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Research Papers

Vibration of Spinning Cantilever Beams With an Attached Rigid Body Undergoing Bending-Bending-Torsional-Axial Motions

[+] Author and Article Information
Christopher G. Cooley

University of Michigan-Shanghai,
Jiao Tong University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: cooley.168@osu.edu

Robert G. Parker

L. S. Randolph Professor and Head
Department of Mechanical Engineering,
Virginia Tech,
Blacksburg, VA 24061

1Corresponding author.

Manuscript received December 22, 2012; final manuscript received October 1, 2013; accepted manuscript posted October 22, 2013; published online December 10, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 81(5), 051002 (Dec 10, 2013) (11 pages) Paper No: JAM-12-1569; doi: 10.1115/1.4025791 History: Received December 22, 2012; Revised October 01, 2013; Accepted October 22, 2013

A linear model for the bending-bending-torsional-axial vibration of a spinning cantilever beam with a rigid body attached at its free end is derived using Hamilton's principle. The rotation axis is perpendicular to the beam (as for a helicopter blade, for example). The equations split into two uncoupled groups: coupled bending in the direction of the rotation axis with torsional motions and coupled bending in the plane of rotation with axial motions. Comparisons are made to existing models in the literature and some models are corrected. The practically important first case is examined in detail. The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system and Galerkin discretization is readily applied where it is not for the original problem. The natural frequencies, vibration modes, stability, and bending-torsion coupling are investigated, including comparisons with past research.

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Figures

Grahic Jump Location
Fig. 1

Schematic of a rotating cantilever beam with an attached rigid body at its free end

Grahic Jump Location
Fig. 2

Eigenvalue loci (the real parts vanish) for a uniform beam (a) with no tip mass (μ = 0), and (b) with a unit tip mass (μ = 1). The beam has γ = 0. The mass offset is d = 0 and the moment of inertia is ζij = 0. The solid lines denote the present model and the circles denote the results from Ref. [10] (the data is taken from Tables 3 and 9 in Ref. [10] for (a) and (b), respectively).

Grahic Jump Location
Fig. 3

(a) Eigenvalue loci (the real parts vanish) for a bending-torsion beam for varying nondimensional rotation speed. The system is from Refs. [1,2] with the parameters in Table 1 and a rectangular cross-section. (b) Low speed region with data from Ref. [1] (diamonds) and from Ref. [2] (circles). The eigenvalue loci including the stiffness contribution from tension N(x3) in Eq. (22) are shown by solid (black) lines. The dashed (red) lines denote eigenvalue loci excluding the stiffness contribution from tension N(x3) = 0.

Grahic Jump Location
Fig. 4

Imaginary (upper) and real (lower) parts of the eigenvalue loci for a bending-torsion beam for varying nondimensional rotation speed. The system is from Ref. [2] with the parameters in Table 1 and a rectangular cross-section. Results from Ref. [2] are shown by circle markers. The eigenvalue loci including the stiffness contribution from tension N(x3) in Eq. (22) are shown by solid (black) lines. Dashed (red) lines denote eigenvalue loci excluding the stiffness contribution from tension N(x3) = 0.

Grahic Jump Location
Fig. 5

Bending-torsion beam eigenvalue loci (the real parts vanish) for varying nondimensional rotation speed. The system has the parameters in Table 1 with a circular tube cross-section. Eigenvalue loci using the tension in Eq. (22) are shown by solid (black) lines and using the tension in Eq. (23) by dashed-dotted (blue) lines. VA denotes veering away of two nearby eigenvalues.

Grahic Jump Location
Fig. 6

Bending-torsion beam vibration modes at Ω = 5. The system has the parameters in Table 1 with a circular tube cross-section. The real part of the complex-valued mode shape is shown by a solid line and the imaginary part is shown by a dashed line. The rigid body at the free end of the beam is not shown.

Grahic Jump Location
Fig. 7

Bending-torsion beam vibration modes at Ω = 15. The system has the parameters in Table 1 with a circular tube cross-section. The real part of the complex-valued mode shape is shown by a solid line and the imaginary part is shown by a dashed line. The rigid body at the free end of the beam is not shown.

Grahic Jump Location
Fig. 8

Ratio of the bending strain energy to the total strain energy for varying speed in the first five bending-torsion beam modes. The system has the parameters in Table 1 with a circular tube cross-section. Modes 1–5 are shown by solid, dashed, dotted, dashed-dotted, and dashed-double-dotted lines, respectively.

Grahic Jump Location
Fig. 9

Imaginary (upper) and real (lower) parts of the bending-torsion beam eigenvalue loci for varying nondimensional rotation speed. The system has the parameters in Table 1 for a circular tube cross-section and no rigid body. Eigenvalue loci using the tension in Eq. (22) are shown by solid (black) lines and using the tension in Eq. (23) by dashed-dotted (blue) lines. VA denotes the veering away of two nearby eigenvalues.

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