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Research Papers

Isochronous Beams by an Inclined Roller Support

[+] Author and Article Information
Stefano Lenci

Department of Civil and
Building Engineering, and Architecture,
Polytechnic University of Marche,
Ancona 60131, Italy
e-mail: lenci@univpm.it

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 23, 2018; final manuscript received May 25, 2018; published online June 18, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 85(9), 091008 (Jun 18, 2018) (11 pages) Paper No: JAM-18-1116; doi: 10.1115/1.4040453 History: Received February 23, 2018; Revised May 25, 2018

The paper addresses the problem of isochronous beams, namely those that oscillate with a frequency that is independent of the amplitude also in the nonlinear regime. The mechanism adopted to obtain this goal is that of having, as a boundary condition, a roller that can slide on a given path. A geometrically exact Euler–Bernoulli formulation is considered, and the nonlinear analysis is done by the multiple time scale method, that is applied directly to the partial differential equations governing the motion without an a priori spatial reduction. The analytical expression of the backbone curve is obtained, up to the third-order, and its dependence on the roller path is addressed. Conditions for having a straight backbone curve, i.e., the isochronous beam, are determined explicitly. As a by-product of the main result, the free and forced nonlinear oscillations of a beam with an inclined support sliding on an arbitrary path have been investigated.

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References

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Lenci, S. , and Clementi, F. , 2018, “ Axial-Transversal Coupling in the Nonlinear Dynamics of a Beam With an Inclined Roller,” (in press).
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Figures

Grahic Jump Location
Fig. 1

The considered mechanical model

Grahic Jump Location
Fig. 4

The coefficients γi for λ = 150. Black = first mode, red = second mode, blue = third mode, green = fourth mode. (Colors in the online version).

Grahic Jump Location
Fig. 2

The functions ξ(α) (β=tan(α)) for λ = 50 (dash line), λ = 100 (solid line), λ = 150 (dashdot line) and λ = 200 (dot line)

Grahic Jump Location
Fig. 6

The coefficients δi/δ7 for λ = 100. Black = first mode, red = second mode, blue = third mode, green = fourth mode. (Colors in the online version).

Grahic Jump Location
Fig. 3

The coefficients γi for λ = 100. Black = first mode, red = second mode, blue = third mode, green = fourth mode. (Colors in the online version).

Grahic Jump Location
Fig. 5

Examples of backbone curves for η = 200 (left, softening), η = 0 (central and vertical, isochronous) and η = −200 (right, hardening).

Grahic Jump Location
Fig. 7

The function coefficients β2(β1) given by Eq. (47) for α = π/4 (β = 1), λ = 100. (a) Large view, (b) zoom. Black = first mode, red = second mode, blue = third mode, green = fourth mode. (Colors in the online version).

Grahic Jump Location
Fig. 8

The path r(d) providing the isochronous beam for β1 = –100 (dot line), β1 = 0 (continuous line) and β1 = 200 (dash line). (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4. α = π/4 (β = 1) and λ = 100.

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