# Energy Storage Flywheel Rotors—Mechanical Design

^{*}

## Definition

**:**

## 1. Introduction

## 2. Applications and Performance

## 3. Manufacture

#### 3.1. Hub Construction

#### 3.2. Rim Construction

#### 3.3. Assembly

## 4. Analytical Modeling

#### 4.1. Energy Storage and Power Capacity

_{o}, r

_{i}, ρ, and h. As the hub cross section increases in complexity it is common to define the energy density (ratio of energy to mass) [13,27,43] of the hub as:

#### 4.2. Material Characterization

^{−1}. Then, the time-dependent compliance matrix for an orthotropic linearly elastic material is as follows:

_{ij}terms with respect to moduli of elasticity, E, and Poison’s ratios, ν:

#### 4.2.1. Hygroscopic Effects

#### 4.2.2. Temperature Effects

_{g}, and non-linear viscoelasticity above. Elevated temperatures facilitate polymer chain mobility, causing a decrease in both moduli and strength [60]. For the TTSP, a trade-off is seen where increasing temperature increases the rate of viscoelastic response, and decreasing temperature decreases this response. By conducting short-term experiments at elevated temperatures, it is possible to predict the long-term behavior of the material at low temperatures. The basic procedure for the TTSP is discussed in [64]. First, material specimens are subjected to constant load at various temperatures in conventional creep testing. These data generate a series of compliance curves when plotted over time in logarithmic scale (log(time)). Second, an arbitrary reference temperature is selected. Third, all compliance curves are shifted along the time axis onto the reference temperature compliance curve to construct a master curve. As a demonstration, consider the data series of tensile experiments in Figure 4. Short-term tensile experiments were conducted on an FRP composite material at various temperatures to collect the viscoelastic data [65]. Data for all temperatures but the reference temperature were shifted along the time axis to construct the master curve at a reference temperature, T

_{r}, of 40 °C.

_{T}. There are several ways to determine the shift factor for each curve, all of which are designed to create a smooth master curve. Brinson [67] studied the time temperature response of Hysol 4290, a common contemporary two-part epoxy. Brinson conducted tensile tests on samples of the material at temperatures between 90 °C and 130 °C and, thus, constructed a master curve covering creep at 90 °C over approximately 6 months. The shift factor was determined using the William-Landel-Ferry (WLF) equation [68], which requires a knowledge of T

_{g}and a set of experimentally determined material constants. While WLF can create a smooth master curve, it is limited to temperatures above T

_{g}, so it may not be suitable for all applications. Another common method is using an Arrhenius’ equation [69,70], which requires knowledge of the activation energy and gas constant. The activation energy is typically determined using dynamic mechanical analysis [71].

_{ij}is the compliance and T is the experimental temperature. Tensile experiments must be conducted to determine [S] for each independent modulus in Equation (10), i.e., E

_{1}, E

_{2}, E

_{3}, etc., and will vary depending on whether the material is isotropic, transversely isotropic, orthotropic, or fully anisotropic.

#### 4.2.3. Aging Effects

_{te}. Then, S

_{ij}becomes the following:

_{e}is the age for which the master curve is created. Under isothermal conditions, the aging shift factor can be calculated as a ratio between a reference aging time and an experimental aging time raised to an experimentally determined thermal shift rate [73]. While it is possible to experimentally determine and account for material aging when modeling flywheel rotors, it is more practical to thoroughly stabilize the flywheel rotor by aging at an elevated temperature under no load conditions until the rotor reaches equilibrium before operation. This stiffens the material, minimizes creep, and provides a more repeatable starting point for designing flywheel rotors. Sullivan [74] showed equilibrium can be achieved by aging epoxy polymers at 115 °C for 1000 h. It is recommended that flywheel rotors be aged to minimize material evolution during operation, which will improve rotor response to applied loads and increase confidence in any simulation or modeling conducted during the design of the rotor.

#### 4.2.4. Stress Magnitude

_{g}[75] and stress to below 50% of the failure strength [76].

#### 4.3. Quasi-Static Analysis

_{r}is the radial displacement and the subscript z signifies the rotor axial direction. Then, Equation (14) can be substituted into Hooke’s law which is further substituted into Equation (13). This yields a second order inhomogeneous ordinary differential equation, which can be solved for the radial displacement and radial stress, yielding the following:

_{1}and C

_{2}are integration constants, detailed in [80], which must be determined by the boundary conditions, see [81].

#### 4.4. Viscoelastic Analysis

^{10}years). Similar to previous work, Tzeng showed that radial stress could decrease by as much as 35%, while circumferential stress could increase by up to 9%. Tzeng also studied flywheels with variable winding angles and found similar though slightly improved results.

^{10}years or the time required to reach full relaxation, viscoelastic behavior of the composite can significantly impact rotor structural health by facilitating either creep rupture, the loss of rotor integrity by the loss of interfacial pressures between hub and rims, or both. However, the expected lifetime for flywheel rotors, as discussed, is between 10 and 20 years [5]. Furthermore, many of these studies occurred on either thick composite disks or arbitrarily long flywheel rotors. Skinner and Mertiny addressed this issue in [16], where a carbon FRP composite flywheel rotor was simulated for up to 10 years. The analytical process they followed to simulate the rotor behavior is similar to that pursued by previous researchers, so it is worth taking a brief aside to discuss this work here.

#### 4.5. Shear Stress

_{rθ}is the in-plane shear stress and α is angular acceleration. Shear strain is defined as:

_{1}and C

_{2}are integration constants. Notice that tangential stress is dependent on a single integration constant because when the strain, Equation (17), is substituted into tangential displacement, the second integration constant, C

_{2}, is eliminated. The integration constants can be found through the boundary conditions as functions of the rotor geometry, density, shear modulus, and angular acceleration. Pérez Aparicio and Ripoll considered a worst-case scenario where peak shear stress is caused by a severe acceleration of 3.6 × 10

^{5}rad/s

^{2}. For this considered worst-case scenario, resulting stress states were described as possibly critical for the hub rather than the rotor.

^{3}, respectively. For an angular velocity of 17,425 rpm (1827.6 s

^{−1}), a supplied power of 1.67 GW is associated with an angular acceleration of 3.6 × 10

^{5}s

^{−2}for 0.005 s. Pérez Aparicio and Ripoll explained that power supplied at this magnitude would occur in specific applications, such as military artillery; however, it is atypical for energy storage systems.

## 5. Failure Analysis

#### 5.1. Failure Criteria

#### 5.2. Maximum Stress Criterion

_{1t}or σ

_{1c}, respectively. In the transverse directions, the material is assumed to be transversely isotropic such that the 2 and 3 directions are congruent; thus, σ

_{2t}= σ

_{3t}and σ

_{2c}= σ

_{3c}. Shear stress is dominated by matrix deformation τ

_{12}and τ

_{23}. With the applied stress tensor as [σ

_{θ}, σ

_{z}, σ

_{r}, τ

_{rθ}], the maximum stress failure criterion is defined as:

#### 5.3. Tsai-Wu Criterion

_{ij}are material coefficients dependent on the tensile and compressive strengths in each direction. A complete list of coefficients is available in [102]. The Tsai-Wu failure criterion can be modified to find the strength ratio (SR), which is the ratio between the applied stress and the failure stress [16,80,100]. Failure is predicted when SR is greater than or equal to unity. This approach provides an intuitive and easily represented term, which facilitates the comparison of combined stresses across the entire flywheel rotor.

#### 5.4. Progressive Failure Analysis

## 6. Conclusions and Prospects

## Author Contributions

## Funding

## Conflicts of Interest

## Entry Link on the Encyclopedia Platform

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**Figure 1.**Cutaway schematic of a flywheel energy storage system for experimental research. Inset shows the actual device [16].

**Figure 2.**Composite flywheel rotor rim at the end of filament winding manufacturing process; (

**a**) fiber payout eye and deposition head on winding machine carriage arm, (

**b**) winding mandrel, and (

**c**) completed aramid fiber/epoxy composite rim.

**Figure 3.**Thermal press-fit accomplished by cooling the aluminum hub with liquid nitrogen before pressing into the composite rims.

**Figure 4.**Time-temperature superposition experimental data, reproduced from [65]. Data were collected from tensile tests for an FRP composite at various temperatures and shifted along the time axis to create a master curve for a reference temperature of 40 °C.

**Figure 5.**Evolution of (

**a**) radial and (

**b**) circumferential stresses at different times of operation (0–10 years) of a flywheel rotor with an aluminum hub and carbon FRP composite rim due to viscoelastic stress relaxation [16].

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

Lifetime [years] | >20 [5] |

Charge/discharge cycles | <10^{7} [5] |

Energy density [Wh/kg] | <130 [17] |

Price [(USD)/kWh] | 400–6960 [5,17] |

Power density [W/kg] | ~1000 [5] |

Shape | Cross Section | k-Value |
---|---|---|

Laval disk | 1.00 | |

Laval disk real | 0.70–0.90 | |

Conical disk | 0.70–0.85 | |

Solid disk | 0.606 | |

Thin ring | 0.50 | |

Thick rim | 0.303 |

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**MDPI and ACS Style**

Skinner, M.; Mertiny, P.
Energy Storage Flywheel Rotors—Mechanical Design. *Encyclopedia* **2022**, *2*, 301-324.
https://doi.org/10.3390/encyclopedia2010019

**AMA Style**

Skinner M, Mertiny P.
Energy Storage Flywheel Rotors—Mechanical Design. *Encyclopedia*. 2022; 2(1):301-324.
https://doi.org/10.3390/encyclopedia2010019

**Chicago/Turabian Style**

Skinner, Miles, and Pierre Mertiny.
2022. "Energy Storage Flywheel Rotors—Mechanical Design" *Encyclopedia* 2, no. 1: 301-324.
https://doi.org/10.3390/encyclopedia2010019