Summary
Free undamped in-plane vibrations of shear undeformable beams around their highly buckled configurations are investigated neglecting rotary inertia effects. The beams are inertially nonuniform since a lumped mass is rigidly clamped along the span. Two mechanical models are considered depending on the boundary conditions in the post-buckling phases. First, the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end, a roller support, both in the buckling and post-buckling phases. In the second model, the beam is extensible in the post-buckling phase because the roller support boundary is changed into a fixed hinged end. Free undamped vibrations are governed, in the first case, by a homogeneous integral-partial-differential equation and, in the second case, by two coupled partial-differential equations with variable coefficients. The solutions of the associated eigenvalue problems are found employing two approaches: a semi-analytical method based on Galerkin discretization and a finite element method. A close agreement in the outcomes is found. The leading differences relating to the natural frequencies and linear normal modes of the two pre-stressed curved beam models are discussed; in particular, the occurrence of the veering phenomenon and the crossovers are outlined.
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References
J. P. Den Hartog (1928) ArticleTitleThe lowest frequency of circular arcs Phil. Mag. 5 400–408 Occurrence Handle54.0856
S. S. Rao V. Sundararajan (1969) ArticleTitleIn-plane flexural vibrations of circular rings J. Appl. Mech. 36 620–625
E. Tufekci A. Arpaci (1998) ArticleTitleExact solution of in-plane vibrations of circular arches with account taken of axial extension, transverse shear and rotatory inertia effects J. Sound Vibr. 209 845–856
P. Chidamparam A. W. Leissa (1995) ArticleTitleInfluence of centerline extensibility on the in-plane free vibrations of loaded circular arches J. Sound Vibr. 183 779–795 Occurrence Handle10.1006/jsvi.1995.0286 Occurrence Handle1055.74528
S. T. Mau A. N. Williams (1988) ArticleTitleGreen's function solution for arch vibration J. Eng. Mech. 114 1259–1264
M. Petyt C. C. Fleischer (1971) ArticleTitleFree vibration of a curved beam J. Sound Vibr. 18 17–30 Occurrence Handle10.1016/0022-460X(71)90627-4 Occurrence Handle0233.73153
G. Prathap (1985) ArticleTitleThe curved beam/deep arch/finite ring element revisited Int. J. Numer. Meth. Engng. 21 389–497 Occurrence Handle0559.73078
S. Y. Yang H. C. Sin (1995) ArticleTitleCurvature-based beam elements for the analysis of Timoshenko and shear-deformable curved beams J. Sound Vibr. 187 569–584
K. Grosh P. M. Pinsky (1996) ArticleTitleIn-plane vibration and stability of shallow circular arches subjected to axial forces Comput. Meth. Appl. Mech. Engng. 132 1–16 Occurrence Handle10.1016/0045-7825(96)01002-X Occurrence Handle0890.73065
K. J. Kang C. W. Bert A. G. Striz (1996) ArticleTitleVibration and buckling analysis of circular arches using DQM Comput. Struct. 60 49–57 Occurrence Handle10.1016/0045-7949(95)00375-4 Occurrence Handle0918.73354
K. Matsunaga (1996) ArticleTitleIn-plane vibration and stability of shallow circular arches subjected to axial forces Int. J. Solids Struct. 33 469–482 Occurrence Handle0929.74039
S.-J. Oh B. K. Lee I.-W. Lee (2000) ArticleTitleFree vibrations of non-circular arches with nonuniform cross-section Int. J. Solids Struct. 37 4871–4891 Occurrence Handle0968.74034
N. C. Perkins (1990) ArticleTitlePlanar vibration of an elastica arch: theory and experiment J. Vibr. Acoust. 112 374–379
S.-J. Hwang N. C. Perkins (1992) ArticleTitleSupercritical stability of an axially moving beam, part I: model and equilibrium analysis J. Sound Vibr. 154 IssueID3 381–396 Occurrence Handle0924.73105
S.-J. Hwang N. C. Perkins (1992) ArticleTitleSupercritical stability of an axially moving beam, part II: vibration and stability analyses J. Sound Vibr. 154 IssueID3 397–409
S.-J. Hwang N. C. Perkins (1994) ArticleTitleHigh speed stability of coupled band/wheel systems: theory and experiment J. Sound Vibr. 169 IssueID4 459–483 Occurrence Handle10.1006/jsvi.1994.1029
Mettler, E.: Dynamic buckling. In: Handbook of engineering mechanics (Flugge, W., ed.). New York: McGraw-Hill 1962.
A. H. Nayfeh W. Kreider T. J. Anderson (1995) ArticleTitleInvestigation of natural frequencies and mode shapes of buckled beams AIAA J. 33 1121–1126 Occurrence Handle10.2514/3.12669 Occurrence Handle0845.73041
W. Lacarbonara A. H. Nayfeh W. Kreider (1998) ArticleTitleExperimental validation of reduction methods for weakly nonlinear distributed-parameter systems: Analysis of a buckled beam Nonlin. Dyn. 17 95–117 Occurrence Handle0968.74503
P. Villaggio (1997) Mathematical models for elastic structures Cambridge University Press Cambridge
Lacarbonara, W., Paolone, A., Yabuno, H.: Parametric resonances of planar pre-stressed curved beams. In: ICIAM, 5th Int. Congress on Industrial and Applied Mathematics, Sydney, Australia, July 7–11 2003.
Lacarbonara, W., Yabuno, H., Okhuma, M.: An experimental investigation of the parametric resonance in a buckled beam. In: 19th Biennial ASME Conf. on Mechanical Vibration and Noise, USA 2003.
J. Winterflood T. Barber B. Slagmolen (2002) ArticleTitleHigh performance vibration isolation using spring in Euler column buckling mode Phys. Lett. A 300 122–130
L. N. Virgin R. B. Davis (2003) ArticleTitleVibration isolation using buckled structures J. Sound Vibr. 260 965–973 Occurrence Handle10.1016/S0022-460X(02)01177-X
Jiang, J., Mockensturm, E.: A novel motion amplifier using an axially driven buckling beam. In: ASME Int. Mechanical Engineering Congress and Exposition, USA 2003.
W. Lacarbonara A. Paolone H. Yabuno (2004) ArticleTitleModeling of planar non-shallow prestressed beams towards asymptotic solutions Mech. Res. Commun. 31 301–310 Occurrence Handle10.1016/j.mechrescom.2003.11.004 Occurrence Handle1079.74568
Addessi, D., Lacarbonara, W., Paolone, A.: Linear vibrations of planar prestressed elastica arches. In: 45th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf. Palm Springs, California, April 19–22, 2004.
D. Addessi W. Lacarbonara A. Paolone (2005) ArticleTitleOn the linear normal modes of planar pre-stressed curved beams J. Sound Vibr. 284 IssueID3–5 1075–1097
S. P. Timoshenko J. M. Gere (1961) Theory of elastic stability McGraw-Hill New York
B. A. Finlayson (1972) The method of weighted residuals and variational principles Academic Press New York Occurrence Handle0319.49020
J. C. Simo L. Vu-Quoc (1986) ArticleTitleA three-dimensional finite strain rod model. Part II: Geometric and computational aspects Comput. Meth. Appl. Mech. Engng. 58 79–116 Occurrence Handle10.1016/0045-7825(86)90079-4 Occurrence Handle0608.73070
J. C. Simo L. Vu-Quoc (1988) ArticleTitleOn the dynamics in space of rods undergoing large motions–a geometrically exact approach Comput. Meth. Appl. Mech. Engng. 66 125–161 Occurrence Handle927417 Occurrence Handle0618.73100
Taylor, R. L.: FEAP–A finite element analysis program, version 7.4. Berkeley, CA: Department of Civil and Environmental Engineering, University of California 2002.
A. W. Leissa (1974) ArticleTitleOn a curve veering aberation Z. Angew. Math. Phys. 25 99–111 Occurrence Handle0293.65083 Occurrence Handle10.1007/BF01602113
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Addessi, D., Lacarbonara, W. & Paolone, A. Free in-plane vibrations of highly buckled beams carrying a lumped mass. Acta Mechanica 180, 133–156 (2005). https://doi.org/10.1007/s00707-005-0259-6
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DOI: https://doi.org/10.1007/s00707-005-0259-6