Effect of Floating Support Parameters on the Load-Sharing Performance of EDPGS Based on Mathematical Statistical Methods

: The encased differential planetary gear system (EDPGS) allows power to be distributed among multiple output paths, enhancing efficiency and reducing weight. Uniform load distribution ensures stable system operation and prolongs service life. However, stochastic manufacturing errors leading to uneven load distribution pose challenges in engineering practice. To investigate the impact of floating support parameters on the load-sharing performance within an acceptable tolerance band, a dynamic model of the EDPGS considering time-varying meshing stiffness and random errors is established using the Monte Carlo method. This study employs the orthogonal experimental design method to analyze the effects of floating support stiffness and clearance on the load-sharing characteristics. The findings indicate that a larger sample size leads to a probability distribution of load-sharing coefficients closer to the Gaussian distribution, with minimal influence on the expectation and variance. Furthermore, this study highlights the significant influence of floating structure parameters on load-sharing characteristics in the encased stage systems compared to the differential stage. Decreasing floating support stiffness or increasing floating clearance proves beneficial in enhancing the load-sharing performance of the system.


Introduction
EDPGSs exhibit exceptional characteristics, such as high efficiency and power density, and are widely used in various fields, including ships, mining machinery, and helicopter transmission systems.The load-sharing performance plays a pivotal role in assessing the overall effectiveness of encased differential planetary gear trains, which hold immense significance in ensuring the efficient, reliable, and stable operation of mechanical systems.By achieving uniform load distribution, these gear trains can not only minimize energy loss and enhance transmission efficiency but also mitigate fatigue damage to crucial components, ultimately extending the service life of the system.Therefore, it is of great significance to study the load-sharing performance of EDPGSs.
There are many scholars that have carried out research on encased differential planetary gear systems.Wang [1] analyzed the effect of a planetary gear/star gear on the transmission efficiency of a closed differential double gear train.Kuznetsova et al. [2] investigated the influence of the parameters on eigenfrequencies of the oscillations of the dynamic model of differential closed planetary gearing.Zhu et al. [3][4][5] analyzed the dynamic floating displacement of center gear and meshing stiffness variation instabilities in an encased differential planetary gear train and studied the nonlinear dynamic behavior.Zhang et al. [6][7][8] studied the dynamic characteristic of a coaxial contrarotating encased differential gear train.Yang et al. [9] investigated the dynamic characteristic of an encased differential gear train with journal bearing.Despite the considerable research conducted

The Physical Model of the EDPGS
The EDPGS in this paper is a single-input and dual-output configuration, and constant velocity reverse rotation of the internal and external rotor shafts can be achieved through structural parameter design.The system consists of the encased stage and the differential stage, as shown in Figure 1.The encased stage system is composed of sun gear s 1 , star gear a i , b i (i = 1, 2, . .., M), and inner ring gear r 1 , and the differential stage system is composed of sun gear s 2 , planetary gear p j (j = 1, 2, . .., N), and inner ring gear r 2 .When the load torque of the internal and external rotor shafts is equal, the power transmission of the system can be divided into three paths.The path 1 route is s 1 -a i -b i -r 1 -r 2 -T r , the path 2 route is s 1 -s 2 -p j -r 2 -T r , and the path 3 route is s 1 -s 2 -p j -c-T c .
study investigates the influence of floating support stiffness and floating clearance on the load-sharing performance of the gear system.These findings offer valuable guidance and references for the design of floating structures.

The Physical Model of the EDPGS
The EDPGS in this paper is a single-input and dual-output configuration, and constant velocity reverse rotation of the internal and external rotor shafts can be achieved through structural parameter design.The system consists of the encased stage and the differential stage, as shown in Figure 1.The encased stage system is composed of sun gear s1, star gear ai, bi(i = 1, 2, ..., M), and inner ring gear r1, and the differential stage system is composed of sun gear s2, planetary gear pj(j = 1, 2, ..., N), and inner ring gear r2.When the load torque of the internal and external rotor shafts is equal, the power transmission of the system can be divided into three paths.The path 1 route is s1-ai-bi-r1-r2-Tr, the path 2 route is s1-s2-pj-r2-Tr, and the path 3 route is s1-s2-pj-c-Tc.

Time-Varying Meshing Stiffness
Since the coincidence of spur gears is generally not an integer and is less than two, there exists alternating meshing of single and double teeth, resulting in periodic fluctuations in gear meshing stiffness.The time-varying meshing stiffness can be calculated by Equation (1) Error!Reference source not found.: where k m a represents the comprehensive meshing stiffness of the gear pair m, 2k m v is the stiffness fluctuation value, an and bn are the Fourier transform coefficient, and ω is the meshing period of the gear pair.The calculation of stiffness parameter values can be referred to in Equations ( 2) and (3):

Time-Varying Meshing Stiffness
Since the coincidence of spur gears is generally not an integer and is less than two, there exists alternating meshing of single and double teeth, resulting in periodic fluctuations in gear meshing stiffness.The time-varying meshing stiffness can be calculated by Equation (1) [12]: where k a m represents the comprehensive meshing stiffness of the gear pair m, 2k v m is the stiffness fluctuation value, a n and b n are the Fourier transform coefficient, and ω is the meshing period of the gear pair.The calculation of stiffness parameter values can be referred to in Equations ( 2) and (3): where k min m denotes the single contact stiffness, k max m denotes the comprehensive meshing stiffness, which can refer to the ISO 6336-1 [23], and ε m denotes the contact ratio of the gear pair.
The meshing damping [24] of the gear pair can be expressed as the following: where ξ m is the meshing damping ratio of the gear, the value ranges from 0.03 to 0.17, I m is the moment of inertia (1 corresponds to the driving gear, 2 corresponds to the driven gear), and r m is the radius of the base circle of the gear.The contact stiffness of the rolling bearing is equal to the tandem of the inner and outer raceway stiffness, which can be expressed as the following: where the contact stiffness between the rolling element and the inner and outer raceways k ci and k co are shown as follows: where ∆F r is the variation in radial load, ∆δ i is the contact deformation between the rolling element and the inner raceway under ∆F r , and ∆δ o is the contact deformation between the rolling element and the outer raceway under ∆F r .
Rolling bearing oil film stiffness can be calculated by the following: where h ′ is the thickness of the oil film between the rolling element and the inner and outer raceways.
Considering the elastic deformation of the bearing and the lubricating oil film, the comprehensive radial stiffness of the bearing can be obtained as the following [25]:

Stochastic Equivalent Meshing Error Based on the Monte Carlo Method
The gear transmission error is a significant parameter in the dynamic model, exerting substantial influence on dynamic behavior [26].It is typically regarded as a constant value [27].However, the error is random in the actual machining process as long as the error is within the required tolerance range and can be regarded as meeting the requirements of machining accuracy.In order to consider the influence of the tolerance band on the load-sharing performance, this paper adopts the following assumptions: (1) The influence of the eccentricity error and installation error is mainly considered, and the error schematic diagram of the sun gear and planetary gear is shown in Figure 2. (2) The Monte Carlo method is used to simulate random eccentricity error and installation error.Assuming that the error machining accuracy in this paper is grade 5, the range of the tolerance band is considered to be between grade 4 and grade 6.
Taking the differential stage system as an example, the eccentricity and installation errors of the external meshing pair are shown as Equation ( 9): where E s0 , A s0 , E p0 , and A p0 are the eccentricity error amplitude and the installation error amplitude of the sun gear and planetary gear under grade 4, respectively, and E s1 , A s1 , E p1 , and A p1 are the eccentricity error amplitude and the installation error amplitude of the sun gear and planetary gear under grade 6, respectively.rand (0,1) represents a random number taken in the interval range [0, 1].ω s and ω c are the angular velocity of the sun gear and the planetary carrier, respectively, and when the system is a fixed shaft gear train, ω c = 0. α w is the meshing angle of the external meshing gear pair, φ i is the position angle of the ith planetary gear relative to the initial position, which can be expressed as φ i = 2π(i − 1)/N p + φ 0 , φ 0 is the initial position angle, and N p is the number of planetary gears.β s and γ s are the initial phase of manufacturing and installation errors of the sun gear, respectively.β pi and γ pi are the initial phase of manufacturing and installation errors of the planetary gears.Taking the differential stage system as an example, the eccentricity and installation errors of the external meshing pair are shown as Equation (9): where Es0, As0, Ep0, and Ap0 are the eccentricity error amplitude and the installation error amplitude of the sun gear and planetary gear under grade 4, respectively, and Es1, As1, Ep1, and Ap1 are the eccentricity error amplitude and the installation error amplitude of the sun gear and planetary gear under grade 6, respectively.rand (0,1) represents a random number taken in the interval range [0, 1].ωs and ωc are the angular velocity of the sun gear and the planetary carrier, respectively, and when the system is a fixed shaft gear train, ωc = 0. αw is the meshing angle of the external meshing gear pair, φi is the position angle of the ith planetary gear relative to the initial position, which can be expressed as φi = 2π(i − 1)/Np + φ0, φ0 is the initial position angle, and Np is the number of planetary gears.βs and γs are the initial phase of manufacturing and installation errors of the sun gear, respectively.βpi and γpi are the initial phase of manufacturing and installation errors of the planetary gears.
The eccentricity and installation errors of the internal meshing pair are shown as Equation (10): where αn is the meshing angle of the inner meshing pair, and βr and γr are the initial phases of the eccentricity error and the installation error of the internal ring gear, respectively.
The equivalent meshing error [28] of the gear pair is as follows: The eccentricity and installation errors of the internal meshing pair are shown as Equation (10): where α n is the meshing angle of the inner meshing pair, and β r and γ r are the initial phases of the eccentricity error and the installation error of the internal ring gear, respectively.
The equivalent meshing error [28] of the gear pair is as follows:

Dynamic Modeling of Support Reaction Forces in Floating Structures
The floating structure refers to the central components floating freely with no radial support within a certain floating range.It can adjust the radial displacement to improve the load-sharing performance automatically, and the essence is to increase the degree of freedom of the components and eliminating or reducing virtual constraints through the floating of the basic component to achieve the purpose of load sharing.The central components can be floated by mechanisms such as cross sliders, couplings, etc.
The reaction force of the floating sun gear is shown in Figure 3, which can refer to Equations ( 12) and ( 13) [29].When −R s1 ≤ x s ≤ R s1 , the sun gear is in an unconstrained state which can float freely in the x direction; when x s > R s1 or x s < −R s1 , the vibration displacement of the sun gear reaches the maximum allowable floating amount, and the reaction force is provided by the bending deformation of the gear shaft.The supporting forces along the y-axis are similar to those in the x-axis direction.

Dynamic Modeling of Support Reaction Forces in Floating Structures
The floating structure refers to the central components floating freely with no radial support within a certain floating range.It can adjust the radial displacement to improve the load-sharing performance automatically, and the essence is to increase the degree of freedom of the components and eliminating or reducing virtual constraints through the floating of the basic component to achieve the purpose of load sharing.The central components can be floated by mechanisms such as cross sliders, couplings, etc.
The reaction force of the floating sun gear is shown in Figure 3, which can refer to Equations ( 12) and ( 13) [29].When , the sun gear is in an unconstrained state which can float freely in the x direction; when x R < − , the vibration displacement of the sun gear reaches the maximum allowable floating amount, and the reaction force is provided by the bending deformation of the gear shaft.The supporting forces along the y-axis are similar to those in the x-axis direction.

Overall System Dynamic Model
Since the gears in this paper are all spur gears, the degrees of freedom in the axial and oscillating directions can be ignored, and the degrees of freedom in the bending and torsion directions can be considered.The EDPGS has (15 + 6M + 3N) DOFs, and its generalized coordinates are as follows: where r1, s1, ai, and bi represent the internal gear, sun gear, first stage star gear, and second stage star gear of the encased stage system.r2, s2, c, and pj represent the internal gear, sun gear, carrier, and planetary gear of the differential stage system.M and N represent the number of star gears and planetary gears, which are both taken as 3 in this paper.x, y, and ϕ represent the horizontal displacement, vertical displacement, and torsional

Overall System Dynamic Model
Since the gears in this paper are all spur gears, the degrees of freedom in the axial and oscillating directions can be ignored, and the degrees of freedom in the bending and torsion directions can be considered.The EDPGS has (15 + 6M + 3N) DOFs, and its generalized coordinates are as follows: where r 1 , s 1 , a i , and b i represent the internal gear, sun gear, first stage star gear, and second stage star gear of the encased stage system.r 2 , s 2 , c, and p j represent the internal gear, sun gear, carrier, and planetary gear of the differential stage system.M and N represent the number of star gears and planetary gears, which are both taken as 3 in this paper.x, y, and ϕ represent the horizontal displacement, vertical displacement, and torsional displacement of the components.ζ and η are the radial and tangential displacements of the star gear or planetary gear.
The relative meshing displacement of the star gear train or planetary gear train meshing pair along the direction of the meshing line can be expressed as follows: δ mn = V mn q mn − e mn (t) (14) where V mn represents the meshing vector, with the external meshing shown in Equation ( 15) and the internal meshing shown in Equation (16).q mn represents the degree of freedom involved in meshing, as shown in Equation ( 17). e mn represents the equivalent meshing error of the meshing pair mn, which can refer to Equation ( 12).When the system is a star gear train, m = s 1 , r 1 , n = a i , b i (i = 1, 2, . .., M).When the system is a planetary gear system, m = s 2 , r 2 , n = p j (j = 1, 2, . .., N).
V mn = sinψ mn cosψ mn r bm −sinψ mn −cosψ mn −r bm V mn = −sinψ mn cosψ mn −r bm sinψ mn −cosψ mn r bm (16) When the meshing form is external meshing, ψ mn = α m − φ i , and when the meshing form is internal meshing, ψ mn = α m + φ i .α m represents the pressure angle of component m.
Figure 4 shows the dynamic model of the EDPGS.The Lagrange equation is used to derive the dynamic equation, and the derivation process can be referred to in [30], which is no longer mentioned in this article.The overall dynamic matrix is the following: where M is the mass matrix, C is the damping matrix, G is the gyroscopic matrix, K b is the support stiffness matrix, K m is the meshing stiffness matrix, K ω is the centrifugal stiffness matrix, and Q is the excitation vector.Figure 5 shows the dynamic matrix assemble rule.
[ ] mn mn mn bm mn mn bm V sin cos r sin cos r [ ] mn mn mn bm mn mn bm When the meshing form is external meshing, ψmn = αm − φi, and when the meshing form is internal meshing, ψmn = αm + φi.αm represents the pressure angle of component m.
Figure 4 shows the dynamic model of the EDPGS.The Lagrange equation is used to derive the dynamic equation, and the derivation process can be referred to in [30], which is no longer mentioned in this article.The overall dynamic matrix is the following: where M is the mass matrix, C is the damping matrix, G is the gyroscopic matrix, Kb is the support stiffness matrix, Km is the meshing stiffness matrix, Kω is the centrifugal stiffness matrix, and Q is the excitation vector.Figure 5 shows the dynamic matrix assemble rule.

Calculation Model of Load-Sharing Coefficient
The load-sharing coefficient (LSC) is used to describe the load distribution of the star gear system or planetary gear system over a period of time, which can be calculated from Equations ( 18) and ( 19).The smaller the LSC, the better the load-sharing performance of the system.

Calculation Model of Load-Sharing Coefficient
The load-sharing coefficient (LSC) is used to describe the load distribution of the star gear system or planetary gear system over a period of time, which can be calculated from Equations ( 18) and ( 19).The smaller the LSC, the better the load-sharing performance of the system.
Here, LSC i s1a , LSC i r1b , LSC i s2p , and LSC i r2p are the maximum LSCs of the sun gear-star gear pair, inner ring gear-star gear pair, sun gear-planetary gear pair, and inner ring gear-planetary gear pair in a meshing cycle in the i th sample trial, respectively.M is the number of star gears and planetary gears, and N is the number of random samples.F i (i = s1a, r1b, s2p, r2p) refers to the dynamic meshing force of i th gear pair.
Load-sharing performance under random samples based on the Monte Carlo method can be properly estimated by mathematical expectations and variances.The expectation and variance of the LSC in a single stage can be obtained by Equations (20) and ( 21):

The Analysis of the Load-Sharing Characteristics of the EDPGS 4.1. Dynamic Parameter of the Encased Differential Planetary System
In order to conduct a dynamic analysis of the encased differential planetary system, structural and dynamic parameters are needed.Table 1 shows the structural parameters of the EDPGS, Table 2 shows the dynamic parameters of the EDPGS, Table 3 shows the error parameters of the EDPGS, and the gear error is given according to the four-grade machining accuracy [31].This paper assumes that the input speed of the system is 1500 r/min and that the input torque is 12,000 Nm.

Dynamic Parameter Value
Support stiffness (N/m) k s = 3.5 × 10 8 , k a = 2.6 × 10 8 , k b = 3.5 × 10 8 , k p = 5.2 × 10 8 , k r = 6.2 × 10 8 Torsional stiffness (Nm/rad) k ts12 = 2.4 × 10 6 , k tab = 8.5 × 10 5 , k tr12 = 5.7 × 10 8 Radial coupling stiffness (N/m) k rs12 = 2.1 × 10 8 , k rab = 1.8 × 10 9 , k rr12 = 6.2 × 10 10 Due to the diversity of parameter combinations, it is difficult to consider the loadsharing performance of the EDPGS under all combinations under a large sample number.Therefore, the orthogonal experimental design method can be used to investigate the influence of floating support stiffness and floating clearance on the load-sharing characteristics of the system.An orthogonal experimental design method is a mathematical statistical method that uses an "orthogonal table" to arrange and analyze multi-factor experiments, which can obtain as much information as possible through effective data combination, and has the characteristics of fewer tests and high efficiency.The experimental steps of orthogonal experimental design are as follows: (1) Determine the experimental factors and levels.
(2) Build the experimental matrix X, ensuring that each level of each experimental factor appears exactly once and that every pair of columns in the experimental matrix is orthogonal.The building method can refer to a Latin hypercube, orthogonal fractional array, etc. (3) Conduct experiments according to the design in the experimental matrix and record the resulting data.
The floating support stiffness considers four parameters, namely encased stage sun gear support stiffness, encased stage inner ring gear support stiffness, differential stage sun gear support stiffness, and differential stage inner ring gear support stiffness, and each parameter is divided into five levels, namely 1 × 10 7 Nm, 5 × 10 7 Nm, 1 × 10 8 Nm, 5 × 10 8 Nm, and 1 × 10 9 Nm.Hence, a four-factor, five-level orthogonal test table is adopted which can be generated by SPSS software, as shown in Table 4.Among them, A, B, C, and D represent the four factors that need to be studied, and 1~5 represents the five levels corresponding to each factor.The rule of variation in LSCs with the changing of support stiffness can be evaluated through 25 trials.Since the ring gear adopts the dual structure and is directly connected to the external rotor shaft, it is not easy to use floating structure.In addition, the large mass of the double inner ring gear results in the small floating amount of the ring gear.Therefore, this paper only considers the influence of the floating clearance of the sun gear of the encased stage and the differential stage on the load-sharing performance of the system.The clearance is divided into four levels, namely 10 µm, 20 µm, 30 µm, and 40 µm.So, a two-factor, five-level orthogonal test table is adopted, as shown in Table 5.The rule of variation in LSCs with the changing of floating clearance can be evaluated through 16 trials.

Analysis Process of EDPGS Based on Mathematical Statistical Methods
Figure 6 shows the flowchart of the analysis process of the EDPGS based on mathematical statistical methods.The parameter allocation values obtained according to the orthogonal experimental design table were substituted, and then the random error excitation e k s(r)pi was generated according to the Monte Carlo method, which was substituted into the dynamic model, and LSC k was obtained by the Newmark method.When the number of iterations k reaches the expected value N, the statistic load-sharing coefficient and floating displacement of the sun gear-including the expectation, standard deviation, and probability distribution-can be obtained.Moreover, the analysis of load-sharing performance of the EDPGS can be conducted.

Analysis Process of EDPGS Based on Mathematical Statistical Methods
Figure 6 shows the flowchart of the analysis process of the EDPGS based on mathematical statistical methods.The parameter allocation values obtained according to the orthogonal experimental design table were substituted, and then the random error excitation  was generated according to the Monte Carlo method, which was substituted into the dynamic model, and LSC k was obtained by the Newmark method.When the number of iterations k reaches the expected value N, the statistic load-sharing coefficient and floating displacement of the sun gear-including the expectation, standard deviation, and probability distribution-can be obtained.Moreover, the analysis of loadsharing performance of the EDPGS can be conducted.

Analysis of Load-Sharing Performance of EDPGS Based on the Monte Carlo Method
The load-sharing performance of the EDPGS can be obtained through the analysis of a large number of random samples based on the Monte Carlo method.The results of the analysis are more accurate when the number of samples tends to infinity under theoretical conditions.However, due to the limitation of calculation conditions, it is necessary to obtain an appropriate sample size for subsequent analysis.Hence, the probability distribution of LSC under the sample sizes of N = 100, N = 300, and N = 500 are compared.the Gaussian distribution under the same stage system.Third, the load-sharing performance of the differential stage system is better than that of the encased stage system.Forth, sample number has less effect on expectations and variance.Therefore, the sample number of this paper is taken as 300, and an ideal expectation can be obtained while the probability distribution is similar to the Gaussian distribution.Moreover, the calculation time is reduced.

The Effect of Floating Support Stiffness on the Load-Sharing Performance of EDPGS
The effect of floating support stiffness on the load-sharing characteristics o encased differential planetary gear train is studied in this section, in which the floa clearance of the central components are 0 µm, respectively.The statistics of the floa support stiffness on the LSC can refer to Appendix A. In Appendix A, the stiff orthogonal testing program and results can be found in Table 1, and the analysis re can be found in Table 2. Figures 10 and 11 can be obtained by collating the data in Ta into a bar chart.
Figures 10 and 11 show the effect of the floating support stiffness of the encased s system and the differential stage system on the LSC, respectively.It can be drawn Figure 10a that LSCs1a and LSCr1b increase as the floating support stiffness of the enc stage sun gear increases; namely, the load-sharing performance of the encased s system becomes unacceptable, and LSCs2p and LSCr2p have no obvious changing law.F Figure 10b, it can be seen that with the increase in the floating support stiffness of the i ring gear in the encased stage, LSCs1a and LSCr1b both show a trend of decreasing first then increasing, and the load-sharing coefficient is the lowest when kr1 = 5 × 10 7 N/m, the variations in LSCs2p and LSCr2p are irregular.Comparing the value of R, it can be fo that the floating support stiffness of the sun gear has a greater influence on the l sharing performance of the system under the encased stage system.
From Figure 11a, it can be found that with the increase in the floating sup stiffness of the differential stage sun gear, LSCs1a and LSCr1b show fluctuation changes no obvious changing law.LSCs2p and LSCr2p show an increasing trend as a whole, bu First, the probability distribution of the differential stage system is more similar to the Gaussian distribution than the encased system under the same sample number.Second, the higher the sample number, the more similar the probability distribution is to the Gaussian distribution under the same stage system.Third, the load-sharing performance of the differential stage system is better than that of the encased stage system.Forth, sample number has less effect on expectations and variance.Therefore, the sample number of this paper is taken as 300, and an ideal expectation can be obtained while the probability distribution is similar to the Gaussian distribution.Moreover, the calculation time is reduced.

The Effect of Floating Support Stiffness on the Load-Sharing Performance of EDPGS
The effect of floating support stiffness on the load-sharing characteristics of the encased differential planetary gear train is studied in this section, in which the floating clearance of the central components are 0 µm, respectively.The statistics of the floating support stiffness on the LSC can refer to Appendix A. In Appendix A, the stiffness orthogonal testing program and results can be found in Table 1, and the analysis results can be found in Table 2. Figures 10 and 11 can be obtained by collating the data in Table 2 into a bar chart.
Figures 10 and 11 show the effect of the floating support stiffness of the encased stage system and the differential stage system on the LSC, respectively.It can be drawn from Figure 10a that LSC s1a and LSC r1b increase as the floating support stiffness of the encased stage sun gear increases; namely, the load-sharing performance of the encased stage system becomes unacceptable, and LSC s2p and LSC r2p have no obvious changing law.From Figure 10b, it can be seen that with the increase in the floating support stiffness of the inner ring gear in the encased stage, LSC s1a and LSC r1b both show a trend of decreasing first and then increasing, and the load-sharing coefficient is the lowest when k r1 = 5 × 10 7 N/m, and the variations in LSC s2p and LSC r2p are irregular.Comparing the value of R, it can be found that the floating support stiffness of the sun gear has a greater influence on the load-sharing performance of the system under the encased stage system.change amplitude is small; when ks2 = 1 × 10 9 N/m, the load-sharing coefficient differential stage shows a slight downward trend, which might result from the infl of the random error.It can be seen from Figure 11b that the effect of the floating su stiffness of the differential stage inner ring gear on the LSC of the system is similar of the encased stage inner ring gear, because of the dual inner ring gear structu addition, it should be mentioned that the load-sharing performance of the diffe stage system is better than that of the encased stage and therefore less affected by cha of the floating support stiffness.From Figure 11a, it can be found that with the increase in the floating support stiffness of the differential stage sun gear, LSC s1a and LSC r1b show fluctuation changes with no obvious changing law.LSC s2p and LSC r2p show an increasing trend as a whole, but the change amplitude is small; when k s2 = 1 × 10 9 N/m, the load-sharing coefficient of the differential stage shows a slight downward trend, which might result from the influence of the random error.It can be seen from Figure 11b that the effect of the floating support stiffness of the differential stage inner ring gear on the LSC of the system is similar to that of the encased stage inner ring gear, because of the dual inner ring gear structure.In addition, it should be mentioned that the load-sharing performance of the differential stage system is better than that of the encased stage and therefore less affected by changing of the floating support stiffness.

The Effect of Floating Clearance of the Sun Gear on the Load-Sharing Performance of the EDPGS
The influence of floating clearance of sun gear on the load-sharing characteris the EDPGS is studied in this section, in which the floating support stiffness of the c components is 3.5 × 10 8 N/m, respectively.The statistics of the floating clearance LSC can refer to Tables A3 and A4 in Appendix A. Figure 12 can be obtained by co the data in Table 4 into a bar chart.
Figure 12 show the effect of floating clearance of the sun gear of the differentia system on the LSC.It can be concluded from Figure 12a that LSCs1a and LSCr1b both a decreasing trend as the floating clearance of the encased stage sun gear increase is, the load-sharing performance of the encased stage system is improving.Conv  Figure 12 show the effect of floating clearance of the sun gear of the differential stage system on the LSC.It can be concluded from Figure 12a that LSC s1a and LSC r1b both show a decreasing trend as the floating clearance of the encased stage sun gear increases, that is, the load-sharing performance of the encased stage system is improving.Conversely, LSC s2p and LSC r2p are increasing.This is because with the increase in the floating clearance of the encased stage sun gear, the sun gear s 1 can adjust the displacement to improve the load-sharing performance of the encased stage system adaptively, but the sun gear s 2 is affected by the displacement of the sun gear s 1 , which aggravates the vibration of the sun gear s 2 ; hence, the load-sharing performance of the differential stage system becomes bad.It can be drawn from Figure 12b that increasing the floating clearance of the sun gear the differential stage system enhances the load-sharing performance of the differential stage system, while the improvement is not as good as that of the encased stage.Simultaneously, the load-sharing coefficient of encased stage system shows an increasing trend.It should be noted that the increase in the floating clearance of the sun gear in a single stage system will improve the load-sharing performance of the system of that stage, while the load-sharing performance of the other stage system becomes bad.
gear s2; hence, the load-sharing performance of the differential stage system becomes It can be drawn from Figure 12b that increasing the floating clearance of the sun g the differential stage system enhances the load-sharing performance of the differ stage system, while the improvement is not as good as that of the encased Simultaneously, the load-sharing coefficient of encased stage system shows an incre trend.It should be noted that the increase in the floating clearance of the sun gea single stage system will improve the load-sharing performance of the system of that while the load-sharing performance of the other stage system becomes bad.The coupling effect of floating support stiffness and floating clearance of the sun gear of the encased stage and differential stage on the load-sharing performance of the system is studied in this section, in which the floating support stiffness of the ring gear is 3.5 × 10 8 N/m, and the floating clearance is 0 µm.Figure 13 show the coupling effect of the floating support stiffness and floating clearance of the encased stage gear s 1 on the LSC s1a , LSC r1b , LSC s2p , and LSC r2p of the system, respectively.The following two rules can be summarized.First, when the floating clearance of the encased stage sun gear s 1 is smaller, the more obvious the influence of the floating support stiffness on the load-sharing coefficient of the system, and the load-sharing coefficient increases with the increase in floating support stiffness.Second, the greater the floating support stiffness of the encased stage sun gear s 1 , the more obvious the influence of the floating clearance on the load-sharing coefficient of the system, and the load-sharing coefficient decreases with the increase in the floating clearance.Figure 14 shows the coupling effect of the floating support stiffness and floating clearance of the differential stage sun gear s 2 on the LSC s1a , LSC r1b , LSC s2p , and LSC r2p of the system, respectively.It can be found that the influence of the floating parameters of the differential stage sun gear on the load-sharing performance of the system is much less obvious than that of the encased stage system.
Machines 2024, 12, x FOR PEER REVIEW 18 of 23 the more obvious the influence of the floating support stiffness on the load-sharing coefficient of the system, and the load-sharing coefficient increases with the increase in floating support stiffness.Second, the greater the floating support stiffness of the encased stage sun gear s1, the more obvious the influence of the floating clearance on the loadsharing coefficient of the system, and the load-sharing coefficient decreases with the increase in the floating clearance.Figure 14 shows the coupling effect of the floating support stiffness and floating clearance of the differential stage sun gear s2 on the LSCs1a, LSCr1b, LSCs2p, and LSCr2p of the system, respectively.It can be found that the influence of the floating parameters of the differential stage sun gear on the load-sharing performance of the system is much less obvious than that of the encased stage system.

Conclusions
In order to investigate the impact of floating support parameters on the dynamic load-sharing behavior of the EDPGS under random error conditions, the dynamic model of the system with floating support parameters is established with the consideration of the influence of time-varying mesh stiffness, and manufacture error and installation error based on the Monte Carlo method and orthogonal experimental design method.The floating support stiffness and floating clearance on the load-sharing performance of the EDPGS were studied, and the following conclusions were obtained: (a) The probability distribution, expectation, and variance under the sample number with 100, 300, and 500 were compared.When the sample number N = 100, the probability distribution significantly deviated from the fitted Gaussian distribution, and the fitting degree improved at N = 300 and 500.However, the expectation and variance were less affected by the sample number.(b) The load-sharing coefficient of the encased stage system increases with the increase in the floating support stiffness, and the load-sharing coefficient of the differential stage system increases with the increase in the floating support stiffness of s2.(c) The load-sharing coefficient of the encased stage system decreased with the increase in the floating clearance of s1.In contrast, the load-sharing coefficient of the differential stage system increased with the increase in the floating clearance of s1 and decreased with the increase in the floating clearance of s2.

Conclusions
In order to investigate the impact of floating support parameters on the dynamic load-sharing behavior of the EDPGS under random error conditions, the dynamic model of the system with floating support parameters is established with the consideration of the influence of time-varying mesh stiffness, and manufacture error and installation error based on the Monte Carlo method and orthogonal experimental design method.The floating support stiffness and floating clearance on the load-sharing performance of the EDPGS were studied, and the following conclusions were obtained: (a) The probability distribution, expectation, and variance under the sample number with 100, 300, and 500 were compared.When the sample number N = 100, the probability distribution significantly deviated from the fitted Gaussian distribution, and the fitting degree improved at N = 300 and 500.However, the expectation and variance were less affected by the sample number.(b) The load-sharing coefficient of the encased stage system increases with the increase in the floating support stiffness, and the load-sharing coefficient of the differential stage system increases with the increase in the floating support stiffness of s 2 .(c) The load-sharing coefficient of the encased stage system decreased with the increase in the floating clearance of s 1 .In contrast, the load-sharing coefficient of the differential stage system increased with the increase in the floating clearance of s 1 and decreased with the increase in the floating clearance of s 2 .(d) The impact of floating support stiffness on load-sharing performance was more pronounced when the floating clearance of s 1 was smaller, leading to an increase in the load-sharing coefficient as the floating support stiffness increased.Conversely, a higher floating support stiffness of s 1 amplified the influence of floating clearance on load-sharing performance, causing the load-sharing coefficient to decrease with the increase in the floating clearance.
In summary, this study provides valuable insights into the dynamic load-sharing behavior of the EDPGS under various floating support parameters.The findings contribute to a better understanding of gear system performance and can guide the optimization of such systems for practical applications.

Data Availability Statement:
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest:
The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this paper.

Figure 1 .
Figure 1.The diagram of the EDPGS.

Figure 1 .
Figure 1.The diagram of the EDPGS.

Machines 2024 , 23 Figure 2 .
Figure 2. Schematic diagram of eccentricity error and installation error of the sun gear and the planetary gear.

Figure 2 .
Figure 2. Schematic diagram of eccentricity error and installation error of the sun gear and the planetary gear.

Figure 3 .
Figure 3. Schematic diagram of the reaction force of the floating sun gear.

Figure 3 .
Figure 3. Schematic diagram of the reaction force of the floating sun gear.

Figure 5 .
Figure 5. Schematic diagram of dynamic matrix assemble rule.

Figure 6 .
Figure 6.Flowchart of analysis process of EDPGS based on mathematical statistical methods.Figure 6. Flowchart of analysis process of EDPGS based on mathematical statistical methods.

Figure 6 .
Figure 6.Flowchart of analysis process of EDPGS based on mathematical statistical methods.Figure 6. Flowchart of analysis process of EDPGS based on mathematical statistical methods.

Figures 7 -
9 show the probability statistic histogram of load-sharing behavior in the EDPGS under N = 100, N = 300, and N = 500, respectively.A Gaussian distribution curve was fitted to match the statistical distribution.It can be seen from the (a) and (b) from Figures 7-9 that the interval ranges of LSC s1a are [1.16,1.31], [1.15, 1.31], and [1.15, 1.32], the expectations are 1.242, 1.231, and 1.235, and the variances are 0.0013, 0.0012, and 0.0012, respectively.And the interval ranges of LSC r1b are [1.15,1.29], [1.14, 1.29], and [1.14, 1.30], the expectations are 1.225, 1.218, and 1.221, and the variances are 0.0011, 0.0011, and 0.0010, respectively.It can be drawn that the probability distributions of LSC s1a and LSC r1b are quite different from the fitted Gaussian distribution when the sample number N = 100, while when the sample number N = 300 or 500, the distributions of LSC s1a and LSC r1b are similar to the fitted Gaussian distribution.From the perspective of expectation and variance, the gap is not very large.From (c) and (d) from Figures 7-9, the interval ranges of LSC s2p are [1.16,1.31], [1.15, 1.31], and [1.15, 1.32], the expectations are 1.242, 1.231, and 1.235, and the variances are 0.0013, 0.0012, and 0.0012, respectively.And the interval ranges of LSC r2p are [1.15,1.29], [1.14, 1.29], and [1.14, 1.30], the expectations are 1.225, 1.218, and 1.221, and the variances are 0.0011, 0.0011, and 0.0010, respectively.In summary, the following four conclusions can be drawn.

Figure 9 .
Figure 9. Probability statistic histogram of load-sharing behavior in the EDPGS under N = 500.

Figure 10 .
Figure 10.The effect of floating support stiffness of the encased stage system on the LSC.

Figure 10 .
Figure 10.The effect of floating support stiffness of the encased stage system on the LSC.

Machines 2024 ,Figure 11 .
Figure 11.The effect of floating support stiffness of the differential stage system on the LSC.

Figure 11 .
Figure 11.The effect of floating support stiffness of the differential stage system on the LSC.

4. 6 .
The Effect of Floating Clearance of the Sun Gear on the Load-Sharing Performance of the EDPGS The influence of floating clearance of sun gear on the load-sharing characteristics of the EDPGS is studied in this section, in which the floating support stiffness of the central components is 3.5 × 10 8 N/m, respectively.The statistics of the floating clearance on the LSC can refer to Tables A3 and A4 in Appendix A. Figure 12 can be obtained by collating the data in Table 4 into a bar chart.

Figure 12 .
Figure 12.The effect of floating clearance of the sun gear of the differential stage system on th

Figure 12 .
Figure 12.The effect of floating clearance of the sun gear of the differential stage system on the LSC.

4. 7 .
The Coupling Effect of Floating Sun Gear Parameter on the Load-Sharing Performance of the EDPGS

Figure 13 .
Figure 13.The coupling effect of floating support stiffness and clearance of the encased stage sun gear on the LSC.

Figure 13 .
Figure 13.The coupling effect of floating support stiffness and clearance of the encased stage sun gear on the LSC.

Figure 14 .
Figure 14.The coupling effect of floating support stiffness and clearance of the differential stage sun gear on the LSC.

Figure 14 .
Figure 14.The coupling effect of floating support stiffness and clearance of the differential stage sun gear on the LSC.

Table 3 .
Error parameters of EDPGS under grades 4 and 6.

Table A1 .
Stiffness orthogonal testing program and results.

Table A2 .
Results of stiffness orthogonal test analysis.1 , k 2 , k 3 , k 4 , and k 5 are the comprehensive averages of the corresponding index values of levels 1, 2, 3, 4, and 5, respectively.R is the difference between the maximum and minimum values in k 1 , k 2 , k 3 , k 4 , and k 5 , and the definitions of k 1 , k 2 , k 3 , k 4 , k 5 , and R are suitable for Table4in the Appendix A. k

Table A3 .
Clearance orthogonal testing program and results.

Table A4 .
Results of clearance orthogonal test analysis.