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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 CRANK-BEAM MECHANISM WITH COMPOUND HINGES WITH FIXED CHANGES IN THE LENGTH OF THE LINKS
Abstract: The article presents a new scheme and principle of operation of the crank and rocker mechanism with composite kinematic pairs with elastic elements. Taking into account the fixed deformations of the elastic elements, the analytical kinematics of the mechanism are obtained, the numerical solution of the problem determines the laws of motion of the rocker mechanism, the main parameters are substantiated.
Keywords: mechanism, crank, connecting rod, rocker, kinematic pair, elastic element, excess coupling, fixed deformation, length, displacement speed.
In the classical theory of machines and mechanisms, sleeve in the joint hinge (kinematic pair) is chosen equal
graphical, graph-analytical and analytical methods for the to the ratio of the length of the next link to the length of
kinematic analysis of plane lever mechanisms [1; 2]. At the previous link multiplied by 1.0 mm. The greater the
the same time, the links are considered rigid and the defor- difference in length between adjacent links, the greater
mations of the elastic elements in the mechanisms are not the reaction force and shock interactions between them,
taken into account. To obtain more complex laws ofmotion so that the value of the thickness of the elastic sleeve will
ofthe output links ofthe mechanisms necessary for techno- also be large. This allows the necessary amortization of
logical machines, we developed a crank-rocker plane mecha- loads and correction in the law of motion of links.
nism with compound hinges and elastic elements [3]. The construction is explained by a drawing, where in
The essence of the recommended crank and rock- (Fig. 1), a is a 1-general diagram of the crank-rocker
er mechanism is that the kinematic pairs (hinges) are mechanism. The crank and rocker mechanism consists
made integral, including an axle, fitted on it with an elastic of a column 1, a crank 2, a connecting rod 3 and a rocker
(rubber) bushing on which the bushing is rigidly con- 4. The joints between the links 2 and 3, 3 and 4, 4 and
nected to the link (crank, connecting rod or rocker). Elas- 1 are made integral, which includes an axis 5, an elastic
tic bushings allow to amortize and smooth shock interac- bush 6, the sleeve 7 rigidly connected to the links 1, 2, 3,
tions in the hinges in the extreme positions of the links, 4 of the mechanism. In this case, the thicknesses of the
and also due to the deformation of the elastic bushings, elastic bushings 6 are chosen:
the lengths of the links change, which allows them to cor- ^ l l
rect the laws of their motion within the necessary limits. Aj = y ■ 1,0 mm; A2 = y ■ 1,0 mm; A3 = y 1,0 mm
For this purpose, the thickness of the elastic (rubber) 1 2 3
bushings of composite balls are chosen according to the where, l1, l2, l , l , - respectively, the lengths of the links
ratio of the lengths ofthe links. The thickness of the elastic 1, 2, 3, 4.
))
b)
Figure 1. a - crank and rocker mechanism with compound hinges and elastic elements; b - calculation scheme of the crank-rocker mechanism with elastic elements
The crank-rocker mechanism works as follows. Crank 2 receives rotational motion from the drive motor (not shown in the figure). Accordingly, the movement from the crank 2 is transferred to the connecting rod 3, then to the rocker arm 4. Movement of the mechanism occurs in the plane. At the extreme positions of the crank 2, connecting rod 3 and rocker 4, impact phenomena occur in the kinematic pairs between the links 2, 3, 4 and
1. At impacts, the elastic sleeve 6 absorbs the impact, the reaction force decreases. In addition, due to the deformation of the elastic sleeve 6, the lengths l1, l2, l, l4. This leads to a change in the trajectory of the points 2, 3, 4. Therefore, choosing the thickness (rigidity) of the elastic sleeve 6, it is possible to regulate its deformations, thus the necessary laws (trajectory) of the movement oflinks
2, 3, 4 of the mechanism.
The thickness of the elastic sleeve 6 in the composite joints (kinematic pairs) is chosen equal to the ratio of the length of the next link to the length of the previou s link and multiplied by 1,0 mm. The greater the difference in length between adjacent links, the greater the reaction force and impact interaction in the kinematic pair. Therefore, the thickness of the deformation of the elastic sle eve 6 will also be large.
In this case, in fact, the elastic sleeves 6 reduce or eliminate the amount of excess bonds in the kinematic pairs.
For the recommended mechanism
q = i - 6n - 5P5 - K = 1 - 6 ■ 3 + 5 ■ 4 - 3 = 0
Hence, in the recommended crank-beam mechan ism there are no excessive connections. This leads to an increase in the resource of work.
The mechanism allows necessary changes (corrections) of the movement of links (points of links) within certain limits necessary for the intensification of technological processes in the machine.
For the considered crank and rocker mechanism, taking into account the deformations of the elastic elements between the crank and the connecting rod, and also between the connecting rod and the rocker, we will compute the design scheme, which is shown in (Fig. 1b). In this case, the deformations of elastic bushes in kinematic pairs are taken into account in the form of fixed values, maximum and minimum. Then the length of the crank, connecting rod and rocker will be increased or decreased due to these values of fixed deformations of the elastic sleeves of the kinematic pairs. To study the kinematics
of the proposed mechanism, taking into account the fixed deformations of the elastic elements in the hinges B and C, we use the classical technique of vector contours [4; 5]. From the calculation scheme in (Fig. 2). It can be seen that the closed contour of the ABCD can be divided into two separate closed triangular contours AABD and ADBC. Taking into account the equilibrium system, the sum of the vectors of closed contours will be zero. In this case we have [3]:
{ + T + I = 0 ; J + I + I = 0 (1)
where, l1, l2, l , l , l- variables modulo vectors.
When designing the vector equations (1), and also taking into account the elongations of the links due to the fixed minimum and maximum deformations of the elastic elements of the kinematic pairs of the crank and beam mechanism on the x and y coordinate axes, we have:
i +Aii)cos(01+A0l) + + (i + Ai)cos(0e -A&) -ZjCOS^ = 0; iicos01 + icos0e - 14cos04 = 0;
(1i -Aii)cos(0i -A0i) + + (i - Ai)cos(0e + ) - iicos^i = 0; (2)
(ii +Aii)sin(0i + A0) -
- (i - Ai )sin(0 - A0) - ii sin 04 = 0 ;
iisin0i + i sin0e-i4sin04 = 0; (ii — Ai1)sin(0i-A0i) -
- (i — Ai)sin(^+A^) — iisin^i = 0.
According to the design scheme (see Figure 1), the fixed values of the elongations of the crank, connecting rod and rocker arm due to deformations of the elastic elements of the kinematic pairs are taken into account by the following expressions
i, = i, + Ai.; i, . = i, -Ai,;
imax i i imin i i >
I = I + AL; I . = L -AL;
2max 2 2 2min 2 2
i3max = i3 +A4 ; i3min = i3 - ^k i
i = i + Ai; i , = i-Ai
max min
where, Al1, Al2, Al3, Al - fixed values of changes in quantities l, ll3, l.
In this case, the maximum and minimum deviations of the angles f2 and f3 are determined from the following expressions
^2max = arcc0s
^min = arCC0S
^min = arCC0S
(li + All)2 + (I2 +AI2)2 -(I3 + AI3)2 +142 + 2(lj + A^cosft
2(l2 + Al2^ + Alj)2 +142 -2(li +Alj)l4Cos^
(l1 + Al,)sinA
+arctg——-^-
(li + Alj)cos-4
(li - All)2 + l - AZ2)2 +14 -(/3 - A4)2 -2(li -Ali)l4 cosfl
2(l2 - Al2)V(li - Ali)2 +14 - 2(li - Ali)l4 cos^i
(l, - Al.)sin0.
+arctg——-^-
(li -Ali)cos0i - /j
(l2 -Al2)2 - (li - Ali)2 - (l3 - Ag2 -1: + 2(li - Aljl cosfl 2(l3-AgVl-Ali)2 +14 -2(li-Al^cos^ (li -Ali)sin0i
(3)
+arctg
(li -Ali)cos0i - /4
In this case, the linear velocity of the hinge axis between the crank and the connecting rod changes within
^max = (li +Al>i ; ^n = (l, - Al,)© (4)
The solution of the kinematics problem of the crank and beam mechanism with the elastic elements of the composite hinges was carried out with the following values of the parameters:©, = 350 s-1; l = 3240 3 m; l2 = 65403 m; l3 = 36403 m; l4 = 62403 m; Al1 = (1.8... 2.1>10-3 m; Al2= (0.06. 1.0)10-3 m; Al3 = (1.7. 2.0>10-3 m.
On the basis of the numerical solution of the kinematics problem of the mechanism under consideration, graphical dependences of the angular velocity of the connecting rod and rocker are plotted without considering the deformations of the elastic elements of the mechanism hinges, which are shown in (Fig. 2). Analysis of the graphs in (Fig. 2), a show that an increase in l3 from 2.7-10-3 m to 4.2-10-3 causes the amplitude of the angular velocity of the rocker to increase from 2.90-10 2 s-1 to 5.05-10 2 s-1. In this case, the amplitude of the oscillations of the angular displacement of the beam increases from 6.85 degrees to 11.8 degrees (see Fig. 2 b). It should be noted that, taking into account the fixed values of Al1, Al2 and Al3, the angular displacement and angular velocity of the connecting rod and rocker the mechanism under consideration. In (Fig. 3) shows the obtained regularities of the change with variation, Al1, Al2 and Al3. By processing the obtained regularities of the change, graphical dependences are constructed, which
are presented in (Fig. 4). It is seen from them that with an increase in the length l3 of the beam of a flat four-link from 2.7-10-2 m to 4.2-10-2 m, the angular velocity of the rocker decreases from 3.35-10-2 s-1 to 0.75-10 2 s 1 for l2 = 6.0-10~2 m (see Figure 4a). With the length of the connecting rod l2 = 7.0 • 10-2 m, the angular frequency of the oscillation of the rocker decreases from 5.0-10 -2 s_1 to 2.95-10 -2 s-1. Analysis of the graphs in Fig. 4 shows that to increase the angular velocity of the rocker arm, it is advisable to increase the length of both the connecting rod and the rocker itself.
Increasing the difference in the lengths of the crank and rocker leads to an increase in the swing angle of the angular oscillations of the connecting rod with its plane-parallel motion. Therefore, to ensure the smallest angular oscillations, it is advisable to decrease the value of l3-l1, and at a value of l3-l1 = 0, the connecting rod moves parallel to the horizontal, <>2 = 0.
Important are the studies with a variation of the fixed values of the deformations of the elastic elements of the kinematic pairs of the mechanism. The changes in Al1, Al2 and Al3 are mutually related and proportional to the respective lengths of the links of the mechanism. Fixed strains according to the above allow you to select the necessary rubber bushings for the corresponding hinges of the mechanism. Analysis of the studies shows that, in the presence of deformations, Al1, Al2 and Al3 lead to some high-frequency oscillations. At the same time, an increase in the values of Al1, Al2, Al3 leads to an increase in the amplitude of high-fre-
quency angular oscillations of the rocker mechanism. The increase in Alt and Al3 leads to an increase in low-frequency oscillations of the beam. This is explained by the fact that an increase in Al3 actually results in an increase in the length of the beam by Al3. At the same
time, the range of oscillations decreases. The increase in Al2 influences insignificantly on the range of oscillations. It should be noted that the greater the difference between Alt and Al3, the greater the amplitude of high-frequency oscillations of the rocker arm.
a) where, 1 - at l3 = 4.2-10"2 m; 2 - at l3 = 3.9-10"2 m; 3 - at l3 = 3.6-10"2 m; 4 - at l3 = 3.3-10"2 m; 5 - at l3 = 3.0-10"2 m; 6 - at l3 = 2.7-10"2 m
b) where, 1 - at 13 = 42-10"3 m; 2 - at 13 = 39-10"3 m; 3 - at ¡3 = 36-10"3 m; 4 - at 13 = 33-10"3 m; 5 - at 13 = 30^10"3 m
Figure 2. a - graphical dependences of the angular velocity variation of the rocker mechanism of the crank angular motion function for various l3 and at Ai1 = 0; Al2 = 0; Al3 = 0; b - graphical dependences of the change in the angular displacement of the rocker arm from the change in the angular displacement of the crank for different I3 and at ^ = 0; AI2 = 0; AI3 = 0
To ensure the necessary amplitudes of high-frequen- Similar regularities were obtained by studying the varia-
cy oscillations of the connecting rod due to fixed defor- tion of Alx, Al2, Al3. In order to increase the amplitude
mations of the elastic elements of the hinges, it is advis- of the high-frequency oscillations of the beam, it is also
able to change the values of Al^ and Al3 rather than A^. advisable to increase the difference A^-A^.
a) where, 1 - at Al3 = 2.5-10"3 m; 2 - atAl3 = 2.0J0-3 m; 3 - at Al3 = 1.540"3 m; 4 - at Al3 = 1,040-3 m;
5 - at AI3 = 0.75-10"3 m
b) where, 1 - at AI3 =4.0J0"3 m; 2 - atA I3 = 3.5-10"3 m; 3 - at AI3 = 3.0J0"3 m; 4 - at Al3=2,0J0"3 m;
5- at AI3 = 1.0J0"3 m
Figure 3. Graphical dependences of the change in the angular movements of the rocker arm
mechanism on the variation of the angular movement of the crank with the variation of Al2 (a) and AI3 (b)
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4. Levitsky N.I. Methods for calculating the kinematic schemes of crank-lever mechanisms. - M.: Mashgiz. - 1963. -204 p.
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