TEKA. An International Quarterly Journal on Motorization, Vehicle Operation, Energy Efficiency and Mechanical Engineering. Lublin-Rzeszow. 2018. Vol. 18. No 1. 31–38 Realization of Optimum Mode of Movement of Roller Forming Installation on Acceleration of Fourth Order Vyacheslav Loveikin1, Konstantin Pochka2 1National University of Life and Environmental Sciences of Ukraine. E-mail: lovvs@ukr.net 2Kyiv National University of Construction and Architecture. E-mail: shanovniy@ukr.net Received February 5.2018: accepted March 22.2018 Summary. For the purpose of increase in reliability and durability of roller forming installation the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order is calculated. Kinematic characteristics of the forming cart at the optimum mode of the movement on acceleration of the fourth order are cal- culated. The design of roller forming installation with the drive from the high-moment step engine which is built in the rolling rollers of the forming cart is offered and pro- vides the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order. Use in installation of the specified driving mechanism leads to improvement of quality of a surface to the processed con- crete mix, reduction of dynamic loadings in elements of the driving mechanism, to disappearance of excess de- structive loads of a frame design and, respectively, to in- crease in reliability and durability of installation in gen- eral. As a result of the conducted researches for the pur- pose of increase in reliability and durability of roller forming installation the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order is calculated. Kinematic characteristics of the forming cart at the optimum mode of back and forth motion on acceleration of the fourth order are calculated. The design of roller forming installation with the drive from the high-moment step engine which is built in the rolling rollers of the forming cart with a possibility of realization of the optimum mode of back and forth motion on acceleration of the fourth order is offered. The design of the drive of installation in a type of the cam mechanism is offered and the cam profile for provid- ing the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order is con- structed. Results of work can be used further for specification and improvement of the existing engineering methods of calculation of driving mechanisms of cars of roller for- mation both at design/designing stages, and in the modes of real operation. Also results of work can be useful at design or improvement of mechanisms with back and forth motion of executive elements. Key words: roller forming installation, mode of movement, step engine, drive. INTRODUCTION In the existing installations of superficial consolida- tion of concrete goods the crank ram or hydraulic drive of back and forth motion of the forming cart with the con- densing rollers is used [1-3]. During the constant starting and brake modes of the movement there are considerable dynamic loadings in elements of the driving mechanism and in elements of the forming cart which can lead to premature getting out of installation of the working condi- tion. THE ANALYSIS OF RECENT RESEARCHES AND PUBLICATIONS In the existing theoretical and pilot studies of cars of roller formation of concrete goods it is proved their de- sign data and efficiency [1-3]. At the same time not enough attention is paid to a research to the operating dynamic loadings and the modes of the movement that considerably influences work of installation and quality of finished goods [4-19]. During the constant starting and brake modes of the movement there are considerable dy- namic loadings in elements of the driving mechanism and in elements of the forming cart that can lead to premature getting out of installation of the working condition [4-19]. In work [20] optimization of the dynamic mode of a re- versal of roller forming installation is carried out. Howev- er in such mode acceleration and acceleration of the sec- ond order (breakthrough) of the cart are of great im- portance in his extreme provisions. By optimization of the breakthrough mode of a reversal of installation [21] ac- celeration of the cart in extreme provisions changes smoothly, however the breakthrough changes sharply and is of rather great importance. Optimization of the mode of a reversal of roller forming installation on acceleration of the third about [22, 23] leads to the fact that in extreme provisions of the cart acceleration and breakthrough mailto:lovvs@ukr.net mailto:shanovniy@ukr.net 32 Realization of Optimum Mode of Movement of Roller Forming Installation on Acceleration of Fourth Order change smoothly, however acceleration of the third order at the same time is of rather great importance and changes sharply from zero to the maximum value. Therefore ur- gent there is a problem of improvement of the driving mechanism of roller forming installation for the purpose of providing such mode of the movement of the forming cart at which dynamic loadings in elements of installation would decrease and its durability increased. OBJECTIVE The purpose of work consists in improvement of a design of the driving mechanism of roller forming instal- lation for increase in her reliability and durability. THE MAIN RESULTS OF THE RESEARCH Coefficients of unevenness of the movement and dy- namism can be criteria of the mode of the movement of mechanisms and cars [24]. In this work as criterion of the mode of the movement the criteria action which is inte- gral on time with sub integral function which expresses a measure of the movement or action of system is used. For the optimum mode of the movement on acceleration of the fourth order we will have criterion of an optimality of the movement in a look: min 1 0 →= ∫ t Z dtQI , (1) where: t − time; 1t − duration of the movement of the cart from one extreme situation in another; Q − energy of accelerations of the fourth order: V xmQ 2 2 1 ⋅⋅= , (2) where: m − mass of the forming cart; V x − acceleration of the fourth order. Poisson's equation is a condition of a minimum of criterion (1): ,05 5 4 4 3 3 2 2 = ∂ ∂ − ∂ ∂ + + ∂ ∂ − ∂ ∂ + ∂ ∂ − ∂ ∂ VIV x Q dt d x Q dt d x Q dt d x Q dt d x Q dt d x Q  (3) where: x , x , x , x , IV x − movement coordinate, speed, acceleration, acceleration of the second order and acceler- ation of the third order of the cart respectively. From expression (3) it is possible to write down: .0 ; ;0 5 5 =⋅= ∂ ∂ ⋅= ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ X V V V IV xm x Q dt d xm x Q x Q x Q x Q x Q x Q  (4) From the last equation (4) we receive the differential equation and its decisions: , 2 1 6 1 24 1 120 1 720 1 5040 1 40320 1 362880 1 ; 2 1 6 1 24 1 120 1 720 1 5040 1 40320 1 ; 2 1 6 1 24 1 120 1 720 1 5040 1 ; 2 1 6 1 24 1 120 1 720 1 ; 2 1 6 1 24 1 120 1 ; 2 1 6 1 24 1 ; 2 1 6 1 ; 2 1 ;;;0 109 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 98 2 7 3 6 4 5 5 4 6 3 7 2 8 1 87 2 6 3 5 4 4 5 3 6 2 7 1 76 2 5 3 4 4 3 5 2 6 1 65 2 4 3 3 4 2 5 1 54 2 3 3 2 4 1 43 2 2 3 1 32 2 1 211 CtCtCtCtC tCtCtC tCtCx CtCtC tCtCtC tCtCtCx CtCtCtCtC tCtCtCx CtCtCtC tCtCtCx CtCtC tCtCtCx CtC tCtCtCx CtCtCtCx CtCtCx CtCxCxx IV V VI VII VIIIIXX +⋅+⋅+⋅+⋅+ +⋅⋅+⋅⋅+⋅⋅+ +⋅⋅+⋅⋅= +⋅+⋅⋅+ +⋅+⋅+⋅+ +⋅+⋅+⋅= ++⋅+⋅+⋅+ +⋅⋅+⋅⋅+⋅⋅= +⋅+⋅⋅+⋅⋅+ +⋅⋅+⋅⋅+⋅⋅= +⋅+⋅⋅+ +⋅⋅+⋅⋅+⋅⋅= +⋅+ +⋅⋅+⋅⋅+⋅⋅= +⋅+⋅⋅+⋅⋅= +⋅+⋅⋅= +⋅===    (5) where: 1C , 2C , 3C , 4C , 5C , 6C , 7C , 8C , 9C , 10C − integration constants which are defined from boundary conditions. Boundary traffic conditions of the cart from one ex- treme situation in another the following: entry conditions Vyacheslav Loveikin, Konstantin Pochka 33 – 0=t , 0xx = , 0=x , 0=x , 0=x , 0= IV x ; final conditions – 1tt = , 1xx = , 0=x , 0=x , 0=x , 0= IV x . Here 0x and 1x – coordinates of extreme pro- visions of the center of mass of the cart. Having substitut- ed boundary conditions in the equations (5), we receive: ;0;0 ;0;0;:0 67 89010 == ==== ÑÑ ÑÑxCt (6)                                      =⋅+⋅⋅+ +⋅⋅+⋅⋅+⋅⋅ =⋅⋅+⋅⋅+ +⋅+⋅+⋅ =⋅⋅+ +⋅⋅+⋅⋅+ +⋅⋅+⋅⋅ =⋅⋅+ +⋅⋅+⋅⋅+ +⋅⋅+⋅⋅ =+⋅⋅+ +⋅⋅+⋅⋅+ +⋅⋅+⋅⋅ = .0 2 1 6 1 24 1 120 1 ;0 2 1 6 1 24 1 120 1 720 1 ;0 6 1 24 1 120 1 720 1 5040 1 ;0 24 1 120 1 720 1 5040 1 40320 1 ; 120 1 720 1 5040 1 40320 1 362880 1 : 15 2 14 3 13 4 12 5 11 2 15 3 14 4 13 5 12 6 11 3 15 4 14 5 13 6 12 7 11 4 15 5 14 6 13 7 12 8 11 10 5 15 6 14 7 13 8 12 9 11 1 tCtC tCtCtC tCtC tCtCtC tC tCtC tCtC tC tCtC tCtC xxtC tCtC tCtC tt (7) Having solved system of the equations (7), we re- ceive integration constants 1C , 2C , 3C , 4C and 5C : ( ) 9 1 01 1 25401600 t xxC − ⋅= ; ( ) 8 1 01 2 12700800 t xxC − ⋅−= ; ( ) 7 1 01 3 2721600 t xxC − ⋅= ; ( ) 6 1 01 4 302400 t xxC − ⋅−= ; ( ) 5 1 01 5 15120 t xxC − ⋅= . (8) We will accept amplitude of movement of the form- ing cart 01 xxx −=∆ . Having substituted certain con- stants of integration (6) and (8) in the equations (5) we receive expressions for definition of kinematic character- istics of the forming cart when moving from one extreme situation to another at the optimum mode of back and forth motion on acceleration of the fourth order: .1 120 9014070 15120 ; 110 303514 15120 ; 320 454214 2520 ; 15 972 2520 ;1464630 ; 126420 54031570 5 1 1 2 1 2 3 1 3 4 1 4 5 1 1 2 1 2 3 1 3 4 1 4 5 1 2 1 2 1 2 3 1 3 4 1 4 5 1 3 1 2 1 2 3 1 3 4 1 4 5 1 4 12 1 2 3 1 3 4 1 4 5 1 5 1 2 1 2 3 1 3 4 1 4 0 t t t t t t t t t xx t t t t t t t t t t xx t t t t t t t t t t xx t t t t t t t t t t xx t t t t t t t t t txx t t t t t t t t t t xxx V IV ⋅               +⋅− −⋅+⋅−⋅ ⋅∆⋅= ⋅               +⋅− −⋅+⋅−⋅ ⋅∆⋅= ⋅               +⋅− −⋅+⋅−⋅ ⋅∆⋅= ⋅               +⋅− −⋅+⋅−⋅ ⋅∆⋅= ⋅        +⋅−⋅+⋅−⋅∆⋅= ⋅               +⋅− −⋅+⋅−⋅ ⋅∆+=    (9) Having accepted amplitude of movement of the form- ing cart mx 4,0=∆ and duration of the movement of the forming cart from one extreme situation to another st 31 = , on the equations (9) kinematic characteristics of the forming cart at the optimum mode of back and forth motion on acceleration of the fourth order have been cal- culated. By results of calculations schedules of the opti- mum mode on acceleration of the fourth order of change of movement (Fig. 1, a), speeds (Fig. 1, b), accelerations (Fig. 1, c), accelerations of the second order (Fig. 1, d), accelerations of the third order (Fig. 1, e), accelerations of the fourth order are constructed (Fig. 1, f) at the move- ment of the forming cart of one extreme situation in an- other. The law of the movement of the cart described by the equations (9) can be carried out by the drive from the high-moment step engine which is built in the rolling roll- ers of the forming cart of installation. At the same time the law of change of angular speed of the driving step engine is described by the equation: 34 Realization of Optimum Mode of Movement of Roller Forming Installation on Acceleration of Fourth Order a b c d e f Fig. 1. Schedules of change of movement – а, speed – b, acceleration – c, accelerations of the second order – d, accel- erations of the third order – e and accelerations of the fourth order – f. at the optimum mode of the movement carts on accelera- tion of the fourth order .0 ,1464630 1 5 1 4 12 1 2 3 1 3 4 1 4 tt t t t t t t t t t t R x ≤≤ ⋅         +−+−⋅ ∆ =ϕ (10) Similarly the law of change of angular speed of the driving step engine at the movement of the forming cart is defined in the opposite direction: ( ) ( ) ( ) ( ) ( ) ;2 , 14 6 4 630 11 5 1 4 1 1 1 2 1 2 1 3 1 3 1 4 1 4 1 ttt t tt t tt t tt t tt t tt R x ≤≤ −                       + − ⋅− − − ⋅+ + − − − ⋅ ∆ −=ϕ (11) Fig. 2. Roller forming installation with the drive from the step engine. Vyacheslav Loveikin, Konstantin Pochka 35 For the purpose of reduction of dynamic loadings in elements of installation and for increase in her reliability the design of roller forming installation with the drive from the high-moment step engine for ensuring back and forth motion of the forming cart with the optimum break- through mode of a reversal (Fig. 2) is offered. Installation consists from the forming cart 1 which is mounted on the portal 2 and carries out back and forth motion in guides 3 over emptiness of a form 4. The forming cart contains the giving bunker 5 and the rolling rollers 6 on axis 7. The cart is set in back and forth motion by means of the high-moment step engine which is built in rollers, and the axis of a roller plays a stator role, and a roller – a rotor [25]. Transforming the first expression (9) for a case when the beginning of coordinates is counted from the average provision of his movement, we will receive:               −⋅               +⋅−⋅+ +⋅−⋅ ⋅ ∆ = 1 126420540 31570 2 2 5 1 5 12 1 2 3 1 3 4 1 4 t t t t t t t t t t xx .(12) The law of the movement of the cart described by the equation (12) can be carried out by the drive with the cam mechanism (fig. 3) of back and forth motion of the cart. At the same time the movement of the cart in one direc- tion is carried out due to turn of a cam 1 on a half of a turn (that is πϕ = ) and in the returnable direction on a half of a turn; a full motion cycle of the cart – for one turn of a cam. It is necessary for implementation of the de- scribed law of the movement of the cart that the increment of radius of a cam corresponded to an increment to movement of the cart. According to it the variable radius of a cam is defined by dependence:                       −                       + +⋅−⋅+ +⋅−⋅ ⋅ ∆ +=ρ 1 126 420540 31570 2 22 5 1 5 12 1 2 3 1 3 4 1 4 t t t t t t t t t t xb , (13) where: b – distance between pushers 2 (Fig. 3). Fig. 3. The scheme of the mechanism with the cam drive of back and forth motion of the cart. Time t can be excluded from dependence (13) as ω ϕ=t , and ω π=1t . Here ϕ – angular coordinate of turn of a cam, and ω – angular speed of a cam. After the corresponding transformations the radius of a cam which describes his profile contacts angular coordinate the fol- lowing expression: π≤ϕ≤                       − π ϕ ⋅                     + + π ϕ ⋅− π ϕ ⋅+ + π ϕ ⋅− π ϕ ⋅ ⋅⋅ ∆ +=ρ 0 ,1 126 420540 31570 2 22 5 5 2 2 3 3 4 4 xb (14) Similarly the cam profile on the site of his turn from π to π2 which is described by the radius changing on dependence is defined: ( ) ( ) ( ) ( ) ( ) π≤ϕ≤π                                   − π π−ϕ                                   + + π π−ϕ − − π π−ϕ + + π π−ϕ − − π π−ϕ ∆ −=ρ 2 ,1 126 420 540 315 70 2 22 5 5 2 2 3 3 4 4 xb . (15) For prevention of blows of a cam about pushers at change of the direction of the movement of the cart (14) and (15) profile of a cam (fig. 4) described by the equa- tions have such appearance that its diameter the d con- stant and is equal to distance between pushers b ( bd = ) in any situation. Fig. 4. The cam profile realizing the optimum mode of the movement on acceleration of the fourth order 36 Realization of Optimum Mode of Movement of Roller Forming Installation on Acceleration of Fourth Order For the purpose of reduction of dynamic loadings in elements of installation and for increase in her reliability it is offered an installation design with the driving mecha- nism for providing the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order (Fig. 5). The driving mechanism is executed in the form of pivotally the cam mechanisms installed on the portal which contact to the pushers which are rigidly at- tached to the forming cart. Installation contains 1 forming cart 2 mounted on the motionless portal which contains in itself the giving bun- ker 3 and the rolling rollers 4 and carries out reciprocating the movement in guides 5 over emptiness of a form 6 [26]. The cart is set in motion by means of two drives 7 attached to the portal 1 in the form of the cam mecha- nisms rotating with a constant angular speed ( const=ω ), but different in the direction and contact to two pushers 8 which are rigidly connected to a cart 2 frame. Existence of two pushers 8 from each party of the forming cart 2 allows to create a rigid power chain at her direct and returnable movement. Fig. 5. Roller forming installation with the cam driving mechanism. When using in installation of the drive from the high- moment step engine which is built in the rolling rollers which law of change of angular speed is described by the equations given above quality of the processed concrete mix increases, dynamic loadings in drive elements de- crease, excess destructive loads of a frame design de- crease and, respectively, durability of installation in gen- eral increases. When using in installation of the cam driving mecha- nism from each party of the forming cart the possibility of her axial distortion is prevented, the quality of the pro- cessed concrete mix increases, dynamic loadings in drive elements decrease, excess destructive loads of a frame design decrease and, respectively, the durability of instal- lation in general increases. CONCLUSIONS 1. As result of conducted researches for purpose of increase in reliability and durability of roller forming in- stallation optimum mode of back and forth motion of the forming cart on acceleration of fourth order is calculated. 2. Kinematic characteristics of the forming cart at the optimum mode of back and forth motion on acceleration of the fourth order are calculated. 3. The design of roller forming installation with the drive from the high-moment step engine which is built in the rolling rollers of the forming cart with a possibility of realization of the optimum mode of back and forth motion on acceleration of the fourth order is offered. 4. The design of the drive of installation in a type of the cam mechanism is offered and the cam profile for providing the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order is con- structed. 5. 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Calculations of the opti- mum modes of the movement of mechanisms and cars. Manual, Kyiv, UMK VO, 168. 25. Lovejkіn V. S., Pochka K. І., Chovnjuk Ju. V., Dikteruk M. G. 2014. Patent of Ukraine for an invention № 105744. Installation for formation of products from concrete mixes. № а201309305 it is stated 25.07.2013; it is published 10.06.2014, Bulletin № 11. РЕАЛИЗАЦИЯ ОПТИМАЛЬНОГО РЕЖИМА ПЕРЕДВИЖЕНИЯ РОЛИКОВОЙ ФОРМОВОЧНОЙ УСТАНОВКИ НА УСКОРЕНИЕ ЧЕТВЕРТОГО ПОРЯДКА Аннотация. В целях увеличения надежности и долговечности ролика формируя установки оптима- льного режима движения вперед и назад в формую- щей тележки на разгон четвертого порядка рассчитывается. Кинематические характеристики фо- рмируя корзину при оптимальном режиме движения на ускорение четвертого порядка рассчитываются. Дизайн роликовые формовочные установки с приво- дом от высокого момента шагового двигателя, ко- торый построен на прокатных роликов, образующих тележку и обеспечивает оптимальный режим движе- ния вперед и назад формующей тележки на разгон четвертого порядка. Использование в установке ука- занного приводного механизма приводит к улучше- нию качества поверхности обрабатываемой бетонной смеси, снижение динамических нагрузок в элементах приводного механизма, к исчезновению лишнего ра- зрушительных нагрузок конструкция рамы и, соо- тветственно, увеличить надежность и долговечность установки в целом. В результате проведенных иссле- дований в целях увеличения надежности и долговеч- ности роликовые формовочные установки оптималь- ного режима движения вперед и назад в формующей тележки на разгон четвертого порядка рассчитывае- тся. Кинематические характеристики формируя кор- зину в оптимальный режим движения вперед и назад на ускорение четвертого порядка рассчитываются. Дизайн роликовые формовочные установки с приводом от высокого момента шагового двигателя, который построен на прокатных роликов, образую- щих тележку с возможностью реализации оптималь- ного режима движения вперед и назад на ускорение четвертого порядка предлагается. Проектирование привода установки в виде кулач- кового механизма и профиля кулачка для обеспечения оптимального режима движения вперед и назад при формировании корзины на ускорение четвертого по- рядка строится. Результаты работы в дальнейшем могут быть ис- пользованы для уточнения и совершенствования су- ществующих инженерных методов расчета привода механизмов машин формирования ролика как на ди- зайн/этапах проектирования, и в режимах реальной эксплуатации. Также результаты работы могут быть https://doi.org/10.21122/2227-1031-2017-16-3-206-214 https://doi.org/10.21122/2227-1031-2017-16-3-206-214 38 Realization of Optimum Mode of Movement of Roller Forming Installation on Acceleration of Fourth Order полезны при проектировании или усовершенствова- нии механизмов с возвратно-поступательное движе- ние исполнительных элементов. Ключевые слова: роликовые формовочные уста- новки, режим движения, шаг двигателя, привод. INTRODUCTION In the existing installations of superficial consolidation of concrete goods the crank ram or hydraulic drive of back and forth motion of the forming cart with the condensing rollers is used [1-3]. During the constant starting and brake modes of the m... THE ANALYSIS OF RECENT RESEARCHES AND PUBLICATIONS In the existing theoretical and pilot studies of cars of roller formation of concrete goods it is proved their design data and efficiency [1-3]. At the same time not enough attention is paid to a research to the operating dynamic loadings and the mode... OBJECTIVE The purpose of work consists in improvement of a design of the driving mechanism of roller forming installation for increase in her reliability and durability. THE MAIN RESULTS OF THE RESEARCH Conclusions 1. As result of conducted researches for purpose of increase in reliability and durability of roller forming installation optimum mode of back and forth motion of the forming cart on acceleration of fourth order is calculated. 2. Kinematic characteristics of the forming cart at the optimum mode of back and forth motion on acceleration of the fourth order are calculated. 3. The design of roller forming installation with the drive from the high-moment step engine which is built in the rolling rollers of the forming cart with a possibility of realization of the optimum mode of back and forth motion on acceleration of th... 4. The design of the drive of installation in a type of the cam mechanism is offered and the cam profile for providing the optimum mode of back and forth motion of the forming cart on acceleration of the fourth order is constructed. 5. Results of work can be used further for specification and improvement of the existing engineering methods of calculation of driving mechanisms of cars of roller formation both at design/designing stages, and in the modes of real operation. Also res... REFERENCES