Abstract
The response of a slender, clastic, cantilevered beam to a transverse, vertical, harmonic excitation is investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Previous work often has neglected the static deflection caused by the weight of the beam, which adds quadratic terms in the governing equations of motion. Galerkin's method is used with three modes and approximate solutions of the temporal equations are obtained by the method of multiple scales. Primary resonance is treated here, and out-of-plane motion is possible in the first and second modes when the principal moments of inertia of the beam cross-section are approximately equal. In Parts II and III, secondary resonances and nonstationary passages through various resonances are considered.
Similar content being viewed by others
References
Wagner, H., ‘Large-amplitude free vibrations of a beam’, Journal of Applied Mechanics 32, 1965, 887–992.
Atluri, S., ‘Nonlinear vibrations of a hinged beam including nonlinear inertia effects’, Journal of Applied Mechanics 40, 1973, 121–126.
Haight, E. C. and King, W. W., ‘Stability of nonlinear oscillations of an elastic rod’, The Journal of the Acoustical Society of America 52, 1972, 899–911.
Hyer, M. W., ‘Whirling of a base-excited cantilever beam’, The Journal of the Acoustical Society of America 65, 1978, 931–939.
Hyer, M. W., ‘Preliminary investigations into the active control of large space structures-whirling motion of a viscously damped, base-excited cantilever beam’, NACA Report, Research Grant NSG 1279, 1978.
Hyer, M. W., ‘The effect of nonlinear curvature and inertia on the predictions for the planar response and stability of an inextensible base-excited cantilever’, Virginia Polytechnic Institute and State University, Report No. VPI-E-78-28, 1978.
Ho, C.-H., Scott, R. A., and Eisley, J. G., ‘Non-planar, non-linear oscillations of a beam - I, forced motions’. International Journal of Non-Linear Mechanics 10, 1975, 113–127.
ho, C.-H., Scott, R.A., and Eisley, J.G., ‘Non-planar, non-linear oscillations of a beam - II, free motions’, Journal of Sound and Vibration 47, 1976, 333–339.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsional dynamics of inextensional beams I. equations of motion’, Journal of Structural Mechanics 6, 1978, 437–448.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsional dynamics of inextensional beams II. forced motions’, Journal of Structural Mechanics 6, 1978, 449–461.
Rosen, A. and Friedmann, P., ‘The nonlinear behavior of elastic slender straight beams undergoing small strains and moderate rotations’, Journal of Applied Mechanics 46, 1979, 161–168.
Hyer, M. W., ‘Nonplanar motions of a base-excited cantilever’, in Recent Advances in Structural Dynamics, ed. M. Petyt, published by Institute of Sound and Vibration Research, University of Southampton, Southampton, England, 2, 1980, 621–630.
Rosen, A., Loewy, R. G., and Mathew, M. B., ‘Nonlinear analysis of pretwisted rods using “principal curvature transformation” part II: numerical results’, AIAA Journal 25, 1987, 598–604.
Pai, P. J. F., ‘Nonlinear flexural-flexural-torsional dynamics of metallic and composite beam’, Ph.D. Dissertation, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1990.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Out-of-plane vibrations of a beam including non-linear inertia and non-linear curvature effects’, Vertica 3, 1979, 261–271.
Crespo da Silva, M. R. M. and Zaretzky, C. L., ‘Non-linear modal coupling in planar and non-planar responses of inextensional beams’, International Journal of Non-Linear Mechanics 25, 1990, 227–239.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Non-linear non-planar resonant oscillations in fixed-free beams with support asymmetry’, International Journal of Solids and Structures 15, 1979, 209–219.
Crespo da Silva, M. R. M., ‘Flexural-flexural oscillations of Beck's column subjected to a planar harmonic excitation’, Journal of Sound and Vibration 60, 1978, 133–144.
Crespo da Silva, M. R. M., ‘Harmonic non-linear response of Beck's column to a lateral excitation’, International Journal of Solids and Structures 14, 1978, 987–997.
Crespo da Silva, M. R. M., ‘Nonlinear resonances in a column subjected to a constant end force’, Journal of Applied Mechanics 47, 1980, 409–414.
Crespo da Silva, M. R. M., ‘Non-linear flexural-flexural-torsional-extensional dynamics of beam I. formulation’, International Journal of Solids and Structures 24, 1988, 1225–1234.
Crespo da Silva, M. R. M., ‘Non-linear flexural-flexural-torsional-extensional dynamics of beam II. response analysis’, International Journal of Solids and Structures 24, 1988, 1235–1242.
Nayfeh, A. H. and Pai, P.-J. F., ‘Non-linear non-planar parametric responses of an inextensional beam’, International Journal of Non-Linear Mechanies 24, 1989, 139–158.
Pai, P.-J. F. and Nayfeh, A. H., ‘Non-linear non-planar oscillations of a cantilever beam under lateral base excitations’, International Journal of Non-Linear Mechanics 25, 1990, 455–474.
Restuccio, J. M., Krousgrill, C. M., and Bajaj, A. K., ‘Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam’, Nonlinear Dynamics 2, 1991, 263–289.
Dowell, E. H., Traybar, J., and Hodges, D. H., ‘An experimental-theoretical correlation study of non-linear bending and torsion deformations of a cantilever beam’, Journal of Sound and Vibration 50, 1977, 533–544.
Rosen, A., Loewy, R. G., and Mathew, M. B., ‘Nonlinear analysis of pretwisted rods using “principal curvature transformation” part I: theoretical derivation’, AIAA Journal 25, 1987, 470–478.
Saito, H., Sato, K., and Yutani, T., ‘Non-linear forced vibrations of a beam carrying concentrated mass under gravity’, Journal of Sound and Vibration 46, 1976, 515–525.
Sato, K., Saito, H., and Otomi, K., ‘The parametric response of a horizontal beam carrying a concentrated mass under gravity’, Journal of Applied Mechanics 45, 1978, 643–648.
Hughes, G. C. and Bert, C. W., ‘Effect of gravity on nonlinear oscillations of a horizontal, immovable-end beam’, ThirdConference on Nonlinear Vibrations, Stability, and Dynamics of Structures and Mechanisms, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1990.
Shih, C.-F., Chen, J. C., and Garba, J., ‘Vibration of a large space beam under gravity effect’, AIAA Journal 24, 1986, 1213–1216.
Rosen, A., Loewy, R. G., and Mathew, M. B., ‘Nonlinear dynamics of slender rods’, AIAA Journal 25, 1987, 611–619.
Crespo da Silva, M. R. M. and Zaretzky, C. L., ‘Effects of approximations on the static and dynamic responses of a cantilever with a tip mass’, International Journal of Solids and Structures 27, 1991, 565–583.
Tseng, W. Y. and Dugundji, J., ‘Nonlinear vibrations of a beam under harmonic excitation’, Journal of Applied Mechanics 37, 1970, 292–297.
Tseng, W. Y. and Dugundji, J., ‘Nonlinear vibrations of a buckled beam under harmonic excitation’, Journal of Applied Mechanics 38, 1971, 467–476.
Minguet, P. J. A., ‘Static and dynamic behavior of composite helicopter rotor blades under large deflection’, Ph.D. Dissertation. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Boston, Massachusetts, 1989.
Kim, T. and Dugundji, J., ‘Nonlinear large amplitude vibration of composite helicopter blade at large static deflection’, Proceedings of the 32nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Baltimore, Maryland, April 8–10, 1991, Part 3, pp. 2071–2081.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1980.
Shyu, I.-M. K., ‘Forced, nonlinear, planar and nonplanar oscillations of a cantilevered beam including static deflection’, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shyu, I.M.K., Mook, D.T. & Plaut, R.H. Whirling of a forced cantilevered beam with static deflection. I: Primary resonance. Nonlinear Dyn 4, 227–249 (1993). https://doi.org/10.1007/BF00046322
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00046322