On some problems of synthesis of spatial five-bar hinged mechanisms with two degrees of freedom

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Internationa Journa of Mechanica Engineering and Appications 014 (6): 104-110 Puished onine ecemer 18, 014 (http://www.sciencepuishinggroup.com/j/ijmea) doi: 10.11648/j.ijmea.014006.

1 Internationa Journa of Mechanica Engineering and Appications 014 (6): Puished onine ecemer 18, 014 ( doi: /j.ijmea ISSN: X (Print) ISSN: (Onine) On some proems of synthesis of spatia five-ar hinged mechanisms with two degrees of freedom Nodar avitashvii *, Otar Geashvii epartment of Transport and Mechanica Engineering of Georgian Technica University, Tiisi, Georgia Emai address: (N. avitashvii), (O. Geashvii) To cite this artice: Nodar avitashvii, Otar Geashvii. On Some Proems of Synthesis of Spatia Five-Bar Hinged Mechanisms with Two egrees of Freedom. Internationa Journa of Mechanica Engineering and Appications. Vo., No. 6, 014, pp doi: /j.ijmea Astract: Soution of the proems of synthesis of spatia five-ar hinged mechanisms with two degrees of freedom task considering the ange of transmission and definition of conditions of existence of cranks is given in the paper. A possiiity of movement of the designed mechanism without seizure that is depended on shape and sizes of the mechanism inks is envisaged. At soution of the proem of synthesis first are determined sizes of the mechanism coupes considering the ange of transmission and then are determined the imits of variation of the ange of transmission depending on sizes of the mechanism inks. The conditions are ascertained which must e met y sizes of the spatia five-ar mechanism inks so that the two inks adjoining to the frame are crank (theorem on existence of cranks in the spatia five-ar mechanism). Keywords: Synthesis, Ange of Transmission, Spatia Five-Bar Mechanism, Couper, Conditions of Existence of Crank 1. Introduction Spatia mechanism with two degrees of freedom rank high among numerous hinged mechanisms used in practice, which in contrast to four-ar mechanisms, can reaize more compex aws of motion favouring at the same time rise of the machines efficiency and optimization of the technoogica processes. A distinctive feature of these mechanisms is their capacity for variation of the couper point ocus from the simpe to the more compex form in the course of operation at variation of anguar veocities of the input inks considering directions of motions and their mutua initia inter-ocation at constant engths of the inks. These circumstances are directy reated with seection of ength of the output inks of mechanisms and predetermine necessary conditions for creation of compact, ightweight mechanisms, especiay in the fieds of space and rootics industry, decreasing at this consumption of materias and energy. The spatia five-ar hinged mechanisms with two degrees of freedom are widey used in various devices, rootics systems as we as in industria practice as operating memers [1, ] and so on. The researches into hinged mechanisms with two degrees of freedom started at the end of XIX and eginning of XX centuries y first studying panar five-ar mechanisms with two input inks. Soution of the proem on ocations for the spatia mechanisms with severa degrees of freedom and spherica five-ar mechanisms with two degrees of freedom was first considered y A.G. Ovakimov [3] ony in the 70-ies of the XX century. The proems of kinematica anaysis of the spatia five-ar mechanism of type RCRCR with one degree of freedom were considered in the work of J. uffi and G.I. Hai-Oahi. [4]. The anaytica formuae were derived with the use of spherica trigonometry. The positions of output inks of the mechanism are determined y the forth order equation and in the work of J. uffi [5] for determination of dispacements of the spatia five-ar mechanism were used the dua equations. Much attention was given to the proems of anaysis, synthesis, kinetostatics and dynamics of mechanisms with two degrees of freedom y Georgian scientists having carried out deep anaysis of the mentioned mechanisms. In shoud e noted in this aspect the works [6, 7] of N.S. avitashvii. In the work [6] is given anaysis of infuence of parameters of the five-ar hinged mechanism on trajectories of the coupers points and in the work [7] are studied proems of kinematics and precision of these mechanisms. Besides the mentioned works numerous researches into anaysis, synthesis, dynamics and precision of hinged

2 105 Nodar avitashvii and Otar Geashvii: On Some Proems of Synthesis of Spatia Five-Bar Hinged Mechanisms with Two egrees of Freedom mechanisms with two degrees of freedom are carried out y first author of these artices [8, 9, 10]. In the work [11] of N.S. avitashvii and O. Geashvii is given synthesis of the five-ar spatia mechanism with two degrees of freedom according to the trajectory. In the presented work is carried out synthesis of the spatia five-ar hinged mechanism with two degrees of freedom taking into account the ange of transmission and conditions of existence of two cranks.. Synthesis of the Spatia Five-Bar Mechanism Taking into Account the Ange of Transmission At soution of the proem of synthesis of spatia five-ar mechanisms with two degrees of freedom it is necessary to consider the foowing requirements: correct structure of the mechanism to e designed, kinematica precision of the movement to e performed, possiiity of reaization of the given movement y mechanism to e designed from the standpoint of dynamics, possiiity of movement of the mechanism under action of appied forces with maximum possie efficiency (taking into account the ange of transmission to avoid their seizure) and finay, ensuring existence of cranks and reproduction of the given movement y the sizes of inks of the synthesized mechanism..1. imensioning of Coupers of the Five-Bar Spatia Mechanism Taking into Account the Given Vaue of the Ange of Transmission Let's consider spatia five-ink hinged everage RSSRR type mechanism ABCE (Fig. 1) with two degrees of freedom and determine the dimensions BC= 3 and C= 4 of coupers with taking into account the given vaues of ange of transmission γ (γ min, γ max ). Figure 1. Spatia five-ar mechanism with two degrees of freedom. For soution of this proem are given: dimensions of inks 1,, 5, designated as 1,, and 5 anges α 0 and β 0, specifying the initia position of the input AB 0 and E 0 inks and 5, with respect to the frame 1. The aws of motion of the input inks and 5 are given as: ϕ = ϕ ) and ϕ = ϕ ), (1) ( t 5 5 ( t ϕ = α + α( ) and ϕ = β + β ( ). () 0 t 5 0 t Here α and β are the variae anges characterizing the motion of input inks. The gear ratios from ink 5 to ink is defined as = = = =±, (3) the sign (+) is taken for the case when input inks and 5 are rotating in the same direction, and the sign ( ) when they are rotating in the opposite directions.. It shoud e noted that y giving functions of dispacements of the input inks, gear ratio is aso =. (4) We seect two fixed coordinate systems for the input inks and 5 so that point A is ocated in the pane X 0 EY 0. The system EX 0Y0 is seected for ink 5, and system AX1Y 1Z1 - for ink. Accordingy are given the coordinates of point A ( x0 y0z0 ) in the system EX 0Y0 considering ange ξ etween axis EX 0 and frame 1: x = 1 cos y = 1 sin z 0 = 0. (5) 0 ξ 0 ξ The transformation of system AX1Y 1Z1 in the system EX Y is carried out due the matrix: = 0, (6) 11, 1,., 33 are directing cosines of axes of the system AX 1 Y 1 Z 1 with respect to the EX 0 Y 0 Z 0 system. It is possie at first to set the four vaues of directing cosines, for exampe, 11, 1, 31 and 3 =0, then for 33, 1, 3, 1 we wi have: =1 = = =1. The variae distance etween movae joints B and, the vaue of that is depended on dimensions of frame 1 and input inks ( ) and 5 ( 5 ), initia position of input inks (α 0,β 0 ), gear ratio etween input inks with taking into account directions of their movement can e determined in the system EX0Y0 with the hep of formua for the distance etween two points in space: (7) =! +# #! +$ $!, (8)

3 Internationa Journa of Mechanica Engineering and Appications 014 (6): x 0 ϕ ϕ. (9) B = yb = cos zb = sin x 0 ϕ ϕ. (10) = y = 5 cos 5 z = 5 sin Considering the ocation of point A at the origin of the system,, the coordinates of point B CAN e written in the system EX0Y0 using matrix (6) x y z B B B = = 1 = 33 cosϕ + cosϕ + sinϕ sinϕ + sinϕ + 1 cosζ 1 sinζ 5 (11) Considering vaues (), (10) and (11) expression (8) wi take a form: =% ' (cos, - +, ' cos/+ sin/+sin, - +, ' cos/+ sin/ sin/cos3 - +, ( cos, - +,cos3 - +,+ ' '+ sin, - +, ' sin3- +,4, (1) β=,. Here α is a variae vaue. Anaysis of expression (1) shows that distance depends on,, 5, α 0, β 0,, α and directing cosines. FROM triange BC taking into account the given maxima (γ max ) and minima (γ min ) vaues of the ange of transmission can written: cos7 89: = 89: + (13) 6 6 cos7 8< = 8<, max and min are the extreme vaues of variae distance. It is known that if derivative = =, exists, then function wi have maximum or minimum ony if > > =0. (14) If second derivative =² =,² exists, then function wi have maximum when and minimum at Accordingy we can write > =0, >² <0, (15) > >² > =0, >² >0. (16) > >² > = C (17) > C >² = CCFF GC >² 6 C H= (cos, - +, ' cos/+ sin/+sin, - +,, (18) ' cos/+ sin/ sin/cos3 - +, ( cos, - +,cos3 - +,+ ' '+ sin, - +,sin3 - +, H = (cos, - +, cos/+ sin/ ' ' sin, - +, cos/+ sin/+ +,+ sin/sin3 - +, ( cos, - +, ' cos3 - +, ',+ sin3 - +, ' sin, - +, sin, - +, ' cos3 - +,+ +,+ sin3 - +, ' cos, - +,+ + cos, - +, ' sin3 - +,+ + ',+ cos3 - +, ' sin, - +,. The vaues k'' wi e found y differentiation of k'. In turn = F G. (19) For determining of such vaue of α that gives the extreme vaues of distance it is necessary that condition (14) e met. From the reations (17) we wi have cos,%,+ sin/sin3 - + ' + sin, - +,( cos3 - + ' +,+ sin3 - ',+ cos cos, - +,(,+ sin3 - ' - ' cos3 - ' sin sin,%,+ sin/sin3 - '+ + sin, - +,( cos3 -,+ ' ' sin3 - +,+ sin , - +,(,+ cos3 - '

4 107 Nodar avitashvii and Otar Geashvii: On Some Proems of Synthesis of Spatia Five-Bar Hinged Mechanisms with Two egrees of Freedom sin3 - ' ' cos3-4 (sin, - +, cos/+ sin/ ' + +cos, - +, ' cos/+ sin/=0. (0) On the ase of known formuae of the Moivre cos, and sin, incuded in the equation (0) can e presented as cos,=cos J, K J cos JG,sin²,+ 6 +K J cos JG6,sin 6, (1) sin,= cos J G,sin, K J cos JG,sin³,+ +K J cos JG,sin, () Taking into account vaues (1) and () equation (0) wi take a form: N ±N N =0, (3) N = Ocos J, K J cos JG,sin²,+ ' + ' 6 K J cos JG6,sin 6, P %,+ sin/sin3 - + ' + sin, - +,( cos3 - + ' +,+ sin3 - ',+ cos cos, - +,(,+ sin3 - ' cos3 - ' ' sin3-4 N = cos JG,sin, ' K J cos JG,sin³,+ + ' K J cos JG,sin, P %,+ sin/sin3 - + sin, - +, ' ( cos3 -,+ sin3 - + ' + ',+ cos3 - + cos, - +, (,+ cos3 - ' sin3 - ' ' cos3-4 N = (sin, - +, cos/+ sin/ ' + + ' cos, - +, cos/+ sin/. Binomia coefficients in turn wi have forms K J = 1! and so on K J = 1 3! K J 6 = 1 3 4! K J K J = ! < = 1 3 O U 1P. U! In equations (3) a sign (+) is taken when gear ratios is a positive numer, and sign ( ) when is a negative numer. The series of inomia coefficients is finite when is a positive or negative integer, and infinite when is a positive or negative fractiona numer. By soution of equations (3) are determined the vaues α i (α 1, α,, α n ), with the hep of which we determine extreme vaues max and min of distance accordingy to formua (1). In practice are mainy encountered cases when the input inks are moving with constant anguar veocity ( =const). In such case expressions (17), (18) and equation (3) wi e significanty simpified. Thus knowing vaues max and min, and transforming the system (13) we otain a vaue of couper 4 (C= 4 ) ength: 6 =± VW ±XW G6W W W, Y =4cos7 8< cos7 89: (4) Y =4( 89: cos7 8< cos7 89: + ' + ' cos7 89: 89: 8< cos7 8< cos7 89: Y = 89: 8<. Accordingy a ength of couper 3 (BC= 3 ) wi e determined from system (13): = Z[\ G Z]^. (5) _ `ac Z]^G`ac Z[\.. etermination of Limits of Anges of Transmission epending on imensions of Links of the Spatia Five-Bar Mechanism At soution of this proem a condition is envisaged according to which an ange of transmission at any position of the mechanism must not e ess that certain given minimum vaue ensuring norma functioning of the mechanism. Seection of aowae anges of transmission is determined y dynamica conditions of the mechanism operation and in the first pace y asence of seizure of

5 Internationa Journa of Mechanica Engineering and Appications 014 (6): output inks. Considering a poygon of inks ABCE and triange BC (Fig. 1) for distance we can write: Suppose 7= h = c Z[\Gc Z]^. (31) = (cos, - +, ' cos/+ sin/+ '+sin, - +, cos/+ sin/ sin/cos3 - +, ( cos, - +,cos3 - +,+ ' + sin, - +, ' sin3 - +,= = cos7. (6) From expressions (6) can e determined the ange of transmission γ: cos7= 6 G % + 6 ' (cos, - +, cos/+ sin/ ' + + ' sin, - +, cos/+ sin/+ + sin/cos3 - +,+ + ( cos, - +,cos3 - +, ' + + sin, - +,cos3 - +, + sin, - +, ' sin3 - +,4. (7) If 0<γ<π, then the extreme vaues of ange γ wi e determined from formua (7): cos7 89: = d _ GZ[\, (8) _ cos7 8< = d _ GZ]^. (9) _ An optima vaue γ=π/ wi e otained at vaue of variae ange α determined from equation + 6 (cos, - +, cos/+ sin/+ ' +sin, - +, ' cos/+ sin/+ + sin/cos3 - +,+ + ( 'cos, - +,cos3 - +,+ + sin, 0 +,sino ,P]=0. (30) The otained resuts aow to determine reations etween dimensions of inks of the spatia five-ar mechanism for which optima vaue of the ange of transmission wi e arithmetica mean of its extreme vaues, i.e. 7 89: = h +i NU= 7 8< = h i, (3) δ is a certain ange. Sustituting of these vaues in (8) and (9) we wi have: 6 sini = : 6 sini= + 6 8<. Excuding δ we wi otain (33) + 6 = 89: + 8<. (34) Thus if dimensions of inks of the spatia five-ar mechanism satisfy condition (34), then deviations of the ange of transmission γ from optima vaue 7 = j wi e identica in oth sides..3. Conditions of Existence of Two Cranks in Spatia Five-Bar Mechanisms At soution of numerous proems of synthesis of spatia mechanisms with two degrees of freedom it is very important to seect reations etween engths of the inks aowing one or two inks to reaize fu or partia turning around chosen points. It is known that condition of existence of cranks in panar and spherica four-ar mechanisms was first formuated y F. Grasgof. The simiar proems were studied y N.G. Bruevich, V.V. orovoski, F. Freidenstein, F. uditsa and others. For spatia four-ar mechanisms conditions of existence of one of two cranks was first formuated y.s. Tavkheidze. By numerous researches into various hinged mechanisms y N.S. avitashvii were formuated theorems of existience of cranks in spherica five-ar, six-ar and seven-ar mechanisms. By studies of structure and kinematics of five-ar mechanisms with two degrees of freedom is ascertained that conditions of turning of input inks are significanty infuenced y dimensions of frame 1 and inks and 5 (Fig. 1), gear ratio u 5 etween input inks considering their directions of movement, initia positions of input inks specified y anges α 0 and β 0 and duration of the mechanism movement cyce. For spatia five-ar mechanism (Fig. 1) we have otained formua (1) for cacuation of variae distance etween movae hinges B and, whose extreme vaues max and min are found on the ase of satisfaction of conditions (15), (16). The found extreme vaues max and min of distance aow us to write inequaities defining conditions of existence of cranks and 5according to the kinematica scheme of spatia five-ar mechanism: 89: + 6 8< 6. (35) Thus we can formuate a theorem of existence of two cranks

6 109 Nodar avitashvii and Otar Geashvii: On Some Proems of Synthesis of Spatia Five-Bar Hinged Mechanisms with Two egrees of Freedom in spatia five-ar hinged mechanisms as foows. In spatia five-ar hinged mechanisms two inks, adjoining to the frame, wi e cranks if and ony if at movement of the mechanism maximum distance etween the hinges created y connection of input inks and corresponding coupers is ess or equa to the sum of engths of coupers and minimum distance is more or equa to the difference of these engths. If the assume that α0=β0=0 and 5=0 the spatia five-ar mechanism ABSE (Fig. 1) wi e converted into spatia four-ar mechanism ABC (Fig. ). ( + + cos, + sin, n + 6 ( + + cos, + sin, n Concusion (41) By accompished researches into five-ar mechanisms with two degrees of freedom are marked out some proems of synthesis of the mentioned mechanisms. For the purpose of ensuring operation of the mechanism without seizure with maximum possie efficiency are determined dimensions of the inks firsty considering a given ange of transmission and then depending on the dimensions of inks of the spatia five-ar mechanism are cacuated imits of variation of the ange of transmission. A theorem of existence of two cranks in spatia five-ar hinged mechanisms is formuated. The otained resuts are suitae for designing new mechanisms and soution of practica proems. Figure. On the conditions of existence of cranks in a spatia four-ar mechanism. In this case variae distance according to expression (1) wi e: =% '+ + (cos, cos/+ sin/ ' + '+ ' sin, cos/+ sin/4 n. (36) The equation for finding α wi e written as: (cos, cos/+ sin/ ' ' sin, cos/+ sin/=0. (37) Accordingy vaue of α wi e found y formua cos,, =± X d = cos/+ sin/ = cos/+ sin/., (38) Taking into account the vaues of α 1 and α from expression (36) we wi have: 89: =( + + ' cos, '+ + ' sin, n, (39) 8< =( + + ' cos, '+ + ' sin, n. (40) Accordingy to expressions (35) conditions of existence of cranks in the spatia four-ar mechanism wi e: References [1] G. Eason, B. Noe, and I. N. Sneddon, On certain integras of Lipschitz-Hanke type invoving products of Besse functions, Phi. Trans. Roy. Soc. London, vo. A47, pp , Apri (references) [] J. Cerk Maxwe, A Treatise on Eectricity and Magnetism, 3rd ed., vo.. Oxford: Carendon, 189, pp [3] A. G. Ovakimov, Task on positions of spatia mechanisms with severa degree of freedom and its soution y cosed vector contour method in Mechanics of Machines, issue 9-3, Moscow: Nauka, 1971, pp (In Russian). [4] J. uffy, H.Y. Hai-Oahi, A dispacement anaysis of spatia five-ink BR-C mechanism. 1. On the cousers of the RCRCR mechanism in Journa of Mechanics, vo. 5, 3, 1971, pp [5] J. uffy, A derivation of dua dispacement equations for five-size-and seven-ink spatia mechanism using spherica trigonometry. 3. eviation of dua equations for seven-ink spatia mechanism s Rev. roum. sci. techn. Scr. mech. app. vo. 17, 197,, pp [6] N.S. avitashvii, Infuence of parameters of five-ar hinged mechanism on trajectory of couper s points, Transactions of Academy of Sciences of GSSR, vo. 86, 1977,, pp , (In Russian). [7] N.S. avitashvii, Issues of kinematics and precision of five-ar hinged mechanism Fifth Word congress on Theory of Machines and Mechanisms, vo. 1, 1979, Canada, Motrea, Juy 8-13, pp [8] N.S. avitashvii, Fundamentas of the theory of synthesis, anaysis and precision of inkage-ever mechanisms, Tiisi: Technica University, p. (In Russian). [9] N.S. avitashvii, Theoretica fundamentas of error and precision of hinged mechanisms with two degrees of freedom, Tiisi: Metsnierea, p. (In Russian).

7 Internationa Journa of Mechanica Engineering and Appications 014 (6): [10] N.S. avitashvii, ynamic investigation of pane five-ink hinged mechanism with two degrees of freedom,tiisi: Georgian Committee of IFToMM, p. (In Russian). [11] N. avitashvii, O. Geashvii, Synthesis of a spatia five-ink mechanism with two degrees of freedom according to the given aws of Motion, in Proceedings of EUCOMES 08, the second European Congress on Mechanism Science, Springer, pp

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