A dynamic modeling method for helical gear systems

Dynamic modeling for gear systems is very important for accurately predicting the dynamic responses during the gear engagements. During the modeling, meshing force must be determined, and it is usually calculated by the product of the relative displacement along the line of action (LOA) and the meshing stiffness. At present, the relative displacement calculation for helical gear systems is very complicated by several existing methods because the complicated geometric relationships need to be derived for determining the spatial positions of two meshing points along LOA. In this study, a simple method for calculating the relative displacement along LOA is presented based on ANSYS software. And on this basis, a new finite element modeling method for a helical gear transmission system with multiple shafts is developed, where the influences of shafts and bearing flexibilities are considered. Moreover, the proposed method is validated by comparing the dynamic characteristics, such as natural characteristics and vibration responses, with those obtained from Kubur’s method and Zhang’s method.


Introduction
Dynamic characteristics of geared systems are closely related to the noise and reliability of the system.The dynamic loads during the gear transmission can lead to larger dynamic stress which may threaten the security operation of the gear systems [1][2][3].Therefore, a good dynamic modeling method for gear systems is essential for accurately predicting the vibration responses during gear engagement.Moreover, such a model can also be used to the dynamic design for the gear pairs, shafts and bearings.
A lot of researches on the dynamic characteristics of the spur gear rotor systems were carried out, such as Iida et al. [4], Choy et al. [4], Kahraman and Singh [6], Lim and Singh [7], Özgüven [8], Kahraman et al. [9], Rao et al. [10], Jia et al. [11], Lee et al. [12] and Ma et al. [13][14][15][16][17]. Except for the dynamics of spur gear systems, many researchers also developed many mathematical models for simulating the dynamic characteristics of spur gear systems.Blankenship and Singh [18] developed a new mesh dynamic model for gear pairs, which can provide a true, three-dimensional representation of the forces and moments during the gear engagement.Velex and Maatar [19] presented a general method for simulating 3D gear meshing, and introduced this meshing model into a lumped mass model of a gear system, and analyzed some parameters such as meshing stiffness and installation error on the system vibration responses.Choi et al. [20] studied the coupled lateral, torsional and axial vibrations of a gear system using finite element method.In their model, the tooth engagement was modeled using distributed spring and the force vibrations caused by unbalances and static transmission errors were also calculated.Baud and Velex [21] developed a finite element model for simulating dynamic tooth loads in gear rotor systems and verified the model by experiments using a single spur and helical gear reducer which includes flexible shafts mounted hydrostatic bearings.Kubur et al. [1,2] proposed a new model for simulating the dynamic characteristics of a helical system with multiple shafts.In their model, the shafts are modeled using finite element method and the helical gear pairs are modeled using a lumped mass model.Based on Kubur's model, Zhang et al. [22] established a general mathematical model of helical gear systems considering the effects of geometric eccentricity, gear meshing, bearing stiffness, and shaft flexibility.Based on the previous works in [1,2,22], Zhang et al. [23] developed a finite element model of a rotor system in centrifugal compressor.In their model, the effects of the constant meshing stiffness of helical gear pairs, the varying stiffness and damping of the bearings on the system vibration responses are considered.
From the above literature analysis, it is clear that the dynamic models of gear pairs are very complicated, especially, in the calculation of the relative displacement along the line of action.This study focuses on a new mathematical model for a rotor system with helical gear pairs and multiple shafts by ANSYS software.This new model can easily calculate the relative displacement and meshing force using the geometric positions of the meshing points.

Dynamic models of geared rotor systems
In this section, a new modeling method for geared rotor systems is developed in ANSYS software.The proposed model is verified by comparing the vibration behaviors with those obtained from Zhang's model [22].A brief introduction to Zhang's model (see Figs.

Dynamic models of geared rotor systems using Zhang's method
A 3D mathematical model of helical gear pair with 12 degrees of freedom (DOFs) is displayed in Fig. 1(b), which includes two helical gears ( and ).Ω , Ω , , , and denote the rotational speeds, geometrical centers, radii of base circles of the gears and .The flexible of gears are ignored and assumed to be the rigid disks.In addition, a spring-damping setting is adopted to simulate the gear meshing and the direction is defined according to the helix angle .The spring stiffness is determined by the average gear mesh stiffness due to the high contact ratio of helical gear.It is worth noting that Zhang's method in Ref. [22] takes the effects of gear geometric eccentricity into account.However, in this paper the eccentric effect is ignored.Helix angle is defined as: = 0, if gear has left hand teeth, = 0, if gear is a spur gear, 0, if gear has right hand teeth. ( As shown in Fig. 1(b), the angle between the connecting line of gear centers and the positive -axis of gear is defined as relative position angle .The angle between the plane of action and positive -axis can be defined by .Under different direction of rotation of driving gear, can be expressed as follows: where is the lateral pressure angle of gear pairs.The meshing position changes under the different direction of rotation of the driving gear.Considering this changing, a sgn function can be defined as follows: The displacement vector of the gear pair can be written as follows: Due to the equilibriums of lateral motion and force, the equations of motion of the gear pair can be written as the following expressions, where the influence of the geometric eccentricity of gear is ignored: where and denote the masses of gear and gear ., , , , and are the moments of inertia about , and axis.
denotes the averaged meshing stiffness of gear pair .
In Ref. [22], the shafts are simulated using Timoshenko beam elements, and some rigid disks on the shaft are simulated using lumped mass elements.By introducing the lumped mass models of gear pairs into the FE models of the shafts, equations of motions of the entire system can be expressed as: where , , , and are the mass, damping, stiffness and gyroscopic matrices.is the external force vector of the system.Here, the stiffness matrix of bearing is expressed as = diag , , , , , .According to the position of bearings and gears, the bearing stiffness matrix and mesh stiffness matrix are added to the stiffness matrix of the shafts.As shown in Fig. 2, total stiffness matrix of system is assembled, in which only two shafts are di splayed.The Rayleigh-type damping is adopted in this study, which can be expressed as: where: here, and denote the first-order and the second-order natural frequencies (Hz), respectively.In this paper, modal damping ratios = = 0.04.

A new mathematical model of gear rotor system
In this section, a new dynamic model is presented using FE method based on ANSYS software.
Gear meshing is modeled as two lumped masses connecting by a liner spring, which is normal to the gear tooth contact surface, as shown in Figs.1(b) and (d).According to the direction of the load, the direction of action plane changes.Here, a driving gear with counterclockwise rotation and left hand teeth is taken as a example, as shown in Figs.1(b) and (d), in which the tangent lines and can be determined according to the geometrical relationship.The length between and can be calculated as (see Fig. 3):

Fig. 2. Schematic of the whole stiffness matrix
The length of and can be expressed as: Considering the symmetry of the gears about plane, and are divided into two equal parts (i.e.= , = ), as shown in Fig. 1d.The coordinates of and can be written as (see Fig. 3): the coordinates of , , and in , and directions.Once these coordinates are determined, the direction of spring can be defined, i.e., the meshing relation of two gears can be determined.Here, for the convenience of the modeling, and are selected as the ends of spring.There are four working modes for the driving gear , i.e., counterclockwise rotation and left-hand teeth, counterclockwise rotation and right-hand teeth, clockwise rotation and left-hand teeth, clockwise rotation and right-hand teeth.Thus the positions of spring will change under different cases.However, these coordinates of the two ends of spring for the four cases can be expressed as: In ANSYS software, the detailed modeling process is briefly introduced as follows (see Fig. 1(c)): (1) According to the physical dimensions of the shafts and the actual gear positions, the FE models of bearing-rotor systems including bearings, shafts, gears and rigid disks are established; (2) The coordinates of gears are determined, and then the coordinates of the two ends of the meshing spring are determined based on Eq. ( 23); (3) Rigid connections are established between the ends of meshing springs and gears by using the command "cerig".Thus, the FE model of the whole gear rotor system will be established.
For the gear static transmission error excitations, it is loaded on both ends of meshing spring in an opposite direction along the direction of spring.And the excitation force due to the static transmission error can be written as:

Model verification and discussion
In order to prove the validity of the present method, two gear rotor systems offered in Refs.[1,9] are adopted.First of all, the present method will be verified by comparing the natural characteristics with those obtained using different methods in published literatures.Subsequently, the vibration responses obtained from present method and previous methods will be compared to verify the present method.

Model verification by comparing the natural characteristics
A spur gear rotor system is taken into account at first (see Fig. 4), and the parameters of system are listed in Table 1, which are offered in Refs.[9,24].Here, the effect of damping is ignored, and the helix angel is equal to zero for the spur gear.The first 11 orders of natural frequencies calculated by present method are compared with the results of Ref. [9] (see Table 2).At the same time, the descriptions of the mode shapes based on the present method are also listed in Table 2.It is obvious that the frequencies obtained by present method agree well with those offered in Ref. [9] except for some higher modes, and the relative error is less than 3.31 %.In this section, a two-stage helical gear rotor system offered in Ref. [1] is taken for further comparison, as shown in Fig. 1(a).Here, the example system with = 0 and = 0 is considered, and the detailed parameters of the system and the STE of gear can be found in Ref. [2].The lumped mass values and the material parameters, which are not provided in Ref. [1], can be found in Ref. [22].The natural frequencies are calculated using Zhang's method and present method, as listed in Table 3.In addition, the typical mode shapes obtained from two methods are shown in Fig. 5, which shows that the natural frequencies and mode shapes obtained from the present method agree well with those obtained from Zhang's method, and the maximum relative error of natural frequency is less than 2.17 % (see Table 3).In Ref. [1], only a mode shape corresponding to 812 Hz is provided, as shown in Fig. 6.This mode corresponds to the first non-rigid mode in present method.It is clear that the natural frequencies and mode shapes obtained by present method also show a good agreement with those provided in Ref. [1].These comparisons show that the present model is effective to calculate the natural characteristics of the geared rotor systems.

Model verification by comparing the vibration responses
In this section, the present dynamic model is further verified by comparing the vibration responses with those in Refs.[1,2] and Zhang's method [22].Considering the excitation force due to the static transmission error (Eq.( 24)), (meshing force of gear pair 12) and (meshing force of gear pair 34) are calculated using the present method, and they are also compared with the results in Ref. [2] (see Fig. 7).In the figure, and denote the meshing frequencies of gear pairs 12 and 34.The figure shows that two peaks ( = and = ) obtained from the present method agree well with those in Ref. [2], however, there are some errors for other peaks.Some reasons may lead to these errors as follows: 1) Some geometric parameters may be different because of the lack in Ref. [2], and some parameters of system are adopted by referring to the Ref. [22].2) The torque and other working condition are not provided in Ref. [2].calculating the displacement of the meshing spring.The new method not only has a good calculation accuracy, but also can greatly reduce the difficulty of programming relative to the conventional method, such as Kubur's method [1, 2] and Zhang's method [6].
2) For the studied geared rotor system provided in Ref. [2], the results show that the first-, second-and fourth-order natural frequencies are more easily excited, especially the first natural frequency.In addition, for the studied geared rotor system, the torsional vibration is easily excited relative to the lateral vibration at a low rotational speed.
1(a) and (b)) is described in Section 2.1, and the proposed model (see Figs. 1(c) and (d)) is introduced in Section 2.2.

Fig. 1 .
Fig. 1. a) Schematic of geared rotor systems, b) 3D model of the helical gear pair , c) FE model of the gear rotor system, d) projection drawing in -direction + sgn × ( ) cos = sgn × , of the two ends of spring in , and directions, respectively.

Fig. 8 .Fig. 9 .
Fig. 8. Gear meshing force amplitudes and maximum values under = 50 Nm: a) meshing force amplitude of , b) maximum values of , c) meshing force amplitude of , d) maximum values of gear pair

Table 1 .
Parameters of the gear system [9].4.Schematic of the spur gear rotor system[9]

Table 2 .
Natural frequencies of the spur gear rotor system

Table 3 .
Natural frequencies of the helical gear rotor system