Whirl frequency of a high speed spindle subjected to different pre-load mechanisms

2.1. Rigid pre-load 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 pre-load 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 pre-load could be adjusted. Fig. 3 shows the relationship of pre-load to displacement during operation. The coordinate system [3] and geometry of Fig. 3 were used.

Fig. 3Rigid pre-load 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 pre-load, 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 pre-load, 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 pre-load. 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 pre-load; only when the pre-load 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 pre-load 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 non-linear 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 Newton-Simpson 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 spindle-bearing 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 pre-load bearing model

The mechanism of the constant pre-load 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 load-deformation relationship and the force balance on the inner and outer rings with the rolling element are consistent with those for a rigid pre-loaded bearing. Fig. 4 shows the displacement relationship. According to the pre-load mechanism, a supplementary force balance equation can be obtained:

where, $Fa$ is the constant pre-load force.

The geometric relationship of the pre-loading process in a constant pre-load bearing is identical to that of the constant pre-load 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 pre-load 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 non-linear model for a constant-pressure 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 Newton-Simpson iteration.

Fig. 4Constant pre-load 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 spindle-bearing system model

Fig. 5 shows the finite element model of the spindle-bearing 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 pre-load 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 ${\epsilon }_1=‖{\left\{q\right\}}_1-{\left\{q\right\}}_0‖$ 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 ${\epsilon }_{k+1}=‖{\left\{q\right\}}_{k+1}-{\left\{q\right\}}_k‖\le \epsilon$; finally, the values of ${\left\{q\right\}}_{k+1}$ are substituted into the sub-models to obtain the system dynamic parameters.

Fig. 5Spindle 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, fatigue-based 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 $2p$ first-order differential equations. Since matrix ${\left[C\right]}^T=-\left[C\right]$, matrix $A$ is also antisymmetric.

Since $\left|\lambda E-A\right|={\left|\lambda E-A\right|}^T=\left|\lambda E-A^T\right|=\left|\lambda E+A\right|$, a couple of conjugate pure imaginary $\lambda$ and $-\lambda$ are the eigenvalues of the system.

4.1. Analysis of bearing stiffness

The support system was analysed under three working conditions:

1. Keeps a constant force on the pre-load bearing (bearing III). While setting the pre-load distance (the sleeve length difference between bearings I and II) on the rigid pre-load 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 pre-load distance on the rigid pre-load bearings (bearings I and II) on the front end at 3.56×10^-5 m. While setting the pre-load force ${Fa}_3$ of the constant pre-load bearing (bearing III) to 200 N, 400 N, 600 N, and 800 N respectively. The rotation speed was zero.

3. Keeps the pre-load distance on the rigid pre-load bearing group (bearings I and II) at 3.56×10^-5 m. Uses a 1,000 N pre-load force on the constant pre-load 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 force-displacement relationship for simultaneous solution. Fig. 2 illustrates the calculation process. Fig. 8 shows the stiffness of the bearing with an increase in the rigid pre-load 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 pre-load force. This resulted in a pre-load force increase on bearing I, and diminished the pre-load force on bearing II. However, as the pre-load distance increased, the stiffness difference between bearings I and II gradually decreased. As the pre-load 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 pre-load distance from the rigid pre-load bearing should be selected. However, an overly high pre-load 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 pre-load 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. 8Bearing stiffness for working condition set 1

Fig. 9Bearing stiffness for the second set of working conditions

Fig. 10Bearing 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 pre-load to the constant-pressure bearing at the rear end, bearings I and II show only a small stiffness difference. As the pre-load 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 pre-load force increased it could enhance the supports stiffness of the spindle back-end while generating a significant stiffness difference in the rigid pre-load bearing group.

Static analysis of the two sets of working conditions suggested that, the pre-load state of the rigid pre-load bearing had no influence on the constant pre-load bearing, whereas the pre-load on the back-bearing exerted a significant influence on the front rigid pre-load bearing group.

Fig. 10 shows the influence of both the centripetal force and gyroscopic moment on the pre-loaded bearings under operating conditions. Analysis of the results reveals that, with an increased spindle rotation speed, the rigid pre-load bearings I and II underwent an increase in stiffness while the constant pre-load 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 pre-load 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 pre-load bearing decreased under high-speed rotation. These results suggest that the rigid pre-loaded bearing unit installed on the front end of the high-speed spindle was more appropriate. It was more favourable to maintain the stiffness of the spindle system; the constant pre-load 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 first-order natural frequencies. These could be validated by impact hammer test at a fixed pre-load distance of 3.56×10^-5 m and a 1,000 N constant pre-load.

Table 3 shows that, when bearing III maintained its high pre-load level, the pre-load on the front bearings exerted little influence on the first-order fundamental frequency of the system. However, when keeping the rigid pre-load distance (bearings I and II) constant, the increase of pre-load (on bearing III) significantly increased the first-order natural frequency of the spindle system. Therefore, when designing the supporting system, the rear constant pre-load bearing should be subjected to a high pre-load.

Ignoring the gyroscopic moment, the natural frequency of spindle could be found. When running, the first-order 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 pre-load 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 first-order critical frequency of the spindle system occurs at 20,520 rpm under the given pre-load conditions. It was deduced that the Siemens CAT40 spindle cannot avoid the first-order critical frequency over its full operating speed domain as it was a flexible spindle.

Fig. 11Whirl frequency of the spindle system