. [39]
    Seemann, W.; Wolf, K.; Hagedorn, P.: Comparison of Refined Beam Theory and FEM for Piezo-Actuated Structures, ASME Design Engineering Technical Conferences, Paper-No. VIB-3838, Sacramento, 1997.Google Scholar
  40. [40]
    Straubli, T.: Untersuchung der oszillierenden Kräfte am querangeströmten, schwingenden Kreiszylinder, Doctoral Thesis, ETH Zürich, 1983.Google Scholar
  41. [41]
    Tiersten, H.F.: Linear Piezoelectric Plate Vibrations, Plenum Press, New York, 1969.CrossRefGoogle Scholar
  42. [42]
    Wallaschek, J.: Piezoelectric Ultrasonic Motors, Journal of Intelligent Material Systems and Structures, Vol. 6: 71–83, 1995.CrossRefGoogle Scholar
  43. [43]
    Weidenhammer, F.: Eigenfrequenzen eines Stabes im zylindrischen Luftraum, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 55, 1975, T187 - T190.CrossRefMATHGoogle Scholar
  44. [44]
    Whittaker, E.T.: A treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, London, 1937MATHGoogle Scholar
  45. [45]
    Yang, C.-I.; Moran, T. J.: Finite Element Solution of Added Mass and Damping of Oscillation Rods in Viscous Fluids, Journal of Applied Mechanics,Vol 46, 1979, 519523.Google Scholar
  46. [46]
    Zampieri, G. and Neto, A. Barone: Attractive Central Forces may yield Liapunov Instability. Relatório Técnico, Dep. Mat. Apl. USP, Dec. 1984.Google Scholar
  47. [47]
    Zharii, O: Thin Piezoelectric Element on an Elastic Half-Space, Private Communication, 1994.Google Scholar

Copyright information

© Springer-Verlag Wien 1998

Authors and Affiliations

  • P. Hagedorn
    • 1
  • W. Seemann
    • 1
  1. 1.Darmstadt University of TechnologyDarmstadtGermany

Personalised recommendations