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Synthesis of the steepest rotation pitch curve design for noncircular gear
Mechanism and Machine Theory 102 (2016) 16–35
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Synthesis of the steepest rotation pitch curve design for noncircular gear Xin Zhang ⁎, Shouwen Fan ⁎ School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, People's Republic of China
a r t i c l e
i n f o
Article history: Received 21 August 2015 Received in revised form 31 January 2016 Accepted 25 March 2016 Available online 13 April 2016 Keywords: Noncircular gear Steepest rotation method Pitch curve Calculus of variations
a b s t r a c t A steepest rotation method for designing pitch curve with fixed boundaries of noncircular gear is proposed by resorting to calculus of variations. A general mathematical model is established to design the steepest rotation pitch curves with integral constraint for noncircular gear pair, including external and internal meshing gear pair. In particular, to achieve the design of pitch curve with epicyclic constraint for driven noncircular gear, the epicyclic constraint conditions are also established. In addition, the unified design algorithm of pitch curves of noncircular gear pair with this steepest rotation characteristic is given in this paper, the pitch curves of desired geometrical and transmission properties can be solved easily by using the proposed algorithm. Examples presented in this paper are implemented in MATLAB, and feasibility and validity of above algorithm are verified. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Since noncircular gears are more compact and better balanced than linkages and cams, they have been already applied to mechanical equipments: Maltese mechanism, packaging machinery, printing machinery, textile machinery, gear shaper and flow meters [1–6]. Dooner D. B. and Seireg A. A. [7] proposed an interactive design approach of noncircular gears. Dooner D. B [8] used noncircular gears to eliminate unnecessary torque and speed fluctuations. Wu L. I. and Chang S. L. [9] obtained the tooth profiles of elliptical gears by combining the involutes generated from base curves with appropriate addendum and dedendum curves. Tong S. H. and Yang D. C. H. [10] obtained identical noncircular conjugate pitch curves with any number of lobes. Wei H. [11] developed an elliptic interpolation algorithm for elliptical gears on the basis of the principle of central angle division of an arc. Hector F. Q. R., Salvador C. F. and et al. [12] designed an approach for generating pitch curves of N-lobed noncircular gears on the basis of Bézier and B-spline nonparametric curves. Mundo D. [13] obtained the planetary gear train with specified variable gear ratio law by combing three noncircular gears. A general generation method of N-lobed elliptical gears from a basic ellipse was proposed in Ref. [14], and the synthesis of N-lobed or high-order elliptical gears and their rack by means of a conjugate shaper cutter with an involute tooth profile was obtained in Refs. [15–22]. Most of pitch curves for these noncircular gears in Refs. [1–22] were constructed by existing smoothed curves as pitch curves of noncircular gears. However, with the rapid development of numerical control machine tool and computer technology, some novel design methods of pitch curves for N-lobed noncircular gears have been proposed in recent studies. Yao W. X. [23] presented a pitch curve design method, which using plane regular N-curved polygon and spiral of Archimedes as the pitch curve for N-lobed noncircular gear. Another new design method of pitch curves with concave for noncircular bevel gears was proposed by Kan S. and Xia J. Q. [24].
⁎ Corresponding authors. E-mail addresses: [email protected] (X. Zhang), [email protected] (S. Fan).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.03.020 0094-114X/© 2016 Elsevier Ltd. All rights reserved.
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
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Although some novel pitch curves are put forward in succession, the high efficiency requirement of noncircular gear transmission still cannot be realized. In view of this, a kind of design method of steepest rotation pitch curves for noncircular gears is presented in this paper, the noncircular gears with steepest rotation pitch curves possess shortest transmission time at a given angular velocity. In addition, the pitch curves with this characteristic are more attractive than previous designs under the high efficient transmission requirement. Several typical design methods of steepest rotation pitch curves for noncircular gears based on calculus of variations are presented. 2. Design of steepest rotation pitch curve with fixed boundaries for noncircular gear As depicted in Fig. 1, points A, B and O are two fixed endpoints and rotation center of plane curve C, respectively. Fixed coordinate system O-xy is attached to rotation center O, θa and θb are polar angles of points A and B, respectively. Assume that the polar equation of plane curve C is rfb(θ), according to the principle of kinematics and differential, the differential dt(θ) of time function t(θ) that noncircular gear through plane curve rfb(θ) can be expressed as:
dsðθÞ ¼ dt ðθÞ ¼ ωr fb
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r fb 2 þ r 0fb 2 dθ ωr fb
ð1Þ
where ds(θ) and rfb′(θ) are small section arc length and first-order derivative for plane curve rfb(θ), respectively. Parameter ω is a given rotation angular velocity. Polar angle θ is measured counterclockwise from the positive direction x-axis. Integrating both sides of Eq. (1) from θa to θb, we obtain: Z T¼
θb θa
Z dtðθÞ ¼
θb θa
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r fb 2 þ r 0fb 2 ωrfb
dθ
ð2Þ
where T is rotation time that noncircular gear through plane curve rfb(θ) at the given angular velocity ω from θ = θa to θ = θb, and 0 ≤ θa b θb ≤ 2π. From the Eq. (2), we know that the value of time T may be different for different plane curve rfb(θ), so the minimum Tmin of rotation time can be expressed as: Z 8 > < T min ¼ minT ¼ min
0 F θ; r fb ; r fb dθ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 0 −1 r fb 2 þ r fb 02 F θ; rfb ; r fb ¼ ðωr fb Þ θb θa
Fig. 1. Pitch curve of noncircular gear with fixed boundaries.
ð3Þ
18
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
According to the calculus of variations [25–26], the solution rfb(θ) of Eq. (3) must satisfy the following condition: ∂F d ∂F − ∂r fb dθ ∂r0fb
! ¼ 0; θ ∈½θa ; θb
ð4Þ
Rearranging Eq. (4) to: d dθ
0
F−rfb
∂F ∂r 0fb
! ¼ 0; θ ∈½θa ; θb
ð5Þ
Substituting Eq. (3) into Eq. (5) and rearranging Eq. (5) to: 0
1 2
d B r fb C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiA ¼ 0; θ∈½θa ; θb @ dθ ωr fb r fb 2 þ r fb 02
ð6Þ
Integrating both sides of Eq. (6), we obtain: r fb ðθÞ ¼ e
H1 ðθ−H 2 Þ
; θ∈½θa ; θb
ð7Þ
where H1 and H2 are integral constants. Referring to Eq. (7), we know that the first-order derivative r′fb(θ) = H1eH1(θ−H2) of plane curve rfb(θ) is a monotonic function about parameter θ. Therefore, in order to satisfy non-circularity of pitch curve, polar radiuses rfb(θa) and rfb(θb) are absolutely unequal when use the plane curve rfb(θ) as pitch curve of noncircular gear, i.e. rfb(θa) ≠ rfb(θb). Substituting the fixed boundaries (θa, rfb(θa)) and (θb, rfb(θb)) of plane curve rfb(θ) into Eq. (7), we obtain: 8 ln ðrfb ðθb ÞÞ− ln ðr fb ðθa ÞÞ > > < H1 ¼ θb −θa ; r fb ðθa Þ≠r fb ðθb Þ > > H ¼ θa ln ðr fb ðθb ÞÞ−θb ln ðr fb ðθa ÞÞ : 2 ln ðr fb ðθb ÞÞ− ln ðr fb ðθa ÞÞ
ð8Þ
Along with Eq. (7), the steepest rotation pitch curve of noncircular gear with fixed boundaries can be expressed as: 8 H ðθ−H2 Þ r fb ð8 θÞ ¼ e 1 ; > > > > θ∈ θ ½ > > a ; θb > > > > r ðθ Þ≠r ðθ Þ > < > > fb a fb b > < ln ðr fb ðθb ÞÞ− ln ðr fb ðθa ÞÞ H ¼ > s:t: > > 1 > θb −θa > > > > > > > > > H ¼ θa ln ðr fb ðθb ÞÞ−θb ln ðr fb ðθa ÞÞ : > : 2 ln ðrfb ðθb ÞÞ− ln ðr fb ðθa ÞÞ
ð9Þ
where s. t. is an abbreviation of “subject to”, it means that contained constraint conditions must be satisfied by the equation. According to Eq. (9), three design examples of this steepest rotation pitch curve with fixed boundaries listed in Table 1 are depicted in Fig. 2(a)–(c), respectively. The polar equations of pitch curves with fixed boundaries and related integral constants are listed in Table 2. The units of parameters listed in Table 2 are adopted by standard international unit (SIU). Without additional explanation, the following discussion relating to parameter in this paper whose unit refers to SIU. 3. Design of steepest rotation pitch curves with integral constraint for noncircular gear pair As depicted in Fig. 3, points O1 and O2 are rotation centers of pitch curves R1 of driving noncircular gear and pitch curve R2 of driven noncircular gear, respectively. Point p is the instantaneous meshing point of pitch curves R1 and R2, its corresponding polar angles are θ1 and θ2, respectively. Parameter ω is a given angular velocity of driving noncircular gear R1, A0 (A0 N 0) is center distance between pitch curves R1 and R2. Polar angles θ1 and θ2 are equal to θ1a and θ2a at the starting position, respectively. Table 1 Design parameters of pitch curves with fixed boundaries. Case number
Left boundary condition (θa, rfb(θa))
Right boundary condition (θb, rfb(θb))
a b c
(0, 2) (π/6, 3) (π/4, 5)
(3π/2, 5) (11π/6, 6) (7π/4, 8)
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 2. Three design examples of steepest rotation pitch curves with fixed boundaries for noncircular gear.
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Table 2 Polar equations of pitch curves with fixed boundaries and related integral constants. Figure number
Integration constant
Fig. 2(a)
H1 H1 H2 H1 H2
Fig. 2(b) Fig. 2(c)
= = = = =
0.1944; H2 = −3.5648 0.1324; -7.7752 0.0997; -15.3513
Pitch curve equation rfb(θ) e0.1944(θ +3.5648) , θ ∈ [0, 3π/2] e0.1324(θ +7.7752) , θ ∈ [π/6, 11π/6] e0.0997(θ +15.3513) , θ ∈ [π/4, 7π/4]
Assuming that O2p = R2 and O1p = R1 = R1(θ1), then the pitch curve equation of driven noncircular gear and closed-law between pitch curves R1 and R2 are listed in Table 3 according to the meshing principle of noncircular gear pair [23]. Polar angle θ2b is the end rotation angle of pitch curve R2 when the polar angle θ1 of pitch curve R1 from θ1a to θ1b. 3.1. Steepest rotation design of pitch curves with integral constraint for external meshing noncircular gear pair According to variational problems with integral constraint [26] and Eq. (3), the minimum Tmin of rotation time for pitch curve R1(θ1) of driving noncircular gear for external meshing noncircular gear pair (EMNGP) can be established as: 8 Z θ 1b > > > T ¼ minT ¼ min Hdθ1 > min < θ1a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R1 2 þ R1 02 > λR1 > > þ :H ¼ ωR1 A0 −R1
ð10Þ
where λ is called an undetermined Lagrange multiplier. Eq. (10) is called an auxiliary function of steepest rotation pitch curve with integral constraint for EMNGP. Because H of Eq. (10) doesn't contain parameter θ1, the first integral of Euler-Lagrange Equation [25–26] can be expressed as: 0
H−R1 H R1 0 ¼ c1
ð11Þ
Substituting H of Eq. (10) into Eq. (11), we obtain: R1 λR1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ c1 A0 −R1 2 02 ω R1 þ R1
ð12Þ
where c1 is an integral constant. Assuming R1′ = R1tanμ and substituting R1′ into Eq. (12), we obtain: 8 A ðωc1 − cosμ Þ > > ; R ðμ Þ ¼ 0 > > ωðλ þ c1 Þ− 8 cosμ < 18 < ωðλ þ c1 ÞN1 < ωðλ þ c1 Þb−1 > > > or ωc1 b−1 s:t: ωc N1 > : : 1 : ωλN0 ωλb0
ð13Þ
Taking the derivative on both sides of Eq. (13), we have: dR1 A0 ωλ sinμ ¼ dμ ½ωðλ þ c1 Þ− cosμ 2
ð14Þ
Along with R1′ = R1tanμ and Eq. (13), we get the differential function θ1′(μ): dθ1 ωλ cosμ ¼ dμ ðωc1 − cosμ Þ½ωðλ þ c1 Þ− cosμ
ð15Þ
Integrating both sides of Eq. (15), we obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2ωc1 ωc1 −1 ωc1 þ 1 μ θ1 ðμ Þ ¼ − arctan tan ωc1 −1 2 ωc1 −1 ωc1 þ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2ωðλ þ c1 Þ ωðλ þ c1 Þ−1 ω ð λ þ c1 Þ þ 1 μ arctan tan þ c2 ωðλ þ c1 Þ−1 ωðλ þ c1 Þ þ 1 ωðλ þ c1 Þ−1 2
ð16Þ
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 3. Geometry relationship of conjugated pitch curves of noncircular gear pair.
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Table 3 Pitch curve equation of driven noncircular gear and closed-law between pitch curves R1 and R2. R2 ¼ A0 −R1 ðθ1 Þ R1 ðθ1 Þ ; θ dθ1 A0 −R1 ðθ1 Þ 0bR1 ðθ1 ÞbA0 R1 ðθ1 Þ s:t:f θ dθ1 θ2b −θ2a ¼ ∫ θ1b 1a A −R ðθ Þ 0 1 1
External meshing (referring to Fig.3(a))
f
Internal meshing
R1 is outer gear pitch curve, R2 is inner gear pitch curve (referring to Fig.3(b))
R1 is inner gear pitch curve, R2 is outer gear pitch curve (referring to Fig.3(c))
θ2 −θ2a ¼ ∫ θ11a
R2 ¼ R1 ðθ1 Þ−A0 R1 ðθ1 Þ ; θ dθ1 R1 ðθ1 Þ−A0 R1 ðθ1 Þ NA0 R1 ðθ1 Þ s:t:f θ θ2b −θ2a ¼ ∫ θ1b dθ1 1a R ðθ Þ−A 1 1 0 R2 ¼ A0 þ R1 ðθ1 Þ R1 ðθ1 Þ f ; θ dθ1 θ2 −θ2a ¼ ∫ θ11a A0 þ R1 ðθ1 Þ R1 ðθ1 Þ N0 R1 ðθ1 Þ s:t:f θ dθ1 θ2b −θ2a ¼ ∫ θ1b 1a A þ R ðθ Þ 0 1 1 f
θ2 −θ2a ¼ ∫ θ11a
where c2 is also an integration constant. Because θ1(μ) must have one-to-one corresponding relationship and monotone increasing property, the range of parameter μ can be expressed as: 8 ½−π; −π=2 > > > > < or μ∈ ½−π=2; π=2 > > > or > : ½π=2; π
ð17Þ
Therefore, the steepest rotation parameter equation of driving noncircular gear pitch curve with integral constraint for EMNGP can be expressed as: 88 A0 ðωc1 − cosμ Þ > > > > < R1 ðμ Þ ¼ ωðλ þ c Þ− cosμ > > 1 ; > > μ μ >> > þ c2 < : θ1 ðμ Þ ¼ B1 arctan B2 tan −B3 arctan B4 tan 2 8 8 2 ωðλ þ c1 Þb−1 ωðλ þ c1 ÞN1 > > > > > > < < > > ωc1 N1 ωc1 b−1 > > or > s:t:> ωλN0 > > ωλb0 > > : > : : μ∈½−π=2; π=2 μ∈½−π; −π=2 or μ∈½π=2; π
ð18Þ
where parameters B1, B2, B3 and B4 are given by: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ωc1 ωc1 −1 > > > ¼ B 1 > > ωc ωc −1 þ1 > 1 > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > > > ωc þ 1 > 1 > > < B2 ¼ ωc −1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ωðλ þ c1 Þ ωðλ þ c1 Þ−1 > > > > B3 ¼ ωðλ þ c Þ−1 ωðλ þ c Þ þ 1 > 1 1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωðλ þ c1 Þ þ 1 > > : B4 ¼ ωðλ þ c Þ−1
ð19Þ
1
Undetermined parameters c1, c2, λ can be calculated by fixed boundaries and integral constraint: Z 8 > > < θ2b −θ2a ¼
μb
cosμ dμ ωðλ þ c1 Þ− cosμ > R ðμ Þ ¼ R1a ; R1 ðμ b Þ ¼ R1b > : 1 a θ1 ðμ a Þ ¼ θ1a ; θ1 ðμ b Þ ¼ θ1b μa
ð20Þ
where μa and μb are undetermined values of parameter μ corresponding to polar angles θ1a and θ1b of two endpoints for pitch curve R1(θ1).
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 4. Three design examples of the steepest rotation pitch curves with integral constraint for EMNGP.
23
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X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Moreover, the pitch curve parameter equation of driven noncircular gear conjugated with driving noncircular gear pitch curve of Eq. (18) for EMNGP can be expressed as: 8 > > < R2 ðμ Þ ¼
A0 ωλ ωðλ þ Z cμ1 Þ− cosμ cosμ > > dμ : θ2 ðμ Þ−θ2a ¼ μ a ωðλ þ c1 Þ− cosμ
ð21Þ
Along with Eq. (18), the transmission ratio parameter equation of EMNGP with the steepest rotation characteristic can be obtained: 8 R ðμ Þ ωλ > < i12 ðμ Þ ¼ 2 ¼ R1 ðμ Þ ωc 1 − cosμ > : θ ðμ Þ ¼ B arctan B tan μ −B arctan B tan μ þ c 1 1 2 3 4 2 2 2
ð22Þ
According to Eqs. (18) and (21), three design examples of this steepest rotation pitch curves with integral constraint for EMNGP are depicted in Fig. 4(a)–(c), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curves with integral constraint for EMNGP are listed in Table 4 and depicted in Fig. 5, respectively. 3.2. Steepest rotation design of pitch curves with integral constraint for internal meshing noncircular gear pair Referring to Table 3 and Eq. (10), the auxiliary function of steepest rotation pitch curve with integral constraint for internal meshing noncircular gear pair (IMNGP) can be expressed as: 8 Z θ 1b > > > T min ¼ minT 1 ¼ min H 1 dθ1 > < θ1a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R1 2 þ R1 02 > λ R > > þ 1 1 : H1 ¼ R1 −A0 ωR1
ð23Þ
8 Z θ 1b > > > T min ¼ minT 2 ¼ min H 2 dθ1 > < θ1a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R1 2 þ R1 02 > λ R > > þ 2 1 : H2 ¼ A0 þ R1 ωR1
ð24Þ
or
where λ1 and λ2 are called the undetermined Lagrange multipliers. Referring to the solving process of the steepest rotation parameter equation of driving gear pitch curve for EMNGP mentioned in Section 3.1, the steepest rotation parameter equation of driving gear pitch curve with integral constraint for IMNGP can be expressed as: 88 A ð cosμ 1 −ωc11 Þ > > < R1 ðμ 1 Þ ¼ 0 > > > ω ðλ1 −c11Þ þ cosμ 1 > > > μ μ > >: > < θ1 ðμ 1 Þ ¼ B11 arctan B12 tan 1 −B13 arctan B14 tan 1 þ c12 2 2 8 8 ωðλ1 −c11 Þb−1 ωðλ1 −c11 ÞN1 > > > > > > < < > > ωc11 b−1 ωc11 N1 > > or s:t: > > ωλ ωλ1 N0 b0 > > > 1 > : > : : μ 1 ∈½−π=2; π=2 μ 1 ∈½−π; −π=2 or μ 1 ∈½π=2; π
ð25Þ
Table 4 Design parameters of the steepest rotation pitch curves with integral constraint for EMNGP. Figure number
Parameter variable μ
Undetermined parameters c1, c2, λ
Fig. 4(a)
μ ∈ [−π/2, π/2]
Fig. 4(b)
μ ∈ [−π, −π/2]
Fig. 4(c)
μ ∈ [π/2, π]
c1 c2 c1 c2 c1 c2
= = = = = =
2, λ = 3, 0.6097 -2, λ=-3, 0.4212 -2, λ = -3, 0.1885
Center distance A0 and angular velocity ω A0 = 20, ω = 1
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
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Fig. 5. Transmission ratio of the steepest rotation pitch curves with integral constraint for EMNGP.
or 88 A ðωc21 − cosμ 2 Þ > > < R1 ðμ 2 Þ ¼ 0 > > > ω ðλ2 −c21Þ þ cosμ 2 > > > > : θ ðμ Þ ¼ B arctan B tan μ 2 −B arctan B tan μ 2 þ c >> < 1 2 21 22 23 24 22 2 2 8 8 ωðλ2 −c21 Þb−1 ωðλ2 −c21 ÞN1 > > > > > > < < > > ωc21 N1 ωc21 b−1 > > or s:t: > > ωλ ωλ2 b−2 N2 > > > 2 > : > : : μ 2 ∈½−π=2; π=2 μ 2 ∈½−π; −π=2 or μ 2 ∈½π=2; π
ð26Þ
where μ1 and μ2 are parameter variable, c11, c12, λ1 and c21, c22, λ2 are undetermined parameters, B11, B12, B13, B14 and B21, B22, B23, B24 are given respectively by: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ωc11 ωc11 −1 > > > B11 ¼ > > ωc ωc −1 > 11 11 þ 1 > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωc11 þ 1 > > > < B12 ¼ ωc −1 11 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ωðλ1 −c11 Þ ωðλ1 −c11 Þ þ 1 > > B ¼ > > > 13 ωðλ1 −c11 Þ þ 1 ωðλ1 −c11 Þ−1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωðλ1 −c11 Þ−1 > > : B14 ¼ ωðλ −c Þ þ 1 1 11
ð27Þ
8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ωc21 ωc21 −1 > > > B ¼ > 21 > ωc21 −1 ωc21 þ 1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωc21 þ 1 > > B ¼ > < 22 ωc21 −1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ω ðλ2 −c21 Þ ωðλ2 −c21 Þ þ 1 > > > > B23 ¼ ωðλ −c Þ þ 1 ωðλ −c Þ−1 > 2 21 2 21 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > > ωðλ2 −c21 Þ−1 > > : B24 ¼ ωðλ −c Þ þ 1 2 21
ð28Þ
and
26
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 6. Three design examples of the steepest rotation pitch curves with integral constraint for R1(θ1) is outer gear pitch curve.
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
27
Table 5 Design parameters of the steepest rotation pitch curves for R1(θ1) is outer gear pitch curve. Figure number
Parameter variable μ1
Undetermined parameters c11, λ1, c12
Fig. 6(a)
μ1 ∈ [−π/2, π/2]
Fig. 6(b)
μ1 ∈ [−π, −π/2]
Fig. 6(c)
μ1 ∈ [π/2, π]
c11 c12 c11 c12 c11 c12
¼ −3; λ1 ¼ −1 ¼ 0:0964 ¼ 4; λ1 ¼ 2 ¼ −0:3830 ¼ 4; λ1 ¼ 2 ¼ 0:5351
Center distance A0 and angular velocity ω A0 = 20, ω = 1
Undetermined parameters c11, c12, λ1 and c21, c22, λ2 can be calculated respectively by: Z 8 > > < θ2b −θ2a ¼
μ 1b
Z 8 > > < θ2b −θ2a ¼
μ 2b
cosμ 1 dμ ωðλ1 −c11 Þ þ cosμ 1 1 > R ðμ Þ ¼ R1a ; R1 ðμ 1b Þ ¼ R1b > : 1 1a θ1 ðμ 1a Þ ¼ θ1a ; θ1 ðμ 1b Þ ¼ θ1b μ 1a
ð29Þ
and
cosμ 2 dμ 2 ω ð λ −c μ 2a 2 21 Þ þ cosμ 2 > R ðμ Þ ¼ R1a ; R1 ðμ 2b Þ ¼ R1b > : 1 2a θ1 ðμ 2a Þ ¼ θ1a ; θ1 ðμ 2b Þ ¼ θ1b
ð30Þ
where μ1a, μ1b, and μ2a, μ2b are undetermined values of parameter μ1 and μ2, respectively. Moreover, the pitch curve parameter equation of driven noncircular gear conjugated with driving noncircular gear pitch curve for IMNGP can be expressed as: 8 > > < R2 ðμ 1 Þ ¼
−A0 ωλ1 ωðλ1Z−c11 Þ þ cosμ 1 μ1 cosμ 1 > > dμ 1 : θ2 ðμ 1 Þ−θ2a ¼ μ 1a ωðλ1 −c11 Þ þ cosμ 1
Fig. 7. Transmission ratio of the steepest rotation pitch curves with integral constraint for R1(θ1) is outer gear pitch curve.
ð31Þ
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X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 8. Three design examples of the steepest rotation pitch curves for R1(θ1) is inner gear pitch curve.
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29
Table 6 Design parameters of the steepest rotation pitch curves for R1(θ1) is inner gear pitch curve. Figure number
Parameter variable μ2
Undetermined parameters c21, λ2, c22
Fig. 8(a)
μ2 ∈ [−π/2, π/2]
Fig. 8(b)
μ2 ∈ [−π, −π/2]
Fig. 8(c)
μ2 ∈ [π/2, π]
c21 c22 c21 c22 c21 c22
¼ 3; λ2 ¼ 5 ¼ 0:8173 ¼ −2; λ2 ¼ −4 ¼0 ¼ −2; λ2 ¼ −4 ¼ 1:2092
Center distance A0 and angular velocity ω A0 = 20, ω = 1
or 8 > > < R2 ðμ 2 Þ ¼
A0 ωλ2 ωðλ2Z−c21 Þ þ cosμ 2 μ2 cosμ 2 > > dμ 2 : θ2 ðμ 2 Þ−θ2a ¼ μ 2a ωðλ2 −c21 Þ þ cosμ 2
ð32Þ
Along with Eqs. (25) and (26), the transmission ratio parameter equation of IMNGP with the steepest rotation characteristic can be obtained: 8 R ðμ Þ ωλ1 > < i12 ðμ 1 Þ ¼ 2 1 ¼ R1 ðμ 1 Þ ωc 11 − cosμ1 > : θ ðμ Þ ¼ B arctan B tan μ 1 −B arctan B tan μ 1 þ c 1 1 11 12 13 14 12 2 2
ð33Þ
8 R ðμ Þ ωλ2 > < i12 ðμ 2 Þ ¼ 2 2 ¼ R1 ðμ 2 Þ ωc 21 − cosμ2 > : θ ðμ Þ ¼ B arctan B tan μ 2 −B arctan B tan μ 2 þ c 1 2 21 22 23 24 22 2 2
ð34Þ
or
For the steepest rotation pitch curves with integral constraint of IMNGP, Eqs. (23), (25), (27), (29), (31) and (33) are applied to R1(θ1) is outer gear pitch curve, while Eqs. (24), (26), (28), (30), (32) and (34) are applied to R1(θ1) is inner gear pitch curve. When R1(θ1) is outer gear pitch curve, according to Eq. (25) and Eq. (31), three design examples of this steepest rotation pitch curves with integral constraint for IMNGP are depicted in Fig. 6(a)–(c), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curves with integral constraint for R1(θ1) is outer gear pitch curve are listed in Table 5 and depicted in Fig. 7, respectively.
Fig. 9. Transmission ratio of the steepest rotation pitch curves with integral constraint for R1(θ1) is inner gear pitch curve.
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Fig. 10. Two design examples of this steepest rotation pitch curves for EMNGP with epicyclic constraint of driven noncircular gear.
When R1(θ1) is inner gear pitch curve, according to Eq. (26) and Eq. (32), three design examples of this steepest rotation pitch curves with integral constraint for IMNGP are depicted in Fig. 8(a)–(c), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curves with integral constraint for R1(θ1) is inner gear pitch curve are listed in Table 6 and depicted in Fig. 9, respectively.
Table 7 Design parameters of the steepest rotation pitch curves with epicyclic constraint of driven noncircular gear for EMNGP Figure number
Parameter variable μ
Undetermined parameters c1, λ, c2
Center distance A0 and angular velocity ω
Fig. 10(a)
μ∈½− π2 ; π2
A0 = 20, ω = 1
Fig. 10(b)
μ∈½− π3 ; π3
c1 ¼ 1:1011; λ ¼ 0:0911; c2 ¼ 1:7611 c1 ¼ 1:1081; λ ¼ 0:0711; c2 ¼ 1:3687
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Fig. 11. Transmission ratio of this steepest rotation pitch curves for EMNGP with epicyclic constraint of driven noncircular gear.
4. Analysis of steepest rotation pitch curve with epicyclic constraint for driven noncircular gear 4.1. Analysis of steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for EMNGP Referring to Section 3.1, for external meshing noncircular gear pair, in order to guarantee the steepest rotation pitch curve R1(θ1) possesses driven noncircular gear pitch curve with epicyclic constraint, Eq. (21) should satisfy the following equations: 8 < θ2 ðμ a Þ ¼ θ2a ¼ 0 θ ðμ Þ ¼ θ2b ¼ 2π : 2 b R2 ðμ a Þ ¼ R2 ðμ b Þ
ð35Þ
According to the constraint conditions of Eqs. (18) and (21), (35) can be established under the following conditions: 8 π > μ a ∈½− ; 0Þ; μ b ¼ −μ a > 8 >Z > 2 > < μb < ωðλ þ c1 ÞN1 cosμ dμ ¼ 2π ; s:t: ωc N1 μ a ωðλ þ c1 Þ− cosμ > : 1 > ωλ N0 > > θ1 ðμ a Þ ¼ θ1a ¼ 0 > : θ1a bθ1 ðμ b Þ ¼ θ1b ≤2π
ð36Þ
Table 8 Epicyclic constraint of driven noncircular gear for IMNGP Internal meshing noncircular gear pair R1 is outer gear pitch curve, R2 is inner gear pitch curve
R1 is inner gear pitch curve, R2 is outer gear pitch curve
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ωðλ1 −c11 Þ ωðλ1 −c11 Þþ1 π þ μ 1a ¼ ωðλ 1 −c11 Þþ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ωðλ1 −c11 Þ−1 μ 1a 1 −c11 Þ−1 ; tan arctanð ωðλ Þ; ωðλ1 −c11 Þþ1 2 μ 1a ∈½− π2 ; 0Þ ωðλ1 −c11 ÞN 1 s:t:f ωc11 b−1 ωλ1 b0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ωðλ2 −c21 Þ ωðλ2 −c21 Þþ1 π þ μ 2a ¼ ωðλ 2 −c21 Þþ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωðλ2 −c21 Þ−1 μ 2a 2 −c21 Þ−1 ; tan Þ; arctanð ωðλ ωðλ2 −c21 Þþ1 2 μ 2a ∈½− π2 ; 0Þ ωðλ2 −c21 Þ N1 s:t:f ωc21 N 1 ωλ2 N2
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X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 12. Design algorithm flow chart of steepest rotation pitch curves for noncircular gear pair.
Along with Eqs. (18)–(21), two design examples of this steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for EMNGP are depicted in Fig. 10(a)–(b), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for EMNGP are listed in Table 7 and depicted in Fig. 11, respectively.
Table 9 Design requirements of the steepest rotation pitch curves design for EMNGP Center distance A0 and angular velocity ω
Polar angles θ1a, θ1b
Polar radiuses R1a, R1b
Polar angles θ2a, θ2b
A0 = 20 ω=1
θ1a = 0 θ1b = 0.5110
R1a = 16 R1b = 16
θ2a = 0 θ2b = 1.1842
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Table 10 Undetermined parameters c1, c2, λ and the range of parameter variable μ Undetermined parameters c1, c2, λ c1 ¼ 2 λ ¼ 0:5 c2 ¼ 0:2555
the range of parameter variable μ μ∈½− π2 ; π2
4.2. Analysis of steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for IMNGP Referring to Section. 3.2 and Section 4.1, for internal meshing noncircular gear pair, in order to guarantee the steepest rotation pitch curve R1(θ1) possesses driven noncircular gear pitch curve with the epicyclic constraint, Eqs. (31) and (32) should satisfy the constraint conditions contained by Eqs. (37) and (38), respectively. 88 > > μ 1a ∈½− π ; 0Þ; μ 1b ¼ −μ 1a > > > >> > μ 1b > cosμ 1 > > dμ 1 ¼ 2π > <> > ω ð λ −c > μ 1 11 Þ þ cosμ 1 1a : θ8 > 1 ðμ 1a Þ ¼ θ1a ¼ 0; θ1a bθ1 ðμ 1b Þ ¼ θ1b ≤2π > > > > < ωðλ1 −c11 ÞN1 > > > s:t: ωc b−1 > > : : ωλ11b0 1
ð37Þ
88 π > > μ ∈½− ; 0Þ; μ 2b ¼ −μ 2a > >> > > < Z 2a 2 > > μ 2b > cosμ 2 > > dμ 2 ¼ 2π > <> > ð ω λ −c > μ 2 21 Þ þ cosμ 2 2a : > θ8 1 ðμ 2a Þ ¼ θ1a ¼ 0; θ1a bθ1 ðμ 2b Þ ¼ θ1b ≤2π > > > ωðλ2 −c21 ÞN1 < > > > > s:t: ωc21 N1 > > : : ωλ N2
ð38Þ
2
The epicyclic constraint of driven noncircular gear for IMNGP is listed in Table 8 by simplifying Eq. (37) and Eq. (38). According to Table 8, we know that the steepest rotation pitch curves for IMNGP with epicyclic constraint of driven noncircular gear are impossible, because the solutions of undetermined parameters c11, c12, λ1 and c21, c22, λ2 don't exist.
Fig. 13. A number example of steepest rotation pitch curve design for EMNGP.
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X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 14. Transmission ratio of this steepest rotation pitch curves depicted in Fig. 13.
5. Algorithm and number example According to Section 3 and Section 4, the steepest rotation pitch curve with epicyclic constraint for driven noncircular gear in Section 4 is particular design of pitch curves with integral constraint when polar angle θ2b of pitch curve R2 is equal to 2π. Therefore, the flow chart depicted in Fig. 12 is a general design algorithm of steepest rotation pitch curves for noncircular gear pair. A number example of steepest rotation pitch curve design for EMNGP is given in the following description. The critical issue is the solution of undetermined parameters c1, c2, λ and the range of parameter variable μ according to the algorithm shown in Fig. 12. Assuming that the design requirements are listed in Table 9, undetermined parameters c1, c2, λ, and the range of parameter variable μ can be calculated by Eqs. (18)–(20), and the solutions are listed in Table 10. Moreover, the steepest rotation pitch curves and its transmission ratio function are depicted in Fig. 13 and Fig. 14, respectively. The solved parameter equations of pitch curves and transmission ratio function can be expressed as: 8 20ð2− cosμ Þ > > > < R1 ðμ Þ ¼ 2:5− cosμ ! pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 4 3 μ 10 21 μ 21 > > > arctan 3 tan − arctan þ 0:2555 tan θ ð μ Þ ¼ : 1 3 2 21 2 3 8 > > < R2 ðμ Þ ¼
10 2:5− Z μ cosμ cosμ > > θ dμ ð μ Þ ¼ : 2 −π2 2:5− cosμ 8 > > > < i12 ðμ Þ ¼
0:5 2− cosμ ! pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 4 3 μ 10 21 μ 21 > > > arctan 3 tan arctan tan θ ð μ Þ ¼ − þ 0:2555 : 1 3 2 21 2 3
ð39Þ
ð40Þ
ð41Þ
Because noncircular gear pair with the steepest rotation pitch curves can meet certain transmission requirements within the shortest time, the promising applications of this noncircular gear pair include rapid braking or start transmission device of automobile, packaging machinery, et al. 6. Conclusion The steepest rotation pitch curve with fixed boundaries can be obtained successfully in this paper. Based on the result, a general design algorithm of steepest rotation pitch curves for noncircular gear pair is proposed by resorting to calculus of variations. In particular, the designer can obtain a pair of conjugate noncircular gears with the steepest rotation characteristic by means of the proposed algorithm. This novel method builds a solid foundation for further researches on other particular geometrical and mechanical properties of noncircular gear pair or noncircular bevel gear pair.
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Acknowledgment The authors are grateful for financial support from National Natural Science Foundation of China [grant number: 51175067].
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Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Synthesis of the steepest rotation pitch curve design for noncircular gear Xin Zhang ⁎, Shouwen Fan ⁎ School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, People's Republic of China
a r t i c l e
i n f o
Article history: Received 21 August 2015 Received in revised form 31 January 2016 Accepted 25 March 2016 Available online 13 April 2016 Keywords: Noncircular gear Steepest rotation method Pitch curve Calculus of variations
a b s t r a c t A steepest rotation method for designing pitch curve with fixed boundaries of noncircular gear is proposed by resorting to calculus of variations. A general mathematical model is established to design the steepest rotation pitch curves with integral constraint for noncircular gear pair, including external and internal meshing gear pair. In particular, to achieve the design of pitch curve with epicyclic constraint for driven noncircular gear, the epicyclic constraint conditions are also established. In addition, the unified design algorithm of pitch curves of noncircular gear pair with this steepest rotation characteristic is given in this paper, the pitch curves of desired geometrical and transmission properties can be solved easily by using the proposed algorithm. Examples presented in this paper are implemented in MATLAB, and feasibility and validity of above algorithm are verified. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Since noncircular gears are more compact and better balanced than linkages and cams, they have been already applied to mechanical equipments: Maltese mechanism, packaging machinery, printing machinery, textile machinery, gear shaper and flow meters [1–6]. Dooner D. B. and Seireg A. A. [7] proposed an interactive design approach of noncircular gears. Dooner D. B [8] used noncircular gears to eliminate unnecessary torque and speed fluctuations. Wu L. I. and Chang S. L. [9] obtained the tooth profiles of elliptical gears by combining the involutes generated from base curves with appropriate addendum and dedendum curves. Tong S. H. and Yang D. C. H. [10] obtained identical noncircular conjugate pitch curves with any number of lobes. Wei H. [11] developed an elliptic interpolation algorithm for elliptical gears on the basis of the principle of central angle division of an arc. Hector F. Q. R., Salvador C. F. and et al. [12] designed an approach for generating pitch curves of N-lobed noncircular gears on the basis of Bézier and B-spline nonparametric curves. Mundo D. [13] obtained the planetary gear train with specified variable gear ratio law by combing three noncircular gears. A general generation method of N-lobed elliptical gears from a basic ellipse was proposed in Ref. [14], and the synthesis of N-lobed or high-order elliptical gears and their rack by means of a conjugate shaper cutter with an involute tooth profile was obtained in Refs. [15–22]. Most of pitch curves for these noncircular gears in Refs. [1–22] were constructed by existing smoothed curves as pitch curves of noncircular gears. However, with the rapid development of numerical control machine tool and computer technology, some novel design methods of pitch curves for N-lobed noncircular gears have been proposed in recent studies. Yao W. X. [23] presented a pitch curve design method, which using plane regular N-curved polygon and spiral of Archimedes as the pitch curve for N-lobed noncircular gear. Another new design method of pitch curves with concave for noncircular bevel gears was proposed by Kan S. and Xia J. Q. [24].
⁎ Corresponding authors. E-mail addresses: [email protected] (X. Zhang), [email protected] (S. Fan).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.03.020 0094-114X/© 2016 Elsevier Ltd. All rights reserved.
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
17
Although some novel pitch curves are put forward in succession, the high efficiency requirement of noncircular gear transmission still cannot be realized. In view of this, a kind of design method of steepest rotation pitch curves for noncircular gears is presented in this paper, the noncircular gears with steepest rotation pitch curves possess shortest transmission time at a given angular velocity. In addition, the pitch curves with this characteristic are more attractive than previous designs under the high efficient transmission requirement. Several typical design methods of steepest rotation pitch curves for noncircular gears based on calculus of variations are presented. 2. Design of steepest rotation pitch curve with fixed boundaries for noncircular gear As depicted in Fig. 1, points A, B and O are two fixed endpoints and rotation center of plane curve C, respectively. Fixed coordinate system O-xy is attached to rotation center O, θa and θb are polar angles of points A and B, respectively. Assume that the polar equation of plane curve C is rfb(θ), according to the principle of kinematics and differential, the differential dt(θ) of time function t(θ) that noncircular gear through plane curve rfb(θ) can be expressed as:
dsðθÞ ¼ dt ðθÞ ¼ ωr fb
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r fb 2 þ r 0fb 2 dθ ωr fb
ð1Þ
where ds(θ) and rfb′(θ) are small section arc length and first-order derivative for plane curve rfb(θ), respectively. Parameter ω is a given rotation angular velocity. Polar angle θ is measured counterclockwise from the positive direction x-axis. Integrating both sides of Eq. (1) from θa to θb, we obtain: Z T¼
θb θa
Z dtðθÞ ¼
θb θa
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r fb 2 þ r 0fb 2 ωrfb
dθ
ð2Þ
where T is rotation time that noncircular gear through plane curve rfb(θ) at the given angular velocity ω from θ = θa to θ = θb, and 0 ≤ θa b θb ≤ 2π. From the Eq. (2), we know that the value of time T may be different for different plane curve rfb(θ), so the minimum Tmin of rotation time can be expressed as: Z 8 > < T min ¼ minT ¼ min
0 F θ; r fb ; r fb dθ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 0 −1 r fb 2 þ r fb 02 F θ; rfb ; r fb ¼ ðωr fb Þ θb θa
Fig. 1. Pitch curve of noncircular gear with fixed boundaries.
ð3Þ
18
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According to the calculus of variations [25–26], the solution rfb(θ) of Eq. (3) must satisfy the following condition: ∂F d ∂F − ∂r fb dθ ∂r0fb
! ¼ 0; θ ∈½θa ; θb
ð4Þ
Rearranging Eq. (4) to: d dθ
0
F−rfb
∂F ∂r 0fb
! ¼ 0; θ ∈½θa ; θb
ð5Þ
Substituting Eq. (3) into Eq. (5) and rearranging Eq. (5) to: 0
1 2
d B r fb C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiA ¼ 0; θ∈½θa ; θb @ dθ ωr fb r fb 2 þ r fb 02
ð6Þ
Integrating both sides of Eq. (6), we obtain: r fb ðθÞ ¼ e
H1 ðθ−H 2 Þ
; θ∈½θa ; θb
ð7Þ
where H1 and H2 are integral constants. Referring to Eq. (7), we know that the first-order derivative r′fb(θ) = H1eH1(θ−H2) of plane curve rfb(θ) is a monotonic function about parameter θ. Therefore, in order to satisfy non-circularity of pitch curve, polar radiuses rfb(θa) and rfb(θb) are absolutely unequal when use the plane curve rfb(θ) as pitch curve of noncircular gear, i.e. rfb(θa) ≠ rfb(θb). Substituting the fixed boundaries (θa, rfb(θa)) and (θb, rfb(θb)) of plane curve rfb(θ) into Eq. (7), we obtain: 8 ln ðrfb ðθb ÞÞ− ln ðr fb ðθa ÞÞ > > < H1 ¼ θb −θa ; r fb ðθa Þ≠r fb ðθb Þ > > H ¼ θa ln ðr fb ðθb ÞÞ−θb ln ðr fb ðθa ÞÞ : 2 ln ðr fb ðθb ÞÞ− ln ðr fb ðθa ÞÞ
ð8Þ
Along with Eq. (7), the steepest rotation pitch curve of noncircular gear with fixed boundaries can be expressed as: 8 H ðθ−H2 Þ r fb ð8 θÞ ¼ e 1 ; > > > > θ∈ θ ½ > > a ; θb > > > > r ðθ Þ≠r ðθ Þ > < > > fb a fb b > < ln ðr fb ðθb ÞÞ− ln ðr fb ðθa ÞÞ H ¼ > s:t: > > 1 > θb −θa > > > > > > > > > H ¼ θa ln ðr fb ðθb ÞÞ−θb ln ðr fb ðθa ÞÞ : > : 2 ln ðrfb ðθb ÞÞ− ln ðr fb ðθa ÞÞ
ð9Þ
where s. t. is an abbreviation of “subject to”, it means that contained constraint conditions must be satisfied by the equation. According to Eq. (9), three design examples of this steepest rotation pitch curve with fixed boundaries listed in Table 1 are depicted in Fig. 2(a)–(c), respectively. The polar equations of pitch curves with fixed boundaries and related integral constants are listed in Table 2. The units of parameters listed in Table 2 are adopted by standard international unit (SIU). Without additional explanation, the following discussion relating to parameter in this paper whose unit refers to SIU. 3. Design of steepest rotation pitch curves with integral constraint for noncircular gear pair As depicted in Fig. 3, points O1 and O2 are rotation centers of pitch curves R1 of driving noncircular gear and pitch curve R2 of driven noncircular gear, respectively. Point p is the instantaneous meshing point of pitch curves R1 and R2, its corresponding polar angles are θ1 and θ2, respectively. Parameter ω is a given angular velocity of driving noncircular gear R1, A0 (A0 N 0) is center distance between pitch curves R1 and R2. Polar angles θ1 and θ2 are equal to θ1a and θ2a at the starting position, respectively. Table 1 Design parameters of pitch curves with fixed boundaries. Case number
Left boundary condition (θa, rfb(θa))
Right boundary condition (θb, rfb(θb))
a b c
(0, 2) (π/6, 3) (π/4, 5)
(3π/2, 5) (11π/6, 6) (7π/4, 8)
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Fig. 2. Three design examples of steepest rotation pitch curves with fixed boundaries for noncircular gear.
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Table 2 Polar equations of pitch curves with fixed boundaries and related integral constants. Figure number
Integration constant
Fig. 2(a)
H1 H1 H2 H1 H2
Fig. 2(b) Fig. 2(c)
= = = = =
0.1944; H2 = −3.5648 0.1324; -7.7752 0.0997; -15.3513
Pitch curve equation rfb(θ) e0.1944(θ +3.5648) , θ ∈ [0, 3π/2] e0.1324(θ +7.7752) , θ ∈ [π/6, 11π/6] e0.0997(θ +15.3513) , θ ∈ [π/4, 7π/4]
Assuming that O2p = R2 and O1p = R1 = R1(θ1), then the pitch curve equation of driven noncircular gear and closed-law between pitch curves R1 and R2 are listed in Table 3 according to the meshing principle of noncircular gear pair [23]. Polar angle θ2b is the end rotation angle of pitch curve R2 when the polar angle θ1 of pitch curve R1 from θ1a to θ1b. 3.1. Steepest rotation design of pitch curves with integral constraint for external meshing noncircular gear pair According to variational problems with integral constraint [26] and Eq. (3), the minimum Tmin of rotation time for pitch curve R1(θ1) of driving noncircular gear for external meshing noncircular gear pair (EMNGP) can be established as: 8 Z θ 1b > > > T ¼ minT ¼ min Hdθ1 > min < θ1a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R1 2 þ R1 02 > λR1 > > þ :H ¼ ωR1 A0 −R1
ð10Þ
where λ is called an undetermined Lagrange multiplier. Eq. (10) is called an auxiliary function of steepest rotation pitch curve with integral constraint for EMNGP. Because H of Eq. (10) doesn't contain parameter θ1, the first integral of Euler-Lagrange Equation [25–26] can be expressed as: 0
H−R1 H R1 0 ¼ c1
ð11Þ
Substituting H of Eq. (10) into Eq. (11), we obtain: R1 λR1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ c1 A0 −R1 2 02 ω R1 þ R1
ð12Þ
where c1 is an integral constant. Assuming R1′ = R1tanμ and substituting R1′ into Eq. (12), we obtain: 8 A ðωc1 − cosμ Þ > > ; R ðμ Þ ¼ 0 > > ωðλ þ c1 Þ− 8 cosμ < 18 < ωðλ þ c1 ÞN1 < ωðλ þ c1 Þb−1 > > > or ωc1 b−1 s:t: ωc N1 > : : 1 : ωλN0 ωλb0
ð13Þ
Taking the derivative on both sides of Eq. (13), we have: dR1 A0 ωλ sinμ ¼ dμ ½ωðλ þ c1 Þ− cosμ 2
ð14Þ
Along with R1′ = R1tanμ and Eq. (13), we get the differential function θ1′(μ): dθ1 ωλ cosμ ¼ dμ ðωc1 − cosμ Þ½ωðλ þ c1 Þ− cosμ
ð15Þ
Integrating both sides of Eq. (15), we obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2ωc1 ωc1 −1 ωc1 þ 1 μ θ1 ðμ Þ ¼ − arctan tan ωc1 −1 2 ωc1 −1 ωc1 þ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2ωðλ þ c1 Þ ωðλ þ c1 Þ−1 ω ð λ þ c1 Þ þ 1 μ arctan tan þ c2 ωðλ þ c1 Þ−1 ωðλ þ c1 Þ þ 1 ωðλ þ c1 Þ−1 2
ð16Þ
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Fig. 3. Geometry relationship of conjugated pitch curves of noncircular gear pair.
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Table 3 Pitch curve equation of driven noncircular gear and closed-law between pitch curves R1 and R2. R2 ¼ A0 −R1 ðθ1 Þ R1 ðθ1 Þ ; θ dθ1 A0 −R1 ðθ1 Þ 0bR1 ðθ1 ÞbA0 R1 ðθ1 Þ s:t:f θ dθ1 θ2b −θ2a ¼ ∫ θ1b 1a A −R ðθ Þ 0 1 1
External meshing (referring to Fig.3(a))
f
Internal meshing
R1 is outer gear pitch curve, R2 is inner gear pitch curve (referring to Fig.3(b))
R1 is inner gear pitch curve, R2 is outer gear pitch curve (referring to Fig.3(c))
θ2 −θ2a ¼ ∫ θ11a
R2 ¼ R1 ðθ1 Þ−A0 R1 ðθ1 Þ ; θ dθ1 R1 ðθ1 Þ−A0 R1 ðθ1 Þ NA0 R1 ðθ1 Þ s:t:f θ θ2b −θ2a ¼ ∫ θ1b dθ1 1a R ðθ Þ−A 1 1 0 R2 ¼ A0 þ R1 ðθ1 Þ R1 ðθ1 Þ f ; θ dθ1 θ2 −θ2a ¼ ∫ θ11a A0 þ R1 ðθ1 Þ R1 ðθ1 Þ N0 R1 ðθ1 Þ s:t:f θ dθ1 θ2b −θ2a ¼ ∫ θ1b 1a A þ R ðθ Þ 0 1 1 f
θ2 −θ2a ¼ ∫ θ11a
where c2 is also an integration constant. Because θ1(μ) must have one-to-one corresponding relationship and monotone increasing property, the range of parameter μ can be expressed as: 8 ½−π; −π=2 > > > > < or μ∈ ½−π=2; π=2 > > > or > : ½π=2; π
ð17Þ
Therefore, the steepest rotation parameter equation of driving noncircular gear pitch curve with integral constraint for EMNGP can be expressed as: 88 A0 ðωc1 − cosμ Þ > > > > < R1 ðμ Þ ¼ ωðλ þ c Þ− cosμ > > 1 ; > > μ μ >> > þ c2 < : θ1 ðμ Þ ¼ B1 arctan B2 tan −B3 arctan B4 tan 2 8 8 2 ωðλ þ c1 Þb−1 ωðλ þ c1 ÞN1 > > > > > > < < > > ωc1 N1 ωc1 b−1 > > or > s:t:> ωλN0 > > ωλb0 > > : > : : μ∈½−π=2; π=2 μ∈½−π; −π=2 or μ∈½π=2; π
ð18Þ
where parameters B1, B2, B3 and B4 are given by: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ωc1 ωc1 −1 > > > ¼ B 1 > > ωc ωc −1 þ1 > 1 > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > > > ωc þ 1 > 1 > > < B2 ¼ ωc −1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ωðλ þ c1 Þ ωðλ þ c1 Þ−1 > > > > B3 ¼ ωðλ þ c Þ−1 ωðλ þ c Þ þ 1 > 1 1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωðλ þ c1 Þ þ 1 > > : B4 ¼ ωðλ þ c Þ−1
ð19Þ
1
Undetermined parameters c1, c2, λ can be calculated by fixed boundaries and integral constraint: Z 8 > > < θ2b −θ2a ¼
μb
cosμ dμ ωðλ þ c1 Þ− cosμ > R ðμ Þ ¼ R1a ; R1 ðμ b Þ ¼ R1b > : 1 a θ1 ðμ a Þ ¼ θ1a ; θ1 ðμ b Þ ¼ θ1b μa
ð20Þ
where μa and μb are undetermined values of parameter μ corresponding to polar angles θ1a and θ1b of two endpoints for pitch curve R1(θ1).
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
Fig. 4. Three design examples of the steepest rotation pitch curves with integral constraint for EMNGP.
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Moreover, the pitch curve parameter equation of driven noncircular gear conjugated with driving noncircular gear pitch curve of Eq. (18) for EMNGP can be expressed as: 8 > > < R2 ðμ Þ ¼
A0 ωλ ωðλ þ Z cμ1 Þ− cosμ cosμ > > dμ : θ2 ðμ Þ−θ2a ¼ μ a ωðλ þ c1 Þ− cosμ
ð21Þ
Along with Eq. (18), the transmission ratio parameter equation of EMNGP with the steepest rotation characteristic can be obtained: 8 R ðμ Þ ωλ > < i12 ðμ Þ ¼ 2 ¼ R1 ðμ Þ ωc 1 − cosμ > : θ ðμ Þ ¼ B arctan B tan μ −B arctan B tan μ þ c 1 1 2 3 4 2 2 2
ð22Þ
According to Eqs. (18) and (21), three design examples of this steepest rotation pitch curves with integral constraint for EMNGP are depicted in Fig. 4(a)–(c), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curves with integral constraint for EMNGP are listed in Table 4 and depicted in Fig. 5, respectively. 3.2. Steepest rotation design of pitch curves with integral constraint for internal meshing noncircular gear pair Referring to Table 3 and Eq. (10), the auxiliary function of steepest rotation pitch curve with integral constraint for internal meshing noncircular gear pair (IMNGP) can be expressed as: 8 Z θ 1b > > > T min ¼ minT 1 ¼ min H 1 dθ1 > < θ1a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R1 2 þ R1 02 > λ R > > þ 1 1 : H1 ¼ R1 −A0 ωR1
ð23Þ
8 Z θ 1b > > > T min ¼ minT 2 ¼ min H 2 dθ1 > < θ1a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > R1 2 þ R1 02 > λ R > > þ 2 1 : H2 ¼ A0 þ R1 ωR1
ð24Þ
or
where λ1 and λ2 are called the undetermined Lagrange multipliers. Referring to the solving process of the steepest rotation parameter equation of driving gear pitch curve for EMNGP mentioned in Section 3.1, the steepest rotation parameter equation of driving gear pitch curve with integral constraint for IMNGP can be expressed as: 88 A ð cosμ 1 −ωc11 Þ > > < R1 ðμ 1 Þ ¼ 0 > > > ω ðλ1 −c11Þ þ cosμ 1 > > > μ μ > >: > < θ1 ðμ 1 Þ ¼ B11 arctan B12 tan 1 −B13 arctan B14 tan 1 þ c12 2 2 8 8 ωðλ1 −c11 Þb−1 ωðλ1 −c11 ÞN1 > > > > > > < < > > ωc11 b−1 ωc11 N1 > > or s:t: > > ωλ ωλ1 N0 b0 > > > 1 > : > : : μ 1 ∈½−π=2; π=2 μ 1 ∈½−π; −π=2 or μ 1 ∈½π=2; π
ð25Þ
Table 4 Design parameters of the steepest rotation pitch curves with integral constraint for EMNGP. Figure number
Parameter variable μ
Undetermined parameters c1, c2, λ
Fig. 4(a)
μ ∈ [−π/2, π/2]
Fig. 4(b)
μ ∈ [−π, −π/2]
Fig. 4(c)
μ ∈ [π/2, π]
c1 c2 c1 c2 c1 c2
= = = = = =
2, λ = 3, 0.6097 -2, λ=-3, 0.4212 -2, λ = -3, 0.1885
Center distance A0 and angular velocity ω A0 = 20, ω = 1
X. Zhang, S. Fan / Mechanism and Machine Theory 102 (2016) 16–35
25
Fig. 5. Transmission ratio of the steepest rotation pitch curves with integral constraint for EMNGP.
or 88 A ðωc21 − cosμ 2 Þ > > < R1 ðμ 2 Þ ¼ 0 > > > ω ðλ2 −c21Þ þ cosμ 2 > > > > : θ ðμ Þ ¼ B arctan B tan μ 2 −B arctan B tan μ 2 þ c >> < 1 2 21 22 23 24 22 2 2 8 8 ωðλ2 −c21 Þb−1 ωðλ2 −c21 ÞN1 > > > > > > < < > > ωc21 N1 ωc21 b−1 > > or s:t: > > ωλ ωλ2 b−2 N2 > > > 2 > : > : : μ 2 ∈½−π=2; π=2 μ 2 ∈½−π; −π=2 or μ 2 ∈½π=2; π
ð26Þ
where μ1 and μ2 are parameter variable, c11, c12, λ1 and c21, c22, λ2 are undetermined parameters, B11, B12, B13, B14 and B21, B22, B23, B24 are given respectively by: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ωc11 ωc11 −1 > > > B11 ¼ > > ωc ωc −1 > 11 11 þ 1 > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωc11 þ 1 > > > < B12 ¼ ωc −1 11 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ωðλ1 −c11 Þ ωðλ1 −c11 Þ þ 1 > > B ¼ > > > 13 ωðλ1 −c11 Þ þ 1 ωðλ1 −c11 Þ−1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωðλ1 −c11 Þ−1 > > : B14 ¼ ωðλ −c Þ þ 1 1 11
ð27Þ
8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ωc21 ωc21 −1 > > > B ¼ > 21 > ωc21 −1 ωc21 þ 1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ωc21 þ 1 > > B ¼ > < 22 ωc21 −1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ω ðλ2 −c21 Þ ωðλ2 −c21 Þ þ 1 > > > > B23 ¼ ωðλ −c Þ þ 1 ωðλ −c Þ−1 > 2 21 2 21 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > > ωðλ2 −c21 Þ−1 > > : B24 ¼ ωðλ −c Þ þ 1 2 21
ð28Þ
and
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Fig. 6. Three design examples of the steepest rotation pitch curves with integral constraint for R1(θ1) is outer gear pitch curve.
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Table 5 Design parameters of the steepest rotation pitch curves for R1(θ1) is outer gear pitch curve. Figure number
Parameter variable μ1
Undetermined parameters c11, λ1, c12
Fig. 6(a)
μ1 ∈ [−π/2, π/2]
Fig. 6(b)
μ1 ∈ [−π, −π/2]
Fig. 6(c)
μ1 ∈ [π/2, π]
c11 c12 c11 c12 c11 c12
¼ −3; λ1 ¼ −1 ¼ 0:0964 ¼ 4; λ1 ¼ 2 ¼ −0:3830 ¼ 4; λ1 ¼ 2 ¼ 0:5351
Center distance A0 and angular velocity ω A0 = 20, ω = 1
Undetermined parameters c11, c12, λ1 and c21, c22, λ2 can be calculated respectively by: Z 8 > > < θ2b −θ2a ¼
μ 1b
Z 8 > > < θ2b −θ2a ¼
μ 2b
cosμ 1 dμ ωðλ1 −c11 Þ þ cosμ 1 1 > R ðμ Þ ¼ R1a ; R1 ðμ 1b Þ ¼ R1b > : 1 1a θ1 ðμ 1a Þ ¼ θ1a ; θ1 ðμ 1b Þ ¼ θ1b μ 1a
ð29Þ
and
cosμ 2 dμ 2 ω ð λ −c μ 2a 2 21 Þ þ cosμ 2 > R ðμ Þ ¼ R1a ; R1 ðμ 2b Þ ¼ R1b > : 1 2a θ1 ðμ 2a Þ ¼ θ1a ; θ1 ðμ 2b Þ ¼ θ1b
ð30Þ
where μ1a, μ1b, and μ2a, μ2b are undetermined values of parameter μ1 and μ2, respectively. Moreover, the pitch curve parameter equation of driven noncircular gear conjugated with driving noncircular gear pitch curve for IMNGP can be expressed as: 8 > > < R2 ðμ 1 Þ ¼
−A0 ωλ1 ωðλ1Z−c11 Þ þ cosμ 1 μ1 cosμ 1 > > dμ 1 : θ2 ðμ 1 Þ−θ2a ¼ μ 1a ωðλ1 −c11 Þ þ cosμ 1
Fig. 7. Transmission ratio of the steepest rotation pitch curves with integral constraint for R1(θ1) is outer gear pitch curve.
ð31Þ
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Fig. 8. Three design examples of the steepest rotation pitch curves for R1(θ1) is inner gear pitch curve.
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Table 6 Design parameters of the steepest rotation pitch curves for R1(θ1) is inner gear pitch curve. Figure number
Parameter variable μ2
Undetermined parameters c21, λ2, c22
Fig. 8(a)
μ2 ∈ [−π/2, π/2]
Fig. 8(b)
μ2 ∈ [−π, −π/2]
Fig. 8(c)
μ2 ∈ [π/2, π]
c21 c22 c21 c22 c21 c22
¼ 3; λ2 ¼ 5 ¼ 0:8173 ¼ −2; λ2 ¼ −4 ¼0 ¼ −2; λ2 ¼ −4 ¼ 1:2092
Center distance A0 and angular velocity ω A0 = 20, ω = 1
or 8 > > < R2 ðμ 2 Þ ¼
A0 ωλ2 ωðλ2Z−c21 Þ þ cosμ 2 μ2 cosμ 2 > > dμ 2 : θ2 ðμ 2 Þ−θ2a ¼ μ 2a ωðλ2 −c21 Þ þ cosμ 2
ð32Þ
Along with Eqs. (25) and (26), the transmission ratio parameter equation of IMNGP with the steepest rotation characteristic can be obtained: 8 R ðμ Þ ωλ1 > < i12 ðμ 1 Þ ¼ 2 1 ¼ R1 ðμ 1 Þ ωc 11 − cosμ1 > : θ ðμ Þ ¼ B arctan B tan μ 1 −B arctan B tan μ 1 þ c 1 1 11 12 13 14 12 2 2
ð33Þ
8 R ðμ Þ ωλ2 > < i12 ðμ 2 Þ ¼ 2 2 ¼ R1 ðμ 2 Þ ωc 21 − cosμ2 > : θ ðμ Þ ¼ B arctan B tan μ 2 −B arctan B tan μ 2 þ c 1 2 21 22 23 24 22 2 2
ð34Þ
or
For the steepest rotation pitch curves with integral constraint of IMNGP, Eqs. (23), (25), (27), (29), (31) and (33) are applied to R1(θ1) is outer gear pitch curve, while Eqs. (24), (26), (28), (30), (32) and (34) are applied to R1(θ1) is inner gear pitch curve. When R1(θ1) is outer gear pitch curve, according to Eq. (25) and Eq. (31), three design examples of this steepest rotation pitch curves with integral constraint for IMNGP are depicted in Fig. 6(a)–(c), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curves with integral constraint for R1(θ1) is outer gear pitch curve are listed in Table 5 and depicted in Fig. 7, respectively.
Fig. 9. Transmission ratio of the steepest rotation pitch curves with integral constraint for R1(θ1) is inner gear pitch curve.
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Fig. 10. Two design examples of this steepest rotation pitch curves for EMNGP with epicyclic constraint of driven noncircular gear.
When R1(θ1) is inner gear pitch curve, according to Eq. (26) and Eq. (32), three design examples of this steepest rotation pitch curves with integral constraint for IMNGP are depicted in Fig. 8(a)–(c), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curves with integral constraint for R1(θ1) is inner gear pitch curve are listed in Table 6 and depicted in Fig. 9, respectively.
Table 7 Design parameters of the steepest rotation pitch curves with epicyclic constraint of driven noncircular gear for EMNGP Figure number
Parameter variable μ
Undetermined parameters c1, λ, c2
Center distance A0 and angular velocity ω
Fig. 10(a)
μ∈½− π2 ; π2
A0 = 20, ω = 1
Fig. 10(b)
μ∈½− π3 ; π3
c1 ¼ 1:1011; λ ¼ 0:0911; c2 ¼ 1:7611 c1 ¼ 1:1081; λ ¼ 0:0711; c2 ¼ 1:3687
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Fig. 11. Transmission ratio of this steepest rotation pitch curves for EMNGP with epicyclic constraint of driven noncircular gear.
4. Analysis of steepest rotation pitch curve with epicyclic constraint for driven noncircular gear 4.1. Analysis of steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for EMNGP Referring to Section 3.1, for external meshing noncircular gear pair, in order to guarantee the steepest rotation pitch curve R1(θ1) possesses driven noncircular gear pitch curve with epicyclic constraint, Eq. (21) should satisfy the following equations: 8 < θ2 ðμ a Þ ¼ θ2a ¼ 0 θ ðμ Þ ¼ θ2b ¼ 2π : 2 b R2 ðμ a Þ ¼ R2 ðμ b Þ
ð35Þ
According to the constraint conditions of Eqs. (18) and (21), (35) can be established under the following conditions: 8 π > μ a ∈½− ; 0Þ; μ b ¼ −μ a > 8 >Z > 2 > < μb < ωðλ þ c1 ÞN1 cosμ dμ ¼ 2π ; s:t: ωc N1 μ a ωðλ þ c1 Þ− cosμ > : 1 > ωλ N0 > > θ1 ðμ a Þ ¼ θ1a ¼ 0 > : θ1a bθ1 ðμ b Þ ¼ θ1b ≤2π
ð36Þ
Table 8 Epicyclic constraint of driven noncircular gear for IMNGP Internal meshing noncircular gear pair R1 is outer gear pitch curve, R2 is inner gear pitch curve
R1 is inner gear pitch curve, R2 is outer gear pitch curve
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ωðλ1 −c11 Þ ωðλ1 −c11 Þþ1 π þ μ 1a ¼ ωðλ 1 −c11 Þþ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ωðλ1 −c11 Þ−1 μ 1a 1 −c11 Þ−1 ; tan arctanð ωðλ Þ; ωðλ1 −c11 Þþ1 2 μ 1a ∈½− π2 ; 0Þ ωðλ1 −c11 ÞN 1 s:t:f ωc11 b−1 ωλ1 b0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ωðλ2 −c21 Þ ωðλ2 −c21 Þþ1 π þ μ 2a ¼ ωðλ 2 −c21 Þþ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωðλ2 −c21 Þ−1 μ 2a 2 −c21 Þ−1 ; tan Þ; arctanð ωðλ ωðλ2 −c21 Þþ1 2 μ 2a ∈½− π2 ; 0Þ ωðλ2 −c21 Þ N1 s:t:f ωc21 N 1 ωλ2 N2
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Fig. 12. Design algorithm flow chart of steepest rotation pitch curves for noncircular gear pair.
Along with Eqs. (18)–(21), two design examples of this steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for EMNGP are depicted in Fig. 10(a)–(b), respectively. The corresponding design parameters and transmission ratio functions of the steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for EMNGP are listed in Table 7 and depicted in Fig. 11, respectively.
Table 9 Design requirements of the steepest rotation pitch curves design for EMNGP Center distance A0 and angular velocity ω
Polar angles θ1a, θ1b
Polar radiuses R1a, R1b
Polar angles θ2a, θ2b
A0 = 20 ω=1
θ1a = 0 θ1b = 0.5110
R1a = 16 R1b = 16
θ2a = 0 θ2b = 1.1842
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Table 10 Undetermined parameters c1, c2, λ and the range of parameter variable μ Undetermined parameters c1, c2, λ c1 ¼ 2 λ ¼ 0:5 c2 ¼ 0:2555
the range of parameter variable μ μ∈½− π2 ; π2
4.2. Analysis of steepest rotation pitch curve with epicyclic constraint of driven noncircular gear for IMNGP Referring to Section. 3.2 and Section 4.1, for internal meshing noncircular gear pair, in order to guarantee the steepest rotation pitch curve R1(θ1) possesses driven noncircular gear pitch curve with the epicyclic constraint, Eqs. (31) and (32) should satisfy the constraint conditions contained by Eqs. (37) and (38), respectively. 88 > > μ 1a ∈½− π ; 0Þ; μ 1b ¼ −μ 1a > > > >>
ð37Þ
88 π > > μ ∈½− ; 0Þ; μ 2b ¼ −μ 2a > >> > > < Z 2a 2 > > μ 2b > cosμ 2 > > dμ 2 ¼ 2π > <> > ð ω λ −c > μ 2 21 Þ þ cosμ 2 2a : > θ8 1 ðμ 2a Þ ¼ θ1a ¼ 0; θ1a bθ1 ðμ 2b Þ ¼ θ1b ≤2π > > > ωðλ2 −c21 ÞN1 < > > > > s:t: ωc21 N1 > > : : ωλ N2
ð38Þ
2
The epicyclic constraint of driven noncircular gear for IMNGP is listed in Table 8 by simplifying Eq. (37) and Eq. (38). According to Table 8, we know that the steepest rotation pitch curves for IMNGP with epicyclic constraint of driven noncircular gear are impossible, because the solutions of undetermined parameters c11, c12, λ1 and c21, c22, λ2 don't exist.
Fig. 13. A number example of steepest rotation pitch curve design for EMNGP.
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Fig. 14. Transmission ratio of this steepest rotation pitch curves depicted in Fig. 13.
5. Algorithm and number example According to Section 3 and Section 4, the steepest rotation pitch curve with epicyclic constraint for driven noncircular gear in Section 4 is particular design of pitch curves with integral constraint when polar angle θ2b of pitch curve R2 is equal to 2π. Therefore, the flow chart depicted in Fig. 12 is a general design algorithm of steepest rotation pitch curves for noncircular gear pair. A number example of steepest rotation pitch curve design for EMNGP is given in the following description. The critical issue is the solution of undetermined parameters c1, c2, λ and the range of parameter variable μ according to the algorithm shown in Fig. 12. Assuming that the design requirements are listed in Table 9, undetermined parameters c1, c2, λ, and the range of parameter variable μ can be calculated by Eqs. (18)–(20), and the solutions are listed in Table 10. Moreover, the steepest rotation pitch curves and its transmission ratio function are depicted in Fig. 13 and Fig. 14, respectively. The solved parameter equations of pitch curves and transmission ratio function can be expressed as: 8 20ð2− cosμ Þ > > > < R1 ðμ Þ ¼ 2:5− cosμ ! pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 4 3 μ 10 21 μ 21 > > > arctan 3 tan − arctan þ 0:2555 tan θ ð μ Þ ¼ : 1 3 2 21 2 3 8 > > < R2 ðμ Þ ¼
10 2:5− Z μ cosμ cosμ > > θ dμ ð μ Þ ¼ : 2 −π2 2:5− cosμ 8 > > > < i12 ðμ Þ ¼
0:5 2− cosμ ! pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 4 3 μ 10 21 μ 21 > > > arctan 3 tan arctan tan θ ð μ Þ ¼ − þ 0:2555 : 1 3 2 21 2 3
ð39Þ
ð40Þ
ð41Þ
Because noncircular gear pair with the steepest rotation pitch curves can meet certain transmission requirements within the shortest time, the promising applications of this noncircular gear pair include rapid braking or start transmission device of automobile, packaging machinery, et al. 6. Conclusion The steepest rotation pitch curve with fixed boundaries can be obtained successfully in this paper. Based on the result, a general design algorithm of steepest rotation pitch curves for noncircular gear pair is proposed by resorting to calculus of variations. In particular, the designer can obtain a pair of conjugate noncircular gears with the steepest rotation characteristic by means of the proposed algorithm. This novel method builds a solid foundation for further researches on other particular geometrical and mechanical properties of noncircular gear pair or noncircular bevel gear pair.
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Acknowledgment The authors are grateful for financial support from National Natural Science Foundation of China [grant number: 51175067].
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