Abstract
The non-conservative instability of a deep cantilever beam subjected to a lateral force with partial distribution has been verified. The governing equations have been derived using the extended Hamilton’s principle, and the Galerkin method has been implemented to approximate the response of system. The influence of system parameters like mass centroid offset, radius of gyration, fundamental frequencies ratio, load distribution model, and the added effect of a free stream with chord-wise velocity has been examined on the instability boundaries of the beam. In addition, the validity of the proposed model has been corroborated in comparison with the available results in the literature.
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Abbreviations
- A :
-
Beam cross sectional area
- b :
-
Beam semi-width
- c :
-
Mass centroid offset from elastic axis
- E :
-
Young’s modulus
- G :
-
Shear modulus
- H(x):
-
Heaviside step function
- \({I_{y}'}\) :
-
Beam’s moment of inertia from y′ axis
- \({I_{z}'}\) :
-
Beam’s moment of inertia from z′ axis
- J :
-
Torsional rigidity
- i, j, k :
-
Unit vectors associated with undeformed beam coordinate system
- \({{\bf i'}, {\bf j'}, {\bf k'}}\) :
-
Unit vectors associated with deformed beam coordinate system
- \({k_{m}}\) :
-
Radius of gyration
- l :
-
Beam length
- \({M_{z'}}\) :
-
Bending moment about the z′ axis
- \({\bar{p}}\) :
-
Intensity of the distributed follower force
- t :
-
Time
- T :
-
Kinetic energy
- u, v, w :
-
Displacements in the x, y, z directions, respectively
- U :
-
Strain energy
- \({U_{\infty}}\) :
-
Free stream velocity
- x, y, z :
-
Mutually perpendicular axis system with x along the undeformed beam
- \({\delta( )}\) :
-
Variational operator
- \({\delta W}\) :
-
Virtual work of the non-conservative forces
- \({\varepsilon_{xx} \quad \varepsilon_{xy}\quad \varepsilon_{xz}}\) :
-
Engineering strains
- \({\sigma_{xx}\quad \sigma_{xy}\quad \sigma_{xz}}\) :
-
Engineering stresses
- \({\omega_\theta}\) :
-
Fundamental torsion frequency
- \({\rho_\infty}\) :
-
Free stream density
- \({\rho}\) :
-
Beam density
- \({\theta}\) :
-
Twist about elastic axis
- \({\kappa_{z'}}\) :
-
Bending curvature about the z′ axis
- \({\left(\;\right)^{\prime}}\) :
-
Derivative with respect to x
- \({(\,^{{\cdot}}\,)}\) :
-
Derivative with respect to t
References
Beck, M.: Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes [The buckling load of the cantilevered, tangentially compressed rod]. ZAMP 3, 225–228 (1952)
Como, M.: Lateral buckling of a cantilever subjected to a transverse follower force. Int. J. Solids Struct. 2, 515–523 (1966)
Ziegler, H.: Principles of Structural Stability. Blaisdell, Waltham (1968)
Leipholz, H.: Die Knicklast des einseitig eingespannten Stabes mit gleichmässig verteilter, tangentialer Längsbelastung [The buckling load of the cantilevered rod with uniformly distributed, tangential longitudinal stress]. ZAMP 13, 581–589 (1962)
Bolotin, V.V.: Non-conservative Problems of Theory of Elastic Stability. Pergamon Press, Oxford (1963)
Simitses, G.J., Hodges, D.H.: Fundamentals of Structural Stability. Elsevier, Burlington (2006)
Langthjem, M.A., Sugiyama, Y.: Dynamic stability of columns subjected to follower loads: a survey. J. Sound Vib. 238, 809–851 (2000)
Hodges, D.H.: Lateral-torsional flutter of a deep cantilever loaded by a lateral follower force at the tip. J. Sound Vib. 247, 175–183 (2001)
Fazelzadeh, S.A., Mazidi, A., Kalantari, H.: Bending-torsional flutter of wings with an attached mass subjected to a follower force. J. Sound Vib. 323, 148–162 (2009)
Feldt, W.T., Herrmann, G.: Bending-torsional flutter of a cantilevered wing containing a tip mass and subjected to a transverse follower force. J. Frankl. I. 297, 467–478 (1974)
Hodges, D.H., Patil, M.J., Chae, S.: Effect of thrust on bending-torsion flutter of wings. J. Aircr. 39, 371–376 (2002)
Mazidi, A., Kalantari, H., Fazelzadeh, S.A.: Aeroelastic response of an aircraft wing with mounted engine subjected to time-dependent thrust. J. Fluids Struct. 39, 292–305 (2013)
Mardanpour, P., Richards, P.W., Nabipour, O., Hodges, D.H.: Effect of multiple engine placement on aeroelastic trim and stability of flying wing aircraft. J. Fluids Struct. 44, 67–86 (2013)
Fazelzadeh, S.A., Kazemi-Lari, M.A.: Stability analysis of partially loaded Leipholz column carrying a lumped mass and resting on elastic foundation. J. Sound Vib. 332, 595–607 (2013)
Fazelzadeh, S.A., Kazemi-Lari, M.A.: Stability analysis of a deep cantilever beam with laterally distributed follower force. J. Eng. Mech. ASCE 140(10), 04014074 (2014)
Anderson, G.L.: The influence of rotatory inertia, tip mass, and damping on the stability of a cantilever beam on an elastic foundation. J. Sound Vib. 43, 543–552 (1975)
Peters, D.A., Karunamoorthy, S., Cao, W.M.: Finite state induced flow models. Part I: Two-dimensional thin airfoil. J. Aircr. 32, 313–322 (1995)
Qin, Z., Librescu, L.: Aeroelastic instability of aircraft wings modelled as anisotropic composite thin-walled beams in incompressible flow. J. Fluids Struct. 18, 43–61 (2003)
Bisplinghoff, R.L., Ashley, H., Halfman, R.L.: Aeroelasticity. Addison Wesley, Cambridge (1955)
Fletcher, C.A.J.: Computational Galerkin Methods. Springer, New York (1984)
Seyranian, A.P., Mailybaev, A.A.: Multiparameter Stability Theory with Mechanical Applications. World Scientific, Singapore (2003)
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Kazemi-Lari, M.A., Fazelzadeh, S.A. Flexural-torsional flutter analysis of a deep cantilever beam subjected to a partially distributed lateral force. Acta Mech 226, 1379–1393 (2015). https://doi.org/10.1007/s00707-014-1258-2
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DOI: https://doi.org/10.1007/s00707-014-1258-2