Roller Forming Unit Dynamic Analysis with Energy Balanced Drive Dissipative Properties Taken into Account

ISSN 0131–2928. Проблеми машинобудування, 2018, Т. 21, No 2 32 1 V. S. Loveikin, D. Sc. (Engineering) 2 K. I. Pochka, Cand. Sc. (Engineering) 1 Yu. O. Romasevych, D. Sc. (Engineering) 1 National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine е-mail: lovvs@ukr.net romasevichyuriy@ukr.net 2 Kyiv National University of Construction and Architecture, Kyiv, Ukraine е-mail: shanovniy@ukr.net UDC 693.546


Problem formulation
During the operation of roller forming units designed for forming reinforced concrete products, considerable dynamic loads appear both in drive and forming trolley elements [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Despite a rather extensive research into the technological process of forming reinforced concrete products by the non-vibration roller forming method [1][2][3][4], the dynamics of a forming trolley movement and its influence on the formation process have not been investigated yet. Little attention has been paid to the modes of a forming trolley movement and the forces arising in drive elements.

Analysis of recent research and publications
In the existing theoretical and experimental studies of roller forming units designed for forming reinforced concrete products, the design parameters and productivity of the units are substantiated [1][2][3][4]. At the same time, insufficient attention is paid to the study of the existing dynamic loads and movement modes, which greatly affect both the operation of the units and the quality of the finished products. During continuous start-stop modes, considerable dynamic loads appear both in the drive and forming trolley elements, which may lead to the premature failure of the units [1][2][3][4][5][6]. Therefore, the task of studying dynamic loads in the elements of the units is actual. In [15][16][17][18][19], loads in the roll forming unit elements were determined, however, the rigidity and dissipation coefficients of the drive were not taken into account.

Purpose of the paper
The purpose of this paper is to determine the loads in the elements of a roll forming unit with an energy-balanced drive, taking into account the drive rigidity and dissipation coefficient.

Statement of the research problem
In order to reduce energy consumption in roller forming machines, a design of a roller forming unit [20,21] was proposed to provide the compaction of reinforced concrete products on a single technological line. It consists of three forming trolleys, located parallel to each other on one side of the drive shaft, which are set in reciprocating movement from one drive. It is composed of three slider-crank mechanisms, whose cranks are tightly fixed on one drive shaft and shifted to each other at the angle 0 120    . Fig. 1 shows a roller forming unit with an energybalanced drive. Each of the forming trolleys 11, 8 and 15 ( Fig. 1, a) is mounted on the gantry 14 and performs reciprocating movement in the guide rails 9 over the cavity of the form 10. The forming trolley 11 consists of the feeding hopper 12 and coaxial sections of the compaction rollers 13. The other two trolleys have the same design. The trolleys 11, 8, and 15 with distribution hoppers are set into reciprocating movement by a drive made in the form of three slidercrank mechanisms, whose cranks 5, 3, and 2 are rigidly fixed on one drive shaft 4 and shifted to each other at the angle 33 connected to the cranks 5, 3, and 2. Such a design of a roller forming unit makes it possible to reduce the dynamic loads in the drive elements, extra devastating loads on the frame structure and, accordingly, increase the unit durability as a whole. Fig. 1, b shows a kinematic scheme of a roller forming unit with an energy-balanced drive for compacting reinforced concrete products on a single technological line. In this kinematic scheme, r is the radius of the cranks 5, 3, and 2; l is the length of the connecting rods 7, 6, and 1;  is the angular coordinate of the first trolley crank position;   is the displacement angle of the cranks 5 and 3, 3 and 2, and 2 and 5 with respect to each other; During the operation of a roller forming unit with an energy-balanced drive, there arise significant dynamic loads in the elements of the transfer mechanism from the electric motor to the cranks, leading to the premature destruction of the drive structural elements. To study these loads, we use a two-mass dynamic model of a roller forming unit (Fig. 2). In this model, the following symbols are used: is the inertia moment of the forming trolleys and crank-and-rod mechanisms, reduced to the crank rotation axis; с is the drive mechanism rigidity, reduced to the crank rotation axis; 1  and 2  are the generalized coordinates of the reduced masses 1 dr J and 2 dr J , respectively.

Fig. 2. Dynamic model of a roller forming unit
The drive mechanism reduced moment of inertia can be determined by the following dependence: can be determined from the second part of the mechanism (Fig. 3), which includes crank-and-rod mechanisms with forming trolleys. In this case, we divide the mass of the connecting rods 7, 6, and 1 conrod m in equal parts in the points 1 A and 1 B , 2 A and 2 B , and 3 A and 3 B . Then the moment of inertia of the cranks can be determined by the dependence  (Fig. 1, b); crank J  is the moment of inertia of each of the cranks relative to their own rotation axes; r is the radius of each of the cranks; crank J is the moment of inertia of each of the cranks with half the mass of the connecting rod relative to their own rotation axes; can be determined from the condition of equality of the kinetic energies of the crank-and-rod mechanisms with the trolleys r T (Fig. 3) and the second disk of the dynamic model ( Fig. 2)

Fig. 3. Calculation scheme of the loads on the elements of a roller forming unit with an energy-balanced drive:
aload on the forming trolley; bload on the drive mechanism We find the kinetic energy of crank-and-rod mechanisms with trolleys  x  are the velocities of the mass centers of the first, second and third forming trolleys, respectively.
Since the three trolleys move progressively, all their points have the same velocity. Therefore, we can accept that . We express the velocities of the points 1 B , 2 B та 3 B through the coordinates of the cranks and their derivatives in time. In order to do this, we use the dependencies Then, taking into account (5), the dependence (4) will have the form The kinetic energy of the second disk in figure 2 is expressed by the dependence Equating the dependences (6) and (7), we have To determine the reduced moment of the resistance forces 2 dr M we use figure 3, on which the following symbols are used: 1 F , 2 F , and 3 F are the forces in the connecting rods necessary for overcoming the resistance forces acting on the trolleys; 1  , 2  , and 3  are the angular coordinates determining the positions of the connecting rods of the first, second and third trolleys relative to the horizontal; 011 F and 012 F are the horizontal forces of interaction of the compaction rollers with a concrete mixture for the first forming trolley; 011 R and 012 R are the vertical forces of interaction of the compaction rollers with a concrete mixture; 11 N and 12 N are the normal reactions of the forming trolley guide rails to the guide rollers; are the friction forces of the guide rollers by the forming trolley guide rails; red f is the reduced friction coefficient of the guide rollers by the forming trolley guide rails; G is the forming trolley gravity force; a , b , p , e are the geometric dimensions of the forming trolley; D is the pressure roller diameter; d is the guide roller diameter; l is the connecting rod length. For the second and third forming trolleys, the force parameters 021 To determine the reactions of the guide rollers 11 N , 12 N , 21 N , 22 N , 31 N , and 32 N as well as the forces in the connecting rods 1 F , 2 F , and 3 F , let us consider the static equilibrium of the first, second and third forming trolleys. We design all the forces acting on each of the trolleys on the coordinate axes x and y , and compute the sum of the moments of these forces relative to the points, 1 B , 2 B , and 3 B (Fig. 3). As a result, for the first forming trolley we obtain  (10) for the second forming trolley we obtain  (11) for the third forming trolley we obtain  (12) Having solved the systems of equations (10) On the basis of the dependences (13)-(15) we find the moments of the resistance forces Mres1, Mres2 and Mres3 from each of the forming trolleys and the total moment of the resistance forces Mdr2, reduced to the crank rotation axis The magnitude of the angles β1, β2, аnd β3 can be determined from the ratios Using the Lagrange equation of the second kind, we make differential equations of the motion of a roller forming unit with an energy-balanced drive, represented by a two-mass dynamic model 37 , dt d ; dt where t is the time; T is the system kinetic energy; Q are the generalized forces corresponding to the coordinates φ1 and φ2; П is the system potential energy that has the form The system kinetic energy, taking into account the expression (9), is expressed by the dependence The generalized forces have the form Here crit M is the critical (maximum) moment on the electric motor shaft; u is the drive mechanism transmission ratio; dr  is the drive mechanism efficiency coefficient; 0  is the synchronous angular speed of the electric motor rotor; crit s is the electric motor critical slip determined by the dependence where  is the electric motor maximum ratio (electric motor overload capacity); nom s is the electric motor nominal slip determined as follows: After substituting the expressions (29), (31)-(33), (13)- (15) and (25) into the system of equations (28) we have

Results of the solution
For a roller forming unit with the parameters 4 to be given below, the functions are determined and the graphs are built, expressing the changes in the reactions of the guide rollers 11 N , 12 N , 21 N , 22 N , 31 N , and 32 N (Fig. 4), forces in the connecting rods 1 F , 2 F , and 3 F (Fig. 5), and moments of the resistance forces  (27). The nominal rated power of the electric motor is determined by the mean value of the reduced moment of resistance for one crank rotation cycle [22,23]. According to these data, a 4A series 4А160М6У3 basic-version asynchronous electric motor with a short-circuit rotor was chosen, having the following parameters were selected.

Fig. 6. Chart of the resistance force changes Mres1 (1), Mres2 (1), Mres3 (3) and Mdr2 (4) depending on the rotation angle of the cranks
The values of the first and second transfer functions of the trolleys are determined from the expressions of the functions of changing the coordinates of the first, second and third forming trolleys in accordance with (Fig. 1, b) The expressions The ratios l r for roller forming units with crank-and-rod drive mechanisms do not exceed ⅓, and the series (41)-(43) converge fairly quickly, so, with satisfactory for the practice accuracy, the third and subsequent members of the series (41)-(43) can be thrown away. Then the dependencies (35)-(37) will look as The values of the first and second transfer functions of all three forming trolleys are determined from the expressions (44) As a result of the numerical experiment, it has been established that the optimal value of the rigidity, reduced to the crank rotation axis, of the drive of a roller forming unit with an energy-balanced drive with the above parameters equals m N с 150000  . The determination of the optimal value of the drive rigidity was carried out according to the method described in papers [26,27]. At this rigidity value, minimum loads are observed in the drive clutches. This rigidity value is used in the following calculations.
To study the movement dynamics of a roller forming unit with one taking into account the dissipation during the start-stop movement modes of forming trolleys, the system of equations (34) was supplemented by the drive dissipation value k