Research Papers

Modeling the Flexural Dynamic Behavior of Axially Moving Continua by Using the Finite Element Method

[+] Author and Article Information
Andrea Tonoli

e-mail: andrea.tonoli@polito.it

Enrico Zenerino

e-mail: enrico.zenerino@polito.it

Nicola Amati

e-mail: nicola.amati@polito.it
Department of Mechanical and
Aerospace Engineering,
Politecnico di Torino,
corso Duca degli Abruzzi 24,
Torino 10129, Italy

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 10, 2012; final manuscript received September 21, 2013; published online October 24, 2013. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 136(1), 011012 (Oct 24, 2013) (8 pages) Paper No: VIB-12-1285; doi: 10.1115/1.4025551 History: Received October 10, 2012; Revised September 21, 2013

Mechanical systems including conveyor belts, band saw blades, and power transmission belts are influenced by the lateral motion of the moving structure. This phenomenon was studied in the literature both using the theory of the continuous linear and nonlinear systems and following the multibody technique. The subject is studied by using the finite element method (FEM) validated with reference to the analytical models described in the literature. The contributions of the Coriolis forces, the negative stiffness linked to the transport speed, and the bending stiffness due to the transverse moment of inertia are discussed. The dynamic behavior of a prototypical belt transmission layout with two fixed pulleys and an automatic tensioner is then analyzed. The results show the effect of the transport speed on the reduction of the flexural natural frequencies of the mode shapes strictly related to the lateral motion of the belt span and evidence the design strategy that needs to be followed for a correct operation of the whole system.

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Fig. 1

Description of the generalized displacements

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Fig. 2

Scheme of a span in the inertial reference frame XY. The degrees of freedom at each node are indicated.

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Fig. 3

First two transverse natural frequencies as function of the transport speed U. FEM model (Itt ≠ 0)—dashdot curve, FEM model (Itt = 0)—circle marks, and Sack [2] analytical model—solid line. Data of Table 1, L = 0.3 m, Fa = 128.7 N.

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Fig. 4

Focus of the eigenvalues in the region where an unstable behavior occurs. (a) Imaginary part of the first two eigenvalues. (b) Real part of the eigenvalues describing an unstable behavior. Data of Table 1, L = 0.3 m, Fa = 128.7 N.

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Fig. 5

Effect of the Coriolis forces on the natural frequencies and on the stability. (a) First two natural frequencies obtained by the complete model (full line) and neglecting the Coriolis forces (G = 0 (dash dot line). (b) Eigenvalue real part obtained by the complete model (full line) and neglecting the Coriolis forces (dash dot line). Data of Table 1, L = 0.3 m, Fa = 128.7 N.

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Fig. 6

Scheme of the belt transmission system. P1 = motor pulley, P2 = automatic tensioner pulley, P4 = driven pulley, J3 = moment of inertia of tensioner arm, Kr = rotational stiffness of tensioner arm, Sk = the kth belt span, and Ψ1,2 = alignment angles between the tensioner arm motion and adjacent belt spans. A single node is considered at the center of each belt span for the sake of simplicity.

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Fig. 7

Campbell diagram for the system described in Fig. 4 considering the frequency range 0–400 Hz. Full line: natural frequencies computed with the proposed FE model. Dashed line with circle marks: first seven natural frequencies computed using the model presented in Ref. [12].

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Fig. 8

Campbell diagram in the speed range of interest

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Fig. 9

Zoom of Fig. 8 in the frequency and rotational speed ranges that highlight the veering phenomena




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