The Kinematics of the Swashplate Engine with Two Rotating Pairs the Kinematics of the Swashplate Engine with Two Rotating Pairs Yu

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect AvailableScienceDirect online at www.sciencedirect.com Procedia Engineering 00 (2017)000–000 Procedia Engineering 00 (2017)000–000 www.elsevier.com/locate/procedia ScienceDirect www.elsevier.com/locate/procedia Procedia Engineering 206 (2017) 1722–1727

International Conference on Industrial Engineering, ICIE 2017 International Conference on Industrial Engineering, ICIE 2017 The Kinematics of the Swashplate Engine with two Rotating Pairs The Kinematics of the Swashplate Engine with two Rotating Pairs Yu. Pogulyaeva, O. Nikishinb,*, A. Zheltova a b, a a South UralYu. State University,Pogulyaev 76, Lenin, O. Ave Nikishinnue, Chelyabinsk*, 454080, A. Zheltov The Russian Federation b Chelyabinska South UralState State University, University, 129, 76,Bratiev Lenin Kashir Avenue,inykh Chelyabinsk st., Chelyabinsk 454080, 454001, The Russian The Russian Federation Federation bChelyabinsk State University, 129, Bratiev Kashirinykh st., Chelyabinsk 454001, The Russian Federation Abstract Abstract The use of axial engines instead of traditional crankshaft mechanisms opens up additional possibilities to improve the dimension characteristics,The use of axial regulation engines instead displacement of tradi tionalvolume, crankshaft and compressi mechanismson ratio. opens The up complex additional study possibilities of motion to of improve all power the mechanism dimension characteristics,elements is required regulation for further displacement engine optimizationvolume, and andcompressi improvements.on ratio. The papercomplex presents study theof motiondesign of theall powermechanism mechanism of the elementsswashplate is axialrequired engine for withfurther two engine rotating optimization pairs. The andbasic improvements. features of the The engine paper design presents are thedescribed. design ofThe the parameters mechanism required of the swashplatefor the theoretical axial engine description with twoof the rotating engine pairs. kinematics The basic are definefeaturesd. Theof the mathematical engine design model are described.describing Thethe engineparameters kinematics required is proposed.for the theoretical The basic description relationship of betweenthe engine the kinematics main kinematic are define variablesd. The is mathematicaldescribed in a model uniform descri motionbing of the the engine swash kinematics plate. is proposed.© 2017 The The Authors. basic relationship Published by between Elsevier the B.V. main kinematic variables is described in a uniform motion of the swash plate. Peer-review© 2017 The underAuthors. responsibility Published by of Elsevierthe scientific Ltd. committee of the International Conference on Industrial Engineering. ©Peer-review 2017 The Authors.under responsibility Published by of Elsevierthe scientific B.V. committee of the International Conference on Industrial Engineering Keywords:Peer-review motion under analysis; responsibility swash-plate of theengine; scientific modeling committee and simulation. of the International Conference on Industrial Engineering. Keywords: motion analysis; swash-plate engine; modeling and simulation.

1. Introduction 1. Introduction The idea of the power mechanism design based on the axial arrangement of cylinders around the output shaft is knownThe ideafor a of long the time.power This mechanism type of desienginegn basedis also on know the axialn as “Barrelarrangement Engine” of cylinders[1]. One aroundof the firstthe output concepts shaft was is knownsuggested for bya longMaxwell time. in This 1905 type [2]. of One engine of the is firstalso Russianknown asaxial “Barrel engine Engine” has been [1]. designed One of theand firsttested concepts in 1918 was by suggestedStechkin andby MaxwellMikulin. inOne 1905 of [2].the Onemost of famous the first modern Russian designs axial engine is the hasengine been offered designed by and the testedcompany in 1918 “Duke by StechkinEngines”. andIn Russia, Mikulin. significant One of resultsthe most were famous obtained modern at the Institutedesigns SSCis the FSUE engine “NAMI” offered [3]. by The the searchcompany in the “Duke field Engines”.of axial engines In Russia, is still significant open [4-7]. results Constructors were obtained are attr at actedthe Institute to the opportunitiesSSC FSUE “NAMI” that can [3]. give The axial search engines in the unlike field traditionalof axial engines crankshaft is still engines. open [4-7]. The Constructorsimprovement are of dimeattractednsion to characteristics, the opportunities regula thattion can displacementgive axial engines volume unlike and traditionalcompression crankshaft ratio, low engines. internal Thefriction improvement and vibration of dimelevelsnsion are amongcharacteristics, them. regulation displacement volume and compressionThis paper ratio, focused low oninternal the kinematics friction and based vibration on the levels new areway among to attach them. the piston to the swash plate. The design featureThis ispaper a cylindrical focused onpusher the kinematics with the free based end on moving the new in waya cylindrical to attach chan the pistonnel of swashto the swashplate. Theplate. pusher The design is not feature is a cylindrical pusher with the free end moving in a cylindrical channel of swash plate. The pusher is not

* Corresponding author. Tel.: +7-908-040-4448. * CorrespondingE-mail address: author. [email protected] Tel.: +7-908-040-4448. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering .

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.704

10.1016/j.proeng.2017.10.704 1877-7058 Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Yu. Pogulyaev et al. / Procedia Engineering 206 (2017) 1722–1727 1723 ScienceDirect 2 Yu. Pogulyaev, O. Nikishin, A. Zheltov / Procedia Engineering 00 (2017) 000–000 Procedia Engineering 00 (2017)000–000 www.elsevier.com/locate/procedia Procedia Engineering 00 (2017)000–000 connected to the piston directly. The piston is rigidly fixed to the rod. The rod consists of two parts: movable and www.elsevier.com/locate/procedia fixed. The fixed part of the rod is rigidly connected to the piston, and its axis coincides with the cylinder axis. The movable part of the rod is attached to the fixed part through the turning pair. The pusher is attached to the movable part of the rod through the another turning pair.

International Conference on Industrial Engineering, ICIE 2017 2. Mathematical model of the kinematics of the power mechanism International Conference on Industrial Engineering, ICIE 2017 The Kinematics of the Swashplate Engine with two Rotating Pairs 2.1. Design description The Kinematics of the Swashplate Engine with two Rotating Pairs The general scheme of the mechanism is presented in figure 1. The crank 2 is connected to the shaft 1 and Yu. Pogulyaeva, O. Nikishinb,*, A. Zheltova interact with swash plate 3 over a surface, which slope angle is denoted by  . The swash plate is fixed on the ball a b, a joint 4. The pusher 5 is inserted into swash plate channel 6 and joined to the movable part of the rod 8 through the a South UralYu. State University,Pogulyaev 76, Lenin, O. Ave Nikishinnue, Chelyabinsk*, 454080, A. Zheltov The Russian Federation b turning pair 7. The movable part of the rod joined to the fixed part of the rod 10 through the turning pair 9. The fixed Chelyabinska South UralState State University, University, 129, 76,Bratiev Lenin Kashir Avenue,inykh Chelyabinsk st., Chelyabinsk 454080, 454001, The Russian The Russian Federation Federation part of the rod is rigidly connected to the piston 11. The angle between the axis of the cylindrical channel and touch bChelyabinsk State University, 129, Bratiev Kashirinykh st., Chelyabinsk 454001, The Russian Federation surface of the crank is denoted by  . The connection point of the pusher and the rod in the general case does not lie Abstract in the median plane of the swash plate. The rotation angle of the shaft is denoted by  . Abstract The general scheme of the swash plate is presented in figure 2a. Some points were fixed on the swash plate. Point The use of axial engines instead of traditional crankshaft mechanisms opens up additional possibilities to improve the dimension O - center of a local coordinate system Ox0 y 00 z . Points A and B lie on the axis of the cylindrical channel 1 of the Thecharacteristics, use of axial regulation engines instead displacement of tradi tionalvolume, crankshaft and compressi mechanismson ratio. opens The up complex additional study possibilities of motion to of improve all power the mechanism dimension swash plate. The point A lies at the beginning of the channel, the point B lies at the end. Points M and N are characteristics,elements is required regulation for further displacement engine optimizationvolume, and andcompressi improvements.on ratio. The papercomplex presents study theof motiondesign of theall powermechanism mechanism of the swashplate axial engine with two rotating pairs. The basic features of the engine design are described. The parameters required fixed on the pusher and lie on its axis. The point M lies on the free end of the pusher. The point N lies at the elements is required for further engine optimization and improvements. The paper presents the design of the mechanism of the center of the turning pair 2, which connect the pusher and the movable part of the rod. The angle between the axis swashplatefor the theoretical axial engine description with twoof the rotating engine pairs. kinematics The basic are definefeaturesd. Theof the mathematical engine design model are described.describing Thethe engineparameters kinematics required is AB and the axis Oz makes the angle  . The pusher MN can move along the axis AB , and rotate around it. The proposed.for the theoretical The basic description relationship of betweenthe engine the kinematics main kinematic are define variablesd. The is mathematicaldescribed in a model uniform descri motionbing of the the engine swash kinematics plate. is 0 proposed.© 2017 The The Authors. basic relationship Published by between Elsevier the B.V. main kinematic variables is described in a uniform motion of the swash plate. rotation angle of the pusher in the channel is denoted by  . ©Peer-review 2017 The Authors.under responsibility Published by of Elsevierthe scientific B.V. committee of the International Conference on Industrial Engineering. The scheme of connection of pusher and rod is presented in figure 2b. The piston 1 is rigidly connected to the Keywords: motion analysis; swash-plate engine; modeling and simulation. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. fixed part of the rod 2. The fixed part of the rod and the movable part of the rod 4 are connected through the turning Keywords: motion analysis; swash-plate engine; modeling and simulation. pair 3. The movable part of the rod is attached to the pusher 6 through the turning pair 5. The point K is fixed at the 1. Introduction center of the turning pair 3. The point K can move only along the cylinder axis, and the y coordinate of the point 1. Introduction K will determine the position of the piston. The piston can rotate around the cylinder axis. The rotation angle of the The idea of the power mechanism design based on the axial arrangement of cylinders around the output shaft is piston is denoted by  . The angle between the cylinder axis and the axis of the movable part of the rod is denoted  knownThe ideafor a of long the time.power This mechanism type of desienginegn basedis also on know the axialn as “Barrelarrangement Engine” of cylinders[1]. One aroundof the firstthe output concepts shaft was is by . knownsuggested for bya longMaxwell time. in This 1905 type [2]. of One engine of the is firstalso Russianknown asaxial “Barrel engine Engine” has been [1]. designed One of theand firsttested concepts in 1918 was by suggestedStechkin andby MaxwellMikulin. inOne 1905 of [2].the Onemost of famous the first modern Russian designs axial engine is the hasengine been offered designed by and the testedcompany in 1918 “Duke by StechkinEngines”. andIn Russia, Mikulin. significant One of resultsthe most were famous obtained modern at the Institutedesigns SSCis the FSUE engine “NAMI” offered [3]. by The the searchcompany in the “Duke field ofEngines”. axial engines In Russia, is still significant open [4-7]. results Constructors were obtained are attr at actedthe Institute to the opportunitiesSSC FSUE “NAMI” that can [3]. give The axial search engines in the unlike field oftraditional axial engines crankshaft is still engines. open [4-7]. The Constructorsimprovement are of dimeattractednsion to characteristics, the opportunities regula thattion can displacementgive axial engines volume unlike and traditionalcompression crankshaft ratio, low engines. internal Thefriction improvement and vibration of dimelevelsnsion are amongcharacteristics, them. regulation displacement volume and compressionThis paper ratio, focused low oninternal the kinematics friction and based vibration on the levels new areway among to attach them. the piston to the swash plate. The design featureThis ispaper a cylindrical focused onpusher the kinematics with the free based end on moving the new in waya cylindrical to attach chan the pistonnel of swashto the swashplate. Theplate. pusher The design is not feature is a cylindrical pusher with the free end moving in a cylindrical channel of swash plate. The pusher is not

* Corresponding author. Tel.: +7-908-040-4448.

* CorrespondingE-mail address: author. [email protected] Tel.: +7-908-040-4448. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Fig. 1. The general scheme of the mechanism. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 2.2. Formulation of the problem

The condition of equality to zero degree of mobility of the kinematic chain connecting the swash plate to the cylinder is satisfied in the present scheme. So the position of all power mechanism units can be determined. 1724 Yu. Pogulyaev et al. / Procedia Engineering 206 (2017) 1722–1727 Yu. Pogulyaev, O. Nikishin, A. Zheltov / Procedia Engineering 00 (2017) 000–000 3 4 Yu. Pogulyaev, O. Nikishin, A. Zheltov / Procedia Engineering 00 (2017) 000–000

The center of fixed coordinate system coincides with the center of the spherical hinge 4, the axis coincides with 1 0 0 cos 0 sin     the axis of the shaft 1 (see fig. 1). The center of fixed coordinate system coincides with the center of coordinate    MMxy( ) 0 cos sin  , ( ) 0 1 0  , system . Ox y z . Right-hand coordinate system is used.    0 00 0 sin cos sin  0 cos  (3) cos sin 0    M z ( ) sin cos 0  0 01

where M x ,,MMyz - rotation matrices about respective axes. The entry of the matrix M w , at row i and column j

will be denoted as mij . In the case of uniform motion of the swash plate, cylinders’ laws of motion are identical, so the only one cylinder will be considered. The axis of the cylinder is located in Oyz plane. When   0 the piston will be at UDP in the cylinder. Since the point N lies on the axis of the pusher, and therefore the axis of the cylindrical channel of swash plate, then its coordinates in the Oxyz system can be expressed as:

Fig. 2. (a) The general scheme of the swash plate; (b) The scheme of connection of pusher and rod. ax  ba x x  a0xx b 0              Nay  b yyw, a M (,) a0 yyw , b M (,)  b sin  (4) Let's define the parameters required for the theoretical description of the mechanism:      az  ba zz  a0zz b  bcos   h - the distance from the cylinder axis to the axis Oy   - the slope angle of the crank (see fig. 1)   - the slope angle of the swash plate channel (see fig. 2a) where  is the quantity characterizing the translational motion of the pusher and should be determined.  b - the length of the swash plate channel (the length of the segment AB ) The coordinates of the point K in the system Oxyz have the form (0,ph , ) , where h is an unchangeable  (a 0, aa , ) - the coordinates of the point A in the system Ox y z parameter; p - coordinate, which determines position of the piston. x yz 0 00 Consider turning pair 5 (see fig. 2b). Its position in space is specified by the coordinates of the point N and the  l - the length of the movable part of the rod NK (see fig. 2b) direction vector of its axis (denoted by the vector n ). When   0 vector n is (1,0,0) . Position of the turning pair  The aim of this paper is to identify the dependence on the angle  of such values as: can be defined in two ways:  the coordinates of the point N in the system Oxyz determining the pusher position in the swash plate channel  through the position of the pusher as a function of values ,,  the coordinates of the point K in the system Oxyz determining the piston position in the cylinder  through the position of the movable part of the rod as a function of values p,,   value of the angle  determining the pusher rotation The expression of the coordinates of the point N through , has already been given in (4). The coordinates of   value of the angle determining the piston rotation the vector n , based on values of , , have the form:  value of the angle  determining the movable part of rod rotation relative to the fixed part

T nX (),(),() Y  Z  M (,)   M (   ) M ()   1,0,0 (5) 2.3. Derivation of motion equations  w xz

Next, consider a uniform motion of the swash plate. As the swash plate makes a complex rotational motion Let’s express the coordinates of the point N through p ,  ,  . N KN  K , where vector T around the point O , at a specified rotation angle of the main shaft  coordinates of any swash plate point in the KN Myz( )  M ( )  (0, l ,0) . system Oxyz can be represented as: 0ll cos  sin  cos   sin         xx0 N p lcos  plcos  (6)            yMw (,) y0 , (1) h lsin sin hl sin sin   zz0 Similarly, the direction vector n is expressed through , as: where (,xyz0 00 ,) - coordinates of the point in the system Ox0 y 00 z , M w - matrix representing the rotation of a rigid cos  cos body in space. For uniform motion of the swash plate elements of matrix M w are known.  nM ( ) M ( ) (1,0,0)T  sin They were obtained in various ways ([3] and [8]). In particular, the matrix M can be represented as: yz  (7) w   sin c os 

Mw( , ) MM yx (  )  (   ) M y (   ), (2) 4 Yu. Pogulyaev,Yu. O. PogulyaevNikishin, A. et Zheltov al. / Procedia / Procedia Engineering Engineering 206 00(2017) (2017) 1722 000–000–1727 1725

1 0 0 cos 0 sin        MMxy( ) 0 cos sin  , ( ) 0 1 0  ,    0 sin cos sin  0 cos  (3) cos sin 0    M z ( ) sin cos 0  0 01

where M x ,,MMyz - rotation matrices about respective axes. The entry of the matrix M w , at row i and column j will be denoted as mij . In the case of uniform motion of the swash plate, cylinders’ laws of motion are identical, so the only one cylinder will be considered. The axis of the cylinder is located in Oyz plane. When   0 the piston will be at UDP in the cylinder. Since the point N lies on the axis of the pusher, and therefore the axis of the cylindrical channel of swash plate, then its coordinates in the Oxyz system can be expressed as:

ax  ba x x  a0xx b 0              Nay  b yyw, a M (,) a0 yyw , b M (,)  b sin  (4)      az  ba zz  a0zz b  bcos  where  is the quantity characterizing the translational motion of the pusher and should be determined. The coordinates of the point K in the system Oxyz have the form (0,ph , ) , where h is an unchangeable parameter; p - coordinate, which determines position of the piston. Consider turning pair 5 (see fig. 2b). Its position in space is specified by the coordinates of the point N and the direction vector of its axis (denoted by the vector n ). When   0 vector n is (1,0,0) . Position of the turning pair can be defined in two ways:  through the position of the pusher as a function of values ,,  through the position of the movable part of the rod as a function of values p,,  The expression of the coordinates of the point N through , has already been given in (4). The coordinates of the vector n , based on values of , , have the form:

T nX(),(),() Y  Z   Mw (,)   M xz (   ) M ()   1,0,0 (5)

Let’s express the coordinates of the point N through p ,  ,  . N KN  K , where vector T KN Myz( )  M ( )  (0, l ,0) .

0ll cos  sin  cos   sin      N p lcos pl cos  (6)     h  lsin  sin hl sin sin

Similarly, the direction vector n is expressed through , as:

cos  cos     T   nMyz( ) M ( ) (1,0,0) sin (7)   sin c os  1726 Yu. Pogulyaev et al. / Procedia Engineering 206 (2017) 1722–1727 Yu. Pogulyaev, O. Nikishin, A. Zheltov / Procedia Engineering 00 (2017) 000–000 5

Equating the corresponding coordinates of the point N given in (4) and (6), and also coordinates of the direction vector n given in (5) and (7), we obtain the system of equations:

lcos sin  abxx  ,    plcos ayy b, hlsin  sin a b,  z z (8) X ( ) cos  cos , Y ( ) sin ,  Z()  sin  cos .

 The system includes five unknown variables: p,,,  ,  (the absolute values of the angles do not exceed ). 2 Expressing them through the engine parameters and angle  , we find the laws of motion of all parts of power mechanism. From the second equation of the system (8) expresses the value of p .

payy  b  lcos (9)

From the fifth equation we obtain the angle depending on the angle  .

sin  Y ( ) (10)

Dividing the sixth equation of the system by the fourth equation we obtain the angle  depending on the angle  .

Z() tan   (11) X ()

Consider the first and third equations of the system (8):

lcos  sin  ab  ,  xx (12) lsins  inhaz bz

Dividing the second equation of the system (12) by the first and using equation (11) we can find the value of  depending on the angle 

aZ()  ( a hX ) ()   xz (13) bXzx() bZ ()

Thus, we have expressed all required variables in the terms of angle  . The angle  can be found from the equation obtained as the sum of squares of the equations in (12). In simplified form it can be written as:

2 2  22    2 D Y() D l ( bXzx () bZ ()) (14)

where D abx zz () a hb x. Equation (14) can be solved numerically. Receiving the value of  for given  using the equations (13), (11), (10), and (9) we can find the position of all elements of the engine in space. Yu. Pogulyaev et al. / Procedia Engineering 206 (2017) 1722–1727 1727 Yu. Pogulyaev, O. Nikishin, A. Zheltov / Procedia Engineering 00 (2017) 000–000 5 6 Yu. Pogulyaev, O. Nikishin, A. Zheltov / Procedia Engineering 00 (2017) 000–000

Equating the corresponding coordinates of the point N given in (4) and (6), and also coordinates of the direction Fig. 3 shows trajectories of the points K and N . vector n given in (5) and (7), we obtain the system of equations: 3. Conclusion

lcos sin  abxx  ,  plcos a b, The kinematics of the axial engine with sloping pusher and two turning pairs has been considered in this paper.  yy The design of the power mechanism and kinematic chain connecting the swash plate with cylinder have been hlsin  sin a b,  z z (8) described. X ( ) cos  cos , Significant parameters required for the theoretical description of the engine kinematics have been determined. Y ( ) sin ,  Z()  sin  cos .

payy  b  lcos (9)

From the fifth equation we obtain the angle depending on the angle  . sin  Y ( ) (10)

Dividing the sixth equation of the system by the fourth equation we obtain the angle  depending on the angle Fig. 3. (a) The value of y coordinate of the point K as a function of angle  ; (b) The projection of the trajectory of the point N on the plane Oxy; (c) . The projection of the trajectory of the point N on the plane Oxz.  . The mathematical model describing the movement of engine components in space has been built. The model- Z() tan   (11) based algorithm for determining the position of the elements of the kinematic chain depending on the rotation angle X () of the shaft has been proposed.

Consider the first and third equations of the system (8): Acknowledgements

This paper is written on the programme 5/100. lcos  sin  abxx  ,  (12) The work was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.0011 lsins  inhaz bz References Dividing the second equation of the system (12) by the first and using equation (11) we can find the value of  depending on the angle  [1] Mc. Lanahan, J. Craig, Barrel Aircraft Engines: Historical Anomaly or Stymied Innovation?, SAE Technical Paper 985597, World Aviation Congress, Anaheim, 1998. [2] L.J.K. Setright, Some Unusual Engines, Mechanical engineering Publication Limited For the Institution of Mechanical Engineers, London, aZxz()  ( a hX ) ()   (13) 1975, pp. 98–108. bXzx() bZ () [3] M.A. Zlenko Theory and practice of an internal combustion engine with variable displacement and compression ratio, PhD Thesis, Moscow, 2005. (in Russian). Thus, we have expressed all required variables in the terms of angle  . The angle  can be found from the [4] C.Mc. Lanahan, S. Srinivasan, A. Shulenberger, Compact Multi-Cylinder Z-Crank Axial Engine, SAE Technical Paper 2005-01-0651, 2005. equation obtained as the sum of squares of the equations in (12). In simplified form it can be written as: [5] W.G. Liang, Z.S. Zhang, R. Zhu, The Motion Modeling and Simulation of the Swash-Plate Engine, Advanced Materials Research. 548 (2012) 672–676. [6] X.D. Han, W.Z. Xu, Analysis on the Cycle Characteristics of Dual Swash Plate Stirling Engine, Advanced Materials Research. 724-725 2 2  22    2 (2013) 946–950. D Y() D l ( bXzx () bZ ()) (14) [7] Y. Qu, S.C. Zheng, Numerical Analysis on the Performance of Miniature Stirling Engine Driven by Double Swash Plates, Applied Mechanics and Materials. 316-317 (2013) 84–90. where D abx zz () a hb x. Equation (14) can be solved numerically. Receiving the value of  for given  using [8] V.N. Yarovoy, Kinematics, dynamics and dimensional characteristics of the internal combustion engine with swash plate, PhD Thesis. the equations (13), (11), (10), and (9) we can find the position of all elements of the engine in space. Bauman MSTU, 1962. (in Russian).