1. Introduction
1.1. Flywheel State-of-the-Art and Article Contributions
Flywheels are well established mechanical components that are able to store kinetic energy (in the form of rotational energy) as well as angular momentum. We define a standard flywheel as a rigid body which rotates around an axis that coincides with its centre of mass (COM) and parallel to one if its principal axes of inertia. Flywheels have many applications, such as smoothing power output, serving as an additional source of energy and generating reaction torques. The smoothing of power output is commonly found in reciprocating engines. Flywheels may serve as an additional auxiliary power source in the form of intermittent pulses (for a forging hammer for example) or as a single impulse (for powering up a tokamak or aircraft catapult for example). The generation of reaction torques comes from the flywheel’s ability to store angular momentum and finds its use in attitude control systems, such as the reaction wheels and control moment gyros of spacecrafts.
A standard flywheel is fixed to its base by means of a rotating bearing, such as ball bearings, gas bearings or magnetic bearings, which permits multiple complete revolutions. This last property is what prevents flexures from being used as the flywheel’s bearing. Indeed, flexures work by using elastic deformation of flexible members in the mechanism. They also have a limited stroke and therefore cannot directly replace the flywheel bearing as the latter must be able to perform multiple complete revolutions. The challenge in using flexures is designing a mechanism that can act like a rotating flywheel with limited stroke in each joint.
This article proposes an alternative design of a flywheel that is compatible with the use of flexures, in the sense that this mechanism stores kinetic energy and generates angular momentum. In addition, a comparative study of performances is drawn between a standard flywheel and the proposed flexure design.
1.2. Requirements for a Flexure-Based Flywheel Equivalent
To establish criteria for a flexure-based flywheel equivalent, the key properties of a standard flywheel need to be highlighted. First, the flywheel’s centre of mass remains fixed during its motion and therefore does export any forces during motion. This property is defined as being
force balanced by Schneegans et al. [
1]. The flywheel is also in equilibrium when at rest (i.e., is not subject to any forces), regardless of its orientation, and therefore has a constant potential energy, a property defined as
statically balanced by [
1]. During its motion, the flywheel generates angular momentum that it constant in amplitude and in direction. Last, if the standard flywheel has a cylindrical shape, as in most cases, the inertia tensor of the flywheel, as seen from its base, does not change. If this is not the case and if the flywheel’s base is also rotating, the flywheel will export torque. This last property, defined as
inertial invariance by [
1], is however regarded as optional as its effect will only be apparent for a flywheel whose base is rotating.
To allow the selection of an appropriate design, the properties described above are now enumerated:
The centre of mass of the reaction mechanism must remain fixed throughout its motion.
The mechanism must be able to generate angular momentum that is constant in direction.
The mechanism must be able to generate angular momentum that is constant in amplitude.
The mechanism must be able to control the amplitude of its angular momentum in order to deliver a reaction torque
The inertia tensor of the flywheel, as seen from its base, must remain constant throughout the motion. (Optional)
1.3. Outline of Paper
Section 2.1 defines the inertial bodies considered in this paper along with the definition of CV-joints and their notational conventions. The flexure-based flywheel and its working principle will then be presented with ideal joints in
Section 2.2. Even though this flywheel uses ideal joints (and therefore not necessarily flexures), it will still be referred to as a flexure-based flywheel to avoid additional terminology. In
Section 2.3.2, the kinetic energy and angular momentum of this mechanism will be discussed, followed by the internal forces that transit through the joints (
Section 2.4). In
Section 3, the comparative study between a standard flywheel and the flexure-based design will then be shown and their performances in terms of kinetic energy, angular momentum and volume will be discussed.
Section 4 shows the influence of the shape ratio of pivoting cylindrical rigid bodies on the generation of angular momentum and storage of kinetic energy. Thereafter, we highlight in
Section 5 the analogy between the proposed mechanism and a falling cat. Last, in
Section 6, an example of a flexure implementation along with the constructed proof-of-concept prototype and experimental measurements will be shown.
3. Comparison of a dumbbell flexure design to a standard flywheel
This section compares the angular momentum and kinetic energy of the flexure-based flywheel to those of a standard flywheel. The standard flywheel selected as a benchmark for this comparison is a hollow cylinder, whose thickness to radius ratio
is fixed to 0.1 (see
Figure 5):
To allow a comparison to be drawn, both the flexure-based and standard designs have the same total mass
M and the dumbbell’s length
matches the flywheel’s diameter
D, as shown in
Figure 5.
The moment of inertia of the standard flywheel is given by:
Moreover, in order to reach the correct mass
M, this standard flywheel has a height
H given by:
resulting in an envelope volume
given by:
For the compared flexure design, the sphere-shaped masses
m and their radii
r are given by:
with
the material’s density. This yields:
Moreover, the cylindrical envelope of this dumbbell design, highlighted in
Figure 5, has a volume
V given by:
The figures of merit of this comparison are
,
and
, defined as:
These figures of merit describe the change in angular momentum, kinetic energy and volume between a standard flywheel and the flexure-based flywheel. These are given by:
and
The three figures of merit
,
and
are plotted in
Figure 6,
Figure 7 and
Figure 8, for masses
kg out of steel (density
kg/m
3). (Note: the range of
D in these plots was selected to avoid interference of sphere shaped masses within dumbbell, as this would not be realistic.)
Eqs. (
33) and (
34) can be further simplified in the case of small masses
m (or elongated dumbbells with large
D). In this case,
, simplifying
and
to:
It can indeed be seen in
Figure 6 and
Figure 7 that, with the exception of the smaller diameters, the diameter
D has little influence on
and
, as they depend mainly on
. The conclusions to draw from the first two graphs are that even though increasing the size of the flexure-base design increases the kinetic energy and angular momentum storage capacity, these keep the same proportion with respect to a standard flywheel. The only way to increase the storage capacity of this dumbbell design is to increase its tilt amplitude
. All curves remain below one, meaning angular momentum and kinetic energy are always lower than for the equivalent flywheel. For the typical range of
for deflection of flexures (as this results in a deflection of 40
o at the ball-joint), the kinetic energy and angular momentum lose almost an order of magnitude. As done for
and
, the same simplification can be applied to
:
and it can be seen that
increases with
, also as seen in
Figure 8. For the typical range of
of deflection on flexures and for a diameter
D of 250 mm, it can be seen that the new design increases its volume by about an order of magnitude.
4. Influence of the Cylindrical Rigid Body Shape
Sect. 3 studied the influence of the flexure-based flywheel’s size and tilt amplitude on its angular momentum and kinetic energy, allowing a comparison to be drawn with a standard flywheel. This section explores instead the effect of the shape of the cylindrical rigid bodies. The dumbbell design, though more realistic, will be left aside and pivoting cylinders (as shown in
Figure 9) are considered instead (As before, these cylinders are linked to the ground by CV-joints located at their centres of mass.). These cylinders have a radius
R, a thickness
t and a fixed mass
m (Even though
m has already been used to denote the sphere-shaped masses in Sect. 3, it will again be used here to avoid complex nomenclature.). Their shape will be parameterised by their
shape ratio :
The thickness
t and radius
R are given by:
The resulting
and
are then (These are the well-known formulas for the moments of inertia of a cylinder around its principal axes.):
The resulting
and
are given by:
The evolution of
and
with respect to
is shown in Figs.
10 and
11 for a selection of elevation angles
and with
kg out of steel (
=8000 kg m
−3):
Figure 10 shows that the most angular momentum is generated with small values of
, which corresponds to elongated slender cylinders. Large values of
, which correspond to thin flat disks, result in little angular momentum, with a minimum for intermediate values. Intuitively explained, long slender cylinders and flat disks have the mass distributed from the centre of rotation, resulting in a larger angular momenta. For cylindrical inertial bodies that are neither slender nor flat, their mass is close to the centre, resulting in a smaller angular momenta. Last, it can be seen that increasing elevation angle
increases the angular momentum.
Figure 11 shows that elongated slender cylinders store some amount of kinetic energy, but that flat disks store much more. There is also the same minimum for the same reason as for
, in which cylindrical rigid bodies which are neither slender nor flat have their mass distribution closer to their centre of rotation. Similarly to
, increasing
also increases the stored kinetic energy.
The most striking result of
Figure 12 is that the equivalent moments of inertia
and
are different. Indeed, for a standard flywheel, the same graph as
Figure 12 would be a flat line
. The fact that the ratio is not one means that the flexure-based flywheel can be seen as having two separate moments of inertia: one for storage of angular momentum and the other for storage of kinetic energy. Elongated cylinders have a ratio of approx. one, while thin flat disks have a ratio of approx. 100. This means that elongated cylinders have matching
and
and therefore behave identically to a standard flywheel. This joins the result of Eq. (
36) because elongated dumbbells fall in the same category as elongated slender cylinders. Thin flat disks on the other hand, for the same amount of stored kinetic energy, generate approx. 100 times less angular momentum than a conventional flywheel would. This result leads to an interesting design consideration: if the goal of the flywheel is to generate angular momentum, then elongated cylinders are more better-suited, and if the goal is to store kinetic energy, then thin flat disks are optimal. This last property is interesting as it leads to reduced gyroscopic forces if the flywheel’s axis is changed. Concerning the elevation angle
, the ratio is larger for smaller elevation angles. This does not mean however that smaller elevation angles store more kinetic energy, but that they generate practically no angular momentum.
5. Analogy with a falling cat
It so happens that this flywheel design follows the mechanics of a falling cat which re-orients itself during its fall in order to land on its feet. Among the different models of a falling cat proposed by the literature [
4,
5,
6,
7], Rademaker [
5] considers a falling cat as two rolling cylinders which keep a constant relative bending angle (
Figure 13). Each cylinder has its own oblique and constant angular momentum (
and
), resulting in a non-zero total angular momentum in the same manner as the proposed mechanism. Indeed, the two cylinders roll against each other without slipping, which implicitly assumes the presence of a CV-joint linking them.
A difference between Rademaker’s model and the analysis presented in this work is the choice of the frame in which the angular momentum is evaluated. In Rademaker’s case, this frame keeps the bend between the two cylinders in the same orientation. If the same falling cat is similarly described in a frame fixed with respect to the cat’s waist, the two cylinders are no longer rolling around their respective axes, but perform the same pivoting motion of the proposed mechanism.
The falling cat phenomenon has already found several applications in robotics attitude control. However, examples of the literature [
8,
9,
10] require motorised joints to provide the pivoting motion whereas this paper relies on passive mechanical means to guarantee the pivoting motion as well as the fixed elevation angle. Moreover, the presented mechanism follows truthfully the rolling cylinder model in contrast to other works in the literature [
8,
9], which use a Hooke joint instead of a CV-joint.
7. Discussion and Conclusion
This article introduces a flexure-based mechanism that successfully meets the essential criteria for a flywheel. Indeed, it is able to store kinetic energy and a constant angular momentum at a constant pivoting rate, it has a fixed centre of mass and has zero stiffness. A flexure-based implementation of the mechanism was shown, and the proof-of-concept prototype was successfully built and characterized in terms of reaction moment generation, which validates the analytical model.
In comparison to standard flywheels that use other types of bearings (such as ball bearings, magnetic bearings, and air bearings), the advantages of this flexure design are the absence of wear and lubrication, its low power consumption, its vacuum compatibility, and the possibility of miniaturisation. The price to pay for these advantages is a six-fold reduction in kinetic energy and angular momentum storage capacity for a given mass, as well as a ten-fold increase in volume.
A particularity of this flexure-based flywheel is that the shape of the pivoting rigid bodies influences the ratio of kinetic energy versus angular momentum stored in the mechanism. Rigid bodies with an elongated cylindrical shape have the same ratio as classical flywheels, whereas those that have the shape of a flat disk store up to 100 times more kinetic energy for the same angular momentum as classical flywheels. This could be advantageous for kinetic energy storage applications where angular momentum is undesired, for example, due to the induced gyroscopic forces.
Hence, although comparatively heavy and cumbersome, the novel flexure-based flywheel offers interesting design features that might be key to specific extreme-environment applications.
Figure 1.
The inertial bodies considered in this paper: spherical inertial body (left), cylindrical inertial body (centre) and dumbbell (right). This dumbbell has its mass concentrated exclusively in the two sphere-shaped masses (the rigid arms linking them are neglected) and the dumbbell effectively acts as a cylindrical inertial body. The centre of mass (COM) of each body is highlighted and it is assumed that principal axes of inertia coincide respectively with the x-, y- and z-axes. It can be seen that the cylinder and dumbbell have the same inertia tensor.
Figure 1.
The inertial bodies considered in this paper: spherical inertial body (left), cylindrical inertial body (centre) and dumbbell (right). This dumbbell has its mass concentrated exclusively in the two sphere-shaped masses (the rigid arms linking them are neglected) and the dumbbell effectively acts as a cylindrical inertial body. The centre of mass (COM) of each body is highlighted and it is assumed that principal axes of inertia coincide respectively with the x-, y- and z-axes. It can be seen that the cylinder and dumbbell have the same inertia tensor.
Figure 2.
Sequential rotations with Euler angles , and , designated respectively as azimuth, elevation and twist. The rotation sequence is first rotation around axis , then rotation around and finally rotation around
Figure 2.
Sequential rotations with Euler angles , and , designated respectively as azimuth, elevation and twist. The rotation sequence is first rotation around axis , then rotation around and finally rotation around
Figure 4.
Rigid body 1 isolated from the rest of the mechanism. The internal forces and are applied respectively to its ball-joint and CV-joint. The net torque , equal to , the time derivative of , is also highlighted.
Figure 4.
Rigid body 1 isolated from the rest of the mechanism. The internal forces and are applied respectively to its ball-joint and CV-joint. The net torque , equal to , the time derivative of , is also highlighted.
Figure 5.
Standard flywheel and flexure-based flywheel used for the performance comparison. The standard flywheel’s diameter D matches the dumbbell’s length (i.e., the distance between the centres of the sphere-shaped masses). The cylindrical envelope of the flexure-based flywheel is also highlighted and encompasses the whole mechanism, including the dumbbells’ spheres.
Figure 5.
Standard flywheel and flexure-based flywheel used for the performance comparison. The standard flywheel’s diameter D matches the dumbbell’s length (i.e., the distance between the centres of the sphere-shaped masses). The cylindrical envelope of the flexure-based flywheel is also highlighted and encompasses the whole mechanism, including the dumbbells’ spheres.
Figure 6.
Angular momentum ratio as a function of flywheel diameter D. The flywheel and dumbbell illustrations show the overall appearance for the different diameters D (not to scale).
Figure 6.
Angular momentum ratio as a function of flywheel diameter D. The flywheel and dumbbell illustrations show the overall appearance for the different diameters D (not to scale).
Figure 7.
Kinetic energy ratio as a function of flywheel diameter D. The flywheel and dumbbell illustrations show the overall appearance for the different diameters D (not to scale).
Figure 7.
Kinetic energy ratio as a function of flywheel diameter D. The flywheel and dumbbell illustrations show the overall appearance for the different diameters D (not to scale).
Figure 8.
Volume ratio as a function of flywheel diameter D. The flywheel and dumbbell illustrations show the overall appearance for the different diameters D (not to scale).
Figure 8.
Volume ratio as a function of flywheel diameter D. The flywheel and dumbbell illustrations show the overall appearance for the different diameters D (not to scale).
Figure 9.
The studied pivoting cylindrical inertial bodies. As before, the cylindrical rigid bodies are fixed to the base via CV-joints and linked together via a ball joint.
Figure 9.
The studied pivoting cylindrical inertial bodies. As before, the cylindrical rigid bodies are fixed to the base via CV-joints and linked together via a ball joint.
Figure 10.
as a function of shape factor . The two illustrations of the cylindrical inertial bodies show the overall appearance of the system for the different values of (not to scale).
Figure 10.
as a function of shape factor . The two illustrations of the cylindrical inertial bodies show the overall appearance of the system for the different values of (not to scale).
Figure 11.
as a function of shape factor . The two illustrations of the cylindrical rigid bodies show the overall appearance of the system for the different values of (not to scale).
Figure 11.
as a function of shape factor . The two illustrations of the cylindrical rigid bodies show the overall appearance of the system for the different values of (not to scale).
Figure 12.
as a function of shape factor . The two illustrations of the cylindrical rigid bodies show the overall appearance of the system for the different values of (not to scale).
Figure 12.
as a function of shape factor . The two illustrations of the cylindrical rigid bodies show the overall appearance of the system for the different values of (not to scale).
Figure 13.
A falling cat seen as two rolling cylinders. In a frame where the cylinders rotate around their axes, they have angular momenta
and
. When in a frame fixed with respect to the cat’s waist, the cylinders pivot and have angular momenta
and
which lead to an overall angular momentum
. (Image adapted from [
11])
Figure 13.
A falling cat seen as two rolling cylinders. In a frame where the cylinders rotate around their axes, they have angular momenta
and
. When in a frame fixed with respect to the cat’s waist, the cylinders pivot and have angular momenta
and
which lead to an overall angular momentum
. (Image adapted from [
11])
Figure 14.
CAD views of the prototype’s implemented flexure-based (a) CV-joint, which enables the upper part to pivot around the x- and y-axes, and (b) ball joint, which connects the two inertial pivoting bodies. Note: the two rods parallel to the -plane lie in two distinct planes separated by a distance d.
Figure 14.
CAD views of the prototype’s implemented flexure-based (a) CV-joint, which enables the upper part to pivot around the x- and y-axes, and (b) ball joint, which connects the two inertial pivoting bodies. Note: the two rods parallel to the -plane lie in two distinct planes separated by a distance d.
Figure 15.
Constructed demonstrator with the different components and joints highlighted.
Figure 15.
Constructed demonstrator with the different components and joints highlighted.
Figure 16.
Measurement setup with the different components highlighted.
Figure 16.
Measurement setup with the different components highlighted.
Figure 17.
Angular velocities of the rotor and the frame with respect to the time. The rotor acceleration, constant speed and braking phases are identified using green, blue and red colors respectively. The changes of angular velocity and are highlighted for the first acceleration phase.
Figure 17.
Angular velocities of the rotor and the frame with respect to the time. The rotor acceleration, constant speed and braking phases are identified using green, blue and red colors respectively. The changes of angular velocity and are highlighted for the first acceleration phase.