## Abstract

We propose a new transmission mechanism that is compatible with high-speed downsizing motors. This mechanism adopts the “pulse drive transmission” (PDT) principle. Similar to the electrical switching converter, the PDT principle allows variable velocity ratios regardless of geometry (cf. the radius relationship is essential for the gear principle as the geometry). According to this similarity, the PDT principle is expected to maintain low inertia even at large velocity ratios and to increase the amount of transmitted power by the dependence of transfer frequency on rotational velocity. Thus, the PDT principle is suitable for high-speed motors. This study employed self-excited vibration in the PDT principle to eliminate the engagement controls that caused problems at high speed in a previous study. Simulations and prototype tests demonstrated that the proposed mechanism, combining self-excited vibrations by magnetic nonlinear springs and one-way clutches, achieves the desired behavior based on the PDT principle and is capable of power transmission at several velocity ratios and rotational speeds. In particular, performance evaluations under steady-state operations showed that the maximum input torque, maximum power transmission, and maximum efficiency were 20.9 ± 0.18 N m, 1.0 kW, and 79.8%, respectively.

## 1 Introduction

In mechanical engineering, a transmission is a device that transmits the driven power between two axes with different velocities and developed as a device to utilize natural energy and livestock labor as well as to amplify human force. The transmission is specifically useful as a device to output work at various speeds while maintaining the cycle of an engine. Thus, various types of transmission systems have been developed along with the commercialization of engines. For example, H. Föttinger’s fluid torque converter developed in 1905 achieved smooth starting; further, Griffith’s two-stage switching gearbox in 1821 and Didier’s continuously variable transmission (CVT) in 1899 achieved a variable velocity ratio. Based on these developments, more compact and efficient technologies have been developed [13]. In particular, CVT technology has expanded to various types (e.g., belt, toroidal [4,5], spherical [6], and hydrostatic [7,8]) and has led to infinitely variable transmission [9] technology.

Transmission systems are still required to extend the driving range and expand the cabin capacity, although electric motors have been replacing engines to reduce carbon dioxide emissions in recent years. Therefore, compact and highly efficient technologies are still required. These requirements apply to motors as well. To achieve a compact motor, studies have been conducted to increase the output power density by increasing the motor’s speed [10,11]. To utilize the advantages of such high-velocity motors, it is necessary to maintain a small size while increasing the rotational velocity. In the field of human–robot interaction (HRI), the “REFLEX torque amplifier,” which is a modification of the planetary gear trains developed by Wolfrim in 1912, and the “Bilateral drive,” which is a modification of the strain wave gearbox developed by Musser in 1955, have been proposed for this purpose [12].

Although many transmission systems have been proposed as described above, the principle has not changed. The general transmission principle is classified into the gear principle and the fluid coupling principle. These principles are not suitable for high speeds. The gear principle is based on the fact that the peripheral velocity of a rotating disk is proportional to its radius. In the case of gears, there are uniformly sized teeth on the circumference, and the velocity ratio corresponds to the ratio of the number of teeth of each gear output to the input gear. The gears must have a certain radius, and their rotational inertia occurs at high speeds. In addition, the contact between gears becomes more intense at high speeds, thus causing damage. The fluid coupling principle utilizes the energy disposal of the velocity difference. In the case of fluid coupling, the fluid is driven by a pump attached to the input shaft, and the kinetic energy of the fluid is output by a turbine attached to the output shaft. The energy of the velocity difference is emitted as heat during mixing of the fluid; thus, the efficiency of the transfer is low. In addition to the issue of lower efficiency, the fluid coupling principle faces other issues, such as cavitation at high-speed rotation.

In a breakthrough development, a unique principle called “pulse drive transmission” (PDT), which aims to realize small and high-efficiency transmission systems, has been introduced [13]. It is completely different from these general principles in that it applies the idea of electrical transmission to kinematics. The use of electrical ideas in mechanical engineering has been reported elsewhere [14] but not for transmission. By considering the velocity in kinematics as an electrical voltage and torque as a current, the transmission is equivalent to a DC–DC conversion circuit. Based on this relationship, the corresponding conversion circuit for general transmission is as follows: the gear is a winding-type conversion circuit, and the fluid coupling is a voltage divider circuit (Table 1). In a winding-type conversion circuit, the ratio of the number of windings is equivalent to the ratio of the voltage, and the principle is based on geometry, as in the case of gears. A voltage divider circuit uses a resistor to release excess energy and an energy disposal principle similar to fluid coupling.

Table 1

Correspondence of principle between general transmission and converter circuit

In addition, the PDT principle corresponds to a method of other conversion circuit. For example, chopper circuits consist of switches and induction coils, while switched-capacitor circuits consist of switches and capacitors as well as their hybrid systems. These switching circuits accumulate the input power briefly and release it intermittently to convert it. Focusing on the behavior of these elements, an electrical capacitor and induction coil are equivalent to mechanical inertia and a spring, respectively. Thus, the PDT structure in a previous study [13] corresponds to these hybrid systems (Table 2). In the switching type, the amount of power transferred is independent of the geometry and is proportional to the amount of temporary charging energy and switching frequency. Furthermore, the energy of the velocity difference is utilized. Thus, the switching-type converter is smaller and more efficient than other methods. These similarities suggest that the PDT principle leads to a small size and high efficiency.

Table 2

Switching converter and pulse drive transmission as its corresponding mechanism

However, in the prototype of the previous study [13], approximately half of the body size is occupied by these switching devices and their measurement parts, which is an obstacle to reducing the body size. In addition, to ensure stable measurement and control, the input velocity of the prototype was 100 rpm during the experiment, and it was difficult to combine this with a high-speed motor spinning at several tens of krpm. The factor responsible for this negative effect is the element with a large motion for switching. Although electrical switching involves a nonmoving element, such as a field-effect transistor (FET), switching in PDT involves an element with large motion for engagement, such as an electromagnetic clutch and brake. A movable element with large motion, such as an electromagnetic clutch and brake, requires extra time and control for switching; thus, it limits downsizing performance and the maximum velocity capacity.

The objective of this study is to design a transmission structure that is more compact and suitable for high-speed motors by increasing the switching frequency of PDT. Therefore, we proposed a new design in which the elements with large motion (clutch and brake) are eliminated, and the switching is driven passively. This design is termed a “passive pulse drive.”

## 2 Idea for Elimination of the Element With Large Motion and Automation of Switching

A roller-type one-way clutch (OWC) is one of the parts that have the most number of nonmovable elements for engagement. Several rollers are located on the circumference of the cylinder that constitutes the clutch, and when the internal shaft rotates, it engages by wedging itself between the cylinder and shaft. This wedging motion of the OWC is completed at a small angle (e.g., less than 3 $deg$ at NCU and NCZ types manufactured by NTN Co., Osaka, Japan), and thus, engagement is highly responsive and efficient. The OWC engages in one direction, and rotation in the opposite direction creates a release state. Therefore, intermittent engagement and release operations can be automated if the direction of the shaft rotation oscillates periodically. However, the input of the transmission does not contain such a vibration component; thus, it is necessary to create a vibration.

In a previous study, free vibration was adopted, which requires careful observation and fine control to maintain vibration. To create vibration passively, we adopted self-excited vibration, which is a type of vibration generated from a nonvibrating input. For example, the brake squeal observed in automobiles is a self-excited vibration called the stick-slip phenomenon [15]. Repeated microscopic slippage and frictional contact of the brake pads with the rotating disc create vibrations. Utilizing this vibration, it is possible to trigger the engagement motion automatically. In practice, several studies have intentionally utilized this vibration (e.g., for energy harvesting [16]). As described in the introduction, considering the losses and inefficiency caused by the friction phenomenon, other phenomena are required for high-speed motors. A nonlinear spring causes torque ripple, and it has the possibility of rotational vibration owing to the rotational dependence of the spring coefficient. We focused on a magnetic coupling structure as a nonlinear spring with rotational dependence. Considering the magnetic coupling as a mechanical spring element, the spring coefficient changes in a cosine waveform, producing a sinusoidal torque against the angle of rotation. This rotational dependence causes intermittent switching between the two states (energy storage and release), resulting in self-excited vibrations. We named this type of spring “periodic inversion spring.” In mechanical elements, the structure combining a tension spring with a clank is also a periodic inversion spring.

Figure 1 shows the initially considered structure. It consists of two shafts for the input and output, two OWCs, and a magnetic-type periodic inversion spring. The magnets are arranged alternately in the circumferential direction to ensure different magnetization directions for the neighboring magnets. This spring is connected to a stator on one side and a middle vibrator (housing of OWCs) on the other. Note that the shaft and stator are coaxial because of the bearing and do not come into contact with each other through the magnetic force. The direction of OWCs satisfies the relation $θ˙i≤θ˙m≤θ˙o$.

Fig. 1
Fig. 1
Close modal

Figure 2 shows the behavior of the middle vibrator that we aim to achieve via this structure. Figure 3 shows the combination of shaft connections and the direction of magnetic force. Assuming that the rotation velocity of the input and output is steady at the initial time, the rotation angle of the middle vibrator is 0 $deg$, and the velocity is slightly faster than the input (the state on the left-hand side of Fig. 2). The middle vibrator is decelerated by a spring, following which it reaches the same velocity as the input. Thereafter, the input shaft and the middle vibrator are engaged. The torque from the input shaft twists the spring, and the velocity is maintained constant (the first dotted line on the left in Fig. 2). Energy accumulates in the spring as it rotates against the magnetic force while engaged with the input shaft (Fig. 3(a)). The spring coefficient becomes negative by rotation, and the middle vibrator accelerates by releasing this energy (the second dotted line on the left in Fig. 2). The middle vibrator velocity reaches the output velocity and engages with the output shaft (the third dotted line on the left in Fig. 2). In engagement with the output shaft, the remaining energy is released to drive the output shaft to rotate in the same direction as the magnetic force (Fig. 3(b)). Most of the accumulated energy is emitted at the end, and the mechanism reaches its initial state. Repeating these actions by rotation allows passive operation for intermittent transmission between two axes at different speeds.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

## 3 Verification of the Proposed Idea Through Dynamic Analysis

To evaluate the power transmission behavior of the mechanism idea in Sec. 2, we performed a dynamic analysis. To simplify the calculation, the motion equation is based on the following assumptions: (1) the system is an undamped rotational vibration system and (2) the velocities of the input and output shafts are constant. The dominant equation is given by Eq. (1), where Ti and To are transmitted torques to the input or output shaft, respectively, via OWC. The spring torque acting on the middle vibrator is expressed as Tm and assumes a sinusoidal torque of amplitude A with N waves per revolution (Eq. (2)). This equation represents the rotational motion about the middle vibrator in the form of a second-order differential equation based on the law of inertia
$Imθ¨m(t)=Tm(θm(t))−Ti−To$
(1)
$Tm(θm(t))=Asin(Nθm(t))$
(2)
As an initial condition (Eq. (3a)), the initial angle and acceleration are set to zero, and the calculation begins with the input axis in the engaged state. The boundary conditions are given by the behavior of the input or output side of the OWCs. An OWC only engages in one direction of rotation and transmits torque only in that direction; hence the torque transmitted in that direction must be positive and the velocity gap during engagement is zero. As mentioned in Sec. 2, the direction of OWCs satisfies the relation $θ˙i≤θ˙m≤θ˙o$. Accordingly, we set the assumption behavior of the OWC on the basis of the kinematic model, as shown in Eqs. (3b) and (3c)
Initial conditions:
${θm(0)=θ¨m(0)=0θ˙m(0)=θ˙i$
(3a)
Boundary conditions:
$Ti={Tm(ifθ˙m=θ˙iandTm<0)0(otherwise)$
(3b)
$To={Tm(ifθ˙m=θ˙oandTm>0)0(otherwise)$
(3c)
An example of the result is shown in Fig. 4. The following conditions were used: the input velocity was $θ˙i=500rpm$, velocity ratio was γ = 2.0, number of vibrations in one rotation was N = 6, and amplitude of the spring was A = 51 N m. The middle vibrator velocity, shown in Fig. 4(a), is the intended waveform shown in Fig. 2. Through passive switching, only negative torque is transmitted to the input, and only positive torque is transmitted to the output (Fig. 4(b)). The spring temporarily accumulates 15.1 J from the input and releases most of them to the output (Fig. 4(c)). Because the spring capacity is 17 J, approximately 2 J is consumed owing to the velocity difference between the input and output.
Fig. 4
Fig. 4
Close modal

We performed a simulation by changing the velocity ratio from 2 to 4 and obtained the intended waveform as well (Fig. 5(a)). The temporary accumulated energy was 7.4 J; hence, the energy consumption due to the velocity difference increased from 2 J to 9.6 J (Fig. 5(c)). Because of this effect, the time-averaged input torque also decreased from 19.6 N m to 13.8 N m (Fig. 5(b)). It can be observed from these results that although the excessive energy excluding the complementary energy for the velocity difference affects the performance, it has no effect on this mechanism’s work.

Fig. 5
Fig. 5
Close modal

Focusing on the time range, the ratio of the engagement time to the input/output shaft changed under both conditions. At γ = 2.0, the ratio of the engagement times was 2:1 (7.8 ms for the input and 3.9 ms for the output). However, the ratio of the engagement times was 4:1 at γ = 4.0 (4.7 ms for the input and 1.2 ms for the output). Because this implies the law of energy conservation, the same relationship naturally holds true for torque.

## 4 Prototype of Passive Pulse Drive Transmission

To confirm the working of the mechanism idea in Sec. 2, we created prototypes. Figure 6 shows a partial cross-sectional view of a prototype. All the input/output shafts and middle vibrators in the prototype have coaxial structures. The input and output shafts are supported by two ball bearings, each from a fixed section. The middle vibrator is supported on the input and output shafts by an OWC (including a bearing) pressed on the inside. The external diameter of the outer case is 80 mm, and the length between the bearing sets on the input and output sides is 151 mm.

Fig. 6
Fig. 6
Close modal

This is a realization of Fig. 1 with the addition of bearings. The only change from Fig. 1 is that the magnetic spring is divided into two parts in the axial direction for torque control by phase difference. Mathematically, the amplitude of a sine wave can be reduced by adding divided waves with a phase difference (Asin (θ) + Asin (θ + ψ) = A′sin (θ + ψ′) ↠ A′ ≤ 2A). Based on this theorem, we divided the magnetic spring to adjust the spring coefficient, resulting in torque control. One part was fixed, and the other part was rotated with the shift lever shown in Fig. 6. The lever angle is defined as 0 $deg$ when the two sets of poles are different from each other. If the magnets do not interact, then the sum of the torques produced by both magnets acts on the middle vibrator. In the opposite phase states of both magnet sets, the torque acting on the middle vibrator is zero because these torques cancel each other. In contrast, for the same phase states, the torques of both magnet sets are combined, and the torque acting on the middle vibrator is twice that produced by one magnet set.

The strength of the prototype spring was designed to be equivalent to that of the simulations in Sec. 2. However, the magnets are expected to be demagnetized because of the prototype’s dimensions. Hence, the prototype shown in Fig. 6 adopted the half-magnet spring in which the magnet in one of the two directions is replaced with iron. This construction reduces the strength of the magnetic spring without making the magnet thinner to prevent demagnetization. A 3-D magnetic field analysis was conducted to obtain the torque characteristics. This analysis was performed using a finite-element method solver, JMAG–Designer. Figure 7 shows the radial magnetic flux density as an example. The magnet replaced with iron had a negative value, which was similar to that of a magnet, despite the absence of magnetomotive force. There is no significant magnetic saturation. The actual peak torque of the prototype was measured by moving the rotor in a quasi-static condition, and the peak torque was compared with that obtained through magnetic field analysis. Figure 8 shows the results. The peak torque of the prototype machine was approximately 90% of that obtained through magnetic field analysis. Although half of the poles in the rotor were not magnets and the torque was not perfectly sinusoidal, the torque was adjusted from approximately 8–46 N m through lever operation.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

## 5 Experimental Verification

Figure 9 shows the environment of the prototype test. Two AC servo motors (HF-JP903B and HG-SR702) manufactured by Mitsubishi Electric Co. (Tokyo, Japan). were used as the power and regenerative devices. Both the torque and speed of the input and output were measured using two rotating torque meters (SS-101) manufactured by Ono Sokki Co. Ltd. (Kanagawa, Japan) A linear motor (LEM4B50M-t) manufactured by Oriental Motor Co. Ltd. (Tokyo, Japan) was used to adjust the magnetic force of the prototype. For the control of each motor and the tabulation of the measured values, a rapid prototyping system (AutoBox) from dSPACE GmbH (Paderborn, Germany) was used.

Fig. 9
Fig. 9
Close modal

### 5.1 Verification of Power Transmission (Under Constant Velocity).

Figure 10 shows the results of the power transmission test of the prototype machine. In this test, the lever angle was operated linearly from 0 $deg$ to 20 $deg$ while controlling the drive and regenerative motors to maintain $θ˙i=500$ rpm and γ = 2.0. When the lever angle was 0 $deg$, the input torque was 3 N m, and the output torque was 0 N m, indicating that the power had not yet been transmitted (Fig. 10(b)). Subsequently, the lever operation increased the input torque from 3 to 17 N m and output torque from 0 N m to 6 N m, and the power transmission started. This result suggests that the objective behavior is achieved, but in this environment, it is not possible to directly confirm the simulation-like behavior owing to the resolution and structure of the prototype.

Fig. 10
Fig. 10
Close modal

### 5.2 Verification of Velocity Ratio Controllability (Under Constant Load).

To confirm the controllability of the velocity ratio, an acceleration test was performed. In this test, the lever angle increased linearly from 0 $deg$ to 20 $deg$ while controlling the drive and regenerative motors to maintain $θ˙i=500$ rpm and an output load of 3 N m. As the lever angle increased, the transmitted torque increased from 5 to 20 N m (Fig. 11(b)), and the output velocity increased from 250 to 900 rpm (Fig. 11(a)). Thus, we confirmed that the passive pulse drive can change the velocity ratio through a shift-lever operation.

Fig. 11
Fig. 11
Close modal

### 5.3 Steady-State Test of Transmission Performance.

To determine the transmission performance, the transfer torque and efficiency were measured under the conditions listed in Table 3. Tests involving 63 conditions (three input velocity conditions, three velocity ratios, and seven lever angles) were performed three times under each set of conditions. The tests were started at a lever angle of 0 $deg$ and then continued until the target angle was attained. The results were averaged over a 4-s period after the condition stabilized.

Table 3

Experimental conditions

NameValue
Input velocity (rpm)250, 500, 750
Velocity ratio (–)1.5, 2.0, 2.5
Lever angle (deg)0, 5, 10, 15, 20, 25, 30
NameValue
Input velocity (rpm)250, 500, 750
Velocity ratio (–)1.5, 2.0, 2.5
Lever angle (deg)0, 5, 10, 15, 20, 25, 30

Figure 12 shows a comparison of the transmission performance under a fixed input velocity. The error bars represent the standard deviation of the results obtained from the three tests. For all velocity ratios, the amount of power transmission increased with an increase in the lever angle (Fig. 12(a)). This result is consistent with that shown in Fig. 10(b) and suggests that the objective behavior can be achieved regardless of the velocity ratio. In this test, the power was transmitted to the output from a lever angle of approximately 10 $deg$, and the power was approximately 0.8 kW at a lever angle of 30 $deg$. There was no change in loss at any velocity ratio nor any change in the lever angle (Fig. 12(b)). Therefore, the transmission efficiency increased under high-power conditions.

Fig. 12
Fig. 12
Close modal

Figure 13 shows a comparison of the transmission performance under a fixed velocity ratio. The error bars represent the standard deviation of the results obtained from the three tests. A similar trend of increasing output power against the lever angle was observed under this condition, indicating the objective behavior (Fig. 13(a)). The increase in the transmission power is greater at a higher input velocity. This trend is similar to that of the switching converter. This is because in the pulse drive principle, although the transfer frequency is increased, the amount of energy stored and released in one cycle is not changed.

Fig. 13
Fig. 13
Close modal

Figure 13(b) indicates that the loss, which does not depend on the lever angle or velocity ratio, increases with increasing input velocity. This loss behavior suggests that the main factor in the loss is the eddy current generated by the rotating permanent magnet. As per Lenz’s law or electromagnetic induction, the time variation of the magnetic flux creates a potential difference in the conductor. The eddy current generated by this potential difference is consumed by the resistance of the conductor. As a result of the estimation by magnetic field analysis, the eddy current loss of the prototype machine was approximately 0.18–0.2 kW at $θ˙i=500rpm$ and γ = 2.0. Therefore, most of the measured losses in this test were eddy current losses. In general, machines with magnetic fluctuations, such as motors, use electromagnetic sheets to reduce eddy currents by increasing electrical resistance while maintaining magnetic permeability. A significant improvement in efficiency is expected with the introduction of such materials in the proposed structure.

Under all test conditions, we observed that the maximum transmission power, maximum input torque, minimum input torque, and maximum transfer efficiency were 1.0 kW, 20.9 ± 0.18 N m, 2.94 ± 0.16 N m, and 79.8%, respectively. When evaluating the structure size of the prototype, we used the torque density to evaluate motors and magnetic gears. The estimated torque density from this result was 69.3 N m/L, considering that the main part without the bearing had an outer diameter of 80 mm and a length of 60 mm. The maximum torque density was 138 N m/L at the full-magnet spring without the replacement of one of the two magnet directions with iron. In a previously published study comparing the performances of magnetic and mechanical planetary gears, the torque density of the mechanical planetary gears was reported to be approximately 45–150 N m/L at a velocity ratio of 2.5 and a Hertz safety factor of 5.6–3.0 [17]. Accordingly, the proposed structure is approximately the same as that of mechanical planetary gears. Studies on magnetic gears with similar shapes have reported that improving the utilization ratio of magnets increases the torque density [18,19]. Thus, the mechanism of the passive pulse drive is expected to allow effective transmission with a high torque density by optimizing the shape of the magnetic components.

## 6 Conclusions

To address the issue at high speed in the pulse drive transmission principle, we proposed a new mechanism that eliminates the elements with large motion (such as electromagnetic clutches and brakes) and drives passively. A combination of self-excited vibrations from a nonlinear spring and one-way clutches can achieve this elimination. Experimental results show that this mechanism allows power transmission at various velocity ratios and rotational speeds regardless of the geometry, and high-speed rotation increases the transfer frequency, resulting in an increase in the transmission power, similar to an electric switching converter. Because of the low inertia and small size, this result suggests that the proposed mechanism is suitable for a high-speed motor.

The transmission design suitable for high speed suggested in this study is expected to affect the development of the electric mobility drive unit in the future without gears. The proposed design is only one realization, and the limits of the transmission in this principle and the optimal scale (e.g., suitable motor and required spring size and power) are unknown. In future work, we will verify the transmission performance and limitations in comparison with previous methods by analyzing the dynamic equations.

## Acknowledgment

The authors thank Toyota Central R&D Labs., Inc., Masaki Ebina, for providing technical support for this research project. We would like to thank Editage (www.editage.com) for English language editing.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

• i =

circuit current

•
• k =

spring coefficient

•
• t =

time

•
• A =

amplitude of vibration

•
• I =

inertia

•
• N =

number of vibrations in one rotation

•
• R =

•
• T =

torque

•
• V =

circuit voltage

•
• γ =

velocity ratio of output to input

•
• θ =

angle

•
• $θ˙$ =

rotational velocity

•
• $θ¨$ =

rotational acceleration

•
• ψ =

phase difference of vibration

•
• $∙˙$ =

first-order time derivative operation

•
• $∙¨$ =

second-order time derivative operation

•
• $∙i$ =

•
• $∙m$ =

•
• $∙o$ =

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