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Research Papers

Influence of Geometric Nonlinearities on the Asynchronous Modes of an Articulated Prestressed Slender Structure

[+] Author and Article Information
Eduardo A. R. Ribeiro

Professor
Escola Politécnica da Universidade de São Paulo,
São Paulo 05508010, Brazil
e-mail: asceduardo@usp.br

Carlos E. N. Mazzilli

Professor
Escola Politécnica da Universidade de São Paulo,
São Paulo 05508010, Brazil
e-mail: cenmazzi@usp.br

Stefano Lenci

Professor
Università Politecnica delle Marche,
Ancona 60131, Italy
e-mail: lenci@univpm.it

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 28, 2017; final manuscript received August 20, 2018; published online October 26, 2018. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 141(2), 021007 (Oct 26, 2018) (9 pages) Paper No: VIB-17-1432; doi: 10.1115/1.4041305 History: Received September 28, 2017; Revised August 20, 2018

Synchronous modal oscillations, characterized by unisonous motions for all physical coordinates, are well known. In turn, asynchronous oscillations lack a general definition to address all the associated features and implications. It might be thought, at first, that asynchronicity could be related to nonsimilar modes, which might be associated with phase differences between displacement and velocity fields. Due to such differences, the modes, although still periodic, might not be characterized by stationary waves so that physical coordinates might not attain their extreme values at the same instants of time, as in the case of synchronous modes. Yet, it seems that asynchronicity is more related to frequency rather than phase differences. A more promising line of thought associates asynchronous oscillations to different frequency contents over distinct parts of a system. That is the case when, in a vibration mode, part of the structure remains at rest, that is, with zero frequency, whereas other parts vibrate with non-null modal frequency. In such a scenario, localized oscillations would explain modal asynchronicity. When the system parameters are properly tuned, localization may appear even in very simple models, like Ziegler's columns, shear buildings, and slender structures. Now, the latter ones are recast, but finite rotations are assumed, in order to verify how nonlinearity affects existing linear asynchronous modes. For this purpose, the authors follow Shaw–Pierre's invariant manifold formulation. It is believed that full understanding of asynchronicity may apply to design of vibration controllers, microsensors, and energy-harvesting systems.

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Figures

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Fig. 1

3DOF linear model: reference configuration with geometric–mechanical properties and applied forces (left); deformed configuration with physical coordinates (right)

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Fig. 2

3DOF nonlinear model: reference configuration with geometric-mechanical properties and applied forces (left); deformed configuration with physical coordinates (right)

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Fig. 3

Linear modes, with α, σ and θ defined: (a) linear modes for θ1 = 0:038179, (b) linear modes for θ2 = 0:063886, and (c) linear modes for θ3 = 0:177935

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Fig. 4

q3 invariant manifold for a D nonlinear mode given by θ1

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Fig. 5

Projection of q3 in function of u and v for a D nonlinear mode given by θ1

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Fig. 6

Frequency–amplitude relationship for a D nonlinear mode given by θ1

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Fig. 7

q3 invariant manifold for a D nonlinear mode given by θ2

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Fig. 8

Projection of q3 in function of u and v for a D nonlinear mode given by θ2

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Fig. 9

Frequency–amplitude relationship for a D nonlinear mode given by θ2

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Fig. 10

q1 invariant manifold for a nearly C nonlinear mode given by θ2

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Fig. 11

q2 invariant manifold for a nearly C nonlinear mode given by θ2

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Fig. 12

q2 invariant manifold for a D nonlinear mode given by θ3

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Fig. 13

q3 invariant manifold for a D nonlinear mode given byθ3

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Fig. 14

Frequency–amplitude relationship for a D nonlinear mode given by θ3

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Fig. 15

q1 invariant manifold for a nearly E nonlinear mode given by θ3

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