Effects of Boundary Conditions on Vibration Characteristics of Planetary Ring Gear

The 14th IFToMM World Congress, Taipei, Taiwan, October 25-3, 215 DOI Number: 1.6567/IFToMM.14TH.WC.OS6.31 Effects of Boundary Conditions on Vibration Characteristics of Planetary Ring Gear Jun Zhang 1 Shiyuan Bian 2 Xianzeng Liu 3 Marco Ceccarelli 4 1,2,3 Anhui University of Technology, 4 University of Cassino Ma anshan, China Cassino, Italy Abstract: A parametric finite element model for planetary ring gear is presented to investigate the effects of boundary conditions on the ring vibration characteristics. The modal analysis indicates that there exists five categories of vibration modes, i.e. rigid-body motion, inplane bending, out-plane bending, torsion and expansion. The effects of internal and external boundary conditions such as constraint stiffness and positions on the ring vibration characteristics are thoroughly explored. The numerical simulations show that the in-plane bending modes are more sensitive to the boundary conditions when compared to other vibration modes. Furthermore, the comprehensive effects of internal and external boundary conditions on ring gear vibration are very complicated. Nevertheless, an optimal combination may be adopted to achieve a high dynamics performance for the ring gear. Keywords: Planetary gear transmission, boundary condition, ring gear, vibration mode, natural frequency I. Introduction Vibration is one of the most critical factors that affect the reliability, operating environment and service durability of planetary gear transmission (PGT) systems. As a key component in a PGT system, ring gear transmits dynamic meshing forces as well as vibration energy to the gearbox case. From this perspective, the vibration characteristics of a planetary ring gear contribute a lot to the overall system dynamics that should be thoroughly investigated. Abundant efforts have been carried out in the past decades to investigate the dynamics of PGT systems, ranging from dynamic modeling, vibration modes classification, dynamic responses prediction, parametric stability analysis and vibration suppression [1]. Among all these studies, the dynamic modeling and vibration modes classification are the basis of the following-up studies of dynamic analysis and vibration suppression. According to the modelling methodology, the dynamic models of the PGT system can be roughly grouped into two types, i.e., the analytical lumped parameter models [2-5] and the numerical distributed parameter models [6-1]. In the first group of models, the gears and carrier are modeled as rigid bodies while treating the gear mesh and bearing supporting as fundamental spring-damping elements. Though this type of models are widely adopted due to its easy modeling and efficient computation, there are still some drawbacks in that too many simplifications are made for the component flexibilities as well as boundary 1 zhang_jun@tju.edu.cn 2 bianshiyuanbiannio@126.com 3 liuxianzeng126@126.com 4 ceccarelli@unicas.it conditions. For example, both theoretical analysis and experimental tests have manifested that the ring gear flexibility has significant influence on the dynamic characteristics of PGT systems so that the ring gear cannot be simply treated as rigid body when modeling the dynamics of a PGT system. Different from the lumped parameter models, the distributed parameter models can calculate the dynamics of PGT system with satisfactory accuracy by including flexibilities of all components as well as boundary conditions between their connections. For example, Kahraman established two finite element models to investigate the effects of ring gear flexibility on gear stresses, load distribution and dynamics for a PGT system [6,7]. Though the ring flexibility was included, the settings of boundary conditions for the ring gear is questionable. In his models, all nodes on the rim surface of ring gear were set to be fixed-only, which idealized the real-life installation conditions of the ring gear [6,7]. Since the analysis accuracy of a finite element model depends on the correct and appropriate handlings of boundary conditions, this simplification will probably reduce the prediction accuracy of the system dynamics. To identify the vibration characteristics of the ring gear, Tanna established a finite element model for the ring structure and predicted four typical free vibration modes [8]. He further discussed the influences of constraints, structural simplifications as well as design parameters on the natural frequencies of ring gear [9,1]. It is worth pointing out that Tanna s model only considered the constrains from ring housing but did not include the constrains exerting by the ring-planet gear mesh. Meanwhile, there were no circumferential constrains for the ring in his model in that he only applied simple radial constrains to the ring gear rim. Obviously, it is much away from the real-life boundary conditions that makes the proposed model not to be directly applied to the planetary gear train dynamic analysis. More recently, Parker established a dynamic model for a planetary ring gear with both meshing and housing constraints [11]. In his model, the ring gear was simplified as a smooth ring structure without teeth and the internal meshing constrains and external housing constraints were represented by two sets of circumferentially distributed orthogonal springs with equivalent stiffness values. Based on this model, he analyzed the in-plane vibrations of the ring as well as the system dynamics. Nevertheless, Tanna proved that simplifying a ring gear structure into a smooth ring would bring significant deviations when its vibration properties are concerned. Besides, the number of internal meshing pairs and that of external constraints such as splines are generally not the same in a real-life PGT system. This indicates that the representing the internal

and external constraints as two sets of orthogonal spring is somewhat questionable. In fact, in a real-life PGT system, the ring gear is often connected to the gearbox case through the following three ways, i.e., splines-coupled, pin-fixed and flange-bolted. Unfortunately, to the authors best knowledge, the influence of the connecting ways of a planetary ring gear on its vibration characteristics has never been explored. Motivated by these considerations, this paper aims to propose a finite element model for a planetary ring gear that includes the internal ring-planet meshing constrains and external housing constrains. Based on this proposed finite element model, the vibration characteristics of the ring gear are investigated and classified. The effects of boundary conditions such as the number and position of internal and external constrains on the ring s vibration properties are further investigated to provide useful information for potential PGT system dynamic analysis and enhancement. refined finite element model for a ring gear without constraint substructure. Fig.1. A finite element model for a ring gear With the proposed finite element modeling method, a general parametric finite element model for a planetary ring gear subject to different installation conditions can be established. Fig. 2 demonstrates the finite element model for a ring gear under three types of conditions. Herein, the pins and bolts are not shown in the pin-fixed and flangebolted installation conditions respectively, for clarity. II. Parametric Finite Element Modeling and Vibration Modes of a Planetary Ring Gear A. Parametric Finite Element Modeling As aforementioned, a planetary ring gear may connect to the case through splines, pins or a flange. From the structural perspective, the ring gear can be regarded as a combination of a thin-rimmed smooth ring, a set of gear teeth and an additional constraint substructure of splines, pins or a flange. Therefore, it is possible to construct a finite element model for the planetary ring gear through substructure superposition. In this paper, the ANSYS APDL is adopted to develop a parametric finite element model of planetary ring gear. In order to guarantee the modeling accuracy, the ring is meshed with an 8-node tetrahedron element. Meanwhile, the gear teeth are refined with node controlling technology to improve the modeling and computation efficiency [12]. Fig. 1 shows a Table 1 Vibration modes of a planetary ring gear without constrains (a) flange-bolted (b) pin-fixed (c) splines-coupled Fig.2. FE model for a ring gear with different installations B. Vibration Modes of a Ring without Constrains In this subsection, a modal analysis for the planetary ring gear shown in Fig. 2 is carried to reveal its vibration characteristics. By using Block Lanczos mode extraction method, the mode shapes of the ring gear can be obtained and classified into four vibration modes, i.e. (a) in-plane bending; (b) out-plane bending; (c) torsion; (d) expansion. The vibration characteristics of the ring gear subject to different constrains are listed in Table 1. Vibration modes In-plane bending Out-plane bending Torsion Expansion Flange-bolted Pin-fixed Splines-coupled From the above analysis, it can be clearly found that although the planetary ring gear may connect to the gearbox case through different manners, there are still four vibration modes in common. From this point of view, it can be roughly assumed that a planetary ring gear with different geometries and/or structures shares the same

vibration modes, thus can be modeled and computed with component mode synthesis (CMS) method as a regular ring structure and then incorporated into the system-level dynamic model of a PGT system. Further investigations show that the in-plane bending vibration mode usually corresponds to lower orders of vibration frequencies followed by relatively higher orders of vibration frequencies corresponding to out-plane bending mode. On the contrary, the vibration modes of torsion and expansion often correspond to the very high orders of vibration frequencies, which generally play less important role in PGT vibration contribution. C. Vibration Modes of a Ring with Constrains C1. Applying boundary conditions In this subsection, the boundary conditions are further applied to the finite element model of planetary ring gear to represent the constrains from ring-planet meshing and housing. For clarity, the ring-planet meshing is defined as the internal constrains and ring housing is defined as the external constrains. The COMBIN14 spring element is adopted to simulate the internal and external constrains. To be specific, a set of one-dimensional COMBIN14 spring elements is applied along the line of action (LOC) of each individual ring-planet meshing pair. Assume the average mesh stiffness of each ring-planet pair is and there are n 1 COMBIN14 spring elements equally distributed along the tooth width, then the equivalent stiffness value of each COMBIN14 spring element can be set as /n 1. Similarly, another set of two-dimensional COMBIN14 spring elements is applied at the corresponding positions of external constrains with equivalent stiffness values in the circumferential and radial directions. Assume the total number of the twodimensional COMBIN14 spring elements is n 2 and the housing constraint stiffness is k s (which can be further decomposed into k sc and k sr denoting stiffness values in circumferential and radial directions respectively), then the equivalent stiffness value of each two-dimensional COMBIN14 spring element can be set as k s /n 2. With these settings, the boundary conditions for the planetary ring gear can be applied as demonstrated in Fig. 3 in which the constrains are indicated as linear springs. (a) Internal constrains (b) External constrains Fig. 3. Boundary conditions for a planetary ring gear C2. Vibration modes By using Block Lanczos mode extraction method again, the mode shapes of the ring gear subject to internal and external constrains can be obtained. Different from the situation in that a ring gear without any constrains, there occurs an additional vibration mode that is defined as rigid body motion in this study. This new vibration mode is depicted as the following in which there is no bending, torsion and expansion. Fig. 4 Rigid body motion mode of a planetary ring gear Once again, among the five vibration modes, the rigid body motion(as shown in Fig. 4), the in-plane bending and out-plane bending modes mainly fall into the low and medium frequency domain while the torsion and expansion modes locate at high frequency domain. Since the low and medium frequency domain vibration modes usually contribute a lot to the PGT system in real-life operations, the following discuss will only focuses on the vibration modes of rigid body motion, the in-plane bending and the out-plane bending. III. Effects of Boundary Conditions In this section, the effects of boundary conditions on the vibration characteristics of a planetary ring gear will be analyzed and discussed to provide useful information for PGT dynamic analysis. Without loss of generality, a pinfixed thin-rimmed planetary ring gear in the first-stage of a helicopter transmission system is taken as an example to demonstrate the problem. The analysis procedure is also applicable to the other two installation situations which are omitted for content limitation. The basic structural sketch of the PGT is depicted in Fig. 5. In this case study, the ring gear is internally constrained by four planets and externally constrained by four pins. Ring gear Sun Planet Fig. 5. Structural sketch of a PGT example system The design parameters of the example system are designated as the followings. The teeth number of the ring z is 117, the module m is 1.322 mm, the pressure angle α is 22.5, the face width b is 25 mm, the rim thickness is 4.5 mm. The ring gear meshes with 4 equally spaced planet gears simultaneously and is fixed to the case with 4 equally distributed pins. Under this situation, the internal constraint stiffness and the external constraint stiffness k s are computed as 8 1 8 N/m and 1 1 9 N/m, respectively through further finite element simulations. Hereafter, all the analysis is carried out based on these basic parameter settings unless specific declarations are made. A. Effects of Internal Constrains The following investigates the effects of internal constraint stiffness values as well as constraint positions (different number of ring-planet gear meshes) on the vibration characteristics of the planetary ring gear. Fig. 6 shows the influence of ring-planet mesh stiffness on the vibration frequencies of the ring gear. For clarity, only the first order frequencies of the three modes of the rigid body Pin

motion, the in-plane bending and the out-plane bending are illustrated. Herein, the horizontal axis represents the internal constraint stiffness value of ( 1 8 N/m), the vertical axis represents the vibration frequency f (Hz). 3 25 15 1 5 8 7 6 5 4 3 1 Three planets Four planets Five planets (a) First order frequency of the rigid body motion mode Three planets Four planets Five planets (b) First order frequency of the in-plane bending mode frequency corresponding to the out-plane bending mode is insensitive to the variation of constraint stiffness as well as the number of constrains. More interestingly, the vibration frequency seems to achieve a minimum value when the number of planet gears is four. In this situation, the four planet gears are equally spaced and opposite to each other, resulting a counteracting effect on the meshing constrains. B. Effects of External Constrains Similarly to the above analysis, one can analyze the effects of external constrains on vibration characteristics of a planetary ring gear. Fig. 7 demonstrates the influence of external constraint stiffness values as well as constraint positions (different number of fixed pins) on the vibration characteristics of the planetary ring gear. 3 25 15 1 5 Two pins Four pins Six pins (a) First order frequency of the rigid body motion mode k s 9 8 7 6 5 4 3 1 Three planets Four planets Five planets (c) First order frequency of the out-of-plane bending mode 1 1 8 6 4 Two pins Four pins Six pins (b) First order frequency of the in-plane bending mode k s Fig. 6. Effects of internal constrains As can be observed from Fig. 6, the vibration frequency of the rigid body motion mode increases monotonously with the increment of internal constraint stiffness value. Meanwhile, the vibration frequency increase monotonously with the increment of internal constraint number, i.e., the more ring-planet mesh pairs the higher vibration frequency. This is coincident with the simple physical fact that a stronger spring constrain of a rigid body results in a higher vibration frequency. Very interestingly, the effects of internal constrains on the inplane and the out-plane bending modes seem to be different. To be specific, the frequency corresponding to the in-plane bending mode is insensitive to the variation of constraint stiffness while sensitive to the number of constrains. This is coincident with the physical fact that the bending stiffness of a ring is primarily decided by its inertial moment of the cross-section. The increasing frequency with respect to the increment of planet gears may be explained as that the meshing of planet gears helps to stiffen the ring gear s bending stiffness. The 95 9 85 8 75 7 Two pins Four pins Six pins (c) First order frequency of the out-of-plane bending mode Fig. 7. Effects of external constrains It can be observed from Fig. 7 that the external constrains have a different influence on the vibration properties of the planetary ring gear. In general, the vibration frequency of the rigid body motion mode and that of the in-plane bending mode are barely affected by the constraint stiffness values. On the contrary, the vibration frequency of the out-plane bending mode k s

increases monotonously with the increment of constraint stiffness when the ring gear is fixed with 4 or 6 pins while almost keeps unchanged when it is fixed with 2 pins. Meanwhile, it can also be found that the number of fixed pin does affect the vibration frequency at a noticeable extent. To be specific, the vibration frequency of the rigid body motion mode decreases when the pin number increases from 4 to 6 while almost keeps the same from 2 to 4. As to the mode of the in-plane bending, the vibration frequency increases monotonously with the increment of pin number. The vibration frequency of the out-plane bending increases firstly and then decrease when the pin number varies from 2 to 6. This is quite interesting and confusing. One possible explanation for this phenomenon may lie in the exactly symmetric distribution of the pins which results in a counteracting effect of external constrains. Nevertheless, the effects of external constrains on the vibration frequency are quite complicated and will be investigated in further effort. C. Compound Effects of constrains In this subsection, the compound effects of the internal and the external constrains will be investigated through numerical simulation. In addition, the constrains are applied to the ring gear in different ways and their corresponding results are analyzed to show the influence of constrain applying manner. Table. 2. Illustration of investigated boundary conditions with pins and planets To demonstrate the compound effects, the vibration frequencies of the ring gear subject to nine kinds of boundary conditions are computed and discussed. The nine cases are marked as A, B, C, D, E, F, G, H, and I. Herein, the boundary conditions are described the as followings: Case A (Two-three): In this case, the ring gear is constrained with two pins and three planets. Case B (Two-four): the ring gear is constrained with two pins and four planets. Case C (Two-five): In this case, the ring gear is constrained with two pins and five planets. Case D (Four-three): the ring gear is constrained with four pins and three planets. Case E (Four-four): In this case, the ring gear is constrained with four pins and four planets. Case F (Four-five): In this case, the ring gear is constrained with four pins and five planets. Case G (Six-three): In this case, the ring gear is constrained with six pins and three planets. Case H (Six-four): the ring gear is constrained with six pins and four planets. Case I (Six-five): In this case, the ring gear is constrained with six pins and five planets. The above nine cases of boundary conditions are illustrated in Table 2 for clarity. Case A: two-three Case B: two-four Case C: two-five Case D: four-three Case E: four-four Case F: four-five Case G: six-three Case H: six-four Case I: six-five Meanwhile, the internal and external constrains are represented by spring elements with different numbers though the summations of stiffness values are the same. Due to content limitation, only three kinds of spring representations are analyzed in this paper. As shown in Fig. 8, each set of the external constrains are represented with four, six and eleven spring elements along the face width of the ring gear. (a) four springs (b) six springs (c) eleven springs Fig. 8. Different external constrains representation infinite element analysis Combined with the nine boundary conditions, there are 27 situations to be calculated to obtain the vibration frequencies of the ring gear. The detailed results are listed in Table 3, 4 and 5. From Table 3, 4 and 5, it can be found that the compound effects of the internal and external constrains on the vibration characteristics of the planetary ring gear are very complicated that no general rule can be drawn from the data. Nevertheless, what can be confirmed is that the vibration frequencies and modes are affected both by the internal and external constraint stiffness values and positions as well. From the perspective of vibration suppression, it may be possible to find an optimal combination of fixed pins and planet gears. In other words, the ring gear can be fixed to case through an optimal number of pins with specific distributions with respect to a give number of planet gears. This,of course, is of great challenge and will be addressed in a separate paper. Another conclusion can be drawn from the comparison of the three tables in that the constrains applying manner affects the finite element simulation results in a quite noticeable extent. Therefore, the boundary conditions of the planetary ring gear need to be dedicatedly treated to yield a satisfactory and accurate result.

Table. 3. The first 9 orders of vibration frequencies of the ring gear when represented with 4 springs (Hz) for the case of Table. 2. Mode number A B C D E F G H I 1 162.9 174.88 146.7 26.28 151.91 151.68 24.43 242.75 15.99 2 313.37 53.52 552.12 43.17 43.51 551.69 494.23 527.27 565.2 3 461.19 72.69 775.16 438.13 565.24 586.5 674.47 95.62 672.66 4 73.81 896.17 939.41 62.11 839.68 95.54 788.99 964.89 927.31 5 761.22 179.2 142.9 659.88 92.25 185 913.64 1127.1 1223.6 6 886.12 187.9 1215.1 1121.1 1177.7 1266.7 1194.4 1372.9 1275 7 1158 121.8 1635.4 1276 128.2 1893.2 1211.7 1947.4 1955.2 8 1244.2 1845.2 1758.9 161.7 1886.1 224.6 238.8 24.5 218.9 9 1667.7 1879.7 2166.8 1681.5 1945.3 2233 27.8 2372.4 223.7 Table. 4. The first 9 orders of vibration frequencies of the ring gear when represented with 6 springs (Hz) for the case of Table. 2. Mode number A B C D E F G H I 1 129.68 134.7 144.42 26.25 29.86 174.1 24.34 241.92 17.29 2 197.34 222.55 379.96 311.3 377.4 418.12 356.38 44.41 429.36 3 322.16 375.41 753.25 39.72 772.96 843.63 671.12 883.29 882.79 4 631.2 82.72 878.89 612.94 816.56 142.6 785.28 97.52 11.6 5 694.72 921.99 937.36 659.39 134.2 115.4 881.79 934.12 1113.9 6 816.7 177.6 987.42 134 185.9 127.6 931.59 1246 1393.7 7 918.14 1277.2 1549.5 136.6 1597.6 2125.9 1193.3 1759.9 276.7 8 1319.4 1753.1 1685.4 1585.1 1892 2154.2 1915 237.1 29.1 9 1458.4 1916.2 271.6 1625.2.2 2191.1 236.7 2274.4 2152.9 Table. 5. The first 9 orders of vibration frequencies of the ring gear when represented with 11 springs (Hz) for the case of Table. 2. Mode number A B C D E F G H I 1 154.17 162.75 166.39 185.59 21.22 197.82 236.25 147.8 148.25 2 294.52 351.9 448.4 326.41 362.39 453.24 365.1 163.39 471.58 3 295.31 527.78 81.97 369.4 761.19 826.7 67.54 265.72 619.48 4 729.28 852.14 915.1 616.42 844.13 924.73 783.57 381.71 88.3 5 736.4 869.46 99.7 631.88 112.8 1134.8 847.98 399.54 197.4 6 854.51 12.1 134.7 119.2 1114.9 1226.3 939.66 51.88 1164.5 7 869.54 1213.3 1931.3 15.1 1659.8 1947.9 1183.7 518.5 1867.5 8 127.4 1776.4 286.5 1567.4 1979.8 285.8 1958.4 535.94 1961 9 1532.7 1877 294.7 1675.3 6.4 2162.7 245.8 763.91 2114.8 IV. Conclusions In this paper, a parametric finite element modeling methodology is proposed for a planetary ring gear in different installation forms by using the substructure superposition. The free vibration analysis indicates that a planetary ring gear subject to flange-bolted, pin-fixed and splines-coupled installation conditions shares four common vibration modes that can be classified as in-plane bending, out-plane bending, torsion and expansion. When the boundary conditions are introduced, another vibration mode occurs with rigid body motion. The effects of internal and external constrains of the planetary ring gear are analyzed to show a complicated influence on vibration characteristics. The present study also indicates an optimal combination of internal and external constrains can be adopted to achieve a high dynamics of the planetary ring gear. The in-depth investigation on this optimal combination will be explored in future work. Acknowledgment This research has been jointly supported by National Natural Science Foundation of China (Grant No. 5137513, 595122), Natural Science Foundation of Anhui Province (Grant No. 12885ME64). References [1] Cooley C. 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