Vibration in a Cracked Machine Tool Spindle with Magnetic Bearings

Due to manufacturing flaws or cyclic loading, cracks frequently appear in a rotating spindle system. These cracks markedly affect the dynamic characteristics in higher modes of the rotating machinery. For faster rotational speeds, especially for super-high-speed cutting, a spindle with magnetic bearings is necessary. However, most investigations into spindle system dynamic characteristics have been confined to ball-bearing-type spindles. The dynamic response of rotating cracked spindle systems with magnetic bearings is examined in this article. A Euler-Bernoulli beam of circular cross section is used to approximate the spindle and the Hamilton principle is employed to derive the equation of motion for the spindle system. The effects of crack depth, rotation speed and bearing length on the dynamic response of a rotating magnetic bearing spindle system are studied. INTRODUCTION Cracks frequently appear in rotating machinery due to manufacturing flaws or cyclic fatigue during operation. Numerous cracks can be observed after severe operating conditions, especially in high speed spindles [1, 2]. Local structural irregularities caused by cracks in the spindle may significantly change the dynamic behavior of a rotating machinery system. The effects of cracks on the dynamic and static behaviors of structures have been studied by a number of researchers [3-5]. The effects of cracks on spindle dynamics, shaft and rotor systems, were also studied by researchers [6-9]. When a spindle rotates, the vibrational response is altered by the crack opening and closing in each cycle. Most investigations were motivated by the hypothesis that only opening cracks markedly change the spindle dynamics. This paper focuses on the dynamics of a spindle with a transverse crack. High speed machining is one of the most modern manufacturing engineering technologies. In a machining system, the spindle is the most critical element that affects the dynamic performance and capabilities of the system in the machining process. However focusing exclusively on the spindle system is insufficient because the bearings can change the dynamics of a machining spindle system. Hence, the bearing effects on the spindle system must also be considered. Bearings are used in many rotating machines to brace the rotating spindles and rotors. In the past, the required rotor speed was low, allowing ball and roller bearings to be used in rotating machinery. High temperatures are generated with ball-bearing spindle systems operating at high speeds. The high temperatures often bring about machine failure. To attain greater complexity and accuracy, modern engineering technologies demand machinery that can be run at high speeds. To avoid the high temperatures generated by the contact between the spindles and bearings, non-contact magnetic bearings are *Address correspondence to this author at the Department of Mechanical Engineering, Cheng Shiu University, Taiwan; E-mail: [email protected] used for the spindle and rotor in high speed rotating machinery. Traditionally, ball bearings have been used to support the spindle systems when the rotational speed was not high. Previous investigations on bearing spindle systems were confined to spindles with ball bearings. In some studies, the focus was on the dynamic response of a spindle supported by bearings [10, 11]. At higher speeds, this bearing changes the stiffness of the entire spindle system and significantly alters the system properties [12-15]. Precise machining requires higher spindle speeds, making the magnetic-bearing spindle necessary. Investigations as [16-20] studied the performance and dynamic properties of magnetic bearings. Most studies deal with a magnetic ring for a radial magnetic bearing used as an unlimited one long magnetic bar for a permanent magnetic bearing. Investigation as [21] studied the bearing capacity and stiffness of radial magnetic bearings. Thus far, most investigations as [22-24] on the dynamic characteristics of a cracked spindle system were limited to ball-bearing-type spindles. This study examines the crack effects on the dynamic response of a rotating spindle system with magnetic bearings. A Euler-Bernoulli beam of circular cross section was used to approximate the spindle model. The equations of motion for the bearing-spindle system were derived using the Galerkin method and Hamilton principle. A model the size of an actual spindle system was used. To simplify the calculations, massless springs were employed to model the stiffness of the magnetic bearings. The effects of crack depth, rotational speed and bearing length on the dynamic response of a spindle system were investigated. a spindle supported by bearings Vibration in a Cracked Machine Tool Spindle with Magnetic Bearings The Open Mechanical Engineering Journal, 2008, Volume 2 33 a simple model of bearing spindle system Fig. (1). A rotating spindle with bearings scheme. Fig. (2). Geometry of a cracked spindle. Theory This paper considers a spindle supported by magnetic bearings, as shown in Fig. (1a), to elucidate the dynamic response of a spindle system. Fig. (1b) presents a simple model for this bearing-spindle system. In this model, massless springs are employed to simulate the stiffness of the magnetic bearing and support the spindle. The rotational speed of the spindle cannot be ignored in the rotating machinery bearing application. In this study, the deflection components (z,t), and u(z,t) denote the two transverse flexible deflections of the spindle system. E and I represent the Young’s Modulus and area inertia of the spindle, respectively. Only the transverse flexible deflections are studied in this article. According to [25], the governing equations of the spindle system are displayed as: Au 2 A v A u + EI u ( ) + k x1 u z z 1 ( ) + kx2 u z z2 ( ) = 0 (1) Av + 2 A u A v + EI v ( ) + k y1 v z z 1 ( ) + ky2 v z z2 ( ) = 0 (2) where kx1 : the bearing stiffness in u deflection at a position z1 , ky1 : the bearing stiffness in v deflection at a position z1 , kx2 : the bearing stiffness in u deflection at a position z2 , ky2 : the bearing stiffness in v deflection at a position z2 , : density , A : cross section area, : rotating speed, ( ) : Dirac delta fuction, z 1 : the first located position of bearings, z 2 : the second located position of bearings. For convenience, the dimensionless equations of motion for this spindle are: u 2 EI AL v + EI AL u + u ( ) +k x1 u z z 1 ( ) + kx2 u z z2 ( )} = 0 (3) v + 2 EI AL u + EI AL v + v ( ) + k y1 v z z 1 ( ) + ky2 v z z2 ( )} = 0 (4) where the dimensionless parameters are given using: z = z L , z 1 = z 1 L , z 2 = z 2 L , = EI AL , (5) u z ( ) = u z ( ) L , v z ( ) = v z ( ) L , k x1 = k x1 L EI , (6) k x2 = k x2 L EI , k y1 = k y1 L EI , k y2 = k y2 L EI (7) and the boundary conditions are: u = u = v = v = 0 , at z = 0 (8) u = u = v = v = 0 , at z = 1 (9) The Galerkin method is employed to derive the spindle equations of motion in matrix form. Therefore, the solutions for Eqs. (3) and (4) can be assumed to be: u z , t ( ) = i i=1 m z ( ) pi t ( ) (10) v z , t ( ) = i i=1 m z ( )qi t ( ) (11) where i z ( ) , i z ( ) are comparison functions for the spindle system, and p i t ( ) , qi t ( ) are the time coefficients to be determined for the system. The exact solution for a beam with free-free boundary conditions is considered, and five comparison function modes are used. i z ( ) = i z ( ) = i z ( ) 2 1 cosh i z cos i z ( ) (12) ξ η