Whirl frequency of a high speed spindle subjected to different preload mechanisms
Ligang Cai^{1} , Yong Yang^{2} , Zhifeng Liu^{3}
^{1, 2, 3}College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing, China
^{3}Corresponding author
Journal of Vibroengineering, Vol. 16, Issue 8, 2014, p. 38273840.
Received 28 September 2014; received in revised form 21 November 2014; accepted 3 December 2014; published 30 December 2014
JVE Conferences
Preloading, both in terms of application method and actual applied force, significantly affects the stiffness and natural frequency of a high speed spindle system. The gyroscopic moment at high speed leads to whirling of the spindle, and the whirl frequency is not equal to the system’s natural frequency. To discover the relationship between preload and whirl frequency, theoretical and experimental research was undertaken. Two numerical models of the angular contact ball bearings, based on rigid and constant preload mechanisms, were established. The shaft is considered as a set of Timoshenko beam elements, and gyroscopic moment and centrifugal force are both considered. Adding bearing stiffness in the form of springs to this finite element system produced a spindlebearing coupled model. Iteration was used to deduce the interactions among bearing groups. The exact whirl frequency of a spindle subjected to different preload mechanisms has been calculated. To validate the proposed theory, frequency analysis was carried out on a Siemens CAT40 spindle. Experimental results agreed with theoretical calculations. The result shows that speed had a great influence on bearing stiffness and spindle whirl frequency. Adopting a reasonable preload method and preload force improved the spindle critical frequency.
Keywords: high speed spindle, preload mechanism, whirl frequency, angular contact ball bearing.
1. Introduction
As the speed and precision requirements increase, the key technologies of CNC machines tools and their main components have attracted increasing research attention. High speed spindles, as a key component of machine tools with rotational speeds greater than 100,000 rpm, present significant challenges for research into their dynamic characteristics. To predict and control the spindle dynamic performance (i.e. natural frequency, critical frequency, whirl and frequency), the relationship between bearing preload mechanism and spindle dynamic parameters should be ascertained: for this, an accurate spindle–bearing system coupled model is required.
However, the stiffness of a bearing varies at high speed and they interact with each other to provide a decisive impact on spindle system natural frequency. Under the action of unbalanced, gyroscopic moment, whirling arises as an intrinsic characteristic of rotating machinery. At high speed, the natural frequency is actually a whirl frequency. The point at which whirl frequency is equal to rotational frequency is the critical frequency of the spindle. Therefore, traditional models are no longer suitable for accurate calculation of high spindle whirl frequency.
The spindle system and its components have been widely studied. Liao et al. [1, 2] calculated bearing parameters using geometric analysis and force equilibrium analysis. Cao et al. [35] proposed a general modelling method for spindle systems comprising angular contact ball bearings, and a housing modelled with a standard nonlinear bearing model and by treating the spindle as a set of Timoshenko beam elements. Bai et al. [6] analysed the influence of an axial preload force on the spindle, while Guo et al. [412] researched bearing behaviour by using a contact finite element model to obtain multigroup bearing stiffnesses. However, these studies focus on singlebearing modelling and calculate the stiffness; they pay less attention to the analysis of the preload mechanism in such systems.
Jedrzejewski [1316] proposed a modelling method for a constant pressure bearing. Cao et al. investigated the rigid preload and constant preload mechanisms of angular contact ball bearings, and analysed the influence of different preload mechanisms on a spindle system in 2011. Gunduz [1720] studied the influence of preload force on a spindlebearing system equipped with a double row angular contact ball bearing. However, this research rarely modelled the same spindle equipped with bearings under different preload methods: thus they failed to analyse the interactions between bearing components adequately.
The arrangement and the preload methods for bearings, and selection of its preload force, are regarded as core technology in the design of spindle systems. The increase of preload force improves the support stiffness of the spindle system, but increases the contact load and heat generation therein. This reduces the fatigue life of a bearing. Besides, interactions among bearing components restricts the increment of the critical frequency of a spindle system. Thus, it was necessary to investigate the effect of bearing preload methods on the critical frequency and upon the interactions between its bearings.
This research revealed the relationship between preloading and whirl frequency of a spindle, and focussed on the interactions between bearing groups. A numerical analysis method based on a coupling spindlebearing model was proposed. Two numerical models of the angular contact ball bearings based on rigid and constant preload mechanisms were established. The bearing support stiffness as modelled by spring elements and the bearing stiffness matrix was substituted into a spindle Timoshenko beam model using the finite element method. Applying preload and speed, the stiffness of bearings under specific working conditions was obtained by iteration. Subsequently, the whirl frequency was calculated. Lastly, the system model was validated by measuring the frequencies of an unloaded Siemens CAT40 spindle on which the preload forces and speed can be altered.
2. Dynamic model of the spindle and its bearing system
A Siemens CAT40 spindle system is shown in Fig. 1. Two groups of angular contact ball bearings are distributed on the two ends of the spindle system. The rigid preload bearing group at the frontend consists of bearings I and II arranged backtoback. The length differences of the sleeves in its inner and outer rings are adjusted to realise the required preload. The length of the sleeve in the outer ring is greater than that in the inner ring. In motion, the outer ring and the spindle system housing are consolidated; the inner ring and shaft are also consolidated. Constant preload bearing III is fitted to the backend of the spindle system. The disk spring applies a constant axial thrust (the preload) to the bearing, when there is thermal elongation of the spindle system; the sliding sleeve was consolidated with the bearing outer rings to allow axial sliding, so as to compensate for any thermal displacement. The axial preloads on the outer and inner rings of the bearing are constant.
Fig. 1. System structure: the highspeed spindles 1. Bearing 1; 2. Sleeve of inner ring; 3. Sleeve of outer ring; 4. Bearing 2; 5. Spindle; 6. Motor rotor; 7. Motor stator; 8. Sliding sleeve; 9. Bearing 3; 10. Sliding zone of sliding sleeve; 11. Disk spring
The coupling relationships among each submodels of the highspeed spindle system are shown in Fig. 2. Based on the interactions between bearing groups with different preload mechanisms, the dynamic performance of a highspeed spindle system was investigated by using a spindle finite element model coupled with each submodel.
Fig. 2. Program flowchart for the spindlebearing system analysis
2.1. Rigid preload bearing model
According to Hertzian contact theory, the relationship of the inner and outer rings of a bearing with the load deformation of the $k$th rolling element is presented [21] as:
where, ${Q}_{ik}$ and ${Q}_{ok}$ are contact loads on the inner and outer rings respectively; ${K}_{i}$ and ${K}_{o}$ are the contact coefficients of the inner and outer rings; while ${\delta}_{ik}$ and ${\delta}_{ok}$ denote the contact loads on the inner and outer rings.
Based on the model proposed by Jones [22], the centripetal force ${F}_{ck}$ and gyroscopic moment ${M}_{gk}$ of a single rolling element are:
In Eq. (2), $m$ is the mass of a single rolling element; ${D}_{m}$ is the pitch diameter of the bearing; ${J}_{b}$ is the rotational inertia of a single rolling element; $\mathrm{\Omega}$ is the rotational speed of the spindle (rad/s); ${\mathrm{\Omega}}_{B}$ is the spinning angular speed of the rolling element (rad/s); ${\mathrm{\Omega}}_{E}$ is the angular speed of the rolling element rotating around the bearing centre; and ${\alpha}_{k}$ is the angle between the spinning axis and the spindle of the rolling element.
The force balancing equation of the $k$th rolling element is expressed [3] as:
where, ${\theta}_{ik}$ and ${\theta}_{ok}$ are the contact angles of the inner and outer rings respectively; and $Dw$ is the diameter of the rolling element. The rigid preload mechanism implied that a set of inner and outer rings of the bearing are consolidated with the spindle housing. By adjusting the size of the sleeve, the preload could be adjusted. Fig. 3 shows the relationship of preload to displacement during operation. The coordinate system [3] and geometry of Fig. 3 were used.
Fig. 3. Rigid preload bearing geometry model
In Fig. 3, B is the centre of curvature of the outer ring of the bearing; D is the centre of curvature of the inner ring of the bearing; and O is the centre of curvature of the rolling element. In the absence of a preload, B, O, and D are collinear and angle $\theta $ between the vertical and the spindle is the initial contact angle of the bearing. With a preload, with the centre of curvature of the outer ring of the bearing B being constant, while that of the inner ring of the bearing D moves to D1 by amount $\mathrm{\Delta}icu$, the axial rigid preload. Meanwhile, the centre of curvature of the rolling element moves from O to O1. In static conditions, B, O1, and D1 are still collinear, while the contact angle between the inner and outer rings increases to ${\theta}_{1}$. During rotation, the centre of curvature of the rolling element moves from O2 to O1 influenced by centripetal force, gyroscopic moment, and external load. When in motion, the force on the back bearing and spindle pulls the centre of curvature of the inner ring of the bearing to D2 as found by iteration. This is the greatest difference between this study of rigid bearings and traditional models. As for the $k$th rolling element, the geometric centre of curvature of the inner ring has displacements $\delta xk$ and $\delta yk$ in the horizontal and vertical directions respectively. Simultaneously, the contact angle of the inner ring ${\theta}_{ik}$ increases, while the contact angle of the outer ring ${\theta}_{ok}$ decreases. According to the geometry:
where:
Meanwhile, in accordance with the geometric relationships, Eq. (7) may be obtained:
where:
In Eqs. (5) to (11), ${f}_{i}$ and ${f}_{o}$ are coefficients for the inner and outer rings of the bearing respectively; ${\phi}_{k}$ is the phase angle of the $k$th rolling element; $\mathrm{\Delta}{\delta}_{x}$, $\mathrm{\Delta}{\delta}_{y}$, $\mathrm{\Delta}{\delta}_{z}$, $\mathrm{\Delta}{\gamma}_{y}$, and $\mathrm{\Delta}{\gamma}_{z}$ are the displacements generated by the preload; only when the preload is axial, $\mathrm{\Delta}{\delta}_{y}$, $\mathrm{\Delta}{\delta}_{z}$, $\mathrm{\Delta}{\gamma}_{y}$, and $\mathrm{\Delta}{\gamma}_{z}$ are zero; $\mathrm{\Delta}{\delta}_{x}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{y}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{z}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\gamma}_{y}^{\text{'}}$, and $\mathrm{\Delta}{\gamma}_{z}^{\text{'}}$ are the displacements induced by the inertial force and load.
By substituting Eqs. (6) to (10) into Eq. (5) separately, the geometric compatibility equation for the rigid preload bearing is established; by substituting Eqs. (1) to (3) into Eq. (4) separately gives the force balance equation. The dynamic contact angle ${\theta}_{ik}$, ${\theta}_{ok}$, ${\delta}_{ik}$, and ${\delta}_{ok}$ of the inner and outer rings of the bearing are set as independent variables. Then the nonlinear model for a rigid bearing is obtained by combining Eq. (4) with Eq. (5):
If $\mathrm{\Delta}{\delta}_{x}$, $\mathrm{\Delta}{\delta}_{y}$, $\mathrm{\Delta}{\delta}_{z}$, $\mathrm{\Delta}{\gamma}_{y}$, $\mathrm{\Delta}{\gamma}_{z}$, and $\mathrm{\Delta}{\delta}_{x}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{y}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{z}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\gamma}_{y}^{\text{'}}$, and $\mathrm{\Delta}{\gamma}_{z}^{\text{'}}$ are known, the dynamic parameters of the bearing ${\theta}_{ik}$, ${\theta}_{ok}$, ${\delta}_{ik}$, and ${\delta}_{ok}$ are found by NewtonSimpson iteration. However, since the displacements $\mathrm{\Delta}{\delta}_{x}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{y}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{z}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\gamma}_{y}^{\text{'}}$, and $\mathrm{\Delta}{\gamma}_{z}^{\text{'}}$ are unknown when establishing a single bearing model, it is necessary to develop the spindlebearing system model. Then the displacements induced by load can be determined iteratively. In turn, the dynamic parameters of the bearing may be found and the supporting stiffness of the bearing determined.
2.2. The constant preload bearing model
The mechanism of the constant preload bearing model is such that the inner ring of the bearing is consolidated with its spindle, while a disk spring applies a constant axial pressure to the outer ring. As the spindle undergoes thermal extension, the sliding sleeve connected to the outer ring of the bearing produces axial displacement. The loaddeformation relationship and the force balance on the inner and outer rings with the rolling element are consistent with those for a rigid preloaded bearing. Fig. 4 shows the displacement relationship. According to the preload mechanism, a supplementary force balance equation can be obtained:
where, $Fa$ is the constant preload force.
The geometric relationship of the preloading process in a constant preload bearing is identical to that of the constant preload bearing. Under the effects of external load, inertial force, and thermal effects on the spindle, the centre of curvature of the inner and outer rings of the bearing would produce relative displacements. Under thermal effects, the centre of curvature of the inner ring moves from B to B1, while that of the outer ring migrates from D1 to D2. In the modelling process of a single bearing, it is difficult to determine the axial displacement $\delta xik$ of the inner ring of the bearing induced by thermal expansion; however, the displacement coordination system of Eq. (5) that is similar to that of the rigid preload method and can be established longitudinally. Similarly, by setting the dynamic contact angles of the inner and outer rings of the bearing ${\theta}_{ik}$, ${\theta}_{ok}$, ${\delta}_{ik}$, and ${\delta}_{ok}$ as dependent variables and combining Eq. (4) with Eq. (5), the nonlinear model for a constantpressure bearing is established:
The symbols in Eq. (14) have the same meanings as those in Eq. (12). Similarly, if $\mathrm{\Delta}{\delta}_{x}$, $\mathrm{\Delta}{\delta}_{y}$, $\mathrm{\Delta}{\delta}_{z}$, $\mathrm{\Delta}{\gamma}_{y}$, $\mathrm{\Delta}{\gamma}_{z}$, and $\mathrm{\Delta}{\delta}_{x}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{y}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\delta}_{z}^{\mathrm{\text{'}}}$, $\mathrm{\Delta}{\gamma}_{y}^{\text{'}}$, and $\mathrm{\Delta}{\gamma}_{z}^{\text{'}}$ are known, the dynamic parameters ${\theta}_{ik}$, ${\theta}_{ok}$, ${\delta}_{ik}$, and ${\delta}_{ok}$ of the bearing may be found by NewtonSimpson iteration.
Fig. 4. Constant preload geometry model
2.3. The stiffness matrix of the bearing
When the dynamic parameters: ${\theta}_{ik}$, ${\theta}_{ok}$, ${\delta}_{ik}$, and ${\delta}_{ok}$ of the bearings are obtained, the stiffness matrix of the bearing can be found according to the model established by Cao. The calculation process is shown as Eqs. (40) and (41) and in [3], Appendix C. By calculation, the stiffness matrix of the bearing $[K{]}_{B}$ may be found.
2.4. The spindlebearing system model
Fig. 5 shows the finite element model of the spindlebearing system. The spindle is treated as a Timoshenko beam with five degrees of freedom. By applying the bearing stiffness at the corresponding nodes, the kinematic equation is obtained [3, 23]:
where, $\left[{M}^{b}\right]$ is the nodal mass matrix; ${\left[{M}^{b}\right]}_{C}$ is the mass matrix related to the centripetal force; $\left[{G}^{b}\right]$ is the mass matrix related to the gyroscopic couple; $\left[{K}^{b}\right]$ is the stiffness matrix of the beam; ${\left[{K}^{b}\right]}_{P}$ is a mass matrix induced by the axial force; $\left[{F}^{b}\right]$ is the nodal load matrix; $\mathrm{\Omega}$ is the rotation speed of the principal spindle; and $\left\{q\right\}={\left\{{\delta}_{x}^{\mathrm{\text{'}}},{\delta}_{y}^{\mathrm{\text{'}}},{\delta}_{z}^{\mathrm{\text{'}}},{\gamma}_{y}^{\text{'}},{\gamma}_{z}^{\text{'}}\right\}}^{T}$ is the nodal displacement vector. In the iteration process, the nodal displacement vector is used as the iterative object. The iteration process went as follows: firstly, the initial value ${\left\{q\right\}}_{0}={\left\{\mathrm{0,0},\mathrm{0,0},0\right\}}^{T}$ is set and the stiffness of the bearing $[K{]}_{B}$ is calculated; afterwards, the stiffness obtained is substituted into Eq. (5) to yield the initial overall model; subsequently, the preload force and external load on the bearing are used as the load matrix to solve for the nodal displacements ${\left\{q\right\}}_{1}$ in the system; and then ${\epsilon}_{1}=\Vert {\left\{q\right\}}_{1}{\left\{q\right\}}_{0}\Vert $ is calculated. If ${\epsilon}_{1}$ is smaller than the designed tolerance on numerical accuracy, the calculation is terminated; otherwise, ${\left\{q\right\}}_{2}={\left\{q\right\}}_{1}\times \mathrm{\Delta}E$ is substituted into the bearing model to recalculate the bearing stiffness ($\mathrm{\Delta}E$ is the convergence coefficient: to guarantee convergence, $\mathrm{\Delta}E=\text{0.5).}$Using this method, the iteration repeats until ${\epsilon}_{k+1}=\Vert {\left\{q\right\}}_{k+1}{\left\{q\right\}}_{k}\Vert \le \epsilon $; finally, the values of ${\left\{q\right\}}_{k+1}$ are substituted into the submodels to obtain the system dynamic parameters.
Fig. 5. Spindle system finite element model
After obtaining the system parameters, Eq. (15) is adjusted to:
where, $\left[M\right]=\left[{M}^{b}\right]$, $\left[C\right]=\mathrm{\Omega}\left[{G}^{b}\right]$, and $\left[K\right]=\left[{K}^{b}\right]+{\left[{K}^{b}\right]}_{P}{\mathrm{\Omega}}^{2}{\left[{M}^{b}\right]}_{C}$.
2.5. The whirling frequency of the spindle system
The spindle undergoes simple harmonic motion at a frequency of ${w}_{n}$ in the two mutually perpendicular directions. Generally, the amplitudes in the two directions are unequal. Since the locus of the circle’s centre of the motion is an ellipse, the motion is called a whirl or a precession. The natural frequency ${w}_{n}$ is the precession angular velocity. It is assumed that the spindle rotates at speed $\mathrm{\Omega}$ and whirls at frequency ${w}_{n}$. Thus the spindle rotates in the bending plane of its axis and the shaft undergoes cyclic tension and compression. As a result, the internal resistance of the material influences the motions of the rotating parts and the fatigue life of the material. Only when $\mathrm{\Omega}={w}_{n}$, can the spindle avoid relative bending and develop synchronous whirl. Therefore, the whirl analysis of the spindle lays a basis for the selection of the bearing material, fatiguebased analysis of the spindle, and the selection of a reasonable working rotation speed.
When matrix $\left[{G}^{b}\right]$, related to the gyroscopic couple, is taken into consideration, the principal spindle model should be converted to get the eigenvalue of the kinematic equation. Let:
where, $\left\{u\right\}$ is a state vector. Eq. (21) can be expressed as:
where:
$E$ is a $p$order unit matrix $p=N\times dof$; $N$ is the node number within the finite element model; and $dof$ is the number of degrees of freedom of a single node.
The eigenvalue of matrix $A$ is the whirl frequency of the system. Through transformation, the system contains $2p$ firstorder differential equations. Since matrix ${\left[C\right]}^{T}=\left[C\right]$, matrix $A$ is also antisymmetric.
Since $\left\lambda EA\right={\left\lambda EA\right}^{T}=\left\lambda E{A}^{T}\right=\left\lambda E+A\right$, a couple of conjugate pure imaginary $\lambda $ and $\lambda $ are the eigenvalues of the system.
3. Experimental verification
To validate the model, a frequency analysis was conducted using impulse analysis [24]. The spindle was excited using an impulse hammer and a laser velocity vibrometer was aimed at the frontend of the shaft to analyse data ultimately collected by an LMS collection and data analysis system. The sample frequency was 2,048 Hz. The frontend rigid preload on the spindle system was set to 3.6×10^{5} m, while the backend constant preload was set to 1,000 N. The experimental system is shown in Fig. 6. The timedomain signal and frequency domain data for a typical spindle (at 18,000 rpm) are shown in Fig. 7. The firstorder frequencies were measured during rotation and key data are shown in Fig. 9 which fully validated the accuracy of the model.
This study investigated the dynamic characteristics of the unloaded spindle system: in real cutting processes, the spindle firstorder frequency is slightly lower than its theoretical value. Since the multiple junction surfaces reduce the stiffness of the system, the cutting load may replicate the preloaded state of the spindle system.
The Siemens CAT40 spindle bearing is a type of hybrid ceramic angular contact ball bearing system. The inner and outer rings are made of alloy steel, while the rolling body is made of Si_{3}N_{4} ceramic (type 7012C/H). Table 1 shows the main parameters of the bearing. The bearings are numbered (I, II, and III from front to back, respectively). The spindle is a hollow stepped spindle with an inner hole of diameter $\varphi $42 mm. Table 2 lists the dimensions of the spindle.
Table 1. Bearing geometry parameters
Item

Symbol

Size

Ball diameter

${D}_{w}$ /$\mathrm{}$mm

15.875

Number of balls

$z$

14

Curvature radius coefficient of inner ring raceway

${f}_{i}$

0.523

Curvature radius coefficient of outer ring raceway

${f}_{o}$

0.525

Initial contact angle

$\alpha $ / (°)

15

Bearing pitch circle diameter

${D}_{m}$ /$\mathrm{}$mm

85.7505

Table 2. Spindle dimension parameters
Spindle segment

Outer diameter / mm

Inner diameter / mm

$\text{0}\le l<\text{240mm}$

60

42

$\text{240}\le l<\text{287mm}$

90

42

$\text{287}\le l<\text{623mm}$

76

42

$\text{623}\le l<\text{802mm}$

58

42

Fig. 6. Spindle testing: the experimental equipment
Fig. 7. The timedomain signal and frequency domain data for a typical spindle (at 18,000 rpm)
4. Analysis of the whirl frequency under mixed preload mechanisms
4.1. Analysis of bearing stiffness
The support system was analysed under three working conditions:
1. Keeps a constant force on the preload bearing (bearing III). While setting the preload distance (the sleeve length difference between bearings I and II) on the rigid preload bearing to 1.29×10^{5} m, 2.02×10^{5} m, 2.6×10^{5} m, and 3.1×10^{5} m, respectively, and setting the rotation speed to zero.
2. Keeps the preload distance on the rigid preload bearings (bearings I and II) on the front end at 3.56×10^{5} m. While setting the preload force ${Fa}_{3}$ of the constant preload bearing (bearing III) to 200 N, 400 N, 600 N, and 800 N respectively. The rotation speed was zero.
3. Keeps the preload distance on the rigid preload bearing group (bearings I and II) at 3.56×10^{5} m. Uses a 1,000 N preload force on the constant preload bearing (bearing III) while setting the rotation speed to 5,000, 10,000, 15,000, 20,000, and 25,000 rpm respectively.
To assess the extent and effects of the interactions between bearing components, the elastic spindle and the bearing model are coupled to establish a forcedisplacement relationship for simultaneous solution. Fig. 2 illustrates the calculation process. Fig. 8 shows the stiffness of the bearing with an increase in the rigid preload distance (working condition 1). The stiffness of bearings I and II were increased. Moreover, the stiffness of bearing I exceeded that of bearing II due to the influence of bearing III’s preload force. This resulted in a preload force increase on bearing I, and diminished the preload force on bearing II. However, as the preload distance increased, the stiffness difference between bearings I and II gradually decreased. As the preload distance reached 3.56×10^{5} m, the difference reduced to 2 %. Therefore, it can be deduced that, to prevent bearing II’s detachment a large preload distance from the rigid preload bearing should be selected. However, an overly high preload may induce sticking as it increases the contact load on the bearings and intensifies the friction between rolling elements and bearing rings.
Under working condition 1, the rotation of the spindle was zero, and the preload force on bearing III was constant. As the spindle generates displacement, the sleeve of the outer ring also shows a corresponding displacement. Therefore, the stiffness of bearing III remained constant.
Fig. 8. Bearing stiffness for working condition set 1
Fig. 9. Bearing stiffness for the second set of working conditions
Fig. 10. Bearing stiffness for the third set of working conditions
Fig. 9 shows the stiffness of each bearing under working condition set 2. In the case of applying a low preload to the constantpressure bearing at the rear end, bearings I and II show only a small stiffness difference. As the preload force increases on bearing III, the stiffness thereof also increases; the stiffness difference between bearings I and II increases significantly; the stiffness of bearing I increased and the stiffness of bearing II decreased. Therefore, it can be seen that, as the preload force increased it could enhance the supports stiffness of the spindle backend while generating a significant stiffness difference in the rigid preload bearing group.
Static analysis of the two sets of working conditions suggested that, the preload state of the rigid preload bearing had no influence on the constant preload bearing, whereas the preload on the backbearing exerted a significant influence on the front rigid preload bearing group.
Fig. 10 shows the influence of both the centripetal force and gyroscopic moment on the preloaded bearings under operating conditions. Analysis of the results reveals that, with an increased spindle rotation speed, the rigid preload bearings I and II underwent an increase in stiffness while the constant preload bearing showed a concomitant stiffness reduction. As the rotation speed increased from 5,000 rpm to 25,000 rpm, the stiffness increased by 19.6 % (Bearing I) and 18.8 % (Bearing II) respectively. This stiffness increase was mainly attributed to the contact load increase on the rolling elements and bearing rings. Meanwhile, the stiffness of the constant preload bearing at the rear end decreased by 5.3 % because of the inertial force increasing the load on the outer ring, while the displacement of the sleeve kept the load on the outer ring constant. The inertial force actually reduced the contact force between rollers and inner ring. Therefore, the stiffness of the constant preload bearing decreased under highspeed rotation. These results suggest that the rigid preloaded bearing unit installed on the front end of the highspeed spindle was more appropriate. It was more favourable to maintain the stiffness of the spindle system; the constant preload bearing installed on the rear end was more conducive to compensating for thermal extension of the spindle system.
4.2. Analysis of the frequency of the spindle system
Table 3 shows theoretical results for the spindle’s firstorder natural frequencies. These could be validated by impact hammer test at a fixed preload distance of 3.56×10^{5} m and a 1,000 N constant preload.
Table 3 shows that, when bearing III maintained its high preload level, the preload on the front bearings exerted little influence on the firstorder fundamental frequency of the system. However, when keeping the rigid preload distance (bearings I and II) constant, the increase of preload (on bearing III) significantly increased the firstorder natural frequency of the spindle system. Therefore, when designing the supporting system, the rear constant preload bearing should be subjected to a high preload.
Table 3. Spindle system frequencies
Frontend preloading distance / 10^{5} m

Backend preload / N

First order frequency / Hz

1.29

1000

382

2.02

1000

394

2.6

1000

396

3.1

1000

397

3.56

200

316

3.56

400

349

3.56

600

364

3.56

800

378

3.56

1000

398

Ignoring the gyroscopic moment, the natural frequency of spindle could be found. When running, the firstorder frequency of the spindle was actually its whirl frequency. The point at which the whirl frequency was equal to the rotational frequency was the critical frequency of spindle. When researching the critical frequency speed of rotating systems, the emphasis is commonly put on investigating the critical speed of synchronous positive whirl. Under unbalanced excitation, the rotator whirls synchronously and positively. Hence, the critical rotation speed generally refers to the critical speed for synchronous positive whirl [10]. Under the effects of the inertial force of the spindle, the positive whirl frequency of the spindle increases, while its negative whirl frequency decreases (the speed becomes negative, i.e. it ceases to exist).
Fig. 11 shows the effects of inertial load and bearing stiffness on the frequency of the spindle. When running, the natural frequency of the spindle decreases significantly due to the rotation speed reduces the supporting stiffness of the constant preload bearing (Fig. 10).
As shown in Fig. 11, when neglecting the whirling effect of the spindle, the calculation error increases with speed. As the rotation speed reaches 25,000 rpm, the theoretical calculation error is 5.3 %. Nevertheless, this can be accepted in practical engineering circumstances, and was more accurate than traditional models. By drawing a straight line at $\mathrm{\Omega}={\omega}_{n}$ in Fig. 11, the intersection point of this line with the positive whirl curve. It gives the critical frequency of the spindle system (at 342 Hz here). Therefore, it can be deduced that the firstorder critical frequency of the spindle system occurs at 20,520 rpm under the given preload conditions. It was deduced that the Siemens CAT40 spindle cannot avoid the firstorder critical frequency over its full operating speed domain as it was a flexible spindle.
Fig. 11. Whirl frequency of the spindle system
5. Conclusions
1) This study reveals the relationship between preloading and whirl frequency of a spindle. Research shows that the bearing stiffness varies at high speed and interacts with that of other components of the system. Its most decisive impact lay upon the spindle system natural frequency while also influencing the whirl frequency, albeit indirectly. The stiffness of the constant preload bearing decreased, as the spindle natural frequency decreased: the whirl frequency also decreased, which therefore reduced the firstorder frequency.
2) This study presented an integrated bearingshaft coupled model. Two numerical bearing models, based on the Harris bearing model, were established for different preload mechanisms. Considering the shaft and housing as Timoshenko beam elements, we modelled the bearing stiffness matrix for the shaft and housing by the finite element method and obtained the bearing stiffness by iteration. The interaction of the bearings under inertial force and preload force were analysed. The proposed theory was validated by frequency test. This model can be used to predict spindle natural frequencies, whirl frequencies, and critical frequencies with acceptable accuracy.
3) In engineering, it is suggested that it is advisable to arrange a rigid preload bearing group in front of the spindle to maintain system stiffness. To increase the critical frequency of a spindle, an effective method was to apply a significant preload force to the constant preload bearing. However, this will increase the stiffness difference so, preload forces should be optimised at the design stage for such high speed spindles.
Acknowledgements
The authors thank the reviewers for their valuable comments and suggestions. This work was supported by the National Hightech R&D Programme (863 Programme): Design and Manufacturing Technology of Precise Horizontal Machining Centre (Grant No. SS2012AA040702) and The Major Specialised Science and Technology Programme: Innovation Ability Platform for Precise and UltraPrecise NC Machine Tools (Grant No. 2011ZX04016011).
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