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Design optimization and fabrication of a hybrid composite flywheel rotor
Composite Structures 94 (2012) 3290–3299
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Design optimization and fabrication of a hybrid composite flywheel rotor Sung K. Ha a,⇑, Seong J. Kim a, Sana U. Nasir a, Sang C. Han b a b
Department of Mechanical Engineering, Hanyang University, 1271, Sa 3-dong, Sangnok-gu, Ansan, Kyeonggi-do 426-791, Republic of Korea Korea Electric Power Research Institute, 103-16 Munji-dong, Yusong-gu, Daejon, 305-380, Republic of Korea
a r t i c l e
i n f o
Article history: Available online 25 April 2012 Keywords: Flywheel rotor Hybrid composite Interference Strength ratio Optimization
a b s t r a c t This paper discusses three different rim design cases of a hybrid composite flywheel rotor using strength ratio optimization. The rotor is composed of four hybrid composite rims. These rims are made from carbon–glass/epoxy with varying volume fractions of hoop wound reinforcements. Optimization is performed to reduce the maximum strength ratio during two rotor states: stationary and the maximum allowable rotational speed. The input specifications for optimization are: maximum useable energy (35 kW h), rotational speed (15,000 rpm), height, and inner radius. In the first case, the rims are wound simultaneously by continuous winding. However, in the second case, the rims are wound separately, and interferences are incorporated for their assembly by press fit. In the third case, a hybrid version of the first two cases is used, whereby two pairs of rims are wound at the same time, and in a secondary operation, the first pair is press fitted to the second pair. Each case has different fabrication costs and different strength ratios. The third case rotor has been successfully manufactured by filament winding with in situ curing, followed by press fit assembly of machined rims. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Composite flywheels are being utilized to provide continuous energy in a variety of applications including spacecrafts, uninterruptable power supplies, and frequency regulation [1–4]. This is because a flywheel made of composite material has distinctively high energy density, long life, and is lightweight. Among various fabrication processes, filament winding incorporating fiber tension with continuous mandrel rotation to ensure axisymmetry is a general procedure for manufacturing composite rotors. Moreover, this method is followed by the stages of heat buildup, curing, and gradual cooling that generate considerable residual stresses due to anisotropic thermal expansion of individual plies [5]. As a result, the performance of composite flywheel rotor is affected prior to operation. Furthermore, the performance is also deteriorated by the inertial stresses at high rotational speed [6,7]. However, several studies have been conducted to reduce the stresses and thus improve the performance of composite flywheel rotors. These efforts include: using high performance constituent materials, adopting rapid and more effective curing techniques, and automating the fabrication processes. In addition, the flywheel has been developed by material hybridization in conjunction with press fitting the multi rim rotor by incorporating interference [7,8]. The hybridization and interference generate pre-stresses that mitigate the net effect of radial stresses on the rotating rotor [7–9]. ⇑ Corresponding author. Tel.: +82 31 400 5249; fax: +82 31 407 1034. E-mail address: [email protected] (S.K. Ha). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.04.015
This reduction in radial stresses subsequently yields a higher rotating speed ensuring large energy storage capacity [9–11]. In general, interference generates compressive stress that reduces the tensile radial stress generated during rotor operation. Therefore, high rotor performance is observed to prevent premature failure by radial delamination, which may occur due to Poisson’s effect. Also, this failure may occur prior to the fiber breakage in the circumferential direction. Moreover, the press fit can induce significant bending and shear stresses along the longitudinal direction of the flywheel rotor. Thus, axial reinforcement may be needed for a rotor to achieve high speed or performance [12]. However, in this study the displacement and stresses in axial direction are neglected. Additionally, in composite hybridization two or more reinforcements are bound in the same matrix to obtain the desired properties [13]. An additional preferred advantage of hybridization is the cost of fabrication which incorporates both the cost of reinforcements and the fabrication process [13]. There are several well known hybridization systems that can be used to attain the desired properties of the structure under consideration including intraply hybridization and interply hybridization. In intraply hybridization, tows of two or more reinforcements are mixed in the same layer [13,14]. However, in interply hybridization, layers of two or more homogeneous reinforcements are stacked [13,14]. Fig. 1 shows the formation of these two hybrid composites. This investigation assumes a similar concept of hybridization for the three different rim design cases. In each rim, tows of carbon and glass fibers are mixed in a single matrix. However, the rims are
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Reinforcement 2
Reinforcement 1
rim 1
rim 2
rim 3
rim 4 Case A
Matrix
δ1
δ2
δ3 Case B
Layer 1
Layer 2
Intraply hybrid
Layer 1
Layer 2
Interply hybrid
Fig. 1. Intraply and interply hybrid material formation.
δ Case C
stacked in such a way that the inner rim has less carbon fiber than the corresponding outer rim. The research in preceding publications [5–8,12,15,16] addresses several fields of composite flywheel rotors including process induced strains, geometric scaling factors, dynamic analysis, stress measurements, and material characterizations. In addition, optimization is also performed for a single design of a multi-rim flywheel rotor. However, optimization of different rim design cases of hybrid composite multi-rim flywheel rotor entails attention that is addressed herein. 2. Rotor design cases A hybrid composite rotor can be made of several hybrid rims. However, as the number of rims increases, the manufacturing process becomes prolonged and complicated because hybridization involves more than one reinforcement. The reinforcement volume fractions can vary in each rim which leads to the addition or subtraction of one type of reinforcement tow/ply during the winding/ layup process. This change in reinforcement tow/ply is difficult to execute during the fabrication process with large number of rims. Accordingly, a few rims are selected for the design of a multi rim hybrid composite rotor. In this study, four hoop wound rims are introduced for each type of rotor design case to ensure an adequate winding process. The design cases are: A, B, and C; and the four hybrid rims are numbered: 1–4. These rims have carbon–glass reinforcements with varying volume fractions in composites. In addition, the reinforcements are hybridized in such a way that the proportion of carbon fiber increases from rim 1 to rim 4 (i.e., from the rotor inner surface to outer surface). This hybridization sequence is selected to maintain low stiffness at the inner radius and relatively high stiffness at the outer radius. Thus, the favorable compressive stresses are developed to counter the tensile radial stress during rotor rotation [7,15]. 2.1. Case A In this case, all the rims are simultaneously wound and cured by continuous filament winding. Initially, rim 1 is wound with carbon–glass tows and carbon fiber tows are then added into the winding process for rims 2–4. The carbon fiber volume fractions of the rims follow the order ‘‘rim 1 < rim 2 < rim 3 < rim 4’’. 2.2. Case B Interference fit is an effective way of reducing the radial stress during rotation of the flywheel rotor [7,12,15]. In this case (unlike the former one), all the rims are wound and cured separately. As in case A, rim 1 is wound initially and the other rims are then wound
Fig. 2. Rotor design cases.
in subsequent order. However, in this design case, the rim winding sequence is unnecessary. The change in reinforcement volume fractions is like in case A, with the least carbon fiber in rim 1 and the most in rim 4. Afterward, these rims are press fitted with the interferences d1, d2, and d3, followed by the machining of rim surfaces. 2.3. Case C The design concepts of the former cases are adopted to some extent in this case of a multi-rim hybrid composite rotor. Here, rims are wound in two sets. Rim set 1 is composed of rims 1 and 2, and rim set 2 is composed of rims 3 and 4. The outer surface of rim 2 and the inner surface of rim 3 are machined and press fitted by an interference d. Furthermore, for this case, the change in reinforcement volume fraction is similar to that in case A or B. For better understanding, Fig. 2 shows schematics of the three design cases. 3. Stress sources of rotor design cases Since the case A, B, and C described hereinbefore have different designs, sources of radial and hoop stress generated in a rotor of height h and inner radius ri are also different. For each case, these stresses are calculated at two rotor states: stationary and rotating at maximum allowable speed xmax. The former state is considered to determine the residual stresses developed during the manufacturing process of a hybrid composite flywheel rotor. These primary stresses are caused by anisotropic thermal expansion of the hybrid plies and/or the assembly of rims incorporating interference. The latter state is taken into account to consider the effect of centrifugal forces during rotor operation. Regarding the stress sources of three cases, for case A, the dominant stress source in the stationary rotor is the temperature difference DT produced during winding and curing of the rims. Since all the rims are wound and cured simultaneously, the effect of thermal residual stress on the rotor performance is worst in this design [5]. Afterward, this stress is accompanied by the inertial stress at xmax. Fig. 3 depicts these stress sources. On the other hand, in case B, all the rims are wound and cured separately, and interferences d1, d2, and d3 are incorporated among the rims. These interferences lead to a compressive stress generation, reducing the effect of centrifugal force during rotor operation.
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ΔT
ω = 0, ω max
h ri Fig. 3. Stress sources of case A.
ΔT
ΔT
ΔT
ΔT δ3
δ1
ω = 0, ω max δ1
δ2
δ3
h ri Fig. 4. Stress sources of case B.
ΔT
ΔT δ
ω = 0, ωmax
h ri Fig. 5. Stress sources of case C.
Furthermore, the effect of thermal residual stress is not as bad as in case A because the thickness of each rim is much smaller. Fig. 4 shows the stress sources of this case. The stress sources in case C are similar to those in case B; however, the magnitude of stress differs due to the manufacturing and assembly process. Since two rims are wound and cured at the same time, thermal residual stress is high. Moreover, only one interference d is integrated for press fit assembly of the rims; the prestress developed is also less than that in case B. These stress sources are depicted in Fig. 5. 4. Stress analysis of a rotor A stress analysis to calculate the deformations and strength ratios of a multi-rim rotor can be found in previous publications [6,7,15]. However, a summary of relevant equations is given below. Eq. (1) shows the governing equation for a rotor rotating at speed x at radius r:
drr rr rh þ þ qrx2 ¼ 0; dr r
ð1Þ
where r and q are stress and density, respectively. Assuming the plane stress state, the stress–strain relationship in cylindrical coordinates is given by
Q 11 Q 12 rh ¼ rr Q 21 Q 22
eh ah DT ; er ar DT
ð2Þ
where Q, e, and a are stiffness matrix, strain, and coefficient of thermal expansion, respectively. Since the rotor satisfies the axisymmetry conditions, strains are defined as a function of radial displacement ur and can be expressed as
ur =r eh ¼ : @ur =@r er
ð3Þ
For the case A rotor type, rims should satisfy the following compatibility conditions:
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procedure, initially calculates thermal residual stresses which are then added to the stresses produced by the interference. Finally, the inertial stress is added to the former stresses to calculate the overall stress. There are several other ways to calculate these stresses, but this option is the most convenient. 5. Design optimization of rotor cases
Fig. 6. Cylindrical coordinate system and symbols used for a multi-rim rotor.
ðjþ1Þ rðjþ1Þ ¼ rðjÞ ¼ uðjÞ ri r o and uri ro :
ð4Þ
For the case B hybrid rotor, rims should satisfy the following compatibility conditions: ðjÞ ðjþ1Þ rðjþ1Þ ¼ rðjÞ uðjÞ ri r o and uri ro ¼ d :
ð5Þ
Two strength ratios are considered for design optimization: the radial strength ratio Rr = rr/Y and the hoop strength ratio Rh = rh/X. Fig. 6 shows the coordinate system and symbols used in these equations. Keeping in mind the stress sources of each rotor case, a simple procedure has been adopted to calculate stresses in a rotor. This
In the design cases, rims may vary with respect to thickness, volume ratio of reinforcement and interference for press fit. Therefore, optimization is essential for the selection of appropriate values. The optimization parameters are: design variables, objective functions and design constraints. Design variables (DVs) include thickness ti, carbon fiber volume fraction Vfi of each rim, and interferences. An objective function is used to reduce the maximum strength ratio in all the cases. This function is subjected to a design constraint that the strength ratio should be less than 1. The purpose of this limitation is to avoid any material failure when the rotor is at rest or rotating at xmax. Moreover, the compressive stress developed in the rims during press fitting in case B or case C is limited to 6 MPa. An optimization flow chart is portrayed in Fig. 7. In this flow chart, Emax and eh are the maximum useable energy and hoop strain at the inner surface respectively. The flow chart in Fig. 7 represents the general outlook of the optimization, in fact for each design case; an individual optimization is performed considering the respective design variables, material properties, and temperature DT. However, given specifications are same for all the cases. During optimization, the stress analysis [6] for each case is performed individually using the given specifications, geometric parameters, material properties, and temperature DT. The strength ratios are then calculated to ensure that the initial and subsequent design variables are selected in such a way that R is less than 1. With the aid of optimization, design variables that satisfy the design constraints are then selected, and analysis is performed to
Given r i, h, ω = 0 & ωmax, Emax, εθ
Stress analysis
Initial value of design variables for case A, B, and C
Strength ratio (Rr, Rθ)
Optimization
Update DV
Find ti, V fi ti, V fi, δe ti, V fi, δ Minimize
for case A for case B for case C
Max. (R*(i))
Subject to R*(i) < 1 at ω = 0, ωmax σr < 6 MPa (compressive) at ω = 0 R*
(i)
= max. (Rri, Rθi)
i = 1, 2, 3, 4; e = 1, 2, 3 No Optimum Yes ti, V fi, δe, δ Fig. 7. Optimization flow chart.
Material properties Temperature ΔT for case A, B, and C
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Table 1 Optimized parameters of hybrid rims. Parameter
Case A
Case B
Case C
Unit
t1 t2 t3 t4 Vf1 Vf2 Vf3 Vf4 d1 d2 d3 d
54.20 35.50 34.30 60.1 11 40 66 100 – – – –
39.90 50.30 51.00 42.30 10 35 72 100 0.13 0.07 0.15 –
36.80 55.00 52.00 40.00 11 30 79 100 – – – 0.20
mm mm mm mm % % % % mm mm mm mm
determine the optimum parameters of hybrid rims. The optimization algorithm is illustrated in Appendix B. 6. Numerical calculations The optimized design variables are obtained using the same input specifications for all the design cases. These specifications are Emax = 35 kW h; xmax = 15,000 rpm; ri = 269.75 mm; h = 1340 mm; and eh = 0.8. The optimized design variables are listed in Table 1. The volume fractions in Table 1 are the percentages of carbon fiber considering only the glass–carbon reinforcements. Table 1 does not contain the percentage of hybrid material that also includes matrix as a constituent material. For example, in case A, the Vf1 value is 11%, which means carbon fiber accounts for 11% and glass fiber 89% of the glass–carbon reinforcement. Although the carbon fiber volume ratio is increased from the inner to the outer rim, each rim hybrid material has an effective reinforcement volume fraction of 67% and a matrix fraction of 33%. Also, the effect of void content is ignored. The properties of constituent materials are listed in Table 2. The effective properties of hybrid composite material are detailed in Table 3. These properties are determined by the rule of mixtures given in Appendix A. Table 3 shows how the hybridization affects the properties. For instance, increasing the volume
fraction of carbon fiber from rim 1 to rim 4, decreases density, but increases longitudinal modulus and tensile strength. However, the transverse modulus decreases due to the anisotropy of the carbon fiber. The effect of hybridization on other properties of interest can also be seen. Primarily, the radial stress distribution along the radial coordinate of the rotor for all cases is shown in Fig. 8. In case A, we first calculate the radial stress due to temperature change DT. The value reaches about 10 MPa, which is greater than in any other case due to simultaneous winding and curing of all four rims. The inertial stress generated due to xmax is then added to the thermal stress to calculate the overall stress of the rotor in case A as shown in Fig. 8a. Since, in case B, all the rims are wound and cured individually, the radial stress produced by the temperature change DT is less than in other cases. The maximum value reaches about 1 MPa for rim 3 because this rim is the thickest. The compressive stress due to interferences de (e = 1, 2, 3) is then added to the thermal residual stress such that the value approaches 6 MPa (optimization constraint). Finally, the radial stress of the rotor in case B rotor is determined by incorporating the centrifugal force effect due to xmax, as shown in Fig. 8b. A radial stress of about 2.5 MPa is generated in the case C rotor during the winding and curing of rim set 1 and rim set 2. The interference d is then incorporated to assemble the rim sets by press fit, and the stress value approaches 6 MPa (optimization constraint). Finally, the radial stress behavior of the rotor in case C is obtained by incorporating inertial stress into the former stresses, as shown in Fig. 8c. The hoop stress is calculated in the way same as the radial stress. First, stress due to DT is considered, and then stress due to interference de (e = 1, 2, 3) or d is added to the thermal residual stress. Finally, inertial stress generated by xmax is incorporated. The hoop stress generated by the temperature change DT is greater in case A than in any other case. Since, in this case, all the rims are wound and cured simultaneously, the value approaches 750 MPa as shown in Fig. 9a. Nevertheless, the thermal residual hoop stress is the least in case B because all the rims are wound and cured separately. However, the overall hoop stress developed by the three
Table 2 Constituent material properties. Property
Symbol
Carbon fiber
Glass fiber
Epoxy
Unit
Density Longitudinal modulus Transverse modulus Poisson’s ratio Longitudinal CTE Transverse CTE Longitudinal tensile strength Transverse tensile strength
q
1800 230 23 0.20 0.90 9.50 4902 77
2540 72 72 0.22 5 5 2068 3103
1190 3.20 3.20 0.37 54 54 67 67
kg/m3 GPa GPa – 106/°C 106/°C MPa MPa
Ex Ey
txy ax ay X Y
Table 3 Effective properties of hybrid rims. Property
q Ex Ey
txy ax ay X Y
Unit
kg/m3 GPa GPa – 106/°C 106/°C MPa MPa
Case A
Case B
Case C
Rim 1
Rim 2
Rim 3
Rim 4
Rim 1
Rim 2
Rim 3
Rim 4
Rim 1
Rim 2
Rim 3
Rim 4
2040 61 19.40 0.27 5 22 1616 15
1896 92 16 0.26 2 24.50 2167 15
1769 119 13 0.26 0.56 26.43 2654 15
1599 155 9 0.26 0.51 28.80 3306 15
2040 61 19.50 0.27 4.20 21.90 1605 15
1922 86 16.50 0.26 2.30 24.10 2067 15
1739 125 12 0.26 0.30 26.90 2770 15
1599 155 9 0.26 0.50 28.80 3307 15
2040 61 19 0.27 4.90 21.90 1616 15
1946 81 17 0.26 2.70 23.70 1977 15
1703 133 11 0.26 0.10 27.40 2908 15
1599 155 9 0.26 0.50 28.80 3306 15
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(a)
Radial stres (MPa)
(b)
ΔT+ωmax
ΔT
15
ΔT
15
ΔT+δe
(c)
ΔT+δe+ωmax
10
10
10
5
5
5
0 0.30
0.35
0.40
0.45
0.50
-5
0.25
0.30
0.35
0.40
0.45
rim 1
rim 2 rim 3
rim 4 Radius (m)
0.25
0.50
ΔT+δ+ωmax
0.30
0.35
0.40
0.45
0.50
-5
-5
-10
ΔT+δ
0
0 0.25
ΔT
15
-10
rim 1
rim 2
rim 3
-10
rim 4
rim 1
rim 2
rim 3
rim 4
Fig. 8. Radial stress distribution in hybrid rims: (a) case A, (b) case B, and (c) case C.
Hoop stress (MPa)
(a)
ΔT
(b)
ΔT+ω max
ΔT
ΔT+δe
ΔT+δe+ω max
(c)
1000
800
800
800
600
600
600
400
400
400
200
200
200
0 -200
0.30
0.35
0.40
0.45
0.50
rim 1 rim 2 rim 3 rim 4 Radius (m)
ΔT+δ
ΔT+δ+ω max
0
0 0.25
ΔT
1000
1000
0.25 -200
0.30
0.35
0.40
0.45
0.25
0.50
0.30
0.35
0.40
0.45
0.50
-200 rim 1 rim 2
rim 3
rim 4
rim 1 rim 2
rim 3
rim 4
Fig. 9. Hoop stress distribution in hybrid rims: (a) case A, (b) case B, and (c) case C.
0.75 when it is at xmax. On the other hand, the hoop strength ratio at xmax shows a downward trend from the inner rim to the outer rim due to the increasing volume fraction of carbon fiber. Furthermore, for a stationary rotor hoop strength ratio is tensile for rims 1 and 2 and compressive for rims 3 and 4, as shown in Fig. 10a. Conversely, the hybrid rotor in case B has the least radial stress because of interferences. Therefore, the radial strength ratio is much lower than in case A or case C at both rotor states, as shown in Fig. 10b. For a stationary rotor, the value is about 0.16, and that at xmax is nearly 0.19. Generally, the hoop strength ratio for a rotor
sources is greater than in the other cases (i.e., about 800 MPa as shown in Fig. 9b). The primary reason for this peak is the interferences that occur when the rotor is assembled by press fit, and the secondary is xmax. A similar effect of interference can also be seen in Fig. 9c when the rotor is assembled by press fit. The strength ratios for all cases have been optimized at x = 0 and xmax = 15,000 rpm. Fig. 10 shows the strength ratios for three cases at different radii. Since case A has the maximum radial stress, the radial strength ratio is greater than those of the other two design cases. The value approaches 0.70 when rotor is stationary and
Strength ratio
(a)
(b)
(c)
0.80
0.80
0.80
0.60
0.60
0.60
0.40
0.40
0.40
0.20
0.20
0.20
radial_ωmax radial_0 rpm
hoop_ωmax hoop_0 rpm
0.00
-0.20
0.00
0.00 0.25
0.30
0.35
0.40
0.45
0.50
rim 1 rim 2 rim 3 rim 4 Radius (m)
0.25 -0.20
0.30 rim 1
0.35 rim 2
0.40 rim 3
0.45 rim 4
0.25
0.50 -0.20
Fig. 10. Strength ratio of hybrid rims: (a) case A, (b) case B, and (c) case C.
0.30
0.35
rim 1 rim 2
0.40
0.45
rim 3 rim 4
0.50
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Table 4 Comparison of cost and strength ratio. Parameter
Case A
Case B
Case C
Maximum radial strength ratio Maximum hoop strength ratio Estimated manufacturing cost ($)
0.75 0.42 25,000
0.19 0.36 44,000
0.39 0.42 31,000
at rest changes from compressive to tensile from the inner radius to the outer radius. However, at xmax, the maximum value reaches 0.36 at the inner rim and the minimum value is approximately 0.19 at the outer rim. The radial strength ratio is higher in case C than in case B, but less than in case A at xmax with a maximum value of about 0.39, as shown in Fig. 10c. In addition, the hoop strength ratio is nearly 0.42 at the inner rim and 0.18 at the outer rim. Furthermore, when the rotor is at rest, the hoop strength ratio is tensile for rims 1 and 2 and compressive for rims 3 and 4. We have found that, in all design cases for a rotor at xmax, the hoop strength ratio is maximal at the inner rim and minimal at the outer rim because of reinforcements changing the volume fractions as described earlier in this study. Moreover, thermal residual stresses have a devastating effect on the rotor performance, particularly when all the rims are wound simultaneously, as in case A. The effect of these stresses can be reduced by winding and curing the rims separately and incorporating the interferences as in cases B and C. Accordingly, each of the three cases has its own manufacturing process, which leads to a tradeoff between cost and strength ratio. For example, rotor strength ratios in case A are higher than in the other cases, but cost is lower due to the simplified fabrication process. Manufacturing requires only one mandrel, on which all the rims are wound and then cured.
On the other hand, the cost of rotor fabrication in case B is higher than in the other cases. To manufacture four rims in case B, mandrels with different outer diameters are required. Subsequently, machining is performed on all the rims to maintain interferences for press fitting, which requires additional arrangements. However, the strength ratio is lower than in case A or case C hybrid rotors. The hybrid rotor in case C can be considered an intermediate between the case A and case B hybrid rotors. Its strength ratio is greater than that of case B, but less than that of case A. The fabrication process is simple and shorter than that of case B which ultimately reduces the cost. However, the press fit is still required to assemble two sets of rims, which increases the cost to be more than that of case A. It has been found that comparing the radial strength ratio within each rotor during rotation, ‘‘case B < case C < case A’’. In terms of manufacturing cost the reverse order is held as ‘‘case B > case C > case A’’. Strength ratios calculated herein are listed in Table 4 along with manufacturing cost. This cost includes mandrels, reinforcing fibers, epoxy, winding, machining, and press fitting assembly. The operational cost is not included in this study. 7. Fabrication of case C hybrid rotor Based on the optimization study and manufacturing process, the hybrid rotor in case C was selected for manufacture using a Hexion 166 epoxy system along with hoop wound carbon–glass reinforcements. Rim set 1 was manufactured on one mandrel by continuous filament winding under in situ curing [16,17] conditions using a slip ring and IR heater. Similarly, rim set 2 was wound on another mandrel as shown in Fig. 11. Both rim sets were wound using similar winding methods; however, different images are shown in Fig. 11 to cover the whole winding process.
Fig. 11. Rim manufacturing by filament winding.
Fig. 12. Penetrant testing of rims.
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Prior to the assembly, all rims were checked for visible surface defects. Penetrant testing was used to inspect for surface cracks, porosity, delamination and a lack of bond at exposed edges. Primarily, a liquid penetrant was applied to the surface of the rotor for a specific period of time to penetrate into the defects present at the surface. After the penetration period, surplus penetrant was removed. An absorbent light colored powdered material called developer was then applied to the surface. This developer acted as a blotter and drew out a portion of the penetrant that previously seeped into the surface cracks. As the penetrant was drawn out, it diffused into a coating of the developer, forming colored indications much wider than the original surface defects (if present) with which they were associated. An inspector then viewed the parts and examined these indications against a background of the developing powder. In this study, using the penetrant method, no visible defects were found on the surface under examination, as shown in Fig. 12.
The rims were then assembled by press fit, incorporating the interference d following the machining of rims. For this assembly, the vertical press fit force Fv was determined using the slope angle b of tapered rims. This angle was determined by considering the machining tolerances and interference. Moreover, this side slope angle was smooth enough not to cause any material cracking during press fit. The vertical force is given by [15]:
F v ¼ F d ðtan b þ lÞ; where Fd is the interior force calculated using Eqs. (1)–(5), and l is coefficient of friction among the rim surfaces (taken as 0.20). The calculation of this force is illustrated in Fig. 13. The outer surface of rim 2 and the inner surface of rim 3 were tapered by CNC machining. These machined rims were then press fitted using a hydraulic press with a vertical force of approximately 1000 tons, as shown in Fig. 14.
Press fit force (Fv)
Fv sin β
Rim 1 Rim 2
β Fv cos β
β
Fv µF δ cos β
Fδ sin β β
Fδ
Fδ cos β
h Rim 3 Rim 4
δ Fig. 13. Calculation of hydraulic press fit force.
Fig. 14. Rim assembly by press fit.
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8. Conclusion
Appendix B. Optimization algorithm
Three cases of composite flywheel rotor design by rim hybridization are presented to investigate the effect on rotor performance by optimizing the strength ratios. Pros and cons are also taken into account for all the designs. We have shown throughout this study that the manufacturing process and strength ratio adequately differentiated the three cases. Due to a rapid and simplified manufacturing process, the hybrid rotor in case A has advantages over the other two hybridization techniques with respect to cost, but the radial strength ratio reaches almost 0.75. The radial strength ratio approaches 0.19 in the hybrid rotor in case B, the lowest of all the cases. However, this design is costly due to a lengthy and extensive fabrication process. Due to interference, the hybrid rotor in case C has a lower radial strength ratio than that in case A (i.e., 0.39), but the cost is higher. Comparing the radial strength ratio in the rotors, ‘‘case B < case C < case A’’. In terms of manufacturing cost the reverse order is held as ‘‘case B > case C > case A’’. Moreover, the strength ratios in all design cases are within the established limits (i.e., less than 1). Furthermore, the rotor performance can be improved by the interference provision, which reduces the radial strength ratio, resulting in higher operational speed. We selected for manufacture one hybrid rotor design that best suits our requirement. The research conducted in this paper can be applied to other hybrid composite flywheel rotors with enhanced performance and energy storage capacity.
The nonlinear optimization in this study requires the constrained minimization to find an optimal solution. Moreover, all the design variables are continuous. Therefore, the sequential quadratic programming (SQP) method [20–23] is used to solve the nonlinear optimization problem in this investigation. Several local optima are possible while considering fiber angles as design variables [21] that affect adversely the accuracy of the optimization method. However, in this study, the probability of several local optima is reduced as the reinforcements are hoop wound in all the three design cases and thus fiber angle is not a design variable. The optimization for each case is performed individually considering the respective design variables, material properties, and temperature DT; however a single algorithm of SQP is shown in Fig. B1 for convenience. The SQP is a well known method to solve nonlinear optimization problems. In this method, the sub-problem for directionfinding is formulated using the Tailor series of the quadratic approximate objective function (Max. R(i)) and linearized conðiÞ straint functions (gj = Rj , rrj ; where j = 1, M). The sub-problem is then solved using the modified method of feasible directions [24]. The B matrix is initially set to identity matrix and then updated to approach the Hessian of the Lagrangian function. Lagrange multipliers kj are then calculated and search is done in S direction using approximate Lagrangian function to minimize /:
Appendix A. Rule of mixtures
/ ¼ FðXÞ þ
M X uj max 0; g j ðXÞ ;
ðB:1Þ
j¼1
The rule of mixtures [14,18,19] is used to calculate the effective properties of hybrid material lamina. These properties are calculated as given by Eqs. (A.1), (A.2), (A.3), (A.4), (A.5), (A.6), (A.7) for a given volume fraction V:
q¼
X
qðpÞ V ðpÞ ;
X
Ex ¼ 1 ¼ Ey
ðA:1Þ
ðpÞ EðpÞ ; x V
V ðmÞ EyðmÞ
þ
gðgf Þ V ðgf Þ Eyðgf Þ
ðA:2Þ
þ
!, gðcf Þ V ðcf Þ Þ Eðcf y
V ðmÞ þ gðgf Þ V ðgf Þ þ gðcf Þ V ðcf Þ ; ðA:3Þ
txy ¼
X
ðpÞ ðpÞ txy V ;
X¼
X
X
ðpÞ ðpÞ 1 þ tðpÞ ax txy ; xy ay V
X ðpÞ V ðpÞ :
Iteration number q q+1
ðA:5Þ
Sub-problem for direction-finding S Max. R*(i) Using modified method of feasible directions
ðA:6Þ
Perform one-dimensional search to minimize
ðA:7Þ
In Eq. (A.3), m, gf, and cf denote matrix, glass fiber and carbon fiber, respectively. For the rest of the equations, p stands for matrix, carbon fiber and glass fiber to consider the hybrid constituents cumulatively. The transverse tensile strength (Y) for the hybrid rims was determined experimentally considering only the matrix effect. The stress partitioning factors
.
Initial design variables X0 = ti0, Vfi0, δe0, δ0 Initial Hessian matrix, B = I
ðA:4Þ
1 X ðpÞ ðpÞ ðpÞ ax ¼ ax Ex V ; Ex
ay ¼
where X = Xq1 + aS, uj ¼ jkj j for first iteration, uj ¼ 1 max½jkj j; 2 ðu0j þ jkj jÞ for subsequent iterations and u0j ¼ uj from the previous iterations. After one-dimensional search, the B matrix is updated using BFGS formula [24,25]:
Xq + α*Sq
Xq+1
Yes Converged No *
Calculate B using BFGS formula
.
Þ Þ gðgf Þ ¼ rðgf rðmÞ and gðcf Þ ¼ rðcf ryðmÞ ; y y y
B were also experimentally determined to have values of 3.78 and 1.48, respectively.
B*
Fig. B1. SQP algorithm.
Exit
S.K. Ha et al. / Composite Structures 94 (2012) 3290–3299
B ¼ B
BppT B ggT þ T ; pT Bp p g
ðB:2Þ
where p ¼ X q X q1 , g ¼ hy þ ð1 hÞBp, P / ¼ FðXÞ þ M j¼1 kj g j ðXÞ, and
(
h¼
1:0
if pT y P 0:2pT Bp
0:8pT Bp pT BppT y
if pT y < 0:2pT Bp
y ¼ rx /q rx /q1 ,
:
References [1] Genta G. Kinetic energy storage: theory and practice of advanced flywheel systems. London: Butterworths; 1985. [2] Rodriguez GE, Studer PA, Baer DA. Assessment of flywheel energy storage for space craft power system. NASA Technical Memorandum 85062; 1983. [3] Caprio MT, Murphy BT, Herbst JD. Spin commissioning and drop test of a 130 kW h composite flywheel. In: Ninth international symposium on magnetic bearings; 2004. [4] Strasik M. Current status of the flywheel electricity system. Superconductivity for electric system 2006 project summary. [5] Ha SK, Kim HT. Measurement and prediction of process-induced residual strains in thick wound composites rings. Compos Mater 2003;37(14):1223–37. [6] Ha Sung Kyu, Kim Dong-Jin, Sung Tae-Hyun. Optimum design of multi-ring composite flywheel rotor using a modified generalized plane strain assumption. Int Mech Sci 2001;43(4):993–1007. [7] Ha Sung K, Kim Jae H, Han Young H. Design of a hybrid composite flywheel multi-rim rotor system using geometric scaling factors. Compos Mater 2008;42(8):771–85. [8] Arvin AC, Bakis CE. Optimal design of press-fitted filament wound composite flywheel rotor. Compos Struct 2006;72(1):47–57. [9] Kulkarni S. Energy and technology review. Lawrence Livermore National Laboratory, UCRL-52000-82-3; 1982. [10] Ries DM. Manufacturing analysis for a composites multi-ring flywheel. M.S. thesis, University of Maryland; 1991.
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[11] Portnov GG. Structures and design. In: Herakovich CT, Tarnopolskii YM, editors. New York: Elsevier Science Publishing Company Inc.; 1989. [12] Jerome Tzeng, Ryan Emerson, Paul Moy. Composite flywheel development for energy storage. ARL-TR-3388; 2005. [13] Pegoretti Alessandro, Fabbri Elena, Migliaresi Claudio, Pilati Francesco. Intraply and interplay hybrid composites based on E-glass and poly (vinyl alcohol) woven fabrics: tensile and impact properties. Polym Int 2004;53(9):1290–7. [14] Daniel Isaac M, Ishai Ori. Engineering mechanics of composite materials. Oxford University Press; 2006. [15] Ha Sung K, Han Hoon H, Han Young H. Design and manufacture of a composite flywheel press-fit multi-rim rotor. Reinf Plast Compos 2008;27(9):953–65. [16] Garbys CW, Bakis CE. Simplified analysis of residual stresses in in situ cured hoop-wound rings. Compos Mater 1998;32(13):1325–43. [17] Abulizi Dilimulati, Duan Yugang, Li Dichen, Lu Bingheng. A new method for glass-fiber reinforced composites manufacturing: automated fiber placement with in situ UV curing. In: IEEE international symposium on assembly and manufacturing; 2011. [18] Voyiadjis George Z, Kattan Peter I. Mechanics of composite materials with MATLAB. Netherlands: Springer; 2005. [19] Govaert LE, D’ Hooghe ELJCJ, Peijs AAJM. A micromechanical approach to the viscoelasticity of unidirectional hybrid composites. Composites 1991;22(2):113–9. [20] Boggs Paul T, Tolle Jon W. Sequential quadratic programming for large-scale nonlinear optimization. J Comput Appl Math 2000;124(1–2):123–37. [21] Ghiasi Hossein, Pasini Damiano, Lessard Larry. Optimum stacking sequence design of composite materials. Part I: constant stiffness design. Compos Struct 2009;90(1):1–11. [22] Wang J, Karihaloo BL. Cracked composite laminates least prone to delamination. Proc Math Phys Sci 1994;444(1920):17–35. [23] George TJ, Herman Shen MH, Huybrechts SM, Meink TE, Wegner PM. Optimal design of composite chambercore structures. Compos Struct 2002;52(3– 4):277–86. [24] Vanderplaats GN. Numerical optimization techniques for engineering design. New York: McGraw-Hill Inc.; 1984. [25] Yuan Gonglin, Lu Xiwen. An active set limited memory BFGS algorithm for bound constrained optimization. Appl Math Model 2011;35(7):3561–73.
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Design optimization and fabrication of a hybrid composite flywheel rotor Sung K. Ha a,⇑, Seong J. Kim a, Sana U. Nasir a, Sang C. Han b a b
Department of Mechanical Engineering, Hanyang University, 1271, Sa 3-dong, Sangnok-gu, Ansan, Kyeonggi-do 426-791, Republic of Korea Korea Electric Power Research Institute, 103-16 Munji-dong, Yusong-gu, Daejon, 305-380, Republic of Korea
a r t i c l e
i n f o
Article history: Available online 25 April 2012 Keywords: Flywheel rotor Hybrid composite Interference Strength ratio Optimization
a b s t r a c t This paper discusses three different rim design cases of a hybrid composite flywheel rotor using strength ratio optimization. The rotor is composed of four hybrid composite rims. These rims are made from carbon–glass/epoxy with varying volume fractions of hoop wound reinforcements. Optimization is performed to reduce the maximum strength ratio during two rotor states: stationary and the maximum allowable rotational speed. The input specifications for optimization are: maximum useable energy (35 kW h), rotational speed (15,000 rpm), height, and inner radius. In the first case, the rims are wound simultaneously by continuous winding. However, in the second case, the rims are wound separately, and interferences are incorporated for their assembly by press fit. In the third case, a hybrid version of the first two cases is used, whereby two pairs of rims are wound at the same time, and in a secondary operation, the first pair is press fitted to the second pair. Each case has different fabrication costs and different strength ratios. The third case rotor has been successfully manufactured by filament winding with in situ curing, followed by press fit assembly of machined rims. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Composite flywheels are being utilized to provide continuous energy in a variety of applications including spacecrafts, uninterruptable power supplies, and frequency regulation [1–4]. This is because a flywheel made of composite material has distinctively high energy density, long life, and is lightweight. Among various fabrication processes, filament winding incorporating fiber tension with continuous mandrel rotation to ensure axisymmetry is a general procedure for manufacturing composite rotors. Moreover, this method is followed by the stages of heat buildup, curing, and gradual cooling that generate considerable residual stresses due to anisotropic thermal expansion of individual plies [5]. As a result, the performance of composite flywheel rotor is affected prior to operation. Furthermore, the performance is also deteriorated by the inertial stresses at high rotational speed [6,7]. However, several studies have been conducted to reduce the stresses and thus improve the performance of composite flywheel rotors. These efforts include: using high performance constituent materials, adopting rapid and more effective curing techniques, and automating the fabrication processes. In addition, the flywheel has been developed by material hybridization in conjunction with press fitting the multi rim rotor by incorporating interference [7,8]. The hybridization and interference generate pre-stresses that mitigate the net effect of radial stresses on the rotating rotor [7–9]. ⇑ Corresponding author. Tel.: +82 31 400 5249; fax: +82 31 407 1034. E-mail address: [email protected] (S.K. Ha). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.04.015
This reduction in radial stresses subsequently yields a higher rotating speed ensuring large energy storage capacity [9–11]. In general, interference generates compressive stress that reduces the tensile radial stress generated during rotor operation. Therefore, high rotor performance is observed to prevent premature failure by radial delamination, which may occur due to Poisson’s effect. Also, this failure may occur prior to the fiber breakage in the circumferential direction. Moreover, the press fit can induce significant bending and shear stresses along the longitudinal direction of the flywheel rotor. Thus, axial reinforcement may be needed for a rotor to achieve high speed or performance [12]. However, in this study the displacement and stresses in axial direction are neglected. Additionally, in composite hybridization two or more reinforcements are bound in the same matrix to obtain the desired properties [13]. An additional preferred advantage of hybridization is the cost of fabrication which incorporates both the cost of reinforcements and the fabrication process [13]. There are several well known hybridization systems that can be used to attain the desired properties of the structure under consideration including intraply hybridization and interply hybridization. In intraply hybridization, tows of two or more reinforcements are mixed in the same layer [13,14]. However, in interply hybridization, layers of two or more homogeneous reinforcements are stacked [13,14]. Fig. 1 shows the formation of these two hybrid composites. This investigation assumes a similar concept of hybridization for the three different rim design cases. In each rim, tows of carbon and glass fibers are mixed in a single matrix. However, the rims are
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Reinforcement 2
Reinforcement 1
rim 1
rim 2
rim 3
rim 4 Case A
Matrix
δ1
δ2
δ3 Case B
Layer 1
Layer 2
Intraply hybrid
Layer 1
Layer 2
Interply hybrid
Fig. 1. Intraply and interply hybrid material formation.
δ Case C
stacked in such a way that the inner rim has less carbon fiber than the corresponding outer rim. The research in preceding publications [5–8,12,15,16] addresses several fields of composite flywheel rotors including process induced strains, geometric scaling factors, dynamic analysis, stress measurements, and material characterizations. In addition, optimization is also performed for a single design of a multi-rim flywheel rotor. However, optimization of different rim design cases of hybrid composite multi-rim flywheel rotor entails attention that is addressed herein. 2. Rotor design cases A hybrid composite rotor can be made of several hybrid rims. However, as the number of rims increases, the manufacturing process becomes prolonged and complicated because hybridization involves more than one reinforcement. The reinforcement volume fractions can vary in each rim which leads to the addition or subtraction of one type of reinforcement tow/ply during the winding/ layup process. This change in reinforcement tow/ply is difficult to execute during the fabrication process with large number of rims. Accordingly, a few rims are selected for the design of a multi rim hybrid composite rotor. In this study, four hoop wound rims are introduced for each type of rotor design case to ensure an adequate winding process. The design cases are: A, B, and C; and the four hybrid rims are numbered: 1–4. These rims have carbon–glass reinforcements with varying volume fractions in composites. In addition, the reinforcements are hybridized in such a way that the proportion of carbon fiber increases from rim 1 to rim 4 (i.e., from the rotor inner surface to outer surface). This hybridization sequence is selected to maintain low stiffness at the inner radius and relatively high stiffness at the outer radius. Thus, the favorable compressive stresses are developed to counter the tensile radial stress during rotor rotation [7,15]. 2.1. Case A In this case, all the rims are simultaneously wound and cured by continuous filament winding. Initially, rim 1 is wound with carbon–glass tows and carbon fiber tows are then added into the winding process for rims 2–4. The carbon fiber volume fractions of the rims follow the order ‘‘rim 1 < rim 2 < rim 3 < rim 4’’. 2.2. Case B Interference fit is an effective way of reducing the radial stress during rotation of the flywheel rotor [7,12,15]. In this case (unlike the former one), all the rims are wound and cured separately. As in case A, rim 1 is wound initially and the other rims are then wound
Fig. 2. Rotor design cases.
in subsequent order. However, in this design case, the rim winding sequence is unnecessary. The change in reinforcement volume fractions is like in case A, with the least carbon fiber in rim 1 and the most in rim 4. Afterward, these rims are press fitted with the interferences d1, d2, and d3, followed by the machining of rim surfaces. 2.3. Case C The design concepts of the former cases are adopted to some extent in this case of a multi-rim hybrid composite rotor. Here, rims are wound in two sets. Rim set 1 is composed of rims 1 and 2, and rim set 2 is composed of rims 3 and 4. The outer surface of rim 2 and the inner surface of rim 3 are machined and press fitted by an interference d. Furthermore, for this case, the change in reinforcement volume fraction is similar to that in case A or B. For better understanding, Fig. 2 shows schematics of the three design cases. 3. Stress sources of rotor design cases Since the case A, B, and C described hereinbefore have different designs, sources of radial and hoop stress generated in a rotor of height h and inner radius ri are also different. For each case, these stresses are calculated at two rotor states: stationary and rotating at maximum allowable speed xmax. The former state is considered to determine the residual stresses developed during the manufacturing process of a hybrid composite flywheel rotor. These primary stresses are caused by anisotropic thermal expansion of the hybrid plies and/or the assembly of rims incorporating interference. The latter state is taken into account to consider the effect of centrifugal forces during rotor operation. Regarding the stress sources of three cases, for case A, the dominant stress source in the stationary rotor is the temperature difference DT produced during winding and curing of the rims. Since all the rims are wound and cured simultaneously, the effect of thermal residual stress on the rotor performance is worst in this design [5]. Afterward, this stress is accompanied by the inertial stress at xmax. Fig. 3 depicts these stress sources. On the other hand, in case B, all the rims are wound and cured separately, and interferences d1, d2, and d3 are incorporated among the rims. These interferences lead to a compressive stress generation, reducing the effect of centrifugal force during rotor operation.
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ΔT
ω = 0, ω max
h ri Fig. 3. Stress sources of case A.
ΔT
ΔT
ΔT
ΔT δ3
δ1
ω = 0, ω max δ1
δ2
δ3
h ri Fig. 4. Stress sources of case B.
ΔT
ΔT δ
ω = 0, ωmax
h ri Fig. 5. Stress sources of case C.
Furthermore, the effect of thermal residual stress is not as bad as in case A because the thickness of each rim is much smaller. Fig. 4 shows the stress sources of this case. The stress sources in case C are similar to those in case B; however, the magnitude of stress differs due to the manufacturing and assembly process. Since two rims are wound and cured at the same time, thermal residual stress is high. Moreover, only one interference d is integrated for press fit assembly of the rims; the prestress developed is also less than that in case B. These stress sources are depicted in Fig. 5. 4. Stress analysis of a rotor A stress analysis to calculate the deformations and strength ratios of a multi-rim rotor can be found in previous publications [6,7,15]. However, a summary of relevant equations is given below. Eq. (1) shows the governing equation for a rotor rotating at speed x at radius r:
drr rr rh þ þ qrx2 ¼ 0; dr r
ð1Þ
where r and q are stress and density, respectively. Assuming the plane stress state, the stress–strain relationship in cylindrical coordinates is given by
Q 11 Q 12 rh ¼ rr Q 21 Q 22
eh ah DT ; er ar DT
ð2Þ
where Q, e, and a are stiffness matrix, strain, and coefficient of thermal expansion, respectively. Since the rotor satisfies the axisymmetry conditions, strains are defined as a function of radial displacement ur and can be expressed as
ur =r eh ¼ : @ur =@r er
ð3Þ
For the case A rotor type, rims should satisfy the following compatibility conditions:
S.K. Ha et al. / Composite Structures 94 (2012) 3290–3299
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procedure, initially calculates thermal residual stresses which are then added to the stresses produced by the interference. Finally, the inertial stress is added to the former stresses to calculate the overall stress. There are several other ways to calculate these stresses, but this option is the most convenient. 5. Design optimization of rotor cases
Fig. 6. Cylindrical coordinate system and symbols used for a multi-rim rotor.
ðjþ1Þ rðjþ1Þ ¼ rðjÞ ¼ uðjÞ ri r o and uri ro :
ð4Þ
For the case B hybrid rotor, rims should satisfy the following compatibility conditions: ðjÞ ðjþ1Þ rðjþ1Þ ¼ rðjÞ uðjÞ ri r o and uri ro ¼ d :
ð5Þ
Two strength ratios are considered for design optimization: the radial strength ratio Rr = rr/Y and the hoop strength ratio Rh = rh/X. Fig. 6 shows the coordinate system and symbols used in these equations. Keeping in mind the stress sources of each rotor case, a simple procedure has been adopted to calculate stresses in a rotor. This
In the design cases, rims may vary with respect to thickness, volume ratio of reinforcement and interference for press fit. Therefore, optimization is essential for the selection of appropriate values. The optimization parameters are: design variables, objective functions and design constraints. Design variables (DVs) include thickness ti, carbon fiber volume fraction Vfi of each rim, and interferences. An objective function is used to reduce the maximum strength ratio in all the cases. This function is subjected to a design constraint that the strength ratio should be less than 1. The purpose of this limitation is to avoid any material failure when the rotor is at rest or rotating at xmax. Moreover, the compressive stress developed in the rims during press fitting in case B or case C is limited to 6 MPa. An optimization flow chart is portrayed in Fig. 7. In this flow chart, Emax and eh are the maximum useable energy and hoop strain at the inner surface respectively. The flow chart in Fig. 7 represents the general outlook of the optimization, in fact for each design case; an individual optimization is performed considering the respective design variables, material properties, and temperature DT. However, given specifications are same for all the cases. During optimization, the stress analysis [6] for each case is performed individually using the given specifications, geometric parameters, material properties, and temperature DT. The strength ratios are then calculated to ensure that the initial and subsequent design variables are selected in such a way that R is less than 1. With the aid of optimization, design variables that satisfy the design constraints are then selected, and analysis is performed to
Given r i, h, ω = 0 & ωmax, Emax, εθ
Stress analysis
Initial value of design variables for case A, B, and C
Strength ratio (Rr, Rθ)
Optimization
Update DV
Find ti, V fi ti, V fi, δe ti, V fi, δ Minimize
for case A for case B for case C
Max. (R*(i))
Subject to R*(i) < 1 at ω = 0, ωmax σr < 6 MPa (compressive) at ω = 0 R*
(i)
= max. (Rri, Rθi)
i = 1, 2, 3, 4; e = 1, 2, 3 No Optimum Yes ti, V fi, δe, δ Fig. 7. Optimization flow chart.
Material properties Temperature ΔT for case A, B, and C
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S.K. Ha et al. / Composite Structures 94 (2012) 3290–3299
Table 1 Optimized parameters of hybrid rims. Parameter
Case A
Case B
Case C
Unit
t1 t2 t3 t4 Vf1 Vf2 Vf3 Vf4 d1 d2 d3 d
54.20 35.50 34.30 60.1 11 40 66 100 – – – –
39.90 50.30 51.00 42.30 10 35 72 100 0.13 0.07 0.15 –
36.80 55.00 52.00 40.00 11 30 79 100 – – – 0.20
mm mm mm mm % % % % mm mm mm mm
determine the optimum parameters of hybrid rims. The optimization algorithm is illustrated in Appendix B. 6. Numerical calculations The optimized design variables are obtained using the same input specifications for all the design cases. These specifications are Emax = 35 kW h; xmax = 15,000 rpm; ri = 269.75 mm; h = 1340 mm; and eh = 0.8. The optimized design variables are listed in Table 1. The volume fractions in Table 1 are the percentages of carbon fiber considering only the glass–carbon reinforcements. Table 1 does not contain the percentage of hybrid material that also includes matrix as a constituent material. For example, in case A, the Vf1 value is 11%, which means carbon fiber accounts for 11% and glass fiber 89% of the glass–carbon reinforcement. Although the carbon fiber volume ratio is increased from the inner to the outer rim, each rim hybrid material has an effective reinforcement volume fraction of 67% and a matrix fraction of 33%. Also, the effect of void content is ignored. The properties of constituent materials are listed in Table 2. The effective properties of hybrid composite material are detailed in Table 3. These properties are determined by the rule of mixtures given in Appendix A. Table 3 shows how the hybridization affects the properties. For instance, increasing the volume
fraction of carbon fiber from rim 1 to rim 4, decreases density, but increases longitudinal modulus and tensile strength. However, the transverse modulus decreases due to the anisotropy of the carbon fiber. The effect of hybridization on other properties of interest can also be seen. Primarily, the radial stress distribution along the radial coordinate of the rotor for all cases is shown in Fig. 8. In case A, we first calculate the radial stress due to temperature change DT. The value reaches about 10 MPa, which is greater than in any other case due to simultaneous winding and curing of all four rims. The inertial stress generated due to xmax is then added to the thermal stress to calculate the overall stress of the rotor in case A as shown in Fig. 8a. Since, in case B, all the rims are wound and cured individually, the radial stress produced by the temperature change DT is less than in other cases. The maximum value reaches about 1 MPa for rim 3 because this rim is the thickest. The compressive stress due to interferences de (e = 1, 2, 3) is then added to the thermal residual stress such that the value approaches 6 MPa (optimization constraint). Finally, the radial stress of the rotor in case B rotor is determined by incorporating the centrifugal force effect due to xmax, as shown in Fig. 8b. A radial stress of about 2.5 MPa is generated in the case C rotor during the winding and curing of rim set 1 and rim set 2. The interference d is then incorporated to assemble the rim sets by press fit, and the stress value approaches 6 MPa (optimization constraint). Finally, the radial stress behavior of the rotor in case C is obtained by incorporating inertial stress into the former stresses, as shown in Fig. 8c. The hoop stress is calculated in the way same as the radial stress. First, stress due to DT is considered, and then stress due to interference de (e = 1, 2, 3) or d is added to the thermal residual stress. Finally, inertial stress generated by xmax is incorporated. The hoop stress generated by the temperature change DT is greater in case A than in any other case. Since, in this case, all the rims are wound and cured simultaneously, the value approaches 750 MPa as shown in Fig. 9a. Nevertheless, the thermal residual hoop stress is the least in case B because all the rims are wound and cured separately. However, the overall hoop stress developed by the three
Table 2 Constituent material properties. Property
Symbol
Carbon fiber
Glass fiber
Epoxy
Unit
Density Longitudinal modulus Transverse modulus Poisson’s ratio Longitudinal CTE Transverse CTE Longitudinal tensile strength Transverse tensile strength
q
1800 230 23 0.20 0.90 9.50 4902 77
2540 72 72 0.22 5 5 2068 3103
1190 3.20 3.20 0.37 54 54 67 67
kg/m3 GPa GPa – 106/°C 106/°C MPa MPa
Ex Ey
txy ax ay X Y
Table 3 Effective properties of hybrid rims. Property
q Ex Ey
txy ax ay X Y
Unit
kg/m3 GPa GPa – 106/°C 106/°C MPa MPa
Case A
Case B
Case C
Rim 1
Rim 2
Rim 3
Rim 4
Rim 1
Rim 2
Rim 3
Rim 4
Rim 1
Rim 2
Rim 3
Rim 4
2040 61 19.40 0.27 5 22 1616 15
1896 92 16 0.26 2 24.50 2167 15
1769 119 13 0.26 0.56 26.43 2654 15
1599 155 9 0.26 0.51 28.80 3306 15
2040 61 19.50 0.27 4.20 21.90 1605 15
1922 86 16.50 0.26 2.30 24.10 2067 15
1739 125 12 0.26 0.30 26.90 2770 15
1599 155 9 0.26 0.50 28.80 3307 15
2040 61 19 0.27 4.90 21.90 1616 15
1946 81 17 0.26 2.70 23.70 1977 15
1703 133 11 0.26 0.10 27.40 2908 15
1599 155 9 0.26 0.50 28.80 3306 15
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(a)
Radial stres (MPa)
(b)
ΔT+ωmax
ΔT
15
ΔT
15
ΔT+δe
(c)
ΔT+δe+ωmax
10
10
10
5
5
5
0 0.30
0.35
0.40
0.45
0.50
-5
0.25
0.30
0.35
0.40
0.45
rim 1
rim 2 rim 3
rim 4 Radius (m)
0.25
0.50
ΔT+δ+ωmax
0.30
0.35
0.40
0.45
0.50
-5
-5
-10
ΔT+δ
0
0 0.25
ΔT
15
-10
rim 1
rim 2
rim 3
-10
rim 4
rim 1
rim 2
rim 3
rim 4
Fig. 8. Radial stress distribution in hybrid rims: (a) case A, (b) case B, and (c) case C.
Hoop stress (MPa)
(a)
ΔT
(b)
ΔT+ω max
ΔT
ΔT+δe
ΔT+δe+ω max
(c)
1000
800
800
800
600
600
600
400
400
400
200
200
200
0 -200
0.30
0.35
0.40
0.45
0.50
rim 1 rim 2 rim 3 rim 4 Radius (m)
ΔT+δ
ΔT+δ+ω max
0
0 0.25
ΔT
1000
1000
0.25 -200
0.30
0.35
0.40
0.45
0.25
0.50
0.30
0.35
0.40
0.45
0.50
-200 rim 1 rim 2
rim 3
rim 4
rim 1 rim 2
rim 3
rim 4
Fig. 9. Hoop stress distribution in hybrid rims: (a) case A, (b) case B, and (c) case C.
0.75 when it is at xmax. On the other hand, the hoop strength ratio at xmax shows a downward trend from the inner rim to the outer rim due to the increasing volume fraction of carbon fiber. Furthermore, for a stationary rotor hoop strength ratio is tensile for rims 1 and 2 and compressive for rims 3 and 4, as shown in Fig. 10a. Conversely, the hybrid rotor in case B has the least radial stress because of interferences. Therefore, the radial strength ratio is much lower than in case A or case C at both rotor states, as shown in Fig. 10b. For a stationary rotor, the value is about 0.16, and that at xmax is nearly 0.19. Generally, the hoop strength ratio for a rotor
sources is greater than in the other cases (i.e., about 800 MPa as shown in Fig. 9b). The primary reason for this peak is the interferences that occur when the rotor is assembled by press fit, and the secondary is xmax. A similar effect of interference can also be seen in Fig. 9c when the rotor is assembled by press fit. The strength ratios for all cases have been optimized at x = 0 and xmax = 15,000 rpm. Fig. 10 shows the strength ratios for three cases at different radii. Since case A has the maximum radial stress, the radial strength ratio is greater than those of the other two design cases. The value approaches 0.70 when rotor is stationary and
Strength ratio
(a)
(b)
(c)
0.80
0.80
0.80
0.60
0.60
0.60
0.40
0.40
0.40
0.20
0.20
0.20
radial_ωmax radial_0 rpm
hoop_ωmax hoop_0 rpm
0.00
-0.20
0.00
0.00 0.25
0.30
0.35
0.40
0.45
0.50
rim 1 rim 2 rim 3 rim 4 Radius (m)
0.25 -0.20
0.30 rim 1
0.35 rim 2
0.40 rim 3
0.45 rim 4
0.25
0.50 -0.20
Fig. 10. Strength ratio of hybrid rims: (a) case A, (b) case B, and (c) case C.
0.30
0.35
rim 1 rim 2
0.40
0.45
rim 3 rim 4
0.50
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Table 4 Comparison of cost and strength ratio. Parameter
Case A
Case B
Case C
Maximum radial strength ratio Maximum hoop strength ratio Estimated manufacturing cost ($)
0.75 0.42 25,000
0.19 0.36 44,000
0.39 0.42 31,000
at rest changes from compressive to tensile from the inner radius to the outer radius. However, at xmax, the maximum value reaches 0.36 at the inner rim and the minimum value is approximately 0.19 at the outer rim. The radial strength ratio is higher in case C than in case B, but less than in case A at xmax with a maximum value of about 0.39, as shown in Fig. 10c. In addition, the hoop strength ratio is nearly 0.42 at the inner rim and 0.18 at the outer rim. Furthermore, when the rotor is at rest, the hoop strength ratio is tensile for rims 1 and 2 and compressive for rims 3 and 4. We have found that, in all design cases for a rotor at xmax, the hoop strength ratio is maximal at the inner rim and minimal at the outer rim because of reinforcements changing the volume fractions as described earlier in this study. Moreover, thermal residual stresses have a devastating effect on the rotor performance, particularly when all the rims are wound simultaneously, as in case A. The effect of these stresses can be reduced by winding and curing the rims separately and incorporating the interferences as in cases B and C. Accordingly, each of the three cases has its own manufacturing process, which leads to a tradeoff between cost and strength ratio. For example, rotor strength ratios in case A are higher than in the other cases, but cost is lower due to the simplified fabrication process. Manufacturing requires only one mandrel, on which all the rims are wound and then cured.
On the other hand, the cost of rotor fabrication in case B is higher than in the other cases. To manufacture four rims in case B, mandrels with different outer diameters are required. Subsequently, machining is performed on all the rims to maintain interferences for press fitting, which requires additional arrangements. However, the strength ratio is lower than in case A or case C hybrid rotors. The hybrid rotor in case C can be considered an intermediate between the case A and case B hybrid rotors. Its strength ratio is greater than that of case B, but less than that of case A. The fabrication process is simple and shorter than that of case B which ultimately reduces the cost. However, the press fit is still required to assemble two sets of rims, which increases the cost to be more than that of case A. It has been found that comparing the radial strength ratio within each rotor during rotation, ‘‘case B < case C < case A’’. In terms of manufacturing cost the reverse order is held as ‘‘case B > case C > case A’’. Strength ratios calculated herein are listed in Table 4 along with manufacturing cost. This cost includes mandrels, reinforcing fibers, epoxy, winding, machining, and press fitting assembly. The operational cost is not included in this study. 7. Fabrication of case C hybrid rotor Based on the optimization study and manufacturing process, the hybrid rotor in case C was selected for manufacture using a Hexion 166 epoxy system along with hoop wound carbon–glass reinforcements. Rim set 1 was manufactured on one mandrel by continuous filament winding under in situ curing [16,17] conditions using a slip ring and IR heater. Similarly, rim set 2 was wound on another mandrel as shown in Fig. 11. Both rim sets were wound using similar winding methods; however, different images are shown in Fig. 11 to cover the whole winding process.
Fig. 11. Rim manufacturing by filament winding.
Fig. 12. Penetrant testing of rims.
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Prior to the assembly, all rims were checked for visible surface defects. Penetrant testing was used to inspect for surface cracks, porosity, delamination and a lack of bond at exposed edges. Primarily, a liquid penetrant was applied to the surface of the rotor for a specific period of time to penetrate into the defects present at the surface. After the penetration period, surplus penetrant was removed. An absorbent light colored powdered material called developer was then applied to the surface. This developer acted as a blotter and drew out a portion of the penetrant that previously seeped into the surface cracks. As the penetrant was drawn out, it diffused into a coating of the developer, forming colored indications much wider than the original surface defects (if present) with which they were associated. An inspector then viewed the parts and examined these indications against a background of the developing powder. In this study, using the penetrant method, no visible defects were found on the surface under examination, as shown in Fig. 12.
The rims were then assembled by press fit, incorporating the interference d following the machining of rims. For this assembly, the vertical press fit force Fv was determined using the slope angle b of tapered rims. This angle was determined by considering the machining tolerances and interference. Moreover, this side slope angle was smooth enough not to cause any material cracking during press fit. The vertical force is given by [15]:
F v ¼ F d ðtan b þ lÞ; where Fd is the interior force calculated using Eqs. (1)–(5), and l is coefficient of friction among the rim surfaces (taken as 0.20). The calculation of this force is illustrated in Fig. 13. The outer surface of rim 2 and the inner surface of rim 3 were tapered by CNC machining. These machined rims were then press fitted using a hydraulic press with a vertical force of approximately 1000 tons, as shown in Fig. 14.
Press fit force (Fv)
Fv sin β
Rim 1 Rim 2
β Fv cos β
β
Fv µF δ cos β
Fδ sin β β
Fδ
Fδ cos β
h Rim 3 Rim 4
δ Fig. 13. Calculation of hydraulic press fit force.
Fig. 14. Rim assembly by press fit.
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8. Conclusion
Appendix B. Optimization algorithm
Three cases of composite flywheel rotor design by rim hybridization are presented to investigate the effect on rotor performance by optimizing the strength ratios. Pros and cons are also taken into account for all the designs. We have shown throughout this study that the manufacturing process and strength ratio adequately differentiated the three cases. Due to a rapid and simplified manufacturing process, the hybrid rotor in case A has advantages over the other two hybridization techniques with respect to cost, but the radial strength ratio reaches almost 0.75. The radial strength ratio approaches 0.19 in the hybrid rotor in case B, the lowest of all the cases. However, this design is costly due to a lengthy and extensive fabrication process. Due to interference, the hybrid rotor in case C has a lower radial strength ratio than that in case A (i.e., 0.39), but the cost is higher. Comparing the radial strength ratio in the rotors, ‘‘case B < case C < case A’’. In terms of manufacturing cost the reverse order is held as ‘‘case B > case C > case A’’. Moreover, the strength ratios in all design cases are within the established limits (i.e., less than 1). Furthermore, the rotor performance can be improved by the interference provision, which reduces the radial strength ratio, resulting in higher operational speed. We selected for manufacture one hybrid rotor design that best suits our requirement. The research conducted in this paper can be applied to other hybrid composite flywheel rotors with enhanced performance and energy storage capacity.
The nonlinear optimization in this study requires the constrained minimization to find an optimal solution. Moreover, all the design variables are continuous. Therefore, the sequential quadratic programming (SQP) method [20–23] is used to solve the nonlinear optimization problem in this investigation. Several local optima are possible while considering fiber angles as design variables [21] that affect adversely the accuracy of the optimization method. However, in this study, the probability of several local optima is reduced as the reinforcements are hoop wound in all the three design cases and thus fiber angle is not a design variable. The optimization for each case is performed individually considering the respective design variables, material properties, and temperature DT; however a single algorithm of SQP is shown in Fig. B1 for convenience. The SQP is a well known method to solve nonlinear optimization problems. In this method, the sub-problem for directionfinding is formulated using the Tailor series of the quadratic approximate objective function (Max. R(i)) and linearized conðiÞ straint functions (gj = Rj , rrj ; where j = 1, M). The sub-problem is then solved using the modified method of feasible directions [24]. The B matrix is initially set to identity matrix and then updated to approach the Hessian of the Lagrangian function. Lagrange multipliers kj are then calculated and search is done in S direction using approximate Lagrangian function to minimize /:
Appendix A. Rule of mixtures
/ ¼ FðXÞ þ
M X uj max 0; g j ðXÞ ;
ðB:1Þ
j¼1
The rule of mixtures [14,18,19] is used to calculate the effective properties of hybrid material lamina. These properties are calculated as given by Eqs. (A.1), (A.2), (A.3), (A.4), (A.5), (A.6), (A.7) for a given volume fraction V:
q¼
X
qðpÞ V ðpÞ ;
X
Ex ¼ 1 ¼ Ey
ðA:1Þ
ðpÞ EðpÞ ; x V
V ðmÞ EyðmÞ
þ
gðgf Þ V ðgf Þ Eyðgf Þ
ðA:2Þ
þ
!, gðcf Þ V ðcf Þ Þ Eðcf y
V ðmÞ þ gðgf Þ V ðgf Þ þ gðcf Þ V ðcf Þ ; ðA:3Þ
txy ¼
X
ðpÞ ðpÞ txy V ;
X¼
X
X
ðpÞ ðpÞ 1 þ tðpÞ ax txy ; xy ay V
X ðpÞ V ðpÞ :
Iteration number q q+1
ðA:5Þ
Sub-problem for direction-finding S Max. R*(i) Using modified method of feasible directions
ðA:6Þ
Perform one-dimensional search to minimize
ðA:7Þ
In Eq. (A.3), m, gf, and cf denote matrix, glass fiber and carbon fiber, respectively. For the rest of the equations, p stands for matrix, carbon fiber and glass fiber to consider the hybrid constituents cumulatively. The transverse tensile strength (Y) for the hybrid rims was determined experimentally considering only the matrix effect. The stress partitioning factors
.
Initial design variables X0 = ti0, Vfi0, δe0, δ0 Initial Hessian matrix, B = I
ðA:4Þ
1 X ðpÞ ðpÞ ðpÞ ax ¼ ax Ex V ; Ex
ay ¼
where X = Xq1 + aS, uj ¼ jkj j for first iteration, uj ¼ 1 max½jkj j; 2 ðu0j þ jkj jÞ for subsequent iterations and u0j ¼ uj from the previous iterations. After one-dimensional search, the B matrix is updated using BFGS formula [24,25]:
Xq + α*Sq
Xq+1
Yes Converged No *
Calculate B using BFGS formula
.
Þ Þ gðgf Þ ¼ rðgf rðmÞ and gðcf Þ ¼ rðcf ryðmÞ ; y y y
B were also experimentally determined to have values of 3.78 and 1.48, respectively.
B*
Fig. B1. SQP algorithm.
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S.K. Ha et al. / Composite Structures 94 (2012) 3290–3299
B ¼ B
BppT B ggT þ T ; pT Bp p g
ðB:2Þ
where p ¼ X q X q1 , g ¼ hy þ ð1 hÞBp, P / ¼ FðXÞ þ M j¼1 kj g j ðXÞ, and
(
h¼
1:0
if pT y P 0:2pT Bp
0:8pT Bp pT BppT y
if pT y < 0:2pT Bp
y ¼ rx /q rx /q1 ,
:
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