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Sensors and Materials, Vol. 32, No. 7 (2020) 2551–2567
MYU Tokyo
S & M 2282

Analytical and Experimental Studyon Gyroscopic Power Generator with Power Feedback
Hiroshi Hosaka* and Yuki Tajima
Department of Human and Engineered Environment, Graduate School of Frontier Sciences,
the University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa-city, Chiba 277-8563, Japan(Received February 28, 2020; accepted June 30, 2020)

Keywords: energy harvesting, gyroscope, power generator, vibration, positive feedback

A gyroscopic power generator that self-accelerates its spinning velocity via positive
power feedback is developed. In our previous study, a power of 1.8 W was generated for agyro-generator, but an external power source was necessary to drive the spinning motor. In
this study, the power generated by the precession movement of a flywheel (FW) is applied tothe spinning motor. This accelerates the spinning velocity by boosting the feedback voltage.
A mathematical model of the generator is presented. Then, the relationships among spin
acceleration, generated power, and boost ratio are studied using approximate solutions andnumerical analyses. Finally, the validity of the theoretical results is verified by experiment.

1.

Introduction

Internet of Things devices are utilized in many fields. Energy supply is the largest problem
in portable applications. To solve this problem, energy harvesting techniques have been studied.(1–3)
Vibration harvesters have been extensively studied because vibration is ubiquitous and contains
much energy. Conventional generators use the inertial force of a simple pendulum. The inertial
force, and thus the power, of low-frequency vibration is very small. The power output of awristwatch generator,(4) for example, is 10 μW. In other wearable generators, the power output
is 10 mW at most.(5–7)
To increase inertial force at low frequencies, generators that use the gyroscopic effecthave been studied. They increase the angular momentum of the vibrating mass by spinning
it at a high angular velocity. There are two types of gyro-generator, namely, those driven by
a motor and friction. Friction-driven gyro-generators, invented by Mishler(8) in 1973, havebeen commercialized as a wrist training tool. A flywheel (FW) is rotated via a friction force
caused by precession, producing a power of about 0.1 W.(9,10) However, such gyro-generators
can operate only under vibration that is synchronized with the precession cycle. Although
theoretical studies on stabilization have been conducted,(11,12) existing devices do not work
under arbitrary vibration. Motor-driven gyro-generators rotate a FW by means of a motor.
They work under arbitrary vibration. This type of gyro-generator was invented by Norden(13) in*Corresponding author: e-mail: hosaka@edu.k.u-tokyo.ac.jp
https://doi.org/10.18494/SAM.2020.2846
ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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Sensors and Materials, Vol. 32, No. 7 (2020)1917 for application as gyrostabilizers for ships. Recently, Kanki et al.(14) fabricated a prototype
of a wave energy converter (WEC) that has two FWs (2.7 t each) and generates 20 kW.
Bracco et al.(15) also fabricated a WEC with a 200 kg FW. This type of gyro-generator is verylarge because the gyro torque, electromagnetic induction, and coil resistance strongly dependon size. Miniaturization greatly reduces the generated power and makes it difficult to harvest
motor spin power. The authors previously developed a motor-driven gyro-generator, whose FW
is 100 mm in diameter with a power output of 1.8 W.(16) To achieve this, precession movement
stabilization by a precession spring and an electromechanical conversion efficiency increase,achieved by using a high-performance motor and a speed-increasing gear with a ratio of 100,
were applied. In this generator, however, the FW spinning energy was supplied by an external
power supply and the spin velocity was constant. Theoretical analyses of the optimization ofmagnetic damping,(17) the nonlinear behavior of gyros,(18) and the application of gyro-generators
to underwater autonomous vehicles(19) have been conducted. However, the spin velocity inthese applications is constant. Spin velocity variation caused by a self-power supply has notbeen researched.
In this study, a gyro-generator that is powered by harvested energy is developed. In thisgenerator, the generated power accelerates the spin velocity and increases the gyro effect, which
in turn increases the output power, creating a positive feedback system. The spin velocity
increases until the generated voltage reaches the level of the countervoltage of the spinningmotor. It is desirable to maximize spin velocity and acceleration. To achieve this, mechanicaland electrical equations for the gyro-generator are obtained. The relationships among time,spin velocity, generated power, and efficiency are clarified using approximate solutions andnumerical analyses. It is shown that spin velocity is efficiently increased by increasing the boost
ratio as the spin velocity increases. Finally, the validity of the theoretical results is verified
experimentally using an apparatus with a FW (diameter: 100 mm) rotating up to 650 rpm.

2.

Nomenclatureex, ey, ez, eY: unit vectors in x, y, z, and Y directions
EC: work done by the viscous damper C in one cycle
Es: work done to the FW spin in one cycle
Ez: work done by the FW precession in one cycle
Jfw, Jfx, Jfz: inertia tensor and principal moment of inertias of FW
Jig, Jiz: inertia tensor and principal moment of inertia of inner gimbal part
Jog: inertia tensor of outer gimbal partIg, Is and Vg, Vs: currents and voltages of generating and spinning motors
τgm, τsm and Ktg, Kts: torques and torque constants of generating and spinning motors
k p: spring constant of precession springnp, ns and τpg, τsg: gear ratios and torques of precession and spin gears
N: boost ratio
Rg, Rs: terminal resistances of generating and spinning motors
VF: voltage drop across diodeVr, Ir: output voltage and current of rectifierSensors and Materials, Vol. 32, No. 7 (2020)

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x, y, z: coordinate system fixed to inner gimbal
x′, Y, z: coordinate system fixed to outer gimbal
X, Y, Z: coordinate system fixed to inertial space
ηp, ηs: efficiencies of precession and spin gears
θY, θYo: input angle and amplitudeθz, θzo: precession angle and amplitude
θzmax: upper limit of precession angle
τs, τz: torques of FW spin and inner gimbal precessionτsb: torque applied from spin gear to FW shaft
τgmf, τsbf : friction torques of generating motor and spin bearing
ωi: excitation frequency
ωfw, ωig, ωog: angular velocity vectors of FW, inner gimbal, and outer gimbal
ωgm, ωsm: angular velocities of generating and spinning motors
ωs, ωY, ωz: spin angular velocity of FW, pitching angular velocity of inner gimbal, andprecession angular velocity of outer gimbal

3.

Configuration of Gyro-generator and Electrical and Mechanical Equations

3.1

Configuration of generator

The configuration of the generator is shown in Fig. 1. The generator consists of a FW,
spinning and generating motors, spin and precession gears, a precession spring, a rectifier,a booster, and outer and inner gimbals. The operation sequence is as follows. (1) The FW
is rotated by a spinning motor. The initial velocity is given by a battery charged in advance.

Fig. 1.

Configuration of gyro-generator.

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Sensors and Materials, Vol. 32, No. 7 (2020)(2) A pitching vibration is applied to the outer gimbal by an external torque. (3) The FW
begins precession owing to the gyro effect. (4) The precession velocity is increased by thegear. (5) Electrical power is generated by the generating motor. (6) The generated electricity is
rectified, and the voltage is boosted and applied to the spinning motor. By this cycle, the spin
angular velocity of the FW, the gyro effect, the precession amplitude, and the power generation
increase. This is a positive feedback system whose power source is the input vibration energy.
The generated electricity is stored as FW kinetic energy.
As the spin angular velocity increases, the countervoltage of the spinning motor increases.
When the countervoltage reaches the level of the generated voltage, the flow of current to thespinning motor stops. Thus, voltage boosting is necessary to increase the spin velocity. If theboost ratio is excessively large, however, the current of the generating motor will increase, thevoltage drop caused by the generating motor resistance will increase, and thus the input voltage
to the booster will drop. In addition, the current of the spinning motor and thus the spin torque
will decrease. When the motor torque is less than the bearing friction torque, the spin velocity
will drop. Therefore, there exists an optimal boost ratio.
In gyroscopes, the gyro torque is generally maximum when the precession angle is 0° (theposition of the inner gimbal shown in Fig. 1) and when the torque is 0 at 90°. These positions
are unstable and stable equilibria, respectively. Thus, when the inner gimbal moves freely,
it moves to the 90° position and the gyro torque and FW spin disappear even if vibration isapplied to the outer gimbal. To avoid this, a torsion spring is attached to the precession shaft tomaintain the precession angle near 0°.
3.2

Assumptions in analysis

Inertial force is considered for the FW, inner and outer gimbals, and motors. Friction isconsidered for the precession and spin gears, spin bearing, and generating motor, all of whichhave large loads and high angular velocities. The gear friction is represented by a constantefficiency for the transmitted power. The bearing friction is represented by a linear equation
for the angular velocity. The motor friction is a constant torque. The electromechanicalcharacteristics of the motors are represented by torque constants and terminal resistances.
The inductances of the motors are neglected. The rectifier consists of a diode bridge, where
the forward voltage drop of the diodes is considered. The booster consists of a diode andan idealized booster that preserves power. The input vibration is a sinusoidal wave and thepitching angle θY is given by Eq. (1). The generator is very light and thus its inertial force does
not affect the input angle.

θY = θYo sin ωit
3.3

(1)

Equations of motion for FW, inner and outer gimbals, and spring

To show the three-dimensional rotation of the rotor, we use the three coordinates shownin Fig. 2. The coordinate system fixed to the inertial space is called the XYZ system, whereSensors and Materials, Vol. 32, No. 7 (2020)

Fig. 2.

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Coordinate systems.

X is the horizontal direction, Y is the pitching axis of the outer gimbal, and Z is the verticaldirection. The coordinate system fixed to the outer gimbal is called the x′Yz system, where x′ isperpendicular to the outer gimbal surface, Y is as defined above, and z is the rotating axis of theinner gimbal. The coordinate system fixed to the inner gimbal is called the xyz system, where x
is the FW spin axis, y is the radial direction of the FW, and z is as defined above. The origin of
all coordinates is the center of the FW. The pitching angle is denoted as θY and the precession
angle is denoted as θz; the corresponding angular velocities are respectively denoted as ωY andωz. The FW spin angular velocity is denoted as ωs. Note that ωs is a relative velocity measured
from the inner gimbal, whereas ωY and ωz are the absolute velocities measured from the inertial
space. The absolute velocity around the x-axis, ωx, is ωs plus the inner gimbal velocity ωYsinθz.
The body consisting of the outer gimbal and the generating motor is referred to as the outergimbal part. The body consisting of the inner gimbal and the spinning motor is referred to as
the inner gimbal part. The inertia tensors of these bodies and the FW are denoted as Jog, Jig,
and Jfw, respectively. The corresponding principal moments of inertia are denoted by adding
the suffix of the axis.
The angular velocity vector of the outer gimbal part, ωog, is the input angular velocity.
The angular velocity vector of the inner gimbal part, ωig, is the sum of ωog and the precession
angular velocity ωzez. The angular velocity vector of the FW, ωfw, is the sum of ωig and thespin angular velocity ωsex. The vectors ωog, ωig, and ωfw are given as follows when expressedin the coordinates fixed to each part.

ωog = ωY eY(2)

ωig =ωY sin θ z e x + ωY cosθ z e y + ω z e z

(3)

ω fw
=

(ωY sin θ z + ωs ) e x + ωY cosθ z e y + ω z e z(4)

Because the coordinate origins are fixed in space, the kinetic energy T of the whole system isgiven by the sum of the rotation energies of these parts.

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Sensors and Materials, Vol. 32, No. 7 (2020)

T=

1
1
1
ωog T J ogωog + ωig T J igωig + ω fwT J fwω fw
2
2
2

(5)

The potential energy of the precession spring U is given as

U = k pθ z 2 / 2 .

(6)

Because the vertical movement of the gravity centers of all parts is small, the effect of gravity
is neglected. By taking the degrees of freedom as θz and ωs and using T and U, the Lagrange
equations of motion are derived. Furthermore, by assuming ωs  ωY, the equations of motionof the precession [Eq. (7)] and spin [Eq. (8)] of the FW are given as follows. The first term in Eq. (7)is the gyro torque.

3.4

τz =
− J fxωsωY cos θ z + ( J iz + J fz ) ω z + k pθ z

(7)

τ s =J fx (ω s + ωY sin θ z + ωY ω z cosθ z )(8)Equations of feedback circuit

The precession velocity ωz and torque τz generated in the inner gimbal shaft become the spin
torque of the FW, τs, after passing through the feedback circuit and mechanical components
shown in Fig. 3. The equations of the components are given as follows.
precession gearω gm = n pω z

Fig. 3.

(9a)

τ gm = η pτ z / n p

Feedback circuit and components.

(9b)

Sensors and Materials, Vol. 32, No. 7 (2020)

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generating motor=
Vg Ktgω gm − I g Rg

(10a)

=
τ gm Ktg I g + τ gmf

(10b)

V
=
Vg − 2VFr(11a)

I g = Ir

(11b)=
Vs NVr − VF(12a)Is = Ir / N

(12b)

(13a)

τ sm = Kts I s

(13b)

(14a)

τ sb = η s nsτ sm(14b)

rectifierboosterspinning motor=
Vs Ktsωsm + I s Rs
spin gearωs = ωsm / ns
spin bearingτ=s τ sb − τ sbf

4.

Estimation of Spin Acceleration Using Approximate Analysis

4.1

Simplification of equations of generating motor and FW precession(15)

To estimate the spin acceleration qualitatively, the approximate solutions of equations givenin the previous section are used. In this analysis, the mechanical and electrical variables areaveraged in a vibration cycle and their long-term characteristics are studied. To simplify the
analysis, the following approximations are introduced. Inertia is considered only for the FW.
The frictions of the generating motor and spin bearing are included in the efficiencies of thegears. The voltage drop across diodes is neglected. The input frequency is the same as the
eigenfrequency of precession. The precession velocity is a sinusoidal function and the spin
velocity is constant in one vibration cycle. The velocities are given as below, where φ is an
unknown constant.

=ω z ω zo sin(ωit + ϕ )ωs = const

(16)
(17)

Here, we consider the period 0 < ωit + φ < 2π. The torques τz and τs are given as follows during
the time periods α < ωit + φ < π − α and π + α < ωit + φ < 2π − α. At other times, they are zero

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Sensors and Materials, Vol. 32, No. 7 (2020)

because the rectifier and the booster pass only one-way current.

N 2 Ktg 2 n p 2


Kts nsωs − sgn(ω z )ωzNKtg n pω zo η p Rg N 2 + Rs 

=
τz

(

=
τs

)

Kts nsη s NKtg n p
2

Rg N + Rs

ω z − sgn(ω z )Kts nsωs
NKtg n pω zo

(18)

(19)

The curves of ωz, τz, and τs in Eqs. (16), (18), and (19), respectively, are shown by continuouslines in Fig. 4. Because they are in phase, τz is approximated by a sinusoidal function
proportional to ωz, as shown by the dashed line in Fig. 4.

τ z = Cω z

(20)

This equation means that the generating motor is approximated by a viscous damper, C. The
value of C is selected so that the work done by the FW precession is the same before and after
the approximation. The work in one cycle before the approximation is given as( 2π −ϕ )/ωiEz ∫=τ zω z dt=−ϕ /ωi

4 N 2 Ktg 2 n p 2ω zo 2(

ωiη p Rg N 2 + Rs)

F ( r ),

(21)

where r is the ratio of the countervoltage of the spinning motor to the boosted voltage of thegenerating motor and F is the drop of the average electric power caused by the countervoltage
and the blocking of the countercurrent in the rectifier and the booster.

r=

Fig. 4.

Kts nsωs
NKtg n pω zo

Curves of simplified torque and velocity.

(22)Sensors and Materials, Vol. 32, No. 7 (2020)

F (r ) = ∫

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π /2

( sin x − r ) sin xdx =
α

π
4

−

α
2

−

r
1− r2
2

(23)

The parameter α, which is the phase angle when the current of the booster becomes zero, isgiven asα = arcsin r .

(24)

The work done by C in one cycle is given as( 2π −ϕ )/ωi 2
πω zo 2
=
EC C=
ω
dt
C
.
z
∫

ωi

−ϕ /ωi(25)

Thus, C is given asC=

4 N 2 Ktg 2 n p 2(πη p Rg N 2 + Rs)

F ( r ).

(26)

Next, the equation of motion of precession is simplified. With the assumption θz  1, Eq. (7)becomes Eq. (27) by using Eq. (20).

J fzθz + Cθz + k pθ z =J fxωsθYoωi cos ωit(27)

Because the excitation frequency is the same as the eigenfrequency, ωi and θz are given as

ωi 2 =

kp
J fz

,

(28)

θ z = θ zo sin ωit ,θ zo =

J fxωsθYo
C

.

(29)
(29)

Also, φ and ωzo are given as φ = π/2 and ωzo = θzoωi, respectively.
4.2

Simplification of equations of spinning motor and FW spin

We obtain the average spin torque in one cycle and denote it as τ s. With this approximation,
the torque variation in one cycle is neglected, but the velocity increment in the cycle ispreserved.

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Sensors and Materials, Vol. 32, No. 7 (2020)

τs =

ωi ( 2π −ϕ )/ωi
τ s dt2π ∫ −ϕ /ωi(31)

Below, we denote τ s as τs, which is given asτs =2 NKtg Kts n p nsη sω zo(πη p Rg N 2 + Rs)

G (r ) ,(32)

where G is the drop in the average voltage caused by the countervoltage and the blocking of thecountercurrent.
G (r ) = ∫
sin x − r ) dx = 1 − r 2 + r  α −
(
α
π /2



π

2

(33)

The curves of F, G, and G/F are shown in Fig. 5. They are monotonically decreasing functions
and approach 0 or 1.
Next, the equation of spin motion is simplified. Equation (8) becomes Eq. (34) when
integrated in one cycle, where Δωs is the increment of ωs in one cycle and Δt is the period of thecycle.

τs

J fx

=

∆ωs
∆t

(34)

The right-hand side of Eq. (34) shows the average acceleration rate of ωs and can thus
approximate ω s . Equation (34) can then be rewritten asτ s = J fxω s .

(35)

The equations of the generating motor [Eq. (26)], spinning motor [Eq. (32)], FW
precession [Eq. (30)], and FW spin [Eq. (35)] are basic equations that represent the long-term
characteristics of the gyro-generator.

Fig. 5.

Countervoltage functions F and G.

Sensors and Materials, Vol. 32, No. 7 (2020)

4.3

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Qualitative analysis of acceleration using approximate solutions

By combining the four basic equations, the differential equation for ωs is obtained asω s =

Kts nsη sη pωYo
2 NKtg n p

ωs .

(36)

The solution is given as below, where ωso is determined by the initial conditions.
 Kts nsη sη pωYo t
 2 NKtg n pωs = ωsoexp (37)

The velocity ωs increases exponentially with time. Note that this solution is valid only under
the condition of θzo  1.
By changing N to θzo in Eq. (36) using Eqs. (26) and (30), Eq. (36) becomes Eq. (38). The
acceleration increases with θzo.

4 Ktg 2 n p 2 F ( r ) Rg
ω s Kts nsη sη pωYo
=
−
θ zo
ωs
2 Ktg n p
J fxπη p RsθYoωs Rs(38)

In the actual generator, there is a mechanical restriction on the precession range. The amplitude
θzo must be smaller than the given upper limit θzmax, which is 45° in the experimental apparatus
described in Sect. 6. Thus, the acceleration becomes maximum when θzo is θzmax. θzo is
controlled by N. The relationship of θzo with N is given as follows using Eqs. (26) and (30). The
optimal N, which sets θzo to θzmax, increases with ωs.
4 Ktg 2 n p 2 F (r ) Rg
1
=
θ
−
zo
π J fxη p RsθYoωs Rs
N2

(39)

The angular acceleration ω s is given as below from Eqs. (32) and (35).

ω s =

2 NKts Ktg n p nsη sω zoG (r )(π J fx Rs + N 2 Rg

)

(40)

The acceleration stops at r = 1 because G = 0 and ω s = 0. Then, ωs becomes maximum and isgiven asωs =NKtg n pω zo
Kts ns

.

(41)

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Sensors and Materials, Vol. 32, No. 7 (2020)

This is the velocity at which the current of the spinning motor becomes 0 owing to thecountervoltage.
Next, we obtain the efficiency of the entire system. The input energy is supplied by the
input vibration and the output energy is stored as the kinetic energy of the FW. Thus, theefficiency η is given by the ratio of the work done to the FW spin, Es, to the work done by the
FW precession, EC. Es and η are given as below from Eqs. (32) and (25).
2π 4 NKts Ktg n p nsη sω zoωsG (r )
Es τ=
=
sωs
ωi
ωi Rs + N 2 Rg

(

=
η

)

Es
= rη sη p
EC

(42)

(43)

Because r is proportional to ωs, η increases with ωs. Because the maximum r is 1, themaximum η is the product of the efficiencies of the precession and spin gears, ηsηp.
The curves of ωs, η, and N are shown in Fig. 6. They increase with time. The maximum
η is ηsηp. When N is constant, the rate of increase in ωs increases with decreasing N, and the
maximum ωs increases with N. By controlling N so that θzo coincides with θzmax, the maximum
ωs reaches its highest value of NKtgnpωzo/Ktsns.

5.

Numerical Analysis

5.1

Calculation procedure

In this section, the equations obtained in Sect. 3 are solved without approximation.
Equations (9a), (9b)–(15) give the relationships among τs, τz, ωs, and ωz. By inserting values

Fig. 6.

Curves of ωs, η, and N obtained from approximate solution.

Sensors and Materials, Vol. 32, No. 7 (2020)

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of ωs and ωz to these equations, τs and τz are obtained. By inserting values of τs and τz
to Eqs. (7) and (8), ωs and ωz after a short period are obtained. By repeating these procedures,
all the parameters are obtained in the time domain. Since Eqs. (9a), (9b)–(15) are algebraic
equations, they can be combined into two algebraic equations that directly give τs and τz from ωsand ωz. For a future expansion to complicated systems, however, these equations are solved one
by one for each component, which enables the simulator to include nonlinear or time-dependent
components.
The procedure is shown in Fig. 7. The numbers in the figure are the equation numbers,
the symbols are physical quantities transferred between components, and the arrows show theorder of calculation. First, initial values are given to ωz, θz, and ωs. The initial ωz is set to 0
to simplify the calculation of Ig, as mentioned later. Next, by inserting ωz into Eq. (9a), ωgm isobtained, and by inserting ωs into Eq. (14a), ωsm is obtained. Then, Vg, Vr, and Vs are obtained
from Eqs. (10a), (11a), and (12a), respectively. In the calculation of Eq. (10a), Ig of the previous
cycle is used. The initial value of Ig is 0 because the rectifier blocks current under the conditionof ωz = 0. Next, by inserting Vs and ωsm into Eq. (13a), Is is obtained. Then, Ir, Ig, τgm, andτz are obtained from Eqs. (12b), (11b), (10b), and (9b), respectively. Also, τsm, τsb, and τs are
obtained from Eqs. (13b), (14b), and (15b), respectively. In the calculation of Eq. (11b), the signof Ig is set to be the same as that of ωz. Since the rectifier transmits energy only from the input (left)
to the output (right), signs of Ig and Vg are the same, and thus the signs of Ig and ωz are the same.
Then, Eqs. (7) and (8) are solved using ωs, ωz, τs, τz, ωY, and ωY where ωY and ωY are given by
Eq. (1), and ωs and ωz after a short period is obtained. Then the above procedures are repeated.
To set N to the optimal value, it is necessary to set θzo to θzmax. In the above procedure,
however, θzo is known only at the moment that θz takes a local minimum or maximum value.
Even if θzo is known, N cannot be obtained directly because the relationship between N and θzo[e.g., Eq. (39)] is given only in the approximate analysis. Thus, in this paper, when θz takes alocal extreme value, θzmax is subtracted from it; if it is positive, N is increased by a certain value
and if it is negative, N is decreased. Then, θzo approaches θzmax.

Fig. 7.

Calculation flow.

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5.2

Sensors and Materials, Vol. 32, No. 7 (2020)

Parameters

In the calculation, the following conditions are applied. The spinning motor is the coreless
DC motor SCL18-3328 (Adamant Namiki Precision Jewel Co., Ltd.) and is called motor A in thisstudy. This motor is small and highly efficient and is used in the experiment described in Sect. 6.
For the generating motor, either motor A or the motor NC-256402 (Citizen Chiba Precision Co.,
Ltd.), which is called motor B, is used. Motor B is larger than motor A and has a higher torqueconstant and a lower resistance than motor A. The characteristics of the motors provided by the
manufacturers are shown in Table 1. Other parameters for the calculation are the same as thosefor the experiment: Jfx = 0.80 gm2, Jfz = 0.45 gm2, Jiz = 1.3 gm2, ωi = 2.5 Hz, kp = 8.2 mNm/deg,θzmax = 45°, θYo = 30°, and initial ωs = 800 rpm. τsbf is given as τsbf = 1.5 × 10−5ωs + 2.6 × 10−4,
where the units for τsbf and ωs are Nm and rad/s, respectively. This is an approximate formulaobtained experimentally using our apparatus. The diodes used for the rectifier and the booster
are SBM1045VSS (PANJIT Semiconductor), whose forward voltage is VF = 0.25 V at 0.1 A.
5.3

Simulation results

The calculated spin angular velocity ωs is shown in Fig. 8 for constant boost ratios of N = 1, 1.6,
and 5 and the controlled boost ratio. The continuous and dashed lines show the spin velocities
obtained from generating motors A and B, respectively. The numbers indicate N. When N isconstant and the precession angle θzo reaches the upper limit θzmax (in this case, 45°), the power
feedback is stopped and ωs is set constant. At t = 250 s, the ωs of motor B reaches 4800 rpm
for N = 1.6 and 9600 rpm for controlled N (data not shown). The rising speed of ωs is higher for
N = 1 than for N = 1.6 for both motors A and B. The maximum value of ωs for N = 1.6 is larger
than that for N = 1. These characteristics are the same as those obtained from the approximate
solution shown in Fig. 6. At N = 5, ωs stalls in both motors A and B. This was not observed in
the approximate solution. The stall is caused by the constant spin friction, which is neglected in
the approximation and becomes dominant in the low-torque region. When N is very large, thecurrent of the spinning motor is so small that the motor torque is smaller than the friction andωs stalls.

Table 1
Characteristics of spinning and generating motors
provided by manufacturers.
Motor type
A
B
21
32
Ktg, Kts (mNm/A)
28
3.75
Rg, R s (Ω)
9.6 × 10 −2
1.6
τgmf (mNm)
105, 3.6
113
np, ns0.61, 0.86
0.73
η p, η s
Fig. 8.

Effect of boost ratio on spin angular velocity.

Sensors and Materials, Vol. 32, No. 7 (2020)

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To control the boost ratio in Fig. 8, the initial value of N was set to 0.8 and θYo was
obtained every half-period. N was increased or decreased by 2% in accordance withwhether or not θYo > θzmax was true. Because N was changed intermittently, this control was
considered long-time-interval sampled-data control, not minimum time control of ωs.
Figure 9 shows the change in N for controlled boosting. N increases with time; the trend is
the same as that in Fig. 6. N converges to certain values because ωs becomes constant and thesystem enters a steady state. The N of motor A rises faster than that of motor B because the Ktg
of motor A is smaller than that of motor B. The N of motor B is initially constant because it is
kept at the initial value until θzo reaches θzmax and the time to reach θzmax is long for motor B
because of its large Ktg.

6.

Experiment

The calculated results are verified by experiment. Figure 10 shows the experimental
apparatus. The FW, made of aluminum, had a diameter of 100 mm and a thickness of 30 mm,
and spun around a horizontal axis. The input vibration was applied manually via a handle. The
rotation of precession was around a vertical axis and the generating motor and the precession
gear were fixed on top of the outer gimbal. The input and precession angles were measured
by encoders. The spin velocity was measured by a photoreflector placed on the hidden sideof the FW. The generating and spinning motors were both motor A. The gear ratios of theprecession and spin gears were 22.22 and 3.6, respectively. The ratio np was set smaller thanthat in Table 1 because the gears easily broke when np was around 100. A torsion spring withk p = 8.2 mNm/deg was fixed below the bottom of the outer gimbal.

Fig. 9.

Time history of boost ratio.

Fig. 10. Experimental apparatus.

Spin angular velocity ωs (rpm)2566

Sensors and Materials, Vol. 32, No. 7 (2020)

800
Control

700
Calc.

600
500

1.4
1.2

400
300
200

N=1.6

Exp.

5.0
0

10

20

1.0

30
Time (s)4050Fig. 11. Experimental and calculated spin angularvelocities.

Fig. 12. Experimental and calculated kinetic energies
and boost ratios.

An input vibration of ωi = 2.5 Hz was applied to the handle with an initial angular velocityof ωs = 300 rpm. When N was constant, θYo was initially set to 8°. After ωs was increased andθzo reached θzmax = 45°, θYo was decreased to keep θzo = θzmax. In the numerical analysis in Sect. 5,
feedback was stopped at θzo = θzmax. In the experiment, however, feedback was continued andθYo was decreased to change ωs in a wide range. When N was controlled, N was increased or
decreased by 5% in accordance with whether θzo was larger or smaller than θzmax, respectively.
θYo was set constant to 8°.
The experimental and calculated spin angular velocities are shown in Fig. 11. The N
and kinetic energy of the FW are shown in Fig. 12. The continuous and dashed lines are
the experimental and calculated results, respectively. In the experiment of controlled N, themaximum spin angular velocity was 650 rpm, the maximum kinetic energy was 4 J, andthe boost ratio changed from 1 to 4.3. In the experiment of constant N, the angular velocity
increased with time and N when N was less than 1.6 and stalled when N was 5.0. Theseexperimental results agree well with the calculated results. The generated power measured
from the slope of the energy was about 0.1 W.
In Fig. 12, both the N and energy from the experiment are smaller than those from thecalculation after 18 s. This is because the input vibration was applied manually and θYo was
smaller than 8° at that time. The N from the experiment approached that from the calculation
after 35 s because θYo was corrected to 8°. From 18 to 40 s, the drop in N was larger than that inenergy. This is because N was changed by 5% by every half-period, and thus, N changed slowly
and the influence of error was expanded and remained for a long time.

7.

Conclusions

To realize a small and high-power vibrational generator using the gyro effect, a methodof increasing the spin velocity of a FW was proposed. This method boosts the generated
voltage and feeds it back to the spinning motor. To analyze spin acceleration characteristics,
the electrical and mechanical equations of the components were derived. The inertial forcesof the FW and gimbals, the electromechanical transformations of the generating and spinningSensors and Materials, Vol. 32, No. 7 (2020)

2567

motors, the efficiency of the gears, the friction of the bearing, and the voltage and current
transformations of the booster and the rectifier were considered in the equations.
Closed-form approximate solutions were obtained by averaging the equations in one cycle.
Strict solutions were also obtained numerically. With these solutions, the spin acceleration
was analyzed. It was shown that there is an upper limit of the spin angular velocity. The limit
is the angular velocity at which the voltage generated by the generating motor is equal to thecountervoltage of the spinning motor. When the precession amplitude is set to the maximum
allowable angle of the device, the acceleration becomes maximum. The precession angle iscontrolled by the boost ratio, which increases with the spin angular velocity.
To confirm the theoretical results, an experimental device with a FW (diameter: 100 mm;
thickness: 30 mm) was developed. The maximum spin angular velocity was 650 rpm, themaximum kinetic energy was 4 J, and the optimal boost ratio changed from 1 to 4.3. Theseexperimental results agree well with the calculated results.

Acknowledgments
This research is supported by the A-STEP Program from the Japan Science and Technology
Agency, JST.

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