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A numerical method for determining positions of links of high-class linkages
Mec~ Madt. T/k,~ryVoL27, No. 4, pp. 403.-41,1,1992 Printedm GreatBri~,, All right*ram-red
0094-114X/92$&00+0.00 CopyrilhtC~1992Perlimm Primlull
A NUMERICAL METHOD FOR DETERMINING POSITIONS OF LINKS OF HIGH-CLASS LINKAGES V. S. KARELIN Tver PolytechnicInstitute, 170035, Tver, Russia (Received 20 August 1990, receivedfor publication 12 August 1991)
Almract--A numerical method for determining positions of high-class linkages in spatial and plane mechanismsis discussed.With the help of one or more clmure links in a systemof four-linkmechanisms the high-classstructural Stoup from high-classlinkagesis transformed.The conditions for the number of closure links are determined. Each four-link mechanism is oriented in accordance with its assembly which in high.class linhaile coincides with the aBembly of the mechanism.The orientation of a four-link mechanismis found by selcctin8 the position's function which corresponds to the assemblyof this mechanism.As a result, the high-class mechanismbecomesoriented in a way that allowssimpledeterminationof the links"positions. The orientation function of a position for basic forms of four-link mechanismsand examples of determination of the positions of individual links for some spatial and plane mechanismsate siren.
High-class linkages have good prospects for mechanizing complicated technological processes. By means of these mechanisms it is possible to achieve especially complicated movements of links. Such mechanisms have unlimited structural possibilities. The wide technical use of high-class linkages is restricted by the fact that kinematic analysis of these mechanisms presents considerable complexity. Analytical expressions of the function of a position are known only for simple (thirdand fourth-class) mechanisms [I-3]. it is apparently impossible to obtain and use analytical expressions for the function of a position for the more complicated high-class mechanisms because of their complexity and lack of practical importance. The numerical method is more prospective for kinematic analysis of high-class linkages. The numerical method has no community with the analytical method; however, it allows simple and precise research of complex multilinks with high-class linkages. At present, the known numerical methods involve definite problems, one of which is related to a mechanism's assembly, and closely connected with this problem is the problem o f branching-off of the function of a position. These problems make the use of the numerical method more difEcult [4, 5].
At present, research is done mainly on plane high-class linkages. There has been little investigation of spatial linkages. This paper presents a numerical method for determining positions of links both for plane and spatial high-class mechanisms. The method is applicable for mechanisms of any complexity and there is no limitation on its use. I. THE S T R U C T U R A L G R O U P OF H I G H CLASS The numerical method for determining the position of links of high-class linkages is based on the possibility of oriented numerical description of a structural group from a singled-out high-class mechanism. Any high-class linkages and any structural groups joined to them and to the frame may be considered part of the first-class mechanisms. The number of first-class mechanisms determines the number of input links for high-class linkages. Links of the structural group joined to the frame are generally the output links. However, there are mechanisms in which output links are not joined to the frame. An arbitrary orthogonal coordinate system, spatial or plane, depends on the type of mechanism analysed. The arbitrary choice of the high-class mechanism's coordinate system is conditional. An expedient choice may reduce the number of parameters of a mechanism to some extent. It is Nm" ~ * - c
403
v. S. Ko.et~ desirable that the origin of the coordinate system coincides with the centre of rotation or spherical kinematic pairs, and that one coordinate axis passes across this pair or is parallel to the guide of the sfider or to the axis of the cylindrical kinematic pair. This condition is not obligatory. Let us take the high-class structural group out of a mechanism, having separated it from the first-class mechanism and from the frame. The position of the structural group is determined by coordinates of outside kinematic pairs at given dimensions of its links and its assembly. The coordinates of outside kinematic pairs, separated from the frame, are constant. The coordinates of outside kinematic pairs, separated from the first-class mechanisms, are variable and dependent on the type of these mechanisms and positions of their links. 2. FIRST-CLASS MECHANISMS There are two known types of first
za = (T +
[i
sin ~p) sin =.
(I)
(2) Movement of link !, of a first-type mechanism takes place on the coordinate plane xOy. The coordinates of point A of link !~ and a point joining it to the outside kinematic pair A, of a structural group are determined by the formulae [Fig. I(b)] xA -- `to + I, cos ¢p
Ya = Y0 + / i sin 4o
where ,to, Y0 are coordinates of the rotation axis of link !,.
1
I
\I
Fig. I
(2)
IX.tamination of positions~ links of hish-dam ti.,t~....
405
(3) Spatial movement of a second-type mechanism takes place on a plane Q. Its position in the
spatial inordinate system is deter-i.ed as in (l) above. The position of the guide on plane Q is determined by the angle ? between guide and coordinate plane x0y, and by the distance L from the point of crossing of the guide with this coordinate plane to coordinate axis y. The position of the tlider on the guide is determined by its distance S to coordinate plane xOy [Fig. l(c)]. The coordinates of point A on a dider and of a point joining it to the outside kinematic pair A~ of a stngtural group are determined by the formulae x,t = (L - S cos ?) sin p - S sin 3' cos ~t cos B y~ "= Y o - ( L
- S cos y)cos p - S siny cosec sin p
:~ = S sin ,, sin ?.
(3)
(4) Movement of a slider of second-type mechanism takes place on the coordinate plane x0y. The coordinates of point A of a slider and of a point joining it to the outside kinematic pair A, of the structural group are determined by formulae [Fig. I(d)] x4=Xo+Scosy
(4)
y,,=Ssiny,
where ,go is the abscissa of the point of crossin8 of the guide with the x-axis. The variable values in these formulae are the position ~ of link it in a mechanism of the first type and the position $ of a slider's point A in the mechanism of the second type. All other values entering these formulae are constant parameters of the mechanism. By changing the position of input links of mechanisms of the tint and second types, keeping the parameters of these mechanisms constant, we shall obtain the corresponding variable positions of the outside mobile kinematic pairs of structural groups which are singled out of a high-class mechanism. 3. T H E T R A N S F O R M A T I O N
OF A HIGH-CLASS
STRUCTURAL
GROUP
The next step in numerical determination of the positions of links of a high-class structural group is its transformation into a system of four-link mechanisms, structural groups of the second class and closure links. Let us isolate four-link contours which have two kinematic pairs corresponding to the outside kinematic pairs of a high-class structural group. Such a contour is named the start-contour. Then we select either four-link contours joined to this contour or structural groups of the second class. By examining the whole structural group in this way we shall obtain either a closure link or a closure point. If the closure link or the closure point is separated from the structural group, then the system of four-link contours will be transformed into a system of four-link mechanisms of the second class. The positions of links of this system will coincide with those of a given structural group only in the case where the distance between kinematic pairs joined by a closure link is equal to the length of this link (or becomes singular at a closure point). To determine the position of the links of a start-mechanism coinciding with the start-contour we take the first rink as an input link and the second link as an output rink. These rinks are joined to the outside kinematic pairs. By giving successive positions to an input link, we may determine positions of all links of the start-four-rink mechanism as well as coordinates of points on its rinks. Then, by the same method, we determine the positions of links of successive four-link mechanisms, either as a mechanism or as structural groups of the second class. As a result, we shall obtain coordinates of kinematic pairs joined by a closure rink (or closure poin0. The positions of an input link of a start-four-rink mechanism are varied until the distance between these kinematic pairs becomes equal to the length of a closure link (or zero for the closure point). This distance is determined using the following formulae: for a spatial mechanism:
! = ~/[(x, - xl)2 + (y, _ yj)Z + (z, - zj~
for a plane mechanism:
! - x/[(x, - xj) 2 + (Yl - Y j ~ ,
where i and j are indices of kinematic pairs on the closure rink.
(5)
406
v.S. KAmu.~
Any four-link contour joined at the frame can be taken as a start-contour that will be further transformed into a start-four-link mechanism. Any link of a structural group may be a closure link, therefore, the transformation of a high-class structural group can be multivariant. The choice of an optimal variant of the transformation may simplify the solution of this problem. In that case, when the closure link becomes a side of the three-joint contour, it is possible to take an angle in this contour as an element of comparison. This variable angle is compared with the given angle until they coincide. When the position of a structural group corresponding to the accepted position of the first-class mechanism (or mechanisms) is determined, its positions changed, and with the help of a given method the position of the structural group links is determined again. The precision in determining the positions of links is found by changing the step in a position of an input link of a start-four-link mechanism. For example, any three (I, II or Ill) of the four-link contours may be taken as a start-four-link mechanism. Links Is-is, 12-Is or 12-15 can be taken as closure links in this structural group of the sixth class [Fig. 2(a)]. If we take the sides BC, DF or RG as closure links of the three joint links then the angles ~, or y respectively can be the parameters of comparison. We must take contours I and II as start-four-link contours in the plane six-link structural group of the third class [Fig. 2(b)]. Excluding kinematic pair E we obtain four-link mechanisms I and II. We choose an input link in these mechanisms and bind successive positions for them. Coordinates of point E, of mechanism I and the coordinates of point E2 of mechanism II are determined for every position until the coordinates coincide. If we complete the four-link contour in this group, the contour with minimum number m of links is chosen, and the position n of input links are given to them. The number of input links is determined from the formula n = m - 3.
(6)
For example, the contours in a 10-link structural group of fourth class [Fig. 2(c)], joined to a frame, have five links as a minimum. These are contours I and II. Taking two input links, for example, in contour I and links ! . and 1~ as closure links, we obtain the following decomposition of this structural group (by contours):
l(12-13-1s-17).-lll(16-1s-lto-lt3)-lV(i,,-I,,-I~s-llg)--II(lu-l,s-12~-123). Positional change of the input links/2 and/3 takes place until the distance between kinematic pairs ut u2 and vt v, becomes simultaneously equal to the length of links !,, and !,3. The problem of determining the links' positions in a spatial mechanism of high class is solved using the same method. 4. ORIENTATION OF SECOND-CLASS MECHANISMS The four-link mechanism in a high-class structural group composition, obtained from the corresponding four-link contours of this group, may be oriented while being assembled. This orientation results from the orientation of the function of the position of corresponding mechanisms. One of the links of the four-link mechanism jointed to the frame is taken as an input link. The coordinates of a point joining an input link to the frame are given and they are constant. If the point joining the links is on the link of the first-class mcchan!sm_~ its coordinates are variable, and they depend on the type of the first-class mechanism and are set when the positions of links of the structural group arc determined. The coordinates of the centre of the kinematic pair As of an input link in a coordinate system of a structural group are determined using formulae (l) or formulae (3) for spatial linkages or by formulae (2) or formulae (4) for plane linkages, depending on the character of movement of these links. The links joined to the frame are taken as output links. The coordinates of kinematic pair A2 of this link are determined using the formulae used for the input link. They differ in their constant parameters. The coordinates of the points joining these links are constant if the links
1
¢,
®
9
g, lt'~b
@
0
@ ®
®
i[
[
408
V.S. KAmu.m
are joined to the frame, and they are variable if they are joined to a point on the link of a first-class mechanism. The function of the position of a four-link mechanism is determined from the formula =
- x 2) +
- yJ
+
- z 2)
(7)
where Iz is the length of thc link which is not joined to the frame. By substituting the coordinates of points A, and Az in this equation after transformation we obtain a square equation for the function of the position of the output link. which depends on the position of the input link.
4. l. The spatial four-bar mechanism By substituting the coordinates of kinematic pairs A, and A2 in equation (7) for their values from (I). we obtain the square equation [Fig. 3(a)]
(s)
Mcos~ +Nsin¢ + P = 0 , where M = A sin p~ - B cos P3; N -- A cos ct3cos p; + B cos ~3 sin P3 - C sin ~3;
P=. ~-~3(AZ+ BZ + C2 + IjZ _ Iz),z" A = (L, - I, cos ~0) sin p, - (7", + l, sin q,) cos ~, cos p, - L3 sin p~ + T3 cos ~3 cos P3; B = Y0, - ( L , - I , cos ¢,)cos O , - ( T , +l, sin ~0)cos ~, sinp, - Y.+L3cos#3+ T3 sin ~3 sin #3; C = (T, + !, sin ¢p) sin ~, - T~ sin ~j. In these relationships the indices o f constant parameters show their ratio to input ia or output /3 links.
~,|
Y,~
Fig. 3
.--,
I~termJnafion of potations of links of high-dam ~
409
The solution of equation (8) may be presented as tan
l "M-P
[N + ~/(M 2 + N 2 - e*)].
(9)
Such an expression of the function of the position is convenknt because it does not require the determination of the quadrant in which the output link is located. The double sign in front of the radical corresponds to two possible assemblies of this mechanism. While analysing the concrete structural group and four-bar mechanism in its composition we choose the sign which corresponds to the assembly of the mechanism. The sign in front of the radical of the function of the position may be determined by tracing the structural group contour: " + " on going clockwise and " - ' " on going counter-clockwise. The position of link/2 executing an intricate movement in space is determined with the help of the angle ~ between the axis of link/2 and the coordinate axes, using the formulae : arcsinfZ4z - z,l~
\
,...,<.,;..,)
(,o)
In H e w of the fact that the sine function is symmetrical itis necessary to determine the quadrant in which this link exists.
4.2. The planefour-barmechanism By substituting the coordinates of points As and Az in equation (7) for their values (zA, = z, 2 - 0) from (I) after its transformation we obtain a square equation which is analogous to equation (8). The values of the coefficients in this equation are as follows [Fig. 3(b)]:
M::Xo3-Xol-llcos~;
N-
¥o3- Yoi-llsin~o;
P-~-~(M2+N2+I~-I~).
(11)
The solution of this equation is in line with solution (9). The orientation of this mechanism is found with the help of a sign placed in front of the radical depending on the direction of following the contour of the structural group in the mechanism. It is possible to use the formula to determine the position of the angle of link /2 which is performing intricate movement and is not joined to the frame in this plane four-hat mechanism: I~ = (x~2 - x f f + (y~, -Ym),
(12)
where x , . Ya are coordinates of the rotation of the axis output link/3. The coordinates of the centre of the rotation axis of the kinematic pair are determined using the formulae x~= = ,go + I~cos cp +/2 cos p;
Ys2" Yo+ I~sin tp +/2 sin #.
(! 3)
By substituting the coordinates of kinematic pair A, for their values after transforming equation (12) we obtain the square equation M cos# + N sing + P, ,- O,
(14)
where P, - ~ (I~- I] - M * - N:).
The solution of this equation is given in the form
tan/~
"M
I
-
P, [N :i:~/(M 2 + N 2 - PD].
(i~
410
V.S.
The orientation of the position of angle/~ of link/2 is obtained by following the contour of the structural group. The sign " + " in front of the radical indicates clockwise and " - ' " indicates counter-clockwise movement along the contour. 4.3. The coordinates o f the point on the link joined to the f r a m e
In a spatial mechanism the coordinates of such a point are determined using the formulae xo, = ILl - !o, cos(rp + P/)]sin p, - [Ti + !~, sin(~ + ~/)]cos ~ cos/~,; Yo, = Yo, - [Lj - !o, cos(~p + q)]cos p, - [7", + lo, sin(cp :i: q)]cos ~1 sin Pl;
to, = [Tt + lot sin((p + p/)]sin %,
(16)
where Io is the distance from the rotation axis of the link to a point on it whose coordinates are determined, and ~/is the position of an angle of the point with reference to the link on the plane. The orientation of point D is found by following the structural group contour of the mechanism; " + " in front of ~/ indicates a clockwise and " - " indicates a counter-clockwise direction. Indices of the constant parameters for the output link are to be replaced and the position of angle (p of the input link is replaced by the position of angle ~ of the output link. The coordinates of a point on the link marking an intricate movement in a spatial mechanism can be easily determined if this point is on the link's axis. The coordinates of this point are determined using the formulae xo = xAI + ~x~4:. I+;t'
'
Y°=
YAI + ~YA" I+A
;
z°=
:~l + ,lzA~ I+A
'
(17)
where ,l is the relationship of the distance to the point and the length of the link. In these formulae we use a " + " sign if a point is between kinematic pairs and " - " if a point is outside the kinematic pairs. If a link making a composite spatial movement is joined to the input and output links by means of spherical kinematic pairs, it is impossible to determine the coordinates of points that are not lying on its axis. Such a link can perform self-motion in regard to its axis. If it is necessary to orient this link in space, then one of the spherical kinematic pairs must be replaced by spherical with a pin kinematic pair. The coordinates of this point will be determined by the position and the form of a slot in this paper. In a plane four-bar mechanism the coordinates of a point on a link joined to a frame are determined using the formulae x~, = xo + locos(~o -+ql);
Yo~ = Yo + losin(~ + ~h).
(18)
The orientation of position point D is found by the method used for a spatial mechanism. The coordinates of the point on a link making a composite movement are determined using the formulae xD: = xA, + Io: cos(p + p/);
yo~ = YA, + l~ sin(p + ~/),
(19)
where io2 is the distance from kinematic pair A~ to a point on a link, and ~/is the position of the angle of the point relative to the axis of the link 12. 4.4. The mechanism with one slider
In a spatial mechanism with one slider we regard the slider as an output link. The function of the position of this mechanism is analogous to (7). Substituting the coordinates of kinematic pairs As and A 2 for their values in these formulae we obtain the function of the position of a slider as a square equation: S 2 + 2 M S + N = O,
(20)
Dmm'minltion of positions of links of high-dam linl~llcs
411
where M = A(cos 3' sin P3 + sin 7 cos =3 cos/13) + B(sin 3' cos ~'3sin P3 - cos y cos P3) - C sin 3' sin %;
N = A2 + B= + C'-12; A = (Lt - I, cos ~)sin Pl - (7"1 + It sin ~)cos ~1 cos Pt - 13sin P3; B = ro, - (/-, - / i cos q,)cos D, - (T, + 11 sin ~ ) o o s "t cos ~, - Yo, + / ~ cos B3;
C = (Tt + It sin ¢p)sin ~'t. The solution of this equation is as follows: S = - M + ~/(M 2 - N2).
(21)
The orientation of the position of a link in this mechanism is determined from the sign in front of the radical; " + " corresponds to tracing the contour of the structural group clockwise, and " - " corresponds to tracing it counter-clockwise.
4.5. The function of the position of a plane mechanism with one slider This is described using a square equation [equation (20)]. The solution of this equation is similar to equation (21). The coefficients of this equation have the following values [Fig. 3(c)]: M = (Xo3- X0t - l~ cos ¢p)cos ~ - (Yo + It sin ¢p)sin y; N = (Xo3- Xot - It cos ¢p)2+ (Y0 + i, sin ~o)2 - I].
(22)
$. THE ORIENTATION OF HIGH-CLASS STRUCTURAL G R O U P S The orientation of high-class structural groups is determined from the orientation of four-link mechanisms in a high-class mechanism. The orientation of the four-link mechanism is determined by its concrete assembly. As a result, the assembly of the high
Theorem The function of a position in a high-class mechanism branches-off only when that of any four-link mechanism in this system branches-off. The position of a point of branching-off of the function of the position in a high-class mechanism coincides with the position of the point of branching-off of function of the position of a four-link mechanism in this system. The possibility of branching-off of the function of the position in a high-class mechanism arises when links of four-link mechanisms in this system in one or many positions may vary their assembly spontaneously. Analysing four-link mechanisms in the high-class structural group, we may come to the conclusion that a structural group, as a whole, has singularity. An example of a mechanism which can change its assembly is shown in Fig. 4. In position "a", the four-link contour I may change spontaneously to its assembly for position "b" or "'c". As a result a high-class mechanism may change its assembly to "B" or "C". 6. EXAMPLES OF DETERMINATION OF P O S I T I O N S OF LINKS OF HIGH-CLASS MECHANISMS
6. !. Spatial linkages of the fourth class In this mechanism, the link r and the three-joint link ABCjoined to it are moving along the plane Q. The plane Q crosses the origin of the coordinate system at an angle ~, to coordinate plane xOy and forms angle ~ with coordinate axis y. The position of the rotation axis of the crank r on the plane Q is determined by the angle 0 and coordinates Tt and L. The three-joint link has sides It and/2, and angle 0 (Fig. 5).
412
V..5. KA/UU.iN
The links 13 and i, join the three-joint fine to the slider by the spherical kinematic pairs B, C, D and £. The guide of the slider coincides with the coordinate axis y. The coordinates of kinematic pairs D and E are as follows: xoffiS;
Yo"0;
zoffiZo,
yeffi0;
x e f f i S + T;
zerO.
Rational sampling of the space coordinate system allows us to reduce the number of constant parameters of the mechanism to a minimum. To determine the position 5 of a slider, the position of a crank ~# is fixed. The three-joint rink rotates around point A, and the coordinates of points D and E are determined using the orientated formulae s =
(z0-
+y,;
where x# ffi x~ - i, (sin ,8 cos ~ + sin fl cos ~, sin ~);
Ya ffi Y,~ + I, (cos fl cos ~, - sin p cos ~ sin ~,); zs ffi zA + !1 sin p sin ~; xc = xA -/z[sin B cos(~ + 6) + cos a cos p sin(~ + 6)]; yc -, y~ +/2[cos P cos(co + 6) - sin B cos a sin(~# + 6)]; z~ ffi z~ + I2 sin p sin(~ + 6); xA ffi (L - r cos of)sin p - (T + r sin ~)cos a cos 0; YA " Y e - ( L
- r cos ¢p)cos p - ( T + r
sin ~0)sin p cosa.
The angle value ~ varies until it coincides with the value 7", within the limits of precision. The next value of ~ is specified and variation of the position of angle ~# is repeated. The function of the position of a slider for this mechanism is shown in Fig. 5. The sizes of links of the mechanism are: •- 4 5 °, I, -. 20 ram;
p - . 6 0 °,
TffilSmm;
l, - 30 ram;
/3 ffi 60 ram;
r f , 12mm;
LffilSmm;
/4 ffi 85 ram;
z0 ffi 30 ram;
Y0ffi0; 6 .- 45%
Figure 5 shows the change of the position ~. The position of other links of mechanism may be easily determined. 6.2. Plane mechanism o f the sixth class
To determine the positions of the links of this mechanism we set the crank's positions and determine the coordinates of the centre of kinematic pair A from the formulae x~ ffi • cos co;
y~ = r sin co.
We rotate the three-joint link AB.I and determine the positions ~ and /~ for links I~ and /7 depending on the position #) of this link, from the orientated formulae
tan
ffi M I- P [N + ~ / ( M ~ + N ~ - P~)];
tan/t2 = M, -I P----~[N, + ~/(M~ + N~ - P~)],
where M' :,,, Xo -- XA-- 11COS~0; N ffi - ( y ~ + !, sin ~); P = ~ ( M ' + N 2 + !| - !,2); M , ffi x r - xA - !, cos(~ + ~'); N, - Yr - Ya - !, sin(~, + ~);
P,
m2~7( M s +
N 2 + 172 - ll). 2
~p
"It!4
g9
~, "a!d £ j
~,~
]
]l.a ~
|
, ~ , , , i ~ mqo-qllpl jo ~
o J..
/~,4
O~
I
/
o9/
J 0U,
,P
£11e
jo mop.rood jo oo.q.emm.m=a
414
V.S.
The three-joint links of this mechanism are orientated in accordance with the kinematic scheme of mechanism "A" (Fig. 4). The coordinates of the centres of kinematic pairs E and G are determined using the formulae x~ = x~ + l, cos(~ - p);
y+ = yo + 1, sin(~ - +e),
xa==xr+i6cos(p +~); Y~--Y~+Iscos(p +7)The value of angle ~p varies until the distance between points E and (7 is equal both to the length of link/5 and to a given value, within the limits of precision. Figure 4 shows the positions of output links 13 and 17 of this mechanism. (Calculations for this mechanism were done by eng. E. A. Barinova.) The sizes of links of this mechanism are as follows: xD--105mm;
x~---60mm; y~--100mm;
r=20mm;
/3 - 45 ram; l+ = 55 mm;/3 --- 16--- 50 mm; ~,=97o;
11=38mm;
12--58mm;
/7 = Is = 62 ram; /9 = 35 mm;
f l = ~ = 5 0 °.
Two cases may arise while determining the positions of links in high-class linkages: (!) Calculation of the function of the position of a four-link mechanism in high-class linkages shows that the expression under the radical has a negative value. In this case, the four-link mechanism is impossible and high-class linkages also are impossible. (2) The closure links are either larger or smaller than their sizes and are not equal to them. In this case, high-class linkages cannot be assembled and are impossible. The described method of numerical determination of the positions of links of high-class mechanisms is universal, and is suitable for all types of these mechanisms without any limitation. REFERENCES I. V. S. Karelin, Proc. 7th World Congr. Tkeory of Machines and Mechanisms, Seville (1987). 2. E. E. Pejsah. Machi~oeedenie 5, 55-61 (1985). 3. P. A. Lebedev and B. O. Marder, Machinouedenie 4, 30-39 (1986). 4. U. A. D g h o l d l s ~ k o v and Gh. Gh. Bajgunchekov. Analiticheskaj kinematika ploskih richaghnih mechanizmov visokikh klas~ov p. 101. AIma-Ala (1980). 5. C. U. Galletti, Meccanica (March). pp. 6--10 (1977).
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0094-114X/92$&00+0.00 CopyrilhtC~1992Perlimm Primlull
A NUMERICAL METHOD FOR DETERMINING POSITIONS OF LINKS OF HIGH-CLASS LINKAGES V. S. KARELIN Tver PolytechnicInstitute, 170035, Tver, Russia (Received 20 August 1990, receivedfor publication 12 August 1991)
Almract--A numerical method for determining positions of high-class linkages in spatial and plane mechanismsis discussed.With the help of one or more clmure links in a systemof four-linkmechanisms the high-classstructural Stoup from high-classlinkagesis transformed.The conditions for the number of closure links are determined. Each four-link mechanism is oriented in accordance with its assembly which in high.class linhaile coincides with the aBembly of the mechanism.The orientation of a four-link mechanismis found by selcctin8 the position's function which corresponds to the assemblyof this mechanism.As a result, the high-class mechanismbecomesoriented in a way that allowssimpledeterminationof the links"positions. The orientation function of a position for basic forms of four-link mechanismsand examples of determination of the positions of individual links for some spatial and plane mechanismsate siren.
High-class linkages have good prospects for mechanizing complicated technological processes. By means of these mechanisms it is possible to achieve especially complicated movements of links. Such mechanisms have unlimited structural possibilities. The wide technical use of high-class linkages is restricted by the fact that kinematic analysis of these mechanisms presents considerable complexity. Analytical expressions of the function of a position are known only for simple (thirdand fourth-class) mechanisms [I-3]. it is apparently impossible to obtain and use analytical expressions for the function of a position for the more complicated high-class mechanisms because of their complexity and lack of practical importance. The numerical method is more prospective for kinematic analysis of high-class linkages. The numerical method has no community with the analytical method; however, it allows simple and precise research of complex multilinks with high-class linkages. At present, the known numerical methods involve definite problems, one of which is related to a mechanism's assembly, and closely connected with this problem is the problem o f branching-off of the function of a position. These problems make the use of the numerical method more difEcult [4, 5].
At present, research is done mainly on plane high-class linkages. There has been little investigation of spatial linkages. This paper presents a numerical method for determining positions of links both for plane and spatial high-class mechanisms. The method is applicable for mechanisms of any complexity and there is no limitation on its use. I. THE S T R U C T U R A L G R O U P OF H I G H CLASS The numerical method for determining the position of links of high-class linkages is based on the possibility of oriented numerical description of a structural group from a singled-out high-class mechanism. Any high-class linkages and any structural groups joined to them and to the frame may be considered part of the first-class mechanisms. The number of first-class mechanisms determines the number of input links for high-class linkages. Links of the structural group joined to the frame are generally the output links. However, there are mechanisms in which output links are not joined to the frame. An arbitrary orthogonal coordinate system, spatial or plane, depends on the type of mechanism analysed. The arbitrary choice of the high-class mechanism's coordinate system is conditional. An expedient choice may reduce the number of parameters of a mechanism to some extent. It is Nm" ~ * - c
403
v. S. Ko.et~ desirable that the origin of the coordinate system coincides with the centre of rotation or spherical kinematic pairs, and that one coordinate axis passes across this pair or is parallel to the guide of the sfider or to the axis of the cylindrical kinematic pair. This condition is not obligatory. Let us take the high-class structural group out of a mechanism, having separated it from the first-class mechanism and from the frame. The position of the structural group is determined by coordinates of outside kinematic pairs at given dimensions of its links and its assembly. The coordinates of outside kinematic pairs, separated from the frame, are constant. The coordinates of outside kinematic pairs, separated from the first-class mechanisms, are variable and dependent on the type of these mechanisms and positions of their links. 2. FIRST-CLASS MECHANISMS There are two known types of first
za = (T +
[i
sin ~p) sin =.
(I)
(2) Movement of link !, of a first-type mechanism takes place on the coordinate plane xOy. The coordinates of point A of link !~ and a point joining it to the outside kinematic pair A, of a structural group are determined by the formulae [Fig. I(b)] xA -- `to + I, cos ¢p
Ya = Y0 + / i sin 4o
where ,to, Y0 are coordinates of the rotation axis of link !,.
1
I
\I
Fig. I
(2)
IX.tamination of positions~ links of hish-dam ti.,t~....
405
(3) Spatial movement of a second-type mechanism takes place on a plane Q. Its position in the
spatial inordinate system is deter-i.ed as in (l) above. The position of the guide on plane Q is determined by the angle ? between guide and coordinate plane x0y, and by the distance L from the point of crossing of the guide with this coordinate plane to coordinate axis y. The position of the tlider on the guide is determined by its distance S to coordinate plane xOy [Fig. l(c)]. The coordinates of point A on a dider and of a point joining it to the outside kinematic pair A~ of a stngtural group are determined by the formulae x,t = (L - S cos ?) sin p - S sin 3' cos ~t cos B y~ "= Y o - ( L
- S cos y)cos p - S siny cosec sin p
:~ = S sin ,, sin ?.
(3)
(4) Movement of a slider of second-type mechanism takes place on the coordinate plane x0y. The coordinates of point A of a slider and of a point joining it to the outside kinematic pair A, of the structural group are determined by formulae [Fig. I(d)] x4=Xo+Scosy
(4)
y,,=Ssiny,
where ,go is the abscissa of the point of crossin8 of the guide with the x-axis. The variable values in these formulae are the position ~ of link it in a mechanism of the first type and the position $ of a slider's point A in the mechanism of the second type. All other values entering these formulae are constant parameters of the mechanism. By changing the position of input links of mechanisms of the tint and second types, keeping the parameters of these mechanisms constant, we shall obtain the corresponding variable positions of the outside mobile kinematic pairs of structural groups which are singled out of a high-class mechanism. 3. T H E T R A N S F O R M A T I O N
OF A HIGH-CLASS
STRUCTURAL
GROUP
The next step in numerical determination of the positions of links of a high-class structural group is its transformation into a system of four-link mechanisms, structural groups of the second class and closure links. Let us isolate four-link contours which have two kinematic pairs corresponding to the outside kinematic pairs of a high-class structural group. Such a contour is named the start-contour. Then we select either four-link contours joined to this contour or structural groups of the second class. By examining the whole structural group in this way we shall obtain either a closure link or a closure point. If the closure link or the closure point is separated from the structural group, then the system of four-link contours will be transformed into a system of four-link mechanisms of the second class. The positions of links of this system will coincide with those of a given structural group only in the case where the distance between kinematic pairs joined by a closure link is equal to the length of this link (or becomes singular at a closure point). To determine the position of the links of a start-mechanism coinciding with the start-contour we take the first rink as an input link and the second link as an output rink. These rinks are joined to the outside kinematic pairs. By giving successive positions to an input link, we may determine positions of all links of the start-four-rink mechanism as well as coordinates of points on its rinks. Then, by the same method, we determine the positions of links of successive four-link mechanisms, either as a mechanism or as structural groups of the second class. As a result, we shall obtain coordinates of kinematic pairs joined by a closure rink (or closure poin0. The positions of an input link of a start-four-rink mechanism are varied until the distance between these kinematic pairs becomes equal to the length of a closure link (or zero for the closure point). This distance is determined using the following formulae: for a spatial mechanism:
! = ~/[(x, - xl)2 + (y, _ yj)Z + (z, - zj~
for a plane mechanism:
! - x/[(x, - xj) 2 + (Yl - Y j ~ ,
where i and j are indices of kinematic pairs on the closure rink.
(5)
406
v.S. KAmu.~
Any four-link contour joined at the frame can be taken as a start-contour that will be further transformed into a start-four-link mechanism. Any link of a structural group may be a closure link, therefore, the transformation of a high-class structural group can be multivariant. The choice of an optimal variant of the transformation may simplify the solution of this problem. In that case, when the closure link becomes a side of the three-joint contour, it is possible to take an angle in this contour as an element of comparison. This variable angle is compared with the given angle until they coincide. When the position of a structural group corresponding to the accepted position of the first-class mechanism (or mechanisms) is determined, its positions changed, and with the help of a given method the position of the structural group links is determined again. The precision in determining the positions of links is found by changing the step in a position of an input link of a start-four-link mechanism. For example, any three (I, II or Ill) of the four-link contours may be taken as a start-four-link mechanism. Links Is-is, 12-Is or 12-15 can be taken as closure links in this structural group of the sixth class [Fig. 2(a)]. If we take the sides BC, DF or RG as closure links of the three joint links then the angles ~, or y respectively can be the parameters of comparison. We must take contours I and II as start-four-link contours in the plane six-link structural group of the third class [Fig. 2(b)]. Excluding kinematic pair E we obtain four-link mechanisms I and II. We choose an input link in these mechanisms and bind successive positions for them. Coordinates of point E, of mechanism I and the coordinates of point E2 of mechanism II are determined for every position until the coordinates coincide. If we complete the four-link contour in this group, the contour with minimum number m of links is chosen, and the position n of input links are given to them. The number of input links is determined from the formula n = m - 3.
(6)
For example, the contours in a 10-link structural group of fourth class [Fig. 2(c)], joined to a frame, have five links as a minimum. These are contours I and II. Taking two input links, for example, in contour I and links ! . and 1~ as closure links, we obtain the following decomposition of this structural group (by contours):
l(12-13-1s-17).-lll(16-1s-lto-lt3)-lV(i,,-I,,-I~s-llg)--II(lu-l,s-12~-123). Positional change of the input links/2 and/3 takes place until the distance between kinematic pairs ut u2 and vt v, becomes simultaneously equal to the length of links !,, and !,3. The problem of determining the links' positions in a spatial mechanism of high class is solved using the same method. 4. ORIENTATION OF SECOND-CLASS MECHANISMS The four-link mechanism in a high-class structural group composition, obtained from the corresponding four-link contours of this group, may be oriented while being assembled. This orientation results from the orientation of the function of the position of corresponding mechanisms. One of the links of the four-link mechanism jointed to the frame is taken as an input link. The coordinates of a point joining an input link to the frame are given and they are constant. If the point joining the links is on the link of the first-class mcchan!sm_~ its coordinates are variable, and they depend on the type of the first-class mechanism and are set when the positions of links of the structural group arc determined. The coordinates of the centre of the kinematic pair As of an input link in a coordinate system of a structural group are determined using formulae (l) or formulae (3) for spatial linkages or by formulae (2) or formulae (4) for plane linkages, depending on the character of movement of these links. The links joined to the frame are taken as output links. The coordinates of kinematic pair A2 of this link are determined using the formulae used for the input link. They differ in their constant parameters. The coordinates of the points joining these links are constant if the links
1
¢,
®
9
g, lt'~b
@
0
@ ®
®
i[
[
408
V.S. KAmu.m
are joined to the frame, and they are variable if they are joined to a point on the link of a first-class mechanism. The function of the position of a four-link mechanism is determined from the formula =
- x 2) +
- yJ
+
- z 2)
(7)
where Iz is the length of thc link which is not joined to the frame. By substituting the coordinates of points A, and Az in this equation after transformation we obtain a square equation for the function of the position of the output link. which depends on the position of the input link.
4. l. The spatial four-bar mechanism By substituting the coordinates of kinematic pairs A, and A2 in equation (7) for their values from (I). we obtain the square equation [Fig. 3(a)]
(s)
Mcos~ +Nsin¢ + P = 0 , where M = A sin p~ - B cos P3; N -- A cos ct3cos p; + B cos ~3 sin P3 - C sin ~3;
P=. ~-~3(AZ+ BZ + C2 + IjZ _ Iz),z" A = (L, - I, cos ~0) sin p, - (7", + l, sin q,) cos ~, cos p, - L3 sin p~ + T3 cos ~3 cos P3; B = Y0, - ( L , - I , cos ¢,)cos O , - ( T , +l, sin ~0)cos ~, sinp, - Y.+L3cos#3+ T3 sin ~3 sin #3; C = (T, + !, sin ¢p) sin ~, - T~ sin ~j. In these relationships the indices o f constant parameters show their ratio to input ia or output /3 links.
~,|
Y,~
Fig. 3
.--,
I~termJnafion of potations of links of high-dam ~
409
The solution of equation (8) may be presented as tan
l "M-P
[N + ~/(M 2 + N 2 - e*)].
(9)
Such an expression of the function of the position is convenknt because it does not require the determination of the quadrant in which the output link is located. The double sign in front of the radical corresponds to two possible assemblies of this mechanism. While analysing the concrete structural group and four-bar mechanism in its composition we choose the sign which corresponds to the assembly of the mechanism. The sign in front of the radical of the function of the position may be determined by tracing the structural group contour: " + " on going clockwise and " - ' " on going counter-clockwise. The position of link/2 executing an intricate movement in space is determined with the help of the angle ~ between the axis of link/2 and the coordinate axes, using the formulae : arcsinfZ4z - z,l~
\
,...,<.,;..,)
(,o)
In H e w of the fact that the sine function is symmetrical itis necessary to determine the quadrant in which this link exists.
4.2. The planefour-barmechanism By substituting the coordinates of points As and Az in equation (7) for their values (zA, = z, 2 - 0) from (I) after its transformation we obtain a square equation which is analogous to equation (8). The values of the coefficients in this equation are as follows [Fig. 3(b)]:
M::Xo3-Xol-llcos~;
N-
¥o3- Yoi-llsin~o;
P-~-~(M2+N2+I~-I~).
(11)
The solution of this equation is in line with solution (9). The orientation of this mechanism is found with the help of a sign placed in front of the radical depending on the direction of following the contour of the structural group in the mechanism. It is possible to use the formula to determine the position of the angle of link /2 which is performing intricate movement and is not joined to the frame in this plane four-hat mechanism: I~ = (x~2 - x f f + (y~, -Ym),
(12)
where x , . Ya are coordinates of the rotation of the axis output link/3. The coordinates of the centre of the rotation axis of the kinematic pair are determined using the formulae x~= = ,go + I~cos cp +/2 cos p;
Ys2" Yo+ I~sin tp +/2 sin #.
(! 3)
By substituting the coordinates of kinematic pair A, for their values after transforming equation (12) we obtain the square equation M cos# + N sing + P, ,- O,
(14)
where P, - ~ (I~- I] - M * - N:).
The solution of this equation is given in the form
tan/~
"M
I
-
P, [N :i:~/(M 2 + N 2 - PD].
(i~
410
V.S.
The orientation of the position of angle/~ of link/2 is obtained by following the contour of the structural group. The sign " + " in front of the radical indicates clockwise and " - ' " indicates counter-clockwise movement along the contour. 4.3. The coordinates o f the point on the link joined to the f r a m e
In a spatial mechanism the coordinates of such a point are determined using the formulae xo, = ILl - !o, cos(rp + P/)]sin p, - [Ti + !~, sin(~ + ~/)]cos ~ cos/~,; Yo, = Yo, - [Lj - !o, cos(~p + q)]cos p, - [7", + lo, sin(cp :i: q)]cos ~1 sin Pl;
to, = [Tt + lot sin((p + p/)]sin %,
(16)
where Io is the distance from the rotation axis of the link to a point on it whose coordinates are determined, and ~/is the position of an angle of the point with reference to the link on the plane. The orientation of point D is found by following the structural group contour of the mechanism; " + " in front of ~/ indicates a clockwise and " - " indicates a counter-clockwise direction. Indices of the constant parameters for the output link are to be replaced and the position of angle (p of the input link is replaced by the position of angle ~ of the output link. The coordinates of a point on the link marking an intricate movement in a spatial mechanism can be easily determined if this point is on the link's axis. The coordinates of this point are determined using the formulae xo = xAI + ~x~4:. I+;t'
'
Y°=
YAI + ~YA" I+A
;
z°=
:~l + ,lzA~ I+A
'
(17)
where ,l is the relationship of the distance to the point and the length of the link. In these formulae we use a " + " sign if a point is between kinematic pairs and " - " if a point is outside the kinematic pairs. If a link making a composite spatial movement is joined to the input and output links by means of spherical kinematic pairs, it is impossible to determine the coordinates of points that are not lying on its axis. Such a link can perform self-motion in regard to its axis. If it is necessary to orient this link in space, then one of the spherical kinematic pairs must be replaced by spherical with a pin kinematic pair. The coordinates of this point will be determined by the position and the form of a slot in this paper. In a plane four-bar mechanism the coordinates of a point on a link joined to a frame are determined using the formulae x~, = xo + locos(~o -+ql);
Yo~ = Yo + losin(~ + ~h).
(18)
The orientation of position point D is found by the method used for a spatial mechanism. The coordinates of the point on a link making a composite movement are determined using the formulae xD: = xA, + Io: cos(p + p/);
yo~ = YA, + l~ sin(p + ~/),
(19)
where io2 is the distance from kinematic pair A~ to a point on a link, and ~/is the position of the angle of the point relative to the axis of the link 12. 4.4. The mechanism with one slider
In a spatial mechanism with one slider we regard the slider as an output link. The function of the position of this mechanism is analogous to (7). Substituting the coordinates of kinematic pairs As and A 2 for their values in these formulae we obtain the function of the position of a slider as a square equation: S 2 + 2 M S + N = O,
(20)
Dmm'minltion of positions of links of high-dam linl~llcs
411
where M = A(cos 3' sin P3 + sin 7 cos =3 cos/13) + B(sin 3' cos ~'3sin P3 - cos y cos P3) - C sin 3' sin %;
N = A2 + B= + C'-12; A = (Lt - I, cos ~)sin Pl - (7"1 + It sin ~)cos ~1 cos Pt - 13sin P3; B = ro, - (/-, - / i cos q,)cos D, - (T, + 11 sin ~ ) o o s "t cos ~, - Yo, + / ~ cos B3;
C = (Tt + It sin ¢p)sin ~'t. The solution of this equation is as follows: S = - M + ~/(M 2 - N2).
(21)
The orientation of the position of a link in this mechanism is determined from the sign in front of the radical; " + " corresponds to tracing the contour of the structural group clockwise, and " - " corresponds to tracing it counter-clockwise.
4.5. The function of the position of a plane mechanism with one slider This is described using a square equation [equation (20)]. The solution of this equation is similar to equation (21). The coefficients of this equation have the following values [Fig. 3(c)]: M = (Xo3- X0t - l~ cos ¢p)cos ~ - (Yo + It sin ¢p)sin y; N = (Xo3- Xot - It cos ¢p)2+ (Y0 + i, sin ~o)2 - I].
(22)
$. THE ORIENTATION OF HIGH-CLASS STRUCTURAL G R O U P S The orientation of high-class structural groups is determined from the orientation of four-link mechanisms in a high-class mechanism. The orientation of the four-link mechanism is determined by its concrete assembly. As a result, the assembly of the high
Theorem The function of a position in a high-class mechanism branches-off only when that of any four-link mechanism in this system branches-off. The position of a point of branching-off of the function of the position in a high-class mechanism coincides with the position of the point of branching-off of function of the position of a four-link mechanism in this system. The possibility of branching-off of the function of the position in a high-class mechanism arises when links of four-link mechanisms in this system in one or many positions may vary their assembly spontaneously. Analysing four-link mechanisms in the high-class structural group, we may come to the conclusion that a structural group, as a whole, has singularity. An example of a mechanism which can change its assembly is shown in Fig. 4. In position "a", the four-link contour I may change spontaneously to its assembly for position "b" or "'c". As a result a high-class mechanism may change its assembly to "B" or "C". 6. EXAMPLES OF DETERMINATION OF P O S I T I O N S OF LINKS OF HIGH-CLASS MECHANISMS
6. !. Spatial linkages of the fourth class In this mechanism, the link r and the three-joint link ABCjoined to it are moving along the plane Q. The plane Q crosses the origin of the coordinate system at an angle ~, to coordinate plane xOy and forms angle ~ with coordinate axis y. The position of the rotation axis of the crank r on the plane Q is determined by the angle 0 and coordinates Tt and L. The three-joint link has sides It and/2, and angle 0 (Fig. 5).
412
V..5. KA/UU.iN
The links 13 and i, join the three-joint fine to the slider by the spherical kinematic pairs B, C, D and £. The guide of the slider coincides with the coordinate axis y. The coordinates of kinematic pairs D and E are as follows: xoffiS;
Yo"0;
zoffiZo,
yeffi0;
x e f f i S + T;
zerO.
Rational sampling of the space coordinate system allows us to reduce the number of constant parameters of the mechanism to a minimum. To determine the position 5 of a slider, the position of a crank ~# is fixed. The three-joint rink rotates around point A, and the coordinates of points D and E are determined using the orientated formulae s =
(z0-
+y,;
where x# ffi x~ - i, (sin ,8 cos ~ + sin fl cos ~, sin ~);
Ya ffi Y,~ + I, (cos fl cos ~, - sin p cos ~ sin ~,); zs ffi zA + !1 sin p sin ~; xc = xA -/z[sin B cos(~ + 6) + cos a cos p sin(~ + 6)]; yc -, y~ +/2[cos P cos(co + 6) - sin B cos a sin(~# + 6)]; z~ ffi z~ + I2 sin p sin(~ + 6); xA ffi (L - r cos of)sin p - (T + r sin ~)cos a cos 0; YA " Y e - ( L
- r cos ¢p)cos p - ( T + r
sin ~0)sin p cosa.
The angle value ~ varies until it coincides with the value 7", within the limits of precision. The next value of ~ is specified and variation of the position of angle ~# is repeated. The function of the position of a slider for this mechanism is shown in Fig. 5. The sizes of links of the mechanism are: •- 4 5 °, I, -. 20 ram;
p - . 6 0 °,
TffilSmm;
l, - 30 ram;
/3 ffi 60 ram;
r f , 12mm;
LffilSmm;
/4 ffi 85 ram;
z0 ffi 30 ram;
Y0ffi0; 6 .- 45%
Figure 5 shows the change of the position ~. The position of other links of mechanism may be easily determined. 6.2. Plane mechanism o f the sixth class
To determine the positions of the links of this mechanism we set the crank's positions and determine the coordinates of the centre of kinematic pair A from the formulae x~ ffi • cos co;
y~ = r sin co.
We rotate the three-joint link AB.I and determine the positions ~ and /~ for links I~ and /7 depending on the position #) of this link, from the orientated formulae
tan
ffi M I- P [N + ~ / ( M ~ + N ~ - P~)];
tan/t2 = M, -I P----~[N, + ~/(M~ + N~ - P~)],
where M' :,,, Xo -- XA-- 11COS~0; N ffi - ( y ~ + !, sin ~); P = ~ ( M ' + N 2 + !| - !,2); M , ffi x r - xA - !, cos(~ + ~'); N, - Yr - Ya - !, sin(~, + ~);
P,
m2~7( M s +
N 2 + 172 - ll). 2
~p
"It!4
g9
~, "a!d £ j
~,~
]
]l.a ~
|
, ~ , , , i ~ mqo-qllpl jo ~
o J..
/~,4
O~
I
/
o9/
J 0U,
,P
£11e
jo mop.rood jo oo.q.emm.m=a
414
V.S.
The three-joint links of this mechanism are orientated in accordance with the kinematic scheme of mechanism "A" (Fig. 4). The coordinates of the centres of kinematic pairs E and G are determined using the formulae x~ = x~ + l, cos(~ - p);
y+ = yo + 1, sin(~ - +e),
xa==xr+i6cos(p +~); Y~--Y~+Iscos(p +7)The value of angle ~p varies until the distance between points E and (7 is equal both to the length of link/5 and to a given value, within the limits of precision. Figure 4 shows the positions of output links 13 and 17 of this mechanism. (Calculations for this mechanism were done by eng. E. A. Barinova.) The sizes of links of this mechanism are as follows: xD--105mm;
x~---60mm; y~--100mm;
r=20mm;
/3 - 45 ram; l+ = 55 mm;/3 --- 16--- 50 mm; ~,=97o;
11=38mm;
12--58mm;
/7 = Is = 62 ram; /9 = 35 mm;
f l = ~ = 5 0 °.
Two cases may arise while determining the positions of links in high-class linkages: (!) Calculation of the function of the position of a four-link mechanism in high-class linkages shows that the expression under the radical has a negative value. In this case, the four-link mechanism is impossible and high-class linkages also are impossible. (2) The closure links are either larger or smaller than their sizes and are not equal to them. In this case, high-class linkages cannot be assembled and are impossible. The described method of numerical determination of the positions of links of high-class mechanisms is universal, and is suitable for all types of these mechanisms without any limitation. REFERENCES I. V. S. Karelin, Proc. 7th World Congr. Tkeory of Machines and Mechanisms, Seville (1987). 2. E. E. Pejsah. Machi~oeedenie 5, 55-61 (1985). 3. P. A. Lebedev and B. O. Marder, Machinouedenie 4, 30-39 (1986). 4. U. A. D g h o l d l s ~ k o v and Gh. Gh. Bajgunchekov. Analiticheskaj kinematika ploskih richaghnih mechanizmov visokikh klas~ov p. 101. AIma-Ala (1980). 5. C. U. Galletti, Meccanica (March). pp. 6--10 (1977).
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