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Composite Structures 55 (2002) 247±259
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Design of carbon ®ber composite shafts for high speed air spindles Kyung Geun Bang, Dai Gil Lee
*
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Mechanical Design Laboratory with Advanced Material, ME3221, 373-1 Kusong-dong, Yusong-gu, Taejon-shi 305-701, South Korea
Abstract For the stable operation of high speed air spindles, the low rotational inertia and high damping ratio of spindle shafts as well as high fundamental natural frequency are indispensable. Conventional steel spindles are not appropriate for high speed operation because of their high rotational inertia and low damping ratio. In this study, a high-speed air spindle composed of a carbon ®ber epoxy composite shaft and two steel ¯anges was designed for maximum critical speed considering both the de¯ection due to bending load and the radial expansion due to centrifugal force and temperature rise during high-speed rotation. The stacking angle and thickness of the composite shaft and the adherend dimensions of the steel ¯anges were selected through vibrational analysis as well as considering the bending stiness and centrifugal characteristics. Ó 2002 Published by Elsevier Science Ltd. Keywords: Composite air spindle; Composite shaft; Air bearing; Critical speed; Natural frequency; Static de¯ection; Radial expansion
1. Introduction These days, high precision products are widely required with job shop type production and small batch production in the ®elds of manufacturing and machining [1]. For this end, high speed and high precision air spindles are widely used as the components of hard disk drives for computers, dental drills and machining of polygon mirrors for laser scanners because high speed rotation with small heat generation is possible for air spindles due to the low viscosity of the air lubricant. However, steel shafts mounted on conventional air spindles may cause either unstable operation due to whirling vibration of the shaft at relatively high rotational speed, or rupture of the air lubricant by radial expansion of the shaft due to centrifugal force. Until now diverse methods for stable operation of the air spindle have been suggested and investigated by many researchers some of which are listed as follows. Hirn [2] suggested the air for working ¯uid of bearings, and Gross [3] found that the existence of start speed of unstable operation in externally pressurized air bearing through experiment. Larson [4] found that the start speed of unstable operation increased as the supply pressure increased and the distance of bearing supply holes *
Corresponding author. Tel.: +82-42-869-3221; fax: +82-42-8693210. E-mail address: [email protected] (D.G. Lee).
decreased, from which he analyzed the spindle characteristics using the lumped parameter method. Taniguchi [5] investigated the operating characteristics of a spindle with respect to supply hole numbers, supply pressure and bearing length through experiment, from which he obtained the optimal bearing length for increasing the start speed of unstable operation. Blondeel et al. [6] modeled the air bearing as a control system with feedback loop and evaluated the dynamic stability of externally pressurized air bearing through frequency analysis. In order to minimize the unstable operation of air spindles, air bearings with non-circular cross-section and active control method were suggested by other researchers [7,8]. However, the diculty of machining precise non-circular cross-sections of the air bearing and additional devices required for active control prohibited their widespread use. Therefore, a spindle shaft made of high speci®c stiness and high damping material will be bene®cial for the improvement of stability of the air spindle. To this end, carbon ®ber composites are appropriate because they have high speci®c stiness E=q, high speci®c strength S=q and good damping property as well as thermal stability due to their low coecient of thermal expansion. There have been several attempts to employ carbon epoxy composite materials as the shaft material of air spindle. Lee and Choi [9±11] investigated the dynamic characteristics of high speed spindles manufactured with carbon composite shaft supported by ball bearings and air bearings.
0263-8223/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 4 6 - 5
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In this work, a high-speed spindle composed of carbon ®ber epoxy composite shaft and steel ¯anges was designed for maximum critical speed considering both the static de¯ection due to bending load, the radial expansion due to centrifugal force and temperature rise during high-speed rotation.
Table 1 Speci®cations of the air spindle for machining of wafers Outer diameter of shaft (mm) Length of shaft (mm) Shaft section shape Bearing clearance (lm) Max. radial load capacity (N) Max. radial stiness (MN/m) Operating speed (rpm)
25 210 Solid 15 50 6 40,000±80,000
2. Objective of the carbon ®ber composite spindle shaft Fig. 1 shows a typical steel air spindle used for machining wafers and drilling printed circuit boards. The speci®cations of the spindle are listed in Table 1. Since the machinability and surface roughness of wafers are improved as the speed of machining is increased, the development and employment of higher speed air spindles has been spurred. Therefore, in this study, the air spindle shaft was designed with carbon ®ber epoxy composite material whose properties are listed in Table 2 instead of the conventional steel shaft which is inappropriate for high speed air spindles due to its heavy weight and low fundamental bending natural frequency. Since conventional air spindles are composed of air bearings, steel shaft and electrical AC motor as shown in Fig. 1(b), they may be modeled as a beam of bending stiness EI supported by several springs of stiness k as shown in Fig. 2. The performance of the air spindle is
Table 2 Properties of the unidirectional carbon ®ber/epoxy composite materiala Longitudinal modulus Transverse modulus Longitudinal CTE Transverse CTE Major poisson's ratio Minor poisson's ratio Density Thickness of one ply Longitudinal tensile strength S1t Longitudinal compressive strength S1c Transverse tensile strength S2t ; S3t Transverse compressive strength (S2c ; S3c 12 and 13-direction shear strength S12 ; S13 23-direction shear strength S23 a
131 GPa 8 GPa 27 lm=m °C 0:9 lm °C 0.28 0.0171 1560 kg=m3 0.15 mm 2000 MPa )1400 MPa 61 MPa )130 MPa 70 MPa 40 MPa
Manufactured by SK Chemical in Korea (USN150).
Fig. 2. Analytic model of the air spindle.
determined by natural frequency and static stiness, which are functions of bending stiness of the shaft, stiness of air bearing and shaft mass. The shaft mass and the stiness of the air bearing are two major factors which in¯uence the fundamental and second natural frequencies of the air spindle. However, the third natural frequency is determined by bending stiness and mass density per unit length of the shaft. The static stiness of the air spindle at the cutting point is determined by the bending stiness of the shaft and the radial stiness of the air bearing. For the air bearing of Fig. 3, the bearing pressure p1 is calculated as follows under the assumption of isothermal condition [12]. pq0 Dh3 2 p 192lLp0 1
p02 sq 2cqs ps 2=c c1=c p1 =ps ; C d At p1 =ps c 1
Fig. 1. Air spindle for machining of wafers: (a) photograph; (b) schematic diagram.
1
where l; q0 ; qs ; Cd ; At ; c; D; h; L; p0 represent the dynamic air viscosity, air density at atmosphere, air density in the ori®ce inlet, ori®ce coecient, area of the ori®ce outlet, speci®c heat ratio, diameter of the shaft, bearing gap, pocket length and pressure at atmosphere, respectively.
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Fig. 3. Aerostatic analysis of the air bearing.
If the bearing pressure is assumed to be a function of bearing gap h, the pressure dierence in the air bearing of Fig. 3 is represented as follows: Dp p1 max
p1 min f C
e
f C e:
2
From Eq. (2), the stiness of the air bearing is represented as follows: R Dp dA Load ; 3 Stiffness k e e where A represents the circumferential area of the shaft. The load capacity and stiness were calculated numerically using Eqs. (1)±(3). Fig. 4 shows the load capacity and stiness of the air bearing of Table 3. In Fig. 4, the stiness of the air bearing increases as the radial clearance increases and reaches a maximum value at 15 lm radial clearance and then decreases for the speci®cations of Table 3. Since the radial clearance of the air bearing changes as the shaft expands radially due to centrifugal force and temperature rise, the stiness of the air bearing is aected by thermo-mechanical characteristics of the shaft. Therefore, the shaft of the air spindle should be designed considering both dynamic and static characteristics. The main objective of this study is to increase the operating rotational speed of the air spindle from 80,000 rpm which is the maximum operational speed of the steel spindle of Fig. 1 to 120,000 rpm as well as to maximize the static stiness of the air spindle. The speci®cations of the carbon composite shaft in Table 4 are the same as in Table 1 except the maximum operating rotational speed. For the carbon ®ber composite spindle, two steel ¯anges adhesively joined to the carbon ®ber shaft as shown in Fig. 5 were employed for mounting a cutting tool and the rotor of an AC electrical motor. Since the bending stiness and mass of the spindle shaft are dependent on the dimensions of the steel ¯anges, the adherend length La and thickness ta except the overhung parts were selected as the design variables in addition to the stacking angle and thickness of the carbon composite shaft as listed in Table 5. The dimensions of the overhung part were ®xed to the same as the steel shaft for mounting a cutting tool and the
Fig. 4. Static characteristics of the air bearing w.r.t. eccentricity ratio e and radial clearance h: (a) load capacity; (b) stiness.
Table 3 Speci®cations of the air bearing Air density q0 and qs Dynamic air viscosity l Speci®c heat ratio c Supply pressure ps Atmosphere pressure p0 Pocket length L Bearing length Radial clearance C Ori®ce coecient Cd Ori®ce outlet diameter At pdt2 =4
1:203 kg=m3 1.8E ) 5 N s/m2 1.4 0.5 MPa 0.1 MPa 20 mm 60 mm 15 lm 0.6 1.5 mm
Table 4 Speci®cations of the carbon composite shaft Outer diameter of shaft (mm) Length of shaft (mm) Shaft section shape Radial stiness of air bearing (MN/m) Max. operating speed (rpm)
25 210 Hollow 6 120,000
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Fig. 5. Schematic diagram of the steel ¯anges and the composite shaft.
Table 5 Design variables of the carbon composite shaft Carbon composite
Stacking angle hc Stacking thickness tc
Steel ¯ange
Adherend length La Adherend thickness ta
Fig. 6. Model of the composite shaft and the air bearing for FE analysis: (a) without steel ¯anges; (b) with steel ¯anges.
rotor of an AC electrical motor used in the conventional steel spindle.
3. Dynamic characteristics of the carbon ®ber composite spindle shaft The maximum operating rotational speed of the air spindle is limited by the natural frequency of the spindle shaft. Therefore, the main design objective of the carbon ®ber composite spindle shaft is to increase the natural frequency of the air spindle. In this study, to ®nd the basic modes of vibration of the air spindle, ®nite element analysis of the shaft without the steel ¯anges was performed ®rst using the model in Fig. 6(a). The commercial FEM software, ANSYS was used for the analysis. The shaft and the air bearings were modeled as 1-D beam elements and 1-D spring elements, respectively. Figs. 7 and 8 represent the mode shapes and natural frequencies of the shaft without steel ¯anges, respectively. In Fig. 7, the ®rst and the second mode shapes represent the translation and conical modes, respectively due to ¯exibility of the air bearing, while the third mode shape represents the bending mode of the shaft. Fig. 8 shows the calculated natural frequencies with respect to thickness of the carbon ®ber composite shaft. In Fig. 8, the natural frequencies in the ®rst and the second modes are little dependent on the shaft axial modulus because
Fig. 7. Mode shapes of the air spindle without steel ¯anges: (a) fundamental mode; (b) second mode; (c) third mode.
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ered as the limit speed where the spindle shaft or air bearing could fail by resonance. Therefore, the limit of operating speed of the air spindle is generally considered as the third natural frequency corresponding to bending mode. When the two steel ¯anges are adhesively bonded to both ends of the shaft, the mode shapes and natural frequencies of the spindle shaft may be dierent from those without steel ¯anges as shown in Figs. 7 and 8. Therefore, the composite spindle shaft with the steel ¯anges in Fig. 6(b) was analyzed again with ®nite element analysis. Figs. 9 and 10 represent the mode shapes and natural frequencies, respectively of the composite spindle shaft with the steel ¯anges. The translation mode of the composite spindle shaft with steel ¯anges disappears as shown in Fig. 9, which was present in the mode shape of the composite shaft without steel ¯anges as shown in Fig. 7(a) due to the overhang of the steel ¯anges on both ends of the shaft. On the contrary, the conical mode of Fig. 7(b) changed from the rigid mode into the mixed modes of Figs. 9(a) and (b), which are composed of conical mode and partial bending mode
Fig. 8. Natural frequencies of the air spindles made of composite material and steel without steel ¯anges w.r.t. thickness of the shaft: (a) fundamental natural frequency; (b) second natural frequency; (c) third natural frequency.
the shaft behaves as a rigid body, while, in the third mode, the natural frequency of the air spindle is much dependent on the shaft modulus because the shaft behaves as a ¯exible beam relative to the air bearing. During the operation of air bearings, the fundamental and second natural frequencies of the spindle shaft can be eliminated by reducing the eccentricity and unbalance of the shaft mass through dynamic balancing of the shaft. However, the third natural frequency occurring in bending mode due to shaft ¯exibility is consid-
Fig. 9. Mode shapes of the air spindle with steel ¯anges: (a) fundamental mode; (b) second mode; (c) third mode.
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the spindle shaft without the steel ¯ange as shown in Figs. 8(a) and (b) in which the fundamental and second natural frequency decrease as the shaft thickness increases. In Fig. 10(c), the third natural frequency of the steel shaft with solid section is about 85,800 rpm (1430 Hz), which is below the design value of 120,000 rpm. While, the third natural frequency of the carbon composite spindle shaft is higher than 120,000 rpm (2000 Hz) when the shaft thickness is less than 3 mm if the axial modulus is larger than 90 GPa. Therefore, in this work, the thickness of the carbon composite spindle shaft was determined to be 3 mm, and the axial modulus of the carbon composite spindle shaft was determined to be from 90 to 131 GPa. Also the natural frequency variation of the carbon composite spindle shaft was investigated with respect to the adherend length and adherend thickness of the steel ¯ange when the axial modulus and thickness of the carbon composite shaft were 105.8 GPa and 3 mm, respectively. Neglecting the stiness and mass of the adhesive, the bending stiness EI and mass per length qA of the overlapped length of the steel adherend and the carbon composite spindle shaft for the ®nite element model of Fig. 6(b) was modi®ed as follows: EIeq EIsteel EIcomposite ;
4
qAeq qAsteel qAcomposite :
5
Fig. 11 shows the ®rst three natural frequencies of the carbon ®ber composite spindle shaft w.r.t. adherend dimensions, in which the second and third natural frequencies decrease as the adherend length and thickness increase.
4. Static characteristics of the carbon ®ber composite spindle shaft
Fig. 10. Natural frequencies of the composite air spindles and the steel spindle w.r.t. thickness of the shaft: (a) fundamental natural frequency; (b) second natural frequency; (c) third natural frequency.
due to the front and rear ¯ange masses. It was found that the fundamental and the second modes of Figs. 9(a) and (b) were the partial bending and the conical modes of the rear and front steel ¯anges, respectively when the composite shaft thickness was less than 3 mm, while both modes became conical mode when the composite shaft thickness was larger than 3 mm. Also, the fundamental and second natural frequencies of the spindle shaft with the steel ¯anges become maximum when the thickness of the composite spindle shaft is around 3 mm as shown in Figs. 10(a) and (b), which is dierent from
The static stiness of the air spindle is dependent on the bending stiness of the shaft and the stiness of the air bearing. Since the stiness of the air bearing is much dependent on the bearing air gap, the radial expansion of the spindle shaft due to centrifugal force and temperature rise, and the bending stiness of the shaft should be considered in the design stage of the composite spindle shaft. From the vibrational analysis, it was found that the stacking angle between 0° and 20° satis®es the natural frequency requirement. However, the carbon composite shaft stacked with just one angle ply is inadequate for the high speed air spindle because the static stiness of the air bearing decreases by the radial expansion of the carbon composite shaft due to centrifugal force. The carbon composite shaft should have both the main stacking angle around 0° from the axial direction and
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Radial expansion (µm)
Fig. 12. Model of the composite shaft for ®nite element analysis.
Fig. 11. Natural frequencies of the composite air spindle w.r.t. length and thickness of the steel adherend when the axial modulus is 105.8 Gpa and the stacking thickness is 3 mm: (a) fundamental natural frequency; (b) second natural frequency; (c) third natural frequency.
the larger subsidiary stacking angle for the enhancement of circumferential modulus. In this work, the static characteristics of the carbon composite shaft stacked with 0° at the outer part of the shaft and 90° at inner part were investigated through ®nite element analysis using a commercial FEM software, ABAQUS under both bending load and centrifugal force as shown in Fig. 12. Since the deformation of the shaft which encapsulates the air bearing is important, the front part of the carbon composite spindle
Fig. 13. Axial modulus, eective bending stiness and radial expansion of the carbon composite shaft w.r.t. the subsidiary ply thickness when the rotational speed is 120,000 rpm: (a) axial modulus; (b) effective bending stiness and radial expansion.
shaft from Fig. 2 where the air bearing is mounted was analyzed with respect to the stacking thickness of the subsidiary angle when the thickness of the spindle shaft was 3 mm and the adherend length and the adherend
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Table 6 Stacking angles and thickness of the carbon composite shaft Stacking angle Stacking thickness (mm)
Main
Subsidiary
58 2.4
904 0.6
Fig. 14. Stacking patterns used in the evaluation of the composite shaft: (a) Group 1 of the stacking pattern; (b) Group 2 of the stacking pattern; (c) Group 3 of the stacking pattern.
thickness of the steel ¯anges were 10 and 0.5 mm, respectively. The static bending stiness of the carbon composite shaft where the air bearing is mounted was estimated by calculating the de¯ection of the node 2 point in Fig. 12 through ®nite element analysis when the load F of 50 N was applied at the end of the steel ¯ange, while ®xing all the axial displacements of cross-section including the
node point 1. Then the eective bending stiness keff of the carbon composite spindle shaft was de®ned as follows: Bending load keff : 6 dNode 2 Also, the maximum radial expansion of the carbon composite shaft was investigated through ®nite element analysis when the rotational speed of the spindle was 120,000 rpm. Only half of the composite shaft was analyzed owing to the symmetry of the shaft with respect to the cross-section including the node point 1. The eective bending stiness and the radial expansion of the carbon composite shaft with respect to stacking thickness of the subsidiary angle are shown in Fig. 13, in which the shafts without (0 mm) subsidiary plies and with 1.2 mm thickness subsidiary plies of 90° have the axial moduli of 131 and 90 GPa, respectively. In Fig. 13, both the radial expansion and the eective stiness of the carbon composite shaft without the subsidiary plies have large values. The eective bending stiness of the carbon composite shaft decreases distinctly and the radial expansion approaches a saturated value when the subsidiary thickness for the carbon composite shaft is larger than 0.6 mm. Therefore, the subsidiary ply thickness was determined to be 0.6 mm with stacking angle of 90°. Also, the main stacking angle was selected to be 5° to avoid the possibility of fracture during grinding process of the outer surface of the composite shaft. The selected stacking angles and thickness of the carbon composite shaft were listed in Table 6. To investigate the static characteristics of the carbon composite shaft w.r.t. the stacking sequences of the main and subsidiary plies and the adherend dimensions of the steel ¯ange, ®nite element analysis for the carbon composite shaft of Fig. 12 was performed with respect to stacking patterns of Fig. 14 under bending and centrifugal forces and temperature rise when the adherend length of the steel ¯ange varies from 10 to 50 mm and the adherend thickness varies from 0.5 to 5 mm. In Fig. 14, the group 1 has one cluster of 90° subsidiary plies, the group 2 has two clusters of 90° subsidiary plies, while the group 3 has more than two clusters 90° sub-
Table 7 Stacking sequence of stacking pattern for FE analysis Group 1
Pattern 1 Pattern 2 Pattern 3
58 = 904 T 904 = 58 T 54 = 904 = 54 T
Group 2
Pattern 4 Pattern 5 Pattern 6
902 = 58 = 902 T 902 = 54 = 902 = 54 T 54 = 902 = 54 = 902 T
Group 3
Pattern 7 Pattern 8 Pattern 9
52 =90= 52 = 902 = 52 =90= 52 T 90= 54 = 902 = 54 =90T 90= 52 =90= 54 =90= 52 =90T
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sidiary plies. The stacking sequences of the nine stacking patterns as shown in Fig. 14 are listed in Table 7. The eective bending stiness and the radial expansion of the carbon composite shaft were analyzed under the same condition as the static analysis mentioned in the previous paragraph. Also, the maximum radial ex-
Fig. 16. Eective stiness of the carbon composite shaft w.r.t. stacking patterns: (a) 10 mm adherend length; (b) 50 mm adherend length.
Fig. 15. Static characteristics of the carbon composite shaft stacked according to pattern 3 w.r.t. adherend length and adherend thickness of the ¯ange: (a) eective stiness; (b) radial expansion when the rotational speed is 120,000 rpm; (c) eective CTE.
Fig. 17. Radial expansion of the carbon composite shaft w.r.t. the stacking patterns when the rotational speed is 120,000 rpm and the adherend length is 20 mm.
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pansion of the shaft was calculated by ®nite element analysis when the temperature dierence DT is 50 °C to investigate the thermal expansion characteristics of the carbon composite spindle shaft. In this work, the eective radial coecient of thermal expansion (CTE) of the carbon composite spindle shaft was de®ned as follows: aeff
dr max ; DTr
7
where dr max and r represent maximum radial expansion and outer radius of the shaft, respectively. The eective bending stiness, the radial expansion and the eective radial CTE of the carbon composite shaft with respect to the adherend length and the adherend thickness of the steel ¯ange show similar trends regardless of the stacking patterns. The typical eective stiness and the radial expansion curves are shown in Fig. 15. In Figs. 15(a) and (b), the eective stiness of the carbon composite shaft becomes saturated beyond 2
Fig. 18. Eective CTE of the carbon composite shaft w.r.t. stacking patterns: (a) 10 mm adherend length; (b) 50 mm adherend length.
mm adherend thickness of the steel ¯ange and the radial expansion has a minimum value between 20 and 40 mm adherend length. Fig. 15(c) shows the typical eective radial CTE of the carbon composite spindle shaft calculated from Eq. (7), in which the eective radial CTE's of the carbon composite shaft increases as the adherend length of the steel ¯ange increases until the adherend length of 30 mm, while they became saturated beyond the adherend length of 30 mm. Figs. 16±18 show the maximum eective stiness, minimum radial expansion and minimum eective CTE of the carbon composite shaft of each stacking group, respectively. In Fig. 16, when the adherend length of the steel ¯ange is 10 mm, the eective stiness of the carbon composite shaft has a maximum value in case of stacking pattern 5 in which the subsidiary plies are stacked on both the inner and middle parts. When the adherend length of the steel ¯ange is 50 mm, the eective stiness of the carbon composite shaft has a maximum value in case of stacking pattern 3 in which the
Fig. 19. Radial clearance and static stiness of the air bearing w.r.t. the stacking patterns when the eccentricity ratio is 0.5: (a) radial clearance; (b) static stiness.
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subsidiary plies stacked on the middle part. Therefore, from Fig. 16, it has been found that the 90° subsidiary plies should be stacked between the inner and middle parts to enhance the bending stiness of the carbon composite shaft. In Fig. 17, the radial expansion of the carbon composite shaft has a minimum value in case of stacking pattern 2 which has the subsidiary angle in the inner part. Therefore, the 90° subsidiary angle should be stacked in the inner part to reduce the radial expansion of the carbon composite shaft due to centrifugal force. In Fig. 18, the eective radial CTE of the carbon composite shaft has a minimum value in case of stacking pattern 1 which has the subsidiary angle in the outer part. Therefore, the 90° subsidiary angle should be stacked in the outer part to reduce the radial thermal expansion of the carbon composite shaft. 5. Design of the carbon composite spindle shaft supported by air bearing The stacking angle and thickness of the composite shaft selected were listed in Table 6 considering the bending stiness and radial expansion of the carbon composite shaft as well as the natural frequencies and mode shapes of the air spindle. In Fig. 11, the natural frequencies of the carbon composite shaft decreased as the adherend length and thickness of the steel ¯ange increased, while the eective bending stiness increased as the adherend length and
257
thickness increased as shown in Fig. 15. Also, the radial expansion due to the centrifugal force has a minimum value when the adherend length has the values between 20 and 40 mm. In Fig. 11, the third natural frequency of the composite spindle shaft is higher than 120,000 rpm either when the adherend thickness of the steel ¯ange was less than 2 mm and the adherend length was 20 mm, or when the adherend thickness of the steel ¯ange was less than 1 mm and the adherend length was 30 mm. Therefore, in this work, the adherend length and thickness of the steel ¯ange were determined to be 30 and 1 mm, respectively. From the results of Figs. 16±18, the 90° subsidiary plies should be stacked between the inner and middle parts to increase bending stiness. In order to reduce the radial expansion and the eective radial CTE of the carbon composite shaft, the 90° subsidiary plies should be stacked in the inner and the outer parts, respectively. Since the heat generation in air bearings is usually negligible during operation, in this study, the design to the stacking sequence of the carbon composite shaft was focused on the eects of the bending of the carbon composite spindle shaft and the radial expansion due to centrifugal force on the static stiness of the air spindle. To investigate the eect of the radial expansion of carbon composite shaft due to centrifugal force on the static stiness of the air bearing, the radial clearance and static stiness of the air bearing were calculated when the air bearing rotational speed was 120,000 rpm as shown in Fig. 19. The radial clearance of the air bearing
Table 8 Results from static analysis of the carbon composite shaft according to stacking patterns Bending stiness of the shaft
Eective CTE of the shaft
Static stiness of the air bearing
Group 1
Pattern 1 Pattern 2 Pattern 3
± ± Medium
Low ± ±
± High ±
Group 2
Pattern 4 Pattern 5 Pattern 6
± High ±
± ± Medium
± Medium ±
Group 3
Pattern 7 Pattern 8 Pattern 9
Low Low Low
± High High
Low Low Low
Table 9 Speci®cations of the designed carbon composite shaft Outer diameter of shaft Length of shaft
25 mm 210 mm
Carbon composite
Stacking thickness Stacking sequence
3 mm 902 = 54 = 902 = 54 T
Steel ¯ange
Adherend length Adherend thickness
30 mm 1 mm
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index of 90° layer due to residual thermal stress is larger than 5 degree layer. Although the maximum failure index of 90° layer stacked on the middle part of the composite shaft is 0.57 above the half of the critical failure index, it was designed in this way because 90° layer stacked on the middle part of the composite shaft is less vulnerable to fracture due to the crack from the surface of the shaft.
Table 10 Maximum failure index of the composite shaft Stress
Plies of composite shaft 902
54
902
54
Residual thermal Bending Centrifugal
0.36 0.02 0.09
0.17 0.13 0.16
0.57 0.01 0.03
0.15 0.17 0.07
of Fig. 19(a) was calculated by adding 15 lm to the radial expansion of Fig. 18 because the static stiness of the air bearing has a maximum value at the radial clearance of 15 lm. Then, the static stiness of the air bearing of Fig. 19(b) was calculated by substituting the initial radial clearance of Fig. 19(a) for the static stiness of the air bearing of Fig. 4(b) when the eccentricity ratio was 0.5. In this work, though the static stiness of the air bearing has a maximum value in case of the carbon composite shaft with the stacking pattern 2 as shown in Fig. 19(b), the pattern 5 of the stacking group 2 was selected for the stacking sequence of the carbon composite shaft because both the bending stiness of the shaft and the static stiness of the air bearing were high for the case of pattern 5 as listed in Table 8. The speci®cations of the designed carbon composite spindle shaft were listed in Table 9. The stresses in the carbon composite shaft were calculated considering fabrication thermal residual stresses. Since the carbon ®ber epoxy composite used in design of the composite shaft was cured at 120 °C, the temperature dierence between curing temperature and room temperature was assumed to be )100 °C. The bending load and rotational speed applied during calculation were 50 N and 120,000 rpm. The failure index was calculated using Tsai±Wu failure criterion as follows [13]: FI Fi ri Fij ri rj
i; j 1; 6;
Acknowledgements This work was supported in part by the NRL project. References
1 1 c; t S1 S1
F11
In this study, the dynamic and static characteristics of the carbon composite high speed air spindle were investigated through ®nite element analysis. The thickness of the carbon composite shaft was determined considering the bending natural frequency and the carbon composite shaft was reinforced in the circumferential direction to enhance the radial stiness of the air spindle. The bending stiness of the carbon composite shaft was signi®cantly improved by enhancement between the inner and middle parts of the shaft and the static stiness of the air bearing was substantially improved by enhancing the inner part of the shaft using the 90° plies. From the analysis results, the stacking sequence was determined to be 902 = 54 = 902 = 54 T considering the bending stiness of the carbon composite shaft and the static stiness of the air bearing. Finally, the safety of the designed carbon composite shaft was evaluated by considering residual thermal stresses, bending load and centrifugal force.
8
where F1
6. Conclusions
1 ; S1t S1c
F2
1 1 c; t S2 S2
F22
1 ; S2t S2c
F3
1 1 c; t S3 S 3
F33
1 1 1 ; F55 2 ; F66 2 ; 2 S23 S13 S12 p p F11 F22 F22 F33 F12 ; ; F23 2 2
1 ; S3t S3c
F44
F13
p F11 F33 : 2
Constants used in Eq. (8) represent strengths of carbon ®ber epoxy composite as listed in Table 2. Table 10 lists the maximum failure indices of the plies calculated through ®nite element analysis using Eq. (8). In Table 10, the eect of residual thermal stress is dominant for the failure index of the composite and the eect due to bending load and centrifugal force is small. The failure
[1] Weck M, Koch A. Spindle-bearing systems for high-speed applications in machine tools. Ann CIRP 1993;42:445±8. [2] Hirn G. Sur les principaux phenomenes qui present les frottementes mediats. Soc Ind Mulhouse Bull 1854;26:188±277. [3] Gross WA. Investigation of whirl in externally pressurized airlubricated journal bearings. J Basic Eng Trans ASME 1962;84:132±8. [4] Larson RH, Richardson HH. A preliminary study of whirl instability for pressurized gas bearings. J Basic Eng Trans ASME 1962;84:511±20. [5] Taniguchi O. Experimental study on instability of externally pressurized air journal bearing. JSME Int J 1967;33:997±1004. [6] Blondeel E, Snoeys R, Devrieze L. Dynamic stability of externally pressurized gas bearings. J Lub Technol Trans ASME 1980;102:511±9. [7] Fuller DD. A review of the state-of-the-art for the design of self acting gas lubricated bearings. J Lub Technol 1969;91:1±16. [8] Mizumoto H, Arii S, Kami Y, Goto K, Yamamoto T, Kawamoto M. Active inherent restrictor for air-bearing spindles. Precision Eng 1996;19:141±7.
K.G. Bang, D.G. Lee / Composite Structures 55 (2002) 247±259 [9] Lee DG, Sin HC, Suh NP. Manufacturing of a graphite epoxy composite spindle for a machine tool. Ann CIRP 1985;34:365±9. [10] Choi JK, Lee DG. Manufacture of a carbon ®ber-epoxy composite spindle-bearing system for a machine tool. Comput Struct 1997;37:241±51. [11] Lee DG, Choi JK. Design and manufacture of an aerostatic spindle bearing system with carbon ®ber-epoxy composites. J Comput Mater 2000;34:1150±75.
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[12] Rowe WB. Hydrostatic and hybrid bearing design. Cambrige: Butterworth & Co; 1983. [13] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Comput Mater 1971;5:58±80.
www.elsevier.com/locate/compstruct
Design of carbon ®ber composite shafts for high speed air spindles Kyung Geun Bang, Dai Gil Lee
*
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Mechanical Design Laboratory with Advanced Material, ME3221, 373-1 Kusong-dong, Yusong-gu, Taejon-shi 305-701, South Korea
Abstract For the stable operation of high speed air spindles, the low rotational inertia and high damping ratio of spindle shafts as well as high fundamental natural frequency are indispensable. Conventional steel spindles are not appropriate for high speed operation because of their high rotational inertia and low damping ratio. In this study, a high-speed air spindle composed of a carbon ®ber epoxy composite shaft and two steel ¯anges was designed for maximum critical speed considering both the de¯ection due to bending load and the radial expansion due to centrifugal force and temperature rise during high-speed rotation. The stacking angle and thickness of the composite shaft and the adherend dimensions of the steel ¯anges were selected through vibrational analysis as well as considering the bending stiness and centrifugal characteristics. Ó 2002 Published by Elsevier Science Ltd. Keywords: Composite air spindle; Composite shaft; Air bearing; Critical speed; Natural frequency; Static de¯ection; Radial expansion
1. Introduction These days, high precision products are widely required with job shop type production and small batch production in the ®elds of manufacturing and machining [1]. For this end, high speed and high precision air spindles are widely used as the components of hard disk drives for computers, dental drills and machining of polygon mirrors for laser scanners because high speed rotation with small heat generation is possible for air spindles due to the low viscosity of the air lubricant. However, steel shafts mounted on conventional air spindles may cause either unstable operation due to whirling vibration of the shaft at relatively high rotational speed, or rupture of the air lubricant by radial expansion of the shaft due to centrifugal force. Until now diverse methods for stable operation of the air spindle have been suggested and investigated by many researchers some of which are listed as follows. Hirn [2] suggested the air for working ¯uid of bearings, and Gross [3] found that the existence of start speed of unstable operation in externally pressurized air bearing through experiment. Larson [4] found that the start speed of unstable operation increased as the supply pressure increased and the distance of bearing supply holes *
Corresponding author. Tel.: +82-42-869-3221; fax: +82-42-8693210. E-mail address: [email protected] (D.G. Lee).
decreased, from which he analyzed the spindle characteristics using the lumped parameter method. Taniguchi [5] investigated the operating characteristics of a spindle with respect to supply hole numbers, supply pressure and bearing length through experiment, from which he obtained the optimal bearing length for increasing the start speed of unstable operation. Blondeel et al. [6] modeled the air bearing as a control system with feedback loop and evaluated the dynamic stability of externally pressurized air bearing through frequency analysis. In order to minimize the unstable operation of air spindles, air bearings with non-circular cross-section and active control method were suggested by other researchers [7,8]. However, the diculty of machining precise non-circular cross-sections of the air bearing and additional devices required for active control prohibited their widespread use. Therefore, a spindle shaft made of high speci®c stiness and high damping material will be bene®cial for the improvement of stability of the air spindle. To this end, carbon ®ber composites are appropriate because they have high speci®c stiness E=q, high speci®c strength S=q and good damping property as well as thermal stability due to their low coecient of thermal expansion. There have been several attempts to employ carbon epoxy composite materials as the shaft material of air spindle. Lee and Choi [9±11] investigated the dynamic characteristics of high speed spindles manufactured with carbon composite shaft supported by ball bearings and air bearings.
0263-8223/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 4 6 - 5
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In this work, a high-speed spindle composed of carbon ®ber epoxy composite shaft and steel ¯anges was designed for maximum critical speed considering both the static de¯ection due to bending load, the radial expansion due to centrifugal force and temperature rise during high-speed rotation.
Table 1 Speci®cations of the air spindle for machining of wafers Outer diameter of shaft (mm) Length of shaft (mm) Shaft section shape Bearing clearance (lm) Max. radial load capacity (N) Max. radial stiness (MN/m) Operating speed (rpm)
25 210 Solid 15 50 6 40,000±80,000
2. Objective of the carbon ®ber composite spindle shaft Fig. 1 shows a typical steel air spindle used for machining wafers and drilling printed circuit boards. The speci®cations of the spindle are listed in Table 1. Since the machinability and surface roughness of wafers are improved as the speed of machining is increased, the development and employment of higher speed air spindles has been spurred. Therefore, in this study, the air spindle shaft was designed with carbon ®ber epoxy composite material whose properties are listed in Table 2 instead of the conventional steel shaft which is inappropriate for high speed air spindles due to its heavy weight and low fundamental bending natural frequency. Since conventional air spindles are composed of air bearings, steel shaft and electrical AC motor as shown in Fig. 1(b), they may be modeled as a beam of bending stiness EI supported by several springs of stiness k as shown in Fig. 2. The performance of the air spindle is
Table 2 Properties of the unidirectional carbon ®ber/epoxy composite materiala Longitudinal modulus Transverse modulus Longitudinal CTE Transverse CTE Major poisson's ratio Minor poisson's ratio Density Thickness of one ply Longitudinal tensile strength S1t Longitudinal compressive strength S1c Transverse tensile strength S2t ; S3t Transverse compressive strength (S2c ; S3c 12 and 13-direction shear strength S12 ; S13 23-direction shear strength S23 a
131 GPa 8 GPa 27 lm=m °C 0:9 lm °C 0.28 0.0171 1560 kg=m3 0.15 mm 2000 MPa )1400 MPa 61 MPa )130 MPa 70 MPa 40 MPa
Manufactured by SK Chemical in Korea (USN150).
Fig. 2. Analytic model of the air spindle.
determined by natural frequency and static stiness, which are functions of bending stiness of the shaft, stiness of air bearing and shaft mass. The shaft mass and the stiness of the air bearing are two major factors which in¯uence the fundamental and second natural frequencies of the air spindle. However, the third natural frequency is determined by bending stiness and mass density per unit length of the shaft. The static stiness of the air spindle at the cutting point is determined by the bending stiness of the shaft and the radial stiness of the air bearing. For the air bearing of Fig. 3, the bearing pressure p1 is calculated as follows under the assumption of isothermal condition [12]. pq0 Dh3 2 p 192lLp0 1
p02 sq 2cqs ps 2=c c1=c p1 =ps ; C d At p1 =ps c 1
Fig. 1. Air spindle for machining of wafers: (a) photograph; (b) schematic diagram.
1
where l; q0 ; qs ; Cd ; At ; c; D; h; L; p0 represent the dynamic air viscosity, air density at atmosphere, air density in the ori®ce inlet, ori®ce coecient, area of the ori®ce outlet, speci®c heat ratio, diameter of the shaft, bearing gap, pocket length and pressure at atmosphere, respectively.
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Fig. 3. Aerostatic analysis of the air bearing.
If the bearing pressure is assumed to be a function of bearing gap h, the pressure dierence in the air bearing of Fig. 3 is represented as follows: Dp p1 max
p1 min f C
e
f C e:
2
From Eq. (2), the stiness of the air bearing is represented as follows: R Dp dA Load ; 3 Stiffness k e e where A represents the circumferential area of the shaft. The load capacity and stiness were calculated numerically using Eqs. (1)±(3). Fig. 4 shows the load capacity and stiness of the air bearing of Table 3. In Fig. 4, the stiness of the air bearing increases as the radial clearance increases and reaches a maximum value at 15 lm radial clearance and then decreases for the speci®cations of Table 3. Since the radial clearance of the air bearing changes as the shaft expands radially due to centrifugal force and temperature rise, the stiness of the air bearing is aected by thermo-mechanical characteristics of the shaft. Therefore, the shaft of the air spindle should be designed considering both dynamic and static characteristics. The main objective of this study is to increase the operating rotational speed of the air spindle from 80,000 rpm which is the maximum operational speed of the steel spindle of Fig. 1 to 120,000 rpm as well as to maximize the static stiness of the air spindle. The speci®cations of the carbon composite shaft in Table 4 are the same as in Table 1 except the maximum operating rotational speed. For the carbon ®ber composite spindle, two steel ¯anges adhesively joined to the carbon ®ber shaft as shown in Fig. 5 were employed for mounting a cutting tool and the rotor of an AC electrical motor. Since the bending stiness and mass of the spindle shaft are dependent on the dimensions of the steel ¯anges, the adherend length La and thickness ta except the overhung parts were selected as the design variables in addition to the stacking angle and thickness of the carbon composite shaft as listed in Table 5. The dimensions of the overhung part were ®xed to the same as the steel shaft for mounting a cutting tool and the
Fig. 4. Static characteristics of the air bearing w.r.t. eccentricity ratio e and radial clearance h: (a) load capacity; (b) stiness.
Table 3 Speci®cations of the air bearing Air density q0 and qs Dynamic air viscosity l Speci®c heat ratio c Supply pressure ps Atmosphere pressure p0 Pocket length L Bearing length Radial clearance C Ori®ce coecient Cd Ori®ce outlet diameter At pdt2 =4
1:203 kg=m3 1.8E ) 5 N s/m2 1.4 0.5 MPa 0.1 MPa 20 mm 60 mm 15 lm 0.6 1.5 mm
Table 4 Speci®cations of the carbon composite shaft Outer diameter of shaft (mm) Length of shaft (mm) Shaft section shape Radial stiness of air bearing (MN/m) Max. operating speed (rpm)
25 210 Hollow 6 120,000
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Fig. 5. Schematic diagram of the steel ¯anges and the composite shaft.
Table 5 Design variables of the carbon composite shaft Carbon composite
Stacking angle hc Stacking thickness tc
Steel ¯ange
Adherend length La Adherend thickness ta
Fig. 6. Model of the composite shaft and the air bearing for FE analysis: (a) without steel ¯anges; (b) with steel ¯anges.
rotor of an AC electrical motor used in the conventional steel spindle.
3. Dynamic characteristics of the carbon ®ber composite spindle shaft The maximum operating rotational speed of the air spindle is limited by the natural frequency of the spindle shaft. Therefore, the main design objective of the carbon ®ber composite spindle shaft is to increase the natural frequency of the air spindle. In this study, to ®nd the basic modes of vibration of the air spindle, ®nite element analysis of the shaft without the steel ¯anges was performed ®rst using the model in Fig. 6(a). The commercial FEM software, ANSYS was used for the analysis. The shaft and the air bearings were modeled as 1-D beam elements and 1-D spring elements, respectively. Figs. 7 and 8 represent the mode shapes and natural frequencies of the shaft without steel ¯anges, respectively. In Fig. 7, the ®rst and the second mode shapes represent the translation and conical modes, respectively due to ¯exibility of the air bearing, while the third mode shape represents the bending mode of the shaft. Fig. 8 shows the calculated natural frequencies with respect to thickness of the carbon ®ber composite shaft. In Fig. 8, the natural frequencies in the ®rst and the second modes are little dependent on the shaft axial modulus because
Fig. 7. Mode shapes of the air spindle without steel ¯anges: (a) fundamental mode; (b) second mode; (c) third mode.
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ered as the limit speed where the spindle shaft or air bearing could fail by resonance. Therefore, the limit of operating speed of the air spindle is generally considered as the third natural frequency corresponding to bending mode. When the two steel ¯anges are adhesively bonded to both ends of the shaft, the mode shapes and natural frequencies of the spindle shaft may be dierent from those without steel ¯anges as shown in Figs. 7 and 8. Therefore, the composite spindle shaft with the steel ¯anges in Fig. 6(b) was analyzed again with ®nite element analysis. Figs. 9 and 10 represent the mode shapes and natural frequencies, respectively of the composite spindle shaft with the steel ¯anges. The translation mode of the composite spindle shaft with steel ¯anges disappears as shown in Fig. 9, which was present in the mode shape of the composite shaft without steel ¯anges as shown in Fig. 7(a) due to the overhang of the steel ¯anges on both ends of the shaft. On the contrary, the conical mode of Fig. 7(b) changed from the rigid mode into the mixed modes of Figs. 9(a) and (b), which are composed of conical mode and partial bending mode
Fig. 8. Natural frequencies of the air spindles made of composite material and steel without steel ¯anges w.r.t. thickness of the shaft: (a) fundamental natural frequency; (b) second natural frequency; (c) third natural frequency.
the shaft behaves as a rigid body, while, in the third mode, the natural frequency of the air spindle is much dependent on the shaft modulus because the shaft behaves as a ¯exible beam relative to the air bearing. During the operation of air bearings, the fundamental and second natural frequencies of the spindle shaft can be eliminated by reducing the eccentricity and unbalance of the shaft mass through dynamic balancing of the shaft. However, the third natural frequency occurring in bending mode due to shaft ¯exibility is consid-
Fig. 9. Mode shapes of the air spindle with steel ¯anges: (a) fundamental mode; (b) second mode; (c) third mode.
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the spindle shaft without the steel ¯ange as shown in Figs. 8(a) and (b) in which the fundamental and second natural frequency decrease as the shaft thickness increases. In Fig. 10(c), the third natural frequency of the steel shaft with solid section is about 85,800 rpm (1430 Hz), which is below the design value of 120,000 rpm. While, the third natural frequency of the carbon composite spindle shaft is higher than 120,000 rpm (2000 Hz) when the shaft thickness is less than 3 mm if the axial modulus is larger than 90 GPa. Therefore, in this work, the thickness of the carbon composite spindle shaft was determined to be 3 mm, and the axial modulus of the carbon composite spindle shaft was determined to be from 90 to 131 GPa. Also the natural frequency variation of the carbon composite spindle shaft was investigated with respect to the adherend length and adherend thickness of the steel ¯ange when the axial modulus and thickness of the carbon composite shaft were 105.8 GPa and 3 mm, respectively. Neglecting the stiness and mass of the adhesive, the bending stiness EI and mass per length qA of the overlapped length of the steel adherend and the carbon composite spindle shaft for the ®nite element model of Fig. 6(b) was modi®ed as follows: EIeq EIsteel EIcomposite ;
4
qAeq qAsteel qAcomposite :
5
Fig. 11 shows the ®rst three natural frequencies of the carbon ®ber composite spindle shaft w.r.t. adherend dimensions, in which the second and third natural frequencies decrease as the adherend length and thickness increase.
4. Static characteristics of the carbon ®ber composite spindle shaft
Fig. 10. Natural frequencies of the composite air spindles and the steel spindle w.r.t. thickness of the shaft: (a) fundamental natural frequency; (b) second natural frequency; (c) third natural frequency.
due to the front and rear ¯ange masses. It was found that the fundamental and the second modes of Figs. 9(a) and (b) were the partial bending and the conical modes of the rear and front steel ¯anges, respectively when the composite shaft thickness was less than 3 mm, while both modes became conical mode when the composite shaft thickness was larger than 3 mm. Also, the fundamental and second natural frequencies of the spindle shaft with the steel ¯anges become maximum when the thickness of the composite spindle shaft is around 3 mm as shown in Figs. 10(a) and (b), which is dierent from
The static stiness of the air spindle is dependent on the bending stiness of the shaft and the stiness of the air bearing. Since the stiness of the air bearing is much dependent on the bearing air gap, the radial expansion of the spindle shaft due to centrifugal force and temperature rise, and the bending stiness of the shaft should be considered in the design stage of the composite spindle shaft. From the vibrational analysis, it was found that the stacking angle between 0° and 20° satis®es the natural frequency requirement. However, the carbon composite shaft stacked with just one angle ply is inadequate for the high speed air spindle because the static stiness of the air bearing decreases by the radial expansion of the carbon composite shaft due to centrifugal force. The carbon composite shaft should have both the main stacking angle around 0° from the axial direction and
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Radial expansion (µm)
Fig. 12. Model of the composite shaft for ®nite element analysis.
Fig. 11. Natural frequencies of the composite air spindle w.r.t. length and thickness of the steel adherend when the axial modulus is 105.8 Gpa and the stacking thickness is 3 mm: (a) fundamental natural frequency; (b) second natural frequency; (c) third natural frequency.
the larger subsidiary stacking angle for the enhancement of circumferential modulus. In this work, the static characteristics of the carbon composite shaft stacked with 0° at the outer part of the shaft and 90° at inner part were investigated through ®nite element analysis using a commercial FEM software, ABAQUS under both bending load and centrifugal force as shown in Fig. 12. Since the deformation of the shaft which encapsulates the air bearing is important, the front part of the carbon composite spindle
Fig. 13. Axial modulus, eective bending stiness and radial expansion of the carbon composite shaft w.r.t. the subsidiary ply thickness when the rotational speed is 120,000 rpm: (a) axial modulus; (b) effective bending stiness and radial expansion.
shaft from Fig. 2 where the air bearing is mounted was analyzed with respect to the stacking thickness of the subsidiary angle when the thickness of the spindle shaft was 3 mm and the adherend length and the adherend
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Table 6 Stacking angles and thickness of the carbon composite shaft Stacking angle Stacking thickness (mm)
Main
Subsidiary
58 2.4
904 0.6
Fig. 14. Stacking patterns used in the evaluation of the composite shaft: (a) Group 1 of the stacking pattern; (b) Group 2 of the stacking pattern; (c) Group 3 of the stacking pattern.
thickness of the steel ¯anges were 10 and 0.5 mm, respectively. The static bending stiness of the carbon composite shaft where the air bearing is mounted was estimated by calculating the de¯ection of the node 2 point in Fig. 12 through ®nite element analysis when the load F of 50 N was applied at the end of the steel ¯ange, while ®xing all the axial displacements of cross-section including the
node point 1. Then the eective bending stiness keff of the carbon composite spindle shaft was de®ned as follows: Bending load keff : 6 dNode 2 Also, the maximum radial expansion of the carbon composite shaft was investigated through ®nite element analysis when the rotational speed of the spindle was 120,000 rpm. Only half of the composite shaft was analyzed owing to the symmetry of the shaft with respect to the cross-section including the node point 1. The eective bending stiness and the radial expansion of the carbon composite shaft with respect to stacking thickness of the subsidiary angle are shown in Fig. 13, in which the shafts without (0 mm) subsidiary plies and with 1.2 mm thickness subsidiary plies of 90° have the axial moduli of 131 and 90 GPa, respectively. In Fig. 13, both the radial expansion and the eective stiness of the carbon composite shaft without the subsidiary plies have large values. The eective bending stiness of the carbon composite shaft decreases distinctly and the radial expansion approaches a saturated value when the subsidiary thickness for the carbon composite shaft is larger than 0.6 mm. Therefore, the subsidiary ply thickness was determined to be 0.6 mm with stacking angle of 90°. Also, the main stacking angle was selected to be 5° to avoid the possibility of fracture during grinding process of the outer surface of the composite shaft. The selected stacking angles and thickness of the carbon composite shaft were listed in Table 6. To investigate the static characteristics of the carbon composite shaft w.r.t. the stacking sequences of the main and subsidiary plies and the adherend dimensions of the steel ¯ange, ®nite element analysis for the carbon composite shaft of Fig. 12 was performed with respect to stacking patterns of Fig. 14 under bending and centrifugal forces and temperature rise when the adherend length of the steel ¯ange varies from 10 to 50 mm and the adherend thickness varies from 0.5 to 5 mm. In Fig. 14, the group 1 has one cluster of 90° subsidiary plies, the group 2 has two clusters of 90° subsidiary plies, while the group 3 has more than two clusters 90° sub-
Table 7 Stacking sequence of stacking pattern for FE analysis Group 1
Pattern 1 Pattern 2 Pattern 3
58 = 904 T 904 = 58 T 54 = 904 = 54 T
Group 2
Pattern 4 Pattern 5 Pattern 6
902 = 58 = 902 T 902 = 54 = 902 = 54 T 54 = 902 = 54 = 902 T
Group 3
Pattern 7 Pattern 8 Pattern 9
52 =90= 52 = 902 = 52 =90= 52 T 90= 54 = 902 = 54 =90T 90= 52 =90= 54 =90= 52 =90T
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255
sidiary plies. The stacking sequences of the nine stacking patterns as shown in Fig. 14 are listed in Table 7. The eective bending stiness and the radial expansion of the carbon composite shaft were analyzed under the same condition as the static analysis mentioned in the previous paragraph. Also, the maximum radial ex-
Fig. 16. Eective stiness of the carbon composite shaft w.r.t. stacking patterns: (a) 10 mm adherend length; (b) 50 mm adherend length.
Fig. 15. Static characteristics of the carbon composite shaft stacked according to pattern 3 w.r.t. adherend length and adherend thickness of the ¯ange: (a) eective stiness; (b) radial expansion when the rotational speed is 120,000 rpm; (c) eective CTE.
Fig. 17. Radial expansion of the carbon composite shaft w.r.t. the stacking patterns when the rotational speed is 120,000 rpm and the adherend length is 20 mm.
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pansion of the shaft was calculated by ®nite element analysis when the temperature dierence DT is 50 °C to investigate the thermal expansion characteristics of the carbon composite spindle shaft. In this work, the eective radial coecient of thermal expansion (CTE) of the carbon composite spindle shaft was de®ned as follows: aeff
dr max ; DTr
7
where dr max and r represent maximum radial expansion and outer radius of the shaft, respectively. The eective bending stiness, the radial expansion and the eective radial CTE of the carbon composite shaft with respect to the adherend length and the adherend thickness of the steel ¯ange show similar trends regardless of the stacking patterns. The typical eective stiness and the radial expansion curves are shown in Fig. 15. In Figs. 15(a) and (b), the eective stiness of the carbon composite shaft becomes saturated beyond 2
Fig. 18. Eective CTE of the carbon composite shaft w.r.t. stacking patterns: (a) 10 mm adherend length; (b) 50 mm adherend length.
mm adherend thickness of the steel ¯ange and the radial expansion has a minimum value between 20 and 40 mm adherend length. Fig. 15(c) shows the typical eective radial CTE of the carbon composite spindle shaft calculated from Eq. (7), in which the eective radial CTE's of the carbon composite shaft increases as the adherend length of the steel ¯ange increases until the adherend length of 30 mm, while they became saturated beyond the adherend length of 30 mm. Figs. 16±18 show the maximum eective stiness, minimum radial expansion and minimum eective CTE of the carbon composite shaft of each stacking group, respectively. In Fig. 16, when the adherend length of the steel ¯ange is 10 mm, the eective stiness of the carbon composite shaft has a maximum value in case of stacking pattern 5 in which the subsidiary plies are stacked on both the inner and middle parts. When the adherend length of the steel ¯ange is 50 mm, the eective stiness of the carbon composite shaft has a maximum value in case of stacking pattern 3 in which the
Fig. 19. Radial clearance and static stiness of the air bearing w.r.t. the stacking patterns when the eccentricity ratio is 0.5: (a) radial clearance; (b) static stiness.
K.G. Bang, D.G. Lee / Composite Structures 55 (2002) 247±259
subsidiary plies stacked on the middle part. Therefore, from Fig. 16, it has been found that the 90° subsidiary plies should be stacked between the inner and middle parts to enhance the bending stiness of the carbon composite shaft. In Fig. 17, the radial expansion of the carbon composite shaft has a minimum value in case of stacking pattern 2 which has the subsidiary angle in the inner part. Therefore, the 90° subsidiary angle should be stacked in the inner part to reduce the radial expansion of the carbon composite shaft due to centrifugal force. In Fig. 18, the eective radial CTE of the carbon composite shaft has a minimum value in case of stacking pattern 1 which has the subsidiary angle in the outer part. Therefore, the 90° subsidiary angle should be stacked in the outer part to reduce the radial thermal expansion of the carbon composite shaft. 5. Design of the carbon composite spindle shaft supported by air bearing The stacking angle and thickness of the composite shaft selected were listed in Table 6 considering the bending stiness and radial expansion of the carbon composite shaft as well as the natural frequencies and mode shapes of the air spindle. In Fig. 11, the natural frequencies of the carbon composite shaft decreased as the adherend length and thickness of the steel ¯ange increased, while the eective bending stiness increased as the adherend length and
257
thickness increased as shown in Fig. 15. Also, the radial expansion due to the centrifugal force has a minimum value when the adherend length has the values between 20 and 40 mm. In Fig. 11, the third natural frequency of the composite spindle shaft is higher than 120,000 rpm either when the adherend thickness of the steel ¯ange was less than 2 mm and the adherend length was 20 mm, or when the adherend thickness of the steel ¯ange was less than 1 mm and the adherend length was 30 mm. Therefore, in this work, the adherend length and thickness of the steel ¯ange were determined to be 30 and 1 mm, respectively. From the results of Figs. 16±18, the 90° subsidiary plies should be stacked between the inner and middle parts to increase bending stiness. In order to reduce the radial expansion and the eective radial CTE of the carbon composite shaft, the 90° subsidiary plies should be stacked in the inner and the outer parts, respectively. Since the heat generation in air bearings is usually negligible during operation, in this study, the design to the stacking sequence of the carbon composite shaft was focused on the eects of the bending of the carbon composite spindle shaft and the radial expansion due to centrifugal force on the static stiness of the air spindle. To investigate the eect of the radial expansion of carbon composite shaft due to centrifugal force on the static stiness of the air bearing, the radial clearance and static stiness of the air bearing were calculated when the air bearing rotational speed was 120,000 rpm as shown in Fig. 19. The radial clearance of the air bearing
Table 8 Results from static analysis of the carbon composite shaft according to stacking patterns Bending stiness of the shaft
Eective CTE of the shaft
Static stiness of the air bearing
Group 1
Pattern 1 Pattern 2 Pattern 3
± ± Medium
Low ± ±
± High ±
Group 2
Pattern 4 Pattern 5 Pattern 6
± High ±
± ± Medium
± Medium ±
Group 3
Pattern 7 Pattern 8 Pattern 9
Low Low Low
± High High
Low Low Low
Table 9 Speci®cations of the designed carbon composite shaft Outer diameter of shaft Length of shaft
25 mm 210 mm
Carbon composite
Stacking thickness Stacking sequence
3 mm 902 = 54 = 902 = 54 T
Steel ¯ange
Adherend length Adherend thickness
30 mm 1 mm
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index of 90° layer due to residual thermal stress is larger than 5 degree layer. Although the maximum failure index of 90° layer stacked on the middle part of the composite shaft is 0.57 above the half of the critical failure index, it was designed in this way because 90° layer stacked on the middle part of the composite shaft is less vulnerable to fracture due to the crack from the surface of the shaft.
Table 10 Maximum failure index of the composite shaft Stress
Plies of composite shaft 902
54
902
54
Residual thermal Bending Centrifugal
0.36 0.02 0.09
0.17 0.13 0.16
0.57 0.01 0.03
0.15 0.17 0.07
of Fig. 19(a) was calculated by adding 15 lm to the radial expansion of Fig. 18 because the static stiness of the air bearing has a maximum value at the radial clearance of 15 lm. Then, the static stiness of the air bearing of Fig. 19(b) was calculated by substituting the initial radial clearance of Fig. 19(a) for the static stiness of the air bearing of Fig. 4(b) when the eccentricity ratio was 0.5. In this work, though the static stiness of the air bearing has a maximum value in case of the carbon composite shaft with the stacking pattern 2 as shown in Fig. 19(b), the pattern 5 of the stacking group 2 was selected for the stacking sequence of the carbon composite shaft because both the bending stiness of the shaft and the static stiness of the air bearing were high for the case of pattern 5 as listed in Table 8. The speci®cations of the designed carbon composite spindle shaft were listed in Table 9. The stresses in the carbon composite shaft were calculated considering fabrication thermal residual stresses. Since the carbon ®ber epoxy composite used in design of the composite shaft was cured at 120 °C, the temperature dierence between curing temperature and room temperature was assumed to be )100 °C. The bending load and rotational speed applied during calculation were 50 N and 120,000 rpm. The failure index was calculated using Tsai±Wu failure criterion as follows [13]: FI Fi ri Fij ri rj
i; j 1; 6;
Acknowledgements This work was supported in part by the NRL project. References
1 1 c; t S1 S1
F11
In this study, the dynamic and static characteristics of the carbon composite high speed air spindle were investigated through ®nite element analysis. The thickness of the carbon composite shaft was determined considering the bending natural frequency and the carbon composite shaft was reinforced in the circumferential direction to enhance the radial stiness of the air spindle. The bending stiness of the carbon composite shaft was signi®cantly improved by enhancement between the inner and middle parts of the shaft and the static stiness of the air bearing was substantially improved by enhancing the inner part of the shaft using the 90° plies. From the analysis results, the stacking sequence was determined to be 902 = 54 = 902 = 54 T considering the bending stiness of the carbon composite shaft and the static stiness of the air bearing. Finally, the safety of the designed carbon composite shaft was evaluated by considering residual thermal stresses, bending load and centrifugal force.
8
where F1
6. Conclusions
1 ; S1t S1c
F2
1 1 c; t S2 S2
F22
1 ; S2t S2c
F3
1 1 c; t S3 S 3
F33
1 1 1 ; F55 2 ; F66 2 ; 2 S23 S13 S12 p p F11 F22 F22 F33 F12 ; ; F23 2 2
1 ; S3t S3c
F44
F13
p F11 F33 : 2
Constants used in Eq. (8) represent strengths of carbon ®ber epoxy composite as listed in Table 2. Table 10 lists the maximum failure indices of the plies calculated through ®nite element analysis using Eq. (8). In Table 10, the eect of residual thermal stress is dominant for the failure index of the composite and the eect due to bending load and centrifugal force is small. The failure
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