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Epicycloid (Hypocycloid) Mechanisms Design Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong Epicycloid (Hypocycloid) Mechanisms Design Meng-Hui Hsu, Hong-Sen Yan, Jen-Yu Liu and Long-Chang Hsieh Abstract—An epicycloid or hypocycloid mechanism is 3-cusped epicycloid. capable of drawing an exact epicycloid or hypocycloid curve. An exact hypocycloid (epicycloids) curve can be also Similar mechanism designs can be found abundantly in produced by using a hypocycloid (epicycloids) mechanism. industrial machines or educational equipments. Currently, the Similar mechanism designs can be found abundantly in major type of epicycloid or hypocycloid configurations is industrial machines. Currently, the major type of epicycloid planetary gear trains, which contain a binary link that has one or hypocycloid configurations is planetary gear trains [5], fixed and one moving pivot, and a singular link adjacent to the which contain a binary link that has one fixed and one moving pivot. The main feature of the configurations is that a point on the singular link may describe an epicycloid or moving pivot, and a singular link adjacent to the moving hypocycloid curve when the binary link is rotated. The main pivot. The main feature of the configurations is that a point aim of this paper is to develop a new design method in designing on the singular link may describe an epicycloid or new configurations of epicycloid (hypocycloid) mechanisms. hypocycloid curve when the binary link is rotated. This paper analyses the characteristics of the topological This paper will analyses the characteristics of the structures of existing planetary gear train type epicycloid topological structures of existing planetary gear train type (hypocycloid) mechanisms. The motion equations and epicycloid (hypocycloid) mechanisms with one degree of kinematical model of the mechanism were derived and freedom. The equation of motion and kinematic model of the appropriate design constraints and criteria were implemented. Finally, using the design constraints and criteria, this work mechanism could be derived and appropriate design designs a new and simple epicycloid (hypocycloid) mechanism. constraints and criteria were implemented. Subsequently, using the design constraints and criteria, this work can design Index Terms—epicycloids, hypocycloid, mechanism, motion a new epicycloid (hypocycloid) mechanism. equation, kinematical model. INTRODUCTION NOMENCLATURE The hypocycloid curve produced by fixed a point P on n Radius ratio, n = ra / rb the circumference of a small circle of radius r rolling b n′ Link length ratio, n′ = rc / rb around the inside of a large circle of radius ra . A 3-cusped ri Radius or length of member i hypocycloid shown in Figure1 is also called a tricuspoid or deltoid [1]. The deltoid was first considered by Euler in 1745 Ti Number of teeth of gear i in connection with an optical problem. It was also x , y X and Y-coordinate of an epicycloid (hypocycloid) investigated by Steiner in 1856 and is sometimes called Steiner's hypocycloid [2]. A 4-cusped hypocycloid which is θ Angular displacement of driving member (the center sometimes also called a tetracuspid, cubocycloid, paracycle, of small circle., carrier, or binary link) or asteroid [3]. The parametric equations of the astroid can be ωi Angular velocity of member i obtained by plugging in n = r / r = 4 into the equations for a b a general hypocycloid. The epicycloids path traced out by a point P on the edge HYPOCYCLOID of a circle of radius rb rolling on the outside of a circle of The hypocycloid curve produced by fixed point P on the radius rA [4]. To get n cusps in the epicycloid, rb = ra / n , circumference of a small circle of radius rb rolling around because then n rotations of rb bring the point on the edge the inside of a large circle of radius ra . The Cartesian back to its starting position. An epicycloid with one cusp is parametric equations of the hypocycloid, path of point P, are: called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid. Figure 2 shows a ⎛ ra − rb ⎞ x = (ra − rb ) cosθ + rb cos⎜ θ ⎟ (1) ⎝ rb ⎠ Meng-Hui Hsu is the corresponding author and now with Department of Mechanical Engineering, Kun Shan University, Yung Kang 71003, Tainan, TAIWAN (phone: +886-6-205-0761/0730; fax: +886-6-205-0509; e-mail: ⎛ ra − rb ⎞ [email protected]). y = (ra − rb ) sinθ − rb sin⎜ θ ⎟ (2) Hong-Sen Yan is with Department of Mechanical Engineering, National ⎝ rb ⎠ Cheng kung University, 71001, Tainan, TAIWAN (e-mail: hsyan@ mail.ncku.edu.tw). Where θ is the angular displacement of the center of Jen-Yu Liu is with Department of Power Mechanical Engineering, National Formosa University, Hu Wei 63201, Yunlin, TAIWAN (e-mail: small circle. n-cusped hypocycloids can also be constructed [email protected]). by beginning with the diameter of a circle, offsetting one end Long-Chang Hsieh is with Department of Power Mechanical Engineering, by a series of steps while at the same time offsetting the other National Formosa University, Hu Wei 63201, Yunlin, TAIWAN (e-mail: end by steps (n −1) times as large in the opposite direction [email protected]). ISBN: 978-988-17012-1-3 IMECS 2008 Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong and extending beyond the edge of the circle. After traveling The parametric equations of the asteroid (4-cusped around the circle once, an n-cusped hypocycloid is produced. hypocycloid) can be obtained by plugging in n = ra / rb = 4 The equation of the deltoid (3-cusped hypocycloid) is into the equations for a general hypocycloid, giving obtained by setting n = ra / rb = 3 in the equation of the parametric equations: hypocycloid, where ra is the radius of the large fixed circle ⎡3 1 ⎤ 3 and rb is the radius of the small rolling circle, yielding the x = ⎢ cosθ + cos3θ ⎥ ra = racos θ (5) ⎣4 4 ⎦ parametric equations: ⎡3 1 ⎤ 3 ⎡2 1 ⎤ y = sinθ − sin 3θ ra = ra sin θ (6) x = ⎢ cosθ + cos 2θ ⎥ ra (3) ⎢4 4 ⎥ ⎣3 3 ⎦ ⎣ ⎦ The following tables summarizes the names given to ⎡2 1 ⎤ y = sinθ − sin 2θ ra (4) this and other hypocycloids with special integer values of ⎢3 3 ⎥ ⎣ ⎦ n = ra / rb . Y Table 1 Hypocycloids with special integer values of n n = ra / rb Hypocycloid 2 line segment (Tusi couple) rb 3 deltoid −2θ ra = 3 rb 4 astroid θ P X O EPICYCLOID The epicycloid path traced out by a point P on the edge of a circle of radius rb rolling on the outside of a circle of radius ra . Epicycloids are given by the parametric equations: ⎛ ra + rb ⎞ x =(ra + rb )cosθ − rb cos⎜ θ ⎟ (7) ⎝ rb ⎠ Figure1 A 3-cusped hypocycloid ⎛ ra + rb ⎞ y =(ra + rb )sinθ − rb sin⎜ θ ⎟ (8) ⎝ rb ⎠ Y Where θ is the angular displacement of the center of small circle. To get n cusps in the epicycloid, rb = ra / n , because then n rotations of rb bring the point on the edge rb back to its starting position. An epicycloid with one cusp is P called a cardioid, one with two cusps is called a nephroid, and 4θ one with five cusps is called a ranunculoid. ra = 3 rb The following tables summarizes the names given to θ X this and other epicycloid with special integer values of n = r / r . O a b Table 2 Epicycloids with special integer values of n n = ra / rb Epicycloid 1 cardioid 2 nephroid Figure 2 A 3-cusped epicycloids 5 ranunculoid ISBN: 978-988-17012-1-3 IMECS 2008 Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong HYPOCYCLOID AND EPICYCLOID DESIGN CRITERIA Substituting ωa = 0 and Ta = 3Tb into Equation (14) and recasting: A. Motion equations of the planetary hypocycloid A hypocycloid mechanism with simple planetary gear ωb = 4ωc (15) train comprises a fixed ring gear, a, a planetary gear, b , and Similarly, in a n-cusped epicycloid planetary gear system, the a planetary carrier, c . In a 3-cusped hypocycloid planetary Cartesian parametric equations of the epicycloid are gear system, Figure 3, the number of teeth in ring gear a is Equations (7) and (8), the relationship among the three Ta , which is thrice that of gear b , i.e., Ta = 3Tb . When the angular velocities is: carrier, c , acts as the input link and rotates one revolution, a ω −ω T r r − r b c = − a = − a = − c b = −n (16) fixed point, P, on the pitch circle of gear b will describe a ωa −ωc Tb rb rb hypocycloid path. and we have: The angular velocity of gear b is ωb and that of the ω = (n +1)ω (17) carrier is ωc . The fixed ring gear a has a angular velocity b c r ωa = 0 and the relationship among the three angular n′ = c = n +1 (18) velocities is [6]: rb ω −ω T Y b c = a (9) ωa −ωc Tb Substituting ωa = 0 and Ta = 3Tb into Equation (9) and recasting: ω = − 2ω (10) b c Planetary gear b Similarly, in a n-cusped hypocycloid planetary gear ra = 3rb Carrier c system, the Cartesian parametric equations of the −2θ hypocycloid are Equations (1) and (2), the relationship rc = 2 rb rb among the three angular velocities is: X θ P ω − ω T r r + r O b c = a = a = c b = n (11) ωa − ωc Tb rb rb Frame 1 and we have: ωc = − ωb 2 ω = −(n −1)ω = −n′ω (12) b c c Ta = 3Tb r n′ = c = n −1 (13) rb Fixed ring gear a Where n′ is link length ratio.
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