SUPERPOSITION OF MOTIONS IN THE SPACE OF LAGRANGE VARIABLES

The technique of superposition of motions in the space of Lagrange variables is described, which allows us to obtain the equations of combined motion by replacing the Lagrange variables of superimposed (external) motion with Euler variables of nested (internal) motion. The components of velocity and acceleration in the combined motion obtained as a result of differentiating the equations of motion in time coincide with the results of vector addition of the velocities and accelerations of the particles involved in the superimposed motions at each moment of time. Examples of motion and superposition descriptions for absolutely solid and deformable bodies with equations for the main kinematic characteristics of motion, including for robot manipulators with three independent drives, pressing with torsion, bending with tension, and cross– helical rolling, are given. Given the fragment of calculation of forces in the kinematic pairs shown the advantages of the description of motion in Lagrangian form for the dynamic analysis of lever mechanisms, allows to determine the required external exposure when performing the energy conservation law at any time in any part of the mechanism.


Features of the description of motion in the form of Lagrange
Any motion of a solid or deformable body can be described in the form of Lagrange ( , ) The different nature of the independent arguments of equations (1) leads to the need to use two differential operators for time (d) and space (δ). The first characterizes the change of any property associated with motion for a particle over an infinitesimal period of time dt, for example coordinates The second one defines increments of motion-related properties for adjacent particles whose In accordance with equations (1), in the general case, the motion is characterized by 12 first derivatives, including the components of the particle velocity The deformation of the particle is characterized by the average value and the standard deviation of the relative lengths of the edges (3) from their average value (4) The quadratic invariant of the tensor (2) remains unchanged when the coordinate axes are rotated, is a generalized measure of elastic deformation [5] In the right-hand sides of the last two equations, the summation rule for the index repeated in the monomial is used.
The determinant of the matrix (2) or the Jacobian of the transformation (1) numerically coincides with the ratio of the particle volumes in the current V  and initial If the values of ep are different from 1, the shape and / or volume of the particles change, the movement is accompanied by deformation. If there is no deformation. Such cases correspond to the motion of absolutely solid bodies.
The body associated with the concept of "material point" differs from a real solid body the domain of the definition of Lagrange variables for which it is contracted to a point. However, this transition is not always appropriate. In particular, the analysis of the kinematics of rigid bodies with the actual scope of Lagrangian variables allows to account for the rotation of the body during its motion, to identify possible errors by the description of the equations of motion in the absence of strain and move by one form of description of motion to another.

The principle of superposition
In accordance with the generally accepted concepts [1][2][3], the superposition of motions is reduced to determining the velocity fields for each of the motions, summing them according to the rules of vector algebra, and then integrating the velocities over time to determine the position of particles in combined motion.
The final result can be obtained much easier if the combined movements are written in the Lagrange form. Then, in order to provide a vector addition of velocities at each moment of time, it is sufficient to replace the Lagrange variables of one motion with the Euler variables of another motion [4][5].
Formally, this replacement can be explained by a simple transition to new Lagrange variables, which take the coordinates of the particles at the current time. Motion can be considered as consisting of a number of stages, each of which can use either a new or a single frame of reference of the observer. In accordance with this, Lagrange variables can be introduced both independently at each stage, and for a sequence of several stages.
Consider two consecutive stages of arbitrary motion of a continuous medium on the time interval 0  t  t2.
Let the first stage be bounded by the time interval 0  t  t1 < t2 for the equations of motion where αp are Lagrange variables, (11) At the second stage t1  t  t2 you can enter a new time frame and new Lagrange coordinates k  that coincide with the current coordinates (11) at time t1 and allow you to describe the further process of movement in the form the time t1 in the second stage should be considered as a constant.
However, as Lagrange variables, you can take not only the initial coordinates, but also any functions of the initial coordinates that are uniquely related to them. In particular, relations (11) or equations of motion (10) can be chosen as such functions for t = t1. Formally, in this case, equations (12) preserve the form ) ), but the Lagrange variables should be considered as functions of the original variables p  in accordance with equations (11).
On the other hand, equations (12) can be extended to the second stage without changing the time frame, it is enough only to limit the time interval of this stage in the original time scale The time t1 remains constant also in this case, if we consider the successive stages of movement. Relations or various components of the movement ui(t) along the coordinate axes during spatial translational motion [4][5].
The considered principle is also applicable to the simultaneous flow of combined movements, since the intervals of stages can be infinitely small and at each of them the movements from both types of movement will be summed up.
The form of the resulting motion (14) depends on the Lagrange coordinates of which of the motions will be replaced by Euler variables. For the sake of definiteness, in the future we will call nested or internal motions , which replace the Lagrangian coordinates of another motion. The movement ( , ) which this replacement occurred, respectively, will be called superimposed or external. External and internal movements are similar to portable and relative ones in classical mechanics, but differ in the mandatory use of a single coordinate system.
in their sequential or simultaneous flow coincide with the equations of external motion after replacing the Lagrange variables with the corresponding equations for the Euler variables of internal motion The difference between nested and superimposed motions can be insignificant when they are equivalent, for example, when rotating relative to two axes, or in processes of deformation, when the displacements from each type where the first term and the left multiplier refer to the portable (superimposed) movement, while the right multiplier is determined by the nested movement.
The components of the velocity of the external motion (the first terms) must be determined at any moment in time at the point in the observer's space that the particle will actually occupy, taking into account the two considered motions. Relations (16) can be interpreted geometrically as a vector sum [7] of the velocities of nested and superimposed motions, as can be seen in concrete examples with different types of motions.
The equations for the acceleration components are not given, but the validity of the general statement about geometric addition is not in doubt, since the derivative of the velocity with respect to the scalar argument (time) does not change the rank of the tensor [7].

Examples of superposition of motions for absolutely rigid bodies
To write down the equations of motion in the form of Lagrange, it is enough to compare the position of the particles of the body at the initial and arbitrary moments of time. In particular, for the translational movement of a solid body along any spatial trajectory, the displacement projections for all particles must be the same matrix (2) takes the form Condition (18) and the current time where Δφ is the rotation angle of the body. Excluding from the last equations the length L0 and the initial value of the angle 0  using equations (19), we obtain the system (the z coordinate does not change) From here, equations for rotation with respect to the origin can be obtained  allows us to obtain equations for two types of motion. If we consider the motion (21) to be external, we obtain the equations of translational motion of a body along a wheel rotating relative to the origin, If we consider the motion (22) to be external, then the equations will correspond to the rotation of a solid body relative to a moving pole ( )cos ( )sin In these equations, the coordinates of the pole xP, yP and the angle of rotation of the body () t  depend on time. If they are differentiated by time, we obtain the relations between the velocities of the pole and an arbitrary particle, in particular for the system (23) Equations (21)-(24) are sufficient to describe any motion of absolutely rigid bodies. For plane-parallel movements in other planes, they can be obtained by circular substitution, and for more complex spatial movementsdue to the superposition principle, which allows us to consider spatial processes as the simultaneous implementation of several plane-parallel movements [6]. In particular, for rotation around the x and y axes, taking into account (23) and replacing the designation of the rotation angles, we obtain Matrix (2) for system (23) takes the form typical for rotations , cos sin 0 sin cos If two motions in the x-y plane are rotational with respect to parallel axes, for example, a nested rotational with respect to a point with coordinates (c, 0) or, after multiplying the matrices, The equation for acceleration can be obtained by repeated differentiation of the components of the velocity. In all the cases considered, the rule of algebraic summation of the same-name projections of velocities and accelerations for nested and portable motions at each moment of time is fulfilled.
If a body rotating around the z axis in accordance with the system (21) receives an additional rotation around the х1 axis passing through the movable pole P parallel to the x axis of the observer's coordinate system in accordance with the system (25), then to describe the joint rotation, it is sufficient to substitute the right part for the Eulerian coordinates x and y from equations (21) instead of the Lagrangian coordinates  and  in equations (25). As a result, we get If the rotation (25) occurs relative to the x-axis of the observer's coordinate system, then 0 P P P P yz = = = = and instead of the previous system, we obtain the equations of combined rotation (simultaneously or sequentially determine the time dependences of the angles Δφ(t) and Δψ(t)) In the case under consideration, the rotations relative to the z and x axes of the observer's coordinate system become equivalent and any of them can be taken as external. If we take the external rotation (21), and the internal rotation (25) at 0 P P P P yz = = = = , then as a result of the superposition, instead of (27) we get a new system of equations cos ( cos sin )sin However, the numerical results of calculations with the coincidence of the corresponding coordinates of the particles and the angles of rotation of the body relative to both axes will be the same. If one of the rotations is stopped, for example, take Δφ = 0, both systems will coincide.
The composite motion (27)  As in other cases, conditions (9) are met, the lengths of edges and particle volume do not change, and rotation occurs without deformation of the particles.

Superposition of movions as a stage of dynamic analysis of mechanisms
The superposition of plane-parallel motions makes it possible to describe the spatial motion of particles of links of hinge-lever, cam and other types of mechanisms of any complexity [6,8], for example, presented in Figures 1 and 2, for subsequent dynamic analysis.
Link 2 rotates relative to the y1 axis, using the drive in a kinematic pair with the point А(αА =0, 0, γА=H), in accordance with equations (26) The subscript after the bracket specifies the number of the link for which they are applicable in the combined movement.
Using rule (15) again, we obtain the equations of motion for link 3, taking into account the internal motion (31) and the external one (32) The block diagram, initial position and coordinate system for the second mechanism are shown in Fig. 2 As noted above, the correctness of the record in all cases of motion of absolutely solid bodies can be checked by fulfilling the condition of constancy of the volume of particles and the absence of deformation (9).
The use of equations of the considered type for mechanisms of various complexity makes it possible to find forces in any cross-sections and kinematic pairs [7] that satisfy the law of conservation of energy at any time on any part of the mechanism from the energy balance whereWe is the power of external forces, Wk and Wp are the rates of change in the kinetic and potential energy of the mechanical system under consideration. The kinetic energy for spatial motions can be determined by the Koenig , , 0,5 0,5( ) The lower indices indicate the point of the block diagram at which the masses mK and moments of inertia  Differentiating the system (37) twice in time and equating the coefficients in both parts of equality (39), for the power factors included in the left part, we obtain From the energy balance for link 2 we find For link 1, which performs translational movement relative to the z axis, the energy balance includes the rate of change of potential and kinetic energy from translational motion, as well as the power of forces () iK Q from an external energy source at point A and transmitted to other links of the mechanism at point B The forces at point A of link 1, taking into account the energy balance (40), are Dynamic analysis should start from the links that are most remote from external energy sources, then the force factors at point B will be calculated beforehand and equation (40)  The energy balance of type (38) can be recorded for any part of the mechanism, including the entire structure from the rack to the working tool at point E. However, in this case, the power of the local drives, as well as the forces and moments in the kinematic pairs of the mechanism remain unknown.
Taking into account that the principle of superposition in the space of Lagrange variables provides for the use of only one common reference system of the observer, such a description eliminates the problems of non-inertial coordinate systems.

Superposition of motions for processes of deformation
The superposition principle is applicable to both absolutely solid and deformable bodies, and in the latter case, taking into account small displacements, the choice of external and internal movements often (except for cases of steady-state deformation processes) does not matter. The resulting equations may take different forms, but the numerical results are almost identical. The characteristics of the deformed state at an arbitrary time interval can be found in terms of the derivatives of the combined motions according to the equations (3)-(8) [5][6].
If the characteristics of the deformed state are known at each stage, the product of the values of the corresponding invariants can be used for the ratio of volumes 2 02 01 12 0 For other characteristics of the deformed state, the additivity condition can be used.
One can also verify the validity of these statements in the field of elastic deformations by the example of joint stretching with equations (41) and twisting with equations Taking into account the peculiarities of the processes under consideration, either Cartesian or cylindrical coordinate systems can be used. The main relations for calculating kinematic and energy parameters during the transition from a Cartesian system to a cylindrical one are given in [5][6].
For example, in the region of large plastic deformations for a homogeneous precipitation of a cylindrical sample (in the absence of friction on the contact surfaces), the equations of motion instead of (41) can be written as [5]  As noted above, uniform deformation provides sufficient conditions to meet the equilibrium conditions. In cases of inhomogeneous deformation, such as bending, additional verification of the energy conservation law is required.
For the combined bending of a wide strip with its stretching, you can use the equations for bending [5] sin( / ) ( 2 ) x r r r =  +  , cos( / ) ( 2 ) y r r r r where r is the radius of curvature of the layer with the coordinate