# Stiffness Model and Experimental Study of Hydrostatic Spindle System considering Rotor Swing

Authors: Runlin Chen; Xingzhao Wang; Chen Du; Jun Zha; Kai Liu; Xiaoyang Yuan
Journal: Shock and Vibration
 (2020)

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## Abstract

For high-end CNC machine tools, the stiffness of the spindle system is one of the most important performance indicators. In this paper, the hydrostatic motorized spindle system of a grinding machine is taken as the research object, and a two-degree-of-freedom stiffness model of the spindle system considering rotor swing is proposed. The stiffness of the spindle system under different excitation frequencies is analyzed, and the contributions of the stiffness of two bearings to the stiffness of the spindle system are evaluated. The vibration test on the spindle system is implemented through the hammering method. The vibration responses of the spindle system are obtained, and the stiffness of the spindle system is identified. The results show that the test results of the stiffness of the spindle system are in good agreement with the theoretical calculation, with an average error of about 14.21%. The research in this paper can provide theoretical and data support for bearing design and stiffness evaluation of a hydrostatic spindle system.


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## Body

## 1. Introduction

With the development of high precision machine tools to high efficiency and high energy [1, 2], precision machine tools require high precision, high efficiency, and high-speed heavy load [3]. Because of the advantages of stable operation, compact structure, high speed, and heavy load, the hydrostatic spindle is more and more widely used in high-end machine tools [4, 5]. As an important performance parameter of the hydrostatic spindle system, the stiffness of the spindle system plays a significant role in determining the service performance of the spindle system [6–8]. Therefore, it is necessary to study the theoretical model of spindle system performance and to analyze the influence rules of the spindle system stiffness, so as to optimize the structure of the spindle system in the design stage of machine tools.At present, scholars at home and abroad have done a lot of research in this field. Xia et al. [9] used FLUENT, a computational fluid dynamics software based on the finite element volume method, to predict the bearing capacity, stiffness, and other performances of the hydrostatic spindle. Guo et al. [10] established a high-speed hydrostatic builtin motorized spindle system model and proposed a multiparameter optimization method of the motorized spindle system based on genetic algorithm. Chen et al. [11] presented a design system for a hydrostatic spindle based on the laws of the fluid mechanics, engineering thermodynamics, and rotor dynamics. Gaber and Hashemi [12] developed a calibrated dynamics stiffness matrix (CDSM) method, which could be effectively applied to predict the vibration characteristics of spindle systems.The test methods of spindle system performance mainly include the time domain analysis method and frequency domain analysis method [13, 14]. The method of time domain analysis is to identify stiffness and damping coefficients by using time domain vibration data of the spindle system. Through exerting sinusoidal excitation to the spindle system, the frequency domain analysis method identifies dynamic characteristics according to the vibration responses at different frequencies [15–17]. The most typical time domain analysis is the hammering method. Through exerting an impulse excitation to the rotor bearing system by a hammer, the stiffness coefficient, damping coefficient, and inherent frequency of the rotor bearing system can be identified by analyzing the free vibration response of the rotor bearing system [3, 18].Taking the motorized system of a grinding machine as the research object, the theoretical stiffness model of the spindle system considering rotor swing is established in this paper, and the contribution degrees of bearings to spindle system stiffness are evaluated. The stiffness of the spindle system is tested by the hammering method, and the test results are compared with the theoretical calculation.


## 2. Stiffness Model of Spindle System considering Rotor Swing

For the motorized spindle system of a grinding machine, the rotor is supported by two hydrostatic bearings, which is shown in Figure1. The mass of the spindle rotor is m, and the rotational inertia round the mass center is J. The stiffness coefficients of two support bearings are k2 and k1. The translational displacement of the mass center of the motorized spindle rotor is xc under the external load F, and the rotation angle is θ. The lengths of the mass center to the bearing at the front and back ends of spindle are L2 and L1, respectively. The length of the overhang end is L3.Figure 1
Schematic diagram of force analysis of the spindle system.When the spindle gravity and bearing damping of the spindle are ignored, the two-degree-of-freedom motion differential equations of the spindle system, which include the translation and swing, are as follows:(1)m00Jx¨cθ¨+k1+k2k2L2−k1L1k2L2−k1L1k1L12+K2L22xcθ=FFL2+L3.If the external load isF=F¯cosωt, where F¯ is the amplitude of the external load, the system responses are as follows:(2)xc=x¯ccosωt+φc,θ=θ¯cosωt+φθ.The amplitudes of the responses are shown as follows by solving equations (1) and (2), simultaneously.(3)x¯c=F¯k1L12+k2L22−Jω2−k2L2−k1L1L2+L3k1+k2−mω2k1L12+k2L22−Jω2−k2L2−k1L12,θ¯=F¯k1+k2−mω2L2+L3−k2L2−k1L1k1+k2−mω2k1L12+k2L22−Jω2−k2L2−k1L12.Ifx¯ is the amplitude of displacement of shaft end, then x¯=x¯c+θ¯L2+L3. The displacement of shaft end under external load F can be obtained as follows:(4)x¯=F¯k1+k2−mω2L2+L32+k1L12+k2L22−Jω2−2k2L2−k1L1L2+L3k1+k2−mω2k1L12+k2L22−Jω2−k2L2−k1L12.The stiffness of the spindle system is(5)K=F¯x¯=k1+k2−mω2k1L12+k2L22−Jω2−k2L2−k1L12k1+k2−mω2L2+L32+k1L12+k2L22−Jω2−2k2L2−k1L1L2+L3.


## 3. Analysis of Influence Factors of Spindle System Stiffness

### 3.1. Parameters of Spindle System

The motorized spindle system of a grinding machine supported by two hydrostatic bearings and the spindle rotor is a hollow shaft. The structural parameters of the rotor of the spindle system are shown in Table1. The hydrostatic bearing parameters at the front and back ends are shown in Table 2.Table 1
Structure and working condition parameters of the spindle system rotor.
ItemSymbolUnitValueMass of rotormkg13.8Length of center of mass to front-end bearingL2mm140.9Length of center of mass to back-end bearingL1mm289.1Length of the overhang endL3mm130Inner diameter of hollow spindlermm55External load at shaft endFN145Speednr/min2600Rotational inertiaJkg·m20.059685Table 2
Structure and working condition parameters of hydrostatic bearing.
ItemSymbolUnitFront-end bearingBack-end bearingDiameter of bearingDmm8569Width of bearingLmm100120Axial width of oil seal surfacel1mm8.56.9Circumferential width of oil seal surfaceb1mm8.56.9Width of oil return grooveb2mm44Radial gap of hydrostatic cavitycμm2518Diameter of throttling orificed0mm1.21Wrap angle of hydrostatic cavity and oil seal surface2θ2°84.683.3Wrap angle of hydrostatic cavity2θ1°61.760.4Dynamic viscosity of lubricating oilηpa·s0.0132Density of lubricating oilρkg·m−3882Supply pressure of lubricating oilPspa2.5 × 106For the hydrostatic bearing with an orifice-type restrictor in the spindle system, the oil film stiffness of the hydrostatic cavity can be calculated as follows [19]:(6)Kj=12PsAeβ−1cosθ1cβ2β−1,where Ae is the effective bearing area, and its calculation equation is as follows [20]:(7)Ae=DL−l1sinθ1+θ22,where β is the throttling ratio of the orifice-type restrictor, which can be calculated by the following equation [19]:(8)β=0.5+128ρPsR2c6ll1/Rb1+2θ129α2π2η2l12d04+1,where α is the flow coefficient of the orifice-type restrictor, which is generally selected from 0.6 to 0.7. In this paper, α is set as 0.7. R is the bearing radius and R = D/2. l is the length of hydrostatic cavity, and l = L − 2l1. The meanings of other symbols are shown in Table 2.Through simultaneous equations from (6) to (8), it can be obtained that the front-end bearing stiffness k2 of the spindle system is 5.40 × 107 N/m, and the back-end bearing stiffness k1 is 7.08 × 107 N/m.


### 3.2. Amplitude and Stiffness of Spindle System under Different Excitation Frequencies

By taking the structural parameters of the spindle rotor in Table1 and the bearing stiffness of the front and back ends into equation (5), the stiffness and amplitude of the spindle system under different excitation frequencies can be obtained, which is shown in Figure 2. The stiffness of the spindle system is not very sensitive to the excitation frequency. In the range of excitation frequency from 0 to 50 Hz, the stiffness of the spindle system decreases slightly with the increase of the excitation frequency, and the reduction amplitude is about 0.66%.Figure 2
Stiffness and amplitude of the spindle system under different excitation frequencies.


### 3.3. Influence Rules of Bearing Stiffness on Spindle System Stiffness

The stiffness of the spindle system is greatly affected by the bearing stiffness, and it is also related to the position distribution of the bearing. Under the different bearing stiffness, the change rules of spindle system stiffness are shown in Figure3, where point A represents the corresponding position in the figure of the bearing stiffness in this case. For the motorized spindle system studied in this paper, the stiffness of the spindle mainly depends on the stiffness of the front-end bearing and almost linearly increases with the increase of the front-end bearing stiffness. The stiffness value of the spindle is about 55% of the front-end bearing in this case.Figure 3
Change rules of spindle system stiffness with different bearing stiffness (2600 r/min).


## 3.1. Parameters of Spindle System

The motorized spindle system of a grinding machine supported by two hydrostatic bearings and the spindle rotor is a hollow shaft. The structural parameters of the rotor of the spindle system are shown in Table1. The hydrostatic bearing parameters at the front and back ends are shown in Table 2.Table 1
Structure and working condition parameters of the spindle system rotor.
ItemSymbolUnitValueMass of rotormkg13.8Length of center of mass to front-end bearingL2mm140.9Length of center of mass to back-end bearingL1mm289.1Length of the overhang endL3mm130Inner diameter of hollow spindlermm55External load at shaft endFN145Speednr/min2600Rotational inertiaJkg·m20.059685Table 2
Structure and working condition parameters of hydrostatic bearing.
ItemSymbolUnitFront-end bearingBack-end bearingDiameter of bearingDmm8569Width of bearingLmm100120Axial width of oil seal surfacel1mm8.56.9Circumferential width of oil seal surfaceb1mm8.56.9Width of oil return grooveb2mm44Radial gap of hydrostatic cavitycμm2518Diameter of throttling orificed0mm1.21Wrap angle of hydrostatic cavity and oil seal surface2θ2°84.683.3Wrap angle of hydrostatic cavity2θ1°61.760.4Dynamic viscosity of lubricating oilηpa·s0.0132Density of lubricating oilρkg·m−3882Supply pressure of lubricating oilPspa2.5 × 106For the hydrostatic bearing with an orifice-type restrictor in the spindle system, the oil film stiffness of the hydrostatic cavity can be calculated as follows [19]:(6)Kj=12PsAeβ−1cosθ1cβ2β−1,where Ae is the effective bearing area, and its calculation equation is as follows [20]:(7)Ae=DL−l1sinθ1+θ22,where β is the throttling ratio of the orifice-type restrictor, which can be calculated by the following equation [19]:(8)β=0.5+128ρPsR2c6ll1/Rb1+2θ129α2π2η2l12d04+1,where α is the flow coefficient of the orifice-type restrictor, which is generally selected from 0.6 to 0.7. In this paper, α is set as 0.7. R is the bearing radius and R = D/2. l is the length of hydrostatic cavity, and l = L − 2l1. The meanings of other symbols are shown in Table 2.Through simultaneous equations from (6) to (8), it can be obtained that the front-end bearing stiffness k2 of the spindle system is 5.40 × 107 N/m, and the back-end bearing stiffness k1 is 7.08 × 107 N/m.


## 3.2. Amplitude and Stiffness of Spindle System under Different Excitation Frequencies

By taking the structural parameters of the spindle rotor in Table1 and the bearing stiffness of the front and back ends into equation (5), the stiffness and amplitude of the spindle system under different excitation frequencies can be obtained, which is shown in Figure 2. The stiffness of the spindle system is not very sensitive to the excitation frequency. In the range of excitation frequency from 0 to 50 Hz, the stiffness of the spindle system decreases slightly with the increase of the excitation frequency, and the reduction amplitude is about 0.66%.Figure 2
Stiffness and amplitude of the spindle system under different excitation frequencies.


## 3.3. Influence Rules of Bearing Stiffness on Spindle System Stiffness

The stiffness of the spindle system is greatly affected by the bearing stiffness, and it is also related to the position distribution of the bearing. Under the different bearing stiffness, the change rules of spindle system stiffness are shown in Figure3, where point A represents the corresponding position in the figure of the bearing stiffness in this case. For the motorized spindle system studied in this paper, the stiffness of the spindle mainly depends on the stiffness of the front-end bearing and almost linearly increases with the increase of the front-end bearing stiffness. The stiffness value of the spindle is about 55% of the front-end bearing in this case.Figure 3
Change rules of spindle system stiffness with different bearing stiffness (2600 r/min).


## 4. Vibration Test and Stiffness Analysis of Spindle System

### 4.1. Test Method and Scheme of Spindle System Stiffness

In general, the hydrostatic bearing rotor system is underdamped. The time domain waveform of its free vibration response is shown in Figure4. The stiffness coefficient, damping coefficient, and inherent frequency of the system can be calculated according to the following equation.(9)ωn=2π⋅f=Km,ωd=ωn1−ζ2,δ=lnMiMi+1,ζ=δ2δ2+4π2,C=2m⋅ζ⋅ωn,where δ is the logarithmic decrement, Mi is the amplitude, ζ is the damping ratio, ωd is the damped inherent frequency, ωn is the undamped inherent frequency, C is the damping coefficient, and K is the stiffness coefficient.Figure 4
Damped free vibration response of the rotor bearing system.The hydrostatic spindle system test bed used in this paper is shown in Figure5. The eddy current displacement sensors are installed in the horizontal and vertical directions of the front and back ends of the spindle. The installation layout of eddy current displacement sensors is shown in Figure 6. The eddy current displacement sensors are installed on an annular bracket connected with the shell, and the installation angle is 90°. The horizontal and vertical vibrations of the front and back ends of the spindle are tested, respectively.Figure 5
Testing scheme of spindle system stiffness by the hammering method.Figure 6
Layout of eddy current displacement sensors at the front and back ends of the spindle.An impulse excitation is applied to the overhanging end of the spindle system through the force hammer in the stiffness experiment. According to the data of the eddy current displacement sensor, the errors caused by the spindle deviation can be eliminated. The stiffness of the spindle system can be identified by analyzing the free vibration response of the spindle system.


### 4.2. Results and Analysis of Stiffness Test

#### 4.2.1. Vibration Test of Shaft End of Spindle System

When the spindle speed is 2600 r/min, the vibration test results of the spindle front end are shown in Figures7–9. The time domain waveform in the vertical direction is shown in Figure 7(a). The maximum amplitude in the vertical direction of the spindle is about 5 μm. The vibration frequency spectrum of the vertical direction is shown in Figure 7(b). The maximum amplitude corresponding to the rotation frequency of the spindle is about 2.2 μm.Figure 7
Vertical vibration test results of spindle front end (2600 r/min). (a) Time domain oscillogram of vertical vibration. (b) Frequency spectrogram of vertical vibration.
(a)(b)Figure 8
Horizontal vibration test results of spindle front end (2600 r/min). (a) Time domain oscillogram of horizontal vibration. (b) Frequency spectrogram of horizontal vibration.
(a)(b)Figure 9
Free attenuation curve in the vertical direction of the spindle front end (2600 r/min).The maximum amplitude of the horizontal direction of the spindle front end shown in Figure8(a) is about 7 μm, and the maximum amplitude corresponding to the spindle rotation frequency shown in Figure 8(b) is about 2.5 μm. The stiffness of the front end of the spindle can be identified as 3.62 × 107 N/m by using the hammering method shown in Figure 9.


#### 4.2.2. Identification of Supporting Stiffness of Spindle System

By changing the spindle speed through the frequency converter, the vibration data of the spindle at different speeds are as shown in Figure10.Figure 10
Stiffness and amplitudes of the frequency multiplication of the spindle front end via the speed.The amplitudes of the frequency multiplication in both directions of the front end of the spindle decrease with the increase of the speed, and the stiffness of the spindle front end also tends to decrease. At about 2300 r/min, the stiffness is about 3.0 × 107 N/m, and the corresponding vertical and horizontal amplitudes are 2.5 μm and 2.9 μm, respectively.


#### 4.2.3. Analysis of Test Results

The stiffness value of the spindle system can be obtained as 3.41 × 107 N/m by averaging the 9 groups of the stiffness values under the speed from 0 to 2600 r/min. Similarly, the average value of the corresponding theoretical stiffness is 3.89 × 107 N/m, as shown in Figure 11. The error between the test identification value and the theoretical calculation value is 14.21%, which means that the two values match each other well.Figure 11
Test value and theoretical value of spindle stiffness via the speed.It should be noted that the natural frequencies of the two-degree-of-freedom spindle system are 427.543 Hz and 1735.68 Hz. The maximum test frequency is 43.33 Hz, so the respective frequency ratios are about 0.101 and 0.024. Therefore, it is shown that the stiffness test values in the lower frequency range are in agreement with the theoretical values, which also indicates that the model established in this paper can better reflect the performance of the spindle in the lower frequency range.


## 4.1. Test Method and Scheme of Spindle System Stiffness

In general, the hydrostatic bearing rotor system is underdamped. The time domain waveform of its free vibration response is shown in Figure4. The stiffness coefficient, damping coefficient, and inherent frequency of the system can be calculated according to the following equation.(9)ωn=2π⋅f=Km,ωd=ωn1−ζ2,δ=lnMiMi+1,ζ=δ2δ2+4π2,C=2m⋅ζ⋅ωn,where δ is the logarithmic decrement, Mi is the amplitude, ζ is the damping ratio, ωd is the damped inherent frequency, ωn is the undamped inherent frequency, C is the damping coefficient, and K is the stiffness coefficient.Figure 4
Damped free vibration response of the rotor bearing system.The hydrostatic spindle system test bed used in this paper is shown in Figure5. The eddy current displacement sensors are installed in the horizontal and vertical directions of the front and back ends of the spindle. The installation layout of eddy current displacement sensors is shown in Figure 6. The eddy current displacement sensors are installed on an annular bracket connected with the shell, and the installation angle is 90°. The horizontal and vertical vibrations of the front and back ends of the spindle are tested, respectively.Figure 5
Testing scheme of spindle system stiffness by the hammering method.Figure 6
Layout of eddy current displacement sensors at the front and back ends of the spindle.An impulse excitation is applied to the overhanging end of the spindle system through the force hammer in the stiffness experiment. According to the data of the eddy current displacement sensor, the errors caused by the spindle deviation can be eliminated. The stiffness of the spindle system can be identified by analyzing the free vibration response of the spindle system.


## 4.2. Results and Analysis of Stiffness Test

### 4.2.1. Vibration Test of Shaft End of Spindle System

When the spindle speed is 2600 r/min, the vibration test results of the spindle front end are shown in Figures7–9. The time domain waveform in the vertical direction is shown in Figure 7(a). The maximum amplitude in the vertical direction of the spindle is about 5 μm. The vibration frequency spectrum of the vertical direction is shown in Figure 7(b). The maximum amplitude corresponding to the rotation frequency of the spindle is about 2.2 μm.Figure 7
Vertical vibration test results of spindle front end (2600 r/min). (a) Time domain oscillogram of vertical vibration. (b) Frequency spectrogram of vertical vibration.
(a)(b)Figure 8
Horizontal vibration test results of spindle front end (2600 r/min). (a) Time domain oscillogram of horizontal vibration. (b) Frequency spectrogram of horizontal vibration.
(a)(b)Figure 9
Free attenuation curve in the vertical direction of the spindle front end (2600 r/min).The maximum amplitude of the horizontal direction of the spindle front end shown in Figure8(a) is about 7 μm, and the maximum amplitude corresponding to the spindle rotation frequency shown in Figure 8(b) is about 2.5 μm. The stiffness of the front end of the spindle can be identified as 3.62 × 107 N/m by using the hammering method shown in Figure 9.


### 4.2.2. Identification of Supporting Stiffness of Spindle System

By changing the spindle speed through the frequency converter, the vibration data of the spindle at different speeds are as shown in Figure10.Figure 10
Stiffness and amplitudes of the frequency multiplication of the spindle front end via the speed.The amplitudes of the frequency multiplication in both directions of the front end of the spindle decrease with the increase of the speed, and the stiffness of the spindle front end also tends to decrease. At about 2300 r/min, the stiffness is about 3.0 × 107 N/m, and the corresponding vertical and horizontal amplitudes are 2.5 μm and 2.9 μm, respectively.


### 4.2.3. Analysis of Test Results

The stiffness value of the spindle system can be obtained as 3.41 × 107 N/m by averaging the 9 groups of the stiffness values under the speed from 0 to 2600 r/min. Similarly, the average value of the corresponding theoretical stiffness is 3.89 × 107 N/m, as shown in Figure 11. The error between the test identification value and the theoretical calculation value is 14.21%, which means that the two values match each other well.Figure 11
Test value and theoretical value of spindle stiffness via the speed.It should be noted that the natural frequencies of the two-degree-of-freedom spindle system are 427.543 Hz and 1735.68 Hz. The maximum test frequency is 43.33 Hz, so the respective frequency ratios are about 0.101 and 0.024. Therefore, it is shown that the stiffness test values in the lower frequency range are in agreement with the theoretical values, which also indicates that the model established in this paper can better reflect the performance of the spindle in the lower frequency range.


## 4.2.1. Vibration Test of Shaft End of Spindle System

When the spindle speed is 2600 r/min, the vibration test results of the spindle front end are shown in Figures7–9. The time domain waveform in the vertical direction is shown in Figure 7(a). The maximum amplitude in the vertical direction of the spindle is about 5 μm. The vibration frequency spectrum of the vertical direction is shown in Figure 7(b). The maximum amplitude corresponding to the rotation frequency of the spindle is about 2.2 μm.Figure 7
Vertical vibration test results of spindle front end (2600 r/min). (a) Time domain oscillogram of vertical vibration. (b) Frequency spectrogram of vertical vibration.
(a)(b)Figure 8
Horizontal vibration test results of spindle front end (2600 r/min). (a) Time domain oscillogram of horizontal vibration. (b) Frequency spectrogram of horizontal vibration.
(a)(b)Figure 9
Free attenuation curve in the vertical direction of the spindle front end (2600 r/min).The maximum amplitude of the horizontal direction of the spindle front end shown in Figure8(a) is about 7 μm, and the maximum amplitude corresponding to the spindle rotation frequency shown in Figure 8(b) is about 2.5 μm. The stiffness of the front end of the spindle can be identified as 3.62 × 107 N/m by using the hammering method shown in Figure 9.


## 4.2.2. Identification of Supporting Stiffness of Spindle System

By changing the spindle speed through the frequency converter, the vibration data of the spindle at different speeds are as shown in Figure10.Figure 10
Stiffness and amplitudes of the frequency multiplication of the spindle front end via the speed.The amplitudes of the frequency multiplication in both directions of the front end of the spindle decrease with the increase of the speed, and the stiffness of the spindle front end also tends to decrease. At about 2300 r/min, the stiffness is about 3.0 × 107 N/m, and the corresponding vertical and horizontal amplitudes are 2.5 μm and 2.9 μm, respectively.


## 4.2.3. Analysis of Test Results

The stiffness value of the spindle system can be obtained as 3.41 × 107 N/m by averaging the 9 groups of the stiffness values under the speed from 0 to 2600 r/min. Similarly, the average value of the corresponding theoretical stiffness is 3.89 × 107 N/m, as shown in Figure 11. The error between the test identification value and the theoretical calculation value is 14.21%, which means that the two values match each other well.Figure 11
Test value and theoretical value of spindle stiffness via the speed.It should be noted that the natural frequencies of the two-degree-of-freedom spindle system are 427.543 Hz and 1735.68 Hz. The maximum test frequency is 43.33 Hz, so the respective frequency ratios are about 0.101 and 0.024. Therefore, it is shown that the stiffness test values in the lower frequency range are in agreement with the theoretical values, which also indicates that the model established in this paper can better reflect the performance of the spindle in the lower frequency range.


## 5. Conclusion

(1)
The force analysis of the hydrostatic spindle system is implemented, and a stiffness model of the spindle system considering rotor swing is established.(2)
The stiffness and amplitude of shaft end under different excitation frequencies are analyzed. The results show that the stiffness of the front-end bearing has a greater impact on the stiffness of the spindle system.(3)
The stiffness of the spindle system is identified by the hammering method and compared with the theoretical calculation stiffness, so that the theoretical model can be preliminarily verified. In the low frequency range, the research results can provide theoretical and data support for the design of hydrostatic bearing and the evaluation of spindle stiffness.



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