Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system

Mechanical Systems and Signal Processing 76-77 (2016) 294–318 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing jou...

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Mechanical Systems and Signal Processing 76-77 (2016) 294–318

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Effects of tooth proﬁle modiﬁcation on dynamic responses of a high speed gear-rotor-bearing system Zehua Hu, Jinyuan Tang n, Jue Zhong, Siyu Chen, Haiyan Yan State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, Hunan, China

a r t i c l e in f o

abstract

Article history: Received 11 November 2015 Received in revised form 15 January 2016 Accepted 28 January 2016 Available online 15 February 2016

A ﬁnite element node dynamic model of a high speed gear-rotor-bearing system considering the time-varying mesh stiffness, backlash, gyroscopic effect and transmission error excitation is developed. Different tooth proﬁle modiﬁcations are introduced into the gear pair and corresponding time-varying mesh stiffness curves are obtained. Effects of the tooth proﬁle modiﬁcation on mesh stiffness are analyzed, and the natural frequencies and mode shapes of the gear-rotor-bearing transmission system are given. The dynamic responses with respect to a wide input speed region including dynamic factor, vibration amplitude near the bearing and dynamic transmission error are obtained by introducing the time-varying mesh stiffness in different tooth proﬁle modiﬁcation cases into the gearrotor-bearing dynamic system. Effects of the tooth proﬁle modiﬁcation on the dynamic responses are studied in detail. The numerical simulation results show that both the short proﬁle modiﬁcation and the long proﬁle modiﬁcation can affect the mutation of the mesh stiffness when the number of engaging tooth pairs changes. A short proﬁle modiﬁcation with an appropriate modiﬁcation amount can improve the dynamic property of the system in certain work condition. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Tooth proﬁle modiﬁcation Dynamic responses High speed Gear-rotor-bearing system

1. Introduction Analysis of vibration and noise behaviors in the dynamic engagement process is one of the hot research issues in the design and application of various gear mechanisms and gear-rotor transmission systems [1–4]. To improve the transmission stability, reduce the dynamic load and eliminate noise in the engagement process, three approaches are available: 1) changing the elasticity of the supporting shafts. This method can decrease the dynamic factor of the gear pair, but its effect is relatively weak; 2) reducing the mesh stiffness of the engaged gear pair to decrease the circular motion dynamic force of the transmission pair by varying the gear body structure. However, the change of the gear body may weaken its strength; 3) tooth proﬁle modiﬁcation, which is a generally used method in the practical condition [5]. For the high speed gear transmission system used in the aircraft engine, the strict requirements on high accuracy, high load carrying capacity and high reliability make it increasingly signiﬁcant to analyze the dynamic characteristics of the gear pair. Manufacturing error, assembly error and the elastic deformation of the supporting structure result in the abnormal engagement of the gear teeth. In order to improve the transmission property of the gear pair, extensive investigations about n

Corresponding author. E-mail addresses: [email protected] (Z. Hu), [email protected] (J. Tang), [email protected] (J. Zhong), [email protected] (S. Chen), [email protected] (H. Yan). http://dx.doi.org/10.1016/j.ymssp.2016.01.020 0888-3270/& 2016 Elsevier Ltd. All rights reserved.

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gear tooth modiﬁcation have been carried out [6–9]. Tavakoli and Houser [10] developed a procedure to calculate the static transmission errors and tooth load sharing both for the low and high contact ratio internal and external spur gears, and investigated the details of optimal proﬁle modiﬁcations for minimizing the static transmission errors. Cai and Hayashi [11] studied the optimum modiﬁcation of tooth proﬁle of a spur gear pair by taking the rotational vibration of the gear pair as the objective function. Litvin [12] developed a program for the design and analysis of modiﬁed asymmetric spur gear pairs, discussing the reduction of the transmission errors and the localization of the bearing contact. Subsequently, Litvin [13] concluded the purposes of modiﬁcation of geometry of spur and helical gears with parallel axes and helical gears with crossed axes and developed the tooth contact analysis computer program for simulation of meshing and contact of shaved pinion-gear tooth surfaces. Wagaj and Kahraman [14,15] investigated the effects of tooth proﬁle modiﬁcation on the helical gear durability and wear by introducing a function of tooth proﬁle modiﬁcation parameters into the gear transmission pair. Mohamad [16,17] studied the inﬂuences of convex tooth ﬂank form modiﬁcation on the general characteristics of transmission error of gear pairs. The actual contact ratio at the minimum point of peak-to-peak value of transmission error was affected obviously by the tooth ﬂank form modiﬁcation. Li [18] presented the calculation method of the transmission error of a spur gear pair with machine errors, assembly errors and tooth modiﬁcations and studied the effects of these factors on loading capacity, load sharing ratio and transmission error by using ﬁnite element method software. Chen [19] proposed a general analytical mesh stiffness model considering different tooth errors and investigated the effects of tooth proﬁle modiﬁcation, applied torque and tooth root crack on the time-varying mesh stiffness. Tang [20] established a threedimensional gear model with tooth proﬁle modiﬁcation and obtained the corresponding mesh stiffness based on the ﬁnite element method. The analytical method to calculate the mesh stiffness of the gear pair with tooth proﬁle modiﬁcation was given and the calculation results were consistent with those of the ﬁnite element method. As mentioned above, numerous researches have been conducted to investigate the effects of tooth proﬁle modiﬁcation on the transmission property of gear pairs in static condition or quasi-static condition. However, in practical dynamic work process, the load, assembly form and work speed are also important to the improvement of the dynamic behaviors. The optimal tooth proﬁle modiﬁcation should be chosen by taking the dynamic engagement process into account in order to adapt to the corresponding work condition. Song [21] proposed a multi-degree-of-freedom spur gear model considering the sliding friction and realistic time-varying stiffness and analytically investigated the effects of proﬁle modiﬁcation and sliding friction on the dynamic transmission error. In order to reduce the vibration and noise of spur gear pairs, Bonori [22] developed an original application of Genetic Algorithms considering the micro-geometric modiﬁcations to the tip and root relief of teeth and studied the features of the static transmission error of the gear pair by applying a ﬁnite element method. Liu [23] developed an analytical model of a multi-mesh spur gear train to study nonlinear gear dynamics due to timevarying mesh stiffness, proﬁle modiﬁcations and contact losses. The closed-form solution was compared with numerical integration and gave guidance for optimizing mesh phasing, contact ratios and proﬁle modiﬁcation amounts. Bahk [24] proposed an analytical model to capture the excitation from tooth proﬁle modiﬁcations at the sun-planet and ring-planet meshes and investigated the impact of tooth proﬁle modiﬁcation on system vibration responses. The tooth proﬁle modiﬁcation parameters that can minimize system responses were given. Eritenel [25] developed an analytical solution for the nonlinear vibration of gear pairs considering tooth surface modiﬁcations that exhibit partial and total contact loss. The analytical results of the model were compared with vibration experiment results. Chen [26] calculated the accurate mesh stiffness and static transmission error of a gear pair with tooth modiﬁcation by using the ﬁnite element method. The effects of modiﬁcation on the nonlinear dynamic characteristics were analyzed by introducing the mesh stiffness and static transmission error into the nonlinear dynamic model directly. Ma [27] developed a mesh stiffness model for the gear pair with addendum modiﬁcations and tooth proﬁle modiﬁcations, and analyzed the vibration behaviors of the ﬁnite element model of the gear pair with different modiﬁcation curves to obtain the optimum proﬁle modiﬁcation curve. Based on the analysis of the aforementioned researches, the main motivation of this paper is to develop a more accurate ﬁnite element model of the high speed gear-rotor-bearing transmission system and study the relationship between the tooth proﬁle modiﬁcation and the dynamic responses of the system. Section 2 gives the calculation of the mesh stiffness and studies the inﬂuences of the tooth proﬁle modiﬁcation on the time-varying mesh stiffness of the gear pair. Section 3 develops a ﬁnite element node model of the gear-rotor system considering the ﬂexible shafts, supporting bearing and meshing gear pair. The time-varying mesh stiffness, backlash, and transmission error excitation are introduced in order to simulate the meshing gear pair in a more reasonable manner. Section 4 mainly presents the analysis of the natural frequencies and mode shapes of the system, and investigates the effects of the tooth proﬁle modiﬁcation on the dynamic responses of the gear-rotor-bearing system. Section 5 draws the main conclusions of the simulation results of the gear-rotor system with different tooth proﬁle modiﬁcations.

2. Tooth proﬁle modiﬁcation and mesh stiffness 2.1. Gear model with tooth proﬁle modiﬁcation Tooth proﬁle modiﬁcation is widely used to avoid the impact of the gear pair during the engagement process, which is realized by removing some material from the tooth surface and keeping the base pitch of the driving pinion equal to the base pitch of the driven gear. Thus, the stability of the transmission behavior can be improved.

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A gear tooth model with tooth proﬁle modiﬁcation is illustrated in Fig. 1. D in Fig. 1 refers to the double-teethengagement region, and S refers to the single-tooth-engagement region. HPSTC is the highest point of single tooth contact, LPSTC is the lowest point of single tooth contact and PP is the pitch point of the tooth. The tooth proﬁle modiﬁcation parameters mainly consist of three factors: the maximum proﬁle modiﬁcation amount Δmax , the length of proﬁle modiﬁcation h and the modiﬁcation curve form. As there is no uniﬁed formula for the calculation of the maximum modiﬁcation amount, the following two methods are usually adopted to obtain the modiﬁcation amount: 1) calculating the modiﬁcation amount based on the deformation of the loaded teeth, and 2) gaining the modiﬁcation amount according to the accuracy level of the pinion and gear. In this paper, different modiﬁcation amounts are introduced into the model to analyze the effects of the proﬁle modiﬁcation on the dynamic responses of the transmission system. The position of the modiﬁcation starting point is determined by the length of proﬁle modiﬁcation. In this research, two different tooth proﬁle modiﬁcation lengths are involved, namely, the long proﬁle modiﬁcation (LPM) and the short proﬁle modiﬁcation (SPM). For the long proﬁle modiﬁcation, the modiﬁcation length is corresponding to the highest point of single tooth contact (HPSTC); for the short proﬁle modiﬁcation, the modiﬁcation length is corresponding to the midpoint of the HPSTC and the highest point of the addendum. The modiﬁcation curve form is mainly determined by x n Δ ¼ Δmax ð1Þ h Here, x refers to the coordinate of the random contact point with the modiﬁcation starting point in the action of line as the origin. n is the exponent of the modiﬁcation curve. n ¼ 1 refers to the straight line modiﬁcation form and n ¼ 2 is the parabola modiﬁcation. In this paper, the teeth of the pinion and gear are modiﬁed with parabola curves. 2.2. Mesh stiffness of the gear pair with modiﬁcation For the gear transmission system, the time-varying mesh stiffness is one of the main excitations of the sound and vibration behaviors. It has an obvious inﬂuence on the dynamic responses of the meshing gear pair. Changing the mesh stiffness of the gear pair with the tooth proﬁle modiﬁcation taken into the teeth can improve the transmission property of the system. In this section, the approaches to calculation of the mesh stiffness with proﬁle modiﬁcation are given and the effects of the tooth proﬁle modiﬁcation on the mesh stiffness are studied in detail. 2.2.1. Finite element method to calculate mesh stiffness As mentioned in Refs. [20,28], there are two methods to obtain the time-varying mesh stiffness of the gear pair with proﬁle modiﬁcation: the ﬁnite element method and the analytical calculating method. First, the mesh stiffness of the gear pair with modiﬁcation can be gained by applying a quasi-static tooth contact analysis with the ﬁnite element method. The design parameters of the gear pair are listed in Table 1. A high accuracy threedimensional geometry model of the gear pair is obtained through a virtual manufacturing process based on the CATIA software as shown in Fig. 2. The ﬁnite element model of the gear pair with ﬁve pairs of meshing teeth is shown in Fig. 3. Fig. 3(a) gives the boundary conditions of the geometry model, where two reference points are attached to the centroid of the gear pair, and coupled with the inner ring. Fig. 3(b) illustrates the meshing grid form of the ﬁnite element model. The hexahedron C3D8R element is adopted to mesh the gear teeth and body to guarantee the accuracy of the solution. The torque is applied to the gear and max

D h S D

SPM HPSTC (LPM) PP LPSTC

Fig. 1. Gear tooth model with tooth proﬁle modiﬁcation.

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the rotational displacement is added to the pinion. Based on the quasi-static analysis of the ﬁnite element model of the gear pair, the time-varying mesh stiffness of the engaged gear pair can be obtained. Ignoring the effect of inertia force and damping force during the quasi-static analysis, the motion equation for the ﬁnite element model can be simpliﬁed as Kδ ¼ F

ð2Þ

where, K is the matrix of structural mesh stiffness, F is the external force and δ is the displacement of ﬁnite element. Taking the errors into account, the no loaded transmission error (NLTE) still exists in spite of the fact that no tooth deformation occurs under torque, and it is unavoidable in the loaded tooth contact analysis because of the errors, i.e. the geometrical error and the numerical calculation error. The displacement of the ﬁnite element which is equal to the deformation can be given as

δ ¼ Rbp θp Rbg θg NLTE

ð3Þ

where, θp and θg refer to the rotation angles of the gear pair; Rbp and Rbg represent the respective radii of the base circle of the driving pinion and the driven gear. Table 1 Design parameters of the gear pair. Parameters

Pinion

Gear

Number of teeth Module (mm) Pressure angle (deg) Tooth width (mm) Addendum coefﬁcient Tip clearance coefﬁcient Load (Nm)

29 3 25 14 1 0.25 127.5

49 3 25 12 1 0.25 215.5

Fig. 2. Three-dimensional geometry model of the gear pair.

Fig. 3. Finite element model of the gear pair. (a) Boundary conditions (b) Meshing grid form.

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Thus, the time-varying mesh stiffness in the ﬁnite element method can be written as km ¼ K ¼

F

δ

¼

F Rbp θp Rbg θg NLTE

ð4Þ

2.2.2. Analytical method to calculate mesh stiffness In Kuang's research [29], a curve ﬁtted tooth stiffness equation was developed to calculate directly the variable gear mesh stiffness which was applicable to both the standard full-depth and addendum modiﬁed involute gears. However, the calculation formula for the mess stiffness proposed by Kuang took no account of the effects of the load on the mesh stiffness. By introducing a load coefﬁcient, Cai and Tang revised the calculation formula and proposed the analytical method to calculate the mesh stiffness considering the load condition and the tooth proﬁle modiﬁcation. The mess stiffness results obtained by applying the ﬁnite element method and the analytical method were compared, and the accuracy of the analytical method was validated [20]. With the use of the curve ﬁtting technique, the variation of stiffness constant for gear i ði ¼ p: pinion; i ¼ g: gearÞ at different loading position r i can be approximated quite adequately by the following equation K i ðr i Þ ¼ ðA0 þ A1 X i Þ þ ðA2 þA3 X i Þ

r i Ri ð1 þ X i Þm

ðN=μm=mmÞ

ð5Þ

where r i is the radius of loading position, Ri is the pitch radius, X i is the addendum modiﬁcation coefﬁcient and zi is the number of teeth for gear and pinion. The coefﬁcients can be expressed as A0 ¼ 3:867 þ 1:612zi 0:0291z2i þ 0:0001553z3i A1 ¼ 17:060 þ0:7289zi 0:01728z2i þ 0:00009993z3i A2 ¼ 2:637 1:222zi þ 0:02217z2i 0:0001179z3i A3 ¼ 6:330 1:033zi þ 0:02068z2i 0:0001130z3i

ð6Þ

The load coefﬁcient taken into account can be expressed as cðFÞ ¼ 0:8497 þ 0:7743F 2 1:1631F 2 þ 0:6637F 3

ð7Þ

where FðkN=mmÞ is the normal contact force of the unit tooth width. Considering the load coefﬁcient, the single stiffness of a single tooth pair is calculated by combining the single tooth stiffness constants K 1 ðr 1 Þ and K 2 ðr 2 Þ as springs connected in series: KN s ¼ K s cðFÞ ¼

K 1 ðr 1 ÞK 2 ðr 2 Þ cðFÞ K 1 ðr 1 Þ þ K 2 ðr 2 Þ

ð8Þ

where the sub-indexes 1 and 2 refer to the driving pinion and the driven gear, respectively. r 1 and r 2 are the radial distances of the contact point. The total mesh stiffness can be written as K m ¼ cðζ A FÞK As þ cðζ B FÞK Bs

ð9Þ

where ζ A and ζ B are the load sharing ratio of the engaged gear pair, which can be simpliﬁed and set as 0.5. The indexes A and B refer to different mating tooth pairs when the contact ratio is larger than 1. And K As ¼

K A1 ðr A1 ÞK A2 ðr A2 Þ K A1 ðr A1 Þ þ K A2 ðr A2 Þ

; K Bs ¼

K B1 ðr B1 ÞK B2 ðr B2 Þ K B1 ðr B1 Þ þ K B2 ðr B2 Þ

ð10Þ

The tooth meshing out process of the gear transmission pair with proﬁle modiﬁcation is shown in Fig. 4. As noted with pb in Fig. 4, for the length of the base pitch of gear teeth with proﬁle modiﬁcation, there exists a small amount of reduction, which is equal to the tooth proﬁle modiﬁcation amount Δ in the corresponding position. Due to the reduction of the length of the base pitch, tooth 1 gets engagement ﬁrst, and then tooth 2 comes into meshing when the deformation of tooth 1 in the line of action is equal to the tooth proﬁle modiﬁcation amount of tooth 2. According to the foregoing meshing out process of the gear pair, the calculation formula of the mesh stiffness of the gear pair with tooth proﬁle modiﬁcation during the tooth meshing out process can be obtained as F k2 ð11Þ k¼ F þ Δðk2 k1 Þ where k2 and k1 are the mesh stiffness of double-tooth-pair engagement and the mesh stiffness of single-tooth-pair engagement in the corresponding meshing position. The analytical results and the ﬁnite element analysis results of the time-varying mesh stiffness with tooth proﬁle modiﬁcation were compared by Cai and Tang, and the relative error of the results was around 3%, through which the accuracy of the analytical method was guaranteed.

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2.2.3. Effects of tooth proﬁle modiﬁcation on mesh stiffness In this sub-section, the time-varying mesh stiffness of the gear pair with tooth proﬁle modiﬁcation is given by applying both the ﬁnite element method and the analytical calculating approach, and the effects of the proﬁle modiﬁcation on the time-varying mesh stiffness are investigated. There is no standard formula to calculate the tooth proﬁle modiﬁcation amount. Generally, the proﬁle modiﬁcation amount can be obtained either by the approach based on the deformation of the gear teeth or by the approach according to the level of the manufacturing accuracy. Taking the manufacturing accuracy into account, the following equation can provide a reference value of tooth proﬁle modiﬁcation amount of the gear in static condition.

Δmax ¼ 2Emse þ δHPSTC

ð12Þ

is the deformation value of the highest point of single tooth contact. Emse is the pitch error, and δ HPSTC For this gear pair, based on its work condition and manufacturing accuracy, Emse ¼ 710μm, δ ¼ 8:9μm, so Δmax ¼ 28:9μm. However, the aforementioned tooth proﬁle modiﬁcation amount is only a reference value, and it may not be the most appropriate modiﬁcation amount considering the dynamic behaviors and the assembly form of the gear pair in practical work condition. In order to study the effects of tooth proﬁle modiﬁcation on the dynamic responses and mesh stiffness of the gear-rotorbearing system in dynamic work condition, different maximum tooth proﬁle modiﬁcation amounts are introduced into the gear pair and studied in detail. The modiﬁcation parameters of the pinion are listed in Table 2. The teeth of the pinion are modiﬁed with the parabola curves with different maximum tooth proﬁle modiﬁcation amounts and different lengths of proﬁle modiﬁcation. In Table 2, N notes the case that the teeth are normal and no proﬁle modiﬁcation is taken into account: S10, S20 and S30 note the cases where the short proﬁle modiﬁcations with a maximum tooth proﬁle modiﬁcation amount of 10μm, 20μm and 30μm are taken into account; L10, L20 and L30 note the cases where the long proﬁle modiﬁcations with a maximum tooth proﬁle modiﬁcation amount of 10μm, 20μm and 30μm are considered. The time-varying mesh stiffness curves of the gear pair with tooth short proﬁle modiﬁcation are illustrated in Fig. 5. FEM refers to the time-varying mesh stiffness obtained with the ﬁnite element method and AM refers to that gained with the analytical method. The amplitude of the AM mesh stiffness in the double-teeth-engagement region is higher than that of the FEM mesh stiffness, while the amplitude of the AM mesh stiffness in the single-tooth-engagement region is relatively close to that of the FEM mesh stiffness. The mutation of the mesh stiffness curve from the double-teeth-engagement region to the single-tooth-engagement region turns into a gentle process when the short proﬁle modiﬁcation is carried out, which is observed both in the FEM result and in the AM result. By comparing the mesh stiffness curves with different maximum tooth proﬁle modiﬁcation amounts in the mutation region, it is observed that the value of the mesh stiffness in this region reduces when the teeth are modiﬁed and the reduction of the mesh stiffness enlarges with the increase of the maximum HPSTC

Δ D

pb C Tooth 1

B

Tooth 2

A

Fig. 4. Illustration of tooth meshing out.

Table 2 Modiﬁcation parameters of the pinion. Parameters

N

S10

S20

S30

L10

L20

L30

Δmax =μm h=mm n

0 0 2

10 1.0994 2

20 1.0994 2

30 1.0994 2

10 2.1226 2

20 2.1226 2

30 2.1226 2

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Fig. 5. Mesh stiffness of gear pair with tooth short proﬁle modiﬁcation.

Fig. 6. Mesh stiffness of gear pair with tooth long proﬁle modiﬁcation.

tooth proﬁle modiﬁcation amount. The curve of mesh stiffness for case S10 is relatively gentler compared with those for case S20 and S30. With the increase of the maximum tooth proﬁle modiﬁcation amount, the rotation angle corresponding to the single-tooth-engagement region enlarges, while the rotation angle corresponding to the double-teeth-engagement region decreases. The time-varying mesh stiffness curves of the gear pair with tooth long proﬁle modiﬁcation are illustrated in Fig. 6. The amplitude of the AM mesh stiffness in the double-teeth-engagement region is higher than that of the FEM mesh stiffness, while the amplitude of the AM mesh stiffness in the single-tooth-engagement region is relatively close to that of the FEM mesh stiffness. The long proﬁle modiﬁcation also has a becoming-gentle effect on the mutation of the mesh stiffness curve. The value of the mesh stiffness in the mutation region reduces when the teeth are modiﬁed and the reduction of the mesh stiffness enlarges with the increase of the maximum tooth proﬁle modiﬁcation amount. Compared to the mesh stiffness curve with short proﬁle modiﬁcation, the effects of the long proﬁle modiﬁcation on the mesh stiffness amplitude of the double-teeth-engagement region are more obvious. However, the length of proﬁle modiﬁcation h corresponding to the long proﬁle modiﬁcation is larger than that of the short proﬁle modiﬁcation, which means that more material needs to be cut from the gear teeth. As analyzed above, both the short proﬁle modiﬁcation and the long proﬁle modiﬁcation can affect the mutation of the mesh stiffness when the number of meshing tooth pairs turn from 2 to 1, and the amplitude of the mesh stiffness reduces with the introduction of the tooth proﬁle modiﬁcation. As one of the main inner excitations of vibration and sound for the gear transmission system, the change of time-varying mesh stiffness may have a great inﬂuence on the dynamic responses of the system. By introducing the time-varying mesh stiffness result with different tooth proﬁle modiﬁcations into a dynamic model of a gear-rotor-bearing system, the effects of the tooth proﬁle modiﬁcation on the dynamic responses of the system are investigated in Section 3.

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28 32 19 22 1718 20 21 23 24 25 26 27 29 30 31 33

B1

B2 D1

L1

B3

G1

B4 G2

D2

L2

B2

B1

B4

B3

16 14 13 12 10 9 8 7 6 5 4 2 1 3 15 11

Input shaft

Output shaft

Fig. 7. Engineering drawing of a high speed gear-rotor-bearing system. (a) Input shaft (b) Output shaft.

3. Dynamic model of a gear-rotor-bearing system An engineering drawing of a high speed gear-rotor-bearing system used in the aircraft engine is illustrated in Fig. 7. In Fig. 7(a), the researched driving pinion G1 and another pinion D1 are attached to the input shaft, which is supported by two ball bearings B1 and B2. In Fig. 7(b), the corresponding researched driven gear G2 and another gear D2 are attached to the output shaft, which is supported by two ball bearings B3 and B4. L1 and L2 are the loads of the hollow input shaft and the hollow output shaft, respectively. Here, the gear transmission pair including the driving pinion G1 in engagement with the driven gear G2 is our research object. D1 and D2 are in contact with other pinions and gears which are not shown. The dynamic model of the high speed gear-rotor-bearing system can be developed by applying the node ﬁnite element method, which consists of four main parts as the gear-rotor disc, the meshing gear pair, the elastic rotor shaft element and the supporting bearings, according to our previous work in Ref [30]. The input shaft is divided into 16 nodes (1 to 16 node), and the output shaft is divided into 17 nodes (17 to 33 node). The corresponding dimension parameters of the shaft element are listed in Table 3. 3.1. Modeling of gear-rotor disc, shaft element and supporting bearing The simpliﬁed illustration of the dynamic model of the gear-rotor-bearing system is shown in Fig. 8. For the input shaft and the output shaft, 33 nodes and six freedoms for each node are considered. The gear-rotor disc is supposed to be rigid with three translational displacements and three rotational displacements. The four gear-rotor disks are located at nodes 7, 11, 22 and 28, respectively. The four supporting bearings are located at nodes 1, 16, 17 and 33, respectively. The governing equations of the gear rotor, shaft element, and supporting bearing are deduced in our previous work [30]. In this paper, only the results of the equations are given as follows: (1) Gear-rotor disc The motion equation of the gear-rotor disc can be written as Mdi q€ i þ Ωi Gdi q_ di ¼ Fdi d

ð13Þ

where T qdi ¼ xi ; yi ; zi ; θxi ; θyi ; αi

ð14Þ

The mass matrix and the gyroscopic matrix can be written as Mdi ¼ diag mi ; mi ; mi ; J Di ; J Di ; J Pi 2

ð15Þ

3

0 60 6 6 60 Gdi ¼ 6 60 6 6 40

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

J Pi

0

0

J Pi

0

07 7 7 07 7 07 7 7 05

0

0

0

0

0

0

ð16Þ

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Table 3 Dimension parameters of the shaft element. Node number

Node number

Length of shaft (m)

Inner diameter of shaft (m)

Outer diameter of shaft (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

0.0075 0.003 0.007 0.0095 0.0095 0.007 0.007 0.0125 0.0125 0.009 0.009 0.016 0.013 0.003 0.00925 0.0075 0.003 0.0135 0.0135 0.006 0.006 0.01325 0.01325 0.01325 0.01325 0.007 0.007 0.009 0.009 0.003 0.007

0.020 0.020 0.020 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.025

0.040 0.046 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.045 0.035 0.040 0.046 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.042 0.035

D2 G2

O4

D1

G1

O3

Fig. 8. Simpliﬁed illustration of the dynamic model.

1 1 d d _ i θxi θ_ yi θyi θ_ xi þ J Pi ðΩi Þ2 þJ Pi Ωi α _i Q~ i qdi ; q_ i ¼ J Pi α 2 2

ð17Þ

The three coordinates x, y and z refer to the translational displacements of the gear-rotor along the radial direction and the axial direction. The two angular coordinates θxi ði ¼ p; gÞ and θyi describe the angle motions of the rotor. θxi is an angular displacement in the xz plane, referring to a small rotation around the x-axis, and θyi is an angular displacement in the yz _ i is the torsional velocity around the z-axis. plane, referring to a small rotation around the y-axis. Ωi is the spin speed and α mi , J Di and J Pi are the mass, transverse mass moment of inertia and polar mass moment of inertia of the pinion and gear, respectively. The parameters of the gear-rotor disc are listed in Table 4. (2) Shaft element As shown in Fig. 9, the ﬁnite element model of the ﬂexible shaft is developed on the Timoshenko beam theory. The generalized coordinate XYZ is ﬁxed to a two-node three-dimensional beam element of length l, where each node has six

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degrees of freedom to describe its motion. The nodal displacement vector qs of the ﬂexible shaft element is given by T qs ¼ xi ; yi ; zi ; θxi ; θyi ; θzi ; xi þ 1 ; yi þ 1 ; zi þ 1 ; θxði þ 1Þ ; θyði þ 1Þ ; θzði þ 1Þ

ð18Þ

where ðxi ; xi þ 1 Þ, ðyi ; yi þ 1 Þ and ðzi ; zi þ 1 Þ are the translational displacements along the radial direction; ðθxi ; θxði þ 1Þ Þ, ðθyi ; θyði þ 1Þ Þ and ðθzi ; θzði þ 1Þ Þ are the rotational displacements. Following a Lagrangian approach, the kinetic energy and potential energy of the shaft element can be introduced into its motion equation without damping effect and the matrix form can be obtained as Ms q€ þ ΩGs q_ s þ Ks qs ¼ 0 s

ð19Þ

The mass matrix, gyroscopic matrix and stiffness matrix of the shaft can be found in Appendix A of Ref. [30]. (3) Supporting bearing As shown in Fig. 7, the shafts of the gear transmission system are supported by four ball bearings, which can be simpliﬁed as a spring–damper system. The shafts of the system are supposed to be able to rotate around the axial line freely, so that the values of the stiffness matrix in the sixth row and the sixth column are zeros. Then, the stiffness matrix of the bearing can be written as 2

kxx

6 6 kyx 6 6k 6 zx B K ¼6 6 kθ x 6 x 6 4 kθy x 0

kxy

kxz

kxθx

kxθy

kyy

kyz

kyθx

kyθy

kzy

kzz

kzθx

kzθy

kθx y

kθ x z

kθx θx

kθx θy

kθ y y

kθ y z

kθ y x

kθy θy

0

0

0

0

0

3

7 07 7 07 7 7 07 7 7 05

ð20Þ

0

Here, kii ði ¼ x; yÞ, kθi θi ði ¼ x; yÞ and kzz are the radial, tilting and axial stiffness of the ball bearing, respectively. The damping matrix of the bearing is assumed to be identical with the bearing stiffness matrix and can be written as 2

cxx 6 cyx 6 6 6 czx CB ¼ 6 6c 6 θx x 6 4 cθy x 0

cxy

cxz

cxθx

cxθy

cyy

cyz

cyθx

cyθy

czy

czz

czθx

czθy

cθ x y

cθx z

cθx θx

cθ x θ y

cθy y

cθy z

cθy x

cθy θy

0

0

0

0

0

3

07 7 7 07 7 07 7 7 05

ð21Þ

0

The bearing parameters of the system obtained with Romax and KISSsoft software are listed in Table 5. Table 4 Parameters of the gear-rotor disc. Node number

Inner diameter of the rotor (m)

Outer diameter of the rotor (m)

Thickness of the rotor (m)

7 11 22 28

0.035 0.035 0.030 0.030

0.087 0.142 0.147 0.093

0.014 0.018 0.012 0.012

Node i+1

Y

θ yi θ zi

θ xi X

θy

Node i

yi xi

θx s

θz y

θ y ( i +1)

θ x (i +1)

θ z (i +1)

yi +1 x i +1

x

Z

l

Fig. 9. Typical ﬁnite ﬂexible shaft element and coordinates.

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3.2. Modeling of the meshing gear pair As shown in Fig. 10, the typical dynamic model of the gear transmission pair is obtained with the pinion and gear (G1 and G2) supposed as rigid bodies linked by line elements that refer to the mesh stiffness and damping effect. For D1 and D2, only the gear-rotor disc effect is considered, and the meshing gear pair effect is ignored in the system. T The coordinates of G1 and G2 xi ; yi ; zi ; θxi ; θyi ; θzi ; ði ¼ p; gÞ are deﬁned as two local Cartesian inertial coordinates, whose origins Op and Og are at the centroid of the pinion and the gear, respectively. xi ; yi and zi are the translational displacements of the gear pair, θzi is the rotational displacement around Z i shaft. θxi and θyi , which are relatively small vibratory quantities, refer to the rotational displacements of the gear pair around X i and Y i shafts. Moreover, all the coordinate shafts are assumed to coincide with the principal axes of inertia of each body in static equilibrium condition. The meshing gear pair is supposed to be forced by the mesh stiffness elastic force and the damping force with the static transmission error excitation (este ðtÞ) considered. The dynamic displacement of the gear pair along the line of action can be expressed as

δm ¼ Vqd este ðtÞ

ð22Þ

Here, " d

q ¼

qp

# ð23Þ

qg

And V¼

h

sin ðαn Þ

cos ðαn Þ

0

0

0

r bp

sin ðαn Þ

cos ðαn Þ

0

0

0

r bg

i

ð24Þ

where αn is the pressure angle of the gear pair, r bi ði ¼ p; gÞ is the radius of the base circle. The static transmission error excitation of the system consists of two main parts, namely, a high frequency short term component which includes the tooth proﬁle error and the base pitch spacing error, and a low frequency long term component which refers to the accumulative pitch error during one revolution of the pinion shaft. The total transmission error excitation of the system can be written as este ðtÞ ¼ e1 sin ð2π f m t þ φ1 Þ þ e2 sin ð2π f s t þ φ2 Þ

ð25Þ

Table 5 Bearing parameters. Stiffness

Bearing 1–3

Bearing 2–4

kxx ; kyy ðN=mÞ

1:10 108 , 1:44 108

2:41 108 ,3:04 108

kzz ðN=mÞ

5:37 106

73:97 106

kθx θx ; kθy θy ðNm=radÞ

2:48 10 ,1:43 10

Damping c ðNs=mÞ

Bearing 1–3

Bearing 2–4

1 103

1 103

3

3

25:32 103 ,15:55 103

Yg

θ yg θ xg Xg Yp

Og

km

θ zg Z g

cm

θ yp

θ xp Xp Op

θ zp

Zp

Fig. 10. Typical dynamic model of gear transmission pair (G1 and G2).

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Here, e1 is the amplitude of the tooth proﬁle error and tooth pitch error and e2 is the amplitude of the accumulative pitch error; f m is the mesh frequency and f s is the input shaft frequency of the pinion. φ1 and φ2 refer to the phase angles, which are assumed to be zero. For the meshing gear pair, the mesh stiffness matrix and the mesh damping matrix can be written as Cm ¼ c m V T V

Km ¼ km VT V;

ð26Þ

where km and cm are the time-varying mesh stiffness and the mesh damping coefﬁcient. Taking the backlash into account, the total mesh force of the meshing gear pair can be written as F m ¼ km ∙ðγ m0 δm þ γ m1 BÞ þ cm ∙γ m0 ∙δ_ m

ð27Þ

where the backlash function is written as 8 > < 1 0 γ m1 ¼ > : 1

δm 4 B

(

else ; γ m0 ¼ δm o B

1 0

δm 4 B

ð28Þ

else

B is half of the backlash. Then, the motion equation in matrix form of the meshing gear pair shown in Fig. 10 can be written as Mm q€ þ Cm γ m0 q_ þ Km γ m0 qd ¼ km γ m1 BVT d

d

ð29Þ

Here, the mass matrix of the gear pair is given as h Mm ¼ diag mp

mp

mp

J Dp

J Dp

J Pp

mg

mg

mg

J Dg

J Dg

J Pg

i

ð30Þ

3.3. Modeling of system Above all, after the gear-rotor disc, meshing gear pair, ﬂexible shaft element and supporting bearing have been modeled, the total matrices M, G, C, K and Q of the gear-rotor system can be gained by assembling the matrices of the gear-rotor disc, meshing gear pair, and discrete bearing onto the matrices of the ﬂexible shaft like the stiffness matrix [31]. The procedure of the development of a gear-rotor-bearing dynamic model is illustrated in Fig. 11. Eventually, the dynamic motion differential equation of the gear-rotor system can be deduced as Mq€ þðΩG þ CÞq_ þ KfðqÞ ¼ Q

ð31Þ

Here, M, G, Cand Kare the mass matrix, the gyroscopic matrix, the damping and stiffness matrix, respectively; q is the generalized displacement of each node and Q is the generalized force excitation. f ðqÞ is the nonlinear displacement function corresponding to gear backlash.

Fig. 11. Flow chart of the development of a high speed gear-rotor-bearing dynamic model.

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4. Numerical analysis and discussion 4.1. Analysis of natural frequency and mode shape The geometrical parameters of the high speed gear-rotor-bearing system corresponding to Figs. 7, 8 and 10 are given in Tables 1, 3, 4 and 5. The amplitudes of the backlash and transmission error excitation of the system are obtained based on the accuracy level and the center distance of the gear pair. In the numerical analysis of the system, the values of the backlash and the transmission error excitation are set as B ¼ 35μm, e1 ¼ 20μm; e2 ¼ 0μm, and φ1 ¼ φ2 ¼ 0. Here, the effects of the accumulative pitch error are ignored. The shaft and the gear pair are steel and have an elastic modulus of 210 GPa, and a density of 7800kg=m3 with Poisson's ratio 0.3. In the analysis of the natural frequency and mode shape of the gear-rotor-bearing system, there are 16 nodes for the input shaft and 17 nodes for the output shaft. Therefore, a total of 33 6 degree-of-freedoms are considered for the system. To obtain the natural frequencies and mode shape of the system, the time-varying mesh stiffness without tooth proﬁle modiﬁcation is assumed to be constant with its mean value as shown in Fig. 5. The Campbell diagram of the gear-rotor-bearing system is shown in Fig. 12. The lateral coordinate is the input speed of the pinion and the vertical coordinate refers to the natural frequency of the system. The Synchronous Whirl Line corresponding to the mesh frequency f m is added to the Campbell diagram for the analysis of the critical speed. The former orders of the natural frequency, and the corresponding damping ratios of the system are listed in Table 6. Fig. 12 (a) gives the whirling speed region that appears in the changing process of the input speed from low to high (0– 10,000 rpm), and Fig. 12 (b) illustrates the Campbell diagram in region (700, 1500) Hz. The former orders of the natural frequency of the gear-rotor-bearing system have no obvious change over the speed range. For modes 1–8, the damping ratio of the system is 1, which is a critical damping ratio, and the vibration displacements of the system in these frequency regions will weaken rapidly. In Fig. 12(c), the Campbell diagram in region (1800, 2400) Hz is given. The natural frequencies change obviously with the increase of the input speed, and an obvious frequency veering frequency phenomenon is observed between the curves of natural frequency 2054.0 Hz and 2090.3 Hz near 2500 rpm. The normal input operating shaft speed of the pinion is Ω ¼ 7500rpm. The Campbell diagram near this input speed (3000, 4000) Hz is illustrated in Fig. 12(d). There is no obvious change in the natural frequency of the gear-rotor-bearing system over the speed range. The corresponding mode shapes of the input and output shafts for different natural frequencies are shown in Fig. 13. The lateral coordinate refers to the node of the input shaft and output shaft, and the vertical coordinate represents the mode shapes of the system. In Fig. 13(a) and (b), the mode shapes for the former two natural frequencies (177 Hz and 185 Hz) are given. The input shaft of the system is in a torsional motion with respect to z-axis, while, the output shaft has both a weak lateral vibration mode and an obvious torsional mode, and the maximum values of the torsional amplitude occur in the ends of the shafts. In Fig. 13(c), for the mode shape corresponding to the natural frequency (2054.0 Hz) where an obvious frequency veering frequency phenomenon is observed, the input shaft is in a torsional motion with respect to x-axis, while the output shaft is in a coupled lateral–torsional vibration mode. The maximum values of the torsional amplitude occur near the gear-rotor disc (node 11 and node 22). As shown in Fig. 13(d) and (e), corresponding to the natural frequencies near the normal input operating shaft speed of the pinion Ω ¼ 7500rpm (3415.9 Hz and 3509.0 Hz), both the input shaft and the output shaft of the system have an obvious lateral–torsional vibration mode shape. The maximum values of the torsional amplitude occur near the gear-rotor disc (nodes 7, 11, 22 and 28). According to the above analysis about the mode shapes, the lateral-torsional vibration may dominate the overall vibration behaviors of the gear-rotor-bearing system. 4.2. Effects of tooth proﬁle modiﬁcation on dynamic responses In the numerical analysis of the dynamic responses, the Newmark's method is used to integrate the dynamic system. In each meshing period, the integration steps related to the input shaft speed are changing and 200 points are calculated. Among the dynamic responses of the system, the dynamic transmission error and the vibration amplitude near the bearing node of the system are chosen as the references to evaluate the effects of the tooth proﬁle modiﬁcation. The dynamic transmission error of the system refers to the error between the actual values and the theoretical values of the driven gear’s torsional angle displacement with respect to z-axis, which can be used to evaluate the stability of the load and transformation of power. The vibration amplitude near the bearing can be taken to analyze the relationship between the vibration of the meshing gear pair and the vibration of the gearbox's supporting bearings. In this section, the time-varying mesh stiffness results of the gear pair with different tooth proﬁle modiﬁcations are obtained with the analytical method. Considering the high speed and heavy load work condition of the system, cutting more material from the gear teeth in long proﬁle modiﬁcation will be a relatively extreme measure that may weaken the strength of the gear pair. Therefore, this section will mainly focus on the effects of the short proﬁle modiﬁcation on the system responses rather than on the inﬂuences of the long proﬁle modiﬁcation. The effects of the tooth proﬁle modiﬁcation on the dynamic responses are investigated by introducing the time-varying mesh stiffness in different tooth proﬁle modiﬁcation cases into the gear-rotorbearing dynamic system. The following are detailed analyses.

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Fig. 12. Campbell diagram (a) Campbell diagram in region (0, 7000) Hz (b) Campbell diagram in region (700, 1500) Hz (c) Campbell diagram in region (1800, 2400) Hz (d) Campbell diagram in region (3000, 4000) Hz.

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Table 6 Natural frequency and damping ratio of the system. Order

Natural frequency (kHz)

Damping ratio

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0 0.0177 0.0185 0.0306 0.0315 0.0315 0.0382 0.0441 0.0491 0.7770 0.8534 1.0337 1.3035 1.3319 1.4110 2.0030 2.0540 2.0903 2.1436 3.4159 3.5090 3.7544 4.2483 6.0492

1 1 1 1 1 1 1 1 0.0617 0.0682 0.0209 0.0253 0.0255 0.0218 0.0110 0.0088 0.0117 0.0095 0.0140 0.0136 0.0127 0.0228 0.0056

4.2.1. Effects of short proﬁle modiﬁcation on dynamic responses Case 1. No tooth proﬁle modiﬁcation The time-varying mesh stiffness of the gear pair without tooth proﬁle modiﬁcation is introduced into the numerical analysis and the dynamic responses of the system are given in Fig. 14. C0 notes that the gear teeth are not modiﬁed. In Fig. 14 (a), the lateral coordinate refers to the increasing process of the rotation speed of the input shaft and the vertical coordinate is the root-mean-square value of the vibration amplitude in x-direction of bearing 1. The ﬁgure shows that some natural frequencies corresponding to Table 6 are excited, including ωn ¼1025.15 Hz, 2035.32 Hz, 3395.42 Hz, 3866.18 Hz, and 4308.92 Hz. In Fig. 14(b), the root-mean-square value of the dynamic transmission error in the overall increasing process of the input speed is illustrated. Some natural frequencies such as ωn ¼1005.33 Hz, 1530.23 Hz, 2151.8 Hz, 3181.78 Hz, 3686.87 Hz and 4172.62 Hz are excited, which is consistent with the vibration of the bearing. As illustrated in the ﬁgure, the main trends for both the vibration of the bearing and the dynamic transmission error are almost consistent, expect that the peaks of the transmission error curve are weaker compared with those of the vibration of the bearing. The effects of the natural frequency and critical characteristics on the vibration of the bearing are stronger than those on the transmission error, which may be related to the coupled lateral–torsional motion of the gear-rotor-bearing system. Case 2. Modiﬁed G1 with short proﬁle modiﬁcation In this case, short proﬁle modiﬁcations with different values are made to the driving pinion G1 and the time-varying mesh stiffness of the gear pair corresponding to tooth proﬁle modiﬁcation is introduced into the numerical analysis of the dynamic responses. The analysis results of the system behavior when the input speed increases from 2000 rpm to 10,000 rpm are shown in Fig. 15. C0 notes that the gear teeth are not modiﬁed. CS10, CS20 and CS30 refer to the cases, where the short proﬁle modiﬁcations with a maximum tooth proﬁle modiﬁcation amount of 10μm, 20μm and 30μm are made to G1, respectively. With the normal input operating shaft speed of G1 as Ω ¼ 7500rpm, a magniﬁcation picture of the system responses near this input speed is illustrated in Fig. 16. As shown in Figs. 15 and 16, for some input speed regions, the introduction of the short proﬁle modiﬁcation can reduce the values of the vibration amplitude of the bearing and the dynamic transmission error. The reduction of the system responses enlarges with the increase of the maximum tooth proﬁle modiﬁcation amount from 10μm to 30μm. However, there exists a limit boundary for the effect of the short proﬁle modiﬁcation on the reduction of the system responses. As observed from the ﬁgures, the differences between the dynamic responses corresponding to 20μm and the behaviors corresponding to 30μm are not obvious. In order to investigate the effects of the short proﬁle modiﬁcation on the dynamic responses in more detail, the accurate values of the dynamic responses of the system when the input speed is under the normal operating condition are given in Fig. 17. The lateral coordinate refers to different short proﬁle modiﬁcation cases and the vertical coordinate is the accurate dynamic values. DF line notes the dynamic factor of the gear pair, which means the ratio of the maximum value of the dynamic force and the static force during the meshing process; DTE line notes the root-mean-square value of dynamic transmission error of the system; BV line notes the root-mean-square value of vibration amplitude near bearing 1. The units

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Fig. 13. Mode shapes for different natural frequencies. (a) 177 Hz (b) 185 Hz (c) 2054.0 Hz (d) 3415.9 Hz (e) 3509.0 Hz.

of BV and DTE are μm and μrad, respectively. As shown in Fig. 17, with the increase of the maximum tooth proﬁle modiﬁcation amount, both the DTE value and the DF value of the system assume a trend to decrease in the beginning and increase at the end. However, the BV value of the system decreases continuously with the increase of the modiﬁcation

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Fig. 14. Dynamic responses of the system without tooth proﬁle modiﬁcation. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

Fig. 15. Dynamic responses of the system corresponding to G1 with short proﬁle modiﬁcation. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

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Fig. 16. Magniﬁcation of system responses near operating input speed. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

Fig. 17. Dynamic responses corresponding to G1 with short proﬁle modiﬁcation when Ω ¼ 7500rpm.

amount. In C0 case, BV¼ 2.0268, DF ¼2.9645 and DTE¼3.1475. As for CS10 case, BV ¼1.8824, DF ¼2.4894 and DTE¼2.6182. Taking the DF value, DTE value and BV value into account, the case CS10 is a priority choice. Case 3. Modiﬁed G2 with short proﬁle modiﬁcation In this case, short proﬁle modiﬁcations with different values are made to the driven gear G2 and the time-varying mesh stiffness of the gear pair corresponding to tooth proﬁle modiﬁcation is introduced into the numerical analysis of the dynamic responses. The modiﬁcation parameters of G2 are listed in Table 7. The analysis results of the system behavior when the input speed increases from 2000 rpm to 10,000 rpm are shown in Fig. 18. C0 notes that the gear teeth are not modiﬁed. CS0S10, CS0S20 and CS0S30 refer to the cases in which the short proﬁle modiﬁcations with a maximum tooth proﬁle modiﬁcation amount of 10μm, 20μm and 30μm are made to G2, respectively. A magniﬁcation picture of the system responses near the normal operating input speed is illustrated in Fig. 19. As shown in Figs. 18 and 19, for most of the input speed regions, the introduction of the short proﬁle modiﬁcation with G2 enlarges the values of the vibration amplitude of the bearing and the dynamic transmission error. The accurate values of the dynamic responses of the system when the input speed is under the normal operating condition are given in Fig. 20. With the

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Table 7 Modiﬁcation parameters of G2. Parameters

N

S10

S20

S30

L10

L20

L30

Δmax =μm h/mm n

0 0 2

10 1.0291 2

20 1.0291 2

30 1.0291 2

10 2.0118 2

20 2.0118 2

30 2.0118 2

Fig. 18. Dynamic responses of the system corresponding to G2 with short proﬁle modiﬁcation. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

increase of the maximum tooth proﬁle modiﬁcation amount, the DF value assumes a trend to decrease in the beginning and increase at the end. However, both the DTE value and the BV value of the system continuously become larger with the increase of the modiﬁcation amount. Taking the DF value, DTE value and BV value all into account, only modifying G2 with short proﬁle modiﬁcation may not be a reasonable measure to improve the dynamic responses of the system. Case 4. Modiﬁed G1 and G2 with short proﬁle modiﬁcation To improve the transmission property of the gear-rotor-bearing system, modifying both the driving pinion G1 and the driven gear G2 with short proﬁle modiﬁcation can also be a reasonable measure. For comparisons with the effects of the short proﬁle modiﬁcation made only to the driving gear or only to the driven gear, short proﬁle modiﬁcations with different values are introduced to the meshing gear pair G1 and G2. In this case, a short proﬁle modiﬁcation with a maximum tooth proﬁle modiﬁcation amount of 10μm is made to G1, which is a priority value discussed in Case 2. Meanwhile, G2 is modiﬁed with different values of short proﬁle modiﬁcation. The dynamic responses of the system when both the driving gear and the driven gear are modiﬁed are illustrated in Figs. 21 and 22. CS10S5, CS10S10, CS10S15, CS10S20, CS10S25, CS10S30, and CS10S35 refer to the cases in which the short proﬁle modiﬁcations with a maximum tooth proﬁle modiﬁcation amount of 5μm, 10μm, 15μm, 20μm, 25μm, 30μm and 35μm are respectively made to G2 and the short proﬁle modiﬁcation with a maximum tooth proﬁle modiﬁcation amount of 10μm is made to G1. As shown in Figs. 21 and 22, when a maximum tooth proﬁle modiﬁcation of 10μm is made to the driving pinion G1, the effects of the short proﬁle modiﬁcation with G2 on the system responses are random for different input speed regions. For some speed regions, the short proﬁle modiﬁcation with G2 increases the amplitudes of the dynamic responses, but, decreases the amplitudes of the system responses in other speed regions. The accurate values of the dynamic responses of the system when the input speed is under the normal operating

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Fig. 19. Magniﬁcation of system responses near operating input speed. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

Fig. 20. Dynamic responses corresponding to G2 with short proﬁle modiﬁcation when Ω ¼ 7500rpm.

condition are given in Fig. 23. With the increase of the maximum tooth proﬁle modiﬁcation amount introduced to G2, the change in the dynamic responses is not obvious. However, the DF value, BV value and DTE value of the system in case CS10S10 are the smallest compared with the results of other cases. For CS10S10 case, BV ¼1.4513, DF¼2.1411 and DTE¼2.4153. That is to say, a maximum tooth proﬁle modiﬁcation amount of 10μm should be introduced to the driven gear G2 when the driving gear G1 is modiﬁed with the same modiﬁcation amount. 4.2.2. Effects of long proﬁle modiﬁcation on dynamic responses As mentioned in Section 2, both the short proﬁle modiﬁcation and the long proﬁle modiﬁcation can affect the mutation of the mesh stiffness when the number of meshing tooth pairs turns from 2 to 1. Considering the high speed and heavy load work condition of the system, cutting more material from the gear teeth in long proﬁle modiﬁcation will be a relatively extreme measure as it may weaken the strength of the gear pair. The long proﬁle modiﬁcation will not be a priority form to modify the gear if the dynamic responses of the system cannot be obviously improved. In order to provide a reference, the numerical analysis results of the transmission system when the driving gear G1 is modiﬁed with long proﬁle modiﬁcation are given in the following case.

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Fig. 21. Dynamic responses of the system corresponding to G1 and G2 with short proﬁle modiﬁcation. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

Fig. 22. Magniﬁcation of system responses near operating input speed. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

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Fig. 23. Dynamic responses corresponding to G1 and G2 with short proﬁle modiﬁcation when Ω ¼ 7500rpm.

Fig. 24. Dynamic responses of the system corresponding to G1 with long proﬁle modiﬁcation. (a) Vibration amplitude with respect to x-direction of bearing 1 (b) Dynamic transmission error (DTE).

Case 5. Modiﬁed G1 with long proﬁle modiﬁcation In this case, long proﬁle modiﬁcations with different values are made to the driving gear G1 and the time-varying mesh stiffness of the gear pair corresponding to tooth proﬁle modiﬁcation is introduced into the numerical analysis of the dynamic responses. The analysis results of the system behavior when the input speed increases from 2000 rpm to 10,000 rpm are shown in Fig. 24. C0 notes that the gear teeth are not modiﬁed. CL10, CL20 and CL30 refer to the cases in which the long proﬁle modiﬁcations with a maximum tooth proﬁle modiﬁcation amount of 10μm, 20μm and 30μm are made to G1, respectively. As shown in Fig. 24, like the short proﬁle modiﬁcation, the introduction of the long proﬁle modiﬁcation can decrease the values of the dynamic responses of the system. Nevertheless, the relationship of the reduction of the system responses and the increase of the maximum tooth proﬁle modiﬁcation amount is nonlinear. The accurate values of the dynamic responses of the system when the input speed is under the normal operating condition are given in Fig. 25. As shown in this ﬁgure, an appropriate long proﬁle modiﬁcation revision to the gear teeth can reduce the amplitude of the dynamic responses. With the increase of the maximum tooth proﬁle modiﬁcation amount, the DTE value, DF value and BV value of the system all assume a trend to decrease in the beginning and increase at the end. For CL10 case, BV ¼1.5259, DF¼2.4974 and DTE¼2.2847. Compared with CS10 case, the reduction trends of the dynamic

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Fig. 25. Dynamic responses corresponding to G1 with long proﬁle modiﬁcation when Ω ¼ 7500rpm.

Table 8 Analysis results of dynamic responses corresponding to normal operating speed. Case

Without tooth proﬁle modiﬁcation C0 G1 with long proﬁle modiﬁcation CL10 CL20 CL30 G1 with short proﬁle modiﬁcation CS10 CS20 CS30 G2 with short proﬁle modiﬁcation CS0S10 CS0S20 CS0S30 G1 and G2 with short proﬁle modiﬁcation CS10S5 CS10S10 CS10S15 CS10S20 CS10S25 CS10S30 CS10S35

Vibration amplitude near bearing (BV/μm)

Dynamic factor (DF) Dynamic transmission error (DTE/μrad)

2.0268

2.9645

3.1475

1.5259 ( 24.5%) 1.3593 1.4779

2.4974 ( 14.9%) 2.4348 2.5013

2.2847 ( 27.5%) 2.3988 2.6720

1.8824 ( 7%) 1.8657 1.8517

2.4894 ( 17%) 2.6277 2.7162

2.6182 ( 17.5%) 2.5474 2.5582

2.0726 (3%) 2.1266 2.1452

2.4103 (–14.7%) 2.4301 2.4291

3.6595 (14.5%) 3.7883 3.8138

1.6209 1.4513 ( 27.5%) 1.5277 1.5381 1.5460 1.5480 1.5621

2.3440 2.1411 ( 27%) 2.1025 2.1224 2.1224 2.1216 2.1195

2.4614 2.4153 ( 23.2%) 2.5596 2.5879 2.6018 2.6067 2.6312

responses of the system in these two cases are quite close. The effects of the long proﬁle modiﬁcation on the dynamic behaviors are not more obvious than those of the short proﬁle modiﬁcation. 4.2.3. Discussion As analyzed above, both the short proﬁle modiﬁcation and the long proﬁle modiﬁcation can affect the amplitudes of the dynamic responses of the gear-rotor-bearing transmission system. The analysis results of the dynamic responses corresponding to the normal operating input speed in different tooth proﬁle modiﬁcation cases are listed in Table 8. For the case of G1 with short proﬁle modiﬁcation, the inﬂuences of a maximum tooth proﬁle modiﬁcation amount of 10μm are obvious. Compared with C0 case, the amplitudes of the BV value, DF value and DTE value have a decrease of 7%, 17% and 17.5%, respectively. For the case of G2 with short proﬁle modiﬁcation, with the increase of the maximum tooth proﬁle modiﬁcation amount, the DF value assumes a trend to decrease in the beginning and increase at the end. However, both the DTE value and BV value of the system continuously become larger with the increase of the modiﬁcation amount. Taking the DF value, DTE value and BV value all into account, only modifying G2 with short proﬁle modiﬁcation may not be a reasonable measure to improve the dynamic responses of the system. As for the case of G1 and G2 with short proﬁle modiﬁcation, with the increase of the maximum tooth proﬁle modiﬁcation amount introduced to G2, the DF value, BV value and DTE value of the system in case CS10S10 are the smallest compared with the results of other cases. Compared with C0 case, the amplitudes of the BV value, DF value and DTE value have a respective decrease of 27.5%, 27% and 23.2%. When the long proﬁle modiﬁcation is introduced into the gear teeth of G1, the effects of the long proﬁle modiﬁcation on the dynamic behaviors are not more obvious than those of the short proﬁle modiﬁcation. To prevent cutting more material from the gear teeth and guarantee the strength of the gear pair, the long proﬁle modiﬁcation should be avoided. Therefore, taking the DF

Z. Hu et al. / Mechanical Systems and Signal Processing 76-77 (2016) 294–318

317

Table 9 Modiﬁcation parameters corresponding to case CS10S10. Modiﬁcation parameters

Driving pinion G1

Driven gear G2

Δmax /μm h/mm n

10 1.0994 2

10 1.0291 2

value, DTE value and BV value into account, the case CS10S10 is the optimum modiﬁcation form for the meshing gear pair in this work condition. The corresponding modiﬁcation parameters of the gear pair are listed in Table 9.

5. Conclusions This paper focuses on the analysis of the effects of the tooth proﬁle modiﬁcation on the dynamic responses of a high speed gear-rotor-bearing transmission system. First, the mesh stiffness of the gear pair with tooth proﬁle modiﬁcation is given by applying the analytical method and the ﬁnite element method, and the effects of the tooth proﬁle modiﬁcation on the mesh stiffness of the gear pair are analyzed. Second, a ﬁnite element dynamic model of the transmission system considering the gear-rotor disc, meshing gear pair, ﬂexible shafts and supporting bearings is given, and the nonlinear factors including time-varying mesh stiffness, backlash and static transmission error excitation are introduced into the dynamic model. Then, the numerical simulation results of the system are discussed. The natural frequencies and mode shapes of the system are given, and the effects of the tooth proﬁle modiﬁcation on the dynamic responses are investigated in detail. The main conclusions are summarized as follows: (1) Both the short proﬁle modiﬁcation and the long proﬁle modiﬁcation can affect the mutation of the mesh stiffness when the number of meshing tooth pairs turns from 2 to 1. The amplitude of the mesh stiffness reduces with the introduction of the tooth proﬁle modiﬁcation. Compared to the mesh stiffness curve with short proﬁle modiﬁcation, the effects of the long proﬁle modiﬁcation on the mesh stiffness amplitude of the double-teeth-engagement region are more obvious. (2) The Campbell diagrams of the system with respect to the gear spin speed are given, and an obvious frequency veering frequency phenomenon is observed between the curves of natural frequency 2054.0 Hz and 2090.3 Hz near 2500 rpm. The normal input operating shaft speed of the pinion is Ω ¼ 7500rpm, and the corresponding natural frequency of the gear-rotor-bearing system has no obvious change near this input speed. The mode shapes of the system are given, and the lateral–torsional vibration may dominate the overall vibration behaviors of the gear-rotor-bearing system. (3) The effects of the tooth proﬁle modiﬁcation on the dynamic responses are investigated by introducing the time-varying mesh stiffness in different tooth proﬁle modiﬁcation cases into the gear-rotor-bearing dynamic system. The dynamic responses including vibration amplitude near the bearing and dynamic transmission error of the system with respect to the input speed region (2000 rpm to 10,000 rpm) are analyzed. The dynamic factor (DF), vibration amplitude nearing the bearing (BV) and dynamic transmission error (DTE) corresponding to the normal operating speed in different tooth proﬁle modiﬁcation cases are given. The introduction of the short proﬁle modiﬁcation can decrease the amplitudes of the dynamic responses. The effects of the long proﬁle modiﬁcation on the dynamic behaviors are not more obvious than those of the short proﬁle modiﬁcation. A short proﬁle modiﬁcation with an appropriate modiﬁcation amount can improve the dynamic property of the system. Nevertheless, a short proﬁle modiﬁcation with an excessive modiﬁcation amount may increase the amplitudes of the dynamic responses. In order to improve the transmission property of the gear-rotor-bearing system in the normal operating input speed and guarantee the strength of the meshing gear pair, a reasonable tooth short proﬁle modiﬁcation case (CS10S10) is recommended and the corresponding modiﬁcation parameters are listed. Compared with the gear pair without modiﬁcation in this speciﬁc work condition, the amplitudes of the BV value, DF value and DTE value have a decrease of 27.5%, 27% and 23.2%, respectively.

Acknowledgments The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NSFC) through Grants nos. 51275530, 51305462, 51535012. The authors also gratefully acknowledge the support of the Fundamental Research Funds for the Central Universities of Central South University (2015zzts039).

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Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system

Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system

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