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a branch of the theory of machines and mechanisms that studies the motion of machines andmechanisms, taking into account the forces acting on them. The dynamics of machines and mechanismsdeals with the following basic problems: definition of the laws of motion of the components ofmechanisms, control of the motion of the components, determination of frictional losses, determination ofthe reactions in kinematic pairs, and balancing of machines and mechanisms.

The laws of motion of the elements of a mechanism are determined from the given characteristics ofexternal forces using differential equations of motion of a mechanical system or of the machine unit,which usually consists of a motor, a transmission mechanism, the working machine, and in some cases, acontrol device. The number of equations is equal to the number of degrees of freedom of the mechanicalsystem. For convenience in solving the problem, all forces and masses in plane mechanisms with a singledegree of freedom are reduced to a single link or point of the mechanism, which is called the reductionlink or reference point. The arbitrary moment applied to the reduction link is called the moment ofreduction. The moment of reduction is equal to the aggregate of all moments and forces applied to thelinks of the mechanism. The arbitrary moment of inertia of the reduction link is called the reduced momentof inertia. The kinetic energy of the reduction link is equal to the sum of the kinetic energies of all links ofthe mechanism. The reduced force and mass of the mechanism at the point of reduction are determinedanalogously (Figure 1, a):

where Mr is the reduced moment; Jr is the reduced moment of inertia; Pr is the reduced force; mr is thereduced mass; M1, M2, P2, and P3 are the moments and forces applied to the

Figure 1. Action of the forces and moments in a crank-and-slide mechanism (a) in the reduction link (b) and at the reference point(c): (1) crank, (2) tie rod, (3) slide, (M) reduced moment Mr, (A) fixed support

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links of the mechanism; 1and 2 are the angular velocities of the links; VH and vc are the velocities ofpoints B and Cof the mechanism; vS2is the velocity of the center of gravity of link 2; VK is the velocity ofpoint Kof application of the force P2; 2 is the angle between the vectors P2and VK; and 3 is the anglebetween the vectors P3 and vc. The equation of motion in this case is

that is, Mr depends in the general case on the time, position, and velocity.

The equations of motion are usually nonlinear. There are no exact methods for the solution of theseequations; there-fore, approximate graphic, graphic-analytical, and numerical integration methods areused. The determination of the law of motion of a mechanical system is more difficult if friction andclearances in the kinematic pairs and the variability of the masses of the links are taken into account. Insome casesfor example, in the study of transient processes in machines some of the external forcescannot be considered as fixed, since the motion of the mechanism may have a reverse effect on thecharacteristics of these forces. For example, under some conditions of high acceleration, it is impossibleto take the speed-torque characteristic of an electric motor as consisting of a fixed dependence of themoment on the motors shaft on the angular velocity, since electromagnetic processes in the motor may

significantly affect the moment. In this case, the differential equation for the electromagnetic processes inthe electric motor is added to the differential equations of motion of the mechanical system, and all theequations are solved simultaneously.

Problems of the regulation of the motion of machine assemblies and of their control are treated in controltheory. A distinction is made among unsteady, transient, and steady-state conditions of motion. Understeady-state conditions, the velocities of the points of the mechanism are either periodic functions of timeor position, or they remain constant. The control of steady-state motion consists in the maintenance of anangular velocity of the reduction link that does not exceed the permissible deviation from its value. Aspecial weight, the flywheel, is designed and installed in the machine for this purpose. The necessity ofcontrolling unsteady motion arises when, in spite of the nonperiodic variation of the external forces ormasses, the mean velocity of the reduction link must be maintained at a constant value. Specialautomatic regulators are installed on the machine for this purpose. In this case, the basic task is the

determination of the stability of motion of the machine-regulator system. However, if the velocity or otherparameter of any link must be changed in accordance with a fixed law (program), a programming deviceis installed in the machine. The programmed control of meat-cutting machine tools may serve as anexample of this type of control. A concrete problem analyzed by the control theory is the identification ofoptimum conditions of motion of the machine (optimum control). Examples are the determination of themotion involving the fastest transient regime at a limited acceleration (the optimum motion with respect tothe speed of operation) or of the motion involving a minimum of energy expenditure in the transientregime (the optimum motion with respect to losses).

The determination of losses other than working losses in machines reduces to the determination offrictional losses, which are basic in nature and affect the efficiency of machines and mechanisms. Thedegree of utilization of energy in a machine is estimated in terms of the mechanical efficiency.

The kinetostatic calculations involving mechanisms, which are performed for known conditions of motionof the mechanism, are performed by determining the reactions in kinematic pairs to all the given externalforces, as well as to the inertial forces of the links and the frictional forces in the kinematic pairs. Thevalues of these reactions are part of the calculation of the strength of the links and are required for theselection of bearings and for the calculation of their lubrication.

Machines and mechanisms are balanced by a rational selection and placement of counterweights, whichreduce the dynamic pressures in the kinematic pairs of mechanisms. In practice, machines are stabilizedby mounting on a foundation (prevention of vibrations) or by balancing the rotating masses. The inertial

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forces in modern high-speed machines attain high values. The forces of inertia, which vary in magnitudeand direction, disrupt the normal operations of the elements of the machine and are sources of vibrationsand noise, which adversely affect the operating personnel and prevent the normal functioning of othermechanisms and devices. In vibration machines, the conditions for the generation of intense vibrations intheir working elements are taken into account. Dynamic studies in machines are directly related to thecalculation of the strength and stiffness of machine elements, which are performed in connection with theselection of the dimensions and the design of parts. Methods for such calculations are usually describedin training disciplines, such as strength of materials, the dynamics of structures, and machine parts.

Dynamic studies are also performed for three-dimensional mechanisms with many degrees of freedom.The operations performed by systems of this type have a high degree of universality.