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Dynamics analysis of a crowned gear transmission system with impact damping: Based on experimental transmission error
Mechanism and Machine Theory 74 (2014) 354–369
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Dynamics analysis of a crowned gear transmission system with impact damping: Based on experimental transmission error Chen Siyu, Tang Jinyuan ⁎, Wu Lijuan State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, Hunan 410083, China
a r t i c l e
i n f o
Article history: Received 7 March 2012 received in revised form 29 August 2013 accepted 7 January 2014 Available online 30 January 2014 Keywords: Gear transmission system Hertz impact damping Experimental TE Dynamic load Transmission error
a b s t r a c t The impact characteristics of a crowned gear transmission system are investigated with experimental static transmission error (TE) and backlash. The effects of load, input speed and impact force exponent coefficient n on the dynamic characteristic of system are investigated. The numerical simulations show that the static TE can be combined with a backlash parameter. Under low speed conditions, the dynamic load is mainly connected with the high frequency component of TE, which can be effectively suppressed with the increase of tooth surface accuracy and the improvement of tooth profile error and roughness. Under high speed conditions, the shaft frequency component of TE has a dominate influence on dynamic load in the resonance area. Parameter n is decreasing and the dynamic load coefficient is increasing especially in the resonance area. When n is 1.0, a loss contact impact area exists in the system and the amplitudes of shaft frequency component in dynamic TE are almost equal, which indicates that dynamic characteristics are little affected by load and parameter n when there is no loss contact. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. Crown Copyright © 2014 Published by Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear dynamic behaviors, with respect to noise and vibration, have been the subject of numerous investigations in recent decades. There is a large amount of literature on gear dynamic models [1,2]. When backlash is considered, the gear pair system is a strong nonlinear system and it can be modeled as a piecewise smooth system. Generally, this type of system can be classified as a vibro-impact system. In recent years, many researchers have been devoted to this issue such as Ibrahim [1], di Bernardo et al. [2] and Stronge [3]. As for the vibro-impact induced by gear backlash, Pfeiffer [4–8], Luo [9–11], Singh and Kahraman [12–14] and Natsiavas et al. [15–17] have conducted relevant studies. As noted in literature [18–20] that “the mechanism of impact damping in impact system should be investigated precisely to analyze the effect of impact damping”. To understand the effect of impact damping on the characteristics of system vibration, the dynamic characteristics of damping system should be exhibited quantitatively and comprehensively. The investigation of impact damping in gear tooth system is preliminary and the issue of energy loss during impact was adduced by Azar and Crossley [21], Herbert and McWhannell [22], Smith [23]. On the basis of the issue, Yang and Lin [24,25] adopted a gear impact damping model with stationary stiffness, and the load variation on the effect of system response was addressed. Subsequently, in 2003 Kim et al. [26] imported an impact model as f δ;δ˙ ¼ δ 1 þ βδ˙ ð1Þ to study the effect of impact damping on resonance, harmonic frequency and super-harmonic frequency. However, the investigation of impact damping on the gear tooth was inadequate, mainly for the neglect of geometric factors of gear tooth face. ⁎ Corresponding author. Tel./fax: +86 731 88877746. E-mail address: [email protected] (J. Tang). 0094-114X/$ – see front matter. Crown Copyright © 2014 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2014.01.003
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Nomenclature B half of gear backlash D impact damping coefficient e the coefficient of restitution es (t) static transmission error f, FK, FD impact force, elastic force and damping force of impact system f (x) nonlinear backlash function Fd fluctuation of load Fv viscous damping force fm meshing frequency frev shaft rotation frequency hi (i = 1,2,…,6) coefficient of ith harmonic components of static transmission error I1, I2 moment of inertia of pinion and gear K the mesh stiffness of gear transmission system k (t), ktp, kr, k0 meshing stiffness function, single tooth stiffness, rth coefficients of stiffness and averaged nonlinear meshing stiffness Kv gear dynamic load m modulus of gear me equivalent gear mass n impact force exponent coefficient N1, N2 line of action Rb1, Rb2 basic radii of the pinion and gear S difference of pinion and gear in line of action T1 (T1 (t)), T2 (T2 (t)) input and output torque of gear transmission system vmax maximum velocity z1 number of gear tooth σ coefficient of load fluctuation ωn approximate natural frequency of gear pair ξ non-dimension viscous damping coefficient φr phase of rth stiffness ϕi (i = 1, 2,…,6) phase of ith harmonic components of static transmission error β impact damping coefficient ð−Þ δ, δ,˙ δ˙ the local relative penetration, velocity and initial velocity of impact θ1, θ2 angle displacement of pinion and gear ICR gear contact ratio DF dynamic load coefficient DMF dynamic mesh force DTE dynamic transmission error DTEppv peak-to-peak values of dynamic transmission error DTErms root-mean-square of dynamic transmission error SMF static mesh force TE transmission error
Moreover, impact damping coefficient β was location constant and the relationship between impact damping and impact velocity —as well as geometric parameters was neglected. In this paper, on the basis of an impact dynamic model, the effects of impact damping parameters, etc. on the system dynamic are considered for the perspective of nonlinear dynamics. In this paper, the impact characteristics in gear system with impact damping are investigated, and the experimental static transmission error and backlash are adopted in the theoretical analyses. The crown modification is performed in the pinion teeth surface. The nonlinear impact damping model is adopted. And that the nonlinear exponent n is non-integer indicates the point contact relative to crown modified gear. The following sections of this paper are organized as follows: in the second section, the impact force model, gear transmission model and consideration of static transmission error and backlash are interpreted. Subsequently, systematic consideration is remarked for a direct comparison between numerical results and experimental dynamic transmission error. The effects of load, input shaft velocity, parameter n on the dynamic characteristics of gear transmission system in terms of frequency response and dynamic load are investigated. Finally, a short remark is exhibited in the last section.
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Fig. 1. The physical model of gear impact model.
2. Gear impact model 2.1. Gear impact force model In this work, the nonlinear impact damping model (Hertz impact force model), n n f ¼ Kδ þ Dδ δ˙
ð2Þ
developed by Hunt and Crossley [27] in 1975 is adopted. According to the Hertz contact theory, the elastic force is represented as [27,28], n
F k ¼ Kδ :
ð3Þ
Here n = 3/2, δ is the local relative penetration and for gear contact, δ ¼ jRb1 θ1 −Rb2 θ2 j−B ¼ jSj−B
ð4Þ
here, S = Rb1θ1 − Rb2θ2, K is time varying mesh stiffness, which differs from the case in Ref. [27,28], and whose detailed calculation process will be described in next subsection. θi and Rbi (i = 1, 2) are angle displacements and basic radii of the pinions and gears respectively, and B is half of the gear backlash. Then the elastic force among meshing can be rewritten as n
F K ðθ1 ; θ2 Þ ¼ K ðθ1 ; θ2 Þδ :
ð5Þ
Generally, it can be assumed that the material structural energy dissipation during tooth contact is caused by the damping function. A hysteretic form for the damping effect is proposed by Hunt and Crossley [27] as, n F D ðθ1 ; θ2 Þ ¼ Dδ δ˙
ð6Þ
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and the damping coefficient is h i 2 ð−Þ ð−Þ 0:36 D ¼ 6K ð1−eÞ= ð2e−1Þ þ 3 δ˙ ¼ K ðθ1 ; θ2 ÞD; e ¼ 1−0:022 δ˙ :
ð7Þ
2.2. Gear pair impact model Backlash measured along the pitch circle is the amount of clearance between mated gear teeth, namely, the difference between tooth space and tooth thickness. Theoretically, backlash should be zero, but in actual practice some backlash must be allowed to prevent the jamming of teeth due to tooth errors and thermal expansion. The physical model of a gear transmission system with backlash is shown in Fig. 1. The mesh sliding friction is ignored and just the torsional vibration of gear transmission system is considered. According to the geometric condition, the motion between two gears can be divided into three cases as Case I : S ¼ Rb1 θ1 −Rb2 θ2 NB; Case II : jSj≤B Case III : Sb−B:
ð8Þ
(1) for Case I A pair of gear teeth meshing along the line of action N2, the contact force combined with two parts, the elastic force and damping force, are described as FK (θ1, θ2) and FD (θ1, θ2) respectively. The dynamic equation of gear pair in this case is I1 θ1 ¼ T 1 −½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb1 I2 θ2 ¼ T 2 þ ½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb2 ;
ð9Þ
where Ii (i = 1, 2) is moment of inertia of gear. (2) for Case II Under this condition, no contact occurs between two gear teeth. Therefore, the contact force is zero and the dynamic equation is, I1 € θ1 ¼ T 1 I2 € θ2 ¼ T 2 :
ð10Þ
(3) for Case III Under this condition, contact occurs along the line of action N1, and then the dynamic equation should be written as, I1 € θ1 ¼ T 1 þ ½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb1 I2 € θ2 ¼ T 2 −½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb2 :
ð11Þ
Substituting Eqs. (5) and (6) into (9)–(11), one can obtain n θ1 ¼ T 1 −Rb1 δ K ðθ1 ; θ2 Þ þ Dδ˙ δ I1 € n θ2 ¼ T 2 þ Rb2 δ K ðθ1 ; θ2 Þ þ Dδ˙ δ ; I2 €
ð12Þ
where δ ¼ jRb1 θ1 −Rb2 θ2 j−B ¼ jSj−B
δ ¼
8 < :
1 SNB 6ð1−eÞ ð−Þ 0:36 0 jSj≤B ; e ¼ 1−0:022 δ˙ ;D ¼ : ð 2e−1 Þ2 þ 3 δ˙ ð−Þ i −1 Sb−B
ð13Þ ð14Þ
Combined with formula (13), the Eq. (12) can be defined as S þ
k0 K ðt Þ n δ 1 þ Dδ˙ δ ¼ F me k0
ð15Þ
F¼
Rb1 T 1 Rb2 T 2 1 R2 R2 þ ; ¼ b1 þ b2 : me I1 I2 I1 I2
ð16Þ
″
where
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We define x ¼ BS; ωn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 =me ; τ ¼ ωn t:
ð17Þ
Then, €x þ
K ðt Þ n−1 F B 1 þ BDωn sign x˙ x˙ f ðxÞ ¼ ; k0 Bω2n
ð18Þ
where, 8 <
n
ðx−1Þ f ðxÞ ¼ 0 : n −ð−x−1Þ
xN1 jxj≤1 xb−1:
ð19Þ
Meanwhile, impact damping force introduced in gear pairs is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k0 u x˙ F v ¼ 2ξt r 2p =I p þ r 2g =Ig
ð20Þ
so that the formula (18) can be redefined as €x þ 2ξ xþ ˙
K ðt Þ n−1 F B 1 þ BDωn sign x˙ x˙ f ðxÞ ¼ : k0 Bω2n
ð21Þ
According to literature [29,30], tooth pairs mesh stiffness k (t) can be simplified as kðt Þ ¼ k0 þ
N X
kr cosð2πr f m t−φr Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2−2 cosð2πr ðICR−1ÞÞ k0 k 1− cosð2πr ðICR−1ÞÞ ; φr ¼ ¼ ICR; r ¼ sinð2πr ðICR−1ÞÞ πr ktp ktp r¼1
Fig. 2. Experimental static transmission error (c) and its spectrum (a, b). A Fourier series expanded is denoted by red line in (c).
ð22Þ
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and the coefficient of load fluctuation which is subjected to load, electrical machine and errors can be simplified as Fd ¼ σ
F sinð2πr f m t Þ Bω2n
ð23Þ
where σ is the coefficient of load fluctuation. The gear tooth impact is imitated by the damping model. The geometric characteristics (crown modification) of gear tooth and material characteristics are included to determine parameter n and damping coefficient D. Parameter n can be set as n = 1, which is not rejected to normal characteristics of non-modified gears. However, gear tooth profiles are modified more or less actually, and line-to-line contact becomes point contact which can be understood as collision impact between spheres. It is a matter of concern as to what value of n can be appropriate. To be precise, the situation of gear pairs should be confirmed experimentally. However, the study starts with the analysis of varying n theoretically for the limits. For parameter D, it is a constant determined by impact transient velocity during single impact while it is varying continuously for continuous impact. Subject to the literature, the effects of parameter n and impact damping coefficient D on gear teeth dynamics are contrasted. 2.3. Considerations of the static transmission error and backlash Generally speaking, the static transmission error is approximated by the grade of gear with certain design and manufacturing errors. And the static transmission error then can be approximated as Fourier series. In this paper, the experimental static transmission error and backlash will be adopted. As mentioned in our previous work [31] (submitted to MMT simultaneously ), the pinion is driven at a very low shaft velocity, and the transmission error combined with assembly error and manufacturing error is obtained as shown in Fig. 2. The main part of static transmission error is the first harmonic component of shaft and the amplitude is 8.4644 × 10−4 rad and the amplitude of mesh frequency component corresponding with the first harmonic component of shaft is very low as shown in Fig. 2(b). For the convenience of theoretical analyses, the experimental static transmission error can be approximated by the sum of three harmonic frequency components related to the shaft velocity of driving gear and mesh frequency, as es ðt Þ ¼ h1 sinðωt þ ϕ1 Þ þ h2 sinð2ωt þ ϕ2 Þ þ h3 sinð3ωt þ ϕ3 Þ þ h4 sinðωt=z1 þ ϕ4 Þ þ h5 sinð2ωt=z1 þ ϕ5 Þ þ h6 sinð3ωt=z1 þ ϕ6 Þ:
ð24Þ
Here, hi, ϕi (i = 1, 2,…, 6) are the amplitudes and rotational displacements of corresponding frequency, zi is the number of driving gear teeth. In practical analyses, the amplitude hi is calculated as,
hðiÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X 2 f j: iω j¼fj f −2π j ≤ 5Bw g
ð25Þ
Here, Bw determined by sample frequency is the frequency resolution. The expression (25) indicates that the amplitude of ith harmonic component is the root-mean-square of the frequency component at iω / 2π, namely f∈(iω/2π − 5Bw,iω/2π + 5Bw). Then the amplitude hi is listed in Table 1 and the corresponding approximate curve is shown in Fig. 2(c) denoted by a red line. Additionally, the phase of experimental static transmission error is mainly determined by initial time and the effect of rotational displacement is less important. Let φi = 0 in the following analyses. As for the backlash, one can fix one shaft of gear pairs and move the other gear side to side in backlash. Although one shaft is fixed, there are still wobbly and elastic deformations and the angular position is recorded by encoders fixed on both shaft ends. The acquired data based on experiment [31] are illustrated in Fig. 3. The nominal static transmission error is shown in Fig. 3(a), and the absolute angle of the gear and pinion denoted by red and blue lines respectively is shown in Fig. 3(b). The dynamic backlash is obtained by comparing the angle difference as Bi ¼ Anglejt iþ1 −Anglejt i ; i ¼ 1; 2; …; 13. The mean value at Bi = 0.0064 is used in the following analyses. Table 1 Component amplitude of experimental static transmission error. Shaft frequencies × 10−4 (rad) 0 1 2 3
h0 h1 h2 h3
Mesh frequencies × 10−6 (rad) −2.9003 −8.4644 0.2553 0.1511
– h4 h5 h6
– 7.9473 1.9474 1.5414
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Fig. 3. Experimental backlash illustration of the (a) nominal static transmission error and (b) absolute angle.
3. Systematic considerations Generally, the static transmission error is a main excitation of gear vibration and noise. Dynamic mesh force generated under operating conditions is transmitted to housing through shafts and bearings. Mathematically, this static transmission error is modeled as an external force which appears at the right hand of dynamics Eq. (21) as in Ref. [12,32–36]. But two deviation methods are applied to Eq. (24), des ðt Þ 2 2 2 ¼ −h1 ω sinðωt þ φ1 Þ−4h2 ω sinð2ωt þ φ2 Þ−9h3 ω sinð3ωt þ φ3 Þ dt h i 2 2 2 2 − h4 ω sinðωt=z1 þ φ4 Þ þ 4h5 ω sinð2ωt=z1 þ φ5 Þ þ 9h6 ω sinð3ωt=z1 þ φ6 Þ =z1 :
ð26Þ
The new amplitudes corresponding with shaft frequency and mesh frequency change via different patterns. Especially the amplitudes of three shaft frequency components are inversely relative to z21 (z1 is the number of gear teeth). Namely the effect of shaft frequency components on dynamic responses is less than the effect of mesh frequency part. In addition, as mentioned in the previous literature [12,32–36], the numerical dynamic transmission error cannot be used to have a direct comparison with the experimental dynamic transmission error. Therefore, a more realistically considered model of the gear transmission system is needed. Recalling the definition of initial static transmission error: “transmission error is the difference between the angular position that the output shaft of a drive would occupy if the drive were perfect and the actual
Table 2 Design data of gears.
Number of teeth Module (mm) Pressure angle (deg) Tooth width (mm) Iterate of moment Modification (mm)
Gear
Pinion
61 4 20 40 107.3344 × 10−3 0.026
61 4 20 40 108.9025 × 10−3 –
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position of the output.” [37]. In this paper, the static transmission error is combined with backlash as 1⇒1 þ
r b1 e ðt Þ ¼ b B s
ð27Þ
and the backlash function (19) can be rewritten as, 8 <
n
ðx−bÞ f ðxÞ ¼ 0 : n −ð−x−bÞ
xNb jxj≤b xb−b:
ð28Þ
4. Numerical analysis Gear design data are shown in Table 2, where the damping coefficient is ξ = 0.07 and backlash B = 0.7337 × 10−3/2m is measured from the test rig. The effect of friction is negative in this work. The mesh stiffness is deduced from gear structural parameters as shown in Table 2. The system dynamic response is obtained through the Runge–Kutta method to integrate the differential Eq. (21). The effects of load, input shaft velocity, parameter n on the dynamic characteristics of gear transmission system in terms of frequency response and dynamic load are investigated. 4.1. Calculation of ISO standard dynamic load coefficient For comparison of the numerical results, the calculation method for the gear dynamic load based ISO standard is illustrated in this subsection. According to the standard [38], the simplified computational formula of gear dynamic load is Kv ¼
−B A pffiffiffiffiffiffiffiffiffiffiffi A þ 200v
ð29Þ
where A = 50 + 56(1.0 − B) and B = 0.25(C − 5.0)0.667. And v is the velocity of gear tooth along the pitch line N2. The maximum velocity is not exceeded by vmax. v max ¼
½A þ ð14−C Þ2 : 200
ð30Þ
2 C=12 1.9 C=11 1.8 C=10 1.7 C=9 1.6 C=8
1.5 1.4
C=7
1.3 C=6 1.2 1.1 1 0
5
10
15
20
25
30
35
Fig. 4. Dynamic load coefficient of gear teeth.
40
45
50
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Fig. 5. Measured error of gear face.
Here, C is the class of gear pair transmission accuracy, which can be deduced with the number of teeth, modules and tooth form errors. C ¼ −0:5048 ln z1;2 −1:144 ln ðmÞ þ 2:852 ln f pt1;2 þ 3:32
ð31Þ
The gear tooth conditions adjusted to the calculation of dynamic load coefficient are 6 ≤C ≤12 6 ≤z≤1200 1:25 ≤m≤50:
ð32Þ
According to the above formula, dynamic load coefficients with different precision values and velocities are shown in Fig. 4. Gear design precision is 6, yet the actual gear precision is lower for big single tooth errors. As shown in Fig. 5, the measuring results of gear face along three coordinates are given representatively. The deviation of a single tooth is related to 17.9 μm. According to the formula (31), the actual class of gear accuracy is DIN 8. 4.2. The effect of load and velocity on dynamic load coefficient Firstly, n = 1.5 is set in the impact force model and the impact damping coefficient is determined by impact velocity. The load torque is 5000 N m and the input velocity varies within the region of (50,3000) rpm. The resonance frequency range should be determined before the analysis of dynamic load characteristics.
Fig. 6. Dynamic load coefficient varying with speed, load torque at 5000 N m.
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Fig. 7. Dynamic load coefficient varying with speed, load torque at 500 N m.
Fig. 8. Frequency response of gear transmission system with load torque at 5000 N m.
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The resonance velocity is defined as sffiffiffiffiffiffiffi 1 km nE1 ¼ ; 2π me
ð33Þ
where me is the equivalent gear mass; km is the mean mesh stiffness of gear tooth. For the parameters shown in Table 2, the corresponding equivalent mass and mesh stiffness of gear pairs are 8
me ¼ 4:1167 kg; k0 ¼ 9:9455 10 N=m:
ð34Þ
The ratio of revolving speed is defined as N¼
n1 nz1 =60 sffiffiffiffiffiffiffi : ¼ nE1 1 km 2π me
ð35Þ
According to the value of N, the gear tooth surface can be divided into four sections: (1) sub-crisis area when N ≤ 0.85; (2) resonance area when 0.85 b N ≤ 1.15; (3) translational area when 1.15 b N ≤ 1.5; and (4) above-critical area when N N 1.5. For the established dynamic model, the dynamic mesh force is n DMF ¼ K ðt Þδ 1 þ Dδ˙ δ
ð36Þ
and then the dynamic load coefficient [39] is DF ¼
maxðDMF Þ SMF
ð37Þ
where SMF is the static mesh force. The dynamic load coefficients of 5000 N m and 500 N m are given (shown in Figs. 6 and7 respectively). In the speed range of (50,2068.2) rpm, the system stays at the sub-critical area. During (2068.2, 2798.2) rpm, the system stays at the resonance area and achieves the maximum value of dynamic load. The system stays at the translational area in (2798.2, 3000) rpm. The theoretical dynamic load coefficient curve based on the gear precision class of ISO 8 is denoted by a black line in Fig. 6. In
Fig. 9. Frequency response of gear transmission system with load torque at 500 N m.
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Fig. 10. The ratios of first three harmonic frequencies with load torque at 5000 N m and 500 N m.
comparison, the dynamic load coefficient of actual manufactured gear system involving tooth surface error is larger than the theoretical dynamic load coefficient. Though large fluctuation of dynamic load coefficient exists at 5000 N m and 500 N m, the tendency of dynamic load coefficient increase with respect to velocity is the same.
Fig. 11. Dynamic load coefficient with variation of speed with load torque at 5000 N m.
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Fig. 12. Dynamic load coefficient with variation of speed with load torque at 500 N m.
The frequency response curves are given (shown in Figs. 8 and9) to further analyze the fluctuation of dynamic load coefficient. The amplitude of the shaft frequency component varies slightly at 5000 N m (shown in Fig. 8). The amplitude variation is not obvious with the increase of velocity. At low speed, the amplitudes of first three mesh frequency components are large, and
Fig. 13. System frequency response with load torque at 5000 N m.
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Fig. 14. System frequency response with load torque at 500 N m.
fundamental mesh frequency and doubling frequency component are principal in system dynamic transmission error. With the increase of speed, the amplitudes of mesh frequency components decrease rapidly, and system dynamic transmission error gives priority to shaft frequency component. When the load decreases to 500 N m, the amplitude of teeth frequency components
Fig. 15. The ratio of prior three harmonic components with load torque at 5000 N m and 500 N m respectively.
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decrease obviously (as shown in Fig. 9). The ratios of first three harmonic frequency components under two load conditions referred are compared (shown in Fig. 10). Apparently, shaft frequency is slightly affected by load while the mesh frequency and doubling frequency components are affected notably. The results show that, (1) Under low speed conditions, the dynamic load is mainly connected with the high frequency component (mesh frequency and harmonic component) of transmission error. The high frequency amplitude of transmission error can be effectively suppressed with the increase of tooth surface accuracy and the improvement of tooth profile error and roughness, so that it is meaningful to dismiss the system dynamic load. (2) Under high speed conditions, the dynamic load is mainly connected with the shaft frequency component of transmission error which exerts a huge influence on the dynamic load, though the gear mesh may be in the resonance frequency area. 4.3. The effect of parameter n on dynamic load coefficient Parameter n is set to 1.0 and others are not modified. The dynamic load coefficients with respect to velocity are shown in Figs. 11 and12. The differences between the system dynamic responses under different values of parameter n can be deduced from Figs. 6 and7, which are mainly expressed as follows. (1) Parameter n is decreasing and dynamic load coefficient is increasing especially in the resonance area. Under the two load conditions, three resonance peak values appear around speed 490 rpm, 790 rpm and 1110 rpm. The corresponding load conditions of frequency response curves are shown in Figs. 13 and14 respectively, which outline that the first resonance peak value is conducted with prior three harmonic frequency components, and the second one and the third one are determined by the third and the second harmonic frequency components respectively. (2) When n is 1.0, the system exhibits loss contact impact which is denoted by a red ‘-o-’and the dynamic load is raised by impact. And corresponding to the frequency response curve, in the loss contact impact area, the amplitudes of the shaft frequency component in dynamic transmission error are almost equivalent (shown in Fig. 15), which explains that dynamic characteristics are slightly affected by load and parameter n with loss contact impact. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. (3) When n is 1.5, loss contact impact does not exist in the system, and the ratio of prior three harmonic components approximates the linear increase of load. When n is 1.0, the system assumes stronger nonlinear characteristics with the appearance of impact. For the engineering perspective, not only should the resonance area be considered to neglect the resonance of the gear transmission system, but also the dynamic impact load should be at low speed. 5. Conclusions On the basis of Hertz impact model, the impact characteristics in a gear system with impact damping are investigated, and the experimental static transmission error under crown modification condition and backlash are adopted in the theoretical analysis. The effects of load, input shaft velocity, parameter n on the dynamic characteristics of gear transmission system in terms of the frequency response and dynamic load are investigated. The numerical simulation shows that, (1) Direct comparison between the analyses of experimental dynamic transmission error and those of static transmission error can be combined with the backlash parameter. (2) Under low velocity conditions, the dynamic load is mainly connected with a high frequency component of transmission error. The high frequency amplitude of transmission error can be effectively suppressed with the increase of tooth surface accuracy and the improvement of tooth profile error and roughness, so as to weaken the system dynamic load. Under high speed conditions, the dynamic load is mainly connected with the shaft frequency component of transmission error, which has a dominate influence on the dynamic load, though the gear pair may be in the resonance frequency mesh area. (3) Parameter n is decreasing and the dynamic load coefficient is increasing especially in the resonance area. When n is 1.0, the system exhibits a loss contact impact area and the amplitudes of shaft frequency components with dynamic transmission error are almost equal, which indicates that dynamic characteristics are slightly affected by load and parameter n under a no loss contact impact condition. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. (4) When n is 1.5, loss contact impact does not exist in the system, and the ratio of prior three harmonic components approximates the linear increase of load. But when n is 1.0, the system assumes stronger nonlinear characteristics with the appearance of impact. Acknowledgements The authors gratefully acknowledge the support of the National Science Foundation of China (NSFC) through Grants Nos. 51305462 and 51275530, as well as the support of the National Basic Research Program of China (2011CB706800).
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
R. Ibrahim, Vibro-Impact Dynamics: Modeling, Mapping and Applications, Springer Verlag, 2009. M. di Bernardo, et al., Piecewise-smooth Dynamical Systems: Theory and Applications, Springer, London, 2008. W. Stronge, Impact Mechanics, Cambridge Univ Pr, 2004. F. Pfeiffer, A. Kunert, Rattling models from deterministic to stochastic processes, Nonlinear Dyn. 1 (1990) 63–74. K. Karagiannis, F. Pfeiffer, Theoretical and experimental investigations of gear-rattling, Nonlinear Dyn. 2 (1991) 367–387. A. Kunert, F. Pfeiffer, Description of chaotic motion by an invariant probability density, Nonlinear Dyn. 2 (1991) 291–304. Q. Feng, F. Pfeiffer, Stochastic model on a rattling system, J. Sound Vib. 215 (1998) 439–453. A. Haj-Fraj, F. Pfeiffer, Optimization of automatic gear shifting, Veh. Syst. Dyn. 35 (2001) 207–222. A.C.J. Luo, D. Oconnor, Periodic motions and chaos with impacting chatter and stick in gear transmission systems, Int. J. Bifurcation Chaos 19 (2009) 1975–1994. A.C.J. Luo, D. O'Connor, Periodic motions with impacting chatter and stick in a gear transmission system, J. Vib. Acoust. 131 (2009)(041013-11). A.C.J. Luo, M.T. Patel, Bifurcation and stability of periodic motions in a periodically forced oscillator with multiple discontinuities, J. Comput. Nonlinear Dyn. 4 (2009) 011011. A. Kahraman, R. Singh, Non-linear dynamics of a spur gear pair, J. Sound Vib. 142 (1990) 49–75. A. Kahraman, R. Singh, Non-linear dynamics of a geared rotor-bearing system with multiple clearances, J. Sound Vib. 144 (1991) 469–506. A. Kahraman, R. Singh, Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system, J. Sound Vib. 146 (1991) 135–156. S. Natsiavas, Stability and bifurcation analysis for oscillators with motion limiting constraints, J. Sound Vib. 141 (1990) 97–102. S. Natsiavas, Modal interactions in self-excited oscillators under external primary resonance, J. Sound Vib. 184 (1995) 261–280. S. Theodossiades, S. Natsiavas, Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash, Chaos Solitons Fractals 12 (2001) 2427–2440. T.C. Kim, et al., Effect of nonlinear impact damping on the frequency response of a torsional system with clearance, J. Sound Vib. 281 (2005) 995–1021. E. Rigaud, J. Perret-Liaudet, Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 1: harmonic excitation, J. Sound Vib. 265 (2003) 289–307. J. Perret-Liaudet, E. Rigaud, Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 2: random excitation, J. Sound Vib. 265 (2003) 309–327. R.C. Azar, F.R.E. Crossley, Digital simulation of impact phenomenon in spur gear systems, J. Eng. Ind. Trans. ASME Ser. B 99 (1977) 792–798. R.G. Herbert, D.C. McWhannell, Shape and frequency composition of pulses from an impact pair, ASME J. Eng. Ind. 99 (1977) 513–518. J.D. Smith, Energy loss in gear tooth impact. ARCHIVE: Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 1983–1988 (vols. 197–202) 198 (1984) 205–208. D.C.H. Yang, J.Y. Lin, Hertzian damping, tooth friction and bending elasticity in gear impact dynamics, J. Mech. Transm. Autom. Des. 109 (1987) 189–196. D. Yang, Z. Sun, A rotary model of spur gear dynamics, J. Mech. Transm. Autom. Des. 107 (1985) 529–535. T.C. Kim, et al., Effect of smoothening functions on the frequency response of an oscillator with clearance non-linearity, J. Sound Vib. 263 (2003) 665–678. K.H. Hunt, F.R.E. Crossley, Coefficient of restitution interpreted as damping in vibroimpact, ASME J. Appl. Mech. 42 (1975) 440–445. H.M. Lankarani, P.E. Nikravesh, A contact force model with hysteresis damping for impact analysis of multibody systems, J. Mech. Des. 112 (1990) 369–376. A. Kahraman, G.W. Blankenship, Effect of involute contact ratio on spur gear dynamics, J. Mech. Des. 121 (1999) 112–118. C. Gill-Jeong, Nonlinear behavior analysis of spur gear pairs with a one-way clutch, J. Sound Vib. 301 (2007) 760–776. C. Siyu, J.Y. Tang, Experimental study of the effect of assembly error on the lightly loaded transmission error of spur gear with crown modification, Mech. Mach. Theory (2014)(submitted for publication MMT). S. He, et al., Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations, J. Sound Vib. 309 (2008) 843–851. W. Kim, et al., Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation, J. Sound Vib. 329 (2010) 4409–4421. S. Baguet, G. Jacquenot, Nonlinear couplings in a gear–shaft–bearing system, Mech. Mach. Theory 45 (2010) 1777–1796. C.-W. Chang-Jian, Nonlinear analysis for gear pair system supported by long journal bearings under nonlinear suspension, Mech. Mach. Theory 45 (2010) 569–583. C. Siyu, et al., Nonlinear dynamic characteristics of geared rotor bearing systems with dynamic backlash and friction, Mech. Mach. Theory 46 (2011) 466–478. D.B. Welbourn, Fundamental knowledge of gear noise: a survey, Inst. Mech. Eng. Conf. Publ. (1979) 9–14. ISO, 6336-1:2006, Calculation of Load Capacity of Spur and Helical Gears—Part 1: Basic Principles, Introduction and General Influence Factors, International Organization for Standardization, 1996. V.K. Tamminana, et al., A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs, J. Mech. Des. 129 (2007) 75–84.
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Dynamics analysis of a crowned gear transmission system with impact damping: Based on experimental transmission error Chen Siyu, Tang Jinyuan ⁎, Wu Lijuan State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, Hunan 410083, China
a r t i c l e
i n f o
Article history: Received 7 March 2012 received in revised form 29 August 2013 accepted 7 January 2014 Available online 30 January 2014 Keywords: Gear transmission system Hertz impact damping Experimental TE Dynamic load Transmission error
a b s t r a c t The impact characteristics of a crowned gear transmission system are investigated with experimental static transmission error (TE) and backlash. The effects of load, input speed and impact force exponent coefficient n on the dynamic characteristic of system are investigated. The numerical simulations show that the static TE can be combined with a backlash parameter. Under low speed conditions, the dynamic load is mainly connected with the high frequency component of TE, which can be effectively suppressed with the increase of tooth surface accuracy and the improvement of tooth profile error and roughness. Under high speed conditions, the shaft frequency component of TE has a dominate influence on dynamic load in the resonance area. Parameter n is decreasing and the dynamic load coefficient is increasing especially in the resonance area. When n is 1.0, a loss contact impact area exists in the system and the amplitudes of shaft frequency component in dynamic TE are almost equal, which indicates that dynamic characteristics are little affected by load and parameter n when there is no loss contact. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. Crown Copyright © 2014 Published by Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear dynamic behaviors, with respect to noise and vibration, have been the subject of numerous investigations in recent decades. There is a large amount of literature on gear dynamic models [1,2]. When backlash is considered, the gear pair system is a strong nonlinear system and it can be modeled as a piecewise smooth system. Generally, this type of system can be classified as a vibro-impact system. In recent years, many researchers have been devoted to this issue such as Ibrahim [1], di Bernardo et al. [2] and Stronge [3]. As for the vibro-impact induced by gear backlash, Pfeiffer [4–8], Luo [9–11], Singh and Kahraman [12–14] and Natsiavas et al. [15–17] have conducted relevant studies. As noted in literature [18–20] that “the mechanism of impact damping in impact system should be investigated precisely to analyze the effect of impact damping”. To understand the effect of impact damping on the characteristics of system vibration, the dynamic characteristics of damping system should be exhibited quantitatively and comprehensively. The investigation of impact damping in gear tooth system is preliminary and the issue of energy loss during impact was adduced by Azar and Crossley [21], Herbert and McWhannell [22], Smith [23]. On the basis of the issue, Yang and Lin [24,25] adopted a gear impact damping model with stationary stiffness, and the load variation on the effect of system response was addressed. Subsequently, in 2003 Kim et al. [26] imported an impact model as f δ;δ˙ ¼ δ 1 þ βδ˙ ð1Þ to study the effect of impact damping on resonance, harmonic frequency and super-harmonic frequency. However, the investigation of impact damping on the gear tooth was inadequate, mainly for the neglect of geometric factors of gear tooth face. ⁎ Corresponding author. Tel./fax: +86 731 88877746. E-mail address: [email protected] (J. Tang). 0094-114X/$ – see front matter. Crown Copyright © 2014 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2014.01.003
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Nomenclature B half of gear backlash D impact damping coefficient e the coefficient of restitution es (t) static transmission error f, FK, FD impact force, elastic force and damping force of impact system f (x) nonlinear backlash function Fd fluctuation of load Fv viscous damping force fm meshing frequency frev shaft rotation frequency hi (i = 1,2,…,6) coefficient of ith harmonic components of static transmission error I1, I2 moment of inertia of pinion and gear K the mesh stiffness of gear transmission system k (t), ktp, kr, k0 meshing stiffness function, single tooth stiffness, rth coefficients of stiffness and averaged nonlinear meshing stiffness Kv gear dynamic load m modulus of gear me equivalent gear mass n impact force exponent coefficient N1, N2 line of action Rb1, Rb2 basic radii of the pinion and gear S difference of pinion and gear in line of action T1 (T1 (t)), T2 (T2 (t)) input and output torque of gear transmission system vmax maximum velocity z1 number of gear tooth σ coefficient of load fluctuation ωn approximate natural frequency of gear pair ξ non-dimension viscous damping coefficient φr phase of rth stiffness ϕi (i = 1, 2,…,6) phase of ith harmonic components of static transmission error β impact damping coefficient ð−Þ δ, δ,˙ δ˙ the local relative penetration, velocity and initial velocity of impact θ1, θ2 angle displacement of pinion and gear ICR gear contact ratio DF dynamic load coefficient DMF dynamic mesh force DTE dynamic transmission error DTEppv peak-to-peak values of dynamic transmission error DTErms root-mean-square of dynamic transmission error SMF static mesh force TE transmission error
Moreover, impact damping coefficient β was location constant and the relationship between impact damping and impact velocity —as well as geometric parameters was neglected. In this paper, on the basis of an impact dynamic model, the effects of impact damping parameters, etc. on the system dynamic are considered for the perspective of nonlinear dynamics. In this paper, the impact characteristics in gear system with impact damping are investigated, and the experimental static transmission error and backlash are adopted in the theoretical analyses. The crown modification is performed in the pinion teeth surface. The nonlinear impact damping model is adopted. And that the nonlinear exponent n is non-integer indicates the point contact relative to crown modified gear. The following sections of this paper are organized as follows: in the second section, the impact force model, gear transmission model and consideration of static transmission error and backlash are interpreted. Subsequently, systematic consideration is remarked for a direct comparison between numerical results and experimental dynamic transmission error. The effects of load, input shaft velocity, parameter n on the dynamic characteristics of gear transmission system in terms of frequency response and dynamic load are investigated. Finally, a short remark is exhibited in the last section.
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Fig. 1. The physical model of gear impact model.
2. Gear impact model 2.1. Gear impact force model In this work, the nonlinear impact damping model (Hertz impact force model), n n f ¼ Kδ þ Dδ δ˙
ð2Þ
developed by Hunt and Crossley [27] in 1975 is adopted. According to the Hertz contact theory, the elastic force is represented as [27,28], n
F k ¼ Kδ :
ð3Þ
Here n = 3/2, δ is the local relative penetration and for gear contact, δ ¼ jRb1 θ1 −Rb2 θ2 j−B ¼ jSj−B
ð4Þ
here, S = Rb1θ1 − Rb2θ2, K is time varying mesh stiffness, which differs from the case in Ref. [27,28], and whose detailed calculation process will be described in next subsection. θi and Rbi (i = 1, 2) are angle displacements and basic radii of the pinions and gears respectively, and B is half of the gear backlash. Then the elastic force among meshing can be rewritten as n
F K ðθ1 ; θ2 Þ ¼ K ðθ1 ; θ2 Þδ :
ð5Þ
Generally, it can be assumed that the material structural energy dissipation during tooth contact is caused by the damping function. A hysteretic form for the damping effect is proposed by Hunt and Crossley [27] as, n F D ðθ1 ; θ2 Þ ¼ Dδ δ˙
ð6Þ
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and the damping coefficient is h i 2 ð−Þ ð−Þ 0:36 D ¼ 6K ð1−eÞ= ð2e−1Þ þ 3 δ˙ ¼ K ðθ1 ; θ2 ÞD; e ¼ 1−0:022 δ˙ :
ð7Þ
2.2. Gear pair impact model Backlash measured along the pitch circle is the amount of clearance between mated gear teeth, namely, the difference between tooth space and tooth thickness. Theoretically, backlash should be zero, but in actual practice some backlash must be allowed to prevent the jamming of teeth due to tooth errors and thermal expansion. The physical model of a gear transmission system with backlash is shown in Fig. 1. The mesh sliding friction is ignored and just the torsional vibration of gear transmission system is considered. According to the geometric condition, the motion between two gears can be divided into three cases as Case I : S ¼ Rb1 θ1 −Rb2 θ2 NB; Case II : jSj≤B Case III : Sb−B:
ð8Þ
(1) for Case I A pair of gear teeth meshing along the line of action N2, the contact force combined with two parts, the elastic force and damping force, are described as FK (θ1, θ2) and FD (θ1, θ2) respectively. The dynamic equation of gear pair in this case is I1 θ1 ¼ T 1 −½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb1 I2 θ2 ¼ T 2 þ ½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb2 ;
ð9Þ
where Ii (i = 1, 2) is moment of inertia of gear. (2) for Case II Under this condition, no contact occurs between two gear teeth. Therefore, the contact force is zero and the dynamic equation is, I1 € θ1 ¼ T 1 I2 € θ2 ¼ T 2 :
ð10Þ
(3) for Case III Under this condition, contact occurs along the line of action N1, and then the dynamic equation should be written as, I1 € θ1 ¼ T 1 þ ½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb1 I2 € θ2 ¼ T 2 −½ F K ðθ1 ; θ2 Þ þ F D ðθ1 ; θ2 ÞRb2 :
ð11Þ
Substituting Eqs. (5) and (6) into (9)–(11), one can obtain n θ1 ¼ T 1 −Rb1 δ K ðθ1 ; θ2 Þ þ Dδ˙ δ I1 € n θ2 ¼ T 2 þ Rb2 δ K ðθ1 ; θ2 Þ þ Dδ˙ δ ; I2 €
ð12Þ
where δ ¼ jRb1 θ1 −Rb2 θ2 j−B ¼ jSj−B
δ ¼
8 < :
1 SNB 6ð1−eÞ ð−Þ 0:36 0 jSj≤B ; e ¼ 1−0:022 δ˙ ;D ¼ : ð 2e−1 Þ2 þ 3 δ˙ ð−Þ i −1 Sb−B
ð13Þ ð14Þ
Combined with formula (13), the Eq. (12) can be defined as S þ
k0 K ðt Þ n δ 1 þ Dδ˙ δ ¼ F me k0
ð15Þ
F¼
Rb1 T 1 Rb2 T 2 1 R2 R2 þ ; ¼ b1 þ b2 : me I1 I2 I1 I2
ð16Þ
″
where
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We define x ¼ BS; ωn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 =me ; τ ¼ ωn t:
ð17Þ
Then, €x þ
K ðt Þ n−1 F B 1 þ BDωn sign x˙ x˙ f ðxÞ ¼ ; k0 Bω2n
ð18Þ
where, 8 <
n
ðx−1Þ f ðxÞ ¼ 0 : n −ð−x−1Þ
xN1 jxj≤1 xb−1:
ð19Þ
Meanwhile, impact damping force introduced in gear pairs is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k0 u x˙ F v ¼ 2ξt r 2p =I p þ r 2g =Ig
ð20Þ
so that the formula (18) can be redefined as €x þ 2ξ xþ ˙
K ðt Þ n−1 F B 1 þ BDωn sign x˙ x˙ f ðxÞ ¼ : k0 Bω2n
ð21Þ
According to literature [29,30], tooth pairs mesh stiffness k (t) can be simplified as kðt Þ ¼ k0 þ
N X
kr cosð2πr f m t−φr Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2−2 cosð2πr ðICR−1ÞÞ k0 k 1− cosð2πr ðICR−1ÞÞ ; φr ¼ ¼ ICR; r ¼ sinð2πr ðICR−1ÞÞ πr ktp ktp r¼1
Fig. 2. Experimental static transmission error (c) and its spectrum (a, b). A Fourier series expanded is denoted by red line in (c).
ð22Þ
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and the coefficient of load fluctuation which is subjected to load, electrical machine and errors can be simplified as Fd ¼ σ
F sinð2πr f m t Þ Bω2n
ð23Þ
where σ is the coefficient of load fluctuation. The gear tooth impact is imitated by the damping model. The geometric characteristics (crown modification) of gear tooth and material characteristics are included to determine parameter n and damping coefficient D. Parameter n can be set as n = 1, which is not rejected to normal characteristics of non-modified gears. However, gear tooth profiles are modified more or less actually, and line-to-line contact becomes point contact which can be understood as collision impact between spheres. It is a matter of concern as to what value of n can be appropriate. To be precise, the situation of gear pairs should be confirmed experimentally. However, the study starts with the analysis of varying n theoretically for the limits. For parameter D, it is a constant determined by impact transient velocity during single impact while it is varying continuously for continuous impact. Subject to the literature, the effects of parameter n and impact damping coefficient D on gear teeth dynamics are contrasted. 2.3. Considerations of the static transmission error and backlash Generally speaking, the static transmission error is approximated by the grade of gear with certain design and manufacturing errors. And the static transmission error then can be approximated as Fourier series. In this paper, the experimental static transmission error and backlash will be adopted. As mentioned in our previous work [31] (submitted to MMT simultaneously ), the pinion is driven at a very low shaft velocity, and the transmission error combined with assembly error and manufacturing error is obtained as shown in Fig. 2. The main part of static transmission error is the first harmonic component of shaft and the amplitude is 8.4644 × 10−4 rad and the amplitude of mesh frequency component corresponding with the first harmonic component of shaft is very low as shown in Fig. 2(b). For the convenience of theoretical analyses, the experimental static transmission error can be approximated by the sum of three harmonic frequency components related to the shaft velocity of driving gear and mesh frequency, as es ðt Þ ¼ h1 sinðωt þ ϕ1 Þ þ h2 sinð2ωt þ ϕ2 Þ þ h3 sinð3ωt þ ϕ3 Þ þ h4 sinðωt=z1 þ ϕ4 Þ þ h5 sinð2ωt=z1 þ ϕ5 Þ þ h6 sinð3ωt=z1 þ ϕ6 Þ:
ð24Þ
Here, hi, ϕi (i = 1, 2,…, 6) are the amplitudes and rotational displacements of corresponding frequency, zi is the number of driving gear teeth. In practical analyses, the amplitude hi is calculated as,
hðiÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X 2 f j: iω j¼fj f −2π j ≤ 5Bw g
ð25Þ
Here, Bw determined by sample frequency is the frequency resolution. The expression (25) indicates that the amplitude of ith harmonic component is the root-mean-square of the frequency component at iω / 2π, namely f∈(iω/2π − 5Bw,iω/2π + 5Bw). Then the amplitude hi is listed in Table 1 and the corresponding approximate curve is shown in Fig. 2(c) denoted by a red line. Additionally, the phase of experimental static transmission error is mainly determined by initial time and the effect of rotational displacement is less important. Let φi = 0 in the following analyses. As for the backlash, one can fix one shaft of gear pairs and move the other gear side to side in backlash. Although one shaft is fixed, there are still wobbly and elastic deformations and the angular position is recorded by encoders fixed on both shaft ends. The acquired data based on experiment [31] are illustrated in Fig. 3. The nominal static transmission error is shown in Fig. 3(a), and the absolute angle of the gear and pinion denoted by red and blue lines respectively is shown in Fig. 3(b). The dynamic backlash is obtained by comparing the angle difference as Bi ¼ Anglejt iþ1 −Anglejt i ; i ¼ 1; 2; …; 13. The mean value at Bi = 0.0064 is used in the following analyses. Table 1 Component amplitude of experimental static transmission error. Shaft frequencies × 10−4 (rad) 0 1 2 3
h0 h1 h2 h3
Mesh frequencies × 10−6 (rad) −2.9003 −8.4644 0.2553 0.1511
– h4 h5 h6
– 7.9473 1.9474 1.5414
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Fig. 3. Experimental backlash illustration of the (a) nominal static transmission error and (b) absolute angle.
3. Systematic considerations Generally, the static transmission error is a main excitation of gear vibration and noise. Dynamic mesh force generated under operating conditions is transmitted to housing through shafts and bearings. Mathematically, this static transmission error is modeled as an external force which appears at the right hand of dynamics Eq. (21) as in Ref. [12,32–36]. But two deviation methods are applied to Eq. (24), des ðt Þ 2 2 2 ¼ −h1 ω sinðωt þ φ1 Þ−4h2 ω sinð2ωt þ φ2 Þ−9h3 ω sinð3ωt þ φ3 Þ dt h i 2 2 2 2 − h4 ω sinðωt=z1 þ φ4 Þ þ 4h5 ω sinð2ωt=z1 þ φ5 Þ þ 9h6 ω sinð3ωt=z1 þ φ6 Þ =z1 :
ð26Þ
The new amplitudes corresponding with shaft frequency and mesh frequency change via different patterns. Especially the amplitudes of three shaft frequency components are inversely relative to z21 (z1 is the number of gear teeth). Namely the effect of shaft frequency components on dynamic responses is less than the effect of mesh frequency part. In addition, as mentioned in the previous literature [12,32–36], the numerical dynamic transmission error cannot be used to have a direct comparison with the experimental dynamic transmission error. Therefore, a more realistically considered model of the gear transmission system is needed. Recalling the definition of initial static transmission error: “transmission error is the difference between the angular position that the output shaft of a drive would occupy if the drive were perfect and the actual
Table 2 Design data of gears.
Number of teeth Module (mm) Pressure angle (deg) Tooth width (mm) Iterate of moment Modification (mm)
Gear
Pinion
61 4 20 40 107.3344 × 10−3 0.026
61 4 20 40 108.9025 × 10−3 –
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position of the output.” [37]. In this paper, the static transmission error is combined with backlash as 1⇒1 þ
r b1 e ðt Þ ¼ b B s
ð27Þ
and the backlash function (19) can be rewritten as, 8 <
n
ðx−bÞ f ðxÞ ¼ 0 : n −ð−x−bÞ
xNb jxj≤b xb−b:
ð28Þ
4. Numerical analysis Gear design data are shown in Table 2, where the damping coefficient is ξ = 0.07 and backlash B = 0.7337 × 10−3/2m is measured from the test rig. The effect of friction is negative in this work. The mesh stiffness is deduced from gear structural parameters as shown in Table 2. The system dynamic response is obtained through the Runge–Kutta method to integrate the differential Eq. (21). The effects of load, input shaft velocity, parameter n on the dynamic characteristics of gear transmission system in terms of frequency response and dynamic load are investigated. 4.1. Calculation of ISO standard dynamic load coefficient For comparison of the numerical results, the calculation method for the gear dynamic load based ISO standard is illustrated in this subsection. According to the standard [38], the simplified computational formula of gear dynamic load is Kv ¼
−B A pffiffiffiffiffiffiffiffiffiffiffi A þ 200v
ð29Þ
where A = 50 + 56(1.0 − B) and B = 0.25(C − 5.0)0.667. And v is the velocity of gear tooth along the pitch line N2. The maximum velocity is not exceeded by vmax. v max ¼
½A þ ð14−C Þ2 : 200
ð30Þ
2 C=12 1.9 C=11 1.8 C=10 1.7 C=9 1.6 C=8
1.5 1.4
C=7
1.3 C=6 1.2 1.1 1 0
5
10
15
20
25
30
35
Fig. 4. Dynamic load coefficient of gear teeth.
40
45
50
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Fig. 5. Measured error of gear face.
Here, C is the class of gear pair transmission accuracy, which can be deduced with the number of teeth, modules and tooth form errors. C ¼ −0:5048 ln z1;2 −1:144 ln ðmÞ þ 2:852 ln f pt1;2 þ 3:32
ð31Þ
The gear tooth conditions adjusted to the calculation of dynamic load coefficient are 6 ≤C ≤12 6 ≤z≤1200 1:25 ≤m≤50:
ð32Þ
According to the above formula, dynamic load coefficients with different precision values and velocities are shown in Fig. 4. Gear design precision is 6, yet the actual gear precision is lower for big single tooth errors. As shown in Fig. 5, the measuring results of gear face along three coordinates are given representatively. The deviation of a single tooth is related to 17.9 μm. According to the formula (31), the actual class of gear accuracy is DIN 8. 4.2. The effect of load and velocity on dynamic load coefficient Firstly, n = 1.5 is set in the impact force model and the impact damping coefficient is determined by impact velocity. The load torque is 5000 N m and the input velocity varies within the region of (50,3000) rpm. The resonance frequency range should be determined before the analysis of dynamic load characteristics.
Fig. 6. Dynamic load coefficient varying with speed, load torque at 5000 N m.
S. Chen et al. / Mechanism and Machine Theory 74 (2014) 354–369
Fig. 7. Dynamic load coefficient varying with speed, load torque at 500 N m.
Fig. 8. Frequency response of gear transmission system with load torque at 5000 N m.
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The resonance velocity is defined as sffiffiffiffiffiffiffi 1 km nE1 ¼ ; 2π me
ð33Þ
where me is the equivalent gear mass; km is the mean mesh stiffness of gear tooth. For the parameters shown in Table 2, the corresponding equivalent mass and mesh stiffness of gear pairs are 8
me ¼ 4:1167 kg; k0 ¼ 9:9455 10 N=m:
ð34Þ
The ratio of revolving speed is defined as N¼
n1 nz1 =60 sffiffiffiffiffiffiffi : ¼ nE1 1 km 2π me
ð35Þ
According to the value of N, the gear tooth surface can be divided into four sections: (1) sub-crisis area when N ≤ 0.85; (2) resonance area when 0.85 b N ≤ 1.15; (3) translational area when 1.15 b N ≤ 1.5; and (4) above-critical area when N N 1.5. For the established dynamic model, the dynamic mesh force is n DMF ¼ K ðt Þδ 1 þ Dδ˙ δ
ð36Þ
and then the dynamic load coefficient [39] is DF ¼
maxðDMF Þ SMF
ð37Þ
where SMF is the static mesh force. The dynamic load coefficients of 5000 N m and 500 N m are given (shown in Figs. 6 and7 respectively). In the speed range of (50,2068.2) rpm, the system stays at the sub-critical area. During (2068.2, 2798.2) rpm, the system stays at the resonance area and achieves the maximum value of dynamic load. The system stays at the translational area in (2798.2, 3000) rpm. The theoretical dynamic load coefficient curve based on the gear precision class of ISO 8 is denoted by a black line in Fig. 6. In
Fig. 9. Frequency response of gear transmission system with load torque at 500 N m.
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Fig. 10. The ratios of first three harmonic frequencies with load torque at 5000 N m and 500 N m.
comparison, the dynamic load coefficient of actual manufactured gear system involving tooth surface error is larger than the theoretical dynamic load coefficient. Though large fluctuation of dynamic load coefficient exists at 5000 N m and 500 N m, the tendency of dynamic load coefficient increase with respect to velocity is the same.
Fig. 11. Dynamic load coefficient with variation of speed with load torque at 5000 N m.
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Fig. 12. Dynamic load coefficient with variation of speed with load torque at 500 N m.
The frequency response curves are given (shown in Figs. 8 and9) to further analyze the fluctuation of dynamic load coefficient. The amplitude of the shaft frequency component varies slightly at 5000 N m (shown in Fig. 8). The amplitude variation is not obvious with the increase of velocity. At low speed, the amplitudes of first three mesh frequency components are large, and
Fig. 13. System frequency response with load torque at 5000 N m.
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Fig. 14. System frequency response with load torque at 500 N m.
fundamental mesh frequency and doubling frequency component are principal in system dynamic transmission error. With the increase of speed, the amplitudes of mesh frequency components decrease rapidly, and system dynamic transmission error gives priority to shaft frequency component. When the load decreases to 500 N m, the amplitude of teeth frequency components
Fig. 15. The ratio of prior three harmonic components with load torque at 5000 N m and 500 N m respectively.
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decrease obviously (as shown in Fig. 9). The ratios of first three harmonic frequency components under two load conditions referred are compared (shown in Fig. 10). Apparently, shaft frequency is slightly affected by load while the mesh frequency and doubling frequency components are affected notably. The results show that, (1) Under low speed conditions, the dynamic load is mainly connected with the high frequency component (mesh frequency and harmonic component) of transmission error. The high frequency amplitude of transmission error can be effectively suppressed with the increase of tooth surface accuracy and the improvement of tooth profile error and roughness, so that it is meaningful to dismiss the system dynamic load. (2) Under high speed conditions, the dynamic load is mainly connected with the shaft frequency component of transmission error which exerts a huge influence on the dynamic load, though the gear mesh may be in the resonance frequency area. 4.3. The effect of parameter n on dynamic load coefficient Parameter n is set to 1.0 and others are not modified. The dynamic load coefficients with respect to velocity are shown in Figs. 11 and12. The differences between the system dynamic responses under different values of parameter n can be deduced from Figs. 6 and7, which are mainly expressed as follows. (1) Parameter n is decreasing and dynamic load coefficient is increasing especially in the resonance area. Under the two load conditions, three resonance peak values appear around speed 490 rpm, 790 rpm and 1110 rpm. The corresponding load conditions of frequency response curves are shown in Figs. 13 and14 respectively, which outline that the first resonance peak value is conducted with prior three harmonic frequency components, and the second one and the third one are determined by the third and the second harmonic frequency components respectively. (2) When n is 1.0, the system exhibits loss contact impact which is denoted by a red ‘-o-’and the dynamic load is raised by impact. And corresponding to the frequency response curve, in the loss contact impact area, the amplitudes of the shaft frequency component in dynamic transmission error are almost equivalent (shown in Fig. 15), which explains that dynamic characteristics are slightly affected by load and parameter n with loss contact impact. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. (3) When n is 1.5, loss contact impact does not exist in the system, and the ratio of prior three harmonic components approximates the linear increase of load. When n is 1.0, the system assumes stronger nonlinear characteristics with the appearance of impact. For the engineering perspective, not only should the resonance area be considered to neglect the resonance of the gear transmission system, but also the dynamic impact load should be at low speed. 5. Conclusions On the basis of Hertz impact model, the impact characteristics in a gear system with impact damping are investigated, and the experimental static transmission error under crown modification condition and backlash are adopted in the theoretical analysis. The effects of load, input shaft velocity, parameter n on the dynamic characteristics of gear transmission system in terms of the frequency response and dynamic load are investigated. The numerical simulation shows that, (1) Direct comparison between the analyses of experimental dynamic transmission error and those of static transmission error can be combined with the backlash parameter. (2) Under low velocity conditions, the dynamic load is mainly connected with a high frequency component of transmission error. The high frequency amplitude of transmission error can be effectively suppressed with the increase of tooth surface accuracy and the improvement of tooth profile error and roughness, so as to weaken the system dynamic load. Under high speed conditions, the dynamic load is mainly connected with the shaft frequency component of transmission error, which has a dominate influence on the dynamic load, though the gear pair may be in the resonance frequency mesh area. (3) Parameter n is decreasing and the dynamic load coefficient is increasing especially in the resonance area. When n is 1.0, the system exhibits a loss contact impact area and the amplitudes of shaft frequency components with dynamic transmission error are almost equal, which indicates that dynamic characteristics are slightly affected by load and parameter n under a no loss contact impact condition. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. (4) When n is 1.5, loss contact impact does not exist in the system, and the ratio of prior three harmonic components approximates the linear increase of load. But when n is 1.0, the system assumes stronger nonlinear characteristics with the appearance of impact. Acknowledgements The authors gratefully acknowledge the support of the National Science Foundation of China (NSFC) through Grants Nos. 51305462 and 51275530, as well as the support of the National Basic Research Program of China (2011CB706800).
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
R. Ibrahim, Vibro-Impact Dynamics: Modeling, Mapping and Applications, Springer Verlag, 2009. M. di Bernardo, et al., Piecewise-smooth Dynamical Systems: Theory and Applications, Springer, London, 2008. W. Stronge, Impact Mechanics, Cambridge Univ Pr, 2004. F. Pfeiffer, A. Kunert, Rattling models from deterministic to stochastic processes, Nonlinear Dyn. 1 (1990) 63–74. K. Karagiannis, F. Pfeiffer, Theoretical and experimental investigations of gear-rattling, Nonlinear Dyn. 2 (1991) 367–387. A. Kunert, F. Pfeiffer, Description of chaotic motion by an invariant probability density, Nonlinear Dyn. 2 (1991) 291–304. Q. Feng, F. Pfeiffer, Stochastic model on a rattling system, J. Sound Vib. 215 (1998) 439–453. A. Haj-Fraj, F. Pfeiffer, Optimization of automatic gear shifting, Veh. Syst. Dyn. 35 (2001) 207–222. A.C.J. Luo, D. Oconnor, Periodic motions and chaos with impacting chatter and stick in gear transmission systems, Int. J. Bifurcation Chaos 19 (2009) 1975–1994. A.C.J. Luo, D. O'Connor, Periodic motions with impacting chatter and stick in a gear transmission system, J. Vib. Acoust. 131 (2009)(041013-11). A.C.J. Luo, M.T. Patel, Bifurcation and stability of periodic motions in a periodically forced oscillator with multiple discontinuities, J. Comput. Nonlinear Dyn. 4 (2009) 011011. A. Kahraman, R. Singh, Non-linear dynamics of a spur gear pair, J. Sound Vib. 142 (1990) 49–75. A. Kahraman, R. Singh, Non-linear dynamics of a geared rotor-bearing system with multiple clearances, J. Sound Vib. 144 (1991) 469–506. A. Kahraman, R. Singh, Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system, J. Sound Vib. 146 (1991) 135–156. S. Natsiavas, Stability and bifurcation analysis for oscillators with motion limiting constraints, J. Sound Vib. 141 (1990) 97–102. S. Natsiavas, Modal interactions in self-excited oscillators under external primary resonance, J. Sound Vib. 184 (1995) 261–280. S. Theodossiades, S. Natsiavas, Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash, Chaos Solitons Fractals 12 (2001) 2427–2440. T.C. Kim, et al., Effect of nonlinear impact damping on the frequency response of a torsional system with clearance, J. Sound Vib. 281 (2005) 995–1021. E. Rigaud, J. Perret-Liaudet, Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 1: harmonic excitation, J. Sound Vib. 265 (2003) 289–307. J. Perret-Liaudet, E. Rigaud, Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 2: random excitation, J. Sound Vib. 265 (2003) 309–327. R.C. Azar, F.R.E. Crossley, Digital simulation of impact phenomenon in spur gear systems, J. Eng. Ind. Trans. ASME Ser. B 99 (1977) 792–798. R.G. Herbert, D.C. McWhannell, Shape and frequency composition of pulses from an impact pair, ASME J. Eng. Ind. 99 (1977) 513–518. J.D. Smith, Energy loss in gear tooth impact. ARCHIVE: Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 1983–1988 (vols. 197–202) 198 (1984) 205–208. D.C.H. Yang, J.Y. Lin, Hertzian damping, tooth friction and bending elasticity in gear impact dynamics, J. Mech. Transm. Autom. Des. 109 (1987) 189–196. D. Yang, Z. Sun, A rotary model of spur gear dynamics, J. Mech. Transm. Autom. Des. 107 (1985) 529–535. T.C. Kim, et al., Effect of smoothening functions on the frequency response of an oscillator with clearance non-linearity, J. Sound Vib. 263 (2003) 665–678. K.H. Hunt, F.R.E. Crossley, Coefficient of restitution interpreted as damping in vibroimpact, ASME J. Appl. Mech. 42 (1975) 440–445. H.M. Lankarani, P.E. Nikravesh, A contact force model with hysteresis damping for impact analysis of multibody systems, J. Mech. Des. 112 (1990) 369–376. A. Kahraman, G.W. Blankenship, Effect of involute contact ratio on spur gear dynamics, J. Mech. Des. 121 (1999) 112–118. C. Gill-Jeong, Nonlinear behavior analysis of spur gear pairs with a one-way clutch, J. Sound Vib. 301 (2007) 760–776. C. Siyu, J.Y. Tang, Experimental study of the effect of assembly error on the lightly loaded transmission error of spur gear with crown modification, Mech. Mach. Theory (2014)(submitted for publication MMT). S. He, et al., Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations, J. Sound Vib. 309 (2008) 843–851. W. Kim, et al., Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation, J. Sound Vib. 329 (2010) 4409–4421. S. Baguet, G. Jacquenot, Nonlinear couplings in a gear–shaft–bearing system, Mech. Mach. Theory 45 (2010) 1777–1796. C.-W. Chang-Jian, Nonlinear analysis for gear pair system supported by long journal bearings under nonlinear suspension, Mech. Mach. Theory 45 (2010) 569–583. C. Siyu, et al., Nonlinear dynamic characteristics of geared rotor bearing systems with dynamic backlash and friction, Mech. Mach. Theory 46 (2011) 466–478. D.B. Welbourn, Fundamental knowledge of gear noise: a survey, Inst. Mech. Eng. Conf. Publ. (1979) 9–14. ISO, 6336-1:2006, Calculation of Load Capacity of Spur and Helical Gears—Part 1: Basic Principles, Introduction and General Influence Factors, International Organization for Standardization, 1996. V.K. Tamminana, et al., A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs, J. Mech. Des. 129 (2007) 75–84.