A. Galip Ulsoy Professor. Fellow ASME Christophe Pierre Professor. Mem. ASME Suhyun Choi Graduate Student. Department of Mactianical Engineering and Applied Mecfianics University of Michigan, Ann Arbor, Ml 48109-2125 Vibration Localization in Rotating Shafts, Part 1: Theory This paper presents a theoretical study of vibration localization in single-span, flexi- ble, rotating shafts. A noncircular cross-section of the rotating shaft, leading to dissimilar lateral moments of inertia, can introduce disorder. Internal coupling be- tween the principal directions of vibration is provided by the rotational speed through the gyroscopic moments. It is shown, both analytically and numerically, that direc- tional vibration localization can occur for certain appropriate combinations of disor- der and coupling. Introduction Shafts rotating about their longitudinal axes are employed for power transmission in industrial machines such as steam and gas turbines, turbogenerators, and internal combustion en- gines. The current trend toward higher speeds and flexibility increases the importance of examining the dynamics oi flexible rotating shafts in mechanical engineering design. The primary concern of the extensive previous research on the dynamics of flexible rotating shafts has been to identify the shaft's critical speeds (at which the angular velocity of the shaft coincides with one of the free vibration natural frequencies) and to investigate stability in inter-critical speed ranges (Dimentberg, 1961; Bo- lotin, 1963; Dimaragonas and Paipeties, 1983; Rao, 1983; Lee 1993). Recent research has shown that irregularities or disorder in nearly periodic structures may lead to free vibration modes which are localized to small geometric regions, and may confine the vibrational energy to the vicinity of the source of excitation. This phenomenon, known as mode localization, may be damag- ing, as it leads to larger amplitudes and consequently to larger stresses, or beneficial, as a means of passive vibration control. All previous research studies show that localization occurs in disordered periodic structures which feature, in some sense, weak internal coupling among their individual components (Ibrahim, 1987; Pierre, 1990). Although much research has been done on the dynamics of flexible rotating shafts and on the localization phenomenon in a number of nearly periodic structures, no attention has been paid to vibration localization in flexible rotating shafts. This paper investigates vibration localization in single-span rotating flexible shafts. A single-span rotating shaft is not a periodic structure in the usual sense, but there does exist symmetry in the cross section of a circular shaft. That symmetry is destroyed in a non-circular shaft, which features dissimilar lateral moments of inertia. Sev- eral researchers have investigated the dynamics of unsymmetric rotors (Crandafl and Brosens, 1961; Genta, 1988; Lee, 1993), however, no one has interpreted such results form the viewpoint of vibration localization. This asymmetry may lead to modes of vibration with bending motion which is primarily confined in one of the two orthogonal principal directions of the cross section. Thus, vibrational energy can be localized, not to a substructure, but to a preferred principal direction. This direc- tional localization of vibration can result in large vibration am- Contributed by the Technical Commitlee on Vibration and Sound for pubhca- tion in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 1995; revised Dec. 1995. Associate Technical Editor: K. W, Wang. plitudes and stresses. Furthermore, in rotating flexible shafts, the mechanism of internal coupling is gyroscopic rather than elastic. The primary objective of this paper is to investigate the mode localization phenomenon in flexible rotating shafts with slightly dissimilar lateral moments of inertia. In the next section, the modeling and analysis of the vibration of rotating shafts with dissimilar lateral moments of inertia is presented using a Ray- leigh beam model. Subsequently, the effects of the ratio of the two lateral moments of inertia, the rotational speed of the shaft, the mode number, and the aspect ratio on the degree of direc- tional vibration localization are investigated. Finally, experi- ments that confirm the existence of vibration localization in rotating shafts are reported, and discussed, in a companion paper (Part 2). Modeling and Analysis The free vibration eigenanalysis for a Rayleigh beam model of a rotating shaft with dissimilar lateral moments of inertia is presented. An exact solution can be easily obtained for a rotating shaft with simply-supported boundary conditions. However, for other boundary conditions, the analysis becomes relatively com- plex, thus, a Galerkin approximation is preferred (Lee et al, 1988). Since the localization of the mode shapes is of primary interest, perturbation methods which yield insights into the oc- currence of the localization phenomenon are also presented (PieiTc, 1988). The presence of irregularities, or disorder, in periodic struc- tures may localize the modes of free vibration (Pierre and Dow- ell, 1987; Pierre, 1988). Although a single-span rotating shaft is not a periodic structure in the usual sense, there does exist a symmetry in the cross section of the circular shaft (i.e., /i = /2). In a noncircular shaft, which has disimilar lateral moments of inertia (i.e., /] =t^ h), that symmetry is destroyed. As will be seen later, this asymmetry may lead to modes of vibration which feature bending motion primarily in one of the two orthogonal principal directions of the cross section. We shall refer to this phenomenon as directional localization of the modes. For a general disordered periodic structure, the degree of localization of the modes typically depends upon the ratio of the disorder to the internal coupling. In a single-span rotating shaft with dissimilar lateral moments of inertia, the disorder term comes from the difference between the two lateral moments of inertia. The coupling between the dynamics in the two orthogonal direc- tions arises from the gyroscopic moments, which depend on the polar moments of inertia, the aspect ratio (the ratio of the radius of gyration to the shaft length) and the rotational speed of the 138 / Vol. 120, JANUARY 1998 Copyright © 1998 by ASME Transactions of the ASME DownloadedFrom:http://vibrationacoustics.asmedigitalcollection.asme.org/on11/21/2014TermsofUse:http://asme.org/terms