# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors

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## Abstract

**:**

## 1. Introduction

#### 1.1. State of the Art

#### 1.2. Outline of the Current Study

## 2. Manufacturing Process

#### 2.1. Filament Winding Process

#### 2.2. Curing Process

#### 2.3. Turning Process and Interference Fits

#### 2.4. Machine Costs

_{operation}. In accordance with [21], 252 days at 8 h per day are assumed, equaling 2016 h per year. K

_{A}are the cost-accounting depreciations. This study considers a linear depreciation over 10 years. After 10 years the machine is replaced by a newer version. K

_{I}are the cost-accounting interests which are not an actual expense but reflect the opportunity costs. These costs are not included in the presented study. K

_{M}are the maintenance costs which are assumed to be 5% of the acquisition price per year. K

_{R}and K

_{E}are room costs and energy costs, respectively. Energy costs are neglected in this study and the room costs are assumed to be 20 CAD/m

^{2}. An example calculation of the MHR for the four-axis filament winding system consisting of a four-axis filament winding machine, a resin mixing system, a resin bath and a fiber tensioner equipped with six creels is shown in Table 2. Similarly, the MHR of the oven, the lathe and the press are calculated. The MHRs of all machines are depicted in Table 3. Please note that MHR are facility and enterprise dependent, and values used in this study should therefore be considered an example.

## 3. Modeling

#### 3.1. Total Stored Energy

_{zz}, the energy can be expressed as

_{j}is the mass of each rotor rim and r

_{j}and r

_{j}

_{+1}are the inner and outer rim radius, respectively. Given the material density ρ

_{j}, the rim mass can be expressed as

#### 3.2. Costs

#### 3.2.1. Material Costs

_{material,j}the calculation is straightforward, i.e.,

#### 3.2.2. Manufacturing Costs

_{conv}. The MHR is part of the calculation for conversion costs because the machine cannot be used for other tasks during conversion. c

_{conv}may represent labor cost or another MHR when a robot performs the conversion. Since a conversion generally happens for each rim, the conversion costs are multiplied by the number of rims N

_{rim}. Equation (5) shows a generic formula for the manufacturing costs.

_{turning}with the tool width d

_{tool}, the process time can be expressed as

#### 3.3. Productivity

#### 3.3.1. Output

#### 3.3.2. Time

_{oven}. The manufacturing times for each process can be written as follows:

#### 3.4. Analytical Stress Calculation and Failure Prediction

#### 3.4.1. Load Cases

#### 3.4.2. Analytical Stress Calculation

_{r}and σ

_{θ}, respectively, the following equation can be written

_{zz}and σ

_{θz}can be neglected. Each of the equations discussed in this subsection can be applied to any rim of the rotor. To calculate stresses and displacements for the entire rotor 2(N

_{rim}− 1), continuity conditions are introduced. The radial stresses are continuous at the rim surfaces, while the radial displacements may differ by an interference δ

^{(i)}due to press fitting.

_{rim}− 1. The continuity of radial stresses is only valid as long as the rims are attached to each other. In case of interference fits the attachment vanishes when the tensile stresses induced by rotation exceed the radial stresses due to the interference fit. Hence, a computed tensile stress would lead to a detachment failure.

_{rim}constraint equations and therefore 2N

_{rim}unknown constants, two additional equations are required for the solution. They can be obtained by predefining the radial stresses at the innermost and outermost rotor radius.

_{in}and p

_{out}, respectively. In this study, p

_{in}and p

_{out}are set to zero.

#### 3.4.3. Failure Criteria

## 4. Optimization

#### 4.1. Formulation and Characterization of the Optimization Problem

_{rim,max}is also externally specified limiting the computation time. Calculating production times and costs requires information about the used fabrication equipment. Related information represents the last design variable. Results of the presented optimization tool are therefore only optimal for the selected fabrication facility configuration. Even though this restricts the validity of the results to a specific configuration, this procedure also highly simplifies the optimization problem. In practice, companies usually have machines in place that cannot be easily replaced due to high acquisition costs, and hence, the optimization tool and design variables would have to be adjusted accordingly.

_{max}, the rim number N

_{rim}as well as the rim thickness t

_{i}(i.e., the radial rotor dimension), the radial displacement due to press fitting δ

_{i}and the material choice x

_{material,i}for each rim i. The rim number and the material choice are the most challenging optimization variables as they tremendously increase the complexity of the optimization problem. The optimization of material properties is discrete. Since a gradient cannot be calculated for a discrete problem, this excludes all local optimization algorithms. Local optimization algorithms are more effective than global algorithms, therefore, the discretization of the problem negatively effects the computation time. To solve discrete optimization problems, they can be transferred into integer problems. For the considered problem, the material choice x

_{material,i}has to be an integer variable:

#### 4.2. Hybrid Optimization Strategy

_{max}. The number of rims considered is increased by one after each iteration. The optimization algorithm terminates if the maximum number of rims is reached. In the result, a sub-problem is created that considers the number of rims as a fixed design variable. Each iteration generates a Pareto-front with the optimization matrix

**X**

_{opt}and the corresponding objective values

**F**

_{opt}. Each solution

**x**

_{s,opt},

**f**

_{s,opt}on this Pareto-front is compared to the previous Pareto-front to find the non-dominated solutions.

_{rim}> 1 in an appropriate time. The number of optimization variables is increased by three for each additional rim. The calculation of the fitness values takes longer with an increased number of rims. Further, a large number of optimization variables makes it more difficult for a stochastic optimization algorithm to find an optimal solution. To gain a better understanding of the problems associated with a GA, an example is analyzed.

_{i}only affect stresses. Second, the angular velocity ω only affects stresses and kinetic energy. After terminating, the GA returns the Pareto-front

**X’**

_{opt},

**F’**

_{opt}, where

**x’**

_{s,opt}is a single solution on this front and is defined as:

**x’**

_{s,opt}on the Pareto-front can be further improved by solving a single-objective sub-problem. This sub-problem only maximizes the kinetic energy by optimizing the angular velocity and the interferences, i.e.,

**x’**

_{s,opt}, the additional computation time is comparatively low. This is due to the fact that the sub-problem is less complex and gradient-based algorithms generally converge faster. To further improve the optimization result, the GA and the local improvement is run a second time. Based on the results of the first optimization, the initial population of the GA is generated.

## 5. Case Studies

_{100%}denotes the solution with the highest SED found. Further, SED

_{85%}represents the solution which achieves 85% of the highest SED possible. Due to the discrete material properties and different numbers of rims, the Pareto-front shows several cost-jumps between the solutions. Comparing the marked solutions, SED

_{100%}achieves a 17% higher stored energy than SED

_{85%}. However, costs also increase by 36%. The Pareto-front creates the possibility to analyze trade-offs, such as the SED-cost trade-off, that are required. Knowing the Pareto-front, the design decision may differ, as the associated trade-offs may be considered too large. This highlights the advantage of a multi-objective optimization approach, which provides more information for decision-makers.

## 6. Conclusions

## 7. Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Examples of rotor rim geometries to facilitate press fitting. Adapted from [15].

**Figure 4.**Two-dimensional Pareto-front of the ACME geometry, AL: aluminum, A: aramid, G: glass, C: carbon.

**Table 1.**Prices, space and maintenance costs of manufacturing machines (based on [21]).

Name | Acquisition Price [CAD] | Space [m^{2}] | Maintenance [CAD/year] |
---|---|---|---|

FW-machine four-axis | 145,000 | 16 | 7250 |

Resin mix-system | 125,000 | 5 | 6250 |

Heated resin bath | 8000 | 3 | 400 |

Fiber tensioner with 4 creels | 71,000 | 10 | 3200 |

Curing oven | 50,000 | 30 | 2500 |

Press | 50,000 | 5 | 2500 |

Lathe | 50,000 | 5 | 2500 |

Time of Operation per Year | 2016 | Hours |
---|---|---|

Acquisition costs | 349,000 | CAD |

Economic lifetime | 10 | years |

Maintenance costs | 17,450 | CAD/year |

Space | 34 | m^{2} |

Room costs | 680 | CAD/month |

K_{A} | 17.30 | CAD/h |

K_{M} | 8.66 | CAD/h |

K_{R} | 4.05 | CAD/h |

MHR | 30.01 | CAD/h |

Name | MHR [CAD/h] |
---|---|

Filament winding system | 30.01 |

Curing oven | 7.29 |

Press | 4.32 |

Lathe | 4.32 |

Parameter | Value | Unit |
---|---|---|

c_{Labor} | 35 | CAD/h |

c_{conv} | 35 | CAD/h |

t_{conv,winding} | 80 | min |

t_{conv,curing} | 25 | min |

t_{conv,turning} | 10 | min |

t_{conv,pressing} | 15 | min |

N_{creels} | 6 | - |

v_{fibers} | 1 | m/s |

A_{roving} | 5 | mm^{2} |

t_{curing} | 420 | min |

C_{oven} | 10 | - |

v_{turning} | 0.1 | m/s |

d_{tool} | 5 | mm |

t_{press} | 15 | min |

Solution | 1 | 2 | 3 | Unit |
---|---|---|---|---|

n | 20,000 | 25,000 | 25,000 | rpm |

t_{1} | 100 | 100 | 100 | mm |

t_{2} | 100 | 100 | 100 | mm |

x_{material,1} | 1 | 1 | 1 | - |

x_{material,2} | 3 | 3 | 3 | - |

δ_{1} | 0.5 | 0.4 | 0.5 | mm |

δ_{2} | 0 | 0 | 0 | mm |

E_{kin} | 1.2 | 1.9 | 1.9 | kWh |

C_{total} | 1154 | 1154 | 1154 | CAD |

P | 0.3 | 0.3 | 0.3 | units/h |

R_{max} | −0.59 | −0.09 | −0.18 | - |

Name | Energy [kWh] | Rotational Speed [rpm] | Rotor Height [mm] | Inner Rotor Radius [mm] | Outer Rotor Radius [mm] |
---|---|---|---|---|---|

ACME flywheel [20] | 0.25 | 30,000 | 50 | 100 | 200 |

Rosseta T2 [24] | 4.0 | 25,000 | 200 | 100 | 300 |

Material Property | Aluminum Alloy | Glass/Epoxy | Aramid/Epoxy | Carbon/Epoxy | Unit |
---|---|---|---|---|---|

E_{1} | 71.7 | 41.0 | 66.7 | 170 | GPa |

E_{2} | 71.7 | 10.4 | 5.5 | 16.36 | GPa |

G_{12} | 26.9 | 4.3 | 2.2 | 7.8 | GPa |

ν_{12} | 0.33 | 0.28 | 0.34 | 0.32 | - |

ν_{23} | 0.33 | 0.49 | 0.5 | 0.41 | - |

ρ | 2810 | 1970 | 1343 | 1568 | kg/m^{3} |

X^{T} | 503 | 1140 | 1400 | 2720 | MPa |

X^{C} | 503 | 620 | 335 | 1689 | MPa |

Y^{T} | 503 | 39 | 30 | 64 | MPa |

Y^{C} | 503 | 39 | 30 | 64 | MPa |

S | 331 | 128 | 158 | 307 | MPa |

Specific cost | 5 | 10 | 22 | 90 | CAD/kg |

Longitudinal CTE | 23.6 | 8.6 | −3.5 | −0.16 | 10^{−6}/K |

Transverse CTE | 23.6 | 22.1 | 35 | 18.9 | 10^{−6}/K |

Parameter | Value | Unit |
---|---|---|

GA: population size (N_{rim} = 1) | 50 | - |

GA: population size (N_{rim} $\ge $ 2) | 200 | - |

GA: minimum number of generations | 100 | - |

GA: maximum number of generations | 400 + 200 × N_{rim} | - |

GA: relative tolerance | 10^{−5} | - |

EA: crossover rate | 20% | - |

SQP: objective tolerance | 10^{−4} | - |

SQP: constraint tolerance | 10^{−5} | - |

Temperature difference | −150 | K |

Maximum number of rims | 5 | - |

Thinnest rim width | 1 | mm |

Minimum angular velocity | 0 | rpm |

Maximum angular velocity | 80,000 | rpm |

Minimum interference between rims | 0 | mm |

Maximum interference between rims | 0.8 | mm |

**Table 9.**Summary of case study results: Optimal material choice and objective values for different rotor designs. (1): lowest costs, (2): highest energy, (3): most cost-efficient composite rotor, AL: aluminum, A: aramid, C: carbon.

Name | Materials | Total Cost [CAD] | Share of Manufacturing Cost | Energy [kWh] | Productivity [units/h] | Cost/Energy [CAD/kWh] |
---|---|---|---|---|---|---|

ACME (1) | AL | 66 | 0% | 0.09 | - | 720 |

ACME (2) | AL,C,C,C,C | 1244 | 49% | 0.46 | 0.04 | 2698 |

ACME (3) | AL,A | 269 | 57% | 0.13 | 0.37 | 2105 |

Rosseta T2 (1) | AL | 706 | 0% | 0.9 | - | 782 |

Rosseta T2 (2) | AL,C,C | 5976 | 6% | 2.96 | 0.12 | 2018 |

Rosseta T2 (3) | AL,G | 995 | 16% | 0.92 | 0.36 | 1079 |

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## Share and Cite

**MDPI and ACS Style**

Mittelstedt, M.; Hansen, C.; Mertiny, P.
Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors. *Appl. Sci.* **2018**, *8*, 1256.
https://doi.org/10.3390/app8081256

**AMA Style**

Mittelstedt M, Hansen C, Mertiny P.
Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors. *Applied Sciences*. 2018; 8(8):1256.
https://doi.org/10.3390/app8081256

**Chicago/Turabian Style**

Mittelstedt, Marvin, Christian Hansen, and Pierre Mertiny.
2018. "Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors" *Applied Sciences* 8, no. 8: 1256.
https://doi.org/10.3390/app8081256