- Email: [email protected]
Non-circular belt transmission design of mechanical press
Mechanism and Machine Theory 57 (2012) 126–138
Contents lists available at SciVerse ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Non-circular belt transmission design of mechanical press Enlai Zheng, Fang Jia, Hongwei Sha ⁎, Suhao Wang School of Mechanical Engineering, Southeast University, Nanjing 211189, China
a r t i c l e
i n f o
Article history: Received 12 May 2011 received in revised form 11 July 2012 accepted 12 July 2012 Available online 21 August 2012 Keywords: Non-circular Press Crank-slider Belt transmission Slack
a b s t r a c t The quality of parts manufactured by metal forming operations depends on the kinematics of the mechanical press in a large degree. Non-circular belt transmission with a rotational angle-dependent speed ratio in the press drive mechanism offers a new way to obtain optimum stroke-time behaviors for specific metal forming operations in terms of manufacturing. In this work, the mathematical model for mechanical press driven by non-circular belt transmission is established with the concept of polar coordinate equation for tangent curves introduced, and a general design method of non-circular belt pulley pitch curves and slack for transmission system is proposed. Compared with traditional crank-slider press, numerical simulation results demonstrate that mechanical press with non-circular belt transmission has a lower speed under deep drawing operation and a quick-return characteristic under non-loading stage, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. Furthermore, the slack of non-circular belt transmission for mechanical press is calculated under operation, which can provide a theoretical basis for the design of take-up mechanism. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The mechanical press, as “less/non-chip finish” high-effective machine tool as well as one of the key industrial equipment, can manufacture metal parts closer to their final shapes and comply better with the requirements of highly advocated clean and green production. Traditional mechanical press commonly consists of belt transmission and crank-slider mechanisms, which are used to transmit the rotary motion of electrical machine into translational movement of slider so as to realize stamping operation. As the conventional belt transmission ratio is constant, the motion law of the slider is sine shape [1] and cannot meet the requirements of blanking and deep stamping products that are made of poor plasticity and embrittlement material. Therefore, to overcome this problem, a type of non-circular belt transmission with variable velocity ratio should be designed and a specific output motion law of slider for mechanical press should be generated, which includes a low speed under loading stage and a high speed under non-loading stage. Many researchers have studied majority kinds of transmission with variable velocity ratio for the special law of motion. The application of non-circular gears in function generating mechanism has been proposed and discussed [2,3]. By designing a pair of non-circular gears, which are able to perform a proper gear ratio function, the output member of a mechanism can be effectively forced to move according to a prescribed law of motion when operated at a constant input velocity [4,5]. For the special law of motion obtained by a pair of meshing elliptical gears, a design solution was given by Wunderlich W and Zenow P [6]. F.S Li and X.T Wu [7–9] proposed the use of non-circular gears to achieve the non-uniform transmission ratio. The use of non-circular gears in the drive of mechanical press offers a new way of meeting the demands on the kinematics. Investigations at the Institute for Metal
⁎ Corresponding author at: School of Mechanical Engineering, Southeast University, 2# Southeast Road, JiangNing District, NanJing 211189, China. Tel.: +86 159 5107 5321(Cell phone); fax: +86 25 52090504. E-mail address: [email protected] (H. Sha). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2012.07.004
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
127
Forming and Metal Forming Machine Tools of Hanover University have shown that in this simple manner all the relevant uninterrupted motions of the ram can be achieved for various forming processes [10]. E Doege and M Hindersmann [11] also came up with the non-circular gears to achieve the optimized kinematics of press, but the specific method of design was not given. For application of cam-follower mechanisms, these devices can also reproduce almost any characteristics of the follower's motion and their manufacturing costs as well as the number and dimensions of moving parts are rather limited [12,13]. F Freudenstein and C.K Chen [14] developed a new mechanical component: variable ratio chain drives with noncircular sprockets and minimum slack, and illustrate applications to the design of an optimum bicycle configuration and a harmonic-motion generator. X.L Cao and J Liu [15] designed a type of chain transmission with a variable velocity ratio and outlined that two spans of a chain are never simultaneous in tension, and the chain slack can undergo moderate periodical changes while the transmission runs. Based on the results [15], J Liu and L.P Feng [16,17] studied the algorithmic computation of Non-circular chain transmission. Carlo Innocenti and Davide Paganelli [18] presented a new procedure to design a two-pulley synchronous belt transmission connecting, with no belt tensioner, two parallel-axis shafts with a variable velocity ratio. The procedure takes into account the limited number of choices for pitch and length of off-the-shelf synchronous belts. Hellmuth Stachel [19] treated the geometry of tooth profiles and pulleys and their algorithmic computation as well as the relation between tooth profiles and ‘strict’ cases without take-up pulley. A geometric solution was proposed by Hoschek J [20], who pointed out that some tangents to the pulley contours were determined through an iterative procedure, and the envelope of these tangents was approximated by a sequence of Bezier splines. The existing approaches mainly apply non-circular gears transmission, belt transmission, chain transmission and cam-follower mechanism to generate specific motion law. Only non-circular gears transmission has been applied to mechanical press in order to optimize its kinematics and meet specific machine requirement. Although non-circular gears have the advantage of lower weight-to-strength ratios and absence of gross separation or decoupling of moving parts [4], non-circular belt transmission is more attractive than gear transmissions if the center distance is relevant or if lubrication is unavailable as well as if the cost of design and manufacture is concerned. Therefore, it is important to design non-circular belt transmission for mechanical press so as to obtain optimized kinematics and improve its manufacturing flexibility. The primary contribution of this work is to design non-circular belt transmission for mechanical press in order to obtain optimized kinematics and meet the machine requirement of deep drawing. Based on the polar equation for tangent curves, the pitch curve of non-circular pulley is designed. Compared with traditional crank-slider press, numerical simulation results demonstrate that mechanical press with non-circular belt transmission has a lower speed under deep drawing operation and a quick-return characteristic under non-loading stage, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. Furthermore, the slack of non-circular belt transmission for mechanical press is calculated under operation, which can provide a theoretical basis for the design of take-up mechanism. This paper is organized as follows: in Section 2, the physical structure and machine requirements of mechanical press are described briefly. In Section 3, the pitch curve of non-circular pulley is designed and slack of belt transmission is obtained. In Section 4, numerical example is simulated and comparative analysis is performed. Section 5 summarizes the design of non-circular belt transmission for mechanical press and the conclusions. 2. Physical description and press machine requirements The physical figure of mechanical press is shown in Fig. 1 and the working principal sketch of press is shown in Fig. 2: Motor (1) passes the movement on to pulley (10) through the belt and then passes motion on to crankshaft by gears (2, 3). The rod (11) connects with crankshaft on the top and with the slider (6) on the bottom, which can translate rotary movement of crankshaft into linear reciprocating motion of slider. The upper die (10) is installed on the slider and lower die (7) on the plate (9). When the sheet metal is put between the upper and lower dies, Press can carry out blanking or other deformation techniques. In order to meet the manufacturing process needs, press is equipped with clutch (4) and brake (12) that can sometimes get the slider to move or stop. The press's working time under load is short throughout the work cycle and most of the time is for empty trip of no-load. In order that motor works under uniform load and makes use of energy with high efficiency, crank-slider press often installs flywheel. As Fig. 2 shows, the big pulley (13) serves as flywheel. One manufacturing cycle, which corresponds to one stroke of the mechanical press, goes through three stages: loading, forming and removing the part. Instead of the loading and removal stages we often find feeding the sheet, especially in sheer cutting. For this, the press slider must have a minimum height for a certain time. During the forming period the slider should have a particular velocity curve, which will be gone into below. The transitions between the periods should take place as quickly as possible to ensure short cycle time. In deep drawing operations, the velocity of impact should be as low as possible to avoid the impact when striking the sheet. On the one hand, velocity must be sufficient for lubrication during forming. On the other hand, we have to consider the rise in the strain rate which creates greater forces and may cause fractures at the transition from the punch radius to the side wall of the part. 3. Design formulation for non-circular belt transmission In typical arrangements for mechanical press shown in Fig. 3, a pair of non-circular pulleys is used to drive a crank-slider mechanism, so that the slider is forced to move according to a specific law of motion. The design of this mechanism consists of two phases, namely the synthesis of the pitch curves, starting from the requested law of motion, and the computation of slack.
128
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
3.1. Tangent equation in polar and rectangular coordinate system As shown in Fig. 4, it is assumed that tangent of the plane curve at point C(x,y) is t, the distance between the original point O and tangent curve is p, and the intersection angle between axis OX and vertical line On is θ. Then, p is the univalent function with respect to θ. p ¼ pðθÞ
ð1Þ
Eq. (1) can be used to describe the plane curve C in the polar coordinate system. For closed plane convex curve, p is the periodic function with respect to variation θ and its period is2π. In the polar coordinate system, the curve equation is r = r(ϕ). Referring to the geometric relation in Fig. 4, the function p can be given. p ¼ r sinðuÞ
ð2Þ
Where the geometric relation between θ, ϕ and u is θ ¼ ϕ þ u−90. Then, the tangent equation of the plane curve can be expressed as follows. x cosðθÞ þ y sinðθÞ ¼ pðθÞ
ð3Þ
Eq. (3) can be considered as linear cluster with respect to variable θ and the plane curve can be regarded as the envelope of the linear cluster. By derivation of Eq. (3), we can obtain ′
−x sinðθÞ þ y cosðθÞ ¼ p ðθÞ:
ð4Þ
By solving the Eqs. (3) and (4) simultaneously, we can transform the equation of plane curve in the polar coordinate system (p, θ) into the equation in the rectangular coordinate system (x, y). (
x ¼ pðθÞcosθ−p′ ðθÞsinðθÞ y ¼ pðθÞsinθ þ p′ ðθÞcosðθÞ
ð5Þ
Fig. 1. Physical figure of mechanical press.
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
129
Fig. 2. Composition and principle sketch of press. 1—motor, 2—small gear, 3—large gear, 4—clutch; 5—crankshaft, 6—slider, 7—Lower die, 8—Foundation, 9—Worktable, 10—Upper die, 11—rod, 12—brake, 13—large pulley, 14—small pulley.
3.2. Design formulation of the pitch curve The non-circular belt transmission, as shown in Fig. 5, consists of two non-circular pulleys and a belt. Pulley 1 is the drive wheel, while pulley 2 is the driven one. It is assumed that the direction of rotation for the two non-circular pulleys is anticlockwise. Therefore, the instantaneous transmission ratio can be given.
i12 ¼
ω1 dϕ1 ¼ ω2 dϕ2
ð6Þ
Where ω1 and ω2 represent the instantaneous angular velocity for pulleys 1 and 2, and ϕ1 and ϕ2 are the instantaneous rotation angles respectively. It is assumed that the transmission ration is differentiable. In an instant, the pitch curve of belt is tangent with two non-circular pulleys at the points of C1 and C2. Then, the line speed of pulley 1 at the point of C1 is equal to r1ω1 and that of pulley 2 at the point of C2 equal to r2ω2. As the belt stretch can be neglected, the sub-speed of r1ω1 and r2ω2 along the direction of belt should be equivalent and can be formulated in the form r 1 ω1 cosβ1 ¼ r 2 ω2 cosβ2 :
ð7Þ
Where β1 and β2 represent the angle between the former vertical line and radius O1C1 and O2C2.
Fig. 3. Schematic diagram of non-circular belt transmission for mechanical press.
130
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
Fig. 4. Tangent of curve in the polar coordinate system.
Then, through Eqs. (6) and (7), the transmission ratio i12 can also be expressed as i12 ¼
ω1 r 2 cosβ2 p2 ¼ ¼ : ω2 r 1 cosβ1 p1
ð8Þ
Where p1 and p2 are the vertical distances from the rotation center of O1 and O2 to belt pitch line. The transmission ratio i12 is variable as a result of non-circular pulleys and can be expressed as a function with respect to the rotation angle ϕ2 of driven pulley. i12 ¼ f ðϕ2 Þ
ð9Þ
When the pitch curve of belt is extended to intersect with the line O1O2, the intersection point is O1O2. Therefore, the transmission ration i12 can be rewritten as i12 ¼
p2 a þ b : ¼ b p1
ð10Þ
Where a is the central distance between two non-circular pulleys, and b is the distance between point O1 and point O3. The Eq. (10) can also be expressed in the form b¼
a : i12 −1
ð11Þ
It is assumed that the coordinate system (O1 − X1Y1) is fixed on pulley 1 and the coordinate system (O2 − X2Y2) is fixed on pulley 2. When the transmission ratio i12 and the distance a between O1 and O2 are determined, ϕ1 and ϕ2 are zero in the starting position and
Fig. 5. Schematic diagram of non-circular belt transmission.
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
131
the direction of axis of X1 and X2 overlap with the line O1O2. If the rotation angle of pulley 1 is ϕ1, the rotation angle of pulley 2 can be evaluated as ϕ
ϕ2 ¼ ∫0 1
1 dϕ : i12 1
ð12Þ
Then, we can obtain the relation as follows in Fig. 4. θ2 ¼ θ1 þ ϕ1 −ϕ2
ð13Þ
Where θ1 is the angle between p1 and the axis of X1 and θ2 is the angle between p2 and X2. When the pitch curve of pulley 1 is assumed to be p1 = p1(θ1)and determined, we can obtain cosðθ1 þ ϕ1 Þ ¼ −
p1 ðθ1 Þ p ðθ Þði −1Þ ¼ − 1 1 12 : b a
ð14Þ
Where (p1, θ1) and (p2, θ2) represent the tangent equations of pitch curves for pulleys 1, 2 in the polar coordinate system. Once rotation angle ϕ1 is given, p1 and θ1 can be obtained from Eqs. (11) and (14). Then the pitch curve of pulley 2 can be expressed as p2 = p2(θ2). In a similar manner, when the pitch curve of driven pulley 2 and transmission ratio i12 are given, the pitch curve of drive pulley 1 can also be obtained. 3.3. Slack of non-circular belt transmission As the pitch curve of pulley is non-circular, the total length of envelop for two pulleys varies with respect to rotation angle. Therefore, the slack calculation of non-circular belt transmission for mechanical press is essential and can provide theoretical basis for design of take-up mechanism as well as the optimization of minimum slack. 3.3.1. Arc length of curve By derivation of Eq. (5) with respect to θ, we can obtain h i 8 0 00 < x ¼ − pðθÞ þ p ðθÞ sinðθÞ h i 0 00 : y ¼ pðθÞ þ p ðθÞ cosðθÞ:
ð15Þ
The curvature radius of plane curve C can be given. ρ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 0 2ffi 0 2 x þ y ¼ pðθÞ þ p ðθÞ
ð16Þ
The arc length of plane curve C can be calculated as θ h i 00 s ¼ ∫ pðθÞ þ p ðθÞ dθ:
ð17Þ
θ0
3.3.2. Length of pitch curve for belt As shown in Fig. 6, the length of pitch curve for belt consists of four parts and can be calculated for every given ϕ2 at any instantaneous time. L ¼ C1 CΙ þ C1 C2 þ C2 CΠ þ CΙ CΠ
h i h i 00 00 θ θ ¼ ∫θΙ1 p1 ðθ1 Þ þ p1 ðθ1 Þ dθ1 þ C 1 C 2 þ ∫θ2Π p2 ðθÞ2 þ p2 ðθ2 Þ dθ2 þ C Ι C Π
ð18Þ
The length of arc C1CΙ can be calculated as h i 00 θ C 1 C Ι ¼ ∫θΙ1 p1 ðθ1 Þ þ p1 ðθ1 Þ dθ1 :
ð19Þ
The length of arc C2CΠ can be calculated as h i 00 θ C 2 C Π ¼ ∫θ2Π p2 ðθ2 Þ þ p2 ðθ2 Þ dθ2 0 dθ θ θ ¼ ∫θ2Π p2 ðθ2 Þ 2 dϕ2 þ p2 θ2Π : dϕ2
ð20Þ
132
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
Fig. 6. Length of pitch curve for belt.
As p2(θ2) and dθ2/dϕ2 are functions with respect to ϕ2, the values of ϕ1, θ1, θ2, p2, p2', x1, y1, x2, and y2 can be obtained from Eqs. (5) and (14) for every given ϕ2. To obtain the length of C1CΙ and C2CΠ, the values of θΙ and θΠ must be calculated in advance. Referring to the schemes in Fig. 5, we can obtain θ1 þ ϕ1 ¼ θ2 þ ϕ2
ð21aÞ
ϕ1 þ θΙ ¼ ϕ2 þ θΠ :
ð21bÞ
For every given ϕ1, in addition that a set of θ1 and θ2 can meet the Eq. (21a), another corresponding group of θΙ and θΠ that satisfies the Eq. (21b) can also be found. If ϕ1 + θΙ and ϕ2 + θΠ belong to the third quadrant or fourth quadrant at the same time, that is
∘
∘
180 bϕ1 þ θΙ b360 180∘ bϕ2 þ θΠ b360∘ :
ð22Þ
' Then, the values of θΙ and θΠ are what we need and ϕΙ,ϕΠ, pΠ, pΠ , xΙ, yΙ, xΠ, yΠ can also be obtained from Eqs. (5) to (14) respectively ' in the same way. By substituting θ1, θ2, θΙ, θΠ, p1, p2, p1', pΠ into Eqs. (19) and (20), the length of C1CΙ and C2CΠ can be obtained. Coordinate values of four points C1(x1, y1), CΙ(xΙ, yΙ), C2(x2, y2) and CΠ(xΠ, yΠ) in the rectangular coordinate system (x, y) can be calculated by Eq. (5) and can be expressed as
8 xΙ ¼ p1 cosðθΙ Þ > > < y ¼ p sinðθ Þ Ι 1 I 0 xΠ ¼ p2 cosðθΠ Þ−p2 sinðθΠ Þ > > : 0 yΠ ¼ p2 sinðθΠ Þ þ p2 cosðθΠ Þ:
ð23Þ
C1(x1, y1) and CΙ(xΙ, yΙ) are the points in the coordinates O1 − X1Y1, while C2(x2, y2) and CΠ(xΠ, yΠ) are points in the coordinatesO2 − X2Y2. In order to calculate the length of C 1 C 2 and C Ι C Π , four points can be transformed to the coordinates of O − XY. Then, C1(X1, Y1), C2(X2, Y2), CΙ(XΙ , YΙ) and CΠ(XΠ, YΠ) can be obtained by means of rotating coordinate system. 8 X > > < 1 Y1 > > X2 : Y2
¼ x1 cosðϕ1 Þ−y1 sinðϕ1 Þ ¼ x1 sinðϕ1 Þ þ y1 cosðϕ1 Þ ¼ x2 cosðϕ2 Þ−y2 sinðϕ2 Þ þ a ¼ x2 sinðϕ2 Þ þ y2 cosðϕ2 Þ
8 X ¼ xΙ cosðϕΙ Þ−yΙ sinðϕΙ Þ > > < Ι Y Ι ¼ xΙ sinðϕΙ Þ þ yΙ cosðϕΙ Þ > > X Π ¼ xΠ cosðϕΠ Þ−yΠ sinðϕΠ Þ þ a : Y Π ¼ xΠ sinðϕΠ Þ þ yΠ cosðϕΠ Þ
ð24Þ
ð25Þ
Then, the length of C 1 C 2 and C Ι C Π can be calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < C C ¼ ðX −X Þ2 þ ðY −Y Þ2 1 2 2 1 2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : : C C ¼ ðX −X Þ2 þ ðY −Y Þ2 Ι Π Π Ι Π Ι
ð26Þ
3.3.3. Slack of belt As the pitch curve of pulley is non-circular, the total length of envelop for two pulleys varies with respect to rotation angle. It is assumed that both sides of the belt for non-circular belt transmission are tight. Then, the difference between maximum length and minimum length of belt during a period is called slack of non-circular belt transmission.
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
133
For every given ϕ21, ϕ22… ϕ2n, length L1, L2… Ln of belt can be obtained respectively. Then, the theoretical slack is given. ΔL ¼ L max −L min
ð27Þ
4. Results and discussion The traditional JB04 crank-slider mechanical press with the non-circular belt transmission applied as front mechanism is taken as a numerical example. It is assumed that the transmission ratio i12 is a function with respect to ϕ2, its period is 2π, and the pitch curve of drive pulley 1 is perfect round. Then, the transmission ration function can be expressed as i12 ¼ A þ B sinðϕ2 Þ ¼ Að1 þ k sinðϕ2 ÞÞ:
ð28Þ
Where k represents the ratio B/A. The transmission ratio function must be positive during the press process, then |k| b 1. When k is equal to zero, transmission ratio is a constant. The pitch curve of driven pulley 2 will also be perfect round, which is the same as the traditional belt transmission. It is assumed that the average transmission ratio is 2 in this work, and then we can obtain i12 ¼ 2ð1 þ k sinðϕ2 ÞÞ:
ð29Þ
By substituting the Eq. (6) into Eq. (29) and integration, the solution of Eq. (29) can be given. ϕ1 ¼ 2ðφ2 −k cosðϕ2 ÞÞ þ C
ð30Þ
The initial condition is that when ϕ1 is equal to zero, ϕ2 is equal to zero. Therefore, C is equal to 2k. It is assumed that the advance-to return-time ratio of crank-slider mechanism in this work is equal to 2. That means the slider's working stroke accounts for two thirds of the total stroke. When ϕ2 is equal to zero, ϕ1 is also equal to zero. When ϕ2 is equal to π, ϕ1 is equal to 2π + 4k. Then, 2π þ 4k 2 ¼ : 4π 3
ð31Þ
The solution of Eq. (34) is k = π/6 and the transmission ratio can also be expressed as π i12 ¼ 2 1 þ sinðϕ2 Þ : 6
ð32Þ
The pitch curve of driven pulley 2 can be calculated as p2 ¼ i12 p1 :
ð33Þ
The rotation angle of drive pulley 1 can be rewritten as ϕ1 ¼ 2ϕ2 −
π π cosðϕ2 Þ þ : 3 3
ð34Þ
Through Eq. (14), θ1 can be calculated as p θ1 ¼ arccos 1 ð1−i12 Þ −ϕ1 : a
ð35Þ
By derivation of pitch curve with respect to θ2 for driven pulley 2 in the polar coordinate system, we can obtain
′
p 2 ðθ2 Þ ¼
dðp1 i12 Þ di12 dθ2 ¼ p1 = : dθ2 dϕ2 dϕ2
ð36Þ
Eq. (34) can also be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1− pa1 ð1−i12 Þ 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q p 2 ðθ2 Þ ¼ p1 2 : p1 di12 a dϕ − 1− a ð1−i12 Þ ′
12 p1 di dϕ
2
ð37Þ
134
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
Then, the pitch curve of driven pulley 2 in the rectangular coordinate system can be given. (
0
x2 ¼ p2 cosðθ2 Þ−p2 sinðθ2 Þ 0 y2 ¼ p2 sinðθ2 Þ þ p2 cosðθ2 Þ
ð38Þ
The structural parameters of traditional JB04 crank-slider mechanical press is: e=150 mm, c=20 mm and central distance abetween two pulleys is equal to 300 mm. It is assumed that the pitch curve of drive pulley 1 is perfect round and its pitch curve equation in the polar coordinate system isp1 =25mm. Substituting the structural parameters into Eq. (41), the pitch curve of driven pulley 2 can be shown in Fig. 7. Fig. 7 shows that the pitch curve of non-circular driven pulley is approximately ellipse, rotation center of which is located at point (0, 0). The pitch curve is cambered outwards and suitable to be used as contour line. The circumference of pitch curve for non-circular driven pulley can be calculated as 2π
2π
L ¼ ∫0 p2 ðθ2 Þdθ2 ¼ ∫0 p2 ðθ2 Þ
dθ2 dϕ : dϕ2 2
ð39Þ
Eq. (39) can be also expressed as p1 π ϕ ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðϕ2 Þdϕ2 −∫ϕ21 L ¼ ∫ϕ21 p i p i dϕ2 : 20 1 12 20 1 12 a 1− pa1 ð1−i12 2 3
ð40aÞ
Where ϕ20 and ϕ21 represent the values of parameter ϕ2 when θ2 = 0 and θ2 = 2π respectively. Then, the circumference of driven pulley 2 can be calculated as ϕ
L ¼ −∫ϕ21 p i dϕ2 ¼ 4πp1 : 20 1 12
ð40bÞ
As can be seen from Eq. (40a), the circumference of non-circular driven pulley 2 is exactly twice than that of drive pulley 1. 4.1. Comparative analysis of kinematics As shown in Fig. 3, the displacement of slider can be expressed as s ¼ c−e þ ccosðϕ2 Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 −c2 sin2 ðϕ2 Þ:
ð41Þ
Where c represents the length of crank, e is the length of linkage. 60
40
Y-axis[mm]
20
0
-20
-40
-60 -40
-20
0
20
40
X-axis[mm] Fig. 7. Pitch curve of driven pulley.
60
80
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
135
By derivation of Eq. (31), the velocity of slider can be obtained. 0
1
ds dϕ2 c2 sinð2ϕ2 Þ B C ffiAω2 ¼ −@c sinðϕ2 Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν¼ dϕ2 dt 2 2 2 2 e −c sin ðϕ2 Þ
ð42Þ
By substituting Eqs. (6) and (29) into Eq. (32), the class speed of slider can be given. ′
ν ¼
! ν 1 c2 sinð2ϕ2 Þ 1 c sinðϕ2 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼− ω1 2 2 e2 −c2 sin2 ðϕ2 Þ 1 þ k sinðϕ2 Þ
ð43Þ
The class speed of slider is shown in Fig. 8 with varying values of k. As shown in Fig. 8, the class speed of slider is low and has little change when deep stamping operation and has a quick-return characteristic during non-load operation. As the value of k increases, average class speed keeps declining with little changes when stamping. However, the class speed during non-loading stage become large and its changes also become dramatical. The value of k can be determined in accordance with permissible speed of material or advance-to return-time ratio of mechanism. When k is equal to zero, the non-circular belt transmission will become the traditional belt transmission. Then, the rotation angle relation between two pulleys is ϕ1 ¼ 2ϕ2 :
ð44Þ
The displacement of slider for traditional mechanical press can be given. s ¼ c−e þ c cosðϕ2 Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 −c2 sin2 ðϕ2 Þ:
ð45Þ
The class speed of slider for traditional mechanical press can be expressed as ′
ν ¼
! 2 ν 1 c sinð2ϕ2 Þ 1 c sinðϕ2 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼− ω1 2 2 e2 −c2 sin2 ðϕ2 Þ 1 þ k sinðϕ2 Þ
ð46Þ
The displacement and class speed of slider for both traditional mechanical press and mechanical press with non-circular belt transmission are shown in Figs. 9 and 10 respectively. Dashed lines represent the displacement and class speed of slider for mechanical press based on non-circular belt transmission while the solid lines represent those of traditional press. Fig. 9 shows the stroke-angle behavior, which is attained by the non-circular belt transmission. The slider class speed in deep drawing can keep almost constant during the sheet metal forming at least over 30 mm before the lower dead center. In addition to the reduction of cycle time, the slider velocity of impact onto the sheet is also considerably reduced. Fig. 10 shows that 22 mm before the lower dead center, the class speed of impact is 10 times than the rotation 10
5
Class speed of slider
0 -5 -10 -15 -20
k=0.5 k=0.7 k=0.2
-25 -30 -35 0
2
4
6
8
10
Rotation angle of drive pulley 1[rad] Fig. 8. Class speed of slider with varying values of k.
12
136
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
40 Traditional mechanical press(k=0) Mechanical press with non-circular belt transmission(k=pi/6)
Displacement of slider[mm]
35 30 25 20 15 10 5 0
0
2
4
6
8
10
12
Rotation angle of drive pulley 1[rad] Fig. 9. Displacement of slider.
speed of driven pulley 2 when using the crank mechanism and only 6.5 times when operated with non-circular belt transmission. Therefore, the class speed of slider for mechanical press with non-circular belt transmission is more uniform and lower than that of traditional press and has a quick-return characteristic, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. 4.2. slack for non-circular belt transmission As the pitch curve of drive pulley is perfect round, Eq. (19) can be also expressed as C 1 C Ι ¼ p1 jθI −θ1 j:
ð47Þ
The main problem of calculating C1CΙ and C2CΠ is to determine the values of θΙ and θΠ. In order to solve this problem, the values of θΙ and θΠ can be singled out by numerical approach and the schematic flow chart is shown in Fig. 11. The step size of is ϕ1 assumed to be π/6 and the numerical results are shown in Table 1. 15 10
Class speed of slider
5 0 -5 -10 -15 Traditional mechanical Traditional mechanical press(k=0) press(k=0) Mechanical press with non-circular non-circular belt transmission (k=pi/6)
-20 -25 0
2
4
6
8
10
Rotation angle of drive pulley 1[rad] Fig. 10. Class speed of slider.
12
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
137
Fig. 11. The schematic flow chart for computation of θΙ and θΠ.
According to numerical results in Table 1, the theoretical maximum slack of non-circular belt transmission can be given. ΔL ¼ L max −L min ¼ 114:4 mm Numerical results demonstrate that large slack exists when non-circular belt transmission operates, and the slack for non-circular belt transmission mainly depends on the pitch curve shape. In order to compensate the excess slack during operation and ensure the operational condition, tensioner should be designed that will not be discussed in this paper. 5. Conclusion In this work, design method of non-circular belt transmission for mechanical press is proposed to generate a specific motion law. The pitch curve of non-circular driven pulley is obtained when the pitch curve of drive pulley is perfect round. Compared to traditional crank-slider mechanical press, numerical results reveal that mechanical press with non-circular belt transmission has a lower speed
Table 1 Numerical results of slack for non-circular belt transmission. ϕ1 (rad)
ϕ2 (rad)
θΙ (rad)
θΠ (rad)
p2 (mm)
p′2 (mm)
x2 (mm)
y2 (mm)
L (mm)
1.1875 2.618 4.1888 5.7596 7.1901 8.3776 9.2845 9.9484 10.472 10.9956 11.6596 12.5664
0.5236 1.0472 1.5807 2.0944 2.618 3.1416 3.6652 4.1888 4.7124 5.236 5.7596 6.2832
0.5106 −0.8876 −2.4466 −4.0292 −5.492 −6.7234 −7.674 −8.3698 −8.9051 −9.417 −10.049 −10.9121
1.1745 0.6832 0.1714 −0.364 −0.92 −1.487 −2.055 −2.61 −3.146 −3.657 −4.149 −4.629
63.09 72.6725 76.1799 72.6725 63.09 50 36.91 27.3275 23.8201 27.3275 36.91 50
−22.6725 −13.09 0 13.09 22.6725 26.1799 22.6725 13.09 0 −13.09 −22.6725 −26.1799
314.4787 301.3754 287.0037 275.5281 269.5007 269.7445 275.88 286.6984 300.0937 312.8776 321.1893 321.9222
65.458 73.8292 75.0632 69.669 59.7007 47.6444 35.9808 27.2251 23.8199 27.4282 37.781 52.0077
889.36 869.39 838.24 804.35 795.42 788.49 791.39 812.51 839.04 864.68 882.64 902.88
138
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
under deep drawing operation and a quick-return characteristic under non-loading stage, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. The slack of non-circular belt transmission for mechanical press is calculated under operation, which can provide a theoretical basis for the design of take-up mechanism. Acknowledgments This paper has been partially supported by the research projects: “Key Scientific and Technological Project”, grant # BE2006036 and “Jiangsu Fund Project for Transformation of Scientific and Technological Achievements”, grant # BA2008030. References [1] W.W. Zheng, K.J. Wu, Mechanical Principal (M), 7th edition Higher Education Press, Beijing, 1997. [2] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory (M), second edition Cambridge University Press, New York, 2004. [3] D.B. Dooner, Function generation utilizing an eight-link mechanism and optimized non-circular gear elements with application to automotive steering (J), Journal of Mechanical Design (215) (2001) 847–857. [4] Erika Ottaviano, Domenico Mundo, et al., Numerical and experimental analysis of non-circular gears and cam-follower systems as function generators (J), Mechanism and Machine Theory 43 (2008) 996–1008. [5] D. Mundo, H.S. Yan, Kinematic optimization of ball-screw transmission mechanisms (J), Mechanism and Machine Theory 42 (2007) 34–47. [6] W. Wunderlich, P. Zenow, Contribution to the geometry of elliptic gears (J), Mechanism and Machine Theory 10 (1975) 273–278. [7] F.S. Li, Non-circular Gear (M), Mechanical Industry Press, Beijing, 1975. [8] F.S. Li, Transmission Design of Non-circular Gear and Special Gear (M), Mechanical Industry Press, Beijing, 1983. [9] X.T. Wu, G.H. Wang, Non-circular Gear and Non-uniform transmission (M), Mechanical Industry Press, Beijing, 1997. [10] E. Doege, M. Hindersmann, Fertigungsgerechte Kurbelpressenkinematik durch Unrundzahnrader (J), VDI-Z Special Antriebstechnik 96 (1996) 74–77. [11] E. Doege, M. Hindersmann, Optimized kinematics of mechanical presses with noncircular gears (J), Annals of the CIBP 46 (1997) 213–216. [12] J.J. Uicker, G.R. Pennock, J.E. Shigley, Theory of Machines and Mechanisms (M), Oxford University Press, New York, 2003. [13] K. Hain, Das Zweikurven-Hebelgetriebe, eine Brucke zwischen gleichmaig und ungleichmaig ubersetzenden Getrieben (J), VDI-Berichte 374 (1980) 43–47. [14] F. Freudenstein, C.K. Chen, Variable ratio chain drives with noncircular sprockets and minimum slack theory and application (J), Journal of Mechanics 113 (1991) 253–262. [15] L.X. Cao, J. Liu, Study of calculation method of transmission ratio's function based on non-circular chain transmission (J), Journal of Mechanical Engineering (1997). [16] L.X. Cao, J. Liu, Study of calculation method of transmission ratio's function based on non-circular chain transmission (J), Journal of Mechanical Engineering 37 (2001) 35–39. [17] L.P. Feng, Study of non-circular chain pulley's pitch curve (J), Mechanical Design and Study (6) (2005) 7–9. [18] Carlo Innocenti, Davide Paganelli, Designing synchronous belt transmissions with variable velocity ratio (J), Journal of Mechanical Design 130 (2008) 1–7. [19] Hellmuth Stachel, Gears and belt drives for non-uniform transmission, [C], In: Proceedings of EQCOMES 08, 2009. [20] J. Hoschek, Konstruktion von kettengetrieben mit veränderlicher Übersetzung mit hilfe von bézier kurven (J), Forschung im Ingenieurwesen 48 (1982) 81–87.
Contents lists available at SciVerse ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Non-circular belt transmission design of mechanical press Enlai Zheng, Fang Jia, Hongwei Sha ⁎, Suhao Wang School of Mechanical Engineering, Southeast University, Nanjing 211189, China
a r t i c l e
i n f o
Article history: Received 12 May 2011 received in revised form 11 July 2012 accepted 12 July 2012 Available online 21 August 2012 Keywords: Non-circular Press Crank-slider Belt transmission Slack
a b s t r a c t The quality of parts manufactured by metal forming operations depends on the kinematics of the mechanical press in a large degree. Non-circular belt transmission with a rotational angle-dependent speed ratio in the press drive mechanism offers a new way to obtain optimum stroke-time behaviors for specific metal forming operations in terms of manufacturing. In this work, the mathematical model for mechanical press driven by non-circular belt transmission is established with the concept of polar coordinate equation for tangent curves introduced, and a general design method of non-circular belt pulley pitch curves and slack for transmission system is proposed. Compared with traditional crank-slider press, numerical simulation results demonstrate that mechanical press with non-circular belt transmission has a lower speed under deep drawing operation and a quick-return characteristic under non-loading stage, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. Furthermore, the slack of non-circular belt transmission for mechanical press is calculated under operation, which can provide a theoretical basis for the design of take-up mechanism. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The mechanical press, as “less/non-chip finish” high-effective machine tool as well as one of the key industrial equipment, can manufacture metal parts closer to their final shapes and comply better with the requirements of highly advocated clean and green production. Traditional mechanical press commonly consists of belt transmission and crank-slider mechanisms, which are used to transmit the rotary motion of electrical machine into translational movement of slider so as to realize stamping operation. As the conventional belt transmission ratio is constant, the motion law of the slider is sine shape [1] and cannot meet the requirements of blanking and deep stamping products that are made of poor plasticity and embrittlement material. Therefore, to overcome this problem, a type of non-circular belt transmission with variable velocity ratio should be designed and a specific output motion law of slider for mechanical press should be generated, which includes a low speed under loading stage and a high speed under non-loading stage. Many researchers have studied majority kinds of transmission with variable velocity ratio for the special law of motion. The application of non-circular gears in function generating mechanism has been proposed and discussed [2,3]. By designing a pair of non-circular gears, which are able to perform a proper gear ratio function, the output member of a mechanism can be effectively forced to move according to a prescribed law of motion when operated at a constant input velocity [4,5]. For the special law of motion obtained by a pair of meshing elliptical gears, a design solution was given by Wunderlich W and Zenow P [6]. F.S Li and X.T Wu [7–9] proposed the use of non-circular gears to achieve the non-uniform transmission ratio. The use of non-circular gears in the drive of mechanical press offers a new way of meeting the demands on the kinematics. Investigations at the Institute for Metal
⁎ Corresponding author at: School of Mechanical Engineering, Southeast University, 2# Southeast Road, JiangNing District, NanJing 211189, China. Tel.: +86 159 5107 5321(Cell phone); fax: +86 25 52090504. E-mail address: [email protected] (H. Sha). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2012.07.004
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
127
Forming and Metal Forming Machine Tools of Hanover University have shown that in this simple manner all the relevant uninterrupted motions of the ram can be achieved for various forming processes [10]. E Doege and M Hindersmann [11] also came up with the non-circular gears to achieve the optimized kinematics of press, but the specific method of design was not given. For application of cam-follower mechanisms, these devices can also reproduce almost any characteristics of the follower's motion and their manufacturing costs as well as the number and dimensions of moving parts are rather limited [12,13]. F Freudenstein and C.K Chen [14] developed a new mechanical component: variable ratio chain drives with noncircular sprockets and minimum slack, and illustrate applications to the design of an optimum bicycle configuration and a harmonic-motion generator. X.L Cao and J Liu [15] designed a type of chain transmission with a variable velocity ratio and outlined that two spans of a chain are never simultaneous in tension, and the chain slack can undergo moderate periodical changes while the transmission runs. Based on the results [15], J Liu and L.P Feng [16,17] studied the algorithmic computation of Non-circular chain transmission. Carlo Innocenti and Davide Paganelli [18] presented a new procedure to design a two-pulley synchronous belt transmission connecting, with no belt tensioner, two parallel-axis shafts with a variable velocity ratio. The procedure takes into account the limited number of choices for pitch and length of off-the-shelf synchronous belts. Hellmuth Stachel [19] treated the geometry of tooth profiles and pulleys and their algorithmic computation as well as the relation between tooth profiles and ‘strict’ cases without take-up pulley. A geometric solution was proposed by Hoschek J [20], who pointed out that some tangents to the pulley contours were determined through an iterative procedure, and the envelope of these tangents was approximated by a sequence of Bezier splines. The existing approaches mainly apply non-circular gears transmission, belt transmission, chain transmission and cam-follower mechanism to generate specific motion law. Only non-circular gears transmission has been applied to mechanical press in order to optimize its kinematics and meet specific machine requirement. Although non-circular gears have the advantage of lower weight-to-strength ratios and absence of gross separation or decoupling of moving parts [4], non-circular belt transmission is more attractive than gear transmissions if the center distance is relevant or if lubrication is unavailable as well as if the cost of design and manufacture is concerned. Therefore, it is important to design non-circular belt transmission for mechanical press so as to obtain optimized kinematics and improve its manufacturing flexibility. The primary contribution of this work is to design non-circular belt transmission for mechanical press in order to obtain optimized kinematics and meet the machine requirement of deep drawing. Based on the polar equation for tangent curves, the pitch curve of non-circular pulley is designed. Compared with traditional crank-slider press, numerical simulation results demonstrate that mechanical press with non-circular belt transmission has a lower speed under deep drawing operation and a quick-return characteristic under non-loading stage, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. Furthermore, the slack of non-circular belt transmission for mechanical press is calculated under operation, which can provide a theoretical basis for the design of take-up mechanism. This paper is organized as follows: in Section 2, the physical structure and machine requirements of mechanical press are described briefly. In Section 3, the pitch curve of non-circular pulley is designed and slack of belt transmission is obtained. In Section 4, numerical example is simulated and comparative analysis is performed. Section 5 summarizes the design of non-circular belt transmission for mechanical press and the conclusions. 2. Physical description and press machine requirements The physical figure of mechanical press is shown in Fig. 1 and the working principal sketch of press is shown in Fig. 2: Motor (1) passes the movement on to pulley (10) through the belt and then passes motion on to crankshaft by gears (2, 3). The rod (11) connects with crankshaft on the top and with the slider (6) on the bottom, which can translate rotary movement of crankshaft into linear reciprocating motion of slider. The upper die (10) is installed on the slider and lower die (7) on the plate (9). When the sheet metal is put between the upper and lower dies, Press can carry out blanking or other deformation techniques. In order to meet the manufacturing process needs, press is equipped with clutch (4) and brake (12) that can sometimes get the slider to move or stop. The press's working time under load is short throughout the work cycle and most of the time is for empty trip of no-load. In order that motor works under uniform load and makes use of energy with high efficiency, crank-slider press often installs flywheel. As Fig. 2 shows, the big pulley (13) serves as flywheel. One manufacturing cycle, which corresponds to one stroke of the mechanical press, goes through three stages: loading, forming and removing the part. Instead of the loading and removal stages we often find feeding the sheet, especially in sheer cutting. For this, the press slider must have a minimum height for a certain time. During the forming period the slider should have a particular velocity curve, which will be gone into below. The transitions between the periods should take place as quickly as possible to ensure short cycle time. In deep drawing operations, the velocity of impact should be as low as possible to avoid the impact when striking the sheet. On the one hand, velocity must be sufficient for lubrication during forming. On the other hand, we have to consider the rise in the strain rate which creates greater forces and may cause fractures at the transition from the punch radius to the side wall of the part. 3. Design formulation for non-circular belt transmission In typical arrangements for mechanical press shown in Fig. 3, a pair of non-circular pulleys is used to drive a crank-slider mechanism, so that the slider is forced to move according to a specific law of motion. The design of this mechanism consists of two phases, namely the synthesis of the pitch curves, starting from the requested law of motion, and the computation of slack.
128
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
3.1. Tangent equation in polar and rectangular coordinate system As shown in Fig. 4, it is assumed that tangent of the plane curve at point C(x,y) is t, the distance between the original point O and tangent curve is p, and the intersection angle between axis OX and vertical line On is θ. Then, p is the univalent function with respect to θ. p ¼ pðθÞ
ð1Þ
Eq. (1) can be used to describe the plane curve C in the polar coordinate system. For closed plane convex curve, p is the periodic function with respect to variation θ and its period is2π. In the polar coordinate system, the curve equation is r = r(ϕ). Referring to the geometric relation in Fig. 4, the function p can be given. p ¼ r sinðuÞ
ð2Þ
Where the geometric relation between θ, ϕ and u is θ ¼ ϕ þ u−90. Then, the tangent equation of the plane curve can be expressed as follows. x cosðθÞ þ y sinðθÞ ¼ pðθÞ
ð3Þ
Eq. (3) can be considered as linear cluster with respect to variable θ and the plane curve can be regarded as the envelope of the linear cluster. By derivation of Eq. (3), we can obtain ′
−x sinðθÞ þ y cosðθÞ ¼ p ðθÞ:
ð4Þ
By solving the Eqs. (3) and (4) simultaneously, we can transform the equation of plane curve in the polar coordinate system (p, θ) into the equation in the rectangular coordinate system (x, y). (
x ¼ pðθÞcosθ−p′ ðθÞsinðθÞ y ¼ pðθÞsinθ þ p′ ðθÞcosðθÞ
ð5Þ
Fig. 1. Physical figure of mechanical press.
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
129
Fig. 2. Composition and principle sketch of press. 1—motor, 2—small gear, 3—large gear, 4—clutch; 5—crankshaft, 6—slider, 7—Lower die, 8—Foundation, 9—Worktable, 10—Upper die, 11—rod, 12—brake, 13—large pulley, 14—small pulley.
3.2. Design formulation of the pitch curve The non-circular belt transmission, as shown in Fig. 5, consists of two non-circular pulleys and a belt. Pulley 1 is the drive wheel, while pulley 2 is the driven one. It is assumed that the direction of rotation for the two non-circular pulleys is anticlockwise. Therefore, the instantaneous transmission ratio can be given.
i12 ¼
ω1 dϕ1 ¼ ω2 dϕ2
ð6Þ
Where ω1 and ω2 represent the instantaneous angular velocity for pulleys 1 and 2, and ϕ1 and ϕ2 are the instantaneous rotation angles respectively. It is assumed that the transmission ration is differentiable. In an instant, the pitch curve of belt is tangent with two non-circular pulleys at the points of C1 and C2. Then, the line speed of pulley 1 at the point of C1 is equal to r1ω1 and that of pulley 2 at the point of C2 equal to r2ω2. As the belt stretch can be neglected, the sub-speed of r1ω1 and r2ω2 along the direction of belt should be equivalent and can be formulated in the form r 1 ω1 cosβ1 ¼ r 2 ω2 cosβ2 :
ð7Þ
Where β1 and β2 represent the angle between the former vertical line and radius O1C1 and O2C2.
Fig. 3. Schematic diagram of non-circular belt transmission for mechanical press.
130
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
Fig. 4. Tangent of curve in the polar coordinate system.
Then, through Eqs. (6) and (7), the transmission ratio i12 can also be expressed as i12 ¼
ω1 r 2 cosβ2 p2 ¼ ¼ : ω2 r 1 cosβ1 p1
ð8Þ
Where p1 and p2 are the vertical distances from the rotation center of O1 and O2 to belt pitch line. The transmission ratio i12 is variable as a result of non-circular pulleys and can be expressed as a function with respect to the rotation angle ϕ2 of driven pulley. i12 ¼ f ðϕ2 Þ
ð9Þ
When the pitch curve of belt is extended to intersect with the line O1O2, the intersection point is O1O2. Therefore, the transmission ration i12 can be rewritten as i12 ¼
p2 a þ b : ¼ b p1
ð10Þ
Where a is the central distance between two non-circular pulleys, and b is the distance between point O1 and point O3. The Eq. (10) can also be expressed in the form b¼
a : i12 −1
ð11Þ
It is assumed that the coordinate system (O1 − X1Y1) is fixed on pulley 1 and the coordinate system (O2 − X2Y2) is fixed on pulley 2. When the transmission ratio i12 and the distance a between O1 and O2 are determined, ϕ1 and ϕ2 are zero in the starting position and
Fig. 5. Schematic diagram of non-circular belt transmission.
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
131
the direction of axis of X1 and X2 overlap with the line O1O2. If the rotation angle of pulley 1 is ϕ1, the rotation angle of pulley 2 can be evaluated as ϕ
ϕ2 ¼ ∫0 1
1 dϕ : i12 1
ð12Þ
Then, we can obtain the relation as follows in Fig. 4. θ2 ¼ θ1 þ ϕ1 −ϕ2
ð13Þ
Where θ1 is the angle between p1 and the axis of X1 and θ2 is the angle between p2 and X2. When the pitch curve of pulley 1 is assumed to be p1 = p1(θ1)and determined, we can obtain cosðθ1 þ ϕ1 Þ ¼ −
p1 ðθ1 Þ p ðθ Þði −1Þ ¼ − 1 1 12 : b a
ð14Þ
Where (p1, θ1) and (p2, θ2) represent the tangent equations of pitch curves for pulleys 1, 2 in the polar coordinate system. Once rotation angle ϕ1 is given, p1 and θ1 can be obtained from Eqs. (11) and (14). Then the pitch curve of pulley 2 can be expressed as p2 = p2(θ2). In a similar manner, when the pitch curve of driven pulley 2 and transmission ratio i12 are given, the pitch curve of drive pulley 1 can also be obtained. 3.3. Slack of non-circular belt transmission As the pitch curve of pulley is non-circular, the total length of envelop for two pulleys varies with respect to rotation angle. Therefore, the slack calculation of non-circular belt transmission for mechanical press is essential and can provide theoretical basis for design of take-up mechanism as well as the optimization of minimum slack. 3.3.1. Arc length of curve By derivation of Eq. (5) with respect to θ, we can obtain h i 8 0 00 < x ¼ − pðθÞ þ p ðθÞ sinðθÞ h i 0 00 : y ¼ pðθÞ þ p ðθÞ cosðθÞ:
ð15Þ
The curvature radius of plane curve C can be given. ρ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 0 2ffi 0 2 x þ y ¼ pðθÞ þ p ðθÞ
ð16Þ
The arc length of plane curve C can be calculated as θ h i 00 s ¼ ∫ pðθÞ þ p ðθÞ dθ:
ð17Þ
θ0
3.3.2. Length of pitch curve for belt As shown in Fig. 6, the length of pitch curve for belt consists of four parts and can be calculated for every given ϕ2 at any instantaneous time. L ¼ C1 CΙ þ C1 C2 þ C2 CΠ þ CΙ CΠ
h i h i 00 00 θ θ ¼ ∫θΙ1 p1 ðθ1 Þ þ p1 ðθ1 Þ dθ1 þ C 1 C 2 þ ∫θ2Π p2 ðθÞ2 þ p2 ðθ2 Þ dθ2 þ C Ι C Π
ð18Þ
The length of arc C1CΙ can be calculated as h i 00 θ C 1 C Ι ¼ ∫θΙ1 p1 ðθ1 Þ þ p1 ðθ1 Þ dθ1 :
ð19Þ
The length of arc C2CΠ can be calculated as h i 00 θ C 2 C Π ¼ ∫θ2Π p2 ðθ2 Þ þ p2 ðθ2 Þ dθ2 0 dθ θ θ ¼ ∫θ2Π p2 ðθ2 Þ 2 dϕ2 þ p2 θ2Π : dϕ2
ð20Þ
132
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
Fig. 6. Length of pitch curve for belt.
As p2(θ2) and dθ2/dϕ2 are functions with respect to ϕ2, the values of ϕ1, θ1, θ2, p2, p2', x1, y1, x2, and y2 can be obtained from Eqs. (5) and (14) for every given ϕ2. To obtain the length of C1CΙ and C2CΠ, the values of θΙ and θΠ must be calculated in advance. Referring to the schemes in Fig. 5, we can obtain θ1 þ ϕ1 ¼ θ2 þ ϕ2
ð21aÞ
ϕ1 þ θΙ ¼ ϕ2 þ θΠ :
ð21bÞ
For every given ϕ1, in addition that a set of θ1 and θ2 can meet the Eq. (21a), another corresponding group of θΙ and θΠ that satisfies the Eq. (21b) can also be found. If ϕ1 + θΙ and ϕ2 + θΠ belong to the third quadrant or fourth quadrant at the same time, that is
∘
∘
180 bϕ1 þ θΙ b360 180∘ bϕ2 þ θΠ b360∘ :
ð22Þ
' Then, the values of θΙ and θΠ are what we need and ϕΙ,ϕΠ, pΠ, pΠ , xΙ, yΙ, xΠ, yΠ can also be obtained from Eqs. (5) to (14) respectively ' in the same way. By substituting θ1, θ2, θΙ, θΠ, p1, p2, p1', pΠ into Eqs. (19) and (20), the length of C1CΙ and C2CΠ can be obtained. Coordinate values of four points C1(x1, y1), CΙ(xΙ, yΙ), C2(x2, y2) and CΠ(xΠ, yΠ) in the rectangular coordinate system (x, y) can be calculated by Eq. (5) and can be expressed as
8 xΙ ¼ p1 cosðθΙ Þ > > < y ¼ p sinðθ Þ Ι 1 I 0 xΠ ¼ p2 cosðθΠ Þ−p2 sinðθΠ Þ > > : 0 yΠ ¼ p2 sinðθΠ Þ þ p2 cosðθΠ Þ:
ð23Þ
C1(x1, y1) and CΙ(xΙ, yΙ) are the points in the coordinates O1 − X1Y1, while C2(x2, y2) and CΠ(xΠ, yΠ) are points in the coordinatesO2 − X2Y2. In order to calculate the length of C 1 C 2 and C Ι C Π , four points can be transformed to the coordinates of O − XY. Then, C1(X1, Y1), C2(X2, Y2), CΙ(XΙ , YΙ) and CΠ(XΠ, YΠ) can be obtained by means of rotating coordinate system. 8 X > > < 1 Y1 > > X2 : Y2
¼ x1 cosðϕ1 Þ−y1 sinðϕ1 Þ ¼ x1 sinðϕ1 Þ þ y1 cosðϕ1 Þ ¼ x2 cosðϕ2 Þ−y2 sinðϕ2 Þ þ a ¼ x2 sinðϕ2 Þ þ y2 cosðϕ2 Þ
8 X ¼ xΙ cosðϕΙ Þ−yΙ sinðϕΙ Þ > > < Ι Y Ι ¼ xΙ sinðϕΙ Þ þ yΙ cosðϕΙ Þ > > X Π ¼ xΠ cosðϕΠ Þ−yΠ sinðϕΠ Þ þ a : Y Π ¼ xΠ sinðϕΠ Þ þ yΠ cosðϕΠ Þ
ð24Þ
ð25Þ
Then, the length of C 1 C 2 and C Ι C Π can be calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < C C ¼ ðX −X Þ2 þ ðY −Y Þ2 1 2 2 1 2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : : C C ¼ ðX −X Þ2 þ ðY −Y Þ2 Ι Π Π Ι Π Ι
ð26Þ
3.3.3. Slack of belt As the pitch curve of pulley is non-circular, the total length of envelop for two pulleys varies with respect to rotation angle. It is assumed that both sides of the belt for non-circular belt transmission are tight. Then, the difference between maximum length and minimum length of belt during a period is called slack of non-circular belt transmission.
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
133
For every given ϕ21, ϕ22… ϕ2n, length L1, L2… Ln of belt can be obtained respectively. Then, the theoretical slack is given. ΔL ¼ L max −L min
ð27Þ
4. Results and discussion The traditional JB04 crank-slider mechanical press with the non-circular belt transmission applied as front mechanism is taken as a numerical example. It is assumed that the transmission ratio i12 is a function with respect to ϕ2, its period is 2π, and the pitch curve of drive pulley 1 is perfect round. Then, the transmission ration function can be expressed as i12 ¼ A þ B sinðϕ2 Þ ¼ Að1 þ k sinðϕ2 ÞÞ:
ð28Þ
Where k represents the ratio B/A. The transmission ratio function must be positive during the press process, then |k| b 1. When k is equal to zero, transmission ratio is a constant. The pitch curve of driven pulley 2 will also be perfect round, which is the same as the traditional belt transmission. It is assumed that the average transmission ratio is 2 in this work, and then we can obtain i12 ¼ 2ð1 þ k sinðϕ2 ÞÞ:
ð29Þ
By substituting the Eq. (6) into Eq. (29) and integration, the solution of Eq. (29) can be given. ϕ1 ¼ 2ðφ2 −k cosðϕ2 ÞÞ þ C
ð30Þ
The initial condition is that when ϕ1 is equal to zero, ϕ2 is equal to zero. Therefore, C is equal to 2k. It is assumed that the advance-to return-time ratio of crank-slider mechanism in this work is equal to 2. That means the slider's working stroke accounts for two thirds of the total stroke. When ϕ2 is equal to zero, ϕ1 is also equal to zero. When ϕ2 is equal to π, ϕ1 is equal to 2π + 4k. Then, 2π þ 4k 2 ¼ : 4π 3
ð31Þ
The solution of Eq. (34) is k = π/6 and the transmission ratio can also be expressed as π i12 ¼ 2 1 þ sinðϕ2 Þ : 6
ð32Þ
The pitch curve of driven pulley 2 can be calculated as p2 ¼ i12 p1 :
ð33Þ
The rotation angle of drive pulley 1 can be rewritten as ϕ1 ¼ 2ϕ2 −
π π cosðϕ2 Þ þ : 3 3
ð34Þ
Through Eq. (14), θ1 can be calculated as p θ1 ¼ arccos 1 ð1−i12 Þ −ϕ1 : a
ð35Þ
By derivation of pitch curve with respect to θ2 for driven pulley 2 in the polar coordinate system, we can obtain
′
p 2 ðθ2 Þ ¼
dðp1 i12 Þ di12 dθ2 ¼ p1 = : dθ2 dϕ2 dϕ2
ð36Þ
Eq. (34) can also be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1− pa1 ð1−i12 Þ 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q p 2 ðθ2 Þ ¼ p1 2 : p1 di12 a dϕ − 1− a ð1−i12 Þ ′
12 p1 di dϕ
2
ð37Þ
134
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
Then, the pitch curve of driven pulley 2 in the rectangular coordinate system can be given. (
0
x2 ¼ p2 cosðθ2 Þ−p2 sinðθ2 Þ 0 y2 ¼ p2 sinðθ2 Þ þ p2 cosðθ2 Þ
ð38Þ
The structural parameters of traditional JB04 crank-slider mechanical press is: e=150 mm, c=20 mm and central distance abetween two pulleys is equal to 300 mm. It is assumed that the pitch curve of drive pulley 1 is perfect round and its pitch curve equation in the polar coordinate system isp1 =25mm. Substituting the structural parameters into Eq. (41), the pitch curve of driven pulley 2 can be shown in Fig. 7. Fig. 7 shows that the pitch curve of non-circular driven pulley is approximately ellipse, rotation center of which is located at point (0, 0). The pitch curve is cambered outwards and suitable to be used as contour line. The circumference of pitch curve for non-circular driven pulley can be calculated as 2π
2π
L ¼ ∫0 p2 ðθ2 Þdθ2 ¼ ∫0 p2 ðθ2 Þ
dθ2 dϕ : dϕ2 2
ð39Þ
Eq. (39) can be also expressed as p1 π ϕ ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðϕ2 Þdϕ2 −∫ϕ21 L ¼ ∫ϕ21 p i p i dϕ2 : 20 1 12 20 1 12 a 1− pa1 ð1−i12 2 3
ð40aÞ
Where ϕ20 and ϕ21 represent the values of parameter ϕ2 when θ2 = 0 and θ2 = 2π respectively. Then, the circumference of driven pulley 2 can be calculated as ϕ
L ¼ −∫ϕ21 p i dϕ2 ¼ 4πp1 : 20 1 12
ð40bÞ
As can be seen from Eq. (40a), the circumference of non-circular driven pulley 2 is exactly twice than that of drive pulley 1. 4.1. Comparative analysis of kinematics As shown in Fig. 3, the displacement of slider can be expressed as s ¼ c−e þ ccosðϕ2 Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 −c2 sin2 ðϕ2 Þ:
ð41Þ
Where c represents the length of crank, e is the length of linkage. 60
40
Y-axis[mm]
20
0
-20
-40
-60 -40
-20
0
20
40
X-axis[mm] Fig. 7. Pitch curve of driven pulley.
60
80
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
135
By derivation of Eq. (31), the velocity of slider can be obtained. 0
1
ds dϕ2 c2 sinð2ϕ2 Þ B C ffiAω2 ¼ −@c sinðϕ2 Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν¼ dϕ2 dt 2 2 2 2 e −c sin ðϕ2 Þ
ð42Þ
By substituting Eqs. (6) and (29) into Eq. (32), the class speed of slider can be given. ′
ν ¼
! ν 1 c2 sinð2ϕ2 Þ 1 c sinðϕ2 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼− ω1 2 2 e2 −c2 sin2 ðϕ2 Þ 1 þ k sinðϕ2 Þ
ð43Þ
The class speed of slider is shown in Fig. 8 with varying values of k. As shown in Fig. 8, the class speed of slider is low and has little change when deep stamping operation and has a quick-return characteristic during non-load operation. As the value of k increases, average class speed keeps declining with little changes when stamping. However, the class speed during non-loading stage become large and its changes also become dramatical. The value of k can be determined in accordance with permissible speed of material or advance-to return-time ratio of mechanism. When k is equal to zero, the non-circular belt transmission will become the traditional belt transmission. Then, the rotation angle relation between two pulleys is ϕ1 ¼ 2ϕ2 :
ð44Þ
The displacement of slider for traditional mechanical press can be given. s ¼ c−e þ c cosðϕ2 Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 −c2 sin2 ðϕ2 Þ:
ð45Þ
The class speed of slider for traditional mechanical press can be expressed as ′
ν ¼
! 2 ν 1 c sinð2ϕ2 Þ 1 c sinðϕ2 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼− ω1 2 2 e2 −c2 sin2 ðϕ2 Þ 1 þ k sinðϕ2 Þ
ð46Þ
The displacement and class speed of slider for both traditional mechanical press and mechanical press with non-circular belt transmission are shown in Figs. 9 and 10 respectively. Dashed lines represent the displacement and class speed of slider for mechanical press based on non-circular belt transmission while the solid lines represent those of traditional press. Fig. 9 shows the stroke-angle behavior, which is attained by the non-circular belt transmission. The slider class speed in deep drawing can keep almost constant during the sheet metal forming at least over 30 mm before the lower dead center. In addition to the reduction of cycle time, the slider velocity of impact onto the sheet is also considerably reduced. Fig. 10 shows that 22 mm before the lower dead center, the class speed of impact is 10 times than the rotation 10
5
Class speed of slider
0 -5 -10 -15 -20
k=0.5 k=0.7 k=0.2
-25 -30 -35 0
2
4
6
8
10
Rotation angle of drive pulley 1[rad] Fig. 8. Class speed of slider with varying values of k.
12
136
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
40 Traditional mechanical press(k=0) Mechanical press with non-circular belt transmission(k=pi/6)
Displacement of slider[mm]
35 30 25 20 15 10 5 0
0
2
4
6
8
10
12
Rotation angle of drive pulley 1[rad] Fig. 9. Displacement of slider.
speed of driven pulley 2 when using the crank mechanism and only 6.5 times when operated with non-circular belt transmission. Therefore, the class speed of slider for mechanical press with non-circular belt transmission is more uniform and lower than that of traditional press and has a quick-return characteristic, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. 4.2. slack for non-circular belt transmission As the pitch curve of drive pulley is perfect round, Eq. (19) can be also expressed as C 1 C Ι ¼ p1 jθI −θ1 j:
ð47Þ
The main problem of calculating C1CΙ and C2CΠ is to determine the values of θΙ and θΠ. In order to solve this problem, the values of θΙ and θΠ can be singled out by numerical approach and the schematic flow chart is shown in Fig. 11. The step size of is ϕ1 assumed to be π/6 and the numerical results are shown in Table 1. 15 10
Class speed of slider
5 0 -5 -10 -15 Traditional mechanical Traditional mechanical press(k=0) press(k=0) Mechanical press with non-circular non-circular belt transmission (k=pi/6)
-20 -25 0
2
4
6
8
10
Rotation angle of drive pulley 1[rad] Fig. 10. Class speed of slider.
12
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
137
Fig. 11. The schematic flow chart for computation of θΙ and θΠ.
According to numerical results in Table 1, the theoretical maximum slack of non-circular belt transmission can be given. ΔL ¼ L max −L min ¼ 114:4 mm Numerical results demonstrate that large slack exists when non-circular belt transmission operates, and the slack for non-circular belt transmission mainly depends on the pitch curve shape. In order to compensate the excess slack during operation and ensure the operational condition, tensioner should be designed that will not be discussed in this paper. 5. Conclusion In this work, design method of non-circular belt transmission for mechanical press is proposed to generate a specific motion law. The pitch curve of non-circular driven pulley is obtained when the pitch curve of drive pulley is perfect round. Compared to traditional crank-slider mechanical press, numerical results reveal that mechanical press with non-circular belt transmission has a lower speed
Table 1 Numerical results of slack for non-circular belt transmission. ϕ1 (rad)
ϕ2 (rad)
θΙ (rad)
θΠ (rad)
p2 (mm)
p′2 (mm)
x2 (mm)
y2 (mm)
L (mm)
1.1875 2.618 4.1888 5.7596 7.1901 8.3776 9.2845 9.9484 10.472 10.9956 11.6596 12.5664
0.5236 1.0472 1.5807 2.0944 2.618 3.1416 3.6652 4.1888 4.7124 5.236 5.7596 6.2832
0.5106 −0.8876 −2.4466 −4.0292 −5.492 −6.7234 −7.674 −8.3698 −8.9051 −9.417 −10.049 −10.9121
1.1745 0.6832 0.1714 −0.364 −0.92 −1.487 −2.055 −2.61 −3.146 −3.657 −4.149 −4.629
63.09 72.6725 76.1799 72.6725 63.09 50 36.91 27.3275 23.8201 27.3275 36.91 50
−22.6725 −13.09 0 13.09 22.6725 26.1799 22.6725 13.09 0 −13.09 −22.6725 −26.1799
314.4787 301.3754 287.0037 275.5281 269.5007 269.7445 275.88 286.6984 300.0937 312.8776 321.1893 321.9222
65.458 73.8292 75.0632 69.669 59.7007 47.6444 35.9808 27.2251 23.8199 27.4282 37.781 52.0077
889.36 869.39 838.24 804.35 795.42 788.49 791.39 812.51 839.04 864.68 882.64 902.88
138
E. Zheng et al. / Mechanism and Machine Theory 57 (2012) 126–138
under deep drawing operation and a quick-return characteristic under non-loading stage, which is suitable for blanking and deep drawing products that are made of poor plasticity and embrittlement material. The slack of non-circular belt transmission for mechanical press is calculated under operation, which can provide a theoretical basis for the design of take-up mechanism. Acknowledgments This paper has been partially supported by the research projects: “Key Scientific and Technological Project”, grant # BE2006036 and “Jiangsu Fund Project for Transformation of Scientific and Technological Achievements”, grant # BA2008030. References [1] W.W. Zheng, K.J. Wu, Mechanical Principal (M), 7th edition Higher Education Press, Beijing, 1997. [2] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory (M), second edition Cambridge University Press, New York, 2004. [3] D.B. Dooner, Function generation utilizing an eight-link mechanism and optimized non-circular gear elements with application to automotive steering (J), Journal of Mechanical Design (215) (2001) 847–857. [4] Erika Ottaviano, Domenico Mundo, et al., Numerical and experimental analysis of non-circular gears and cam-follower systems as function generators (J), Mechanism and Machine Theory 43 (2008) 996–1008. [5] D. Mundo, H.S. Yan, Kinematic optimization of ball-screw transmission mechanisms (J), Mechanism and Machine Theory 42 (2007) 34–47. [6] W. Wunderlich, P. Zenow, Contribution to the geometry of elliptic gears (J), Mechanism and Machine Theory 10 (1975) 273–278. [7] F.S. Li, Non-circular Gear (M), Mechanical Industry Press, Beijing, 1975. [8] F.S. Li, Transmission Design of Non-circular Gear and Special Gear (M), Mechanical Industry Press, Beijing, 1983. [9] X.T. Wu, G.H. Wang, Non-circular Gear and Non-uniform transmission (M), Mechanical Industry Press, Beijing, 1997. [10] E. Doege, M. Hindersmann, Fertigungsgerechte Kurbelpressenkinematik durch Unrundzahnrader (J), VDI-Z Special Antriebstechnik 96 (1996) 74–77. [11] E. Doege, M. Hindersmann, Optimized kinematics of mechanical presses with noncircular gears (J), Annals of the CIBP 46 (1997) 213–216. [12] J.J. Uicker, G.R. Pennock, J.E. Shigley, Theory of Machines and Mechanisms (M), Oxford University Press, New York, 2003. [13] K. Hain, Das Zweikurven-Hebelgetriebe, eine Brucke zwischen gleichmaig und ungleichmaig ubersetzenden Getrieben (J), VDI-Berichte 374 (1980) 43–47. [14] F. Freudenstein, C.K. Chen, Variable ratio chain drives with noncircular sprockets and minimum slack theory and application (J), Journal of Mechanics 113 (1991) 253–262. [15] L.X. Cao, J. Liu, Study of calculation method of transmission ratio's function based on non-circular chain transmission (J), Journal of Mechanical Engineering (1997). [16] L.X. Cao, J. Liu, Study of calculation method of transmission ratio's function based on non-circular chain transmission (J), Journal of Mechanical Engineering 37 (2001) 35–39. [17] L.P. Feng, Study of non-circular chain pulley's pitch curve (J), Mechanical Design and Study (6) (2005) 7–9. [18] Carlo Innocenti, Davide Paganelli, Designing synchronous belt transmissions with variable velocity ratio (J), Journal of Mechanical Design 130 (2008) 1–7. [19] Hellmuth Stachel, Gears and belt drives for non-uniform transmission, [C], In: Proceedings of EQCOMES 08, 2009. [20] J. Hoschek, Konstruktion von kettengetrieben mit veränderlicher Übersetzung mit hilfe von bézier kurven (J), Forschung im Ingenieurwesen 48 (1982) 81–87.