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MODELING APPLICATIONS

Bond Graph Modeling of an Internally Damped Nonideal Flexible Spinning Shaft

[+] Author and Article Information
A. K. Samantaray1

Department of Mechanical Engineering, Systems, Dynamics, and Controls Laboratory, Indian Institute of Technology, Kharagpur 721302, Indiasamantaray@lycos.com

S. S. Dasgupta, R. Bhattacharyya

Department of Mechanical Engineering, Systems, Dynamics, and Controls Laboratory, Indian Institute of Technology, Kharagpur 721302, India

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(6), 061502 (Oct 29, 2010) (9 pages) doi:10.1115/1.4002483 History: Received January 21, 2009; Revised November 13, 2009; Published October 29, 2010; Online October 29, 2010

The rotating internal damping or nonconservative circulatory force in a rotor shaft system causes instability beyond a certain threshold rotor spinning speed. However, if the source loading of the drive is considered, then the rotor spin is entrained at the stability threshold and a stable whirl orbit is observed about the unstable equilibrium. As we move toward the use of more and more lightweight rotor dynamic components such as the shaft and the motor, overlooking this frequency entrainment phenomenon while sizing the actuator in the design stage may lead to undesirable performance. This applies to many emerging areas of strategic importance such as in vivo medical robots where flexible probes are used and space robotics applications involving rotating tools. We analyze this spin entrainment phenomenon in a distributed parameter model of a spinning shaft, which is driven by a nonideal dc motor. A drive whose dynamics is influenced by the dynamics of the driven system is called a nonideal source and the whole system is referred to as a nonideal system. In particular, we show the advantages of representing such nonideal drive-system interactions in a modular manner through bond graph modeling as compared to standard equation models where the energetic couplings between dynamic variables are not explicitly shown. The developed modular bond graph model can be extended to include rotor disks and bearings placed at different locations on the shaft. Moreover, the power conserving property of the junction structure of the bond graph model is exploited to derive the source loading expressions, which are then used to analytically derive the steady-state spinning frequency and whirl orbit amplitude as functions of the drive and the rotor system parameters. We show that the higher transverse modes may become unstable before the lower ones under certain parametric conditions. The shaft spinning speed is entrained at the lowest stability threshold among all transverse modes. The bond graph model is used for numerical simulation of the system to validate the steady-state results obtained from the theoretical study.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic diagram of a flexible shaft with internal and external damping, which is driven by a dc motor

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Figure 2

Forces and moments acting in the x-y plane on a small segment of the shaft

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Figure 3

Bond graph submodel for a segment of a flexible spinning shaft

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Figure 4

Integrated bond graph model of the rotor-motor system

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Figure 5

Transient response of the system showing convergence to the predicted steady-state values

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Figure 6

Steady-state transverse vibrations showing whirl frequency and amplitude

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