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Path generating five-bar mechanisms
Jnl. MechanismsVolume 4, pp. 17-30/Pergamon Press1969/Printed in Great Britain
Path Generating Five-Bar Mechanisms J. A. Brewer, II1"
M a r v i n Dixon," and G. D. Whitehouse; Received 2 May 1968 Abstract The synthesis of a linearly coordinated five-bar mechanism using six precision points to represent the trace curve is a complex problem involving a large number of unknowns and causing a multiplicity of roots in the solution. A modified Newton Method is utilized in this paper to solve the resulting equations. A continuation of the six-precision-point problem is accomplished by releasing the linear coordination of the input cranks and then designing for the non-linear coordination. This design consisted of a non-linear band mechanism. The procedure is fully demonstrated by means of an example problem. A discussion of the inaccuracies, uniqueness of solution, and the extension of this method to the seven-precision-point problem is presented. Zusammenfassung--Band koordinierung von ffinfgliedrigen KoppelkurvenGetrieben : J. A. Brewer, Marvin Dixon und G. D. Whitehouse. Die Synthese eines linear koordinierten ffinfgliedrigen Getriebes ffir die Anwendung von sechs Pr~zisionspunkten zur Darstellung einer Koppelkurve ist ein kompliziertes Problem, das eine grosse Anzahl von Unbekannten enth~lt und deren LSsung eine Vielfalt yon Wurzeln ben6tigt. In der vorliegenden Arbeit ist eine modifizierte Methode von Newton zur L6sung der resultierenden Gleichungen benutzt. Eine Fortsetzung des Sechs-Pr~zisionspunkt-Problems wird erreicht durch das Ausschalten der Linearkoordinaten der Antriebskurbeln und nachtr~gliche Konstruktion der nichtlinearen Koordination. Diese Vorrichtung bestand aus einem nichtlinearen Bandmechanismus. Das Verfahren ist durch ein Beispiel ausfLihrlich veranschaulicht. Eine Fehlerbetrachtung eine Diskussion der Einzigartigkeit der L6sung, sowie eine Erweiterung dieser Methode zum Sieben-Punkt Pr~zisions-Problem ist durchgefuhrt. Pe3mMe--J'IeHTO~IHaa KOOp~lHattH~l rlaTH3BCHH~IX MexaHH3MOB~n~ BocnpoI,I3BC~eHI4~ maTym~ix KpnnbiX: I'. A. Bpmep, Mapmm ~Hxcoa ~t F. A. BaTxyc. CaHTe3 naae~lHoKoop~anapoaanaaro n.qTaaBcm~aro Mexaaa3Ma ~rm aocnpoa3Be~eaa~ maryawLx Kpt4ablX no tuecTa TO~IHblMTO*IlcaMsaaaeTca caoxool~ npo6aeMofi, Tpe6y~omeii 6om~moe ~acno H C H 3 B e C T H I ~ I X I~ MHOFOqFIC.r[eHHblX K o p n e f l ypaBHenmi~. B nacToameR pa6oTe nph'MeHaeTCR MO~I4q~HraIIRg MeTO~a Hh~OTOHa ~ p e m e n I ~ [ COOTBCTCTByK)IIIRX y p a B H e R I ~ , l-[po~oYDKeHne n p o 6 n e M ~ meCTH. TO~IHbIX TOtlCK ocyKIeCTBYlgeTC~ HCK.rIIO~IeHHeM .rlRHe~HOfl r o o p ~ I a H a u m ~ Be~rttlRX Kph-BOmm1OB H
nocne~ty~oLue~ rOHCTpyrm4efi Hennne~IHofi roop~4Hat~H. 3TO ycTpofiCTBOCOCTOIITH3 Hemme~maro neHTO~lHaro MexaRH3Ma. ~TOT IIpHeM Bl~Onlle o n p a B ~ I B a e T C R pa3CMOTpeRI~IM ~IpRMepOM. O6cy~c~leHI~e HeTOtlHOCT¢I~, e~HHCTBeHHOCTR p e m e H H ~ H p a c n p o c T p a H e H H e 3TOrO MeTO~a Ha CeM~ TO~I~tX To~eK Ta k"~e I'IpHBO~HTC~I.
*Instructor, Engineering Graphics, Louisiana State University, Baton Rouge, Louisiana 70803, U.S.A. tlnstructor, Mechanical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803. ~.Assistarit Professor, Mechanical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803.
17
18 1. Introduction CERTAIN kinematic requirements call for a pointer or stylus to traverse a definite and sometimes irregular path. A mechanical linkage system which may be offered as a solution to this type of problem is usually referred to as a "path-generator". The five-bar mechanism with its two degrees of freedom offers lucrative possibilities as a path-generator. It, for a specified five-bar, the "pointer" is forced to exactly follow a desired path, generally a non-linear relation between input-crank rotations will result. If the input cranks can be coordinated to rotate precisely in this non-linear manner, a path-generating mechanism with an infinite number of precision points, exact points specified by the path, would be developed. The five-precision-point solution for linearly coordinated five-bar mechanisms [l] has been known for some time. This solution requires the solution of five non-linear, algebraic equations in five unknowns. Many iteration techniques have been developed [2] and applied to the above case with limited success. With all of the techniques, failure to converge to a root of the set is quite common. A modification of Newton's Method, formulated by Freudenstein and Roth [4] is particularly effective in the solution of linkage problems and has been shown to widen the domain of convergence considerably. This method is used to obtain the solution of the six-precision-point problem by the authors. As a sequel to Freudenstein's paper on path generation by geared five-bars, Hoffpauir [5] investigated the five-bar mechanism utilizing non-circular gearing to coordinate the input cranks. The object of the investigation was to increase the number of precision points on the desired path. The object of this investigation is similar, in that nonlinear coordination is used to increase the number of available precision points. In developing an approach to the synthesis of the five-bar mechanism to generate a given path, the selection of a coordinating mechanism to control the input-crank relationships must be considered. In this paper, the mechanism selected to coordinate the inputcrank rotations is the "non-linear band mechanism" (Fig. I).
~'"~--~------r~DE.~REDPATH
~
CIRCULAR PULLEY
INEXTENS]BLEB~ND Figure 1. Non-linear band mechanism. Design equations developed by McPhate [1] are used to derive the non-linear band mechanism. Using this method a band device can be found to generate nearly any nonlinear rotational relationship. The prime restriction is the developed range of rotation. For a relationship that is quasi-linear, the effective range of rotation will be increased. It is therefore desirable to keep the rotational relationship as nearly linear as possible.
19 2. Synthesis of the Six-Precision-Point, Linearly Coordinated Five-Bar Mechanism A discussion of the five-precision-point solution for a linearly coordinated, five-bar mechanism follows for comparison purposes. The equations of constraint are given by equation (l), and a layout of the mechanism is shown in Fig. 2.
c
POINTER
~
CURVE
03
~ (INI'rlAL)
Figure 2. Linearly coordinated five-bar.
Ct+ r t e i(°t('" ~+o,)+ RE = F C 2..~ r3ei(°3( i.tt )+o~)+R4=F 03=KIOt+K 2
(1)
where: Ct, C2, R 2, R4 and F are complex numbers; Ct and C2 locate the axes of rotation of the input cranks; rt and r3, and K2 are scalar constants;
K1
R, and R4 are the remaining links of the five-bar; F defines the path of the "pointer" on the trace curve, and 0 t and 03 are the input-crank angles. The last expression of (1) insures linear coordination of the input cranks. Assuming 0 t, K~, Kz and F are known, the above set of equations contains twelve unknowns. We can reduce this set to two equations with ten unknowns. If 01 and F are incremented four times, the number is increased to ten equations and the resulting set yields five coordinated precision points. In order to increase the generality of the equations, thereby facilitating a greater number of precision points, extension vectors PI and P2 are introduced. The equations of constraint for a system of this type (Fig. 3) are given by equations (2). The tip of the extension vector PL, rather than the junction of R 2 and R4, will define the location of the trace point or "pointer".
20 ~OINTER t.-
/
2/"°,
2-'°
X
Figure 3. Five-bar with extension vectors P~ and P2. The equations now become: Ct + r j °~ + Rz q- Pt = F
C2 "~ r3 el°3 + R4 q"P2 "4"Pt = F
(2)
03=KtOI+K2
where Pt and P2 are extension vectors. Since Pt and P2 are complex numbers they add four unknowns, while eliminating 0t(i,itiat) and 0a(i,mao by specifying these quantities, thus keeping the complexity of the equations at a minimum, and reducing the number of unknowns by two. The result is an overall increase of two unknowns, or two equations in twelve unknowns. Incrementing the known parameters five times yields a set of twelve equations in twelve unknowns. A total of six coordinated precision points is therefore obtained. Considering equations (2) and the "left-half" of the five-bar (see Fig. 3), we can write in complex notation: C t + rl ei°' + r2 ei02 nt"Pl ei~' = F
(3)
where Cl=u+iv,
We define angle above substitutions,
F=x+iy,
R 2 = r 2 el°2,
and P t = p x e i~ .
(~=02"~-kI"/, where W is a constant. Affixing Pt
to R 2 and making the
u + iv + r ~ e i°' 4- r 2 e I02 -I- p 1e~(°~+v) = x + iy
(4)
and rewriting equations (4), we obtain r2ei°2+ p~ei(°2+V)=x + i y - - u -- iv-- r~e i°' .
We reduce the expression to a simple form using complex algebra; thus r22 + p12 + 2r2pl cos ~ = x2 + y2 + u2 + v2 + rt2 -2xu-2yv-2xQ
cos 0~ - 2 y r I sin 0t + 2 u r t cos 0~ + 2 v Q sin 0t •
(5)
21 Substituting Pt cos ~ =Plx, we obtain x z + y2 + u2 + v2 + rt z _ r2 z _ Pt z _ 2r2pt. ~ - 2xu -2yv-2xr
t cos 0t - 2 y r ~ sin 0~ + 2urt cos 0t + 2vr t sin 0t = 0.
(6)
Equation (6) represents one equation with nine unknowns. The path to be followed, and the corresponding input-crank angle, 0~, will be specified, thus reducing the number of unknowns to six: namely, ,t, v, r t, r z, Pt and pt.,. If x, y and 0t are incremented five times, equation (6) becomes the set (7).
F=x+iy,
xZ + y2 +u2 +v2 + r t 2 - - r z 2 - - p t Z - - 2 r 2 P t x -- 2XkU -- 2 ygv -- 2Xkr 1 COSOt k -- 2 ykr t sin Ot k + 2ur I cosOtk + 2vrt sin Olk=O
(7)
where k = 0, 1, 2, 3, 4, 5.
The right half of the five-bar is considered in a similar manner as the previous derivation, but with x and y (or F) effectively transformed to X and Y. In other words, the trace path for the right half is defined by the head of R,. One can observe that X and Y are given by the solution of the left half of the mechanism.
Xk2-~ Yk2-1-g2 +he + r a 2 - r 4 2 - p 2 2 - 2 r 4 P 2 x - 2Xk#-- 2 Y k h - 2Xkr3 COSOk - 2 Ykr3 sin Ok
+ 29r 3 COS 03k -1-2 h r 3 sin 03k = 0
(8)
where k=O, 1, 2, 3, 4, 5. The following quantities are defined: X= x - I ;
Y = i(y - m)
and Pl=l+im.
Given any set of six coordinated points, all dimensions of a linearly coordinated, five-bar mechanism which will move a pointer through each point exactly, can theoretically be found by solving equations (7) and (8) in that order. Each is a set of six non-linear, algebraic equations; the solution of such a set will now be discussed.
3. Solution of the Simultaneous, Non-Linear, Algebraic Equations Newton's method is one of the best techniques to solve non-linear algebraic equations because of its rapid convergence. Newton's method, as with all iteration techniques, requires a " g o o d " initial estimate of a root to insure convergence. Freudenstein and Roth [4] have developed a modification of Newton's method which circumvents this
22 difficulty; their method is especially adaptable to the solution of linkage-type equations. An outline of it is as follows: Equations (7) and (8) may be written in the form, f.(x) =
P.o
+ pnl X I 2 + Pn2X 22 + Pn3X 32 + pn4X-t.2
+ pnsX52 + pn6X4X6 + pnTXt + pnsX2 + pn9X3 + P.loXIX3
+ P.i
1XzX3
(9)
where n = 1, 2, 3, 4, 5, 6. In the above set, the coefficients P,k will be determined by specifying six coordinated precision points which the five-bar is desired to trace and the six corresponding input-crank positions. A graphically determined, derived set 9(,°)(x) is used in the solution off,(x). g(.°)(x) =
q.o + q . l X t 2 + q n z X 2 2
+ qn3X32qngX42
-t- qnsX5 2 + qn6X4X6 + qnvX1 + qnsX2 + qn9X3 + qnloXtS3 + q.t
1X2X3
(10)
where n = 1, 2, 3, 4, 5, 6.
The coefficients q , k of the derived set are first incremented toward the coefficients P,k of the unknown set. This gives a new unknown set 9~1)(x), which has coefficients close to that of the derived set. Using the known root, x, of the derived set as a first approximation to the root of g~,t)(x), Newton's method is used to determine a root of g~l)(x). The coefficients are incremented once more and the process repeated until finally 9(~J)(x) =f,(x) with the final root being one of the unknown set, f,(x). The above method considerably expands the range of convergence. This effectively eliminates the need for a close initial guess of an unknown root. 4. Non-Linear Coordination of Input Cranks by Band Mechanisms A five-bar mechanism with linearly coordinated input cranks was the result of solving equations (7) and (8). We must release the linear coordination in order to alter the trace path of the pointer. The rotational relationship of the input cranks, (01 vs. 03), developed by altering the path, can be determined analytically by the use of equations (4) and (5). Rewriting equation (4) u + iv + r I e i°= + r 2e i°' + p t el(02 + V) = X + i y
(1 l)
and rearranging, we obtain r l e i°= +
(r 2 +
plei~r)e I°2 = x - u + i ( y - v ) .
(12)
Let us define, W=we"=r2+ple Z = z e ip = x -
iv
u + i(y-
and v),
(13)
and after substituting and dividing by e ~p w e i(°2 + "-B) = z - r l e i(°~-a)
(14)
23 Separating 01 , we can reduce the expression to: 01 =/3 + arc cos[-(z 2 + r t z _
wZ)/2zrl],
(i~)
where/3, z, r t and w are all known quantities defined by the position of the pointer of a given five-bar. From equation (14), and in a similar fashion, as the solution for 0 I, we obtain the solution for 02;
02 = / ~ -
r 4- arc cos[(z z + w2 -rlZ)/2"_w].
(16)
The remaining loop of the five-bar is considered in a similar manner to determine 03. 03 = + arccosq + r 3- - r ~ -
([7)
2qr3
A graphical interpretation of equations (15), (16) and 17) is shown in Fig. 4.
i¥
r4
.Z Figure 4. Graphical layout of input-output crank angles•
The input-crank angles can be expressed in tabular form after solving equations (15) and (17). Since derivatives of 03 with respect to 01 are required in the design of the nonlinear band mechanism, crank-angle data were then fitted to a polynomial which enabled the higher derivatives of the function to be obtained. The design equations for a non-linear band system have been developed by McPhate [1]. The developed equations are as follows* (See Fig. 5): qbb= arc sin(rz~b~) + ~b1
~b;= + r2~I-1 - ( r 2 ~ ) 22-÷-t- 1 • ~ , i,~ , ZI = l-(q~,- 1)el*' + tr2(~bb-q~2)e b)lq~b.
*NoT~--Primes denote derivatives with respect to the input angle ~x.
([8)
24
~
2
Figure 5. Non-linear band mechanism nomenclature.
5. Example Problem Consider a five-bar mechanism that is to trace the path shown in Fig. 6. Pt, P2, P3, Pc, Ps, and P6 are six precision points, specified by the given coordinates. The pointer
Figure 6. Desired trace path with six precision points.
25 of the five-bar must travel from Pt to P6 in one-half of a revolution of the driving input crank. Furthermore, the intervals between Pl and P6 must be as follows" Pt to Pz--~-~ revolution of the driving crank; P2 to P 3 -
7Lfz revolution of the driving crank;
P3 to P.~ t
revolution of the driving crank;
P.~ to P5 - - ~o revolution of the driving crank; P5 to P 6 - - ~ revolution of the driving crank. The first phase of the problem is to establish a five-bar mechanism with linearly coordinated inputs which will trace six precision points on the given path. This involves the solution of equations (7) and (8). A "derived" mechanism is utilized to solve the equations using the methods discussed previously. The derived mechanism must be of the same form as the unsolved mechanism, but all parameters, including the trace path, must be known. The derived mechanism used in the solution of the example problem was determined graphically and is shown in Fig. 7. ( 1.70, 2.71 )
(.95, 2.66)
iY
/
/i
/'
(.26, 2.35)
1
/ 3I, .851[
/
I
'~
~.'Ox I
\ I ",,
/
I \\
i/
/
•
,
I
~. / Jx', /
/.,', /
/
," \ X
",,,",
Y ,,,
,
P,x = R cos B~ B, ='45 °
\/X" \
"\'x/
(2.78, 2.11) CONSIDERING THE "LEFT-HALF" ONLY V= 0.0 U= Rl=Rz =P~ :1.0 P=x= 0.7266
/
/
;~. /
,
/
/
I
I
(2.36, 2.51 )
jl
iI
i
\ \
/
/
/
\f /\\
/
i
I I
lJ--u F i g u r e 7. Derived mechanism for linear coordination.
,/ /
/
26 The following values were used to determine the coefficients, qnk. of the equations of constraint for the known mechanism: Coordinates of the trace path x t =2'78 Yt=2'22 x,=2'36 3'2=2'51 x 3= 1'70 3'3=2'71 x,=0"95 y_~= 2"66 x~ =0-26 y~=2"35 x 6=0-01 .v6= 1"85
Input-crank angle 01t=30 ° 0~2=60 ° 0t3=90 ° 0t.~= 120 ~ 0 t s = 150 ° 016= 180 ~
The root of that known set is u = r l = r z = p t = l . O , v = 0 , and q~=45L u and v are the x and y coordinates of the input-crank axis, r t and r 2 are the lengths of the first two links, Pt is the length of the extension vector and ~P is the angle of Pt with respect to R z shown in Fig. 7 as B t. The six pairs of coordinates on the traced path, the six input-crank angles and the root of the derived mechanism serve as input data for a digital computer program to solve the simultaneous equations. Complete input data also includes six pairs of coordinates for a given or desired path and the six corresponding input-crank angles of the unknown five-bar. The accuracy of the computer program was tested by substituting the values of precision points and crank angles of the known or derived five-bar for the values of precision points and crank angles of the unknown mechanism. Theoretically, the root of the unknown set of equations should be equal to that known root of the derived mechanism. The results were as follows: Root of u t, rI r, Pt Ptx
derived set 1.0000 0.0000 1.0000 l'0000 1.0000 0"707l
Root of unknown set 0.9979 -0-0014 1"0003 1"002l 1.0049 0"7052
Error 0.0021 -0.0014 0.0003 0-0021 0'0049 0-0019
The coordinates of tile precision points and input-crank angles for the left half of the example problem were: Coordinates of the trace path x t =0"700 Yt =0"804 .X'2 = 0 " 5 0 0 y2=0"900 x3 =0"200 y3 =0"984 .x'.~= --0"200 y~=0"996 .x'5=--0"500 y5=0-975 x 6 = --0"700 y6=0"951
Input-crank angle Ott = O(i.itlao 0t2=011+30 ° 0t3 = 0 1 2 + 3 0 ° 0t4=01.3+60 °
0ts=0t.~+36 ~ 016=0t5+24 °
where 0(i~m=t ) was arbitrarily specified. Only at 0 t , . m , o = 0 did the program fail to converge to a root. The solution at 0t(~,m,o=50° is shown in Fig. 8 with results given by u =0.2119, v=0.1236, r t = - 0 " 4 6 6 7 , r2 =0-6470, pt =0.7897, and pt.~=0.5002.
27
~¥
//'/
v
7 Figure 8. Com puter designed mechanism.
The computer program was utilized again to obtain a matching right half of the five-bar mechanism. Control variables specified by the operator determine which half is to be calculated. Analytical checks for jamming are made by the computer after a right half is determined. A common cause of jamming is the two intermediate links becoming collinear. Three of ten attempts to find a right half for the five-bar converged to roots of the equations. Two of these were found to be physically compatible with the left half, while one was not because jamming occurred between precision points No. 2 and No. 3. The solution was selected on the basis of the physical limitations of the mechanism. The results are indicated in Fig. 8. G = - 0"2562
R~ = 0" 1033
H = - 0"030 l
P2 = 1.0328
R3= -0.6807
P2x = 0.0707
The maximum error of 0-0180 occurred at precision point No. 2. One should note that the vector P2 is fixed to the tip of vector R2. The selection of that pivot point is arbitrary. A modification was made in which P2 was moved half-way along the length of Pt for the purpose of demonstrating flexibility. The results of the modified design are depicted in Fig. 9. The maximum error for the modified solution was 0.0090. This error occurred at precision point No. 3. Requirements for designing the non-linear band mechanism include the derivatives, dOt~d03,d2Ot/dO32,and d3Ot/d033,as well as 03 = f ( 0 t ) . The above derivatives are internally calculated by curve fitting 03=f(Ot) to a higher order polynomial. For the example problem, the data (03 vs. 01) were fitted to a fifth order polynomial (see Fig. 10) as follows: 03 = - 1"6290+0"6276 0 t +0"1168 where 0~ and 03 are in radians.
012-0"0511
013-0"0751 0t 4-0"0117 015 (19)
28 IY
Figure 9. Modified design of mechanism.
,,,
-,o
rr
u.]
-90
=_
-,so
32
F--170
-tSO --140
ll[r ~
/
DATA POINTS CALCULATED POINTS USING E(:(JATION (=9)
J
ii
.y/r i
i
iI
"~ [ -~20
-iO0
-80
-60
~40
-20
0
20
40
6O
THETA ONE (e,) ,. DEGREES
F i g u r e 10. Polynomial fit of input-crank relation.
The digital computer was used to calculate an irregular pulley profile and the radius of a circular pulley by solving equations (17) and (19). The irregular profile, linked with the circular pulley, generates the angular relation between the input cranks, specified by equation (19). The results are displayed in Fig. 11. The driving crank of the computer designed band-mechanism rotates through 158 °, or 87.9 per cent of the specified range of 180 ° before the curvature of the irregular pulley becomes concave. A concave shape cannot be tolerated since the band must stay in contact with both pulley surfaces.
29
LLEY
F i g u r e 11. Non-linear band mechanism design.
Figure 12 demonstrates the accuracy achieved in tracing the specified path by the non-linearly coordinated, five-bar mechanism designed by the technique presented in this paper. iY
'
I --
t
i
KEY
!
i
i i
I !I i
,
i
p
i
I
j P
J
i
-
I
r
!
I I
I t
'
I
ill -0.5
1
0.5
! : I I
i
i
SPECIFIED PATH' DEVELOPED PATH -
I
-I0
'
i
0.0
i '
l i
!
i
i
t
i
~5
I
IO
Figure 12. Comparison of specified and developed trace paths.
6. Conclusion
The five-precision-point solution for a linearly coordinated five-bar mechanism has been extended to six. The initial input-crank angles, 0t(i,l,i~l) and 03o,m=l ), were arbitrarily specified parameters in the six-precision-point solution. A seventh precision point could be gained by allowing the input angles to be unknown quantities, for the two additional unknowns in the equations would increase the number of precision points to seven. The increase in unknowns in the equations of constraint would increase the complexity and the resulting solutions, and physically compatible solutions might be more difficult to find. A variety of roots are possible in the six-precision-point solution simply by specifying different values for 01(initial) and 03(i,itial). The angles would not be specified in the sevenpoint problem. If a root is not found or if the root found is not physically compatible, the seven precision points could possibly be altered in order to try again for a workable solution, still maintaining the desired trace path. However, the precision points may be fixed and no deviation allowed in the solution.
30 The six-precision-point solution for the linearly coordinated five-bar mechanism has been established in this paper. The constraint equations specified by a given set of six points on a trace path have been found to contain a multiplicity of roots. A number of the roots constituted physically compatible solutions to the problem. Therefore, one selects the most workable solution based on the dynamic and physical limitations of the system. Non-linear coordination of input cranks by a band mechanism is feasible. Problems appear to be restricted, however, to those of repeating input and output amplitude of less than two revolutions.
References [I] MCPHATEA. J. Non-tmiform Motion Band tt4rechanism. ASME, Publication 64-Mech-17 (1964). [2] RorH B. and FP~UDENSTEt~ F. Synthesis of path generating mechanisms.by numerical methods. Trans. AS,~/[E August 1963, 298-306. [3] TRAUBJ. F. Iterative ~,[ethodsfor the Sohltion of Eqttations. Prentice-Hall, New Jersey (1964). [4] FREUDENSTEINF. and R.o'ra B. Numerical solution of systems of nonlinear equations. Association for Compttting Machines Journal 10, 550 (1963). [5] HoF~t,aum C. R. Path Generation by five-bar Mechanisms. Master's Thesis, Louisiana State University (1964). [6] BREWERJ. A. Non-linear Band Coordination of Path generating five-bar l~Iechanisms. Master's Thesis, Louisiana State University (1968). [7] DIXON M. Kinematic Synthesis of Functional Band Mechanisms. Masier's Thesis, Louisiana State University (1965).
Path Generating Five-Bar Mechanisms J. A. Brewer, II1"
M a r v i n Dixon," and G. D. Whitehouse; Received 2 May 1968 Abstract The synthesis of a linearly coordinated five-bar mechanism using six precision points to represent the trace curve is a complex problem involving a large number of unknowns and causing a multiplicity of roots in the solution. A modified Newton Method is utilized in this paper to solve the resulting equations. A continuation of the six-precision-point problem is accomplished by releasing the linear coordination of the input cranks and then designing for the non-linear coordination. This design consisted of a non-linear band mechanism. The procedure is fully demonstrated by means of an example problem. A discussion of the inaccuracies, uniqueness of solution, and the extension of this method to the seven-precision-point problem is presented. Zusammenfassung--Band koordinierung von ffinfgliedrigen KoppelkurvenGetrieben : J. A. Brewer, Marvin Dixon und G. D. Whitehouse. Die Synthese eines linear koordinierten ffinfgliedrigen Getriebes ffir die Anwendung von sechs Pr~zisionspunkten zur Darstellung einer Koppelkurve ist ein kompliziertes Problem, das eine grosse Anzahl von Unbekannten enth~lt und deren LSsung eine Vielfalt yon Wurzeln ben6tigt. In der vorliegenden Arbeit ist eine modifizierte Methode von Newton zur L6sung der resultierenden Gleichungen benutzt. Eine Fortsetzung des Sechs-Pr~zisionspunkt-Problems wird erreicht durch das Ausschalten der Linearkoordinaten der Antriebskurbeln und nachtr~gliche Konstruktion der nichtlinearen Koordination. Diese Vorrichtung bestand aus einem nichtlinearen Bandmechanismus. Das Verfahren ist durch ein Beispiel ausfLihrlich veranschaulicht. Eine Fehlerbetrachtung eine Diskussion der Einzigartigkeit der L6sung, sowie eine Erweiterung dieser Methode zum Sieben-Punkt Pr~zisions-Problem ist durchgefuhrt. Pe3mMe--J'IeHTO~IHaa KOOp~lHattH~l rlaTH3BCHH~IX MexaHH3MOB~n~ BocnpoI,I3BC~eHI4~ maTym~ix KpnnbiX: I'. A. Bpmep, Mapmm ~Hxcoa ~t F. A. BaTxyc. CaHTe3 naae~lHoKoop~anapoaanaaro n.qTaaBcm~aro Mexaaa3Ma ~rm aocnpoa3Be~eaa~ maryawLx Kpt4ablX no tuecTa TO~IHblMTO*IlcaMsaaaeTca caoxool~ npo6aeMofi, Tpe6y~omeii 6om~moe ~acno H C H 3 B e C T H I ~ I X I~ MHOFOqFIC.r[eHHblX K o p n e f l ypaBHenmi~. B nacToameR pa6oTe nph'MeHaeTCR MO~I4q~HraIIRg MeTO~a Hh~OTOHa ~ p e m e n I ~ [ COOTBCTCTByK)IIIRX y p a B H e R I ~ , l-[po~oYDKeHne n p o 6 n e M ~ meCTH. TO~IHbIX TOtlCK ocyKIeCTBYlgeTC~ HCK.rIIO~IeHHeM .rlRHe~HOfl r o o p ~ I a H a u m ~ Be~rttlRX Kph-BOmm1OB H
nocne~ty~oLue~ rOHCTpyrm4efi Hennne~IHofi roop~4Hat~H. 3TO ycTpofiCTBOCOCTOIITH3 Hemme~maro neHTO~lHaro MexaRH3Ma. ~TOT IIpHeM Bl~Onlle o n p a B ~ I B a e T C R pa3CMOTpeRI~IM ~IpRMepOM. O6cy~c~leHI~e HeTOtlHOCT¢I~, e~HHCTBeHHOCTR p e m e H H ~ H p a c n p o c T p a H e H H e 3TOrO MeTO~a Ha CeM~ TO~I~tX To~eK Ta k"~e I'IpHBO~HTC~I.
*Instructor, Engineering Graphics, Louisiana State University, Baton Rouge, Louisiana 70803, U.S.A. tlnstructor, Mechanical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803. ~.Assistarit Professor, Mechanical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803.
17
18 1. Introduction CERTAIN kinematic requirements call for a pointer or stylus to traverse a definite and sometimes irregular path. A mechanical linkage system which may be offered as a solution to this type of problem is usually referred to as a "path-generator". The five-bar mechanism with its two degrees of freedom offers lucrative possibilities as a path-generator. It, for a specified five-bar, the "pointer" is forced to exactly follow a desired path, generally a non-linear relation between input-crank rotations will result. If the input cranks can be coordinated to rotate precisely in this non-linear manner, a path-generating mechanism with an infinite number of precision points, exact points specified by the path, would be developed. The five-precision-point solution for linearly coordinated five-bar mechanisms [l] has been known for some time. This solution requires the solution of five non-linear, algebraic equations in five unknowns. Many iteration techniques have been developed [2] and applied to the above case with limited success. With all of the techniques, failure to converge to a root of the set is quite common. A modification of Newton's Method, formulated by Freudenstein and Roth [4] is particularly effective in the solution of linkage problems and has been shown to widen the domain of convergence considerably. This method is used to obtain the solution of the six-precision-point problem by the authors. As a sequel to Freudenstein's paper on path generation by geared five-bars, Hoffpauir [5] investigated the five-bar mechanism utilizing non-circular gearing to coordinate the input cranks. The object of the investigation was to increase the number of precision points on the desired path. The object of this investigation is similar, in that nonlinear coordination is used to increase the number of available precision points. In developing an approach to the synthesis of the five-bar mechanism to generate a given path, the selection of a coordinating mechanism to control the input-crank relationships must be considered. In this paper, the mechanism selected to coordinate the inputcrank rotations is the "non-linear band mechanism" (Fig. I).
~'"~--~------r~DE.~REDPATH
~
CIRCULAR PULLEY
INEXTENS]BLEB~ND Figure 1. Non-linear band mechanism. Design equations developed by McPhate [1] are used to derive the non-linear band mechanism. Using this method a band device can be found to generate nearly any nonlinear rotational relationship. The prime restriction is the developed range of rotation. For a relationship that is quasi-linear, the effective range of rotation will be increased. It is therefore desirable to keep the rotational relationship as nearly linear as possible.
19 2. Synthesis of the Six-Precision-Point, Linearly Coordinated Five-Bar Mechanism A discussion of the five-precision-point solution for a linearly coordinated, five-bar mechanism follows for comparison purposes. The equations of constraint are given by equation (l), and a layout of the mechanism is shown in Fig. 2.
c
POINTER
~
CURVE
03
~ (INI'rlAL)
Figure 2. Linearly coordinated five-bar.
Ct+ r t e i(°t('" ~+o,)+ RE = F C 2..~ r3ei(°3( i.tt )+o~)+R4=F 03=KIOt+K 2
(1)
where: Ct, C2, R 2, R4 and F are complex numbers; Ct and C2 locate the axes of rotation of the input cranks; rt and r3, and K2 are scalar constants;
K1
R, and R4 are the remaining links of the five-bar; F defines the path of the "pointer" on the trace curve, and 0 t and 03 are the input-crank angles. The last expression of (1) insures linear coordination of the input cranks. Assuming 0 t, K~, Kz and F are known, the above set of equations contains twelve unknowns. We can reduce this set to two equations with ten unknowns. If 01 and F are incremented four times, the number is increased to ten equations and the resulting set yields five coordinated precision points. In order to increase the generality of the equations, thereby facilitating a greater number of precision points, extension vectors PI and P2 are introduced. The equations of constraint for a system of this type (Fig. 3) are given by equations (2). The tip of the extension vector PL, rather than the junction of R 2 and R4, will define the location of the trace point or "pointer".
20 ~OINTER t.-
/
2/"°,
2-'°
X
Figure 3. Five-bar with extension vectors P~ and P2. The equations now become: Ct + r j °~ + Rz q- Pt = F
C2 "~ r3 el°3 + R4 q"P2 "4"Pt = F
(2)
03=KtOI+K2
where Pt and P2 are extension vectors. Since Pt and P2 are complex numbers they add four unknowns, while eliminating 0t(i,itiat) and 0a(i,mao by specifying these quantities, thus keeping the complexity of the equations at a minimum, and reducing the number of unknowns by two. The result is an overall increase of two unknowns, or two equations in twelve unknowns. Incrementing the known parameters five times yields a set of twelve equations in twelve unknowns. A total of six coordinated precision points is therefore obtained. Considering equations (2) and the "left-half" of the five-bar (see Fig. 3), we can write in complex notation: C t + rl ei°' + r2 ei02 nt"Pl ei~' = F
(3)
where Cl=u+iv,
We define angle above substitutions,
F=x+iy,
R 2 = r 2 el°2,
and P t = p x e i~ .
(~=02"~-kI"/, where W is a constant. Affixing Pt
to R 2 and making the
u + iv + r ~ e i°' 4- r 2 e I02 -I- p 1e~(°~+v) = x + iy
(4)
and rewriting equations (4), we obtain r2ei°2+ p~ei(°2+V)=x + i y - - u -- iv-- r~e i°' .
We reduce the expression to a simple form using complex algebra; thus r22 + p12 + 2r2pl cos ~ = x2 + y2 + u2 + v2 + rt2 -2xu-2yv-2xQ
cos 0~ - 2 y r I sin 0t + 2 u r t cos 0~ + 2 v Q sin 0t •
(5)
21 Substituting Pt cos ~ =Plx, we obtain x z + y2 + u2 + v2 + rt z _ r2 z _ Pt z _ 2r2pt. ~ - 2xu -2yv-2xr
t cos 0t - 2 y r ~ sin 0~ + 2urt cos 0t + 2vr t sin 0t = 0.
(6)
Equation (6) represents one equation with nine unknowns. The path to be followed, and the corresponding input-crank angle, 0~, will be specified, thus reducing the number of unknowns to six: namely, ,t, v, r t, r z, Pt and pt.,. If x, y and 0t are incremented five times, equation (6) becomes the set (7).
F=x+iy,
xZ + y2 +u2 +v2 + r t 2 - - r z 2 - - p t Z - - 2 r 2 P t x -- 2XkU -- 2 ygv -- 2Xkr 1 COSOt k -- 2 ykr t sin Ot k + 2ur I cosOtk + 2vrt sin Olk=O
(7)
where k = 0, 1, 2, 3, 4, 5.
The right half of the five-bar is considered in a similar manner as the previous derivation, but with x and y (or F) effectively transformed to X and Y. In other words, the trace path for the right half is defined by the head of R,. One can observe that X and Y are given by the solution of the left half of the mechanism.
Xk2-~ Yk2-1-g2 +he + r a 2 - r 4 2 - p 2 2 - 2 r 4 P 2 x - 2Xk#-- 2 Y k h - 2Xkr3 COSOk - 2 Ykr3 sin Ok
+ 29r 3 COS 03k -1-2 h r 3 sin 03k = 0
(8)
where k=O, 1, 2, 3, 4, 5. The following quantities are defined: X= x - I ;
Y = i(y - m)
and Pl=l+im.
Given any set of six coordinated points, all dimensions of a linearly coordinated, five-bar mechanism which will move a pointer through each point exactly, can theoretically be found by solving equations (7) and (8) in that order. Each is a set of six non-linear, algebraic equations; the solution of such a set will now be discussed.
3. Solution of the Simultaneous, Non-Linear, Algebraic Equations Newton's method is one of the best techniques to solve non-linear algebraic equations because of its rapid convergence. Newton's method, as with all iteration techniques, requires a " g o o d " initial estimate of a root to insure convergence. Freudenstein and Roth [4] have developed a modification of Newton's method which circumvents this
22 difficulty; their method is especially adaptable to the solution of linkage-type equations. An outline of it is as follows: Equations (7) and (8) may be written in the form, f.(x) =
P.o
+ pnl X I 2 + Pn2X 22 + Pn3X 32 + pn4X-t.2
+ pnsX52 + pn6X4X6 + pnTXt + pnsX2 + pn9X3 + P.loXIX3
+ P.i
1XzX3
(9)
where n = 1, 2, 3, 4, 5, 6. In the above set, the coefficients P,k will be determined by specifying six coordinated precision points which the five-bar is desired to trace and the six corresponding input-crank positions. A graphically determined, derived set 9(,°)(x) is used in the solution off,(x). g(.°)(x) =
q.o + q . l X t 2 + q n z X 2 2
+ qn3X32qngX42
-t- qnsX5 2 + qn6X4X6 + qnvX1 + qnsX2 + qn9X3 + qnloXtS3 + q.t
1X2X3
(10)
where n = 1, 2, 3, 4, 5, 6.
The coefficients q , k of the derived set are first incremented toward the coefficients P,k of the unknown set. This gives a new unknown set 9~1)(x), which has coefficients close to that of the derived set. Using the known root, x, of the derived set as a first approximation to the root of g~,t)(x), Newton's method is used to determine a root of g~l)(x). The coefficients are incremented once more and the process repeated until finally 9(~J)(x) =f,(x) with the final root being one of the unknown set, f,(x). The above method considerably expands the range of convergence. This effectively eliminates the need for a close initial guess of an unknown root. 4. Non-Linear Coordination of Input Cranks by Band Mechanisms A five-bar mechanism with linearly coordinated input cranks was the result of solving equations (7) and (8). We must release the linear coordination in order to alter the trace path of the pointer. The rotational relationship of the input cranks, (01 vs. 03), developed by altering the path, can be determined analytically by the use of equations (4) and (5). Rewriting equation (4) u + iv + r I e i°= + r 2e i°' + p t el(02 + V) = X + i y
(1 l)
and rearranging, we obtain r l e i°= +
(r 2 +
plei~r)e I°2 = x - u + i ( y - v ) .
(12)
Let us define, W=we"=r2+ple Z = z e ip = x -
iv
u + i(y-
and v),
(13)
and after substituting and dividing by e ~p w e i(°2 + "-B) = z - r l e i(°~-a)
(14)
23 Separating 01 , we can reduce the expression to: 01 =/3 + arc cos[-(z 2 + r t z _
wZ)/2zrl],
(i~)
where/3, z, r t and w are all known quantities defined by the position of the pointer of a given five-bar. From equation (14), and in a similar fashion, as the solution for 0 I, we obtain the solution for 02;
02 = / ~ -
r 4- arc cos[(z z + w2 -rlZ)/2"_w].
(16)
The remaining loop of the five-bar is considered in a similar manner to determine 03. 03 = + arccosq + r 3- - r ~ -
([7)
2qr3
A graphical interpretation of equations (15), (16) and 17) is shown in Fig. 4.
i¥
r4
.Z Figure 4. Graphical layout of input-output crank angles•
The input-crank angles can be expressed in tabular form after solving equations (15) and (17). Since derivatives of 03 with respect to 01 are required in the design of the nonlinear band mechanism, crank-angle data were then fitted to a polynomial which enabled the higher derivatives of the function to be obtained. The design equations for a non-linear band system have been developed by McPhate [1]. The developed equations are as follows* (See Fig. 5): qbb= arc sin(rz~b~) + ~b1
~b;= + r2~I-1 - ( r 2 ~ ) 22-÷-t- 1 • ~ , i,~ , ZI = l-(q~,- 1)el*' + tr2(~bb-q~2)e b)lq~b.
*NoT~--Primes denote derivatives with respect to the input angle ~x.
([8)
24
~
2
Figure 5. Non-linear band mechanism nomenclature.
5. Example Problem Consider a five-bar mechanism that is to trace the path shown in Fig. 6. Pt, P2, P3, Pc, Ps, and P6 are six precision points, specified by the given coordinates. The pointer
Figure 6. Desired trace path with six precision points.
25 of the five-bar must travel from Pt to P6 in one-half of a revolution of the driving input crank. Furthermore, the intervals between Pl and P6 must be as follows" Pt to Pz--~-~ revolution of the driving crank; P2 to P 3 -
7Lfz revolution of the driving crank;
P3 to P.~ t
revolution of the driving crank;
P.~ to P5 - - ~o revolution of the driving crank; P5 to P 6 - - ~ revolution of the driving crank. The first phase of the problem is to establish a five-bar mechanism with linearly coordinated inputs which will trace six precision points on the given path. This involves the solution of equations (7) and (8). A "derived" mechanism is utilized to solve the equations using the methods discussed previously. The derived mechanism must be of the same form as the unsolved mechanism, but all parameters, including the trace path, must be known. The derived mechanism used in the solution of the example problem was determined graphically and is shown in Fig. 7. ( 1.70, 2.71 )
(.95, 2.66)
iY
/
/i
/'
(.26, 2.35)
1
/ 3I, .851[
/
I
'~
~.'Ox I
\ I ",,
/
I \\
i/
/
•
,
I
~. / Jx', /
/.,', /
/
," \ X
",,,",
Y ,,,
,
P,x = R cos B~ B, ='45 °
\/X" \
"\'x/
(2.78, 2.11) CONSIDERING THE "LEFT-HALF" ONLY V= 0.0 U= Rl=Rz =P~ :1.0 P=x= 0.7266
/
/
;~. /
,
/
/
I
I
(2.36, 2.51 )
jl
iI
i
\ \
/
/
/
\f /\\
/
i
I I
lJ--u F i g u r e 7. Derived mechanism for linear coordination.
,/ /
/
26 The following values were used to determine the coefficients, qnk. of the equations of constraint for the known mechanism: Coordinates of the trace path x t =2'78 Yt=2'22 x,=2'36 3'2=2'51 x 3= 1'70 3'3=2'71 x,=0"95 y_~= 2"66 x~ =0-26 y~=2"35 x 6=0-01 .v6= 1"85
Input-crank angle 01t=30 ° 0~2=60 ° 0t3=90 ° 0t.~= 120 ~ 0 t s = 150 ° 016= 180 ~
The root of that known set is u = r l = r z = p t = l . O , v = 0 , and q~=45L u and v are the x and y coordinates of the input-crank axis, r t and r 2 are the lengths of the first two links, Pt is the length of the extension vector and ~P is the angle of Pt with respect to R z shown in Fig. 7 as B t. The six pairs of coordinates on the traced path, the six input-crank angles and the root of the derived mechanism serve as input data for a digital computer program to solve the simultaneous equations. Complete input data also includes six pairs of coordinates for a given or desired path and the six corresponding input-crank angles of the unknown five-bar. The accuracy of the computer program was tested by substituting the values of precision points and crank angles of the known or derived five-bar for the values of precision points and crank angles of the unknown mechanism. Theoretically, the root of the unknown set of equations should be equal to that known root of the derived mechanism. The results were as follows: Root of u t, rI r, Pt Ptx
derived set 1.0000 0.0000 1.0000 l'0000 1.0000 0"707l
Root of unknown set 0.9979 -0-0014 1"0003 1"002l 1.0049 0"7052
Error 0.0021 -0.0014 0.0003 0-0021 0'0049 0-0019
The coordinates of tile precision points and input-crank angles for the left half of the example problem were: Coordinates of the trace path x t =0"700 Yt =0"804 .X'2 = 0 " 5 0 0 y2=0"900 x3 =0"200 y3 =0"984 .x'.~= --0"200 y~=0"996 .x'5=--0"500 y5=0-975 x 6 = --0"700 y6=0"951
Input-crank angle Ott = O(i.itlao 0t2=011+30 ° 0t3 = 0 1 2 + 3 0 ° 0t4=01.3+60 °
0ts=0t.~+36 ~ 016=0t5+24 °
where 0(i~m=t ) was arbitrarily specified. Only at 0 t , . m , o = 0 did the program fail to converge to a root. The solution at 0t(~,m,o=50° is shown in Fig. 8 with results given by u =0.2119, v=0.1236, r t = - 0 " 4 6 6 7 , r2 =0-6470, pt =0.7897, and pt.~=0.5002.
27
~¥
//'/
v
7 Figure 8. Com puter designed mechanism.
The computer program was utilized again to obtain a matching right half of the five-bar mechanism. Control variables specified by the operator determine which half is to be calculated. Analytical checks for jamming are made by the computer after a right half is determined. A common cause of jamming is the two intermediate links becoming collinear. Three of ten attempts to find a right half for the five-bar converged to roots of the equations. Two of these were found to be physically compatible with the left half, while one was not because jamming occurred between precision points No. 2 and No. 3. The solution was selected on the basis of the physical limitations of the mechanism. The results are indicated in Fig. 8. G = - 0"2562
R~ = 0" 1033
H = - 0"030 l
P2 = 1.0328
R3= -0.6807
P2x = 0.0707
The maximum error of 0-0180 occurred at precision point No. 2. One should note that the vector P2 is fixed to the tip of vector R2. The selection of that pivot point is arbitrary. A modification was made in which P2 was moved half-way along the length of Pt for the purpose of demonstrating flexibility. The results of the modified design are depicted in Fig. 9. The maximum error for the modified solution was 0.0090. This error occurred at precision point No. 3. Requirements for designing the non-linear band mechanism include the derivatives, dOt~d03,d2Ot/dO32,and d3Ot/d033,as well as 03 = f ( 0 t ) . The above derivatives are internally calculated by curve fitting 03=f(Ot) to a higher order polynomial. For the example problem, the data (03 vs. 01) were fitted to a fifth order polynomial (see Fig. 10) as follows: 03 = - 1"6290+0"6276 0 t +0"1168 where 0~ and 03 are in radians.
012-0"0511
013-0"0751 0t 4-0"0117 015 (19)
28 IY
Figure 9. Modified design of mechanism.
,,,
-,o
rr
u.]
-90
=_
-,so
32
F--170
-tSO --140
ll[r ~
/
DATA POINTS CALCULATED POINTS USING E(:(JATION (=9)
J
ii
.y/r i
i
iI
"~ [ -~20
-iO0
-80
-60
~40
-20
0
20
40
6O
THETA ONE (e,) ,. DEGREES
F i g u r e 10. Polynomial fit of input-crank relation.
The digital computer was used to calculate an irregular pulley profile and the radius of a circular pulley by solving equations (17) and (19). The irregular profile, linked with the circular pulley, generates the angular relation between the input cranks, specified by equation (19). The results are displayed in Fig. 11. The driving crank of the computer designed band-mechanism rotates through 158 °, or 87.9 per cent of the specified range of 180 ° before the curvature of the irregular pulley becomes concave. A concave shape cannot be tolerated since the band must stay in contact with both pulley surfaces.
29
LLEY
F i g u r e 11. Non-linear band mechanism design.
Figure 12 demonstrates the accuracy achieved in tracing the specified path by the non-linearly coordinated, five-bar mechanism designed by the technique presented in this paper. iY
'
I --
t
i
KEY
!
i
i i
I !I i
,
i
p
i
I
j P
J
i
-
I
r
!
I I
I t
'
I
ill -0.5
1
0.5
! : I I
i
i
SPECIFIED PATH' DEVELOPED PATH -
I
-I0
'
i
0.0
i '
l i
!
i
i
t
i
~5
I
IO
Figure 12. Comparison of specified and developed trace paths.
6. Conclusion
The five-precision-point solution for a linearly coordinated five-bar mechanism has been extended to six. The initial input-crank angles, 0t(i,l,i~l) and 03o,m=l ), were arbitrarily specified parameters in the six-precision-point solution. A seventh precision point could be gained by allowing the input angles to be unknown quantities, for the two additional unknowns in the equations would increase the number of precision points to seven. The increase in unknowns in the equations of constraint would increase the complexity and the resulting solutions, and physically compatible solutions might be more difficult to find. A variety of roots are possible in the six-precision-point solution simply by specifying different values for 01(initial) and 03(i,itial). The angles would not be specified in the sevenpoint problem. If a root is not found or if the root found is not physically compatible, the seven precision points could possibly be altered in order to try again for a workable solution, still maintaining the desired trace path. However, the precision points may be fixed and no deviation allowed in the solution.
30 The six-precision-point solution for the linearly coordinated five-bar mechanism has been established in this paper. The constraint equations specified by a given set of six points on a trace path have been found to contain a multiplicity of roots. A number of the roots constituted physically compatible solutions to the problem. Therefore, one selects the most workable solution based on the dynamic and physical limitations of the system. Non-linear coordination of input cranks by a band mechanism is feasible. Problems appear to be restricted, however, to those of repeating input and output amplitude of less than two revolutions.
References [I] MCPHATEA. J. Non-tmiform Motion Band tt4rechanism. ASME, Publication 64-Mech-17 (1964). [2] RorH B. and FP~UDENSTEt~ F. Synthesis of path generating mechanisms.by numerical methods. Trans. AS,~/[E August 1963, 298-306. [3] TRAUBJ. F. Iterative ~,[ethodsfor the Sohltion of Eqttations. Prentice-Hall, New Jersey (1964). [4] FREUDENSTEINF. and R.o'ra B. Numerical solution of systems of nonlinear equations. Association for Compttting Machines Journal 10, 550 (1963). [5] HoF~t,aum C. R. Path Generation by five-bar Mechanisms. Master's Thesis, Louisiana State University (1964). [6] BREWERJ. A. Non-linear Band Coordination of Path generating five-bar l~Iechanisms. Master's Thesis, Louisiana State University (1968). [7] DIXON M. Kinematic Synthesis of Functional Band Mechanisms. Masier's Thesis, Louisiana State University (1965).