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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Skutch, R., 1897, “Uber die Bewegungeines Gespannten Fadens (On the Motion of a Tensioned String),” Ann. Phys. Chem., 61, pp. 190–195.
Sack, R. A., 1954, “Transverse Oscillations in Traveling Strings,” Br. J. Appl. Phys., 5, pp. 224–226. [CrossRef]
Wickert, J. A., and Mote, C. D., 1988, “Current Research on the Vibration and Stability of Axially-Moving Materials,” Shock Vib. Dig., 20(5), pp. 3–13. [CrossRef]
Wickert, J. A., and Mote, C. D., 1988, “On the Energetics of Axially Moving Continua,” J. Acoust. Soc. Am., 85(3), pp. 1365–1368. [CrossRef]
Wickert, J. A., and Mote, C. D., 1990, “Classical Vibration Analysis of Axially Moving Continua,” ASME J. Appl. Mech., 57, pp. 738–744. [CrossRef]
Wickert, J. A., and Mote, C. D., 1991, “Response and Discretization Methods for Axially Moving Materials,” ASME Appl. Mech. Rev., 44(11), pp. S279–S284. [CrossRef]
Wickert, J. A., 1991, “Travelling Load Response of an Axially Moving String,” J. Sound Vib., 149(2), pp. 267–284. [CrossRef]
Wickert, J. A., and Mote, C. D., 1988, “Linear Transverse Vibration of an Axially Moving String–Particle System,” J. Acoust. Soc. Am., 84(3), pp. 963–969. [CrossRef]
Hwang, S. J., Perkins, N. C., Ulsoy, A. G., and Meckstroth, R. J., 1994, “Rotational Response and Slip Prediction of Serpentine Belt Drive Systems,” ASME J. Vibr. Acoust., 116, pp. 71–78. [CrossRef]
Beikmann, R. S., Perkins, N. C., and Ulsoy, A. G., 1996, “Free Vibration of Serpentine Belt Drive Systems,” ASME J. Vibr. Acoust., 118, pp. 406–413. [CrossRef]
Beikmann, R. S., Perkins, N. C., and Ulsoy, A. G., 1996, “Nonlinear Coupled Vibration Response of Serpentine Belt Drive Systems,” ASME J. Vibr. Acoust., 118, pp. 567–574. [CrossRef]
Zhang, L., and Zu, J. W., 1999, “Modal Analysis of Serpentine Belt Drive System,” J. Sound Vib., 222(2), pp. 259–279. [CrossRef]
Zhang, L., Zu, J. W., and Hou, Z., 2001, “Complex Modal Analysis of Non-Self-Adjoint Hybrid Serpentine Belt Drive Systems,” ASME J. Vibr. Acoust., 123, pp. 150–156. [CrossRef]
Brake, M. R., and Wickert, J. A., 2010, “Modal Analysis of Continuous Gyroscopic Second-Order System With Nonlinear Constraints,” J. Sound Vib., 329, pp. 893–911. [CrossRef]
Kong, L., and Parker, R. G., 2003, “Equilibrium and Belt-Pulley Vibration Coupling in Serpentine Belt Drives,” ASME J. Appl. Mech., 70, pp. 739–750. [CrossRef]
Kong, L., and Parker, R. G., 2004, “Coupled Belt-Pulley Vibration in Serpentine Drives With Belt Bending Stiffness,” ASME J. Appl. Mech., 71, pp. 109–119. [CrossRef]
Lin, Y., DePauw, T., and Jiang, Y., 1997, “Analysis of the Dynamic Effects of an Elastic Belt in a General Mechanical System,” Proceedings of the 12th European ADAMS User's Conference, Marburg, Germany, November 18–19, pp. 1–9.
Hashimoto, Y., 2007, “Finite Element Vibration Analysis of Axially Moving Belt,” Trans. Jpn. Soc. Mech. Eng., Ser. C, 73, pp. 655–660. [CrossRef]
Amati, N., Tonoli, A., and Zenerino, E., 2005, “Dinamica Flessionale di Sistemi Continui Traslanti: Modellazione Agli Elementi Finiti (Flexural Dynamics of Continuum Translating Systems: Finite Element Modeling),” Proceedings of the 34th Convegno Nazionale AIAS, Milano, Italy, September 14–17, pp. 1–10.
Zen, G., and Muftu, G., 2006, “Stability of an Axially Accelerating String Subjected to Frictional Guiding Forces,” J. Sound Vib., 289, pp. 551–576. [CrossRef]
Przemieniecki, J. S., 1968, Theory of Matrix Structural Analysis, McGraw-Hill, New York.
Genta, G., 1999, Vibration of Structures and Machines: Practical Aspects, Springer, New York.
BelingardiG., 1995, Il Metodo Degli Elementi Finiti Nella Progettazione Meccanica (The Finite Elemet Method in Machine Design), Levrotto and Bella, Torino.
Genta, G., and Tonoli, A., 1996, “A Harmonic Finite Element for the Analysis of Flexural, Torsional and Axial Rotordynamic Behaviour of Discs,” J. Sound Vib., 196, pp. 13–43. [CrossRef]
Perkins, N. C., and Mote, C. D., 1986, “Comments on the Curve Veering in Engineering Problems,” J. Sound Vib., 106, pp. 451–463. [CrossRef]
Amati, N., Tonoli, A., and Zenerino, E., 2006, “Dynamic Modeling of Belt Drive Systems: Effects of the Shear Deformations,” ASME J. Vibr. Acoust., 128, pp. 555–567. [CrossRef]


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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. 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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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. 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. 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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