Stability Analysis of a Spinning and Precessing Viscoelastic Rotor Model
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IGINAL CONTRIBUTION
Stability Analysis of a Spinning and Precessing Viscoelastic Rotor Model S. Bose • A. Nandi • S. Neogy
Received: 3 November 2012 / Accepted: 30 September 2013 / Published online: 7 November 2013 Ó The Institution of Engineers (India) 2013
Abstract The present work deals with stability analysis of a spinning and precessing gyroscopic systems, where the spin axis and precession axis intersect at right angle. The nutation speed is zero, the spin and precession speeds are considered to be uniform and the precession axis is located at one end of the shaft. The properties of the shaft material correspond to a four element type linear viscoelastic model. The shaft disk system is assumed to be axially and torsionally stiff. For analysis, a simple rotor has been considered with the rigid disk placed on a massless viscoelastic shaft at specified locations from one end of the shaft. The governing parametric equations for such a rotor are derived in the simultaneously spinning and precessing frame. A stability analysis is performed considering both two- and four-degree of freedom models. The stability borderlines are computed considering spin and precession speeds as parameters. It is shown that though viscoelastic material may appear attractive for its large material damping, for gyroscopic systems it may lead to unstable vibrations. Keywords Precession Stiffness matrix Viscoelastic Loss coefficient Storage modulus Gyroscopic effect
S. Bose (&) A. Nandi S. Neogy Mechanical Engineering Department, Jadavpur University, Kolkata 700032, India e-mail: [email protected] A. Nandi e-mail: [email protected]
Introduction Rotor on maneuvering base has presently become a topic of considerable interest. Lin and Meng [1] reported on dynamics of an elastic rotor on a maneuvering support. They considered a two-degree of freedom accelerating Jeffcott rotor model primarily in a pitching aircraft. Lai and Chow [2] analyzed a rotating thin elastic pipe carrying a flowing fluid. They reported loss of stability of the pipe with rotation. Transverse vibrations and stability analysis of flexible rotor shaft systems mounted on a moving vehicle like a ship or aircraft was analyzed by Das et al. [3, 4]. Noise and vibration control is a major concern in several industries such as aerospace and automobiles. The reduction of noise and vibration is a major requirement for performance and customer satisfaction. Passive damping technology using viscoelastic materials is traditionally used to control vibrations of structures [5]. The growing use of such structures has motivated many authors to study stability analysis of viscoelastic spinning rotor models [6]. Spinning and precessing motion of a rotor with a constant nutation angle can be encountered in practice in a motor-shaft-disk system on a turntable, a helicopter rotor in pitching flight, a special purpose machine tool etc. It is known that a simultaneously spinning and precessing elastic rotor becomes unstable beyond a certain precession speed. This was demonstra
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