An innovative speed reduction mechanism with self-adaptability to variable transmission angles

Mechanism and Machine Theory 48 (2012) 41–51

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An innovative speed reduction mechanism with self-adaptability to variable transmission angles Xin-Bo Chen a, Bin Wang a,⁎, Mei-fang Chen b, Yan Li b a b

School of Automotive Engineering, Tongji University, Shanghai 201804, PR China School of Mechanical Engineering, Tongji University, Shanghai 201804, PR China

a r t i c l e

i n f o

Article history: Received 26 September 2010 Received in revised form 27 July 2011 Accepted 20 September 2011 Available online 28 October 2011 Keywords: Speed reduction mechanism Self-adaptability Planetary transmission Spatial linkage mechanism Kinematics

a b s t r a c t An innovative speed reduction mechanism featuring self-adaptability to variable transmission angles is being proposed, the purpose of which is to be able to maintain expected reduction ratio by adapting itself to the variable intersection angles between input shaft and output shaft. The synchronous planetary pulley-based mechanism is combined with the features of RCRCR transmission structure and K–H planetary transmission mechanism. Structure analysis and force analysis of the mechanism were demonstrated to explain the working principle and kinematic feasibility of the mechanism. The NX model of the mechanism was drawn and simulated to prove the realistic kinematic results of mechanism were the same as the theoretic one. The mechanism significantly lowers the accuracy requirements of relative position between the prime mover and the working machine while shortening the transmission chain at the same time. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Speed reducers are widely used in various applications for speed and torque conversion purposes [1]. A common speed reducer can realize speed reduction between input shaft and output shaft which occupy fixed relative positions. For example, studies on the cylindrical gear mechanism between two parallel axes were published by Yang et al. [2], Wang et al. [3], and Yang et al. [4]; cone gear mechanism between two intersection shafts was researched by Tang et al. [5], Fang and Liu [6], and Wang et al. [7,8]; research on worm mechanism or screw gear mechanism between two skewed shafts was carried out by Bishop et al. [9], Wan et al. [10], Jiang and Li [11], Wang et al. [12], Zhou and Zheng [13], and Mundo and Yan [14]; pinwheel joint that allows the motion to be transmitted between parallel, non-coincident axes with high values of the center distance and non-constant transmission ratio was researched by Bagnoli et al. [15], and devised by Fantoni [16]. When the reducer is necessary to realize speed reduction transmission between two intersecting shafts, of which the spatial angle is variable, usually a universal drive shaft is installed between two shafts to help the transmission mechanism adapt to the changes of the spatial angle between input and output shafts. However this method will definitely lengthen the size of the transmission chain and complicate the structure of shaft bearings and linkages. Though the patent [16] can realize transmission between two intersecting shafts without a universal drive shaft- and it likely also achieves reduction function by changing the circumference of one pin-wheel- there exists a large amount of sliding friction between pins, so transmission efficiency of the pinwheel is worthy of study. The transmission angle is an important criterion for the design of mechanisms by means of which the quality of motion transmission in a mechanism can be judged at its design stage [17]. A new type of speed reduction mechanism is introduced in this paper which features self-adaptability and achieves speed reduction transmission in addition to option of changing the angle between input shaft and output shaft, as indicated in Fig. 1. This mechanism has been manufactured and granted an invention patent by the ⁎ Corresponding author at: Tongji University, Detailed permanent address: Room 313, Clean Energy Automotive Engineering Center, School of Automotive Engineering, Tongji University, No. 4800 Cao'an Highway, Shanghai, 201804, PR China. Tel.: +86 15005106262; fax: +86 21 69589842. E-mail address: [email protected] (B. Wang). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.09.004

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Nomenclature C F1, F2 K H i M0 M n P0 R T vi ωi ωH θi θ˙ i xi(yi)zi α l

Cylindric pair of the spatial linkage mechanism Tensile force of stretched side and slack side (N) Center gear Planet carrier Transmission ratio Driving torque of driving link 1 (N·m) Output resistance torque of synchronous center pulley (N·m) Speed of synchronous pulley (r/s) Input power (W) Revolute pair of the spatial linkage mechanism Torque (N·m) Linear velocity of component i(i = 0, 1, 2, 3, 4)(m/s) Angular velocity of componenti(i = 0, 1, 2, 3, 4) (rad/s) Angular velocity of planet carrier (rad/s) Rotating angle of the corresponding component i(i = 0, 1, 2, 3, 4) (deg) Derivate of θi, wherei(i = 0, 1, 2, 3, 4) Hand rectangular coordinate system built on each component i(i = 0, 1, 2, 3, 4) Angle of contact (deg) Distance (m)

People's Republic of China [18]. As a universal joint, this new type of speed reduction mechanism directly connects the prime mover shaft to the working machine shaft, as well as freeing the transmission ratio from the angle changing influence between the two shafts. This allows for a greater decrease in the positioning accuracy requirement between the prime mover and the working machine, as well as allowing for angle changing during the working period. Unlike the transmission ratio of a universal joint which is equal to 1.0 on average, the speed reduction mechanism itself has a speed-reducing transmission ratio. So the transmission chain between the prime mover and the working machine can be designed to be shorter and more compact. In this paper we intend to demonstrate the performance of the above mentioned speed reduction mechanism which has been proven by theoretical analysis and kinematical simulation. In addition, the prototype of the mechanism has been successfully manufactured. 2. Theoretical analysis 2.1. Structural analysis Planetary gear trains offer the possibility of achieving a given speed ratio with a smaller weight and size than would be required with an ordinary gear train [19,20]. The principle of K–H (K—center gear, H—planet carrier) planetary gear train-one of three major categories of planetary gear trains classified by former Soviet Union Scholar Б.Б.Кудpявцeв (B.B.Kudryavtsev)- [21] is the basis of the speed reduction mechanism proposed in this paper. Suppose that the K–H planetary gear train is composed of a center gear with tooth

Fig. 1. Photo of the speed reduction mechanism.

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number z0, a planet gear with tooth number z1 and a planet carrier H (Fig. 2). The relationship among the angular velocities of these three components can be expressed by the following equation (based on Fig. 2). ω0 −ωH z ¼þ 1 ω1 −ωH z0

ð1Þ

Where ω0, ω1and ωH are angular velocities of center gear, planet gear and planet carrier respectively, symbol “+” means that the center gear and the planet gear have the same rotation direction (planet carrier H acts as the reference object). This means the center gear and the planet gear should be a pair of inside engaged gears, a pair of synchronous pulleys, or two pairs of series-wound external gears, etc. Symbol “–” means that the center gear and the planet gear have the opposite rotation direction. If a spatial mechanism RCRCR is introduced into the planetary gear train, the rotation movement of the planet gear will be restricted, and the gear train only translates (ω1 ≡ 0). Thus Eq. (1) can be converted into the following equation: ωH z0 ¼ ω 0 z 0− z 1

ð2Þ

Fig. 3 shows that spatial mechanism RCRCR is added to the original K–H planetary gear train mechanism. This RCRCR mechanism is a one DOF (degree of freedom) spatial closed-loop mechanism-ABCDEFGHA, which consists of input shaft 1, connecting shaft 2 and 3, output shaft 4 and stander. Revolute pair A on input shaft 1 is parallel with the axis of cylindric pair B on connecting shaft 2, and the setover is equal to r1; revolute pair F on output shaft 4 is parallel to the axis of cylindric pair E on connecting shaft 3, and the setover is equal to r4, so r4 = r1. The axis of connecting shafts 2 and 3 are skewed lines (the dip separation between them is equal to d1), which are connected by revolute pair D. The axis of revolute pair A and F are also skewed lines (the dip separation between them is equal to d0), so d0 = d1. For spatial mechanism RCRCR, we can now see that angular velocity of input shaft 1 is equal to that of output shaft 2 (ω4 ≡ ω1). The connecting shafts 2 and 3 continually translate in space (planet carrier H acts as the reference object) and are independent of the intersection angle and dip separation between their axes [22,23]. As shown in Fig. 3, planet gear z1 is connected to the above-mentioned traversing component 3 by prismatic pair P, and center gear z0 is supported on output shaft 4 with revolute pair R. Thus, planet gear z1 is only able to translate, and here output shaft 4 acts as a planet carrier, so ωH in Eq. (1) can be replaced by ω4, which can be defined by the following equation:

i¼

ω1 z0 ¼ : ω 0 z 0 −z 1

ð3Þ

Eq. (3) is not related to the intersection angle and dip separation (d0 = d1) between the axes of the input shaft and the output shaft, which shows that the mechanism can maintain constant reduction ratio by self-adapting to the changes of the intersection angle. When d0 = d1 = 0, the axes of the input shaft and the output shaft intersect. Fig. 4 shows an example of the synchronous planetary pulley transmission mechanism, in which the inside engaged center gear and planet gear (based on Fig. 3) are replaced by a pair of synchronous pulleys. This new mechanism demonstrates functions of universal transmission and decreasing velocity (or increasing velocity in a reverse direction), which is designed on the basis of kinematic characteristics and planetary transmission principles of universal shaft coupling. In order to distribute load evenly and improve supporting capacity of the mechanism, three pairs of synchronous planetary pulleys are needed.

ω1

ωH

ω0 H

z1 z0

Fig. 2. Model of the K–H planetary gear train.

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Fig. 3. Speed reduction mechanism based on RCRCR structure and inside gearing planet gear transmission structure.

2.2. Kinematic analysis Fig. 5 shows a RCRCR spatial mechanism (R—revolute pair, C—cylindric pair), which is made up of components 1, 2, 3, 4 and stander 0. In this mechanism, the angular velocities of shaft 1 and shaft 4 are the same. For convenience, a corresponding right-hand rectangular coordinate system xi(yi)zi based on Denavit Hartenberg convention is built on each component i(i= 0, 1, 2, 3, 4), in which zi coincides with the axis of each kinematic pair, and xi coincides with the common perpendicular of two axes of the adjacent kinematic pairs (e.g. zi and zj) [24,25]. In each coordinate system as shown in Fig. 3, z0//z1 and z3//z4 are the significant features of the mechanism. The relationship between structure parameters can be expressed as: α34 = α01 = 0°, α23 = α12 = − 90°, h4 = h1, l2 = h0, l4 = l0. Position parameters of the mechanism includes: θ0, θ1, θ2, θ3, θ4 and l1, l2, l3, l4. The direction cosine matrix [Cij] can be used to express the exchanging relationships between coordinate system xi(yi)zi and xj(yj)zj, which can written by the following equation [26]:

h

Cij

i

2 cos θj h iT ¼ Cji ¼ 4 sin θj 0

− sin θj cos αij cos θj cos αij sin αij

3 sin θj sin αij − cos θj sin αij 5 cos αij

ð4Þ

Where i, j = 0, 1, 2, 3, 4, θi is the rotating angle of the corresponding component i.

3 2

4

M M0 1

Fig. 4. Speed reduction mechanism based on RCRCR structure and synchronous planetary gear transmission structure.

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Fig. 5. RCRCR spatial mechanism with equal angular velocity.

Along the axes z2 and z3, the closed loop can be separated into two parts: fixed chain 3–4–0–1–2 and float chain 2–3. According to the geometric equivalence property of the kinematics [26], the direction cosine of the axes z2 and z3 in fixed chain 3–4–0–1–2 equals that in float chain 2–3, which can be defined by the following equation: cos ð z2 ; z3 Þ 3−4−0−1−2 ¼ cos ð z2 ; z3 Þ2−3 ¼ cos α 23

ð5Þ

Where cos(z2, z3)3−4−0−1−2 = [0, 0, 1][C34][C40][C01][C12][0, 0, 1] T, and the calculation result of Eq. (5) is cos(θ1 + θ2) = 0. And, in a similar way, we can easily calculate the following kinematic parameters: 8 > > > > > > > <

∘ •

•

θ0 ¼ θ1 −180 ; θ0¼ θ1 • • θ2 ¼ 270B−θ1 ; θ2¼ − θ1 • • θ4 ¼ 90B−θ1 ; θ4¼ −θ1 > > l ¼ l −h 0 1 ð1 þ cos α40 Þ sin θ1 = sin α40 > 1 > > > l ¼ l4 −h1 ð1 þ cos α40 Þ sin θ1 = sin α40 > : 3 θ3 ¼ 180B þ α40

ð6Þ

On the basis of Fig. 5, Eq. (6) indicates that angular velocities of output shaft 4 and input shaft 1 of the RCRCR mechanism are equal (θ˙ 0 ¼ θ˙ 1 Þ. Relative to input shaft 1, the angular velocity of connecting shaft 2 is θ˙ 2 ¼ − θ˙ 1 (Relative to the stander, the angular velocity of input shaft 1 is θ˙ 1 ). So connecting shaft 2 only translates in space, and its angular velocity is always equal to 0 relative to the stander. For the same reason, relative to input shaft 4, the angular velocity of connecting shaft 3 is θ˙ 4 ¼ − θ˙ 0 , and the angular velocity of input shaft 4 is θ˙ 0 (relative to the stander). Thus, relative to the stander, connecting shaft 3 only translates in space. However, the features of the motion are never correlated to the intersection angle α40 of the input and output shafts. In addition, the relative rotation angle θ3 between connecting shafts 2 and 3 is a constant, which only varies with α40. Therefore, the mechanism in Fig. 5 is a type of constant speed drive (CSD) with an intersection angle self-adapting feature and two traversing connecting shafts. If the mechanism (Fig. 5) is introduced to the K–H planetary gear train (Fig. 2), and the planet gear is constrained by two traversing connecting shafts that only translate (the angular velocity of the planet gear is equal to constant 0). Thus, we get a new type of spatial speed reduction mechanism, whose intersection angle of axes is variation self-adapted, and its transmission ratio is defined by Eq. (3). The structure of this mechanism is shown in Figs. 3 and 4. 2.3. Force analysis 2.3.1. Revolute and prismatic pair A universal speed reduction mechanism of the synchronous planetary pulley is composed of a synchronous pulley mechanism and a RCRCR double crank mechanism. To simplify the mechanism, the mass of each component is not considered. In other words, the inertia force and the inertia torque of each component are neglected. M0 is defined as the driving torque of driving link 1, and M is defined as the output resistance torque of the sychronous center pulley (Fig. 4). So the relationship between M0 and M can be written as: M ¼ iM0

ð7Þ

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By studying the revolute pair consisting of components 2 and 3 (Fig. 4), we can simplify the reaction force of component 3 on component 2 as force vector R32 and torque vector M32, and the start point of two vectors is the crossing point of X2 and Y3. As action and reaction are opposite and equal, force vector R32 and torque vector M32 can be expressed as: (

R32 ¼ −R23 ¼ ðr32 ; s32 ; t32 ÞT T M32 ¼ −M23 ¼ ðu32 ; v32 ; w32 Þ

ð8Þ

Where r32, s32, t32 are three force vector components along the three coordinate axis of RS2; and u32, v32, w32 are three force torque components along the three coordinate axis of RS2. RS2 (X2, Y2, Z2) is fixed on the point C' of component 2 (Fig. 6). As the rotating freedom of component 3 around axes Z2 is not restricted, ω32 = 0. And Y2-axis consides with the direction of the axes of the cylindric pair (Z1). Hence, we can easily construct the following equations: 

s32 ¼ 0 v32 ¼ 0

ð9Þ

This is because component 3 and the synchronous planetary pulley are connected by a prismatic pair (Z3-axis), and the out force and torque acted on it are expressed by r, s and T (Fig. 6). r32, v32 and u32 are calculated by following force and torque balance equations of component 3 and prismatic pair: 8 0 1 0 19 0 1 0 r = r32 < ð0; 0; 1Þ −½C32 @ 0 A þ @ s A ¼ @ 0 A : ; 0 0 t32

ð10Þ

8 1 0 1 0 1 0 0 19 0 1 0 u32 r r32 = 0 < ð0; 0; 1Þ −½C32 @ v32 A þ @ 0 A þ ½rCD0 ×@ s A−½rCC 0 ×@ 0 A ¼ @ 0 A : ; −T 0 t32 0 0

ð11Þ

Where C'(0,− l2, 0) and D' (0, 0, l3) are two points located in RS3, and 2

−l3 0 0

0 ½rCD0 ×¼4 l3 0

3 2 0 0 5 0 ; ½rCC 0 ×¼4 0 0 l2

3 0 −l2 0 0 5 0 0

In Eqs. (10) and (11), equilibrium of the component 3, forces and torques are expressed in RS3. So matrix [C32] is employed to pass from RS2 to RS3. From Eq. (4), we can get the following equation: 2

cos θ3 ½C23  ¼ 4 sin θ3 0

− sin θ3 cos α23 cos θ3 cos α23 sin α23 2

½C32  ¼ ½C23  ¼ 4 T

cos θ3 0 − sin θ3

sin θ3 0 cos θ3

3 2 cos θ3 sin θ3 sin α23 − cos θ3 sin α23 5 ¼ 4 sin θ3 cos α23 0

0 0 −1

3 − sin θ3 cos θ3 5 0

3 0 −1 5 0

ð12Þ

ð13Þ

l3

Fig. 6. Force analysis of component 3.

l2

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Where α23 is the angle between the axes Z2 and Z3. And viewed from the positive direction of X3, the axes Z rotates from Z2 to Z3 in clockwise direction, hence the symbol of α23 is “–”, otherwise the symbol of α23 is “+”. Eqs. (10) and (11) can be simplified as:  r32 sin θ3 ¼ 0 ð14Þ u32 sin θ3 −T ¼ 0 From the above equations, we can get the following parameters: r32 ¼ 0;

ð15Þ

u32 ¼ T= sin θ3

Because components 1 and 2 both have the degree of freedom (DOF) of rotating around axis Z0 of the revolute pair, the sum of all torques on axis Z0 acts upon the two components that are equal to zero, which can be written as follows: 8 1 2 30 3 20 13 2 0 19 0 1 0 1 0 0 u32 0 = −u10 0 −l1 0 0 l1 0 < ð16Þ ð0; 0; 1Þ½C01  ½C12 4@ 0 A þ 4 l1 0 0 5@ 0 A5 þ 4 −l1 0 −h1 5½C12 @ 0 A þ @ −v10 A ¼ @ 0 A : ; t32 0 0 t32 M0 0 0 0 0 h1 0 Hence, one component of the constraining force t32 can be written as: t32 ¼ −M0 =h1 cos θ2

ð17Þ

The cylindric pair consisting of components 2 and 1 has four constraint reactions. According to the balance relation of component 2, we can easily express the following equations: 1 0 1 0 1 0 0 r21 @ A @ − s21 þ ½C12  0 A ¼ @ 0 A 0 t32 0 0

80 1 2 1 u21 0 < u32 −@ v21 A þ ½C12  @ 0 A þ 4 l1 : 0 0 0 0

ð18Þ −l1 0 0

30 19 0 1 0 0 0 = 0 5@ 0 A ¼ @ 0 A ; t32 0 0

The components of the constraining force can be expressed as: 8 r ¼ −t32 sinθ2 > > < 21 s21 ¼ t32 cosθ2 > > u21 ¼ u32 cosθ2 : v21 ¼ u32 sinθ2

Fig. 7. NX model of the speed reduction mechanism.

ð19Þ

ð20Þ

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The cylindric pair consisting of components 4 and 3 also has four constraint reactions. On the same reason, we can express the following equations: 0 1 0 1 0 1 r 0 0 @ s A−½C32 @ 0 A ¼ @ 0 A 0 0 t32 1 2 1 0 0 u32 0 −½C32 @ 0 A þ @ 0 A−4 0 −T 0 l2 0

ð21Þ 3 0 1 2 0 0 −l2 0 0 0 5½C32 @ 0 A þ 4 l3 t32 0 0 0

−l3 0 0

30 1 0 1 0 r 0 0 5@ s A ¼ @ 0 A 0 0 0

ð22Þ

Fig. 8. Angular velocities of different working components. a). Angular velocities of input shaft (120°/s) and output shaft (20°/s). b). Angular velocities of input shaft (90°/s) and output shaft (15°/s). c). Angular velocities of two connecting shafts ≡ 0.

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Fig. 8. (continued)

From Eq. (21) the following parameters can easily be calculated: r = 0, and s = −t32, and from Eq. (22) the following parameters can easily be calculated: u32 = t32l3/cos θ3, and T = u32 sin θ3, respectively. Ignoring friction and energy loss, the above theoretical analysis validates the transmission principle of the universal speed reduction mechanism. The force and torque calculated on each joint indicates that mechanism can realize decelerating (or accelerating) transmission of two crossed shafts, and the transmission performance is unrelated to the degree of the cross angle.

3. Analysis of kinematic simulation As shown in Fig. 7, a mechanism kinematic simulation model based on UG NX 6.0 is established for the synchronous planetary gear speed reduction mechanism to verify the correctness of the transmission principle of the above-mentioned mechanism. According to the schematic diagram of the universal speed reduction mechanism, connecting shafts are created to coordinate the positions of bearing supports, shafts, carriers, etc. The input shaft is combined with a revolute pair and operates at a constant rotating speed. A cylindric pair is added to the joint between the connecting shaft and the rotating shaft, and a revolute pair is added to the output shaft. To verify the efficiency of the universal speed reduction mechanism, the angular velocity of the input shaft is set at 90°/s and 120°/s, where z1/z0 =5/6, i.e. reduction ratio i =ω1/ω0 =6. To test if the transmission ratio i of the mechanism changes along with the variation of the intersection angle α between two shafts, intersection angle α is given two different values: 120° and 150° respectively. The results of the kinematic simulation show that the angular velocities of the output shaft are 20°/s and 15°/s respectively, which are shown in the Fig. 8 a) and b). The results of kinematic simulation indicate that however the intersection angle between the input and output shafts changes, their angular velocity ratio remains constant. And the relative angular velocities of the three pairs of connecting shafts (eg. #13 and #15 in Fig. 9) are equal to 0 as shown in Fig. 8 c).”

4. Mechanism prototype design and manufacture On the basis of above-mentioned mechanism principle and kinematic simulation analysis, the prototype of transmission mechanism shown in Fig. 4 is designed and manufactured in a way which adopts three pairs of equispaced planetary pulleys (Figs. 9 and 1). When the prototype was designed and manufactured, we made the dip separation between two skewed shafts equal to 0 (i.e. d0 = d1 = d2 = 0 in Fig. 3) to simplify the manufacturing process. Although the prototype is a special case of Fig. 3, practical results demonstrate it performs successfully. It can achieve the expected reduction transmission between two intersected shafts while the transmission ratio is unrelated to the variation of the two crossed axes. The mechanism is primarily made up of shafts, pulleys, copper bushes and additional standard parts (e.g. straight keys, spring circlips, etc.), which are expressed in Table 1.

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Fig. 9. Assembly drawing of the speed reduction mechanism.

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Table 1 The names of the main parts of the assembly figure. No.

Name

No.

Name

1 2 3 4 5 6 7 8 9 10 11 12

Output shaft Connecting shaft-3 Straight key Spring circlip for shaft Synchronous center pulley-1 Synchronous planetary pulley-1 Copper bush Rotary table-2 Spring circlip for shaft Straight key Copper bush Synchronous center pulley-2

13 14 15 16 17 18 19 20 21 22 23 24

Connecting shaft-2 Synchronous planetary pulley-2 Connecting shaft-1 Spring circlip for shaft Copper bush Rotary table-1 Input shaft Bearing block Copper bush Straight key Washer Pin with hole

5. Conclusions The goal of this study is to analyze the feasibility of an innovative speed reduction mechanism with self-adaptability to variable transmission angles based on synchronous planetary pulleys, which is integrated with RCRCR-based equal speed transmission structures and K–H planetary transmission mechanisms. The research work begins with structural analysis and force analysis of this mechanism, followed by the development and simulation of the NX model of the mechanism. The results of theoretical analysis and kinematic simulation proves that this innovative speed reduction mechanism can flexibly adapt itself to variations of the intersection angles between the input and output shafts. Thus this mechanism is fully capable of lowering the accuracy requirements of relative position between prime mover and working machine, and decreasing the transmission chain at the same time. Judging from the results of the working trials of the prototype, this mechanism performs successfully and satisfies industry requirements. References [1] J.H. Shin, S.M. Kwon, On the lobe profile design in a cycloid reducer using instant velocity center, Mechanism and Machine Theory 41 (5) (2006) 596–616. [2] F.C. Yang, X.J. Zhou, H.W. Hu, Nonlinear vibration characteristics of two-stage gear reducer, Journal of Zhejiang University (Engineering Science) 43 (7) (2009) 1243–1248. [3] X.S. Wang, J.G. Yang, Q.J. Guo, H.C. Tong, Building of virtual prototype of cylinder gear decelerator based on computer, Journal of Nanjing University of Aeronautics & Astronautics 37 (Supp 1) (2005) 130–133. [4] Y.M. Yang, J.X. Kuang, Lubrication design of two-steps expanding gear reducer, Lubrication Engineering 4 (2005) 178–179. [5] J.Y. Tang, Y.F. Lu, C. Zhou, Error tooth contact analysis of spiral bevel gears transmission, Chinese Journal of Mechanical Engineering 44 (7) (2008) 16–23. [6] Z.D. Fang, T. Liu, Tooth contact analysis of spiral bevel gears based on the design of transmission error, Journal of Aeronautics 23 (3) (2002) 226–230. [7] L.H. Wang, Y.Y. Huang, R.F. Li, T.J. Lin, Study on nonlinear vibration characteristics of spiral bevel transmission system, China Mechanical Engineering 18 (3) (2007) 260–264. [8] L.H. Wang, R.F. Li, T.J. Lin, Y.Y. Huang, Analysis for coupling vibration of a spiral bevel gear system, China Mechanical Engineering 17 (14) (2006) 1431–1434. [9] G. Bishop et al. Seal for worm gear speed reducer, Sealing Technology 2006 (5) (2006) 14 Patent number: WO 2006/0009720. [10] F. Wan, Y.Q. Wang, G.H. Zhang, Research and application of conditioning of tooth surface of ZC1 worm-wheel of large rolling mill for mesh control, Iron and Steel 42 (3) (2007) 76–78. [11] P.D. Jiang, S.L. Li, The improving of worm reduction gears’ seal types, Lubrication Engineering 3 (2005) 141–142. [12] Z. Wang, H.J. Wang, J.M. Wang, Application of simulated annealing in the product layout of reducers, Journal of Tianjin University 38 (2) (2005) 150–153. [13] Z.H. Zhou, H. Zheng, Skill of using CAD software for 3D profile of worm-gear-screw, Journal of Jianghan Petroleum Institute 25 (6) (2003) 136–137. [14] D. Mundo, H.S. Yan, Kinematic optimization of ball-screw transmission mechanism, Mechanism and Machine Theory 42 (1) (2007) 34–47. [15] L. Bagnoli, G. Fantoni, A. Filippeschi, M. Guiggiani, Kinematic analysis of a novel pin-wheel joint, Meccanica 42 (5) (2007) 495–502. [16] G. Fantoni. Sistema per la trasmissione di moto rotatorio equiverso fra assi aralleli, incidenti e sghembi. Italian Patent No PI20020000023 (2001). [17] S.S. Balli, S. Chand, Transmission angle in mechanism (Triangle in mech), Mechanism and Machine Theory 37 (2) (2002) 175–195. [18] X. B. Chen, Y. Li. Speed Up/Down Mechanism of Universal Transmission Based on Planetary Synchronous Pulley, Chinese Patent No. 200410024617.3 (2004). [19] J.M. Castillo, The analytical expression of the efficiency of planetary gear trains, Mechanism and Machine Theory 37 (2) (2002) 197–214. [20] R. Mathis, Y. Remond, Kinematic and dynamic simulation of epicyclic gear trains, Mechanism and Machine Theory 44 (2) (2009) 412–424. [21] Б.Б. Кудpявцeв, Плaнeтapныe пepeдaчи cпpaвoчник (Translated by Q. S. Chen), Metallurgical Industry Press, Beijing, 1982. [22] X.B. Chen, Y. Li, Y.F. Le, The constructure and analysis of a new type constant speed ratio coupling with variable crossed axes angle lower pair, Mechanical Science and Technology for Aerospace 18 (3) (1999) 430–431. [23] X.B. Chen, Research on the composition of equal angular velocity linkage mechanism, Journal of Tongji University (Natural Science) 25 (6) (1997) 704–708. [24] J. Denavit, R. Hartenberg, A kinematic notation for lower-pair mechanisms based onmatrices, ASME Journal of Applied Mechanics 22 (1955) 215–221. [25] C.R. Rocha, C.P. Tonetto, A. Dias, A comparison between the Denavit–Hartenberg and the screw-based methods used in kinematic modeling of robot manipulators, Robotics and Computer-Integrated Manufacturing 27 (2011) 723–728. [26] Q.X. Zhang, Analysis and Synthesis of Spatial Mechanisms, China Machine Press, Beijing, 1984. Xinbo Chen, male, born in 1962, professor. Research direction: drive mechanism, electric vehicle. Bin Wang, male, born in 1975, Ph.D.Candidate of Tongji University. Research direction: drive mechanism, CAD/CAE. Meifang Chen, female, born in 1981, postgraduate student of Tongji University. Research direction: mechanical transmission. Yan Li, female, assistant professor. Research direction: mechanical drive.