- Email: [email protected]
A variable-ratio line gear mechanism
Mechanism and Machine Theory 98 (2016) 151–163
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
A variable-ratio line gear mechanism Yang-zhi Chen ⁎, Huai Huang, Yueling Lv School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China
a r t i c l e
i n f o
Article history: Received 5 July 2015 Received in revised form 19 November 2015 Accepted 14 December 2015 Available online xxxx Keywords: Line gear (LG) Variable-ratio Contact curves
a b s t r a c t Based on the previous studies of line gear (LG), which is also named as space curve meshing wheel (SCMW) in the published papers, a variable-ratio LG mechanism is proposed in this article. The formulae of contact curves of a variable-ratio LG pair were derived. Then prototypes of variable-ratio LG pairs were designed by these formulae and manufactured by 3D printing. Afterwards, kinematics experiments were conducted by measuring the transmission ratio. The result indicates that a variable transmission ratio is feasible to a variable-ratio LG mechanism designed by these derived formulae, and the change of the transmission ratio is steady according to a prescribed kinematic rule. It is capable of being employed to devices on demand of variable-ratio mechanical transmission mechanism in compact structure. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Variable-ratio gear pair refers to the one which could produce a variable rotational motion between two connected drive shafts in a transmission. It is mainly applicable to meet the demand of special mechanical transmission, such as continuously variable transmissions (CVTs) in vehicle, oval gear flowmeters, etc. The most widely used gear that provides periodically variable transmission is non-circular gear [1], due to its advantage in offering special motion designed by its equation of motion. Noncircular gear first appeared in the beginning of the 20th century [2] and then this kind of gear was rapidly developed from the late 1940s to the early 1960s [3,4]. At present, non-circular gears used for variable-ratio transmission mainly include noncircular cylindrical gears and non-circular bevel gears. Non-circular cylindrical gears were studied more in depth than noncircular bevel gears, while the latter possess better applicability because they transmit between intersecting axes. In this research field, D. Mundo et al. [5] combined non-circular gears with a five-bar linkage in a mechanism which is capable of precisely moving a coupler point along a prescribed trajectory. One of the typical non-circular bevel gear is elliptical bevel gear [6], proposed by C. Lin. Orthogonal variable-ratio face gear, another novel gear transmission mechanism, was proposed by combining the advantages of non-circular cylindrical gear, non-circular bevel gear and face gear [7,8], thus making it easier to design and manufacture than non-circular gear. Additionally, a variable ratio could also be achieved by a planetary gear set [9], a continuously variable-ratio magnetic gear mechanism that consists of three concentric rotors, and a stator to accommodate a three-phase winding, whose transmission ratio between the output rotor and input rotor can be varied by controlling the speed of the control rotor [10]. Line gear (LG) [11], proposed by YZ Chen, is a novel gear mechanism based on space conjugate curve meshing theory instead of space surface meshing theory [12]. It is applicable to micro mechanical systems due to its small size, light weight and large transmission ratio. At present, line gear pairs could be applied to transmissions with perpendicular shafts [13], intersecting shafts [14] or skew shafts [15], and it is developed in the design equations, contact ratios, strength criterion, micro reducers and manufacturing technology [11].
⁎ Corresponding author. Tel./fax: +86 20 87114915. E-mail address: [email protected] (Y. Chen).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.12.005 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
152
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Nomenclature e1 ,e2 Overall averages of the dynamic measurement errors on the variable-ratio line tooth i Transmission ratio Demanded transmission ratios in a LG pair ia,ib Instantaneous transmission ratio at point k ik Distance from point op to axis z l1 Distance from point op to axis x l2 m Helix radius of driving contact curve n Pitch parameter of driving contact curve o − xyz,o1 − x1y1z1 Fixed cartesian coordinate and rotational cartesian coordinate of driving gears op − xpypzp,o2 − x2y2z2 Fixed cartesian coordinate and rotational cartesian coordinate of driven wheels p Pitch of driving contact curve t Scope parameter of helix contact curve Relative velocity v12 A, B ,C , D Coefficients of angular acceleration function of variable-ratio line tooth Unit normal vector of driving contact curve β(1) Angular velocity of driving line gear ω1 Angular velocity of driven line gear ω2 Rotation angle of driving line gear φ1 Rotation angle of driven line gear φ2 The begin angle and the end angle of the driving line gear φa,φb θ Included angle between angular velocity vectors Overall standard deviations of the dynamic measurement errors on the variable-ratio line tooth σ1,σ2
The previous study of LG mainly focuses on designing gear pairs that provide constant ratios. In this article, a design method for a novel variable-ratio LG mechanism [16] is proposed based on those previous theories on demand of variable-ratio mechanical transmission mechanism in compact structure. 2. The design equations of variable-ratio LG In the scale of millimeter-level mechanical system, a requirement of a periodically variable ratio could not always be satisfied by means of embedded systems or non-circular gears. Therefore the formulae of LGs which could provide a periodically variable ratio are developed in this section, which is based on the theory of LG. A variable-ratio LG pair also consists of a driving line gear and a driven gear. The driving line gear rotates with constant angular velocity, and then the driven line gear rotates with the required angular velocities and increases or decreases the angular velocity steadily. Therefore, the line teeth of the driven line gear can be divided into two parts, the constant ratio line teeth and the variable-ratio line teeth. Their design equations are deduced in the following section.
Fig. 1. Space-curve meshing coordinates.
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
153
2.1. The design fundamental equations of LG for producing a constant ratio line tooth On the basis of the theory of LG [12], the motion at the meshing point should satisfy the kinematical equation as Eq. (1) v12 β ¼ 0
ð1Þ
where v12 is the relative velocity at the meshing point between the driving and driven line teeth, and β is the unit normal vector of the driving contact curve at the meshing point. The equations of the driving and driven contact curves for a LG pair are denoted in the space-curve meshing coordinates, as shown in Fig. 1. The driving gear shaft z1 and the driven gear shaft z2 are fixed on axis z and zp, respectively. The transmission ratio is denoted as i, the distance from op to z axis as l1, the distance from op to x axis as l2, and the included angle between the angular speeds of the driving and driven wheels as θ, 0 ≤ θ ≤π. The contact curve equation along the driving line tooth is given as Eq. (2) in o1 − x1y1z1 [17] 8 ð1Þ > < xM ¼ m cos t ð1Þ y ¼ m sin t > : ðM1Þ zM ¼ nπ þ nt
π −π ≤ t ≤ − : 2
ð2Þ
From Eqs. (1) and (2), the contact curve equation of a constant-ratio driven line tooth can be deduced as following Eq. (3) in o2 − x2y2z2 [17] 8 ð2Þ > ¼ ½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ cos φ2 > > xM < ð2Þ yM ¼ −½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ sin φ2 > > zðM2Þ ¼ −ðm−l1 Þ sin θ−ðnπ þ nt−l2 Þ cos θ > : φ2 ¼ iφ1
ð3Þ
where m is the helix radius of the driving contact curve; n is the pitch parameter of the driving contact curve, denoting the pitch p ; t is the parameter indicating the scope of the driving contact curve. −π ≤t ≤ π2 means a quarter circle of the driving as p, n ¼ 2π contact curve. If t = − π, then the driving and driven line teeth begin to mesh; if t ¼ − π2, then the two line teeth begin to separate. The lengths of the driving and the driven contact curves are directly controlled by the scope of t on demand [17]. A contact curve is the essential factor of a line tooth of a LG. As shown in Fig. 2, The driving and the driven line teeth are produced along the contact vector direction and at the opposite sides of the meshing point. Cylindrical cantilever beam have been adopted to be the line teeth in previous papers [17]. 2.2. The derivation of design equations for variable-ratio line tooth of variable-ratio LG This section focuses on deriving the equations of the driven contact curve, which increases or decreases the transmission ratio steadily while gear driving, meeting the kinematic requirement. Namely, the angular acceleration of the driven line gear should not jump or fall dramatically when it is driven by the driving line gear.
Fig. 2. Contact curves and cylindrical line teeth [17].
154
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 3. The transmitting process when the transmission ratio changes from ia to ib.
As shown in Fig. 3, suppose that the driving line gear 1 is meshing with the driven line gear 2 from the begin location a to the end location b, where the transmission ratio changes from ia to ib. During this process, the jerk j and the angular acceleration α should both not change rapidly. 8 3 d φ2 > > > < j¼ dt 31 ; > d 2 φ2 > > :α ¼ 2 dt 1
φ t 1 ¼ 1 ; i∈½ia ; ib ; φ1 ∈½φa ; φb ω1
ð4Þ
where ia and ib are two demanded transmission ratios; φ1 and φ2 are rotation angles of the driving and driven line gears, respectively; φa and φb are the begin angle and the end angle of the driving line gear, respectively. It is known that the instantaneous transmission ratio is i, 1 ω2 dφ2 =dt dφ2 ¼ ¼ ; φ ∈½φa ; φb : ¼ i ω1 dφ1 =dt dφ1 1
ð5Þ
It is a function from the rotation angle φ1 of the driving line gear to the rotation angle φ2 of the driven line gear. Then, the rotation angle φ2, depending on the rotation angle φ1, is denoted as φ2(φ1). Therefore 1 dφ2 ðφ1 Þ 1 : ¼ ¼ i dφ1 iðφ1 Þ For actual application, iðφ1 Þ is expected to be a monotonic increasing function where 1 where i1a N i1b , if the transmission ratio changes from ia to ib. Namely, 8 1 > > d > > > 1 1 iðφ1 Þ > > ≥0; b < ia ib dφ1 ; φ1 ∈½φa ; φb 1 > > > d > > 1 1 iðφ1 Þ > > ≤0; N : ia ib dφ1
ð6Þ 1 ia
b i1b , or a monotonic decreasing function
ð7Þ
Eq. (7) can also be written as 8 2 d φ2 ðφ1 Þ 1 1 > > > ≥0; b < 2 ia ib dφ1 ; φ1 ∈½φa ; φb 2 > d φ ð φ Þ 1 1 > 2 1 > ≤0; N : ia ib dφ21
ð8Þ
where 1 1 ¼ ¼ ia iðφa Þ
dφ2 ðφ1 Þ dφ1 φ1 ¼φa
ð9Þ
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
1 1 ¼ ¼ ib iðφb Þ
155
dφ2 ðφ1 Þ : dφ1 φ1 ¼φb
ð10Þ
If a function from φ1 to φ2 satisfies Eq. (8) when the transmission ratio changes from one to another, it will offer a variableratio transmission on the geometry. However, it should obey the kinematic rules for a steady and reliable variable-ratio transmis2 ðφ1 Þ had better to be continuously differentiable in the domain φ1 ∈ [φa2,φb]. Meanwhile sion. Therefore the angular velocity v ¼ dφdφ 1 d3 φ2 ðφ1 Þ d φ2 ðφ1 Þ is equal to the value of the jerk j ¼ dφ3 is equal to 0 at φ1 =φa or φ1 = φb, and the value of the angular acceleration a ¼ dφ 2 1 1 0 at φ1 = φa or φ1 = φb. The functions of the jerk and the angular acceleration can be produced by any function that has the characteristics as mentioned above, such as trigonometric functions or quadratic functions. Here the jerk function is defined as Eq. (11), where A and B are undetermined. j¼
d3 φ2 ðφ1 Þ ¼ A sin B : dφ31
ð11Þ
In order to make the jerk a continuous function in the domain φ1 ∈ [φa, φb], where the values at the left and right end points are both equal to 0, and the values of the angular acceleration function are both equal to 0, B is defined as B¼2
ðφ1 −φa Þ π; φ1 ∈½φa ; φb : φb −φa
ð12Þ
Therefore the codomain of B is B ∈ [0, 2π]. Then the angular acceleration can be obtained as a¼
d2 φ2 ðφ1 Þ ¼− dφ21
A cos B þ C; φ1 ∈½φa ; φb : 2π φb −φa
ð13Þ
In addition, the angular velocity of the driven line gear, depending on φ1, is v¼
dφ2 ðφ1 Þ ¼ − dφ1
A
2π φb −φa
2 sin B þ Cφ1 þ D; φ1 ∈½φa ; φb
where A, C and D are undetermined. Coefficients A, C and D can be obtained from the boundary conditions, 2 ðφb Þ ¼ 1=ib . =ia and dφdφ 1 Therefore the angle φ2, depending on φ1, can be obtained as Eq. (14). φ2 ðφ1 Þ ¼
A
2π φb −φa
3 cos B þ
C 2 φ þ Dφ1 þ E; φ1 ∈½φa ; φb 2 1
d2 φ2 ðφb Þ dφ21
¼ 0,
dφ2 ðφa Þ dφ1
¼1
ð14Þ
where E can be defined as a begin rotation angle in meshing of the driven gear, like location a shown in Fig. 3, and it can be set as 0. Actually, when an angular velocity of the driving gear is given as ω1, Eq. (15) can be obtained by substituting φ1 = ω1t1 into Eq. (14). φ2 ðt 1 Þ ¼
3 Aðφb −φa Þ πðω1 t 1 −φa Þ C φa φb 2 cos þ ð t Þ þ Dω t ; t ∈ ; ω 1 1 1 ω1 ω1 φb −φa 2 1 1 8π3
ð15Þ
This is a common angular displacement function, depending on the time. Similarly, its jerk, depending on the time, is in the form of Eq. (16). ‴
φ2 ¼ A sin B; B ∈ ½0; 2π
ð16Þ
That means the jerk is continuous; the angular acceleration is continuous and the angular velocity is continuously differentiable. Generally, this will satisfy the requirement of a gradual changing transmission according to the law of motion predetermined.
156
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
The final result is then obtained by substituting the Eq. (14) into Eq. (3). The driven contact curve equation of a variable-ratio driven line gear tooth is 8 ð2Þ xM ¼ ½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ cos φ2 > > > > ð2Þ > y ¼ −½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ sin φ2 > > < M ð2Þ zM ¼ −ðm−l1 Þ sin θ−ðnπ þ nt−l2 Þ cos θ 3 > > Aðφb −φa Þ π ðω1 t 1 −φa Þ C > 2 > φ2 ¼ cos þ ðω1 t 1 Þ þ Dω1 t 1 > > 3 > φb −φa 2 8π : φ1 ¼ t þ π
ð17Þ
where the coefficients A, C and D can be obtained from the following conditions. d2 φ2 ðφb Þ ¼0 dφ21
ð18Þ
dφ2 ðφa Þ ¼ 1=ia dφ1
ð19Þ
dφ2 ðφb Þ ¼ 1=ib dφ1
ð20Þ
Therefore, if a circular helix, as expressed by Eq. (2), is adopted to be the driving contact curve, the driven contact curve equation for the part with a constant-ratio would be expressed by Eq. (3), and the driven contact curve equation for the part with a variable-ratio would be expressed from Eqs. (17) to (20). 3. A design example The necessary parameters are adopted as follows to build a model of a LG pair, m= 10, n = 8, l1 = 40, l2 =52, θ = 120°, ia = 5, ib = 6. Then the driving contact curve equation would be 8 ð1Þ > < xM ¼ 10 cos t ð1Þ yM ¼ 10 sin t : > : ð1Þ zM ¼ 8π þ 8t
ð21Þ
The domain of t is set as −π ≤t ≤− π2. Because the rotation angle of the driving line gear is φ1 = t + π, the codomain is 0 b φ1 b 2π, which means the driving line gear rotates π2 radians in a meshing process with a single driving line tooth. Then φb −φa ¼ π2. The driven contact curve equation for the part in ia = 5 is " # 8 pffiffiffi > tþπ 3 ð2Þ > > ð 8π þ 8t−52 Þ cos ¼ 15− x > M > 2 5 > > > " # pffiffiffi < tþπ : 3 ð2Þ y ¼ − 15− ð8π þ 8t−52Þ sin > > M 2 5 > > pffiffiffi > > > 30 3 1 > ð 2 Þ :z ¼ þ ð8π þ 8t−52Þ M 2 2
ð22Þ
The driven contact curve equation for the part in ib = 6 is " # 8 pffiffiffi > tþπ 3 ð2Þ > > xM ¼ 15− ð8π þ 8t−52Þ cos > > 6 2 > > > " # pffiffiffi < tþπ : 3 ð2Þ yM ¼ − 15− ð8π þ 8t−52Þ sin > > 6 2 > > pffiffiffi > > > > ð2Þ 30 3 1 : zM ¼ þ ð8π þ 8t−52Þ 2 2
ð23Þ
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
157
The driven contact curve equation for the part in which the transmission ratio changes from ia = 5 to ib = 6 is " # 8 pffiffiffi > 3 ð2Þ > > xM ¼ 15− ð8π þ 8t−52Þ cos φ2 > > > 2 > > " # pffiffiffi > > > > < yð2Þ ¼ − 15− 3ð8π þ 8t−52Þ sin φ 2 M 2 : > pffiffiffi > > > 30 3 1 > > zðM2Þ ¼ þ ð8π þ 8t−52Þ > > 2 2 > > > −1 1 1 > 2 :φ ¼ cos ð4t þ 4πÞ− ðt þ πÞ þ ðt þ πÞ 2 240π 30π 5
ð24Þ
The driven contact curve equation for the part in which the transmission ratio changes from ib = 6 to ia = 5 is " # 8 pffiffiffi > 3 ð2Þ > > > xM ¼ 15− ð8π þ 8t−52Þ cos φ2 > > 2 > > " # pffiffiffi > > > > < yð2Þ ¼ − 15− 3ð8π þ 8t−52Þ sin φ 2 M 2 : > pffiffiffi > > > 30 3 1 > > zðM2Þ ¼ þ ð8π þ 8t−52Þ > > 2 2 > > > 1 1 1 > 2 :φ ¼ cos ð4t þ 4πÞ þ ðt þ πÞ þ ðt þ πÞ þ π 2 240π 30π 6
ð25Þ
4. Prototypes of a variable-ratio LG pair designed The line teeth could be obtained by producing the volumes on the opposite sides of the driving and driven contact curves. The driven line gear is built based on the contact curve equations from Eqs. (21) to (25), as shown in Fig. 4. As presented, the line teeth on the left side are shorter and more than those on the right side. Additionally, there are line teeth that change the transmission ratio from 6 to 5 on the upside and 5 to 6 at the bottom. In transmission, only the precisions of the contact curves are required to ensure the accuracy of meshing [17], so the specific shapes of the line teeth could be designed on the actual application demand providing the sufficient high strength [17,18]. Fig. 5(a) and (b) present two types of line tooth shape with different cross-sections. The prototypes were manufactured by 3D printing using VisiJet plastic with ProJet 3510 HD &Plus machine which precision is 0.02 mm, as shown in Fig. 6. It is worth mentioning that the black objects on the surface of the line gear teeth are carbon dust which used to be lubricant. 5. A kinematics experiment of a variable-ratio LG pair To measure the transmission ratio of the designed variable-ratio LG pair, a kinematics experiment was conducted by use of a test rig with a 4-DOF positioning stage, as shown in Fig. 7.
Fig. 4. A driven line gear with transmission ratios ia =5 and ib =6.
158
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 5. Line tooth shapes with different cross-sections: (a) Line teeth with sector cross-sections; (b) line teeth attached to a conical surface.
The driving line gear rotates with an input angular velocity ω1, while the rotation angular of the driven line gear is measured by a rotary encoder with a resolution of 2000 pulses/rev. The measurement system records the interval time tp every np pulses. Therefore, the rotation angle of the driving line gear would be φ1 ¼ ω1 t p :
ð26Þ
And the rotation angle of the driven line gear would be φ2 ¼
2π ∑np : 2000
ð27Þ
Fig. 6. Prototypes of variable-ratio line gear mechanism manufactured by 3D printing: (a) Line teeth with sector cross-sections; (b) line teeth attached to a conical surface.
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
159
Fig. 7. Prototypes installed on the test rig.
Then a function from φ1 to φ2 would be built and a relationship plotting of the rotation angle φ2 − φ1 would be obtained. The line graph of transmission ratio could also be demonstrated by appropriate calculation. 5.1. An overview of transmission of a variable-ratio LG pair Firstly, the interval time tp was recorded every 3 pulses. The relationship between the angle φ2 and the time could be obtained according to Eq. (27) and tp. Therefore the relationship between an angle φ2 and an angle φ1 comes out according to Eq. (26). Mark the computed points and joined with straight line segments, as shown in Fig. 8. As the line graph presented, the driven line gear experienced approximate 1.5 period of rotation. At first the rotation angle φ2 increased steadily. Then the upward trend became slightly faster and finally got resumed. They had gradual transitions during the two changes. The instantaneous transmission ratio is i¼
dφ2 Δφ2 ≈ : dφ1 Δφ1
ð28Þ
According to Eqs. (26)–(28), the transmission ratio at the data point k is given: ik ¼
h
πnp =1000
i : ω1 t p ðk þ 1Þ−t p ðkÞ
ð29Þ
Therefore the line graph of transmission ratio can be obtained from Eqs. (29) and (26), as presented in the first graph in Fig. 9, where the transmission ratio mainly fluctuates around 15 or 16.
Fig. 8. Line graph of the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear.
160
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 9. The computed result of the instantaneous transmission ratio and its filtering result.
Therefore the line graph of transmission ratio can be obtained from Eqs. (29) and (26), as presented in the first graph in Fig. 9, where the transmission ratio small fluctuates around 15 or 16. In order to see the main trend of the instantaneous transmission ratio, median filter algorithm is employed in smoothing the line: the value at the point k is substituted for the median among the nearest several points. The filtering result is illustrated in Fig. 9, where the median is adopted among the nearest 6 points. Apparently, there are two separated transmission ratios in a cycle period.
5.2. The variable transmission on the variable-ratio line tooth The variable transmission parts are studied in the following content, where the transmission ratio decreases or increases. To see more details, the datasets of the transmission ratio decreases from 15 to 16 and from 16 to 15 received from the rotary encoder, which interval time was also recorded every 3 pulses, are separated out and analyzed. Three datasets from experiments presented in Fig. 10 show the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear, where the transmission ratio decreases from 15 to 16.
Fig. 10. Line graph of the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear (where the transmission ratio decreases from 1 1 5 to 6).
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 11. The computed result of the transmission ratio (where the transmission ratio decreases from
161
1 5
to 16).
However, there is a complicated fluctuation of the transmission ratio in a small scale caused by various errors, including test rig accuracy, installation error, 3D-printing error and vibration error. Therefore the instantaneous transmission ratio ik at point k is calculated via point k − n0 and point k + n0, instead of the adjacent points. Therefore, ik ¼
πnp =1000 h i : ω1 t p ðk þ n0 Þ−t p ðk−n0 Þ
ð30Þ
In this case, it also makes effective use of the data. The computed result of the transmission ratio is presented in Fig. 11, where n0 = 12. As illustrated, the transmission ratio declines steadily from 15 to 16, with the data points distributing around the designed curve of the transmission ratio function. Analogously, Fig. 12 illustrates the experiment result of the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear, where the transmission ratio increases from 16 to 15. Fig. 13 illustrates its computed result of the transmission ratio, where n0 = 8. 5.3. Dynamic measurement error analysis on the variable-ratio line tooth In Figs. 10 and 12, the data points distribute around the designed curves of the functions from φ1 to φ2. The two overall averages of the dynamic measurement errors computed by Matlab are e1 ¼
n 1X eðiÞ ¼ 0:0017 rad ¼ 0:0974 n i¼1
Fig. 12. The rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear (where the transmission ratio increases from
ð31Þ
1 6
to 15).
162
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 13. The computed result of the transmission ratio (where the transmission ratio increases from
e2 ¼
n 1X eðiÞ ¼ −0:0035 rad ¼ −0:201 : n i¼1
1 6
to 15).
ð32Þ
These two overall averages of the dynamic measurement errors indicate that the results are relatively good and they are caused by the system errors of the experiment rig and prototypes, such as vibration error, the positioning accuracy of test rig and installation error. In addition, the two overall standard deviations of the dynamic measurement errors are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X 2 ½eðiÞ−e ¼ 0:0015 rad ¼ 0:0859 σ1 ¼ t n−1 i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X 2 σ2 ¼ t ½eðiÞ−e ¼ 0:0015 rad ¼ 0:0859 : n−1 i¼1
ð33Þ
ð34Þ
These two standard deviations of the dynamic measurement errors indicate the amount of dispersion of the datasets of the two groups of experiment and their results indicate that the datasets are relatively concentrated. 6. Conclusions In this article, a design method for a novel variable-ratio LG pair is proposed for a periodically variable transmission ratio, which can be employed in some special requirements. The formulae of contact curves of a variable-ratio LG pair are derived and are applied to design the prototypes. The result of a kinematics experiment indicates that variable-ratio LGs produced from these formulae will be feasible. The transmission ratio of the variable-ratio line teeth will increase or decrease steadily while gear meshing, which means the angular acceleration will change from 0 to a certain value and then return to 0 again. Acknowledgment We thank the National Natural Science Foundation of China (items no. 51175180, no. 51575191) for supporting the research expressed in the paper. It is our honor to thank the reviewers and editors for their valuable criticisms and comments. References [1] [2] [3] [4]
D. Mundo, Geometric design of a planetary gear train with non-circular gears, Mech. Mach. Theory 41 (4) (2006) 456–472. S. Dunkerly, Mechanism, Longmans, Green, and Co., London, New York, 1910. F.L. Litivin, Noncircular Gear, National Machine Science and Tech-nology Press, Moscow, 1956 (in Russian). E. Ottaviano, D. Mundo, G.A. Danieli, M. Ceccarelli, Numerical and experimental analysis of non-circular gears and cam-follower systems as function generators, Mech. Mach. Theory 43 (8) (2008) 996–1008. [5] D. Mundo, G. Gatti, D. Dooner, Optimized five-bar linkages with noncircular gears for exact path generation, Mech. Mach. Theory 44 (2009) 751–760.
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
163
[6] C. Lin, Y.J. Hou, Q.L. Zeng, et al., The design and experiment of oval bevel gear, The International Conference on Power Transmission, Applied Mechanics and Materials, vol. 86 2011, pp. 297–300. [7] D.W. Liu, T.Z. Ren, X. Jin, Geometrical model and tooth analysis of undulating face gear, Mech. Mach. Theory 86 (2015) 140–155. [8] C. Lin, Y.J. Hou, H. Gong, G. Liu, Design and analysis of transmission mode for high-order deformed elliptic bevel gears, J. Mech. Eng. 47 (13) (2012) 131–139. [9] S. Sasaki, Toyota's newly developed hybrid power train, Proc. IEEE Int. Symp. Power Semicond, Devices ICs 1997, pp. 17–22. [10] C.C. Huang, M.C. Tsai, D.G. Dorrell, B.J. Lin, Development of a magnetic planetary gearbox, IEEE Trans. Magn. 44 (3) (2008) 403–412. [11] Y.Z. Chen, Line Gear, Science Press, Beijing, 2014. [12] Y.Z. Chen, G.Q. Xing, X.F. Peng, The space curve mesh equation and its kinematics experiment, 12th IFToMM World Congress, Besançon, 2007. [13] Y.Z. Chen, X.Y. Xiang, L. Luo, A corrected equation of space curve meshing, Mech. Mach. Theory 44 (7) (2009) 1348–1359. [14] Z. Chen, Y.Z. Chen, J. Ding, A generalized space curve meshing equation for arbitrary intersecting gear, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 227 (7) (2013) 1599–1607. [15] Y.Z. Chen, Y.L. Lv, J. Ding, Z. Chen, Fundamental design equations for space curve meshing skew gear mechanism, Mech. Mach. Theory 70 (2013) 175–188. [16] Y.Z. Chen, H. Huang, Y.L. Lv, A variable-ratio line gear mechanism, Patent Cooperation Treaty, PCT/CN2015/100289 12 (2015). [17] J. Ding, Y.Z. Chen, Y.L. Lv, Design of space-curve meshing-wheels with unequal tine radii, Strojniški vestnik 58 (11) (2012) 633–641. [18] Y.Z. Chen, S.K. Liang, J. Ding, The equal bending strength design of space curve meshing wheel, J. Mech. Des. ASME 136 (6) (2013) 061001.1–061001.10.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
A variable-ratio line gear mechanism Yang-zhi Chen ⁎, Huai Huang, Yueling Lv School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China
a r t i c l e
i n f o
Article history: Received 5 July 2015 Received in revised form 19 November 2015 Accepted 14 December 2015 Available online xxxx Keywords: Line gear (LG) Variable-ratio Contact curves
a b s t r a c t Based on the previous studies of line gear (LG), which is also named as space curve meshing wheel (SCMW) in the published papers, a variable-ratio LG mechanism is proposed in this article. The formulae of contact curves of a variable-ratio LG pair were derived. Then prototypes of variable-ratio LG pairs were designed by these formulae and manufactured by 3D printing. Afterwards, kinematics experiments were conducted by measuring the transmission ratio. The result indicates that a variable transmission ratio is feasible to a variable-ratio LG mechanism designed by these derived formulae, and the change of the transmission ratio is steady according to a prescribed kinematic rule. It is capable of being employed to devices on demand of variable-ratio mechanical transmission mechanism in compact structure. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Variable-ratio gear pair refers to the one which could produce a variable rotational motion between two connected drive shafts in a transmission. It is mainly applicable to meet the demand of special mechanical transmission, such as continuously variable transmissions (CVTs) in vehicle, oval gear flowmeters, etc. The most widely used gear that provides periodically variable transmission is non-circular gear [1], due to its advantage in offering special motion designed by its equation of motion. Noncircular gear first appeared in the beginning of the 20th century [2] and then this kind of gear was rapidly developed from the late 1940s to the early 1960s [3,4]. At present, non-circular gears used for variable-ratio transmission mainly include noncircular cylindrical gears and non-circular bevel gears. Non-circular cylindrical gears were studied more in depth than noncircular bevel gears, while the latter possess better applicability because they transmit between intersecting axes. In this research field, D. Mundo et al. [5] combined non-circular gears with a five-bar linkage in a mechanism which is capable of precisely moving a coupler point along a prescribed trajectory. One of the typical non-circular bevel gear is elliptical bevel gear [6], proposed by C. Lin. Orthogonal variable-ratio face gear, another novel gear transmission mechanism, was proposed by combining the advantages of non-circular cylindrical gear, non-circular bevel gear and face gear [7,8], thus making it easier to design and manufacture than non-circular gear. Additionally, a variable ratio could also be achieved by a planetary gear set [9], a continuously variable-ratio magnetic gear mechanism that consists of three concentric rotors, and a stator to accommodate a three-phase winding, whose transmission ratio between the output rotor and input rotor can be varied by controlling the speed of the control rotor [10]. Line gear (LG) [11], proposed by YZ Chen, is a novel gear mechanism based on space conjugate curve meshing theory instead of space surface meshing theory [12]. It is applicable to micro mechanical systems due to its small size, light weight and large transmission ratio. At present, line gear pairs could be applied to transmissions with perpendicular shafts [13], intersecting shafts [14] or skew shafts [15], and it is developed in the design equations, contact ratios, strength criterion, micro reducers and manufacturing technology [11].
⁎ Corresponding author. Tel./fax: +86 20 87114915. E-mail address: [email protected] (Y. Chen).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.12.005 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
152
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Nomenclature e1 ,e2 Overall averages of the dynamic measurement errors on the variable-ratio line tooth i Transmission ratio Demanded transmission ratios in a LG pair ia,ib Instantaneous transmission ratio at point k ik Distance from point op to axis z l1 Distance from point op to axis x l2 m Helix radius of driving contact curve n Pitch parameter of driving contact curve o − xyz,o1 − x1y1z1 Fixed cartesian coordinate and rotational cartesian coordinate of driving gears op − xpypzp,o2 − x2y2z2 Fixed cartesian coordinate and rotational cartesian coordinate of driven wheels p Pitch of driving contact curve t Scope parameter of helix contact curve Relative velocity v12 A, B ,C , D Coefficients of angular acceleration function of variable-ratio line tooth Unit normal vector of driving contact curve β(1) Angular velocity of driving line gear ω1 Angular velocity of driven line gear ω2 Rotation angle of driving line gear φ1 Rotation angle of driven line gear φ2 The begin angle and the end angle of the driving line gear φa,φb θ Included angle between angular velocity vectors Overall standard deviations of the dynamic measurement errors on the variable-ratio line tooth σ1,σ2
The previous study of LG mainly focuses on designing gear pairs that provide constant ratios. In this article, a design method for a novel variable-ratio LG mechanism [16] is proposed based on those previous theories on demand of variable-ratio mechanical transmission mechanism in compact structure. 2. The design equations of variable-ratio LG In the scale of millimeter-level mechanical system, a requirement of a periodically variable ratio could not always be satisfied by means of embedded systems or non-circular gears. Therefore the formulae of LGs which could provide a periodically variable ratio are developed in this section, which is based on the theory of LG. A variable-ratio LG pair also consists of a driving line gear and a driven gear. The driving line gear rotates with constant angular velocity, and then the driven line gear rotates with the required angular velocities and increases or decreases the angular velocity steadily. Therefore, the line teeth of the driven line gear can be divided into two parts, the constant ratio line teeth and the variable-ratio line teeth. Their design equations are deduced in the following section.
Fig. 1. Space-curve meshing coordinates.
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
153
2.1. The design fundamental equations of LG for producing a constant ratio line tooth On the basis of the theory of LG [12], the motion at the meshing point should satisfy the kinematical equation as Eq. (1) v12 β ¼ 0
ð1Þ
where v12 is the relative velocity at the meshing point between the driving and driven line teeth, and β is the unit normal vector of the driving contact curve at the meshing point. The equations of the driving and driven contact curves for a LG pair are denoted in the space-curve meshing coordinates, as shown in Fig. 1. The driving gear shaft z1 and the driven gear shaft z2 are fixed on axis z and zp, respectively. The transmission ratio is denoted as i, the distance from op to z axis as l1, the distance from op to x axis as l2, and the included angle between the angular speeds of the driving and driven wheels as θ, 0 ≤ θ ≤π. The contact curve equation along the driving line tooth is given as Eq. (2) in o1 − x1y1z1 [17] 8 ð1Þ > < xM ¼ m cos t ð1Þ y ¼ m sin t > : ðM1Þ zM ¼ nπ þ nt
π −π ≤ t ≤ − : 2
ð2Þ
From Eqs. (1) and (2), the contact curve equation of a constant-ratio driven line tooth can be deduced as following Eq. (3) in o2 − x2y2z2 [17] 8 ð2Þ > ¼ ½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ cos φ2 > > xM < ð2Þ yM ¼ −½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ sin φ2 > > zðM2Þ ¼ −ðm−l1 Þ sin θ−ðnπ þ nt−l2 Þ cos θ > : φ2 ¼ iφ1
ð3Þ
where m is the helix radius of the driving contact curve; n is the pitch parameter of the driving contact curve, denoting the pitch p ; t is the parameter indicating the scope of the driving contact curve. −π ≤t ≤ π2 means a quarter circle of the driving as p, n ¼ 2π contact curve. If t = − π, then the driving and driven line teeth begin to mesh; if t ¼ − π2, then the two line teeth begin to separate. The lengths of the driving and the driven contact curves are directly controlled by the scope of t on demand [17]. A contact curve is the essential factor of a line tooth of a LG. As shown in Fig. 2, The driving and the driven line teeth are produced along the contact vector direction and at the opposite sides of the meshing point. Cylindrical cantilever beam have been adopted to be the line teeth in previous papers [17]. 2.2. The derivation of design equations for variable-ratio line tooth of variable-ratio LG This section focuses on deriving the equations of the driven contact curve, which increases or decreases the transmission ratio steadily while gear driving, meeting the kinematic requirement. Namely, the angular acceleration of the driven line gear should not jump or fall dramatically when it is driven by the driving line gear.
Fig. 2. Contact curves and cylindrical line teeth [17].
154
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 3. The transmitting process when the transmission ratio changes from ia to ib.
As shown in Fig. 3, suppose that the driving line gear 1 is meshing with the driven line gear 2 from the begin location a to the end location b, where the transmission ratio changes from ia to ib. During this process, the jerk j and the angular acceleration α should both not change rapidly. 8 3 d φ2 > > > < j¼ dt 31 ; > d 2 φ2 > > :α ¼ 2 dt 1
φ t 1 ¼ 1 ; i∈½ia ; ib ; φ1 ∈½φa ; φb ω1
ð4Þ
where ia and ib are two demanded transmission ratios; φ1 and φ2 are rotation angles of the driving and driven line gears, respectively; φa and φb are the begin angle and the end angle of the driving line gear, respectively. It is known that the instantaneous transmission ratio is i, 1 ω2 dφ2 =dt dφ2 ¼ ¼ ; φ ∈½φa ; φb : ¼ i ω1 dφ1 =dt dφ1 1
ð5Þ
It is a function from the rotation angle φ1 of the driving line gear to the rotation angle φ2 of the driven line gear. Then, the rotation angle φ2, depending on the rotation angle φ1, is denoted as φ2(φ1). Therefore 1 dφ2 ðφ1 Þ 1 : ¼ ¼ i dφ1 iðφ1 Þ For actual application, iðφ1 Þ is expected to be a monotonic increasing function where 1 where i1a N i1b , if the transmission ratio changes from ia to ib. Namely, 8 1 > > d > > > 1 1 iðφ1 Þ > > ≥0; b < ia ib dφ1 ; φ1 ∈½φa ; φb 1 > > > d > > 1 1 iðφ1 Þ > > ≤0; N : ia ib dφ1
ð6Þ 1 ia
b i1b , or a monotonic decreasing function
ð7Þ
Eq. (7) can also be written as 8 2 d φ2 ðφ1 Þ 1 1 > > > ≥0; b < 2 ia ib dφ1 ; φ1 ∈½φa ; φb 2 > d φ ð φ Þ 1 1 > 2 1 > ≤0; N : ia ib dφ21
ð8Þ
where 1 1 ¼ ¼ ia iðφa Þ
dφ2 ðφ1 Þ dφ1 φ1 ¼φa
ð9Þ
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
1 1 ¼ ¼ ib iðφb Þ
155
dφ2 ðφ1 Þ : dφ1 φ1 ¼φb
ð10Þ
If a function from φ1 to φ2 satisfies Eq. (8) when the transmission ratio changes from one to another, it will offer a variableratio transmission on the geometry. However, it should obey the kinematic rules for a steady and reliable variable-ratio transmis2 ðφ1 Þ had better to be continuously differentiable in the domain φ1 ∈ [φa2,φb]. Meanwhile sion. Therefore the angular velocity v ¼ dφdφ 1 d3 φ2 ðφ1 Þ d φ2 ðφ1 Þ is equal to the value of the jerk j ¼ dφ3 is equal to 0 at φ1 =φa or φ1 = φb, and the value of the angular acceleration a ¼ dφ 2 1 1 0 at φ1 = φa or φ1 = φb. The functions of the jerk and the angular acceleration can be produced by any function that has the characteristics as mentioned above, such as trigonometric functions or quadratic functions. Here the jerk function is defined as Eq. (11), where A and B are undetermined. j¼
d3 φ2 ðφ1 Þ ¼ A sin B : dφ31
ð11Þ
In order to make the jerk a continuous function in the domain φ1 ∈ [φa, φb], where the values at the left and right end points are both equal to 0, and the values of the angular acceleration function are both equal to 0, B is defined as B¼2
ðφ1 −φa Þ π; φ1 ∈½φa ; φb : φb −φa
ð12Þ
Therefore the codomain of B is B ∈ [0, 2π]. Then the angular acceleration can be obtained as a¼
d2 φ2 ðφ1 Þ ¼− dφ21
A cos B þ C; φ1 ∈½φa ; φb : 2π φb −φa
ð13Þ
In addition, the angular velocity of the driven line gear, depending on φ1, is v¼
dφ2 ðφ1 Þ ¼ − dφ1
A
2π φb −φa
2 sin B þ Cφ1 þ D; φ1 ∈½φa ; φb
where A, C and D are undetermined. Coefficients A, C and D can be obtained from the boundary conditions, 2 ðφb Þ ¼ 1=ib . =ia and dφdφ 1 Therefore the angle φ2, depending on φ1, can be obtained as Eq. (14). φ2 ðφ1 Þ ¼
A
2π φb −φa
3 cos B þ
C 2 φ þ Dφ1 þ E; φ1 ∈½φa ; φb 2 1
d2 φ2 ðφb Þ dφ21
¼ 0,
dφ2 ðφa Þ dφ1
¼1
ð14Þ
where E can be defined as a begin rotation angle in meshing of the driven gear, like location a shown in Fig. 3, and it can be set as 0. Actually, when an angular velocity of the driving gear is given as ω1, Eq. (15) can be obtained by substituting φ1 = ω1t1 into Eq. (14). φ2 ðt 1 Þ ¼
3 Aðφb −φa Þ πðω1 t 1 −φa Þ C φa φb 2 cos þ ð t Þ þ Dω t ; t ∈ ; ω 1 1 1 ω1 ω1 φb −φa 2 1 1 8π3
ð15Þ
This is a common angular displacement function, depending on the time. Similarly, its jerk, depending on the time, is in the form of Eq. (16). ‴
φ2 ¼ A sin B; B ∈ ½0; 2π
ð16Þ
That means the jerk is continuous; the angular acceleration is continuous and the angular velocity is continuously differentiable. Generally, this will satisfy the requirement of a gradual changing transmission according to the law of motion predetermined.
156
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
The final result is then obtained by substituting the Eq. (14) into Eq. (3). The driven contact curve equation of a variable-ratio driven line gear tooth is 8 ð2Þ xM ¼ ½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ cos φ2 > > > > ð2Þ > y ¼ −½ðm−l1 Þ cos θ−ðnπ þ nt−l2 Þ sin θ sin φ2 > > < M ð2Þ zM ¼ −ðm−l1 Þ sin θ−ðnπ þ nt−l2 Þ cos θ 3 > > Aðφb −φa Þ π ðω1 t 1 −φa Þ C > 2 > φ2 ¼ cos þ ðω1 t 1 Þ þ Dω1 t 1 > > 3 > φb −φa 2 8π : φ1 ¼ t þ π
ð17Þ
where the coefficients A, C and D can be obtained from the following conditions. d2 φ2 ðφb Þ ¼0 dφ21
ð18Þ
dφ2 ðφa Þ ¼ 1=ia dφ1
ð19Þ
dφ2 ðφb Þ ¼ 1=ib dφ1
ð20Þ
Therefore, if a circular helix, as expressed by Eq. (2), is adopted to be the driving contact curve, the driven contact curve equation for the part with a constant-ratio would be expressed by Eq. (3), and the driven contact curve equation for the part with a variable-ratio would be expressed from Eqs. (17) to (20). 3. A design example The necessary parameters are adopted as follows to build a model of a LG pair, m= 10, n = 8, l1 = 40, l2 =52, θ = 120°, ia = 5, ib = 6. Then the driving contact curve equation would be 8 ð1Þ > < xM ¼ 10 cos t ð1Þ yM ¼ 10 sin t : > : ð1Þ zM ¼ 8π þ 8t
ð21Þ
The domain of t is set as −π ≤t ≤− π2. Because the rotation angle of the driving line gear is φ1 = t + π, the codomain is 0 b φ1 b 2π, which means the driving line gear rotates π2 radians in a meshing process with a single driving line tooth. Then φb −φa ¼ π2. The driven contact curve equation for the part in ia = 5 is " # 8 pffiffiffi > tþπ 3 ð2Þ > > ð 8π þ 8t−52 Þ cos ¼ 15− x > M > 2 5 > > > " # pffiffiffi < tþπ : 3 ð2Þ y ¼ − 15− ð8π þ 8t−52Þ sin > > M 2 5 > > pffiffiffi > > > 30 3 1 > ð 2 Þ :z ¼ þ ð8π þ 8t−52Þ M 2 2
ð22Þ
The driven contact curve equation for the part in ib = 6 is " # 8 pffiffiffi > tþπ 3 ð2Þ > > xM ¼ 15− ð8π þ 8t−52Þ cos > > 6 2 > > > " # pffiffiffi < tþπ : 3 ð2Þ yM ¼ − 15− ð8π þ 8t−52Þ sin > > 6 2 > > pffiffiffi > > > > ð2Þ 30 3 1 : zM ¼ þ ð8π þ 8t−52Þ 2 2
ð23Þ
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
157
The driven contact curve equation for the part in which the transmission ratio changes from ia = 5 to ib = 6 is " # 8 pffiffiffi > 3 ð2Þ > > xM ¼ 15− ð8π þ 8t−52Þ cos φ2 > > > 2 > > " # pffiffiffi > > > > < yð2Þ ¼ − 15− 3ð8π þ 8t−52Þ sin φ 2 M 2 : > pffiffiffi > > > 30 3 1 > > zðM2Þ ¼ þ ð8π þ 8t−52Þ > > 2 2 > > > −1 1 1 > 2 :φ ¼ cos ð4t þ 4πÞ− ðt þ πÞ þ ðt þ πÞ 2 240π 30π 5
ð24Þ
The driven contact curve equation for the part in which the transmission ratio changes from ib = 6 to ia = 5 is " # 8 pffiffiffi > 3 ð2Þ > > > xM ¼ 15− ð8π þ 8t−52Þ cos φ2 > > 2 > > " # pffiffiffi > > > > < yð2Þ ¼ − 15− 3ð8π þ 8t−52Þ sin φ 2 M 2 : > pffiffiffi > > > 30 3 1 > > zðM2Þ ¼ þ ð8π þ 8t−52Þ > > 2 2 > > > 1 1 1 > 2 :φ ¼ cos ð4t þ 4πÞ þ ðt þ πÞ þ ðt þ πÞ þ π 2 240π 30π 6
ð25Þ
4. Prototypes of a variable-ratio LG pair designed The line teeth could be obtained by producing the volumes on the opposite sides of the driving and driven contact curves. The driven line gear is built based on the contact curve equations from Eqs. (21) to (25), as shown in Fig. 4. As presented, the line teeth on the left side are shorter and more than those on the right side. Additionally, there are line teeth that change the transmission ratio from 6 to 5 on the upside and 5 to 6 at the bottom. In transmission, only the precisions of the contact curves are required to ensure the accuracy of meshing [17], so the specific shapes of the line teeth could be designed on the actual application demand providing the sufficient high strength [17,18]. Fig. 5(a) and (b) present two types of line tooth shape with different cross-sections. The prototypes were manufactured by 3D printing using VisiJet plastic with ProJet 3510 HD &Plus machine which precision is 0.02 mm, as shown in Fig. 6. It is worth mentioning that the black objects on the surface of the line gear teeth are carbon dust which used to be lubricant. 5. A kinematics experiment of a variable-ratio LG pair To measure the transmission ratio of the designed variable-ratio LG pair, a kinematics experiment was conducted by use of a test rig with a 4-DOF positioning stage, as shown in Fig. 7.
Fig. 4. A driven line gear with transmission ratios ia =5 and ib =6.
158
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 5. Line tooth shapes with different cross-sections: (a) Line teeth with sector cross-sections; (b) line teeth attached to a conical surface.
The driving line gear rotates with an input angular velocity ω1, while the rotation angular of the driven line gear is measured by a rotary encoder with a resolution of 2000 pulses/rev. The measurement system records the interval time tp every np pulses. Therefore, the rotation angle of the driving line gear would be φ1 ¼ ω1 t p :
ð26Þ
And the rotation angle of the driven line gear would be φ2 ¼
2π ∑np : 2000
ð27Þ
Fig. 6. Prototypes of variable-ratio line gear mechanism manufactured by 3D printing: (a) Line teeth with sector cross-sections; (b) line teeth attached to a conical surface.
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
159
Fig. 7. Prototypes installed on the test rig.
Then a function from φ1 to φ2 would be built and a relationship plotting of the rotation angle φ2 − φ1 would be obtained. The line graph of transmission ratio could also be demonstrated by appropriate calculation. 5.1. An overview of transmission of a variable-ratio LG pair Firstly, the interval time tp was recorded every 3 pulses. The relationship between the angle φ2 and the time could be obtained according to Eq. (27) and tp. Therefore the relationship between an angle φ2 and an angle φ1 comes out according to Eq. (26). Mark the computed points and joined with straight line segments, as shown in Fig. 8. As the line graph presented, the driven line gear experienced approximate 1.5 period of rotation. At first the rotation angle φ2 increased steadily. Then the upward trend became slightly faster and finally got resumed. They had gradual transitions during the two changes. The instantaneous transmission ratio is i¼
dφ2 Δφ2 ≈ : dφ1 Δφ1
ð28Þ
According to Eqs. (26)–(28), the transmission ratio at the data point k is given: ik ¼
h
πnp =1000
i : ω1 t p ðk þ 1Þ−t p ðkÞ
ð29Þ
Therefore the line graph of transmission ratio can be obtained from Eqs. (29) and (26), as presented in the first graph in Fig. 9, where the transmission ratio mainly fluctuates around 15 or 16.
Fig. 8. Line graph of the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear.
160
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 9. The computed result of the instantaneous transmission ratio and its filtering result.
Therefore the line graph of transmission ratio can be obtained from Eqs. (29) and (26), as presented in the first graph in Fig. 9, where the transmission ratio small fluctuates around 15 or 16. In order to see the main trend of the instantaneous transmission ratio, median filter algorithm is employed in smoothing the line: the value at the point k is substituted for the median among the nearest several points. The filtering result is illustrated in Fig. 9, where the median is adopted among the nearest 6 points. Apparently, there are two separated transmission ratios in a cycle period.
5.2. The variable transmission on the variable-ratio line tooth The variable transmission parts are studied in the following content, where the transmission ratio decreases or increases. To see more details, the datasets of the transmission ratio decreases from 15 to 16 and from 16 to 15 received from the rotary encoder, which interval time was also recorded every 3 pulses, are separated out and analyzed. Three datasets from experiments presented in Fig. 10 show the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear, where the transmission ratio decreases from 15 to 16.
Fig. 10. Line graph of the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear (where the transmission ratio decreases from 1 1 5 to 6).
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 11. The computed result of the transmission ratio (where the transmission ratio decreases from
161
1 5
to 16).
However, there is a complicated fluctuation of the transmission ratio in a small scale caused by various errors, including test rig accuracy, installation error, 3D-printing error and vibration error. Therefore the instantaneous transmission ratio ik at point k is calculated via point k − n0 and point k + n0, instead of the adjacent points. Therefore, ik ¼
πnp =1000 h i : ω1 t p ðk þ n0 Þ−t p ðk−n0 Þ
ð30Þ
In this case, it also makes effective use of the data. The computed result of the transmission ratio is presented in Fig. 11, where n0 = 12. As illustrated, the transmission ratio declines steadily from 15 to 16, with the data points distributing around the designed curve of the transmission ratio function. Analogously, Fig. 12 illustrates the experiment result of the rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear, where the transmission ratio increases from 16 to 15. Fig. 13 illustrates its computed result of the transmission ratio, where n0 = 8. 5.3. Dynamic measurement error analysis on the variable-ratio line tooth In Figs. 10 and 12, the data points distribute around the designed curves of the functions from φ1 to φ2. The two overall averages of the dynamic measurement errors computed by Matlab are e1 ¼
n 1X eðiÞ ¼ 0:0017 rad ¼ 0:0974 n i¼1
Fig. 12. The rotation angle φ2 of the driven line gear versus the rotation angle φ1 of the driving line gear (where the transmission ratio increases from
ð31Þ
1 6
to 15).
162
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
Fig. 13. The computed result of the transmission ratio (where the transmission ratio increases from
e2 ¼
n 1X eðiÞ ¼ −0:0035 rad ¼ −0:201 : n i¼1
1 6
to 15).
ð32Þ
These two overall averages of the dynamic measurement errors indicate that the results are relatively good and they are caused by the system errors of the experiment rig and prototypes, such as vibration error, the positioning accuracy of test rig and installation error. In addition, the two overall standard deviations of the dynamic measurement errors are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X 2 ½eðiÞ−e ¼ 0:0015 rad ¼ 0:0859 σ1 ¼ t n−1 i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X 2 σ2 ¼ t ½eðiÞ−e ¼ 0:0015 rad ¼ 0:0859 : n−1 i¼1
ð33Þ
ð34Þ
These two standard deviations of the dynamic measurement errors indicate the amount of dispersion of the datasets of the two groups of experiment and their results indicate that the datasets are relatively concentrated. 6. Conclusions In this article, a design method for a novel variable-ratio LG pair is proposed for a periodically variable transmission ratio, which can be employed in some special requirements. The formulae of contact curves of a variable-ratio LG pair are derived and are applied to design the prototypes. The result of a kinematics experiment indicates that variable-ratio LGs produced from these formulae will be feasible. The transmission ratio of the variable-ratio line teeth will increase or decrease steadily while gear meshing, which means the angular acceleration will change from 0 to a certain value and then return to 0 again. Acknowledgment We thank the National Natural Science Foundation of China (items no. 51175180, no. 51575191) for supporting the research expressed in the paper. It is our honor to thank the reviewers and editors for their valuable criticisms and comments. References [1] [2] [3] [4]
D. Mundo, Geometric design of a planetary gear train with non-circular gears, Mech. Mach. Theory 41 (4) (2006) 456–472. S. Dunkerly, Mechanism, Longmans, Green, and Co., London, New York, 1910. F.L. Litivin, Noncircular Gear, National Machine Science and Tech-nology Press, Moscow, 1956 (in Russian). E. Ottaviano, D. Mundo, G.A. Danieli, M. Ceccarelli, Numerical and experimental analysis of non-circular gears and cam-follower systems as function generators, Mech. Mach. Theory 43 (8) (2008) 996–1008. [5] D. Mundo, G. Gatti, D. Dooner, Optimized five-bar linkages with noncircular gears for exact path generation, Mech. Mach. Theory 44 (2009) 751–760.
Y. Chen et al. / Mechanism and Machine Theory 98 (2016) 151–163
163
[6] C. Lin, Y.J. Hou, Q.L. Zeng, et al., The design and experiment of oval bevel gear, The International Conference on Power Transmission, Applied Mechanics and Materials, vol. 86 2011, pp. 297–300. [7] D.W. Liu, T.Z. Ren, X. Jin, Geometrical model and tooth analysis of undulating face gear, Mech. Mach. Theory 86 (2015) 140–155. [8] C. Lin, Y.J. Hou, H. Gong, G. Liu, Design and analysis of transmission mode for high-order deformed elliptic bevel gears, J. Mech. Eng. 47 (13) (2012) 131–139. [9] S. Sasaki, Toyota's newly developed hybrid power train, Proc. IEEE Int. Symp. Power Semicond, Devices ICs 1997, pp. 17–22. [10] C.C. Huang, M.C. Tsai, D.G. Dorrell, B.J. Lin, Development of a magnetic planetary gearbox, IEEE Trans. Magn. 44 (3) (2008) 403–412. [11] Y.Z. Chen, Line Gear, Science Press, Beijing, 2014. [12] Y.Z. Chen, G.Q. Xing, X.F. Peng, The space curve mesh equation and its kinematics experiment, 12th IFToMM World Congress, Besançon, 2007. [13] Y.Z. Chen, X.Y. Xiang, L. Luo, A corrected equation of space curve meshing, Mech. Mach. Theory 44 (7) (2009) 1348–1359. [14] Z. Chen, Y.Z. Chen, J. Ding, A generalized space curve meshing equation for arbitrary intersecting gear, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 227 (7) (2013) 1599–1607. [15] Y.Z. Chen, Y.L. Lv, J. Ding, Z. Chen, Fundamental design equations for space curve meshing skew gear mechanism, Mech. Mach. Theory 70 (2013) 175–188. [16] Y.Z. Chen, H. Huang, Y.L. Lv, A variable-ratio line gear mechanism, Patent Cooperation Treaty, PCT/CN2015/100289 12 (2015). [17] J. Ding, Y.Z. Chen, Y.L. Lv, Design of space-curve meshing-wheels with unequal tine radii, Strojniški vestnik 58 (11) (2012) 633–641. [18] Y.Z. Chen, S.K. Liang, J. Ding, The equal bending strength design of space curve meshing wheel, J. Mech. Des. ASME 136 (6) (2013) 061001.1–061001.10.