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Analytical determination of back-side contact gear mesh stiffness
Mechanism and Machine Theory 78 (2014) 263–271
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Short communication
Analytical determination of back-side contact gear mesh stiffness Yichao Guo a, L. S. Randolph Professor and Head Robert G. Parker b,⁎ a b
Rochester Hills, MI 48306, United States Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, United States
a r t i c l e
i n f o
Article history: Received 10 February 2014 Received in revised form 15 March 2014 Accepted 24 March 2014 Available online 26 April 2014 Keywords: Gear mesh Back-side contact Mesh stiffness Mesh phase Scissor gear Anti-backlash gear
a b s t r a c t Back-side gear tooth contact happens when anti-backlash (or scissor) gears are used, tooth wedging or tight mesh occurs, or vibration amplitudes are high enough that teeth separate and pass the backlash zone. An accurate description of the back-side gear tooth mesh stiffness is needed to study gear mechanics in such cases. This work studies the time-varying back-side mesh stiffness and its correlation with backlash by analyzing the relationship between the drive-side and back-side mesh stiffnesses. Results of this work yield the general form of the back-side mesh stiffness in terms of the known drive-side mesh stiffness for an arbitrary gear pair. The analytical results are confirmed by simulation results from gear contact analysis software that precisely tracks drive- and back-side gear tooth contact. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Back-side contact in a gear mesh refers to contact on the surfaces of a gear that are not used to transmit power. Recent studies on gear dynamics [1–4] show that it is possible for tooth wedging (or tight mesh), that is, simultaneous drive-side and back-side contact, to happen in applications such as wind turbine gearboxes. Tooth wedging in wind turbines results from the combined effect of gravity and bearing clearance nonlinearity, and it proved a likely source of gearbox bearing failures in a particular case. For better understanding of the impact of tooth wedging on gearbox failures, it is necessary to have a model that includes accurate description of the back-side contact mesh stiffness. Besides tooth wedging, anti-backlash (or scissor) gears are another case for back-side contact to occur. To minimize the undesirable characteristics caused by backlash, anti-backlash gears eliminate the backlash by using a preloaded spring to force the fixed part of the driving gear to contact the drive-side of the driven gear teeth and, simultaneously, force the free part of the driving gear to contact the back side of the driven gear [5]. Accurate modeling of back-side contact mesh stiffness is necessary to analyze such systems. Mesh stiffness variation and its impact on gear mechanics have been extensively investigated. Mesh stiffness variation is the source of static transmission error fluctuations. Munro and his team experimentally investigated gear tooth mesh stiffness throughout and beyond the path of contact [6]. Blankenship and Kahraman experimentally and analytically studied a single degree of freedom gear pair driven by time-varying mesh stiffness variation; they showed contact loss and back-side contact that are subject to a symmetric backlash condition [7]. The same system was investigated with analytical and finite element models [8]. Lin, Liu, and Parker analyzed mesh stiffness variation instabilities in two-stage gear systems [9,10], as well as in simple planetary gear systems [11]. Their studies showed that parametric excitation from time-varying mesh stiffness causes instability and severe vibration under certain operating conditions. They applied a perturbation method to analytically determine the instability conditions. Velex and Flamand extended the ⁎ Corresponding author. E-mail address: [email protected] (R.G. Parker).
http://dx.doi.org/10.1016/j.mechmachtheory.2014.03.011 0094-114X/© 2014 Elsevier Ltd. All rights reserved.
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research scope to planetary gear trains and studied their dynamic responses with varying mesh stiffness [12]. Wu and Parker [13] extended the study on parametric instability to planetary gears with elastic continuum ring gears. Sun and Hu [14] investigated mesh stiffness parametric excitation and clearance nonlinearity for simple planetary gears. Bahk and Parker [15] derived closed-form solutions for the dynamic response of planetary gears with time-varying mesh stiffness and tooth separation nonlinearity based on a purely torsional planetary gear model. They extended this to systems with tooth profile modifications [16]. Guo and Parker [1] modeled and analyzed a simple planetary gear with time-varying mesh stiffness, tooth wedging, and bearing clearance nonlinearity. Although back-side contact is included in their model, the average value of the periodic mesh stiffness on the drive-side is used to approximate the back-side mesh stiffness, which is a simplified description of the back-side mesh stiffness. Despite the abundance of literature on mesh stiffness variation and gear dynamics, no studies have derived the back-side mesh stiffness in their analytical model. One possible reason is that the usual symmetry of the gear teeth ensures that the contact ratios, mesh periods, and average mesh stiffnesses over the mesh period are the same for drive- and back-side contact. This may lead to the mistaken conclusion that the back-side mesh stiffness is the same as the drive-side one. For example, Kahraman and Blankenship performed experiments on the nonlinear response of spur gear pairs with varying involute contact ratios [17,18]. The back-side contact is assumed to be the same as the drive-side contact in their study. In the tight mesh case shown in Fig. 1, the back-side mesh stiffness, however, is not equivalent to the drive-side one, because the back-side contact is along the back-side line of action (the dashed line in Fig. 1) and the number of gear teeth in contact along the back-side line of action is not always equal to that along the drive-side line of action (the solid line in Fig. 1). Fig. 2 illustrates one such case (the simulation results are from Calyx [19], a multi-body finite element/contact mechanics program with precise gear tooth contact capability). There are two pairs of gear teeth in contact along the back-side line of action, while only one pair of teeth is in contact along the line of action. Therefore, the back-side mesh stiffness differs from the drive-side mesh stiffness at this moment. 2. Derivation of back-side mesh stiffness The drive-side mesh stiffness refers to the stiffness of the nominally contacting teeth at a mesh in the direction of power transmission. It varies as the number of teeth in contact fluctuates with the gear rotation. The stiffness acts along the line of action. The period of its variation is known for the given rotation speed. Mesh stiffness variation functions are often approximated by Fourier series in analytical studies. They can be accurately calculated by finite element software. The drive-side mesh stiffness function is critical for analytical studies on gear dynamics [8,15,20–22]. Similar to the drive-side mesh stiffness, the back-side mesh stiffness is the stiffness of the changing number of contacting teeth along the back-side line of action. In order to have simultaneous drive-side and back-side contacts at all times, we first investigate an ideal gear pair (one that operates at the nominal center distance and has zero backlash tooth thickness). The results of the ideal gear pair are then extended to arbitrary gear pairs with backlash and to anti-backlash gears. 2.1. Back-side mesh stiffness for an ideal gear pair Fig. 3 illustrates the drive-side and back-side contacts for an ideal gear pair. The gear details are not needed for what follows. The tooth numbers of the driving and driven gears are Zdr and Zdn, respectively. T is the mesh period. At t = 0, the pitch point at the drive-side of the driving gear is in mesh (Fig. 3a). The dashed line in the middle of each sub-figure is the center line between the two gears. After one mesh period T, the driving gear tooth moves one driving gear circular pitch pdr ¼ 2πr Z , and the driven gear (rdr and rdn are the pitch radii). After one fourth of the mesh period (t ¼ T4), the driving gear tooth moves one tooth moves pdn ¼ 2πr Z fourth of its circular pitch. Because there is no backlash along the pitch circle, the circular tooth thickness of the driving gear qdr is dr
dr
dn
dn
qdr ¼
1 p : 2 dr
ð1Þ
Back_side line of action
Driving direction
Driving Gear
Driven Gear
Line of action
Fig. 1. Drive-side gear contact (solid line) and back-side gear contact (dashed line) in a tight mesh case (both drive and back-sides are in contact).
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Driving Driven Gear Gear
Fig. 2. Numerical simulation of Calyx on an ideal gear pair with tight mesh. One pair of teeth (marked by a circle) is in contact along the drive-side line of action, and two pairs of teeth (marked by two rectangles) are in contact along the back-side line of action.
Because the driving gear rotates 2qdr in one mesh period T, the movement of the driving gear in T4 is half of its circular tooth thickness, or 12qdr . Thus, at t ¼ T4 the center line passes through the middle of the tooth tip of the driving gear (Fig. 3b). At this moment both gears are symmetric about the center line, and the number of gear teeth in contact along the drive-side line of action equals that along the back-side line of action. The back-side and drive-side mesh stiffnesses are equal at this moment. Furthermore, at t ¼ T2, the driving gear tooth moves qdr relative to its position at t = 0, and the center line passes directly through the pitch point at the back-side of the driving gear tooth (Fig. 3c). At t ¼ 3T4 , the driving gear moves 32qdr relative to its position at t = 0, and the driven gear rotates 32qdn relative to its initial position at t = 0 (qdn is the circular tooth thickness of the driven gear). At this moment, the center line passes through the middle point of the tooth tip of the driven gear, and both gears are symmetric about the center line. Therefore, the back-side and drive-side mesh stiffnesses are equal at t ¼ 3T4 . To clearly illustrate the drive-side and back-side mesh stiffness relationship, the finite element/contact mechanics software Calyx [19] is used in this investigation to precisely track the tooth contact for precise tooth surface geometry as the gears rotate. Its tooth surface geometry representation is effectively error-free; it does not rely on the node points to define the surface. Calyx's ability to accurately monitor the contact conditions makes it a reliable benchmark. In the simulations that follow Calyx tracks the contact for specified gear kinematics under unloaded conditions. The parameters of the example gear pair are listed in Table 1 and the Calyx gear model is shown in Fig. 4. Fig. 5 shows the variations of the number of teeth in contact for the drive and back sides by tracking the numbers of gear teeth in contact along the drive- and back-side lines of action using the Calyx model. All the simulation results in this study use sufficient resolution of the time axis to capture changes with needed accuracy. At t = 0 (as shown in Fig. 3a), the pitch point at the drive-side of the driving gear is in contact. O and O′ are the points on the drive- and back-side tooth number variation functions at t = 0. Due to the symmetry of both gears about the center line at t ¼ T4 , the drive-side tooth number variation function from this moment on by rotating the gears in the nominal driving direction is identical to the back-side tooth number variation function from this moment on by rotating the gears in the opposite direction. This is verified by the simulation results in Fig. 5. Let A and A′ be the points on the drive- and back-side tooth number variation functions at t ¼ T4 . The drive-side tooth number variation function after point A is identical to the back-side gear tooth number variation function before point A′. To mathematically describe the above relations, the varying drive-side mesh stiffness is kI(t), and the matching back-side mesh stiffness is kIb(t). Rotating the gears in the nominal driving direction until the moment of t ¼ T4 is identical to shifting the I phase of kIb(t) by T4. This generates a phase-shifted function k ðt−T4Þ that is the drive-side gear tooth number variation function in I Fig. 5 with the moment A being its origin of time axis. Replacing t−T4 with τ in k ðt−T4Þ yields kI(τ) whose origin is at τ = 0 or t ¼ T4 when the time axis is t. Noting that the back-side mesh stiffness function that matches with kI(τ) is kIb(τ) and applying the symmetry relation that the drive-side tooth number variation function after τ = 0 is equal to the back-side gear tooth number variation function before τ = 0, the mesh stiffnesses have the relation of I
I
kb ðτ Þ ¼ k ð−τ Þ:
ð2Þ
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a) t =0, T
c) t =T/2
b) t =T/4
d) t =3T/4
Fig. 3. The gear mesh contacts for an arbitrary external gear pair with no backlash. The dashed line in the middle of each sub-figure is the center line. The driving gear is at the right hand side of each subplot and the driving direction is counter-clockwise.
The insertion of τ ¼ t−T4 into Eq. (2) yields T I I kb ðτ Þ ¼ k −t þ : 4
ð3Þ
I
Because τ ¼ t−T4; kb ðτÞ, is a phase-shifted function of kIb(t), and they have the relation T I I kb ðτ Þ ¼ kb t− 4
ð4Þ
insertion of Eq. (4) into Eq. (3) yields T T I I kb t− ¼ k −t þ 4 4 T T I I I ¼ k −t− ⇒kb ðt Þ ¼ k −t þ 2 2
ð5Þ
Eq. (5) reveals that kIb(t) is uniquely determined once the drive-side mesh stiffness function kI(t) is known. According to Eq. (5), the number of gear teeth in contact at the back-side at t = 0 equals the number of gear teeth in contact at the drive-side at t ¼ T2. This is confirmed by the simulation results in Fig. 3. There are two pairs of teeth in contact at the back-side in Fig. 3a (t = 0), Table 1 Gear parameters for the example system shown in Fig. 4.
Number of Teeth Diametrical pitch Pressure angle (deg) Outer diameter (m) Root diameter (m) Mesh period (sec)
Driving gear
Driven gear
41 10.34 25 0.105 0.090 0.293
32 10.34 25 0.085 0.073 0.293
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Fig. 4. Calyx finite element model of the example ideal gear pair.
and there are exactly two pairs of teeth in contact at the drive-side in Fig. 3c (t ¼ T2). In addition, Fig. 3 provides further numerical validation of Eq. (5) that the drive-side tooth number variation function before point O″ is symmetrical to the back-sidegear tooth number variation function after point O′ where O″ in Fig. 5 is the point on the drive-side tooth number variation function at t ¼ T2. 2.2. Back-side mesh stiffness for a gear pair with nominal backlash
Back-side gear tooth number variation nb (t)
Drive-side gear tooth number variation n(t)
In applications, gears always include backlash to allow lubrication, manufacturing errors, deflection under load, and thermal expansion. It is typically created by slightly increasing the center distance of the gear pair or reducing the circular tooth thickness.
3 2.5
-t
*
2 1.5
O''
*
*
1
O
0.5 0
3 2.5 2
O'
t
1.5 1 0.5 0 -T
- 3T/4
- T/2
- T/4
0
T/4
T/2
3T/4
T
Time
Fig. 5. The drive- and back-side gear tooth number variation functions for the example ideal gear pair in Fig. 4. ‘x’ indicates the time that the pitch point of the drive-side of the driving gear is in contact, ‘A’ indicates the time that the middle point of the drive gear tooth tip is aligned with the center line, ‘+’ indicates the time that the pitch point of a driven gear tooth is in contact, and ‘*’ indicates the time that the middle point of a driven gear tooth tip is aligned with the center line.
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Radial direction Matching ideal gear pair Tooth of the driven gear Back-side line of action
Line of action 2b
Pitch circle
Base circle
Tooth of the driving gear
Fig. 6. The drive-side and back-side gear mesh contacts for a gear pair with 2b backlash and its matching ideal gear pair.
Back-side gear tooth number variation nb (t)
Drive-side gear tooth number variation n(t)
Fig. 6 illustrates the backlash for an external gear pair. The nominal backlash for the gear pair is 2b, the circular pitch is p, and the mesh period is T. The case that the center distance of the gear pair remains unchanged and there is no relative radial motion between the gears is investigated first. The impacts of changing the center distance and relative radial motion are studied subsequently. Once a gear pair is installed with its nominal center distance, the backlash remains at its nominal value 2b if there is no relative radial motion between the gears. The thick dashed line in Fig. 6 represents the case of back-side contact. If the varying drive-side mesh stiffness is k(t), Eq. (5) does not give the desired back-side mesh stiffness variation function. Instead, Eq. (5) gives kIb(t), the
3 2.5
-t
O
2 1.5
-T/2+bT/p 1 0.5 0
3 2.5 2
O'
t
1.5 1 0.5 0
-T
- 3T/4
- T/2
- T/4
0
T/4
T/2
3T/4
T
4
Time: t (s) Fig. 7. Back-side and drive-side gear tooth number variation functions the example gear pairs with zero center distance change and 2b nominal backlash (b satisfies bq ¼ 0:05).
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back-side mesh stiffness variation function for the matching ideal gear pair (the thin dashed lines in Fig. 6) that has the same center distance and pitch circle as the original gear pair in Fig. 6 but different tooth thicknesses. To ensure zero backlash, the tooth thickness of the matching ideal gear pair equals the thickness of the original gear pair increased by b. The slight increase in tooth thickness of the matching ideal gear pair only causes the associated mesh stiffness variation function to differ from the original gear pair by a phase shift. Thus, the drive-side mesh stiffness variation function of the matching ideal gear pair equals that of the original gear pair, because the gear contacts of the matching ideal gear pair at the drive-side always coincide with the original gear pair. The phase shift between the matching ideal gear pair and the original gear pair at the back-side is derived as follows. If the matching ideal gear pair moves b along the pitch circle in the reverse driving direction in Fig. 6, the back-side contact of the ideal gear mesh in the thin dashed line will coincide with the actual back-side contact in the thick dashed line. The phase lag between the back-side mesh stiffness functions of the matching ideal gear pair and the original gear pair for the above process is bpT. In other words, the actual back-side mesh stiffness variation function kb(t) of a gear pair is the back-side mesh stiffness variation function for the matching ideal gear kIb(t) with pbT phase lag, that is, b I kb ðt Þ ¼ kb t− T : p I
ð6Þ
I
The insertion of kb ðt Þ ¼ k ð−t−T2Þ (from Eq. (5)) into Eq. (6) and application of the condition that the drive-side mesh stiffness function of the matching ideal gear pair equals that of the original gear pair (Fig. 6) yield T b kb ðt Þ ¼ k −t− þ T 2 p
ð7Þ
where k(t) is the drive-side mesh stiffness variation of the original gear pair. To verify the back-side mesh stiffness variation function for the gear pairs with 2b nominal backlash in Eq. (7), the tooth thickness of the example gear pair in Fig. 4 is reduced by 10% such that bp ¼ 0:05 . The tracking results of the drive-side and back-side tooth number variations from Calyx are shown in Fig. 7. O in Fig. 7 is the point on the drive-side gear tooth number variation function at t ¼ −T2 þ pbT . The drive-side gear tooth number variation function before point O is symmetrical to the back-side gear tooth number variation function after point O′. Thus, the results in Fig. 7 agree with Eq. (7). 2.3. Back-side mesh stiffness for an arbitrary gear pair with changing backlash In real applications, the actual operating center distance of a gear pair can differ from the nominal one due to manufacturing errors of the axes, deflections under load, and misalignments. In addition, vibration of the gears in the relative radial direction changes the instantaneous center distance [1].
a)
Radial direction Tooth of the driven gear
Δ
Line of action
Tooth wedging happens when
2Δ tanα=2 b Back-side line of action
α 2Δ sinα
Pitch circle
Tooth of the driving gear after Δ change of center distance Base cicle Tooth of the driving gear
b)
Back-side line of action
Line of action Δα
α
Δ sinα
c) Pitch circle
2Δ tanα
α
Back-side line of action
2Δ sinα
Fig. 8. Back-side tooth contact for the case of tooth wedging when 2Δctanα = 2b. b is half of the backlash along the pitch circle, Δc is the change of the central distance, and α is the pressure angle.
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Fig. 8a shows that a change of center distance Δc (positive Δc indicates a reduction of center distance) reduces the backlash by 2Δc tan α, where α is the pressure angle. Fig. 8b and c explains this relationship geometrically. When the driving gear moves Δc toward the driven gear (i.e., a Δc change in center distance), the gaps between the two gears along the drive- and back-side lines of action are simultaneously reduced by Δc sin α (Fig. 8b). Because the driving gear is pressed against the driven gear along the line of action and does not move toward the driven gear along the line of action, the total gap reduction along the back-side line of action becomes 2Δc sin α. The gap change along the pitch circle is approximated by the associated cord length shown in Fig. 8c. By applying trigonometric relations to the triangle shown in Fig. 8c, the length of the chord that approximates the gap change along the pitch circle is 2Δcossinα α ¼ 2Δc tan α. When Δc ¼ tanb α such that 2Δc tan α = 2b (2b is the nominal backlash), the back-side of the driving gear (the dashed line in Fig. 8a) is in contact with the back-side of the driven gear and tooth wedging (tight mesh) occurs. When Δc b tanb α, the actual backlash of the gear pair 2b′ is 2(b − Δc tan α). This case is equivalent to the case in Fig. 6 with the exception that the backlash is 2b′ instead of the nominal value 2b. Replacing b with b′ in Eq. (7) gives c
0 T b kb ðt Þ ¼ k −t− þ T 2 p T b−Δc tan α ¼ k −t− þ T 2 p
ð8Þ
which is the back-side mesh stiffness for a general external gear pair with 2b nominal backlash and Δc center distance change. In most applications, the backlash and the change of center distance of a gear pair are much smaller than the circular pitch p. Therefore, the phase lag term b−Δ ptan αT in Eq. (8) is small, and Eq. (5) may suffice to estimate the back-side mesh stiffness of a gear pair with small backlash and center distance change. c
2.4. Back-side tooth number variation function for an anti-backlash gear pair with changing backlash
Back-side gear tooth number variation nb (t)
Drive-side gear tooth number variation n(t)
For anti-backlash gear pairs, the driving gear consists of a fixed part (in contact with the drive-side of driven gear) and a free part (in contact with the back-side of the driven gear). The drive- and back-side mesh stiffnesses are not only determined by the number of teeth in contact, but also by other design parameters of the fixed and free parts, such as their facewidths and modulus 3 2.5
-t
O
2 1.5 1
-T/2+(b- Δ c tanα)T/p 0.5
-T/2+b/p
0
1
0.
0.
4
3 2.5 2
t
O'
1.5 1 0.5 0
-T
- 3T/4
- T/2
- T/4
0
T/4T/2
3T/4
T
Time: t (s)
Fig. 9. Back-side and drive-side gear tooth number variation functions the example gear pairs with 2b nominal backlash (b satisfies bq ¼ 0:05) and Δc change in the ¼ 0:025). center distance (Δ tanα p c
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of elasticity. To exclude the impact from parameters other than the number of teeth in contact, the number of teeth in contact variation functions of the drive (n(t)) and back (nb(t)) sides are used to investigate the back-side mesh stiffness of anti-backlash gears. Because the derivation of Eqs. (5)–(8) relies only on the number of teeth in contact and the phase relations between the drive and back-sides, replacing the mesh stiffness functions k(t) and kb(t) with the number of teeth in contact variation functions n(t) and nb(t) and applying the same analytical process in the above three sub-sections yields T b−Δc tan α nb ðt Þ ¼ n −t− þ T 2 p
ð9Þ
which is the back-side tooth number variation function for an arbitrary anti-backlash gear or a general external gear pair with 2b nominal backlash and Δc center distance change. In order to verify the general back-side tooth variation function in Eq. (9), the center distance between the two gears in Fig. 4 is α ¼ 0:025 in addition to the 10% tooth thickness reduction. The tracking results for the drive-side and back-side reduced such that Δ tan p e in Fig. 9 is the point on the drive-side gear tooth number variation function at tooth number variation functions are shown in Fig. 9. O e is symmetrical to the back-side gear tooth t ¼ − T2 þ b−Δ ptan αT . The drive-side gear tooth number variation function before point O e Comparing with Fig. 7, the phase lag of the back-side tooth number variation function in Fig. 9 number variation function after point O. is reduced by 0.025T. This phase lag reduction is caused by the reduction of the center distance, and this result matches Eq. (9). c
c
3. Conclusion This study investigates the relationships between the drive- and back-side mesh stiffnesses for arbitrary gear pairs and anti-backlash gear pairs. The impact of backlash and center distance changes on the phase lag in the back-side mesh stiffness variation function is analytically determined. The resulting formulae are useful for the static and dynamic analysis of gear systems that involve back-side gear tooth contacts, including anti-backlash gears. References [1] Y. Guo, R.G. Parker, Dynamic modeling and analysis of a spur planetary gear involving tooth wedging and bearing clearance nonlinearity, Eur. J. Mech. A Solids 29 (2010) 1022–1033. [2] T. Larsen, K. Thomsen, F. Rasmussen, Dynamics of a wind turbine planetary gear stage, Technical Report risoe-i-2112 (en), Risoe National Laboratory, Denmark, 2003. [3] F. Rasmussen, K. Thomsen, T. Larsen, The gearbox problem revisited. Risoe fact sheet aed-rb-17 (en), Risoe National Laboratory, Denmark. [4] A. Hansen, F. Rasmussen, T. Larsen, Gearbox loads caused by double contact simulated with hawc2, European Wind Energy Conference and Exhibition, Poland, 2010. [5] J. Brauer, Investigation of Transmission Error, Friction, and Wear in Anti-backlash Gear Transmissions: A Finite Element Approach, Royal Institute of Technology, Sweden, 2003. (PhD thesis). [6] R.G. Munro, D. Palmer, L. Morrish, An experimental method to measure gear tooth stiffness throughout and beyond the path of contact, IMechE C 215 (7) (July. 2001) 793–803. [7] G.W. Blankenship, A. Kahraman, Steady state force response of a mechanical oscillator with combined parametric excitation and clearance type nonlinearity, J. Sound Vib. 185 (5) (May. 1995) 743–765. [8] R.G. Parker, S.M. Vijayakar, T. Imajo, Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons, J. Sound Vib. 237 (3) (Oct. 2000) 435–455. [9] J. Lin, R.G. Parker, Mesh stiffness variation instabilities in two-stage gear systems, J. Vib. Acoust. 124 (1) (Jan. 2002) 68–76. [10] G. Liu, R.G. Parker, Nonlinear dynamics of idler gearsets, Nonlinear Dyn. 53 (2008) 345–367. [11] J. Lin, R.G. Parker, Planetary gear parametric instability caused by mesh stiffness variation, J. Sound Vib. 249 (3) (Jan. 2002) 1. [12] P. Velex, L. Flamand, Dynamic response of planetary trains to mesh parametric excitations, J. Mech. Des. 118 (1) (Mar. 1996) 7–14. [13] R.G. Parker, X. Wu, Parametric instability of planetary gears with elastic continuum ring gears, ASME J. Vib. Acoust. 134 (4) (2012) 041011–041021. [14] T. Sun, H. Hu, Nonlinear dynamics of a planetary gear system with multiple clearances, Mech. Mach. Theory 38 (12) (Dec. 2003) 1371–1390. [15] C.-J. Bahk, R.G. Parker, Analytical solution for the nonlinear dynamics of planetary gears, ASME J. Comput. Nonlinear Dyn. 6 (2) (April. 2011) 021007. [16] C.-J. Bahk, R.G. Parker, Analytical investigation of tooth profile modification effects on planetary gear dynamics, Mech. Mach. Theory 70 (December 2013) 298–319. [17] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, part-ii: effect of involute contact ratio, ASME Power Transmission and Gearing Conference, San Diego, CA, 1996. [18] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, part-i: characterization of forced response, ASME Power Transmission and Gearing Conference, San Diego, CA, 1996. [19] S. Vijayakar, Calyx Programmers Manual, Advanced Numerical Solutions, Hilliard, OH, 2006. [20] R. August, Dynamics of Planetary Gear Trains, (PhD thesis) Cleveland State University, 1983. [21] R.G. Parker, V. Agashe, S.M. Vijayakar, Dynamic response of a planetary gear system using a finite element/contact mechanics mode, J. Mech. Des. 122 (3) (Sept. 2000) 304–310. [22] V.K. Ambarisha, R.G. Parker, Nonlinear dynamics of planetary gears using analytical and finite element models, J. Sound Vib. 203 (2007) 577–595.
Nomenclature T Mesh period Pitch radius of the driving gear rdr Pitch radius of the driven gear rdn Tooth number of the driving gear Zdr Tooth number of the driven gear Zdn k(t) Time-varying drive-side mesh stiffness Time-varying back-side mesh stiffness kb(t) 2b Nominal backlash along the pitch circle p Circular pitch
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Short communication
Analytical determination of back-side contact gear mesh stiffness Yichao Guo a, L. S. Randolph Professor and Head Robert G. Parker b,⁎ a b
Rochester Hills, MI 48306, United States Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, United States
a r t i c l e
i n f o
Article history: Received 10 February 2014 Received in revised form 15 March 2014 Accepted 24 March 2014 Available online 26 April 2014 Keywords: Gear mesh Back-side contact Mesh stiffness Mesh phase Scissor gear Anti-backlash gear
a b s t r a c t Back-side gear tooth contact happens when anti-backlash (or scissor) gears are used, tooth wedging or tight mesh occurs, or vibration amplitudes are high enough that teeth separate and pass the backlash zone. An accurate description of the back-side gear tooth mesh stiffness is needed to study gear mechanics in such cases. This work studies the time-varying back-side mesh stiffness and its correlation with backlash by analyzing the relationship between the drive-side and back-side mesh stiffnesses. Results of this work yield the general form of the back-side mesh stiffness in terms of the known drive-side mesh stiffness for an arbitrary gear pair. The analytical results are confirmed by simulation results from gear contact analysis software that precisely tracks drive- and back-side gear tooth contact. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Back-side contact in a gear mesh refers to contact on the surfaces of a gear that are not used to transmit power. Recent studies on gear dynamics [1–4] show that it is possible for tooth wedging (or tight mesh), that is, simultaneous drive-side and back-side contact, to happen in applications such as wind turbine gearboxes. Tooth wedging in wind turbines results from the combined effect of gravity and bearing clearance nonlinearity, and it proved a likely source of gearbox bearing failures in a particular case. For better understanding of the impact of tooth wedging on gearbox failures, it is necessary to have a model that includes accurate description of the back-side contact mesh stiffness. Besides tooth wedging, anti-backlash (or scissor) gears are another case for back-side contact to occur. To minimize the undesirable characteristics caused by backlash, anti-backlash gears eliminate the backlash by using a preloaded spring to force the fixed part of the driving gear to contact the drive-side of the driven gear teeth and, simultaneously, force the free part of the driving gear to contact the back side of the driven gear [5]. Accurate modeling of back-side contact mesh stiffness is necessary to analyze such systems. Mesh stiffness variation and its impact on gear mechanics have been extensively investigated. Mesh stiffness variation is the source of static transmission error fluctuations. Munro and his team experimentally investigated gear tooth mesh stiffness throughout and beyond the path of contact [6]. Blankenship and Kahraman experimentally and analytically studied a single degree of freedom gear pair driven by time-varying mesh stiffness variation; they showed contact loss and back-side contact that are subject to a symmetric backlash condition [7]. The same system was investigated with analytical and finite element models [8]. Lin, Liu, and Parker analyzed mesh stiffness variation instabilities in two-stage gear systems [9,10], as well as in simple planetary gear systems [11]. Their studies showed that parametric excitation from time-varying mesh stiffness causes instability and severe vibration under certain operating conditions. They applied a perturbation method to analytically determine the instability conditions. Velex and Flamand extended the ⁎ Corresponding author. E-mail address: [email protected] (R.G. Parker).
http://dx.doi.org/10.1016/j.mechmachtheory.2014.03.011 0094-114X/© 2014 Elsevier Ltd. All rights reserved.
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research scope to planetary gear trains and studied their dynamic responses with varying mesh stiffness [12]. Wu and Parker [13] extended the study on parametric instability to planetary gears with elastic continuum ring gears. Sun and Hu [14] investigated mesh stiffness parametric excitation and clearance nonlinearity for simple planetary gears. Bahk and Parker [15] derived closed-form solutions for the dynamic response of planetary gears with time-varying mesh stiffness and tooth separation nonlinearity based on a purely torsional planetary gear model. They extended this to systems with tooth profile modifications [16]. Guo and Parker [1] modeled and analyzed a simple planetary gear with time-varying mesh stiffness, tooth wedging, and bearing clearance nonlinearity. Although back-side contact is included in their model, the average value of the periodic mesh stiffness on the drive-side is used to approximate the back-side mesh stiffness, which is a simplified description of the back-side mesh stiffness. Despite the abundance of literature on mesh stiffness variation and gear dynamics, no studies have derived the back-side mesh stiffness in their analytical model. One possible reason is that the usual symmetry of the gear teeth ensures that the contact ratios, mesh periods, and average mesh stiffnesses over the mesh period are the same for drive- and back-side contact. This may lead to the mistaken conclusion that the back-side mesh stiffness is the same as the drive-side one. For example, Kahraman and Blankenship performed experiments on the nonlinear response of spur gear pairs with varying involute contact ratios [17,18]. The back-side contact is assumed to be the same as the drive-side contact in their study. In the tight mesh case shown in Fig. 1, the back-side mesh stiffness, however, is not equivalent to the drive-side one, because the back-side contact is along the back-side line of action (the dashed line in Fig. 1) and the number of gear teeth in contact along the back-side line of action is not always equal to that along the drive-side line of action (the solid line in Fig. 1). Fig. 2 illustrates one such case (the simulation results are from Calyx [19], a multi-body finite element/contact mechanics program with precise gear tooth contact capability). There are two pairs of gear teeth in contact along the back-side line of action, while only one pair of teeth is in contact along the line of action. Therefore, the back-side mesh stiffness differs from the drive-side mesh stiffness at this moment. 2. Derivation of back-side mesh stiffness The drive-side mesh stiffness refers to the stiffness of the nominally contacting teeth at a mesh in the direction of power transmission. It varies as the number of teeth in contact fluctuates with the gear rotation. The stiffness acts along the line of action. The period of its variation is known for the given rotation speed. Mesh stiffness variation functions are often approximated by Fourier series in analytical studies. They can be accurately calculated by finite element software. The drive-side mesh stiffness function is critical for analytical studies on gear dynamics [8,15,20–22]. Similar to the drive-side mesh stiffness, the back-side mesh stiffness is the stiffness of the changing number of contacting teeth along the back-side line of action. In order to have simultaneous drive-side and back-side contacts at all times, we first investigate an ideal gear pair (one that operates at the nominal center distance and has zero backlash tooth thickness). The results of the ideal gear pair are then extended to arbitrary gear pairs with backlash and to anti-backlash gears. 2.1. Back-side mesh stiffness for an ideal gear pair Fig. 3 illustrates the drive-side and back-side contacts for an ideal gear pair. The gear details are not needed for what follows. The tooth numbers of the driving and driven gears are Zdr and Zdn, respectively. T is the mesh period. At t = 0, the pitch point at the drive-side of the driving gear is in mesh (Fig. 3a). The dashed line in the middle of each sub-figure is the center line between the two gears. After one mesh period T, the driving gear tooth moves one driving gear circular pitch pdr ¼ 2πr Z , and the driven gear (rdr and rdn are the pitch radii). After one fourth of the mesh period (t ¼ T4), the driving gear tooth moves one tooth moves pdn ¼ 2πr Z fourth of its circular pitch. Because there is no backlash along the pitch circle, the circular tooth thickness of the driving gear qdr is dr
dr
dn
dn
qdr ¼
1 p : 2 dr
ð1Þ
Back_side line of action
Driving direction
Driving Gear
Driven Gear
Line of action
Fig. 1. Drive-side gear contact (solid line) and back-side gear contact (dashed line) in a tight mesh case (both drive and back-sides are in contact).
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Driving Driven Gear Gear
Fig. 2. Numerical simulation of Calyx on an ideal gear pair with tight mesh. One pair of teeth (marked by a circle) is in contact along the drive-side line of action, and two pairs of teeth (marked by two rectangles) are in contact along the back-side line of action.
Because the driving gear rotates 2qdr in one mesh period T, the movement of the driving gear in T4 is half of its circular tooth thickness, or 12qdr . Thus, at t ¼ T4 the center line passes through the middle of the tooth tip of the driving gear (Fig. 3b). At this moment both gears are symmetric about the center line, and the number of gear teeth in contact along the drive-side line of action equals that along the back-side line of action. The back-side and drive-side mesh stiffnesses are equal at this moment. Furthermore, at t ¼ T2, the driving gear tooth moves qdr relative to its position at t = 0, and the center line passes directly through the pitch point at the back-side of the driving gear tooth (Fig. 3c). At t ¼ 3T4 , the driving gear moves 32qdr relative to its position at t = 0, and the driven gear rotates 32qdn relative to its initial position at t = 0 (qdn is the circular tooth thickness of the driven gear). At this moment, the center line passes through the middle point of the tooth tip of the driven gear, and both gears are symmetric about the center line. Therefore, the back-side and drive-side mesh stiffnesses are equal at t ¼ 3T4 . To clearly illustrate the drive-side and back-side mesh stiffness relationship, the finite element/contact mechanics software Calyx [19] is used in this investigation to precisely track the tooth contact for precise tooth surface geometry as the gears rotate. Its tooth surface geometry representation is effectively error-free; it does not rely on the node points to define the surface. Calyx's ability to accurately monitor the contact conditions makes it a reliable benchmark. In the simulations that follow Calyx tracks the contact for specified gear kinematics under unloaded conditions. The parameters of the example gear pair are listed in Table 1 and the Calyx gear model is shown in Fig. 4. Fig. 5 shows the variations of the number of teeth in contact for the drive and back sides by tracking the numbers of gear teeth in contact along the drive- and back-side lines of action using the Calyx model. All the simulation results in this study use sufficient resolution of the time axis to capture changes with needed accuracy. At t = 0 (as shown in Fig. 3a), the pitch point at the drive-side of the driving gear is in contact. O and O′ are the points on the drive- and back-side tooth number variation functions at t = 0. Due to the symmetry of both gears about the center line at t ¼ T4 , the drive-side tooth number variation function from this moment on by rotating the gears in the nominal driving direction is identical to the back-side tooth number variation function from this moment on by rotating the gears in the opposite direction. This is verified by the simulation results in Fig. 5. Let A and A′ be the points on the drive- and back-side tooth number variation functions at t ¼ T4 . The drive-side tooth number variation function after point A is identical to the back-side gear tooth number variation function before point A′. To mathematically describe the above relations, the varying drive-side mesh stiffness is kI(t), and the matching back-side mesh stiffness is kIb(t). Rotating the gears in the nominal driving direction until the moment of t ¼ T4 is identical to shifting the I phase of kIb(t) by T4. This generates a phase-shifted function k ðt−T4Þ that is the drive-side gear tooth number variation function in I Fig. 5 with the moment A being its origin of time axis. Replacing t−T4 with τ in k ðt−T4Þ yields kI(τ) whose origin is at τ = 0 or t ¼ T4 when the time axis is t. Noting that the back-side mesh stiffness function that matches with kI(τ) is kIb(τ) and applying the symmetry relation that the drive-side tooth number variation function after τ = 0 is equal to the back-side gear tooth number variation function before τ = 0, the mesh stiffnesses have the relation of I
I
kb ðτ Þ ¼ k ð−τ Þ:
ð2Þ
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a) t =0, T
c) t =T/2
b) t =T/4
d) t =3T/4
Fig. 3. The gear mesh contacts for an arbitrary external gear pair with no backlash. The dashed line in the middle of each sub-figure is the center line. The driving gear is at the right hand side of each subplot and the driving direction is counter-clockwise.
The insertion of τ ¼ t−T4 into Eq. (2) yields T I I kb ðτ Þ ¼ k −t þ : 4
ð3Þ
I
Because τ ¼ t−T4; kb ðτÞ, is a phase-shifted function of kIb(t), and they have the relation T I I kb ðτ Þ ¼ kb t− 4
ð4Þ
insertion of Eq. (4) into Eq. (3) yields T T I I kb t− ¼ k −t þ 4 4 T T I I I ¼ k −t− ⇒kb ðt Þ ¼ k −t þ 2 2
ð5Þ
Eq. (5) reveals that kIb(t) is uniquely determined once the drive-side mesh stiffness function kI(t) is known. According to Eq. (5), the number of gear teeth in contact at the back-side at t = 0 equals the number of gear teeth in contact at the drive-side at t ¼ T2. This is confirmed by the simulation results in Fig. 3. There are two pairs of teeth in contact at the back-side in Fig. 3a (t = 0), Table 1 Gear parameters for the example system shown in Fig. 4.
Number of Teeth Diametrical pitch Pressure angle (deg) Outer diameter (m) Root diameter (m) Mesh period (sec)
Driving gear
Driven gear
41 10.34 25 0.105 0.090 0.293
32 10.34 25 0.085 0.073 0.293
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Fig. 4. Calyx finite element model of the example ideal gear pair.
and there are exactly two pairs of teeth in contact at the drive-side in Fig. 3c (t ¼ T2). In addition, Fig. 3 provides further numerical validation of Eq. (5) that the drive-side tooth number variation function before point O″ is symmetrical to the back-sidegear tooth number variation function after point O′ where O″ in Fig. 5 is the point on the drive-side tooth number variation function at t ¼ T2. 2.2. Back-side mesh stiffness for a gear pair with nominal backlash
Back-side gear tooth number variation nb (t)
Drive-side gear tooth number variation n(t)
In applications, gears always include backlash to allow lubrication, manufacturing errors, deflection under load, and thermal expansion. It is typically created by slightly increasing the center distance of the gear pair or reducing the circular tooth thickness.
3 2.5
-t
*
2 1.5
O''
*
*
1
O
0.5 0
3 2.5 2
O'
t
1.5 1 0.5 0 -T
- 3T/4
- T/2
- T/4
0
T/4
T/2
3T/4
T
Time
Fig. 5. The drive- and back-side gear tooth number variation functions for the example ideal gear pair in Fig. 4. ‘x’ indicates the time that the pitch point of the drive-side of the driving gear is in contact, ‘A’ indicates the time that the middle point of the drive gear tooth tip is aligned with the center line, ‘+’ indicates the time that the pitch point of a driven gear tooth is in contact, and ‘*’ indicates the time that the middle point of a driven gear tooth tip is aligned with the center line.
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Radial direction Matching ideal gear pair Tooth of the driven gear Back-side line of action
Line of action 2b
Pitch circle
Base circle
Tooth of the driving gear
Fig. 6. The drive-side and back-side gear mesh contacts for a gear pair with 2b backlash and its matching ideal gear pair.
Back-side gear tooth number variation nb (t)
Drive-side gear tooth number variation n(t)
Fig. 6 illustrates the backlash for an external gear pair. The nominal backlash for the gear pair is 2b, the circular pitch is p, and the mesh period is T. The case that the center distance of the gear pair remains unchanged and there is no relative radial motion between the gears is investigated first. The impacts of changing the center distance and relative radial motion are studied subsequently. Once a gear pair is installed with its nominal center distance, the backlash remains at its nominal value 2b if there is no relative radial motion between the gears. The thick dashed line in Fig. 6 represents the case of back-side contact. If the varying drive-side mesh stiffness is k(t), Eq. (5) does not give the desired back-side mesh stiffness variation function. Instead, Eq. (5) gives kIb(t), the
3 2.5
-t
O
2 1.5
-T/2+bT/p 1 0.5 0
3 2.5 2
O'
t
1.5 1 0.5 0
-T
- 3T/4
- T/2
- T/4
0
T/4
T/2
3T/4
T
4
Time: t (s) Fig. 7. Back-side and drive-side gear tooth number variation functions the example gear pairs with zero center distance change and 2b nominal backlash (b satisfies bq ¼ 0:05).
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back-side mesh stiffness variation function for the matching ideal gear pair (the thin dashed lines in Fig. 6) that has the same center distance and pitch circle as the original gear pair in Fig. 6 but different tooth thicknesses. To ensure zero backlash, the tooth thickness of the matching ideal gear pair equals the thickness of the original gear pair increased by b. The slight increase in tooth thickness of the matching ideal gear pair only causes the associated mesh stiffness variation function to differ from the original gear pair by a phase shift. Thus, the drive-side mesh stiffness variation function of the matching ideal gear pair equals that of the original gear pair, because the gear contacts of the matching ideal gear pair at the drive-side always coincide with the original gear pair. The phase shift between the matching ideal gear pair and the original gear pair at the back-side is derived as follows. If the matching ideal gear pair moves b along the pitch circle in the reverse driving direction in Fig. 6, the back-side contact of the ideal gear mesh in the thin dashed line will coincide with the actual back-side contact in the thick dashed line. The phase lag between the back-side mesh stiffness functions of the matching ideal gear pair and the original gear pair for the above process is bpT. In other words, the actual back-side mesh stiffness variation function kb(t) of a gear pair is the back-side mesh stiffness variation function for the matching ideal gear kIb(t) with pbT phase lag, that is, b I kb ðt Þ ¼ kb t− T : p I
ð6Þ
I
The insertion of kb ðt Þ ¼ k ð−t−T2Þ (from Eq. (5)) into Eq. (6) and application of the condition that the drive-side mesh stiffness function of the matching ideal gear pair equals that of the original gear pair (Fig. 6) yield T b kb ðt Þ ¼ k −t− þ T 2 p
ð7Þ
where k(t) is the drive-side mesh stiffness variation of the original gear pair. To verify the back-side mesh stiffness variation function for the gear pairs with 2b nominal backlash in Eq. (7), the tooth thickness of the example gear pair in Fig. 4 is reduced by 10% such that bp ¼ 0:05 . The tracking results of the drive-side and back-side tooth number variations from Calyx are shown in Fig. 7. O in Fig. 7 is the point on the drive-side gear tooth number variation function at t ¼ −T2 þ pbT . The drive-side gear tooth number variation function before point O is symmetrical to the back-side gear tooth number variation function after point O′. Thus, the results in Fig. 7 agree with Eq. (7). 2.3. Back-side mesh stiffness for an arbitrary gear pair with changing backlash In real applications, the actual operating center distance of a gear pair can differ from the nominal one due to manufacturing errors of the axes, deflections under load, and misalignments. In addition, vibration of the gears in the relative radial direction changes the instantaneous center distance [1].
a)
Radial direction Tooth of the driven gear
Δ
Line of action
Tooth wedging happens when
2Δ tanα=2 b Back-side line of action
α 2Δ sinα
Pitch circle
Tooth of the driving gear after Δ change of center distance Base cicle Tooth of the driving gear
b)
Back-side line of action
Line of action Δα
α
Δ sinα
c) Pitch circle
2Δ tanα
α
Back-side line of action
2Δ sinα
Fig. 8. Back-side tooth contact for the case of tooth wedging when 2Δctanα = 2b. b is half of the backlash along the pitch circle, Δc is the change of the central distance, and α is the pressure angle.
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Fig. 8a shows that a change of center distance Δc (positive Δc indicates a reduction of center distance) reduces the backlash by 2Δc tan α, where α is the pressure angle. Fig. 8b and c explains this relationship geometrically. When the driving gear moves Δc toward the driven gear (i.e., a Δc change in center distance), the gaps between the two gears along the drive- and back-side lines of action are simultaneously reduced by Δc sin α (Fig. 8b). Because the driving gear is pressed against the driven gear along the line of action and does not move toward the driven gear along the line of action, the total gap reduction along the back-side line of action becomes 2Δc sin α. The gap change along the pitch circle is approximated by the associated cord length shown in Fig. 8c. By applying trigonometric relations to the triangle shown in Fig. 8c, the length of the chord that approximates the gap change along the pitch circle is 2Δcossinα α ¼ 2Δc tan α. When Δc ¼ tanb α such that 2Δc tan α = 2b (2b is the nominal backlash), the back-side of the driving gear (the dashed line in Fig. 8a) is in contact with the back-side of the driven gear and tooth wedging (tight mesh) occurs. When Δc b tanb α, the actual backlash of the gear pair 2b′ is 2(b − Δc tan α). This case is equivalent to the case in Fig. 6 with the exception that the backlash is 2b′ instead of the nominal value 2b. Replacing b with b′ in Eq. (7) gives c
0 T b kb ðt Þ ¼ k −t− þ T 2 p T b−Δc tan α ¼ k −t− þ T 2 p
ð8Þ
which is the back-side mesh stiffness for a general external gear pair with 2b nominal backlash and Δc center distance change. In most applications, the backlash and the change of center distance of a gear pair are much smaller than the circular pitch p. Therefore, the phase lag term b−Δ ptan αT in Eq. (8) is small, and Eq. (5) may suffice to estimate the back-side mesh stiffness of a gear pair with small backlash and center distance change. c
2.4. Back-side tooth number variation function for an anti-backlash gear pair with changing backlash
Back-side gear tooth number variation nb (t)
Drive-side gear tooth number variation n(t)
For anti-backlash gear pairs, the driving gear consists of a fixed part (in contact with the drive-side of driven gear) and a free part (in contact with the back-side of the driven gear). The drive- and back-side mesh stiffnesses are not only determined by the number of teeth in contact, but also by other design parameters of the fixed and free parts, such as their facewidths and modulus 3 2.5
-t
O
2 1.5 1
-T/2+(b- Δ c tanα)T/p 0.5
-T/2+b/p
0
1
0.
0.
4
3 2.5 2
t
O'
1.5 1 0.5 0
-T
- 3T/4
- T/2
- T/4
0
T/4T/2
3T/4
T
Time: t (s)
Fig. 9. Back-side and drive-side gear tooth number variation functions the example gear pairs with 2b nominal backlash (b satisfies bq ¼ 0:05) and Δc change in the ¼ 0:025). center distance (Δ tanα p c
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of elasticity. To exclude the impact from parameters other than the number of teeth in contact, the number of teeth in contact variation functions of the drive (n(t)) and back (nb(t)) sides are used to investigate the back-side mesh stiffness of anti-backlash gears. Because the derivation of Eqs. (5)–(8) relies only on the number of teeth in contact and the phase relations between the drive and back-sides, replacing the mesh stiffness functions k(t) and kb(t) with the number of teeth in contact variation functions n(t) and nb(t) and applying the same analytical process in the above three sub-sections yields T b−Δc tan α nb ðt Þ ¼ n −t− þ T 2 p
ð9Þ
which is the back-side tooth number variation function for an arbitrary anti-backlash gear or a general external gear pair with 2b nominal backlash and Δc center distance change. In order to verify the general back-side tooth variation function in Eq. (9), the center distance between the two gears in Fig. 4 is α ¼ 0:025 in addition to the 10% tooth thickness reduction. The tracking results for the drive-side and back-side reduced such that Δ tan p e in Fig. 9 is the point on the drive-side gear tooth number variation function at tooth number variation functions are shown in Fig. 9. O e is symmetrical to the back-side gear tooth t ¼ − T2 þ b−Δ ptan αT . The drive-side gear tooth number variation function before point O e Comparing with Fig. 7, the phase lag of the back-side tooth number variation function in Fig. 9 number variation function after point O. is reduced by 0.025T. This phase lag reduction is caused by the reduction of the center distance, and this result matches Eq. (9). c
c
3. Conclusion This study investigates the relationships between the drive- and back-side mesh stiffnesses for arbitrary gear pairs and anti-backlash gear pairs. The impact of backlash and center distance changes on the phase lag in the back-side mesh stiffness variation function is analytically determined. The resulting formulae are useful for the static and dynamic analysis of gear systems that involve back-side gear tooth contacts, including anti-backlash gears. References [1] Y. Guo, R.G. Parker, Dynamic modeling and analysis of a spur planetary gear involving tooth wedging and bearing clearance nonlinearity, Eur. J. Mech. A Solids 29 (2010) 1022–1033. [2] T. Larsen, K. Thomsen, F. Rasmussen, Dynamics of a wind turbine planetary gear stage, Technical Report risoe-i-2112 (en), Risoe National Laboratory, Denmark, 2003. [3] F. Rasmussen, K. Thomsen, T. Larsen, The gearbox problem revisited. Risoe fact sheet aed-rb-17 (en), Risoe National Laboratory, Denmark. [4] A. Hansen, F. Rasmussen, T. Larsen, Gearbox loads caused by double contact simulated with hawc2, European Wind Energy Conference and Exhibition, Poland, 2010. [5] J. Brauer, Investigation of Transmission Error, Friction, and Wear in Anti-backlash Gear Transmissions: A Finite Element Approach, Royal Institute of Technology, Sweden, 2003. (PhD thesis). [6] R.G. Munro, D. Palmer, L. Morrish, An experimental method to measure gear tooth stiffness throughout and beyond the path of contact, IMechE C 215 (7) (July. 2001) 793–803. [7] G.W. Blankenship, A. Kahraman, Steady state force response of a mechanical oscillator with combined parametric excitation and clearance type nonlinearity, J. Sound Vib. 185 (5) (May. 1995) 743–765. [8] R.G. Parker, S.M. Vijayakar, T. Imajo, Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons, J. Sound Vib. 237 (3) (Oct. 2000) 435–455. [9] J. Lin, R.G. Parker, Mesh stiffness variation instabilities in two-stage gear systems, J. Vib. Acoust. 124 (1) (Jan. 2002) 68–76. [10] G. Liu, R.G. Parker, Nonlinear dynamics of idler gearsets, Nonlinear Dyn. 53 (2008) 345–367. [11] J. Lin, R.G. Parker, Planetary gear parametric instability caused by mesh stiffness variation, J. Sound Vib. 249 (3) (Jan. 2002) 1. [12] P. Velex, L. Flamand, Dynamic response of planetary trains to mesh parametric excitations, J. Mech. Des. 118 (1) (Mar. 1996) 7–14. [13] R.G. Parker, X. Wu, Parametric instability of planetary gears with elastic continuum ring gears, ASME J. Vib. Acoust. 134 (4) (2012) 041011–041021. [14] T. Sun, H. Hu, Nonlinear dynamics of a planetary gear system with multiple clearances, Mech. Mach. Theory 38 (12) (Dec. 2003) 1371–1390. [15] C.-J. Bahk, R.G. Parker, Analytical solution for the nonlinear dynamics of planetary gears, ASME J. Comput. Nonlinear Dyn. 6 (2) (April. 2011) 021007. [16] C.-J. Bahk, R.G. Parker, Analytical investigation of tooth profile modification effects on planetary gear dynamics, Mech. Mach. Theory 70 (December 2013) 298–319. [17] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, part-ii: effect of involute contact ratio, ASME Power Transmission and Gearing Conference, San Diego, CA, 1996. [18] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, part-i: characterization of forced response, ASME Power Transmission and Gearing Conference, San Diego, CA, 1996. [19] S. Vijayakar, Calyx Programmers Manual, Advanced Numerical Solutions, Hilliard, OH, 2006. [20] R. August, Dynamics of Planetary Gear Trains, (PhD thesis) Cleveland State University, 1983. [21] R.G. Parker, V. Agashe, S.M. Vijayakar, Dynamic response of a planetary gear system using a finite element/contact mechanics mode, J. Mech. Des. 122 (3) (Sept. 2000) 304–310. [22] V.K. Ambarisha, R.G. Parker, Nonlinear dynamics of planetary gears using analytical and finite element models, J. Sound Vib. 203 (2007) 577–595.
Nomenclature T Mesh period Pitch radius of the driving gear rdr Pitch radius of the driven gear rdn Tooth number of the driving gear Zdr Tooth number of the driven gear Zdn k(t) Time-varying drive-side mesh stiffness Time-varying back-side mesh stiffness kb(t) 2b Nominal backlash along the pitch circle p Circular pitch