内圆磨床毕业设计翻译原文.doc

Investigation on critical speed and vibration mode of high speed grinder
Abstract
The influences of the grinder spindle’s major structural parameters on its vibration mode were investigated. Based on the transfer-matrix method and taking into consideration the gyroscopic couple, the shear, the variable cross-section and other influential factors, a dynamic model was established for the multi-disk rotor of the rotor-bearing system of the grinder spindle. The critical speeds of first three orders, the modes of variation and other dynamic characteristic parameters of the grinder spindle were programmed and calculated. The influences of the axial pre-tightening force of the bearing, the span of the fulcrum bearing as well as the changes in the front and rear overhangs on the critical speed of the rotor-bearing system on the grinder spindle and their pattern of changes were analyzed. The results showed that the working speed of the spindle system is much lower than the primary critical speed and can therefore stay away the resonance range effectively.
1. Introduction
The spindle of high speed grinding machine tool is a typical high speed bearings-rotor dynamics system, and its rotating speed far surpasses the critical speed of the low order system. The realization of dynamic balance is associated with the performance and stability of the whole equipment. The research of rotor dynamic balance mainly involved the rigid rotor before the 1950s [1-4]. With the increase of the rotating speed and the flexibility of, the research on the method of flexible rotor’s dynamic balance appeared. The early stage method of flexible rotor’s dynamic balance was put forward by Feder, which was later called vibration balance method or modal balance method. The number of times of start-up of high-speed balance was relatively less, and is of high sensitivity. The low order mode would not be affected when the balancing of high order mode was conducted, but identification was not balanced[5-7]. The influence of bearing characteristics was especially great under large damping. It was not easy to achieve single mode
when used near critical speed in shafting balance. Later the American Goodman formally presented influence coefficient method based on least square method, and its advantages were as follows: electronic computer could be used to assist dynamic balance; the identification of unbalance factors was not affected by bearing characteristics; it could balance several vibration models simutaneously, which was more convenient especially for the balance of the shafts. However, the number of times of start-up in high speed balance was more, and the susceptibility of high order modes was reduced [8-10]. Based on the basic theories of rotor dynamics and structural system dynamics, dynamics theory model of the spindle system of high speed grinder was established using the transfer matrix method and the overall transfer matrix method in this study, and the simulation study was conducted on the influence of critical speed, vibration mode on the dynamic characteristics of main shaft systems of high speed grinder.
2. Mathematical model of critical speed and vibration mode
2.1 Model simplification
High speed spindle system is a continuous elastomer. Based on the concentration mass dynamic modeling theory of the transfer matrix method, it was necessary to simplify the quality continuous actual rotor into the rotor with a series of concentrated mass and rigid disks, which were connected by massless and flexible shaft sections in between[11-13]. Ultra-high speed grinding spindle system mainly consisted of bearings, spindle, motor rotor, and additional parts (locking nut, bearing retainer ring, encoder, sealing rings, etc.). The structural diagram is shown in Fig. 1 (1-Spindle 2- Motor rotor 3-Bearing).

Figure 1. The structural diagram of spindle system
For sliding bearing, the oil film’s dynamic coefficient matrix was as follows:

The coefficient matrix of the dynamic performance of bearing block was as follows:

When there was not too much difference in the stiffness and the equivalent mass of the bearing block in the direction of x, y, and coupling was relatively weak, the damping effect was negligible and the bearing was considered as isotropic:

The model was converted to a fixed in-plane, and the rigidity coefficient of elastic bearing K could be calculated according to the formula below:

K was the total bearing rigidity coefficient of the sliding bearing, which reflected the dynamic characteristics of the oil film, the bearing block and the foundation in a comprehensive way. The total bearing rigidity coefficient of the bearing was not constant, which was different from that of the common elastic bearing of equivalent stiffness, which was related to the whirling angular velocity of the rotor. For rolling bearing, the radial stiffness of the two bearings could form a larger angular stiffness Kθ, which was calculated according to the formula below:

Where Kr was the radial stiffness, l was the center distance between series bearings. At this time, the bearing point was selected as of the midpoint of the line connected the centers of series bearings. When the mass of the bearing itself was taken into account, it could be treated as a concentrated mass point. After the above treatment, the computational model of the dynamics of ultra-high speed grinding spindle system is shown in Fig. 2. The spindle system was divided into several typical components or parts, such as concentrated mass, disc, shaft section and bearing.
The two ends of the i unit was labeled with i and i+1, respectively, and the concentrated mass, disk and bearing etc. were focused on i point.Qi, Mi, θi, Xi represented shear, bending, angle, deflection of the left end of i unit, respectively. Likewise, Qi, Mi, θi, Xi represented shear, bending, angle, deflection of the right end of the i unit, respectively. The state vector could be expressed as [Z]i = [Q M θ X]I T , and the relation between the state vectors of any two ends is shown below:

Figure 2. The analysis model of the dynamics of spindle system

Where [U]iwas a transfer matrix, which satisfied mechanic equilibrium and connected the state vectors of the two sections of i and i+1.
2.2 Transfer matrix for the typical unit
Fig. 3 shows a typical transfer matrix unit which consisted of a massless shaft of uniform section, a disk and an elastic bearing.

Figure 3. The typical transfer matrix unit

Where l was the length of unit shaft; m was the total mass of the nodes (except the mass of shaft section, the roulette mass should be added.); _ was angular velocity of precession of the rotor; was angular velocity of rotation of the rotor; jd was the diameter moment of inertia of the rotor; jp was the polar moment of inertia of the rotor; EI was the bending rigidity of the cross section of the shaft; v=6EI/ktGAl2,G was the shear elastic modulus; A was the area of the cross section of the shaft;kt was the coefficient of the shape of cross section, which was chosen as 2/3 when it was hollow circular cross section, and 0.9 when it was a solid one.
2.3 Solution of critical speeds and vibration mode
The state vector of section of the spindle terminal was as follows:

Where [U] was 4×4 order matrix. The critical rotational speed during the synchronized process when Ω = ω was usually calculated. Therefore, the element in matrix [U] was the function of Ω.
The boundary conditions that both sides were free ends were substituted, then:

The following result could be obtained:

For the homogeneous equation to have non-trivial solutions, the following requirement must be satisfied:

f(Ω) was residual formula, and the solutions of equation f(Ω) = 0 were the desired critical rotational speeds. The rope root law was applied to seek solutions in this paper. The corresponding vibration mode could be calculated after the critical rotational speeds were obtained. A proportion solution could be obtained according to the homogeneous linear equation (10). For example, if θ0 = 1, then X0 could be calculated, and the initial state vector was thus obtained. According to Formula (8), the state vector of each computing website could be obtained. As a result, all the vibration modes in every order were known.
3. Calculation of critical speeds and vibration mode
The main parameters of spindle system of HSG150 type are shown in Tab. 1.The span of the front and the rear fulcrum bearings on the grinder spindle was 215mm, the overhang at the front end was 85mm, and the overhang at the rear end was 60mm. Bearings are HCB71909-E angular contact ceramic ball bearings from FAG (Germany), with the actual preload of F being 216N. Matlab was used to write the computation program which was solved iteratively with the Prower method. The critical speeds of the first three orders obtained are shown in Tab. 2, and the major modes of vibration are shown in Fig. 4.
Table 1. Main parameters of HSG150 type spindle system

Table 2. Computational results of critical speed and natural frequency in the first three orders of spindle system.

Figure 4. Modes of vibration of different orders
It could be seen from the Fig.4 that the deformations at the middle and front end of the spindle were relatively large, as well as the back-end deformation. The main vibration mode was the first order bending. The main vibration mode at the second order critical speed that the deformation of spindle at the front end was relatively great, while the deformation at middle and back end was slight, with almost no deformation at the back bearing. The main vibration mode was swing vibration mode of the front overhang. Moreover, it could be seen from the Fig.4 that the deformation at the back end of spindle was relatively great, while the deformation at the middle and front end was slight, with almost no deformation at front bearing. The main vibration mode was the swing vibration mode of back overhang.
To ensure safe and reliable operation of the rotating machinery, it was usually required, that if the rotor’s rotational speedwas below the first order critical speed nc1, then n£º0.75nc1; if the rotor’s rotational speed was above the first order critical speed, then 1.4nck£ºn£º0.75nck+1. The maximum rotational speed of spindle of this type was 15000r/min, which was far less than that of the first order critical speed of the spindle system. Therefore, during the operation of the spindle system, resonance problem wouldn’t appear. In this study, structural factors which influenced critical speed of the spindle system were mainly discussed, and the analysis is shown as follows:
3.1 Shear effect and gyroscopic effect

Gyroscopic effect referred to the influence of gyroscopic moments on the critical speed of rotor. When the rotor rotated, the momentum to centroid produced by angle motion of rotator changes direction constantly, thus generating the moment of inertia. The influence of gyroscopic torque on the critical speed of rotor was that the forward precession promoted critical speed, while the backward precession decreased critical speed. When the modeling of spindle was simplified, continuous shaft could be simplified as a massless elastic shaft segment and an equivalent rigidity disk with mass and moment of inertia. Therefore, it was also necessary to investigate the influence of gyroscopic moments on critical speed of spindle.Figs. 5 show the influence of shear effect and gyroscopic effect on critical speed of spindle in the first five orders. It could be seen from the Fig.5 that shear effect reduced the critical speed in various orders, and its influence on higher orders was more severe than on lower orders. This was because the shear deformation intensified the deformation of the spindle, and shear force increased with the rise in rotational speed, which was even more significant at high speed. It could be known from Fig. 5 that the reduction of fifth order critical speed was 16.5% due to shear effect.

Figure 5. Changes of influence of shear effect (considering the gyroscopic effect) on critical speed.
3.2 Preload of bearing
Fig. 6 shows variation curves of the critical rotational speed of the spindle system in the first three orders during the changes of the preload (50 ~ 550N). It could be known from Fig. 6 that the critical rotational speeds in the first three orders of the spindle system increased with the increase of the preload of bearing, which was especially significant in high orders; then the increase gradually became less significant. It could be known from the variation curve of the first order critical speed to the preload that the critical rotational speed tended to become fixed with the increase of the preload. Therefore, when the preload of bearing was at low or medium mode, the critical rotational speed of spindle system could be improved by increasing the bearing preload on the basis of good bearing lubrication.

Figure 6. Influence curves of preload on the critical rotational speed of the spindle system
3.3 Length of overhang
The effects of the length of the front overhang on critical rotational speed in different orders were manifested within a certain range, as shown in Fig.7. The critical rotational speed would be reduced greatly with the increase of overhang length within the impact range.

Figure 7. Influence of the length of front overhang on the critical speed of the spindle system
4. Conclusions
(1) The main vibration mode was the first order bending. The main vibration mode at the second order critical speed that the deformation of spindle at the front end was relatively great, while the deformation at middle and back end was slight, with almost no deformation at the back bearing. The main vibration mode was swing vibration mode of the front overhang.
(2) Shear effect reduced the critical speed in various orders, and its influence on higher orders was more severe than on lower orders.
(3) Increasing the pre-tightening force of bearings can improve the critical speeds of different orders of the spindle and is also highly sensitive to the influence on high orders. The comparison of the variation rates of the pre-tightening forces and corresponding critical speeds reveals that the increase in pre-tightening force can only improve the critical speeds to a limited extent.
(4) Increasing the front overhangs has similar influences on critical speeds of different orders to increasing the span of fulcrum bearings, the reason of which is that increasing the span of fulcrum bearings equates decreasing the front and rear overhangs.