Study of Geometries and Stability for Energy Density in Electromechanical Battery Flywheels with a Gaussian Shape

A study on flywheels materials and geometries is presented here with the use of finite element modeling (FEM) simulations. The study analyzes the stress behavior of flywheel rotors subject to the rotational speed that creates the maximum stress that the flywheel can support, and then calculates the rotational energy mass ratio for each geometry. The obtained geometry was Gaussian section for the Flywheel rotor. The material used in this case study is carbon fiber Hexcel UHM 12000. The best results were a rotational speed of ≈ 279.000 rpm and rotational energy density of ≈ 440 Wh/kg. Then, to increase the total energy, Flywheel with 2 and 3 rotors were analyzed. Then the stability under rotation of these Flywheels were analyzed, some instability was found and a solution was presented.


Introduction
Flywheels is a rotational device that stores rotational energy, when this energy can be converted into electrical energy, it is called an electromechanical battery [1][2][3], in the simplest way it is referred to as the first generation.The 2nd generation incorporates an AC generator, an inverter and a rectifier.If it reaches the power grid and the flow of energy can be reversed, it becomes a Flywheel Energy Storage System (FESS) [2,3].
The 3rd generation [3][4][5][6] includes more characteristics (magnetic bearings, vacuum, etc).New and expensive materials can improve the range of energy storage, but as they are expensive it is desirable to increase the energy density of its geometry.
To reach such a goal, many simulations in FEM (finite elements method) of a variety of geometries.were made that calculates the stress of the Flywheel using the von Mises criterion, then simulations of the normal vibration mode were made to verify the rotational stability of the system.The rotational energy density in a flywheel is estimated by the following formula [8,9]: (1) where E is the rotational energy stored, m is the rotating mass of the Flywheel, max is the material ultimate tensile stress and is its specific mass.Using equation 1, table 1 exemplifies energy densities for different materials [8,10].The material used in the simulations appears in the last line of table 1.Studies from Coppede [11] several simulations were performed using a Finite Element Analysis of flywheels made of two materials: Maraging steel (AISI 18 Ni 350) and carbon fiber (Hexcel UHM 12000), with carbon fiber as a reinforcement.The main result obtained indicates the best option is to make the flywheel only with the carbon fiber as can be seen in figure 1.

2.
Research Methodology The authors have experience with instrumentation [12][13][14],that experience helped this work.A variety of flywheels were simulated by finite element modeling using SOLIDWORKS [15] using geometries studied by Coppede and Nogueira [11,16].The used FEM simulation characteristics are shown in table 2. The characteristics of carbon fiber Hexcel UHM 12000 appear in reference [17].

Table 2. Finite Elements characteristics
Type-of-Analysis static analysis

Type-of-Meshing solid
Thermal-effect activated

Load-type centrifugal
Connector-Support-type bearing

MODELS ANALYZED BY FINITE ELEMENT MODELING
The simulations presented below are characterized as solid cylindrical bodies.All geometries have in general a cylindrical geometry with 200 mm of external diameter and about 100 mm of height.The results of the simulations are shown in table 3.As the Q model presented a better behavior and there was an indication that to increase the energy density the model needed to present a more uniform distribution in tensile stress then a model with 180 mm of external diameter and 130 mm of height with a shape defined by the revolution of a Gaussian shape.The Gaussian parameter was the one defined by Stodola [36], which is the formulation for a disk with constant stress, where the exponential squared parameter is the ratio of the squared rotational speed times specific mass divided by 2 times the ultimate tensile stress.This model is denominated R where the central shaft with 20 mm diameter extends a further 20 mm on each side of the Flywheel, see figure 3.In table 3 the results of all simulations can be seen.The new model R (figure 3) shows for highest rotation and energy density the best results, as the total energy and mass depends on the size of the Flywheel .

Edges for Increasing Energy Density
It is a fact now that, the more uniform the distribution of the tensile stress and the higher its value, the more energy can be stored in the Flywheel.As in the edges of the Flywheel rotor there is a region where the tensile stress is small, that is not surprising as there is no mass to force the inner mass outward and create stress, remembering that the expression for a constant stress comes from an approximation.Then to increase the stress on the edge, a small mass was inserted on the edges, as shown in figure 4. Various mass shapes were inserted on edges.Table 4 shows the energy densities for these trials.The gaussian shape rotor is called CURVE01.
The mass inclusion increases the energy density but for a tiny amount.The energy increases for a big factor but the reason is that there is more mass, and some of this mass is not well used.Also there is a decrease in the maximum rotation.

Double and Triple Flywheel Rotors
As the mass of these rotors are small the total energy stored in these Flywheels is also small.To increase the total stored energy of each Flywheel, each of them can have two or three rotors with a gaussian shape.In figure 5 there is an example Flywheel with a double gaussian rotor and in figure 6 there is an example of a Flywheel with a triple gaussian rotor.In table 5 there is an example of total energy, total mass, maximum rotation and energy density for a triple gaussian rotor.

Vibration Normal Modes
One of the main concerns when working at very high rotational speeds is the stability of the rotor.As the rotor rotates, small variations in its shape create torques that could excite the rotor normal modes of vibration that, in the resonant condition, could destroy the rotor, and the Flywheel .To avoid such a condition, the vibrational normal modes of the Flywheels are simulated to have their frequencies found.An example is shown in figure 7, its first normal mode has a frequency of 0.086 Hz, this mode is characterized by the added masses added on the edges of the gaussian rotor vibrating up and down in the system axial direction.This characteristic is a very bad one, since all the harmonics of this frequency could resonate with the rotation torque.These masses do not help much, increasing the energy density, then the best option seems to be to redraw them from the Flywheel.Then all Flywheel rotors now are formed only by the gaussian shape ring.A double rotor vibration mode can be seen in figure 8 with only the gaussian disks and shown no signal of vibration as this model presents a frequency of zero for this mode with no amplitude of vibration.The normal modes then are found in the simulations for the double and double Flywheel rotors.The respective frequencies and their harmonics could be avoided by the Flywheel control system of the electromechanical battery, this job is reasonable as there are few of these modes in the Flywheel operational range, figure 9 and figure 10 shows these results.For the double rotor the first normal mode appears at a frequency of 1151 Hz, for a single rotor this frequency appears at 3341 Hz, these are the normal modes that appear in the operational range up to 4000 Hz in both cases.In order to avoid the instability of the double rotor Flywheel a reinforcement was added connecting the both disks as shown in figure 11.The normal mode first frequency calculated is of 2500 Hz which can be changed to a higher value making the bearing axes shorter.It was not done in the simulation as this length is necessary for the simulation to run.There is also the possibility of improvement making the reinforcement wider.Figure 12 shows the simulation of this double rotor reinforced and presents an energy density of 403 Wh/kg or 1.45 MJ/kg, with some room for improvements as the point of maximum von Mises stress stress is located on the surface of the gaussian curve of the shaft.

Summary and Conclusions
This work shows a case study of many flywheel rotors geometries by the use of FEM simulations.All of the Flywheel rotors are made of carbon fiber Hexcel UHM 12000.There have been compared 18 different geometries, comparing: maximum rotational speed, total stored energy, total mass and specific stored energy.An analysis of the von Mises stress was made in all models, showing the critical stress points.The model R with the Gaussian shape rotor presents the highest value for the specific energy, with a value of 387 Wh/kg and the biggest rotational speed of 279180 rpm.
As for the triple rotor it reached an energy density of 431 Wh/kg, this value rotor is about 56% higher than the one found by eq. ( 1) for the same Hexcel UHM 12000 carbon fiber ( 277 Wh/kg).
Possible instability by the normal rotational modes were found on the operational frequency; one mode for each kind of Flywheel rotor, these frequencies must be avoided at all cost in the Flywheel operation, it can be easily implemented by the Flywheel control system that can accelerate or decelerate the device as passing by these frequencies redirecting energy from one device to another as these devices operates in pairs.
The double and triple rotors can have their instability problem improved by the use of a reinforcement tube connecting the two gaussian disks that does not degrade much the specific rotational energy, as for an example the reinforced double rotor has an energy density of 403 Wh/kg with room for improvement.
The results in this article are a reference for future works in the next steps for the manufacturing of the actual flywheel.It can also be used in the future as reference for the manufacturing Flywheels made of fused silica or carbon nanotubes, which will be very expensive materials.

Figure 1 .
Figure 1.Several FEM simulations were performed using flywheels made of Maraging steel, carbon fiber and Maraging steel revested with carbon fiber.The best result was achieved with pure carbon fiber.Source: The Authors.

Figure 2 .
Figure 2. Models are labeled from A to Q.The bearings are located in the very top and in the very button of the Flywheels.Source: The authors.

Figure 3 .
Figure 3. Von Mises stress for the R Model.Source: the authors.

Figure 4 .
Figure 4. Masses added on Flywheel edges.Source: the authors.

Figure 7 .
Figure 7. First vibration mode of a double rotor.Source: the authors.

Figure 8 .
Figure 8. Double rotor with the only gaussian disks frequency zero of vibration with no vibration amplitudes.Source: the authors.

Figure 9 .
Figure 9. Single rotor first and only normal mode in the operational range up to 4000 Hz at a frequency of 3453 Hz.Source: the authors.

Figure 10 .
Figure 10.Double rotor first and only normal mode in the operational range up to 4000 Hz at a frequency of 1151 Hz, must be considered the harmonics.Source: the authors.

Figure 11 .Figure 12 .
Figure 11.First and only normal mode up to operational frequency of the reinforced Flywheel double rotor at a frequency of 2500 Hz.Source: the authors

Table 3 .
Results of Flywheels simulations.The best results are bolded.

Table 4 .
Simulated models of the Flywheel rotors.The best values are bolded.

Table 5 .
Maximum rotation, total mass, total energy and energy density for a triple rotor.