Abstract
The multipower coupling transmission system (MPCTS) is an electromechanical coupling system, which typically use two or more induction motors to provide its driving torque. It is used in various industrial applications, such as cutter header driving systems of tunnel boring machinery, and yaw systems of wind turbines. In practical engineering applications, the loads borne by the MPCTS are very complex, including both torque loads and nontorque loads. Its dynamic characteristics are affected by both the gear transmission system and the drive motors. For this study, an electromechanical coupling dynamic model of the MPCTS, coupled with a gear transmission system, induction motor, and nontorque load, was constructed to analyze its dynamic characteristics under a nontorque load condition. The results showed increased offsets from the gear vibration equilibrium position, as well as deterioration in system loadsharing characteristics from increases in nontorque load and rotational speed. However, the influence of the nontorque load was significantly greater than that of the rotational speed.
Highlights
 An electromechanical coupling dynamic model of a multipower coupling transmission system (MPCTS) is established.
 The influence of nontorque load and rotational speed on the dynamic characteristics of MPCTS are analyzed.
 With an increase in nontorque load, the offsets of vibration equilibrium positions of the central members almost increased linearly, the loadsharing characteristics of gear pairs deteriorated further.
 With an increase in rotational speed, the offsets of the vibration equilibrium position of the central members exhibit a paraboliclike trend. The loadsharing characteristics of gear pairs exhibit a very slow deterioration trend but accelerates further when the speed exceeds 1200 rpm.
 The influence of the nontorque load on the vibration equilibrium position and loadsharing characteristics is significantly greater than that of increased rotational speed.
1. Introduction
The multipower coupling transmission system (MPCTS) is an electromechanical coupling system, which typically uses two or more induction motors to provide its driving torque. It is used in various industrial applications, such as cutter header driving systems of tunnel boring machinery, and yaw systems of wind turbines, because of its high dynamic performance, robustness, low cost, reliability, and efficiency [1, 2]. In practical engineering applications, MPCTS systems are used to provide high driving torque. However, the effects of loading from both the torque load and the nontorque, axial and radial loads, can be complex. During MPCTS operation, the nontorque load can lead to unequal loading between gears, and an increase in the dynamic meshing force among gear pairs, which negatively affect the dynamic characteristics and loadsharing properties of the system. This indicates that nontorque load is a major contributing factor in the safety and reliability of MPCTS operation, because it contributes to premature gear transmission system failures, and internal permanent damage [3]. Therefore, it is necessary to investigate the effects of nontorque load on MPCTS systems, in order to provide better theoretical guidance for system design. Researchers have recently begun to pay attention to the influence of nontorque loads on the dynamic characteristics of gear transmission systems. Tan et al. created a dynamic model of a wind turbine to investigate the effects of these loads on the performance of its drivetrain, bearing in mind the effects of nontorque loading caused by platform motion. The results suggest that the nontorque loads not only introduce additional excitation frequencies, but also increase the vibration amplitude of the drivetrain [4]. Likewise, Li et al. constructed a coupling model of a wind turbine drivetrain to analyze its dynamic characteristics under nontorque loads, caused by blade gravity, wind shear, tower shadow, and yawed inflow. Those results indicate that nontorque loads can lead to unequal planet load sharing, which are commonly carried by carrier 1 bearings [5]. Zhao et al. created a dynamic model of a helical gear system, to investigate the dynamic behaviors influenced by gear geometric eccentricity. It was found that there are sideband frequencies around the meshing frequency of gear pairs with geometric eccentricities [6]. Cao et al. built a dynamic model of a planetary gear set with four planets to analyze dynamic performance under nontorque loads caused by the gear eccentricities. This study points out that the nontorque loads caused by gear eccentricity error not only affect the dynamic response of the planetary gear set, but also deteriorate the loadsharing performance of the planetary gear set [7]. Zhai et al. studied the dynamic characteristics of one stage helical planetary gear system, under the influence of nontorque loading caused by errors in carrier assembly. The results show that those errors tend to increase the dynamic meshing forces of the gear pairs [8]. Park et al. developed two analysis models of a wind turbine gearbox, consisting of two different load configurations: one with both torque and nontorque loading, and the another with torque only loading. After investigating the influence of wind fluctuation on the load distribution of gear tooth flanks, along with the load sharing of planets, it was concluded that the influence of nontorque forces are very important to consider, to accurately determine the design load for a wind turbine gearbox [9]. Subsequently, Park et al. built a 3D analysis model of a threepoint suspension gearbox for a 2MW class wind turbine, to study the effects of axial force, radial force, and bending moment, on the load sharing of planet gears. The results show that the load sharing of planet gears is mainly affected by the nontorque factors of moment and radial force [10]. Choudhury and Tandon created a theoretical analysis model of a rotor bearing system to study its dynamic response under radial load conditions. They concluded that vibration amplitudes in both healthy and defective bearings increase with increasing speed but that those amplitudes are greater in defective bearings owing to radial load [11].
Qiu et al. developed a rotationaltranslationalaxial dynamic model of a planetary gear transmission system, to investigate the effects of gravity, ring support stiffness, and bedplate tilt angle on the loadsharing characteristics of a planetary gear transmission system [12]. Zhu et al. created a dynamic model of a multifloating gear system using the lumped parameter method, and analyzed the effects of a flexible pin on gear natural frequency and dynamic meshing force [13]. Wei et al. proposed a dynamic modeling method for a multistage planetary gear system by combining the lumped parameter and finite element methods (FEM). The dynamic response calculated by the proposed method was compared with the individual results calculated the individual methods. The results show that the accuracy of the proposed combination method is higher than that of the lumped parameter method, and lower than that of FEM, although the calculation efficiency is higher than that of FEM [14]. Frabco et al. developed a dynamic model of a planetary gearbox by combining the lumped parameter method and FEM, yielding very similar modal property results from simulation versus experimentation [15]. Zeng et al. built a nonlinear dynamic model of a gearshafthousing system by using FEM, to study the influence of the meshing and bearing support stiffness on vibration response. It was concluded that if the bearing support and meshing stiffness increase, the vibration acceleration and velocity of the housing also increase. However, the influence from the bearing stiffness is greater than that of the meshing stiffness [16]. Wang et al. created a coupled dynamic model of a spur, gearshaftbearing system, to investigate the effects of the shaft flexibility on bearing response and radiated noise [17].
In the aforementioned research studies, the focus is mainly on either dynamic characteristic of a system affected by nontorque loads, or a dynamic modeling method which only considers the characteristics of the gear transmission system itself. Subsequently, the effects of the timevarying excitations caused by the induction motor are ignored, although the mechanical and electric assemblies of the MPCTS cannot be divided into two noncorrelated parts. Since the vibration caused by the timevarying meshing stiffness of the gear transmission system can be transmitted to the induction motor through the coupling, and the vibration caused by the speed or torque fluctuations of the motor can to be transmitted to the gear transmission system through the coupling. Therefore, in this study, an electromechanical coupling dynamic model of MPCTS with a gear transmission and induction motor, was used to analyze dynamic characteristic under nontorque loads. Section 2 presents the development of the MPCTS model used, while Section 3 details its resulting dynamic characteristics under a nontorque load condition. Section 4 presents the conclusions drawn on the basis of the analysis of the results.
2. Electromechanical coupling dynamic model for MPCTS
The MPCTS, as shown in Fig. 1, is composed of a singlestage planetary gear system and a parallel shaft gear system. In the parallel shaft gear system, the three pinions are driven separately by three induction motors, and the wheel is driven by the three pinions, which drive the wheel. The sun shaft is connected to the wheel via a spline, and the power from the wheel is transmitted to the singlestage planetary gear system through the sun shaft.
Fig. 1Threedimensionalmodel of MPCTS
2.1. Dynamic model of Timoshenko beam
The dynamic model of the rotating shafts and gears was developed using the Timoshenko beam theory proposed by Nelson [18], and is shown in Fig. 2. The subscripts $pii$ and $piiw$ represent the $i$th pinion and ith gear pair formed by the $i$th pinion and wheel, respectively. The subscript $w$ represents the wheel. The mathematical model of the twonode Timoshenko beam can be written as:
where ${M}_{j,j+1}$ denotes the mass matrix of the Timoshenko beam. ${C}_{j,j+1}$ and ${K}_{j,j+1}$ denote the damping matrix and stiffness matrix of the Timoshenko beam, and their detailed matrix forms are shown in Ref. [19]. ${X}_{j,j+1}$ and ${F}_{j,j+1}$ denote the displacement and exciting force column vectors of the Timoshenko beam.
Fig. 2Timsoshenko beam
2.2. Dynamic model of parallel shaft gear system
In the parallel shaft gear system, all the gears and shafts are equivalent to the Timoshenko beam. The basic parameters of the parallel shaft gear system are listed in Table 1. The dynamic model of the parallel shaft gear system can be obtained using the node finite element method, which is formed from the bearing, meshing gear pair, and shaft, as shown in Fig. 3(a). Nodes $i$ and $j$ are assumed to be the meshing nodes of the pinionwheel gear pair. Nodes $i2$, $i+2$, and $i+3$ are the bearing supporting points of the pinion and wheel, respectively. As shown in Eq. (2), the dynamic model of the supporting bearing can then be expressed in matrix form, using $i2$ as the bearing supporting point in this example:
where ${C}_{i2}$ and ${K}_{i2}$ denote the damping matrix and stiffness matrix of the bearing, respectively. ${X}_{i2}$ denotes the displacement vector of bearing supporting point $i2$. The detailed expressions of the matrices can be found in Ref. [20].
Fig. 3a) Finite element node diagram of parallel shaft gear system; b) meshing pair model of parallel shaft gear system
a)
b)
The meshing pair model of the parallel shaft gear system is shown in Fig. 3 (b), where the gear meshing pair is modeled as a stiffnessdamping spring system, with the spring acting along the line of action between the two gears. The deformation along the line of action can be written as:
${y}_{w}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha {\varphi}_{piiw}\right)\mathrm{c}\mathrm{o}\mathrm{s}\beta +{r}_{pii}{\theta}_{pii}\mathrm{c}\mathrm{o}\mathrm{s}\beta {r}_{w}{\theta}_{w}\mathrm{c}\mathrm{o}\mathrm{s}\beta +{z}_{pii}\mathrm{s}\mathrm{i}\mathrm{n}\beta {z}_{w}\mathrm{s}\mathrm{i}\mathrm{n}\beta {e}_{piiw},$
where ${x}_{j}$, ${y}_{j}$ and ${z}_{j}\mathrm{}(j=pii,w)$ denote the displacement of the pinion and wheel along the $x$, $y$, and $z$axes, respectively. ${\theta}_{j}$ and ${r}_{j}\mathrm{}(j=pii,w)$ denote the angle displacement around the $z$axis and the base circle radius of the pinion and wheel, respectively. $\alpha $ is the pressure angle, and ${\varphi}_{piiw}$ is the position angle of the pinion. $\beta $ denotes the helix angle of the gear. ${e}_{piiw}$ is the transmission error, which can be expressed as:
where ${\widehat{e}}_{piiw}$ and ${\stackrel{}{e}}_{piiw}$ are the $l$th harmonic term amplitude and mean transmission error, respectively, and ${\gamma}_{i}$ is the $i$th pinion phase angle.
Hence, the differential equation of the pinionwheel pair can be deduced as:
where ${m}_{j}$ and ${J}_{j}\mathrm{}(j=pii,w)$ denote the mass and inertia of the pinion and wheel, respectively. ${T}_{mi}$ and ${T}_{l}$ denote the input torque and load torque, while ${c}_{piiw}$ and ${k}_{piiw}$ denote the meshing damping and stiffness of the pinionwheel pair, respectively. They can be expressed as Eq. (6):
where ${\widehat{k}}_{piiw}$ and ${\stackrel{}{k}}_{piiw}$ are the $l$th harmonic term amplitude and mean mesh stiffness, respectively. $\zeta $ is the meshing damping ratio, ${m}_{eq,i}={I}_{xi}/{r}_{i}^{2}(i=pii,w)$ represents the equivalent quality. Therefore, the matrix form of the pinionwheel pair dynamic model can be written as:
where, ${V}_{piiw}$ is the meshing vector of the pinionwheel pair, which can be written as:
where ${M}_{piiw}$, ${C}_{piiw}$, and ${K}_{piiw}$ denote the mass matrix, meshing damping matrix, and meshing stiffness matrix of the pinionwheel pair, respectively. ${X}_{piiw}$ and ${F}_{piiw}$ denote the displacement and exciting force vectors, respectively. In Eq. (8), $\mathrm{\Phi}=\alpha {\varphi}_{piiw}$.
Table 1Basic parameters of parallel shaft gear system
Tooth number  Module/mm  Face width/mm  Pressure angle/°  Supporting stiffness/Nm^{1}  Diameter/mm  
Pinion  19  4  146  20  1e8  D1 = 100; D2 = 110; D3 = 160; D4 = 55; D5 = 50 
Wheel  99  4  146  20  1e8 
2.3. Dynamic model of singlestate planetary gear system
The dynamic model of a singlestage planetary gear system can also be obtained using the node finite element method, consisting of the bearing, sunplanet meshing pair, planetring meshing pair, pin, and carrier. The carrier, ring, planet, sun, and pin are modeled as different crosssection types of Timoshenko beams, and can be divided into different element nodes, as shown in Fig. 4(a). The basic parameters of the planetary stage are listed in Table 2. Herein, the dynamic model of the supporting bearing in a singlestage planetary gear system can be expressed using Eq. (2) as well.
Table 2Basic parameters of singlestage planetary gear system
Tooth number  Module / mm  Face width / mm  Pressure angle /°  Helix angle /°  Supporting stiffness / Nm^{1}  Diameter/mm  
Sun  18  6  146  20  25.5  1e8  D6 =260; D7 = 174.6; D8 = 268; D9 = 320 
Planet  52  6  146  20  25.5  1e9  
Ring  123  6  146  20  25.5  1e10 
The meshing pair model of the singlestage planetary gear system is shown in Fig. 4(b). The subscripts $s$, $r$, $c$, and ${p}_{i}$ represent the sun, ring, carrier, and $i$th planet, respectively. There are two types of meshing pairs existing in the model: an external, sunplanet meshing pair, and an internal, planetring meshing pair. Without loss of generality, the external and internal meshing pairs are also modeled as a stiffnessdamping spring system, with the spring acting along the line of action between the two gears.
For the sunplanet $i$ meshing pair, the deformation along the line of action can be written as:
$+{y}_{pi}\mathrm{c}\mathrm{o}\mathrm{s}\left({\alpha}_{t}\right)\mathrm{c}\mathrm{o}\mathrm{s}\beta {r}_{s}{\theta}_{s}\mathrm{c}\mathrm{o}\mathrm{s}\beta {r}_{pi}{\theta}_{pi}\mathrm{c}\mathrm{o}\mathrm{s}\beta +{z}_{pi}\mathrm{s}\mathrm{i}\mathrm{n}\beta {z}_{s}\mathrm{s}\mathrm{i}\mathrm{n}\beta {e}_{spi},$
where ${x}_{j}$, ${y}_{j}$ and ${z}_{j}\mathrm{}(j=s,pi)$ denote the displacement of the sun and planet $i$ along the $x$, $y$, and $z$axes, respectively. ${\theta}_{j}$ and ${r}_{j}\mathrm{}(j=s,pi)$ denote the angle displacement around the $z$axis and the base circle radius of the sun and planet $i$, respectively. ${\alpha}_{t}$ represents the pressure angle, and ${\varphi}_{n}$ represents the position angle of the planet. ${e}_{spi}$ is the transmission error of the sunplanet $i$ meshing pair, which can be expressed as Eq. (15).
The differential equation of the sunplanet $i$ meshing pair can be deduced as follows:
where ${m}_{j}$ and ${J}_{j}\mathrm{}(j=s,pi)$ denote the mass and inertia of the sun and planet $i$, respectively. ${c}_{spi}$ and ${k}_{spi}$ denote the meshing damping and stiffness of the sunplanet $i$ meshing pair, and can be expressed as Eq. (15). ${T}_{in}$_{}denotes the input torque of the sun. The matrix form of the dynamic model of the sunplanet $i$ meshing pair can be written as:
where ${M}_{spi}$, ${C}_{spi}$, and ${K}_{spi}$ denote the mass matrix, meshing damping matrix, and meshing stiffness matrix of the sunplanet $i$ pair, respectively. ${X}_{spi}$ and ${F}_{spi}$ denote the displacement and excitation force vectors, respectively. Moreover, ${V}_{spi}$ is the meshing vector of the sunplanet $i$ meshing pair, which can be written as:
where $\mathrm{\Psi}={\alpha}_{t}+{\varphi}_{n}$.
Fig. 4a) dynamic model of singlestage planetary gear system; b) meshing pair model of singlestage planetary gear system
a)
b)
For the planet $i$ring meshing pair, the deformation along the line of action can be written as:
${y}_{r}\mathrm{c}\mathrm{o}\mathrm{s}\left({\varphi}_{n}{\alpha}_{t}\right)\mathrm{c}\mathrm{o}\mathrm{s}\beta +{r}_{pi}{\theta}_{pi}\mathrm{c}\mathrm{o}\mathrm{s}\beta {r}_{r}{\theta}_{r}\mathrm{c}\mathrm{o}\mathrm{s}\beta +{z}_{pi}\mathrm{s}\mathrm{i}\mathrm{n}\beta {z}_{r}\mathrm{s}\mathrm{i}\mathrm{n}\beta {e}_{pir},$
where ${x}_{r}$, ${y}_{r}$ and ${z}_{r}$ denote the displacement of the ring along the $x$, $y$, and $z$axes, respectively. ${\theta}_{r}$ and ${r}_{r}$ denote the angle displacement around the $z$axis and the base circle radius of the ring, respectively. ${e}_{pir}$ is the transmission error of the planet $i$ring meshing pair, which can be expressed as Eq. (15).
The differential equation of the planet $i$ring meshing pair can be deduced as follows:
where ${m}_{r}$ and ${J}_{r}$ denote the mass and inertia of the ring, respectively. ${c}_{pir}$ and ${k}_{pir}$ denote the meshing damping and stiffness of the planet $i$ring meshing pair, respectively, and can be expressed as Eq. (15):
${\widehat{e}}_{i}$ (or ${\widehat{k}}_{i}$, $i=spil$ and $pirl$) and ${\stackrel{}{e}}_{j}$ (or ${\stackrel{}{k}}_{j}$, $j=spil$ and $pirl$) represent the $l$th harmonic term amplitude and mean transmission error (or mean mesh stiffness), respectively. ${Z}_{r}$ and ${Z}_{s}$ represent the number of teeth in the ring and sun, respectively. ${\gamma}_{sr}$ represents the phase angle between planet $i$ (or ring) and sun (or planet $i$) meshing pair. ${p}_{bs}$ and ${p}_{bpi}$ are the base pitch of the sun and planet $i$, respectively. $\mathrm{\Gamma}$ represents the phase difference between ${e}_{j}$ and ${k}_{j}$. ${z}_{r}{\varphi}_{n}$ and ${z}_{s}{\psi}_{m}$ are the phase relationships between the meshing pairs. ${m}_{eq,i}={I}_{xi}/{r}_{i}^{2}(i=s,pi,r)$ represents the equivalent quality. The matrix form of the dynamic model of the planet $i$ring meshing pair can be written as follows:
where ${M}_{pir}$, ${C}_{pir}$, and ${K}_{pir}$ denote the mass matrix, meshing damping matrix, and meshing stiffness matrix of the planet $i$ring meshing pair, respectively. ${X}_{pir}$ and ${F}_{pir}$ denote the displacement and excitation force vectors, respectively. Further, ${V}_{pir}$ is the meshing vector of the planet $i$ring pair, which can be expressed as follows:
where $\mathrm{\Theta}={\varphi}_{n}{\alpha}_{t}$.
If it is assumed that the number of planets is $n$ in the planetary gear system (Fig. 5), it can be concluded that the number of pins and connections are also $n$, respectively, according to the structural features of the planetary gear system. The stiffnessdamping spring system is used to represent the coupled relationship between the side plate, pin, and connecting plate; ${c}_{c1}$ and ${k}_{c1}$ represent the coupling damping and stiffness between the side plate and pin, while ${c}_{c2}$ and ${k}_{c2}$ represent the coupling damping and stiffness between the side plate and connecting plate. ${c}_{i}$ and ${k}_{i}$$(i=pin,cp)$ represent the damping and stiffness of the pin and connecting plate, respectively.
The matrix form of the dynamic model of the carrier pin can be expressed as:
where ${M}_{c}$, ${C}_{c}$ and ${K}_{c}$ denote the mass matrix, damping matrix, and stiffness matrix of the carrier pin, respectively. ${X}_{c}$ and ${F}_{c}$ denote the displacement and exciting force vector of the carrier pin, respectively, and can be obtained using Eq. (13) and (14) in Ref. [19].
Fig. 5Dynamic model of carrier pin
2.4. Model of induction motor
Alternating current (AC) induction motors are frequently used in industrial applications, to provide the driving force for mechanical systems. Therefore, a threephase AC induction motor with a rated power of 100 kW was used in this study. The basic parameters of the motor are shown in Table 3. The mathematical model of such a motor in the $d$$q$ coordinate system can be described as follows [21, 22].
Table 3Basic parameters of motor
Resistance / Ω  Rated speed / rpm  Voltage / V  Current / A  Leakage inductance / H  Rotating inertial / kg·m^{2}  Magnetizing inductance / H  
Stator  0.1749  —  1140  59.58  0.0048  —  0.2001 
Rotor  0.1954  1475  —  660  0.0048  10  0.2001 
Eq. (19) shows the voltage and flux linkage equations:
$\left[\begin{array}{l}{\psi}_{sd}\\ {\psi}_{sq}\\ {\psi}_{rd}\\ {\psi}_{rq}\end{array}\right]=\left[\begin{array}{cccc}{L}_{s}& 0& {L}_{m}& 0\\ 0& {L}_{s}& 0& {L}_{m}\\ {L}_{m}& 0& {L}_{r}& 0\\ 0& {L}_{m}& 0& {L}_{r}\end{array}\right]\left[\begin{array}{l}{i}_{sd}\\ {i}_{sq}\\ {i}_{rd}\\ {i}_{rq}\end{array}\right],$
where, ${u}_{i}(i=sd,sq,rd,rq)$ represents the component of the stator/rotor voltage in the $d$/$q$axis. ${i}_{k}$$(k=sd,sq,rd,rq)$ represents the component of the stator/rotor current in the $d$/$q$axis. ${R}_{i}$$(i=s,r)$ denotes the resistance of the stator/rotor. ${L}_{j}(j=s,r,m)$ denotes the stator/rotor selfinductance, and the mutual inductance between the stator and rotor, respectively. ${\omega}_{i}$$(i=dqs,dqr)$ denotes the magnetic speed of the stator/rotor. Lastly, $q$ denotes the differential operator, while ${\psi}_{i}(i=sd,sq,rd,rq)$ denotes the component of the stator/rotor flux linkage in the $d$/$q$axis.
Eq. (20) shows the equations for both torque and motion:
where ${T}_{e}$ denotes the electromagnetic torque of the motor, ${n}_{p}$ denotes the number of winding pole pairs, $\sigma $ denotes the coefficient of leakage inductance, and $\theta $ denotes the included angle between the stator flux leakage and rotor flux leakage. Lastly, ${T}_{L}$ denotes the load torque, while $J$ denotes the inertia of the rotor.
2.5. Electromechanical coupling dynamic model
Using the node finite element method and the structural features of MPCTS, the system can be divided into 88 nodes with 352 degrees of freedom. The entire stiffness matrix of the MPCTS can be obtained using the coupled method described in Ref. [20], and the distribution of nodes illustrated in Fig. 6; which shows the stiffness of the Timoshenko beam supporting the bearing and gear meshing.
Fig. 6Dynamic model of MPCTS
The MPCTS dynamic model can be developed after obtaining the mass matrix, stiffness matrix, damping matrix, and exciting force vector of each subelement. The components of each subelement matrix can be superimposed on the corresponding position of the total matrix [20], based upon the total number of system nodes, and the relationship between each node. In addition, the input torques ${T}_{mi}(i=1,2,3)$ act on nodes 1, 11, and 21, respectively, as the driving torques of the system. The radial loads ${F}_{X}$, ${F}_{Y}$, and the torque load ${T}_{L}$ act on node 87. The masterslave control method was used to control the three induction motors, and the torque signal was selected as the control signal between the induction motors. Finally, the electromechanical coupling dynamic model of the MPCTS was be deduced as:
where $M$, $C$, and $K$ denote the mass, damping, and stiffness matrices of the MPCTS, respectively. $X$ and $F$ denote the displacement and exciting force column vectors of the MPCTS, respectively.
3. Result and discussion
Two load conditions were selected to investigate the influence of nontorque load on the dynamic characteristics of MPCTS: (a) the torque load ${T}_{L}$, and (b) the nontorque, radial loads of ${F}_{X}$ and ${F}_{Y}$ in addition to the torque load ${T}_{L}$, which is shown in Fig. 7.
The motion trajectories of the pinions and the wheel are shown in Fig. 8. Geometrically, the motion trajectories of the pinions are linear, as shown in Fig. 8(a) and (b). Marked changes were observed in the angle ($\gamma $) between trajectories and the length ($L$) of the trajectories. The difference between the angles is small under the torque load condition, where ${\gamma}_{1}=$ 120.0020°, ${\gamma}_{2}=$ 120.0007°, and ${\gamma}_{3}=$ 119.9973°, and the maximum angle difference is between ${\gamma}_{1}$ and ${\gamma}_{3}$, namely, ${\gamma}_{1}{\gamma}_{3}=$ 0.0047°. A similar phenomenon also occurred in the length of trajectory under torque load condition, where ${L}_{1}=$ 729.17 μm, ${L}_{2}=$ 732.25 μm and ${L}_{3}=$ 749.50 μm. The maximum length difference was between ${L}_{1}$ and ${L}_{3}$, namely, ${L}_{1}{L}_{3}=$ 20.33 μm, as shown in Fig. 8(a). With regards to the effects of nontorque load, both the angle and the length vary significantly, but the length varied more, as shown in Fig. 8(b). Owing to the radial loads ${F}_{X}$ and ${F}_{Y}\text{,}$ the difference between the angles was further increased, where ${\gamma}_{1}=$ 120.7559°, ${\gamma}_{2}=$ 120.2356°, and ${\gamma}_{3}=$ 119.0085°. Notably, the maximum difference was still betwee n ${\gamma}_{1}$ and ${\gamma}_{3}$, which is ${\gamma}_{1}{\gamma}_{3}=$ 1.7474°. Meanwhile, the lengths of trajectories increased to ${L}_{1}=$1000 μm, ${L}_{2}=$ 1200 μm and ${L}_{3}=$ 791.44 μm, and increased by 270.89 μm, 467.75 μm and 41.94 μm, respectively, as compared to the torque load effects. However, some difference occurred when the maximum length difference was between ${L}_{2}$ and ${L}_{3}\text{;}$ reaching ${L}_{2}{L}_{3}=$ 408.56 μm.
Fig. 7Load: a) torque load; b) nontorque load
a)
b)
As shown in Fig. 8(c) and (d), the shape of the trajectory was close to a circle under either the torque load condition or the nontorque load condition. When nontorque load condition was considered, the vibration equilibrium positions of the wheel in $X$ and $Y$ directions went from –0.83 μm to 1.22 μm, and from 3.80 μm to 6.02 μm – an increase of 2.47 and 0.58 times, respectively. Moreover, the loadsharing characteristics of the pinionwheel gear pairs also changed significantly because of the change in the vibration equilibrium position, as shown in Fig. 9.
Fig. 8Motion trajectories: a) and b) motion trajectories of the pinions under torque load and nontorque load (the blue solid line, red dotted line and green dashdotted line are the motion trajectories of pinion 1, pinion 2 and pinion 3, respectively); c) and d) motion trajectories of the wheel under torque load and nontorque load, respectively
a)
b)
c)
d)
In this study, the maximum loadsharing coefficient (LSC) of the gear pair was selected to represent the LSC of the corresponding gear pair [23]. The LSC of the pinionwheel gear pairs was 1.072 under torque load conditions, and 1.179 under nontorque load conditions – an increase of 9.98 %. In summary, nontorque load has a significant influence on both the vibration equilibrium positions of the pinions and the wheel, and the loadsharing characteristics of the pinionwheel gear pairs.
The motion trajectory of the sun was analogous to a circle, as shown in Fig. 10(a) and (b). Significant changes occurred in the vibration equilibrium position of the sun. When the nontorque load wass considered, the vibration equilibrium positions of it in both $X$ and $Y$ directions were markedly offset, as compared with the torque load condition. They changed from –0.04 μm to 3.69 μm, and from 0.25 μm to 14.74 μm – increases of 93.25 and 57.96 times, respectively. In contrast, the motion trajectories of the planets were analogous to an ellipse under torque load condition, and their offsets of vibration equilibrium positions in $X$ and $Y$ directions were about 0.20 μm and –10.52 μm, respectively, as shown in Fig. 10(c). In Fig. 10(d), there was an obvious difference, as the shapes of the motion trajectories of the planets were analogous to a rectangle under nontorque load condition, and their vibration equilibrium positions in the $X$ direction were offset in two different directions. For planet 3, the vibration equilibrium position was shifted in the negative direction of the $X$axis, with an offset of approximately $\mathrm{\Delta}{X}_{3}=$ 24.46 μm; increasing by 123.42 times. However, the vibration equilibrium positions for the other two planets were shifted in the positive direction of the $X$axis, and their offsets were $\mathrm{\Delta}{X}_{1}=$ 20.33 μm and $\mathrm{\Delta}{X}_{2}=$ 11.25 μm – increasing by 63.78 and 56.78 times, respectively. In addition, the vibration equilibrium positions of the planets in the $Y$ direction were shifted in the negative direction of the $Y$axis and change from approximately –10.52 μm to –163.66 μm – increasing by 14.56 times.
Fig. 9LSC: a) LSC of pinionwheel gear pairs under torque load; b) LSC of pinionwheel gear pairs under nontorque
a)
b)
Fig. 10Motion trajectories: a) and b) motion trajectories of the sun under torque load and nontorque load, respectively; c) and d) motion trajectories of the planets under torque load and nontorque load (the blue solid line, red dotted line and green dashdotted line are the motion trajectories of planet 1, planet 2 and planet 3, respectively)
a)
b)
c)
d)
Fig. 11Motion trajectories: a) and b) motion trajectory of the carrier under torque load and nontorque load, respectively; c) and d) motion trajectory of the ring under torque load and nontorque load, respectively
a)
b)
c)
d)
When the nontorque load was considered, the same held true in that the motion trajectories of the carrier and ring, and their vibration equilibrium positions, were hardly affected by it, as shown in Fig. 11. In addition, their motion trajectories were also analogous to a circle. Given that the ring is a fixed part, its restraint stiffness is set to be relatively large. As a result, there was an obvious difference in the offsets of the vibration equilibrium positions of the carrier in the $X$ and $Y$ directions, as they were markedly larger than those of the ring. Compared with the torque load condition, the center of motion trajectory of the carrier changes from (0.009, 0.038) to (12.79, 20.52) when the nontorque load is considered. In other words, the offsets of their vibration equilibrium positions in the $X$ and $Y$ directions are increased by 1420 and 539 times, respectively, as shown in Fig. 11(a) and (b). However, compared with Fig. 10(c) and (d), the offsets of the vibration equilibrium positions of the ring in the $X$ and $Y$ directions are significantly reduced, and its center of motion trajectory goes from (–0.009, –0.008) to (–0.44, 0.36); increasing by 47.89 and 46 times.
The loadsharing coefficients of the sunplanet gear pairs and planetring gear pairs are shown in Fig. 12. SP$i$ ($i=$ 1, 2, and 3) in Fig. 12(a) and (b) represent the external meshing pair formed by the sun and the $i$th planet, and P$i$R ($i=$ 1, 2, and 3) in Fig. 12 (c) and (d) denote the internal meshing pair formed by the $i$th planet and ring. The vibration equilibrium positions of the sun, planets, ring, and carrier deviated significantly under the action of the radial loads ${F}_{X}$ and ${F}_{Y}$ resulting in a deterioration of the loadsharing characteristics of the sunplanet gear pairs and planetring gearpairs. The loadsharing coefficients of the sunplanet gear pair and the planetring gear pair increased from 1.087 to 1.237, and from 1.062 to 1.189, increased by 13.80 % and 11.96 %, respectively. Obviously, the influence of the nontorque load on the loadsharing characteristics of the sunplanet gear pairs is greater than that of the planetring gear pairs. This is mainly caused by the offset of the vibration equilibrium position of the sun being larger than that of the ring.
Fig. 12Loadsharing coefficient (LSC): a) and b) LSC of sunplanet gear pairs under torque load and nontorque load, respectively; c) and d) LSC of planetring gear pairs under torque load and nontorque load, respectively
a)
b)
c)
d)
The variations in the vibration equilibrium position of each central member are listed in Table 4. Obviously, in addition to the ring, the offsets of the other central members in the $Y$ direction are significantly greater than those in the $X$ direction, when the nontorque load is considered. This is because the radial load acting in the $X$ direction is significantly less than that in the $Y$ direction, as shown in Fig. 7. However, it is noteworthy to mention that the increased multiple of the offset of the vibration equilibrium position in the $X$ direction is obviously higher than that of the $Y$ direction. Moreover, the increased proportion of the offset of the vibration equilibrium position of the carrier is the largest, whether in the $X$ or $Y$ direction, closely followed by the sun and the wheel. This is because the carrier is closest to the point of the load application, followed by the sun and wheel. Similar variations can be seen in the LSC of each gear pair, as listed in Table 5. This indicates that the closer the central members are to the point of the load application, the more their vibration equilibrium positions and loadsharing characteristics are affected by it.
The offsets of the vibration equilibrium position of the central members at different nontorque loads are shown in Fig. 13. For convenience, the relationship between the resultant force ${F}_{R}$ and the offset, shown in the figure, where ${F}_{R}=\sqrt{{F}_{X}^{2}+{F}_{Y}^{2}}$. It is clear that the offsets of the vibration equilibrium position increase almost linearly with the increase in the resultant force ${F}_{R}$. Whether in the $X$ or $Y$ direction, the vibration equilibrium positions of the wheel, sun, and carrier are shifted in the positive direction of either the $X$axis or $Y$axis, with an increase in the resultant force ${F}_{R}$, as shown in Fig. 13(a), (b), and (c). However, with the increase in the resultant force ${F}_{R}$, the vibration equilibrium position of the ring in the $X$ direction is shifted in the negative direction of the $X$axis, but the vibration equilibrium position of the ring in the $Y$ direction is shifted in the positive direction of the $Y$axis, as shown in Fig. 13(d). As a result, the loadsharing characteristics of all gear pairs deteriorate further with the increase in the resultant force ${F}_{R}$, as shown in Fig. 14.
Table 4Variation of vibration equilibrium position of each central member
Offset of vibration equilibrium position in $X$ direction / μm  Offset of vibration equilibrium position in $Y$ direction / μm  
Central member  Central member  
Load condition  Wheel  Sun  Carrier  Ring  Wheel  Sun  Carrier  Ring 
Torque load  –0.83  –0.04  0.009  –0.009  3.80  0.25  0.038  –0.008 
Nontorque load  1.22  3.69  12.79  –0.44  6.02  14.74  20.52  0.36 
Increased multiple  2.47  93.25  1420  47.89  0.58  57.96  539  46 
Table 5Variation of loadsharing coefficient of each gear pair
Loadsharing coefficient  
Load condition  Pinionwheel gear pairs  Sunplanet gear pairs  Planetring gear pairs 
Torque load  1.072  1.087  1.062 
Nontorque load  1.179  1.237  1.189 
Increased proportion  9.98 %  13.80 %  11.96 % 
Fig. 13Offset of vibration equilibrium position at different nontorque loads: a) wheel; b) sun; c) carrier; and d) ring
a)
b)
c)
d)
Fig. 14Root mean square (RMS) of LSC at different nontorque loads: a) pinionwheel gear pairs; b) sunplanet gear pairs; c) planetring gear pairs
a)
b)
c)
Fig. 15Offset of vibration equilibrium position at different speeds: a) wheel; b) sun; c) carrier; d) ring
a)
b)
c)
d)
Fig. 16RMS of LSC at different speeds: a) pinionwheel gear pairs; b) sunplanet gear pairs; c) planetring gear pairs
a)
b)
c)
The offsets of the vibration equilibrium position of the central members at different speeds are shown in Fig. 15. When compared with Fig. 13, similarities can be observed. However, with the increase in the rotational speed, the offsets of the vibration equilibrium position of all the central members exhibit an approximate parabolic trend. When the rotational speed changes reaches the range of 8001200 rpm, the offsets of the vibration equilibrium position of all central members increase slowly. The loadsharing characteristics of all gear pairs exhibit a very slow deterioration trend as well. Once the rotational speed exceeds 1200 rpm, the offsets of the vibration equilibrium position of the central members increase rapidly with the increase in speed, and the deterioration trend accelerates further, as shown in Fig. 16. In addition, the comparisons between Fig. 13 with Fig. 15, and Fig. 14 with Fig. 16, shows that the influence of nontorque load on the vibration equilibrium position and loadsharing characteristics is significantly greater than that of the rotational speed.
4. Conclusions
In this study, an electromechanical coupling dynamic model of the MPCTS, based upon the node finite element method, was established considering motor characteristics. The dynamic characteristics of the MPCTS are investigated under nontorque load conditions, and include a detailed study of the resulting motion trajectories and loadsharing characteristics. Based on the simulations and discussions, the following conclusions were drawn:
1) The motion trajectories of the pinions are linear. The angles and the lengths between the trajectories increase in different degrees when the nontorque load is considered. Here, the maximum angle deference and the maximum length deference increased from 0.0047° to 1.7474°, and from 20.33 μm to 408.56 μm, respectively. The motion trajectories of the wheel, sun, carrier, and ring are analogous to a circular shape. The motion trajectory of the planet is almost elliptical in shape, under the torque load condition, but is analogous to more rectangular under nontorque load condition. In addition, when the nontorque load is considered, the offsets of the vibration equilibrium position of the central members increase significantly, and the offsets in the $Y$ direction are noticeably greater than those in the $X$ direction.
2) For the central members, under the nontorque load condition, the increased multiples of their offsets of vibration equilibrium positions in the $X$ direction are clearly higher than those in the $Y$ direction. The closer the central members are to the applied load, the greater the increase in offset of the vibration equilibrium position of the carrier, closely followed by the sun and wheel. Similar characteristics were also observed in the loadsharing coefficient of each gear pair.
3) With an increase in the nontorque load, the offsets of the vibration equilibrium positions of the central members almost increased linearly, and the loadsharing characteristics of all gear pairs deteriorated further. However, with an increase in rotational speed, the offsets of the vibration equilibrium position of the central members exhibit a paraboliclike trend. When rotational speed is within the 8001200 rpm range, the offsets increase slowly. However, once the rotational speed exceeds 1200 rpm, the offsets increase rapidly. Similarly, when the rotational speed is within the 8001200 rpm range, the loadsharing characteristics of all gear pairs exhibit a very slow deterioration trend but accelerates further when the speed exceeds 1200 rpm. Moreover, the influence of the nontorque load on the vibration equilibrium position and loadsharing characteristics is significantly greater than that of increased rotational speed.
Acknowledgements
This research is supported by the National Natural Science Foundation of China (51905060/52005067/51905062 /51905064/51805060/51875068); China Postdoctoral Science Foundation (2021M700619); National Key Research and Development Project of China (2020YFB2010100); Natural Science Foundation Project of Chongqing Science and Technology Commission (cstc2020jcyjmsxmX0346/cstc2019jcyjmsxmX0733); Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN201901107/KJQN201901112/KJQN201901115). We also want to thank the support from the Scientific Research Foundation of Chongqing University of Technology, China.
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