 Research Article
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Switching backdrivability of a planetary gear by vibration: design parameter setting and excited force estimation
ROBOMECH Journal volume 10, Article number: 25 (2023)
Abstract
Switching backdrivability according to use status is important to attain energysaving and compliance in coworker robots. Authors have proposed switching backdrivability by exciting gear surface, eliminating the need for sensors while switching. We have confirmed that exciting 2KH planetary gear can switch its backdrivability. To systematically design the switching backdrivable 2KH planetary gear, this study reveals less backdrive condition and the required torque for a vibrator. In the lessbackdrivable condition, the tooth number difference between the fixed and output gears is dominant. Furthermore, the required torque for the vibrator increases as the load of the output shaft and the vibration frequency increase. The experimental results of the lessbackdrivable condition and the required torque for the vibrator are almost entirely consistent with simulation result of the mechanical model
Introduction
Reducers without backdrive are used in industrial robots to reduce energy consumption of the manipulator while maintaining its posture. Alternatively, to manually drive a manipulator during an emergency stop, reducers for coworker robots (e.g., wearable robots and exoskeleton robots) need compliance control. Switching compliance according to the use status is significant to attain energysaving and compliance for robots. General soft robots [1,2,3,4] can easily attain compliant motion but have difficulty in reducing compliance compared with general industrial robots. To switch compliance of general industrial robots, the use of backdrivable reducers with lock mechanisms (e.g., clutch and brake) is significant, but requires complicated control. Alternatively, controlling the backdrivability of nobackdriven reducers with sensors [5] is still complex, resulting in low reliability. Thus, switching backdrivablity without complex control for nobackdriven reducers is required to enhance reliability.
Because backdrivability depends on the friction caused by the reducer gears, our studies focused on reducing friction by vibration. Researchers have developed devices which reduced friction by vibration [6,7,8]; however, these studies do not focus on backdrivability of reducers. Our research group has proposed switching backdrivability by exciting the gear surface [9], eliminating the need for sensors. Apart from the clutch mechanism, the proposed method does not shut down the power transmission path, reducing the risks of secondary disaster. Takayama et al. examined the switching backdrivability in worm gears [9,10,11].
However, the skew output shaft of a worm gear limits its application, and the vibrator slight actuation to the output shaft could cause secondary disaster. To apply these methods to general robots more safely, the types of applicable reducers should be increased. To gain a reduction ratio of –1/100 other than the worm gear, planetary gears are generally used for industrial robots. Thus, this study focuses on planetary gears for lessbackdriven reducers for switching backdrivability. There exists a lesslikelybackdriven planetary gear using spur gear [12]. In addition, we had confirmed that exciting 2KH planetary gear [13, 14] can switch its backdrivability [15, 16]. However, because the design parameters of the proposed system are determined by trial and error, the planetary gear switching backdrivability phenomenon is not obvious.
To systematically design the switching backdrivable 2KH planetary gear, this study reveals less backdrive condition and the required torque for a vibrator. We first discuss the lesslikelybackdriven structure of the 2KH planetary gear [13] and then estimates the excitation force for switching backdrivability to select the vibrator. The dynamic model enables calculation of the excitation force for switching backdrivability. This study does not focus on reducing the vibrator’s energy consumption because it is used only in emergencies.
The remaining paper is organized as follows: “Modeling of the vibrating planetary gear” section discusses the mechanical analysis of the lessbackdriven structure and the excitation force; “Simulation and Experiments” section verifies the effectiveness of the proposed mechanical model; “Discussion” section describes the limitations of the proposed method and “Conclusion” section concludes this study.
Modeling of the vibrating planetary gear
Figure 1 shows the concept of the switching system for a planetary gear. The carrier is excited in the axial direction to reduce friction between gear surfaces. This study uses standard gears and coulomb friction model. Let \(G_i\) be the ith gear as shown in Fig. 1, and \(Z_i\) be the tooth number of gear \(G_i\). The reduction ratio \(\rho\) is given by
where the sign of \(\rho\) denotes the rotational direction: if the sign is positive, the rotational direction of the motor and output side is the same. Figure 2 shows a front view of the proposed planetary gear and an expanded view of the tooth meshing of the proposed planetary gear. The spur gear parameters include the following:
\(\alpha\): pressure angle (\(=20^{\circ }\))
n: tooth number difference between \(G_1\) and \(G_4\)(\(n=Z_1  Z_4\))
\(F_{ijn}\): normal force to to \(G_i\) and \(G_j\) surface
\(F_{ijt}\): tangential force to \(G_i\) and \(G_j\) surface
\(F_{\text{rot}}\): force acting on \(G_2\) and \(G_3\) axis (see Fig. 2)
\(m_{ij}\): module of \(G_i\) and \(G_j\)
\(R_i\): radius of the pitch circle of \(G_i\)
\(\ell\): difference between \(R_1\) and \(R_4\)(\(=R_1 R_4\))
\(b_{\text{sign}}\): sign of \(\ell\) (\(=\text{sgn} ( \ell )\))
\(M_1\): mass of the input shaft and carrier
\(I_1\): inertia moment of the input shaft and carrier
\(M_{23}\): mass of \(G_2\) and \(G_3\)
\(I_{23}\): inertia moment of \(G_2\) and \(G_3\)
\(I_{4}\): inertia moment of the output shaft
L: radius of the carrier
\(\omega _t\): angular velocity of \(G_4\)
\(\omega _n\): angular velocity of vibration
\(\tau _\text{rot}\): torque acting on \(G_4\) axis
\(\varepsilon _{i}\): approach or recess contact ration of \(G_i\) and is calculated as [17]
\(\mu\): coefficient of friction
\(\mu _{ij}\): coefficient of friction between \(G_i\) and \(G_j\) (detail is described in the later section; static coefficient of friction = 0.271, dynamic coefficient of friction = 0.174) Because the material and manufacturing of \(G_1\) to \(G_4\) is identical, this study assumes \(\mu = \mu _{12} = \mu _{34}\).
\(\mu _d\): dynamic coefficient of friction
\(\mu _s\): static coefficient of friction
Lesslikelybackdrive condition of the proposed planetary gear
Although backdrivability depends on real engagement situations, this paper discusses backdrivability tendency based the average efficiency engagement situations. Therefore, this paper defines lesslikelybackdrive condition as nobackdrive tendency based on the average efficiency engagement situations. This section describes the lesslikelybackdrive condition of the proposed planetary. Let \(\Lambda\) be the constant defined by
Because the planetary gear is static in this state, this section uses static analysis. From motion equation analysis as described in “Appendix” section, the proposed mechanism is less likely to backdrive when
Herein, we consider the value of \(\Lambda\). Let \(k'\) be the design parameter of \(Z_2\) defined by
Fig. 3 shows the relationship between \(\Lambda\) and \(Z_1\). \(\Lambda\) increases as \(Z_1\), \(k'\), and \(\frac{1}{n}\) increase. However, compared with \(Z_1\) and \(k'\), n causes more changes in \(\Lambda\), making it the dominant parameter. Hence, to generate lesslikelybackdriven condition in the 2KH planetary gear, the adjustment of n is significant.
Behavior of the output shaft during backdriven state
This section states behavior of the output shaft during backdriven state using the dynamic model. The motion equation of the proposed planetary gear is calculated by
where \(\widehat{I}\) denote the equivalent moment of inertia given by
Detail of the derivation is described in “Appendix” section. Equation (6) indicates that if the career is excited, then \(\frac{d \omega _t}{dt}>0\) and the planetary gear is backdriven because \(\omega _{t} (0) =0\).
Estimation of the excitation force
This section estimates the excitation force using the dynamic model. Let the vibration of amplitude e be given by \(x= e \cos \omega _n t\), and \(F_\text{ex}\) be the excitation force. \(F_\text{ex} \) can be derived, as follows:
where \(\widehat{M}= M_1 +M_{23}\) denotes the total mass of the carrier, and \(\mu _\text{ax}\) denotes the coefficient of friction in the axial direction. Equation (8) shows that \(F_\text{ex} \) increases as either \(\tau _\text{rot}\) or \(\omega _n\) increases. Detail of the derivation is described in “Appendix” section. Let us consider the appropriate vibrator for the device. Piezo elements can output sufficient excitation force for the proposed device but have difficulty in outputting sufficient amplitude compared with the backlash. To gain enough amplitude (\(\ge 0.3\) mm), this study selects an eccentric cam for the vibrator. The required torque for the actuator \(\tau _\text{cam}\) is calculated by
Herein, let us consider the estimation method of the required \(\tau _\text{cam}\). Because negative power consumption does not compensate for the positive power consumption[18], the simulation value of Eq. (9) is expected to be smaller than the experimental value in actuators. Moreover, because the purpose of estimation is to select the actuator, the estimated value would be slightly larger than the experimental value. Therefore, to estimate the approximately required \(\widehat{\tau _\text{cam}}\), this study uses the average \( \tau _\text{cam}\) during the acceleration phase \((0 \le t \le \frac{\pi }{2 \omega _n})\). Assuming that \(\mu _\text{ax} \approx \mu\), then \(\widehat{\tau _\text{cam}}\) is calculated by
Simulation and experiments
Experimental setup
Figure 4(a) shows the developed planetary gear whose structure is illustrated in Fig. 4(b). Equation (4) indicates that the size of the device hardly affects the backdrivability. Thus, the size of the device is determined as shown in Fig. 4(a), for ease of assembly and disassembly. The range of \(\rho \) is from 7.84 to 25.5, which is consistent with that of general planetary gear box. L and module m are set as 100 and 1 mm, respectively. Gears are fixed to the shaft via ETP bushes (Miki Pulley: ETPA15B), for ease of assembling and disassembling. A DC motor (MABUCHI RS735, reduction ratio: 8.5:1) is installed to drive the eccentric cam. Quenched stainless shafts (SUS440C) are used in the vibration part to prevent damage to the shafts. Moreover, the gears’ surfaces are lubricated with molybdenum grease to prevent wearing.
The rest of this section is organized as follows: “Switching backdrivability of the developed device” section simply verifies whether backdrivability of the developed device can be switched; “Measurement of \(\mu _s\) and \(\mu _d\)” section experimentally estimates \(\mu _s\) and \(\mu _d\) for the backdrivability condition calculation of Eq. (4); “Backdrivability condition” section confirms the validity of the backdrivability condition calculation of Eq. (4); “Torque constant identification of the vibrator” section identifies the torque constant of the vibrator and “Required vibrator torque” section estimates the excitation force from the vibrator current to verify the validity of Eq. (10) (i.e., whether \(F_\text{ex} \) increases as either \(\tau _\text{rot}\) or \(\omega _n\) increases).
Switching backdrivability of the developed device
To verify whether backdrivability of the developed device can be switched, the input shaft is excited. Figure 5 shows the experimental results for switching backdrivability where \(m_\text{out} =1.0\) kg, \(r_\text{out}\) = 75 mm and (\(Z_1\), \(Z_4\)) = (50, 49). The load remained motionless between 0 to 14.5 s of the novibration state and, fell down after 14.5 s of the vibration state. These results indicate that vibration can switch the backdrivability of the proposed device.
Measurement of \(\mu _s\) and \(\mu _d\)
To calculate backdrivability condition of Eq. (4), \(\mu _s\) and \(\mu _d\) are experimentally estimated. Because \(\eta _{ij} \ge 0.9\), estimation of \(\eta _{ij}\) from a pair of gears requires accurate force measurement, causing difficulty in estimating \(\mu _s\) and \(\mu _d\). Therefore, this study estimates \(\mu\) from total efficiency \(\eta _\text{gross}\) of the device in the forward drive. \(\eta _\text{gross}\) is calculated by Ryokaku’s method[13] as follows:
Assuming \(\mu ^2 \approx 0\) and \(\mu \frac{n}{Z_i} \approx 0\), then Eqs. (19) and (11) can derive \(\mu\) as follows:
where \(h_{ij}\) is defined by
Figure 6(a) shows the experimental setup for \(\mu _s\) measurement. To measure \(\eta _{\text{gross}}\) at the beginning of the forward drive, we gradually increase the load of the input shaft while applying a constant load (29.4 N) to the output pulley. \(\eta _{\text{gross}}\) is calculated by
In the experiments, \(\eta _\text{gross}\) was 0.423, which resulted in \(\mu _s=0.271\) where \(m_\text{out} =3.0\) kg, (\(r_\text{in}\), \(r_\text{out}\)) = (100, 75) mm and (\(Z_1\), \(Z_4\)) = (50, 49).
Figure 6(b) shows the experimental setup for \(\mu _d\) estimation. A motor pulls a wire; the tensile force of the wire is measured by a force gauge (AIKOH, RZ10). \(\eta _\text{gross}\) was 0.533, which resulted in \(\mu _d=0.174\); the other conditions were the same as those of \(\mu _s\) measurement experiments.
Backdrivability condition
We performed backdrivability experiments to verify the backdrivability conditions of Eq. (4). Within \(1 \le n \le 3\), \(Z_1\) and \(Z_4\) are set as shown in Table 1. To approximate \(\xi \) to \(\overline{p_{ij}}\), \(Z_1\) and \(Z_4\) are set as \(3i \pm 1\) (\(i=1, 2\ldots\)). A load is applied to the output pulley at static conditions, and the carrier rotation is confirmed. The load is adjusted so that the maximum \(\tau _\text{rot}\) becomes \(3.68\,\hbox {N} \cdot \hbox {m}\).
Table 1 summarizes the experimental results of the backdrivability. The reducer was less likely to be back driven when \(n=1\), while it was back driven when \(n \ge 2\). In simulation, \(\mu _s \Lambda\) \(<1\) when \(n \ge 2\) and \(\mu _s =0.271\). Thus, the experimental results are consistent with the simulation result of Eq. (4).
Note: The maximum \(\tau _\text{rot}\) is determined depending on tooth face and bending strength.
Torque constant identification of the vibrator
To estimate the torque from the speed and current of the vibrator, torque constant identification of the vibrator is performed. An arm, in contact with a force gauge (AIKOH, RZ20), is attached to the rotational axis of the vibration part. A DC stabilization power unit (Kikusui: PWR401L) changes the input voltage \(V_\text{in}\) and measures the current. This study calculates \(\overline{ \tau _\text{cam}}\) from the input current \(i_\text{in}\), as follows:
The vibrator contains viscous resistance \(\overline{ \tau _\text{loss}}\) [19, 20], identified by the following equation
To derive Eq. (16), we confirm the relationship between \(\overline{ i_\text{loss}}\) and \(\omega _n\) where no load is applied to the vibrator. A DC stabilization power unit (Kikusui: PWR401L) changes \(V_\text{in}\) (1.0\(\)1.9 V), and measures \(i_\text{loss}\). Referring to [20], the following approximate expression was used:
Fig. 7(a) shows the currenttorque relationship, and Fig. 7(b) shows the currentangularvelocity relationship. These results identifies \(C_1\)–\(C_6\) as summarized in Table 2.
Note 1: This identification is significant because individual difference and assembling cause torque constant variation.
Note 2: Because the vibrator did not rotate when \(V_\text{in} <1.0\) V, \(V_\text{in}\) was set as \(\ge 1.0\) V.
Required vibrator torque
To verify the validity of Eq. (10), the excitation force of the vibrator is measured. To estimate the excitation force, we measure the vibrator current. Figure 4(b) shows the experimental setup of the vibrating planetary gear carrier. \(Z_1\) and \(Z_4\) are set as 49 and 50, respectively, and e is designed as 1 \(\hbox {mm}\). \(\widehat{M}\) became 3.82 kg, and \(\widehat{I}\), calculated by CAD Software (SolidWorks 2021), is \(2.79\,\hbox {kgm}^{2}\). A DC stabilization power unit (Kikusui: PWR401L) changes the value of \(V_\text{in}\), and an encoder (CUIDEVICE: AMT102V) measures the value of \(\omega _n\). \(V_\text{in}\) is set as 3.6, 6.2, and 8.8 V so that the approximate frequency of the vibration will become 5, 10, and 15 Hz respectively. \(i_\text{in}\) is measured by a current sensor(Pololu: ACS714), whose signal is converted by an analogdigitalconverter (Elmos: RAI16, sampling rate: 10 kHz). \(\tau _\text{rot}\) varies from 0.368 to \(3.68\,\hbox {N} \cdot \hbox {mm}\), and each load test is performed five times.
Figure 8 shows the measurement results of required current and angular velocity, and Fig. 9 shows the estimated result of \(\overline{\tau _\text{cam}}\). \(i_\text{in}\) is the average value during 0.5 s of the beginning phase of the eccentriccam’s constant rotation. \(i_\text{in}\) data at \(\tau _\text{rot}=0.368\) \(\hbox {N} \cdot \hbox {mm}\), and \(V_\text{in} = 3.6, 6.2\) V does not exist because the output shaft was not back driven. Figure 9 also illustrates the simulation result of \(\widehat{\tau _\text{cam}}\). \(\widehat{\tau _\text{cam}}\) is simulated at an average angular velocity \(\overline{\omega _n}\). The graphs of Fig. 9 indicate that \(\overline{\tau _\text{cam}}\) increases with increasing \(\tau _\text{rot}\) and \(\omega _n\). Though the required torque is smaller than that simulated at \(V_\text{in} =3.6\) V, the experimental results of the required torque are mostly consistent with the simulation results of Eq. (10). Note: The beginning phase of the eccentriccam’s constant rotation phase was determined by the time history of the current and angular velocity.
Discussion
Certainty of nonbackdrivability
In the backdrivability condition experiments, the developed device became lessbackdrivable when the tooth number difference was 1. Thus, the value of \(\mu \Lambda\) are consistent with the backdrivability condition of Eq. (4). In practical condition, the value of \(\mu\) depends on temperature, humidity and wearing of the gear surfaces[21]. Furthermore, because \(\Lambda\) is estimated in the average efficiency engagement, the value of \(\Lambda\) will fluctuate with real engagement situations. Although \(\mu\) and \(\Lambda\) varies, the experimental results indicate that the proposed method can approximately judge the lesslikelybackdrivability of the 2KH planetary gear.
Estimation of the excitation force
The experimental results show that \(\overline{\tau _\text{cam}}\) increases as \(\tau _\text{rot}\) and \(\omega _n\) increase. Although the required torque became smaller than estimated \(V_\text{in} =3.6\) V, the experimental and simulation results are mostly consistent. Because the decrease of estimated value at \(V_\text{in} =3.6\) V was small, the proposed method can estimate the required torque for switching backdrivability of 2KH planetary gear.
The decrease can be attributed to a discrepancy between \(\mu _\text{ax}\) and \(\mu\). Because \(\mu _\text{ax}\) decreases as \(\overline{\omega _n}\) decreases, \(\mu _\text{ax}\) became much smaller than \(\mu\), resulting in \(\overline{\tau _\text{cam}}\) decrease.
The cause of the nonbackdriven output shaft at \(\tau _\text{rot}=0.368\) \(\hbox {N} \cdot \hbox {mm}\), and \(V = 3.6, 6.2\) V can be attributed to the friction at the linear bush. The coefficient of friction at linear bush will be similar to Eq. (39) as described in “Appendix” section. Therefore, the switching backdrivability will likely occur as \(\omega _{n}\) increases.
Output shaft motion after quitting vibration
In the experiments, the backdrive motion continued after quitting the vibration. Because \(\mu\) becomes \(\mu _d\) after quitting the vibration, Eq. (6) can derive \(I \frac{d \omega _t}{dt} > (1  \mu _d \Lambda ) \tau _\text{rot}\). Thus, these results are consistent with the estimation of Eq. (6).
\(\mu _d \Lambda\) is expected to be \(>1\) to stop the backdrive motion after quitting the vibration. Referring to [17], the use of profileshifted gears is effective to increase \(\mu _d \Lambda\).
Miniaturization of the vibrator
This study assumes that the backdrive motion is only performed in emergency cases and is not frequently used. Hence, this study does not focus on reducing the energy consumption of the vibrator. If reducing the expected energy is expected, using a resonance will be effective.
Conclusion
To systematically design switching backdrivable 2KH planetary gear, this study reveals less backdrive condition and the required torque for a vibrator. We then discuss the lessbackdrivable condition and required force for the vibrator. Based on the simulation and experiments, we confirm the following:

In the lessbackdrivable condition, the tooth number difference between the fixed and output gears are dominant. It was lessbackdrivable when the tooth number difference was 1.

The required torque for the vibrator increases as the output shaft load and vibration frequency increases. The experimental results are almost entirely consistent with simulation result of dynamic model.
These results indicate that the dynamic model of 2KH planetary gear can predict the switching backdrivability phenomenon. Profile shifted gears to stop the backdrive motion after quitting the vibration will be considered in our future work.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.
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Acknowledgements
We acknowledge Yuji Fujita and Shogo Yoshida for designing the experimental device prototype. We would like to thank Editage (www.editage.com) for English language editing.
Funding
This work was supported by JSPS KAKENHI Grant Number JP20K14704.
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SN and KY invented the proposed mechanisms. SN, RI and KE conducted the examinations. TT refined the proposed mechanism and improved the quality of principles. All authors read and approved the final manuscript.
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Appendix
Appendix
Derivation of Eq. (4)
We consider the mesh efficiency of the gear pair to discuss the lesslikelybackdrive condition of the proposed planetary gear. Let \(\xi\) denote the distance from the pitch point. Referring to [12], the efficiency between \(G_i\) and \(G_j\) at \(\xi\) is given by
\(\xi\) depends on the rotational angle of \(G_i\), causing difficulty in discussing the backdrivability of the proposed planetary gear.
Herein, let us consider the expected value of \(\xi\). This study focuses on Niemann’s equations given by
Comparing (18) and (19), this study defines expected value \(\overline{p_{ij}}\) of \(\xi \) as follows:
\(F_{ijt}= \mu _{ij} F_{ijn}\) when gears begin to backdrive; thus, the translation motion equation of \(G_2\) and \(G_3\) is given by
\(A_{12}\) and \(A_{34}\) denote
where double signs are in random order. A rotational motion equation of \(G_2\) and \(G_3\) at the center of mass is given by
The motion equation of the output shaft is given by
Because the planetary gear is static in this state, static analysis is significant. If \(\frac{d \omega _t}{dt}=0\), then \(F_{34n}\), \(F_\text{rot}\) and \(F_\text{12n}\) are derived as follows:
If the planetary gear is backdrived without vibration, \(F_\text{rot}\) must satisfy
thus
Let m be \(m = m_{12} = m_{34}\). Assuming \(\mu \ll 1\), \(\mu ^2 \approx 0\) and \(\frac{\mu }{Z_i} \approx 0\), then Eq. (30) become
Alternatively, the proposed mechanism is less likely to backdrive when \(\mu \Lambda >1\).
Preparation for derivation of Eqs. (6) and (8)
To derive Eqs. (6) and (8), this section discusses \(\mu _\text{rad}\) and \(\mu _\text{ax}\), the coefficient of friction in the radial and axial direction, respectively. Figure 10 illustrates the force acting on the gear surface when the carrier is excited. Because friction force acts in the x direction, total friction force \(\mu F_{ijn}\) becomes:
The dynamic friction force is assumed to act in the relative velocity direction of the tooth surface. \(v_\text{rel}\), which denotes the relative velocity in the tangential direction between \(G_i\) and \(G_j\), is calculated by
where \(\omega _i\) denotes the angular velocity of \(G_i\).
Let \(v_\text{rad}\) and \(v_\text{ax}\) be the relative velocity of the radial and axial direction of the tooth surface, respectively. Thus, the relative velocity \(\overline{v_\text{rad}^{p12}}\) in the tangential direction between \(G_1\) and \(G_2\) and \(\overline{v_\text{rad}^{p34}}\) is calculated by
Let us consider the numerical value of \(\overline{p_{ij}}\). Because \(h_{12} \approx h_{34} \approx \frac{n \Lambda }{2 \pi }\) (see “Mechanical analysis for backdrivability” section and Fig. 3 ), this study assumes \(\overline{p_{12}} \approx \overline{p_{34})} \approx \overline{p}\); \(\overline{p}\) is defined by
This study also assumes \(\overline{v_\text{rad}^{p12}} \approx \overline{v_\text{rad}^{p34}}\), and \(\overline{v_\text{rad}}\) is defined by
Let \(\omega _n\) be the angular velocity of vibrator. Because the period of the vibrator T is given by \(T = \frac{2 \pi }{ \omega _n}\), \(\overline{v_\text{ax}}\) is given as follows:
Hence, \(\mu _\text{rad}\) and \(\mu _\text{ax}\) are denoted by
where \(\lambda\) is defined by
Derivation of Eqs. (6) and (8)
This section derive Eqs. (6) and (8) from motion equations. The dynamic equation of \(G_1\) is given by
Assuming \(\mu \frac{I_4}{\widehat{I}} \approx 0\), \(\mu \frac{1}{Z_i} \approx 0\) and \(\mu = \mu _d\), then Eqs. (21), (24), (25) and (42) can derive
Let x be the displacement of the carrier in axial direction, and the dynamic motion equation is given by
Assuming \(\mu \frac{I_4}{\widehat{I}} \approx 0\), \(\mu \frac{1}{Z_i} \approx 0\) and \(\mu = \mu _d\), then Eqs. (21), (24), (25) and (43) can derive
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Noda, S., Igari, R., Endo, K. et al. Switching backdrivability of a planetary gear by vibration: design parameter setting and excited force estimation. Robomech J 10, 25 (2023). https://doi.org/10.1186/s4064802300264w
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DOI: https://doi.org/10.1186/s4064802300264w