Contemporary problems in the theory of machines and mechanisms

Jnl. MechanismsVolume 2, pp.233-252/Pergamon Press 1967/Printed in Great Britain

Reviews Books Contemporary Problems in the Theory of Machines and Mechanisms. Edited by the Academy of Sciences of the U.S.S.R., Moscow, 1965 (in Russian).

THIS volume was written to honor Academician I. I. Artobolevskii on his 60th birthday; it contains forty-one articles contributed by authors throughout the world, twenty-nine of whom are from the Soviet Union. In the following brief reviews, the name of the author is given first followed by the title of the paper: 1. V. A. Zinoviev: L L Artobolevskii on his 60th Birthday and after 40 Years of Scientific and Teaching Activity An account is given of Artobolevskii's career and can be briefly summarized as follows: 1905--Born, the son of a professor at Moscow Agricultural Institute. 1926---Graduated from the Machine Building Faculty of the Moscow Agricultural Institute and began teaching in the Universities and Institutes. 1928--Docent in the department of Applied Mechanics in the Moscow ElectroMechanical Institute. 1929--Professor, Chair of Technical Mechanics at Moscow Chemical-Technological Institute; also lecturer at Moscow University. 1932--Professor of the Theory of Mechanisms and Machines, Military Air Academy. 1942 to date--Chair of Theory of Mechanisms and Machines, Moscow Aviation In ~titute. His honors include: honorary degree of Doctor of Technical Science, 1937; in 1939 elected a corresponding member and in 1946 a member of the Academy or Sciences. A list is given of over 500 books and articles published by Artobolevskii and also an account of his scientific activities and the seminars and conferences he has organized. 2. A. A. Tchekanov: The Work of Academician L I. Artobolevskii in the History of Science and Technology Artobolevskii has thoroughly investigated the historical development of the Theory of Mechanisms. He is of the opinion that whereas Russian scientists were ahead of Western scientists in synthesis, structure and classification, the backward machine-building industry in Russia could not make use of the advanced Russian theories. Foreign scientists and technologists, on the other hand, were more advanced in the practical applications of mechanism theory. 3. A. J. Ilinsky, I. I. Kapustin : On the Design of Optimal Structural Schemes of Technological Automatic Machinery 233

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The authors derive an expression for the efficiency of a working machine and find that that this depends on (1) the time of actual working; (2) time of idling and auxiliary operations such as transport from one position to another; (3) time of auxiliary operations such as loading and unloading; (4) non-operational down time, i.e. change and adjustment of cutter. They stress the value of computers in arriving at maximum efficiency. 4. M. L. Bychovski and A. A. Vishnevski: Cybernetic Systems in Medicine In medicine, cybernetics can be used as an aid to the management of the patient's organs in the process of the treatment of disease and also with the use of electronic computers in order to investigate or describe the action of functional systems within the organ. In 1960 a cybernetics laboratory was established, firstly to create a dynamical diagnostic system which could use the results of an investigation of the patient to obtain a clear picture of the state of his health, and also to create an automatic medical information system in which every medical case-history can be stored, by which symptoms and diagnoses can be compared. Accuracy of diagnosis has now reached 90 per cent. 5. V. R~Sssner (Magdeburg, East German D.R.):

On the Classification of Kinematic

Chains Examples of plane chains with freedom - 1, 0, I and 2 are given. Spatial chains based on the six degrees of freedom of the free body are also discussed and in addition, the chains of Rittershaus and Goldberg are discussed in which one of these freedoms is absent.

On the Range of Application of the Principle of Transforms of Kotelnikov-Study to the Mechanics of a Rigid Body

6. F. M. Dimentberg:

The principle of transforms is discussed with particular reference to the vector and the screw. In 1873, Clifford established a relationship between Quaternions and Biquaternions. In 1895 Kotelnikov applied the principle of transforms to vectors and screws in order to replace the real coefficients in quaternions by complex coefficients, which enabled him to proceed from vector to screw theory. Simultaneously in 1901 a German scientist, E. Study, developed the application of complex numbers to the screw; this principle was known as the Study principle. Later this has been applied to the analysis of space mechanisms. In his conclusion the author states that this principle cannot be applied in Dynamics, but the application is limited to kinematics. 7. F. L. Litvin : Development of the Theory of Gearing The classical method for the determination of an enveloping surface of a number generating surfaces, using Differential Geometry, is rather complicated. A simpler alternative technique using matrices is presented here. The author derives the law of motion of gears and the curvature of the tooth surfaces and investigates the conditions for undercutting. 8. L. N. Borisenko and J. L. Geronimus: On Some Methods of Choice of Optimal Laws

of Motion The choice of optimal laws of motion generally requires use of the variational calculus, but some cases may be solved more simply.

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For a f(x) which is valid within an interval [0, l], the following notations are introduced :

~,~(f)=folf(x)Jdx l,,_(f)=~[filf(x)lSd.x] and

t,(f) -- m a x [ f (x)[,

0 ___x _< 1

Here ~q(f) is the arithmetic mean, and #2(f) is the mean square of the modulus of the function within limits [0, 1]; # ( f ) is the maximum modulus of the f ( x ) through the interval [0, 1], and is called its "peak". Denote further:

G,(x)=a,x" +a,_ Ix ~- t + . . . + a t x +ao m ( G , ) = a , s , + a , _ t s , _ t + . . . +ats1+aoSo where G,(z) is a polynomial of the nth power with arbitrary coefficients, the linear combination of which is donoted by co(G~), where So, s t . . . s, are known. Using these notations the authors derive the optimal laws of motion for several specific cases.

9. V. I. Sergejev: Some Questions on the Accuracy and Certainty of the Simplest Calculating Devices In a general case in a calculating device there may be m inputs and m* outputs, which may be determined by variables: xl, x2 • • • xm and Yt, Y2

•

•

•

ym,

The functional relationship between the variables y and x with one input and one output may be expressed as:

y=A{x}

(1)

where A is a given operator. The operator A depends on a number of constructional parameters and constants q,. Therefore in an ideal case:

y o = A o ( q O . . , q O . . . qO, x)=Ao{qO, x};

l
(2]

Building up the device, designed for an accurate realization of (2), there may be difficulties of a constructional or technological type. In connection with that the device realizes another operator A t close to A0; therefore we will have instead of equation (2) another equation

yt=A1{q~, x}

(3)

The difference Ayt =Yl - Y o

(4)

represents the error of the calculating device. Choosing various operators, the author arrives at percentage errors for particular cases. 10. G. G. Baranov: On the Calculation of Accuracies

For the solution of this problem, Lagrange's method is applied.

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Denoting by ¢5 the overall output accuracy,

6"-=(Kta1~l)2+(Kza2~z)2+... + (K,a,(Jn) 2

(I)

where 6~ is the accuracy of element i; a~ is the partial derivative of the accuracy of element i; K~ is the coefficient of relative dispersion of the law of distribution of the accuracy of element i. Also, define Bt = (K~afi~)z. It is assumed that all partial derivatives are constant. This is true for mechanical chains with constant transmissions and for linear electric chains. Further, it is considered that the relationship between the value of each element R and its accuracy is known: Thus R1 =FI(61) ;

R., =F2(d2) ;

. . . R,=F,(6,)

(2)

Consider a case when each of these functions is continuous and differentiable within certain limits. The summary value R = R ~ + R 2 + . . . +R n should then be minimized. It is shown that for the minimal value of the whole, the derivatives of each R~ with respect to each Bt must be equal. Further derivations for particular cases are carried out graphically. 11. N. I. Levitsky: Determination of Fundamental Dimensions of Plane Cam Mechanisms with a Roller Follower The three fundamental dimensions are taken to be: r, the radius of the roller, b the width of the roller and the cam, and Ro the minimal radius vector of the cam. R 0 is determined (1) by the contact stresses using the Hertz formula, and (2) by the prescribed condition of normal wear. b and r, are determined from the condition of equal strength of the axis of the roller for bending and pressure. The influence of the pressure angle in determining the initial radius vector of the cam is discussed, and the way to determine the fundamental dimensions for an oscillating follower given. 12. J. S. Beggs (University of California): Cams with Frame-Followers The use of frame-followers with contact on two or four sides of a positive cam for reciprocating and oscillating motion is discussed. Profiles of cams composed of circular arcs are shown, and the design of the profile is explained. An analysis of existing constructions is made. Cams with frame-followers are used in "Mercedes-Benz" automobiles in order to eliminate valve springs. 13. L. V. Petrokas: On the Design of Friction-Cam Mechanisms In automatic machinery, cam-and-follower mechanisms are widely used. Among the numerous types of plane cam-and-follower mechanisms are also friction-cam mechanisms, in which transfer of rotational motion through the higher pair is maintained by frictional forces and a toggle action. Such mechanisms are force-closed and, to eliminate slip, the driving cylinder is usually made of rubber and the cam of metal with a knurled surface. Examples are given of mechanisms of this sort used in linotypes, the type-bar drive mechanism in electric typewriters, and automatic record changing. Angular velocities and angular accelerations are derived by assuming pure rotation without slip in the higher pairs.

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14. J. Voimer (Karl-Marx-Stadt, East German D.R.): Geometrical Synthesis of a Transferring Mechanism Designers seldom apply the theory of synthesis for a number of reasons: (1) lack of time to study scientific publications which are rarely relevant to his immediate problem, (2) unfamiliarity with the published work makes it seem complicated. In order to draw the attention of designers to synthesis it would be very useful to publish "Construction sheets" dealing with typical problems and indicating required sizes in the form of diagrams and curves. Some such sheets are already published by the V.D.I. As an illustration of methods of synthesis, the author develops the design of a six-link mechanism which transfers a link through 5 prescribed positions in translational motion. 15. Jan Oderfeld (Warsaw, Poland): On the Accuracy of Graphical Methods in the Theory of Machines and Mechanisms Despite the increasing use of computers, the variety of problems in Dynamics of Machinery requires that graphical methods will still be useful for at least another 20 years; therefore the accuracy of these solutions is important. As the fundamental of his investigations, the author uses the method of the total differential. Five standard operations used in the drawing of vector diagrams are considered and the errors involved in each of these are enumerated. In conclusion the author indicates that the accuracy of a graphical construction may be increased by (1) avoiding intersections of lines under small angles, (2) eliminating repetitive operations, (3) reducing the number of simple graphical and analytical operations. 16. S. A. Cherkudinov: Some Properties of Burmester's Curves Geometrical methods of synthesis of hinged mechanisms are based upon the application of Burmester's curves--the centre and the circle-point curves. Therefore, the study of these curves has not only a theoretical significance, but may open new possibilities for the application of these curves to the solution of various problem sofsynthesis. Chapter 37 of the book: Synthesis of Plane Mechanisms by Artobolevskii, Levitski and Cherkudinov, which deals with four positions of the plane, has been written by the author. In the present article he expands Burmester's theory by introducing the cross of opposite points, determining tangents to Burmester's centre curves and showing the construction of asymptotes to the Burmester curves. 17. C. Pelecudi (Bucharest, Roumania): An Analogy between Higher Accelerations and Velocities in Plane Kinematic Chains Three theorems are presented and proved. (l) The projections of combined higher acceleration vectors of points .4 and B on to the line joining the points, are equal. (2) The polygon formed by the extremities of the higher accelerations is similar to the polygon formed by the points, but turned with respect to it by a fight angle. (3) The instantaneous centre H is at the intersection of the lines perpendicular to the combined higher acceleration vectors. In addition a new property is discovered for the plane relative motion of two rigid links.

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18. K. Hain (Braunschweig, West German F.R.): Rotational and Oscillating Relative Motions in Four-Link Kinematic Chains with a Sliding Pair The author discusses a closed kinematic chain having three hinges and one sliding pair, known as the slider-crank chain. The criteria determining whether this is a rotational or oscillating chain are established and the dead-point angles for oscillating links are found. The author states that mechanisms with non-uniform transfer are not used enough in practice. 19. F. E. Crossley (Atlanta, U.S.A.): Simulation of Motion of Plane Four-Link Mechanisms on an Electronic Analogue Computer Computer circuits are described by which the motion of the following mechanisms can be simulated: 1. Elliptical trammel; 2. Crank and rocker mechanism; 3. Oscillating slider-crank mechanism; 4. Double-crank mechanism; 5. Whitworth quick-return mechanism. 20. I. V. Klimov and O. S. Koshelev: Some General Questions onthe Analytical Determination of the Path, Velocity and Acceleration of Particular Mechanisms The authors consider a group of mechanisms that transform sliding motion to rotational but not circular motion, or vice versa. Expressions are derived for the displacement of the slider as a function of the crank angle and the link lengths. The velocity and acceleration of the slider are found by differentiation. 21. D. Mangeron and V. V. Topentcharov (Jassy, Roumania and Sophia, Bulgaria): Dual Problems of Euclid's Kinematics The authors discuss the general theory of reduced accelerations, in rectangular or polar coordinates; spatial motion is described in vector matrix terms. Five general theorems are derived. General questions of duality of kinematic fields are also discussed, and two theorems are derived. 22. D. P. Adams and D. S. Nokes (M.I.T., U.S.A.): Application of Quaternions to the Analysis of a Spatial Four-Link Mechanism The available methods for the analysis of spatial mechanisms are reviewed; the analysis of a spatial four-link mechanism is carried out as an example. Quaternion algebra is used, but the method does not give a satisfactory result; another solution is given using an iteration method. Velocities are determined by considering the motion of each link as a screw motion. Accelerations are determined for two cases: (1) The driving link having an initial acceleration, but no velocity; (2) the driving link having a constant velocity. For a particular case, the results of a relationship between angles are obtained using a computer and the results presented graphically. 23. F. Freudenstein (Columbia University, New York): Type of Spherical Four-Link Mechanisms

On the Determination of the

Grashof's conditions for plane mechanisms are not always valid for spherical mechanisms. It is demonstrated the sides of a spherical four-link mechanism may be considered as the arcs of great circles; by producing the planes of these great circles,

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8 spherical mechanisms are obtained having different lengths of sides. For these mechanisms Grashof's conditions are valid only for those, of which the sides, except the coupler, are less than or equal to 90 °. 24. N. M. Guseinov and S. I. Gamrekeli: On a Method of Simulation of Spatial Four-

Link Mechanisms The question of determining the position of a four-link spatial mechanisms is the most important synthesis. The configuration of a closed four-link spatial mechanism is dependent on 9 independent parameters: the lengths of four links, the turning angles of the driving and driven links and the coordinates of the relative centre of rotation of the crank and follower crank. The simulator arrangement consists of four interconnected spatial four-bar mechanisms enabling the variation of all 9 parameters. From a mathematical point of view the method of simulation consists in variation of all parameters excepting one, which should keep a more or less constant value. The mechanical variation of the parameters by the use of electrical measuring arrangements show their influence on the value of the free parameter, and enables one to know which parameters should be varied in order to keep the free parameter constant. 25. D. Mangeron, D. S. Tavldaelidze, R. Shalea (Jassy, Roumania; Tbilisi, G.S.S.R.; Besanqon, France): A General Theory of Successive Approximations for the Investigation

of Mechanisms and Machines by a New Matrix-Tensor Method Besides the screw method, the matrix method has also been used, as this is very suitable for programming on computers. The present investigation is based on the previous work of D. Mangeron, and the results are related to the tensor of the position, tensor of velocity and tensors of acceleration fields of various orders. 26. S. N. Kozhevnikov: Dynamics of Mechanisms having Two Degrees of Freedom Considering mechanisms having more than one degree of freedom, the author distinguishes between stationary and non-stationary constraints. Stationary constraints are pure geometrical constraints which limit relative displacements between links and the number of degrees of freedom. If the motion of driving links is given, it is then possible to determine displacements, velocities and accelerations. Non-stationary constraints have certain dynamical characteristics such as elastic links. An elastic link may be considered as a translator which translates motion from one link to another. In this case, the kinetic energy of the mechanism is used to derive the equations of motion, using Lagrangian differential equations of the second form. 27. A. E. Kobrinsky: Some Problems of the Theory of Shock- Vibratory Systems Shock vibration may be an unavoidable part of the working process, or it may be due to inaccuracies in kinematic pairs. Two elements of a shock vibratory system may be termed a shock pair. Differential equations of a multi-mass system with one shock pair are derived. The integration of such a system is shown, and further, the motion of a system with two shock pairs including transient processes is discussed. 28. A. P. Bessonov: On the Equilibrium of Mechanisms with Variable Masses of Links. Previously the author has shown that the stability of motion of mechanisms is convenienfly analyzed by using auxiliary points of the equation of motion. In this paper

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the same method is applied to mechanisms with variable masses of links by setting up the differential equations of motion, and using the phase plane to note the form of their singularities. The results indicate that the conditions of stability depend on the characteristics of applied forces, resistance forces and reactive forces. The motion of an automatic pulsator type of conveyor is considered as an example. 29. A. V. Shliachtin: Some Peculiarities of Vibration Calculations of Elastic Links in Complex Systems The method of initial parameters introduced by A. N. Krylov has been applied. Differential equations of motion for elastic parts contained between two concentrated masses are discussed for two cases: (I) For bending-axial vibrations of links with variable cross-sections; (2) For bending-torsional vibrations. The mechanism of a shaking-trough is considered as an example. 30. M. M. Gernet: Theory and Calculation of Machines Connected With the Motion of a Grain in an Air Flow This is the study of the behavior of grain in agricultural machinery. In practice grain describes a plane curve in free flight and it is therefore sufficient to use three differential equations of motion. The coefficients of aerodynamical moments are necessary, and these were found by the use of a wind tunnel. It was also necessary to determine the ellipsoid of inertia of a grain: to do this, the grain was fixed on the balance wheel of a watch, and the change of time observed during a 24 hour period. Final calculations show grain spins up to 1250 r.p.m. 31. B. V. Edelstein: A Dynamical Evaluation of the Coefficient of Fluctuation of Speed of the Driving Link of a Machine Assembly Using the well-known coefficient of fluctuation of speed: 6=(co~,,x-wm~,)/~o, the dynamical coefficient of a machine is given as:

f= 8max~O)2 where e~x is the maximum angular acceleration between a~m~. and ogmax. The author then obtains another expression

f=2a Kn

where K=A_._t" T 32. V. I. Nebesnov: An Investigation of the Regime of Motion of Ship Engines The question of investigating the transient and stable regime of the motion of a force aggregate is a most difficult problem in the dynamics of machinery. Some analytical, grapho-analytical, and numerical methods have been developed for the approximate solution of the differential equations of motion. The problem is even more complicated when the system is placed on a moving object and the relationship between

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the engine and the object is not simple. In such a case, two related non-linear differential equations need to be integrated. As an example a system consisting of the engine, screw, and hull of a ship is considered. 33. N. P. Rajevski: A Recording Apparatus for the Third Derivative of the Displacement with Respect to Time This apparatus consists of a small beam of uniform strength fixed vertically at the upper end, which carries at its lower end a small inertia element of rectangular section. The lower part of the element has a cylindrical surface with a radius equal to the length of the beam. The base of the frame has a cylindrical cut-out. Between the two cylindrical surfaces is a small gap which may be varied. By putting some drops of silicon into the gap and changing its size, a desired damping may be achieved for direct measurement of linear accelerations, since the deflections of the beam are proportional to the inertia force. A further arrangement consists of a magnet fixed to the inertia element, which moves between reels carrying a winding. The motion of the magnet induces electric potential in the wiring proportional to the velocity of the element, i.e. to the third derivative of the displacement. Using the apparatus, clear diagrams of ~, ~ and ~ can be obtained. 34. A. K. Bakshis and K. M. Ragulskis: On the Synthesis of Quick-action Transfer Mechanisms For mechanization and automation of industrial processes, various transfer mechanisms are used. Any complex system of mechanisms may be reduced essentially to three links: Link 1--the source of energy which usually acts with rotational motion; link 2--the transfer mechanism with a variable transfer ratio, and link 3--the working arrangement having rectilinear motion and connected with the transferable mass m upon which a resisting force F is acting. In the synthesis, the transfer ratio should be determined so that the mass m is moved from the initial position into the final position in a minimum time. To solve this problem, the following Lagrangian equation is established:

(I s + my2)~p+ m y . y,•2 = M , ( $ ) - Fy where /.. is the moment of inertia of rotating masses, and M, is the turning moment of the engine. The equation is a non-linear differential equation, which can be solved by the use of a computer. A numerical example is given. 35. J. Ko~e~nik: Vibrations of a Steam Turbine and Turbo-compressor Blades Vibrations of the blades of a turbine or compressor are induced mainly by the intermittent action of steam or air flow. In order to calculate the frequencies of free flexural and torsional vibrations of a separate blade, it is considered as a cantilever beam. Flexural vibrations are considered in two perpendicular planes passing through the principal axes of inertia of all sections perpendicular to longitudinal axis. The influence of centrifugal forces on the frequency of free vibrations f~ is considered negligible, so long as W
242

Frequently, however, turbine blades are made connected into packets bound together by a shroud, or by wires. Such construction complicates the calculations, even for the case of two connected blades. The author shows a method of solving the differential equations for a packet of n blades. A simplified solution is also shown for the case of an infinite number of blades. 36. A. Morecki, J. Golinski, Z. Vasserstrum: An Investigation of a Vibrating System with

two Degrees of Freedom and a Resistance Proportional to the Square of the Velocity Difference by the Monte-Carlo Method Dynamic analysis of some machines may be represented by a chain of two masses and three springs, with damping between the two masses proportional to their velocity difference. One mass receives a simple harmonic force excitation. The variational equations of Galerkin are applied to the differential equations, and solution is by the Monte Carlo Method. Statistical methods of solution are used for the systems of equations, and two numerical examples are worked out. 37. A. G. Burgvitz and T. A. Savjalov: On the Theory of Vibrations of High-speed Lightly

Loaded Shafts on an Oil Film High-speed lightly loaded shafts in bearings with oil films work with small values of relative eccentricity. In this state, the existence of stable equilibrium may be explained only if, in the equations of motion of the lubricant, inertia forces are taken into account. The authors start by integrating the equations of Navier-Stokes for plane flow of the lubricant. Special cases of vibrations of a rigid shaft and a flexible shaft are considered and the results discussed. 38. E. V. Hertz: Dynamics of Pneumatic Arrangements Considering Heat Exchange Investigating some pneumatic arrangements, it is necessary to take into account the heat exchange with the surroundings. The paper is dedicated to questions of heat exchange in fundamental equations describing thermodynamical processes in pneumatic arrangements of automatic machinery. This requires introducing into the system two additional equations which enable temperatures in both parts of the working cycle to be determined. Due to the complexity of non-linear differential equations obtained, it is necessary to use computers for the solution. 39. E. A. Tsuchanova: A Simplified Method for Determining the Time of the Working

Process of a Hydraulic Cylinder Automatic machinery often contains hydrosystems composed of several cylinders. In order to set up cyclograms of the machine function, it is necessary to know the time spent in motion by each hydrocylinder (either the piston or the cylinder itself). Assuming that during one cycle the temperature and density of the fluid remain unaltered, it is possible to determine the relationship between the pressures in the two sections by using the equation of energy for a linear non-stationary motion of a viscous fluid. Final results are shown as graphs for particular cases. 40. L. N. Reshetov: Some Relationships in Spiral Gearing An orthogonal spiral gear pair with involute teeth is considered for which ~t +/~z = 90°The axes are skew, and the velocity ratio is i. Assuming a certain angle ill, the author

243 derives the condition for a minimum sliding velocity as r2: r t = i z i.e. a hyperboloid gear which is suitable for a gear ratio near to 1. For a minimum force the author obtains r2: r~ = i ÷. A minimum of friction power loss is obtained if fl~ =32=45 °. If both gears with a right spiral are replaced by gears with a left spiral, the direction of rotation of the driven wheel will be opposite, keeping the direction of the driver the same. 41. V. F. Malsev: On the Dynamics of Jamming of Rollers in a Free-Wheel Clutch Roller or sprag devices are used in free-wheel clutches of many machines. The process of jamming is very short and the acting loads become very high. The differential equation of the relative motion is established and integrated, and the value of the relative turning angle of the enclosing part during jamming is determined for three cases of the contact surface of the centre piece: a plane contact surface, a cylindrical contact surface with a direction line in the form of a log spiral and a circular cylindrical surface. Experimental results are presented graphically showing the relationship between the turning moment and the angle. N. ROSENAUER F. E. CROSSLEY

F. M. D i m e n t b e r g : Screw Calculus and its Applications to Mechanics (Vintovoe lstchislenye i iego Prilojheniya v Mekhanike): [zdat "Nauka", Moscow, 1965. 200 pp., paperback. A scaew is a motor whose vector parts are collinear. It occurs naturally in mechanics, for example, as the resultant of a system of forces or as a rigid-body displacement. This monograph is a compact and self-contained exposition of the algebra and calculus of screws and its application to rigid-body mechanics and mathematics. The mathematics of screw operations--associated with the names of W. K. Clifford, A. P. Kotelnikov and E. Study--is developed with the aid of dual-number algebra (a+tob, co2=0). The applicatiorts include the principle of transference, the reduction of any rigid-body displacement to a screw motion, displacement analysis of a spatial four-•k mechanism, elements of differential geometry, reciprocal screws, linear complexes and congruences, affinors and a generalization of the latter ("vintovoi binori," or screw binors, involving the product of a dual affinor and a screw--a concept introduced by S. G. Kislitsin), and applications to statics and dynamics, including the equations of motion in screw form, the inertia screw binor, and the theory of small vibrations. Although the treatment of each topic is necessarily brief, much information is contained within a limited amount of space. It should help familiarize mechanical engineers with an elegant and powerful tool in mechanics, which is finding increasing application in recent times, ranging from gyrodynamics to mechanisms. This reviewer knows of no equivalent text, the nearest approximation, perhaps, comprising several chapters in L. Brand's Vector and Tensor Analysis (J. Wiley, New York, 1947). This fine monograph, therefore, falls a gap in the literature. It is recommended to all those active or interested in applied mechanics and engineering analysis. F. FREUDENSTEIN