IAENG International Journal of Applied Mathematics, 38:4, IJAM_38_4_06 _______________________________________________________________________________ Planetary Hypocycloid (Epicycloid) Mechanisms Design Meng-Hui Hsu Abstract—An epicycloid or hypocycloid mechanism is Similar mechanism designs can be found abundantly in capable of drawing an exact epicycloid or hypocycloid curve. industrial machines. Currently, the major type of epicycloid Similar mechanism designs can be found abundantly in or hypocycloid configurations is planetary gear trains [5], industrial machines or educational equipments. Currently, the which contain a binary link that has one fixed and one major type of epicycloid or hypocycloid configurations is moving pivot, and a singular link adjacent to the moving planetary gear trains, which contain a binary link that has one fixed and one moving pivot, and a singular link adjacent to the pivot. The main feature of the configurations is that a point moving pivot. The main feature of the configurations is that a on the singular link may describe an epicycloid or point on the singular link may describe an epicycloid or hypocycloid curve when the binary link is rotated. hypocycloid curve when the binary link is rotated. The main This paper will analyses the characteristics of the aim of this paper is to develop a new design method in designing topological structures of existing planetary gear train type new configurations of planetary epicycloid (hypocycloid) epicycloid (hypocycloid) mechanisms with one degree of mechanisms. This paper analyses the characteristics of the topological structures of existing planetary gear train type freedom. The equation of motion and kinematic model of the epicycloid (hypocycloid) mechanisms. The motion equations mechanism could be derived and appropriate design and kinematical model of the mechanism were derived and constraints and criteria were implemented. Subsequently, appropriate design constraints and criteria were implemented. using the design constraints and criteria, this work can design Finally, using the design constraints and criteria, this work a new epicycloid (hypocycloid) mechanism. designs new epicycloid (hypocycloid) mechanisms. Index Terms—epicycloids, hypocycloid, mechanism, motion equation, kinematical model. NOMENCLATURE n Radius ratio, n = ra / rb INTRODUCTION n′ Link length ratio, n′ = rc / rb The hypocycloid curve produced by fixed a point P on ri Radius or length of member i the circumference of a small circle of radius rb rolling Ti Number of teeth of gear i around the inside of a large circle of radius ra . A 3-cusped x , y X and Y-coordinate of an epicycloid (hypocycloid) hypocycloid shown in Figure 1 is also called a tricuspoid or deltoid [1]. The deltoid was first considered by Euler in 1745 θ Angular displacement of driving member (the center in connection with an optical problem. It was also of small circle., carrier, or binary link) investigated by Steiner in 1856 and is sometimes called ωi Angular velocity of member i Steiner's hypocycloid [2]. A 4-cusped hypocycloid shown in Figure 2 is sometimes also called a tetracuspid, cubocycloid, paracycle, or asteroid [3]. The parametric equations of the deltoid and astroid can be obtained by plugging in HYPOCYCLOID The hypocycloid curve produced by fixed point P on the n = ra / rb = 3 and n = ra / rb = 4 into the equations for a general hypocycloid, respectively. circumference of a small circle of radius rb rolling around The epicycloids path traced out by a point P on the edge the inside of a large circle of radius ra . The Cartesian of a circle of radius rb rolling on the outside of a circle of parametric equations of the hypocycloid, path of point P, are: radius rA [4]. To get n cusps in the epicycloid, n = ra / rb , ⎛ r − r ⎞ because then n rotations of rb bring the point on the edge x =(r − r )cosθ + r cos⎜ a b θ ⎟ a b b ⎜ r ⎟ back to its starting position. An epicycloid with one cusp is ⎝ b ⎠ (1) called a cardioid, one with two cusps is called a nephroid, and ⎛ 1 1 ⎞ one with five cusps is called a ranunculoid. Figure 3 and 4 = ⎜ (1− )cosθ + cos[](n −1)θ ⎟ ra ⎝ n n ⎠ show a 3-cusped and a 4-cusped epicycloid, respectively. An exact hypocycloid (epicycloids) curve can be also ⎛ r − r ⎞ produced by using a hypocycloid (epicycloids) mechanism. ⎜ a b ⎟ y =(ra − rb )sinθ − rb sin ⎜ θ ⎟ ⎝ rb ⎠ (2) Meng-Hui Hsu is the corresponding author and now with Department of ⎛ 1 1 ⎞ = ⎜ (1− )sinθ − sinθ [](n −1)θ ⎟ r Mechanical Engineering, Kun Shan University, Yung Kang 71003, Tainan, n n a TAIWAN (phone: +886-6-205-0761/0730; fax: +886-6-205-0509; e-mail: ⎝ ⎠ [email protected]). (Advance online publication: 20 November 2008) IAENG International Journal of Applied Mathematics, 38:4, IJAM_38_4_06 _______________________________________________________________________________ Where n = r / r is radius ratio, θ is the angular a b Y displacement of the center of small circle. n-cusped hypocycloids can also be constructed by beginning with the diameter of a circle, offsetting one end by a series of steps while at the same time offsetting the other end by steps (n −1) times as large in the opposite direction and extending rb beyond the edge of the circle. After traveling around the circle once, an n-cusped hypocycloid is produced. −2θ The equation of the 3-cusped hypocycloid is obtained by θ P X setting n = ra / rb = 3 in the equation of the hypocycloid, ra = 3rb O where ra is the radius of the large fixed circle and rb is the radius of the small rolling circle, yielding the parametric equations: ⎡2 1 ⎤ x = ⎢ cosθ + cos 2θ ⎥ ra (3) ⎣3 3 ⎦ ⎡2 1 ⎤ y = ⎢ sinθ − sin 2θ ⎥ ra (4) ⎣3 3 ⎦ Figure1 A 3-cusped hypocycloid The parametric equations of the 4-cusped hypocycloid Y can be obtained by plugging in n = ra / rb = 4 into the equations for a general hypocycloid, giving parametric equations: rb P ⎡3 1 ⎤ −3θ x = ⎢ cosθ + cos3θ ⎥ ra (5) ⎣4 4 ⎦ r = 4 r ⎡3 1 ⎤ a b y = sinθ − sin 3θ r (6) θ X ⎢4 4 ⎥ a ⎣ ⎦ O The following tables summarizes the names given to this and other hypocycloids with special integer values of n = ra / rb [2-4]. Table 1 Hypocycloids with special integer values of n n = ra / rb Hypocycloid Figure2 A 4-cusped hypocycloid 2 line segment (Tusi couple) Y 3 deltoid rb 4 astroid P 4θ r = 3r EPICYCLOID a b θ X The epicycloid path traced out by a point P on the edge O of a circle of radius rb rolling on the outside of a circle of radius ra . Epicycloids are given by the parametric equations: ⎛ r + r ⎞ x =(r + r )cosθ − r cos⎜ a b θ ⎟ a b b ⎜ r ⎟ ⎝ b ⎠ (7) ⎛ 1 1 ⎞ = ⎜ (1+ )cosθ − cos[](n +1)θ ⎟ ra ⎝ n n ⎠ Figure 3 A 3-cusped epicycloid (Advance online publication: 20 November 2008) IAENG International Journal of Applied Mathematics, 38:4, IJAM_38_4_06 _______________________________________________________________________________ Y The following tables summarizes the names given to this and other epicycloid with special integer values of P n = ra / rb [2-4]. rb 5θ Table 2 Epicycloids with special integer values of n n = r / r ra = 4 rb a b Epicycloid θ X O 1 cardioid 2 nephroid 5 ranunculoid HYPOCYCLOID AND EPICYCLOID DESIGN CRITERIA Figure 4 A 4-cusped epicycloid A. Motion equations of the planetary hypocycloid A hypocycloid mechanism with simple planetary gear train comprises a fixed ring gear, a, a planetary gear, b , and ⎛ r + r ⎞ a planetary carrier, c . In a 3-cusped hypocycloid planetary y =(r + r )sinθ − r sin⎜ a b θ ⎟ a b b ⎜ ⎟ gear system, Figure 5, the number of teeth in ring gear a is ⎝ rb ⎠ (8) Ta , which is thrice that of gear b , i.e., Ta = 3Tb . When the ⎛ 1 1 ⎞ carrier, c , acts as the input link and rotates one revolution, a = ⎜ (1+ )sinθ − sinθ [](n +1)θ ⎟ ra ⎝ n n ⎠ fixed point, P, on the pitch circle of gear b will describe a hypocycloid path. Where θ is the angular displacement of the center of The angular velocity of gear b is ωb and that of the small circle. To get n cusps in the epicycloid, rb = ra / n , carrier is ωc . The fixed ring gear a has a angular velocity because then n rotations of rb bring the point on the edge ωa = 0 and the relationship among the three angular back to its starting position. An epicycloid with one cusp is velocities is [6]: called a cardioid, one with two cusps is called a nephroid, and ω −ω T one with five cusps is called a ranunculoid. b c = a (13) The equation of the 3-cusped epicycloid is obtained by ωa −ωc Tb setting n = r / r = 3 in the equation of the epicycloid, a b Substituting ωa = 0 and Ta = 3Tb into Equation (13) and where ra is the radius of the large fixed circle and rb is the recasting: radius of the small rolling circle, yielding the parametric ω = − 2ω (14) equations: b c Similarly, in a n-cusped hypocycloid planetary gear system, the Cartesian parametric equations of the ⎡4 1 ⎤ x = ⎢ cosθ − cos 4θ ⎥ ra (9) hypocycloid are Equations (1) and (2), the relationship ⎣3 3 ⎦ among the three angular velocities is: ω − ω T r r + r ⎡4 1 ⎤ b c = a = a = c b = n (15) y = ⎢ sinθ − sin 4θ ⎥ ra (10) ω − ω T r r ⎣3 3 ⎦ a c b b b and we have: The parametric equations of the 4-cusped epicycloid can ωb = −(n −1)ωc = −n′ωc (16) be obtained by plugging in n = ra / rb = 4 into the equations rc for a general epicycloid, giving parametric equations: n′ = = n −1 (17) rb ⎡5 1 ⎤ Where n′ is link length ratio.