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Djuraev Anvar Djuraevich, technical sciences associate, professor, Tashkent institute of textile and light industry, Republic of Uzbekistan E-mail: [email protected] Kenjaboev Shukurjon Sharipovich, dotsent, candidate for technical sciences, Namangan engineering and construction institute,
Republic of Uzbekistan E-mail: [email protected]
KINEMATIC ANALYSIS OF THE FOUR-LINK LEVER MECHANISM IN ACCORDANCE WITH THE LIMITS OF ELASTIC ELEMENTS IN SHARNIR
Abstract: The paper presents the scheme, the principle of operation, as well as the kinematic analysis of the four-link hinged planar mechanism, taking into account the limiting values of the deformations of the elastic element of the joint hinge between the connecting rod and the rocker. By the kinematics problem, regularities in the movement of the rocker mechanism are obtained, graphical dependences of the parameters are constructed, along which the necessary values can be selected.
Keyword: Lever mechanism, four-link, hinge, compound, elastic element, crank, connecting rod, rocker, kinematics, motion law, graphs, parameters angle, speed, swing, length change.
In modern engineering, flat lever mechanisms are widely used, in particular hinged four-link mechanisms [1,2]. These mechanisms allow obtaining the necessary complex laws of rocker motion [3]. But, if necessary, these mechanisms do not allow the required changes in the laws of motion during the movement of the mechanism. Therefore, it is important to develop new schemes of four-link hinged mechanisms, which allow obtaining the laws of motion of the output link with the necessary correction. We have developed a number of variants of schemes of lever mechanisms including compound hinges with elastic elements that allow the required necessary laws of motion of the output links [4, 5].
The developed mechanism is a four-link lever-hinged planar mechanism, the scheme of which is shown in Fig. 1a. In the proposed mechanism, an elastic element is used: in the kinematic pair between the connecting rod and the rocker in the form of a rubber bushing. In kinematic analysis, the modes of motion under the action of forces are not considered. During the operation of the mechanism, the crank 1 rotates the movement of the connecting rod 2, and further the rocker 3. In this case, the joint is made integral, including a rubber elastic sleeve. In this case, due to the deformation of the elastic element, the lengths of the connecting rod and the rocker arm 3 change. This leads to a change in the law of motion of the rocker arm 3. By choosing the characteristics of the elastic element in the hinge between the connecting rod 2 and the rocker 3, it is possible to change its motion to a certain extent to the desired extent.
We shall study the kinematics of the mechanism according to the existing analytical method of vector contours [6, 7, 8]. According to the scheme in Fig. 1a and the closed contour of the ABCD is divided into two separate contours of the internal combustion engine DBC and DAB.
In this case, the sum of a closed contour of vectors from the system equilibrium condition equals zero. For the contours under consideration, we have:
_ _ T+T-X = 0, I-I-T = o, (1)
where, l - l2 - l3 - are variables modulo vectors.
From the origin we draw the coordinate axes X and Y. We project the vectors on the X and Y axes:
l, cos^ +l cosq>e -l4 = 0 Zjsin^jj +l sin (pe = 0 . (2)
From the vector equation (2) we can determine the value l, and from the first equation the value of the angle q>e:
; • A —l, Sin V
l4 — l!C0S^! y
Considering the closed contour A DBC of the material transport mechanism and using the cosine theorem [9], we obtain:
l2 = l2 +^ -2ll3cos^3e; ll = l2 +1,2 - 2ll2cos^2e. (3)
From the obtained equations (3) we have:
, , sin«,
l = —l,-—; cpe = arctg
sin 9e
93 =9e+93e = arccos"
L2 -12 - H
2ll
- + 9e
92 =9e + arccos-
l2 -12 - i
2ll
(4)
Section 15. Technical science
Figure 1. Scheme of a four-link planar mechanism with allowance for the limiting values of the deformation of the elastic element in the hinge C
Using the cosine theorem in A DAB:
l2 = + ¡1 - 2l4l1 cos ft, then finally we have:
<p3 = arccos
K - K -12 - li + 2l4l1 cos<
2l3-yj + l, — 2l4l, cos<, +arctg-
l1 sin ft l1 cosft -14
ft =arccos
l2 + ¡i +11 -1\ - 2l4l1 cosft 2l2 +l,2 - 2l4l1 cosft
+arctg
l, sin ft
(5)
lj cos ft -14
According to the scheme in (Fig. 1 b), the changes in the length of the connecting rod and rocker arm due to the deformation of the elastic element in the kinematic pair are taken into account by the expressions:
L = L +AL; I = L + AL;l . = l-AL;l, . = L-AL ,(6)
2max 2 2' 3max 3 3' 2min 2 2' 3min 3 3
where, Al2 and Al3 - are the changes in the quantities l2 and l3.
In this case, the angular values f2 and f3 have the following limiting values:
m _ arr Cos(l2 +Al2)2 - (l3 +M3)2 - l2 - l2 + 2^ CoSft +
T 3min "rr COS 1--+
2(l3 + A^)^/ +arctg
2 + l,2 - 2l4l, Cosft l, sin ft,
l, cosft -14
^ l42 +1,2 + (L + AL)2 -(l3 + Al3)2 -2l4l, Cosft
^2max = arC C°sJ-,-2- ■ 3 3 4 1 -L +
2(l2 + Al2)Jl.
+arctg
2 +12 -2l4l, Cosft, l, sin ft
l, cosft -14
ft3m
= arcCos(l2 -Al2)2 -(l3 -M3)2 -14 -l, + Cosft +
2(Al3 -Al3)^/l42 + l,2 -2l4l, Cosft, +arctg-
l, sin ft, l, cosft, -14
p2min = arc Cos
l42 + lj + (l2 - Al2)2 -(l3 - Al3)2 -2l4l, Cosft
2(l2 - Al2)^/l42 +1,2 - 2l4l, Cosft +arctg-
l, sin ft
(7)
l1 cosft -14
In addition, according to Fig. 1.b can be written:
= ^2 max = ?2 -^2mta; A%3 =^3ma, - % = -%3min ,
or ft =
ft
max + ft2 m
ft3 =
ft
max + ft3 m
2 2 Numerical solution ofthe kinematics problem of the mechanism of material movement taking into account the change in the length ofthe connecting rod and rocker was performed with the following initial values of the parameters l1 = 30-10-3 m; l2 = 225-10-3 m; l3 = 28-10-3 m; l4 = 21-10-3 m; a = 10,l = 28-10-3 m ; Al2 = (2.0...4.0)-10-3 m; Al3 = (1,0.. .2,0)-10-3 m; ^ = 303,5 s-1.
But, a significant increase Af3 leads to undesirable phenomena. Therefore, it is advisable to use additional angular oscillations within the established limits. Taking into account the technology taking into account the calculated values of the parameters of the mechanism, the recommended values of the parameters are: l2 = (220...225)-10-3 m; l3 = (23...30>10-3 m; Al2 = (3.0...4.0)-10-3 m; Al3 = (0.75...1.2)-10-3 m.
The scheme of a four-link flat hinged mechanism with a joint hinge between the connecting rod and the rocker was developed. Analytic methods are used to determine the angular displacement of the connecting rod and rocker arm. Numerical solution of the problem obtained the laws of the movement of links and graphical dependencies parameters on which you can choose the necessary value.
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