Effects of Bearing Locations on the Characteristics of a Spindle System Supported by Tapered Roller Bearings

2.1 TRB Stiffness Model

Fig. 2 presents a five DOF model for a TRB. The inner ring of the TRB is loaded by a load vector {F} and is displaced by a displacement vector {δ}, where:

Fig. 2

Free-body diagram and two solving procedures for bearing subject to initial displacements or external loading

There are two types of computational procedure for the TRB model according to the input and output conditions as shown in Fig. 2. The first computational procedure is related with the stiffness matrix and displacements of the TRB subject to bearing loading. Fig. 3 shows the calculation flowchart of the first procedure. This procedure is based on a two-loop iteration: The global loop for the inner equilibrium and the inner loop for local equilibrium of each roller. The calculation process starts by assuming the initial displacements of the inner ring and rollers. Subsequently, the contact loads between the rollers and raceways are calculated using the slicing method.¹⁶ Next, the new displacements of rollers are determined by solving the equilibrium equations of rollers using the Newton-Raphson method. The free-body diagram of a roller with all applied loads is demonstrated in Fig. 2. After the equilibriums of individual rollers are solved, the inner ring equilibrium equation can be solved for the unknown inner ring displacements. At the end of the local or global loop, the convergence should be checked, in which the current displacements of rollers or inner ring are compared to those of the previous iteration. A new iteration is needed if the differences between the current and previous displacements are larger than a given tolerance. Details of the roller loads, roller and inner ring equilibrium equations, and stiffness matrix formulation may be referred from several papers by the authors.^16,17

Fig. 3

Computational procedure for bearing equilibrium

The second computational procedure is such that the displacement vector of inner ring is given a priori, and the applied load and stiffness matrix of the bearing are determined. This procedure may be a simplified version of the first procedure, because only the equilibriums of rollers are needed to solve. Accordingly, the computational procedure is similar to Fig. 2 except excluding the outer loop.

Both the two computation procedures provide the stiffness matrix and roller-race contact loads, which are essential for the estimation of the natural frequency, static stiffness, and bearing fatigue life of the spindle-bearing system. The TRB stiffness matrix is determined by Eq. (3).

which is a [5 × 5] matrix represented as Eq. (4):

Fig. 4 shows the numerically obtained stiffness of a TRB as a function of axial and radial loads. The bearing geometric data are taken from.¹⁶ The computed stiffness values are compared with those determined by a commercial code.¹⁸ It can be observed that the calculated results match well with those from the commercial code.

Fig. 4

Effect of external loads on the TRB stiffness

2.2 Bearing Fatigue Life

In this section, the fatigue life of TRB is estimated in terms of reference rating life with roller loads. The reference rating life (L_10r) is defined as the number of revolutions (in 10⁶ Revolutions) such that a bearing may survive with a probability of 90%. Determination of L_10r is outlined in ISO standard¹⁹ and briefly summarized below.

First, the basic dynamic load ratings of inner and outer ring are determined as:

where the subscripts i and e represent the inner and outer races, respectively. λν = 0.83 represents the reduction factor considering the effect of stress concentration and variation of exponent.¹⁹ Parameter c depends on the TRB geometry including the roller mean diameter (D_a), pitch diameter (d_m), and contact angle (α) as Eq. (7):

C_r is the basic dynamic radial load rating of the TRB, which is determined by Eq. (8):¹⁹

where b_m = 1.1 indicates the contemporary improvement factor of bearing quality. f_c = 88.02 denotes a factor representing the accuracy regarding the geometry and material of manufactured bearing components.

Next, the basic load ratings of a slice are calculated as Eqs. (9) and (10):

Considering the TRB with rotating inner ring and stationary outer ring, the dynamic equivalent loads on the k-th slice of inner and outer rings are calculated by Eqs. (11) and (12):

where p_ikj and p_ekj indicate the maximum contact pressures at the k-th slice and the j-th roller of inner and outer race (denoted by i and e), which are found by means of contact pressure analysis. In this calculation, the roller and race contact pressure is determined by using a three dimensional elastic contact model.^20,21 Z is the total number of rollers in the TRB. Finally, the basic reference rating life of TRB is estimated as follows:

2.3 Spindle Natural Frequency

To determine the natural frequency, the spindle system is modelled using Timoshenko beam elements.²² The shaft is discretized into a number of finite elements and two TRBs are located at nodes as shown in Fig. 5. The shaft internal damping and bearing damping are neglected here. The equation of motion for a shaft element is expressed as follows:

Fig. 5

Finite element model of spindle-bearing system

where m^s and g^s are the [4 × 4] mass and gyroscopic matrices of shaft element, respectively. {y^s} and {z^s} represent the [4 × 1] displacement vectors in xOy and xOz planes, respectively. The equation of motion for a bearing is represented as Eq. (15):

where $$ {\left\{f_y^b f_z^b\right\}}^T$$ denotes the force vector of the bearing. $$ k_{yy}^b, k_{yz}^b, k_{zy}^b \mathrm{a}\mathrm{n}\mathrm{d} k_{zz}^b$$ represent [2 × 2] bearing stiffness matrix obtained from the bearing model. Assembling the shaft element and bearing equations gives the equation of motion for the whole spindle system as follows:

The state space form of the system equation of motion is as follows:

where

The eigenvalue problem in association with Eq. (17) can be written as

where α and {h} are the eigenvalue and corresponding eigenvector, respectively. Then the natural frequencies can be obtained from the imaginary parts of the eigenvalues.

2.4 Calculation Process

When the spindle shaft is subjected to radial loading, the resulting reaction loads at TRBs commonly include radial, axial and, moment loads. Likewise, the multi-stepped spindle shaft is elastically bent under loading. As a result, the equilibrium equation of the spindle system becomes indeterminate and could not be directly calculated using force equilibrium equations.^22,23 Moreover, because the bearing stiffness depends nonlinearly on the applied loads as shown in Fig. 4, the shaft equilibrium cannot be solved separately with assuming a given bearing stiffness. Therefore, the equilibrium equation of spindle system coupled with bearing stiffness and the reaction forces should be solved simultaneously. In order to overcome this problem, an iterative procedure is used in this study as shown in Fig. 6. First, the stiffness matrices of TRBs are calculated under initial axial preload using the first iteration of bearing calculation procedure. Having obtained the bearing stiffness matrices, the combined stiffness matrix of the whole spindle-bearing system [K] can be constructed using the FE model presented in previous section. Then, the displacement vector of all nodes {q} is obtained by using the following static relationship:

Fig. 6

Calculation flowchart for spindle-bearing system equilibrium

where {f} is the load vector of spindle-bearing system. In this study, because only one radial load F_r is applied on the spindle nose as shown in Fig. 1, the load vector {f} contains only one non-zero element F_r at the correspondent node, while the other elements of {f} are all zero. Obtaining the displacement vector {q}, the reaction loads on the TRBs and the TRB stiffness matrices are re-calculated using the next bearing calculation procedure. In the next step, the convergence for the bearing loads is checked. If the differences in the bearing loads between the current and previous steps are higher than a pre-assigned tolerance value, then the bearing stiffness matrices are updated and a new iteration begins. Otherwise, the iterative process is stopped. Finally, the fatigue lives of TRBs and the spindle natural frequency are calculated.

3.1 Bearing Load

Fig. 7 shows the bearing reaction loads on the shaft, which are also equal to the loads sustained by the bearings but in the opposite directions. The TRBs are all preloaded by the constant force preload method, and the varying loads to each TRB include a radial load and a bending moment as shown in Fig. 7. Fig. 8 shows the radial and moment loads on the shaft determined at TRB locations as a function of bearing location variables, L₁ and L₂. It is observed that the left TRB endures higher load compared to that of the right TRB. The loads of TRBs all increase with increasing L₁, i.e., increasing the distance between the left TRB and the radial loading point. Moreover, the left bearing location (L₁) manifests a higher influence on the bearing load variation than the right bearing location (L₂). This could be because the left TRB is closer to the loading point than the right one.

Fig. 7

Bearing loads include radial and moment loads

Fig. 8

Effect of TRB locations on the reaction loads in the bearings

Fig. 9 shows the vertical deflections of TRBs in y direction (See Fig. 7) with varying the TRB locations. The deflections of TRBs vary significantly with changing the TRB locations along the shaft. The relationship between the deflections and TRB locations shows a high degree of nonlinearity. Owing to the above-mentioned changes in the TRB loads and deflections, the TRB fatigue life and shaft deflection are varied accordingly. Moreover, the spindle natural frequency is also influenced by the nonlinear dependence between the bearing loads and stiffness. The behavior of fatigue life, natural frequency and static stiffness as a function of bearing locations will be discussed in the next section.

Fig. 9

Effect of bearing locations on vertical deflections at bearings

3.2 Bearing Fatigue Life

As shown in Fig. 8, the loads on the left TRB are significantly higher than those of the right TRB. Therefore, we consider only fatigue of the left TRB. Fig. 10 indicates that the maximum fatigue life is reached when L₁ = 11 mm and L₂ = 106 mm. On the other hand, the minimum bearing fatigue life is obtained when L₁ = 106 mm and L₂ = 11 mm. This demonstrates that the maximum fatigue life is obtained when the distance between the two TRBs is longest as shown in Fig. 11(a). To the contrary, when the distance between two TRBs is minimized, the fatigue life is deteriorated as shown in Fig. 11(b). The behavior of the TRB fatigue life in Fig. 10 can be explained by the correspondent variation of bearing loads already shown in Fig. 8. The bearing fatigue life generally reveals an opposite tendency to the load applied to bearing.

Fig. 10

Effect of bearing location on the fatigue life of the left TRB

Fig. 11

Configurations of spindle-bearing system with maximum or minimum fatigue life

3.3 Natural Frequency of Spindle-Bearing System

Fig. 12 shows the influence of bearing locations on the natural frequency of spindle system. In this study, we consider only the first natural frequency. Fig. 12 illustrates that the maximum natural frequency is 1,312.7 Hz when L₁ = 66 mm and L₂ = 11 mm. The minimum natural frequency is 648.44 Hz when L₁ = 11 mm and L₂ = 106 mm. This result demonstrates that the bearing locations significantly influence the spindle natural frequency. The bearing locations at which the natural frequency becomes minimal and maximal are shown in Fig. 13. It should be noted from Figs. 11(a) and 13(b) that the bearing locations of L₁ = 66 mm and L₂ = 11 mm maximize the bearing fatigue life but minimize the spindle natural frequency. Hence, the bearing locations should be carefully determined to consider such trade-off between the bearing fatigue life and spindle natural frequency.

Fig. 12

Effect of bearing location on the spindle natural frequency

Fig. 13

Configuration of spindle-bearing system with maximum and minimum natural frequencies

Fig. 14 shows the effects of TRB locations on the stiffnesses of the TRBs. The bearing stiffnesses show a contrary behavior against the external radial load shown earlier in Fig. 8(a). This is because increasing the radial load reduces the number of contacted rollers under loading. The stiffnesses of the left TRB vary significantly with bearing locations; the maximum stiffnesses are increased by more than 31% when compared to the minimum stiffnesses. On the other hand, the stiffnesses of the right TRB have only slight dependency on the bearing locations. Here, the differences between the maximum and minimum stiffnesses are approximately 4.5%. It is also noted that the bearing stiffnesses converge to their maximum values when the distances L₂ reaches its upper limit of 106 mm, and L₁ reaches its lower limit of 11 mm. Although the bearing stiffnesses become maximum, they could not ensure the highest spindle natural frequency as can be seen in Fig. 12.

Fig. 14

Effect of bearing location on the bearing stiffnesses

To evaluate the spindle natural frequency change with respect to the TRB stiffnesses variation caused by the bearing locations, Fig. 15 plots the natural frequency ratios that are defined as f_i divided by f_t, in which f_i is the natural frequency determined with invariant bearing stiffnesses, and f_t is the natural frequency determined with actual bearing stiffnesses. First, f_i is calculated with intentionally excluding the effect of bearing locations on the bearing stiffnesses. Secondly, f_i is calculated with the minimum values of bearing stiffnesses, i.e., the stiffnesses under pure axial preload. The resulting frequency ratios are shown in Figs. 15(a) and 15(b). Fig. 15(a) shows that the minimum value of natural frequency ratio is approximately 0.96. This means that the natural frequency estimated with the minimal bearing stiffnesses would be reduced by approximately 4% compared to the current natural frequency. On the other hand, Fig. 15(b) shows that under pure axial preload, we can obtain a maximum increase of the spindle natural frequency by approximately 7.5% compared to the current natural frequency. Through Figs. 14 and 15, changes in bearing locations lead to the subsequent change in the bearing stiffnesses, and therefore the spindle natural frequency.

Fig. 15

Frequency ratios as a function of bearing locations

3.4 Static Stiffness

Fig. 16 shows the static deflection of spindle determined at the loading point. Unlike the spindle natural frequency and bearing fatigue life, the deflection at the loading point mainly depends on the location of the left TRB. The effect of the right TRB location on the loading point deflection is very limited. Fig. 17 illustrates the minimum and maximum static deflections of the spindle calculated at L₁ = 11 mm and L₁ = 106 mm, respectively, while L₂ is kept as 11 mm. The deflection shape of the whole spindle, as shown in Fig. 17(a), demonstrates that most of the deflection stems from the elastic cutting tool. In order to view more details of spindle deflection, Fig. 17(b) illustrates the deflections without the tool part. The left TRB location L₁ = 11 mm, which leads to the minimal static deflection, does not necessarily guarantee the highest spindle natural frequency.

Fig. 16

Deflection of spindle at loading point

Fig. 17

Static deflections of the spindle system