a new method of calculating the equation of the connecting rod curve

STRUCTURAL AND KINEMATICAL PROBLEMS OF A LIGHT INDUSTRY MACHINE
MECHANISM
Daniela Cristiana MÎLCOMETE *, Traian PREOTEASA **
*Bondia Tricot SRL Craiova, milcomete@yahoo.com, **CFR Bucureşti
The structural and kinematical analysis of a spatial knitting machine mechanism is made. The
mechanism has a seeming family and uses two planar cams, couplings and spatial components. The
kinematical calculation relations are written and results are provided, and then checked
experimentally.
Keywords: Kinematical analyse, spatial trajectory, cam – lugs, motions parasit.
MECHANISM DESCRIPTION
It is well known that very complicated machines, based on mechanical systems, are used in the light
industry. Such machines are, more often than not, built empirically, the designers having worked on older
variants, which they have developed. The analysis of these machines proved to be difficult and the designers
needed solid knowledge of the technological processes involved.
The mechanism is presented in figures 1 and 2.
Fig. 1

Daniela Cristiana MÎLCOMETE , Traian PREOTEASA 244
Fig. 2
As shown, this spatial mechanism is complex indeed. It is designed to displace a “catch member”
observing a number of laws so that its point draws the trajectory required in the technological process.
MECHANISM STRUCTURE
The kinematical mechanism scheme is shown in figure 3.
Fig. 3
The structural solution was found by taking into consideration “seeming families” [Antonescu, 1973].
In respect to this idea, the scheme in figure 4 was drawn.

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Structural and kinematical problems of a light industry machine mechanism
The kinematical couplings – structurally parasitical motions being eliminated - are the following
(figure 4):
A (0,1) – class V;
G (4,5) – class IV;
B (0, 2) – class V;
H (5,7) – class III;
C (1, 2) – class IV; K (6,0) – class V;
D (2,3) – class IV;
L (3,6) – class III;
E (0,4) – class V; M (6,7) – class V.
F (1,4) – class IV;
Results: n = 7, C5 = 5, C4 = 4, C3 = 2.
Using the relation N = c - n, where N is the number of free outlines, n is the number of mobile elements and
c – the number of couplings, we have: N = 11 – 7 = 4, i.e. there are 4 free outlines. Figure 4 shows these
outlines (Roman numerals are used). Looking simultaneously at figures 3 and 4, we conclude that:
- the I (AFE) outline has f = 3 and is planar;
- the II (ACB) outline has f = 3 and is planar;
- the III (BDLK) outline has f = 0 and the DL connecting rod has all motions;
- the IV (DLMHGFC) outline has f = 0 and and the HM component has all motions.
The seeming family is the average of real families: f a = (3+3+0+0)/4 = 6/4
The mobility degree is (in our case):
M = (6 – fa) n – (5 – fa) C5 – (4 – fa) C4 – (3 – fa) C3 = ( 6 – 6 / 4) .7 – (5 – 6 / 4).5 – (4 – 6 / 4 ).4 – ( 3 – 6 /
4).2 = 1.
Fig. 4

Daniela Cristiana MÎLCOMETE , Traian PREOTEASA 246
BAR KINEMATICAL MECHANISMS
In the kinematical analysis, the laws governing the two main elements AB and EF are assumed to be
known; it is about motions received from the cam shaft.
Experimentally, the lengths of the components and the coordinating points attached to the base are
determined in relation with the system of axes originating in E. Therefore, XE, YE, ZE, XA, YA, ZA, XD,
YD, ZD and the lengths of AB, BC, CD, DH, GH, EF, FG and HM are found.
Applying the outline and distance method, we have the final relation for the ABCDH outline:
h sin ϕ+ h cosϕ+ h sin ψ+ h cosψ+ h −2acsin ϕsin ψ= 0 , where:
2 3 4 5 1
2 2 2 2 2
h1 = ( xA − xD) + ( yA − yD) + c + a + ( zA − zD)
−b
2
h2 = 2 a( zA
− zD)
; h3 = 2 a( yA
− yD)
h4 =−2( c zA
− zD)
; h5 = −2( c xA
− xD)
Similarly, for the EFGH outline, we write:
e + e sin ψ + e cosψ + e cosϕ + e sin ϕ −2ac cosϕ cosψ −2a h sin ϕ sin ψ = 0 , where:
1 2 1 3 1 4 1 5 1 1 1 1 1 1 9 1 1
e = a + c + h + y + h −b
2
2 2 2 3 2
1 1 1 6 H 7 1
2 = 2 8 6 + 2 7 9 ; e3 = 2c1yH
e4 = − 2a1yH
; e5 = − 2a1h
7
e h h h h
THE CAM-LUG CHAIN KINEMATICS
The profiles of the two cams and the motion law of the cam shaft are known in the mechanism
analysis. The cam outline equations are determined (in fact the equations of the profiles equidistant to them
at ray rollers) as polar coordinates – r and θ- and then the cam follower inclinations are calculated (fig. 5).
Figure 6 shows the oscillatory cam follower, whose point is articulated in B 0 (initially) and then on the
cam in C.
By applying the reverse motion principle, i.e. by ensuring an exterior motion having –ω 1 , the cam stops
and the cam follower is spinning around the cam at –ω 1 in B, B’, etc.
Fig. 5

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Structural and kinematical problems of a light industry machine mechanism
Fig. 6
The cam profile equation in polar coordinates is r = f (φ). The current angle of rotating B is the angle γ.
The following relations are written:
h= ( x − x ) + ( y − y )
2 2
BO
A BO
A
x = rcosθ= x + acosϕ
C
yC
= rsin
θ= yB
+ asin
ϕ
x = hcos
γ+ acosϕ
C
B
yC
= hsin
γ+ asin
ϕ
The variable θ value is assigned and r (in the equation r = f (θ)), XC and YC are calculated. γ is
eliminated from the last equations and we have:
2 2 2
( yC
−asin ϕ ) + ( xC
−acos ϕ)
− h = 0
After successive development, we have:
2 2 2 2
( xC + yC + a −h ) −2ayC sinϕ−2axC
cosϕ=
0
φ, i.e. the cam follower inclination, is calculated in this equation. The cam outline equations are
determined (in fact the equations of the profiles equidistant to them at ray rollers) as polar coordinates – r
and θ - and then the cam follower inclinations are calculated.

Daniela Cristiana MÎLCOMETE , Traian PREOTEASA 248
The mechanism positions are determined by using the polynomials φ (t) and φ 1 (t) and by calculating
the time in relation to the cam shaft revolution.
The spatial trajectory of the characteristic point (M) is shown in figure 7, against a remote system of axes in
order to have a spatial image.
Fig. 7
CONCLUSIONS
- The mechanism is very complex both structurally and kinematically and different calculations have
been necessary to complete the calculation programme.
- The algorithm and the mechanism kinematics were proved to be right in comparison with
experimental results;
- The mechanism is not the best possible, having parasitic motions and cams which work only part of
the loop; it is obvious that it was designed empirically, on the basis of previous design and exploitation of
other variants;
- The above mentioned algorithm can also be used for real mechanism synthesis.
REFERENCES
1. ANTONESCU, P., Extinderea formulei structurale Dobrovolski la mecanismele complexe de familie aparentǎ, SYROM'73, vol.
B, pg. 1-10.
2. ANTONESCU, P., Sinteza mecanismelor, I.P.B., 1983.
3. PARIKIAN, T.,F. – Multi-generation of cupler curves of spatial linkages, Mechanism and Machine Theory, 1/1997, pg. 103-110.
4. POPESCU, I., Mecanisme. Noi algoritmi şi programe, Repr. Univ. Craiova, 1997.
5. VOINEA, R., VOICULESCU, D., SIMION, F., P., Introducere în mecanica solidului cu aplicaţii în inginerie, Ed. Academiei, 1989.