Simulation Modelling and Parametric Synthesis of Metal Working Machine Tools Mechatronic Systems of Surface Generation

This paper presents the principles of the designing metal working machine tools mechatronic systems of surface generation which are optimal from the point of view of reproducing given part profiles. It is shown that the degree of their optimality is determined by the proximity of their characteristics to the frequency characteristics of an ideal low-pass filter. Mathematical methods for building compact mathematical models of the drive hardware, adequately describing the inertial and elastic characteristics of the elements of its structure, is described. These models are convenient for simulating control loops of the machine tools feed drives. A method for the parametric synthesis of a two-mass mathematical model of a subordinate control system for a feed drive with an elastic-dissipative coupling between equivalent lumped masses which simulate a moving unit and rotating elements of the drive mechanism is proposed. The synthesis problem is reduced to solving a system of nonlinear equations. Examples to demonstrate the proposed methods and models capabilities are given. 1 Feed drives design optimization criterion Increasing productivity and ensuring high precision machining of parts are priority areas for improving metalworking machines. Part profile is the result of coordinated motions of tool and workpiece. In the CNC machine tools mechatronic systems of surface generation, the feed drives are the crucial elements which ensure coordination of their motions. They receive their job via information network of the controller. The change from mechanical links to data communication allowed to increase the equipment flexibility and part profile variety significantly, while simultaneously increasing the machining accuracy, due to the exclusion of the influence of the error of the mechanical links of kinematic chains which provide mortions coordinations in the manually operated machine tools [1–4]. However, axis drives errors and the information network latency have direct impact on the part accuracy. The relationship between movements of the machine components along generalized coordinates and paths of tool and workpiece relative motion is determined by dependencies: ( ) ( ) { } ( ) д вых r t f t r =  (1) and ( ) { } ( ) ( ) 1 вх o t f r t r − =  , (2) which are formalism for direct and inverse kinematics. Here ( ) , , д i i i r x y z  and ( ) , , o j j j r x y z  are radius-vectors of corresponding points of a part and an image in the global coordinate system Oxyz, ( ) { } ( ) ( ) 1 ,..., T вх вх вх m t q t q t r   =   and ( ) { } вых t r = ( ) ( ) 1 ,..., T вых вых m q t q t   =   are input and output vectors of the machine generalized coordinates associated with drives 1,...,m, t is a scalable parameter (time). The relation between input and output vectors of the machine generalized coordinates is: ( ) { } ( ) { } ( ) вых вх t f t r r = . (3) The analysis of expressions (1) to (3) allows us to identify the main directions of improving the accuracy of contouring. We can vary system of surface generation, transfer functions of drives and the control signal. However, all of these are by some means or other depend on the characteristics of the machine tool drives. The biggest challenge in designing a machine drive is the search for the optimality criterion of its design, since as well as in servo drive, the law of variation of the master control is unknown in advance. The CNC program is loaded into the machine tool immediately prior to part machining, therefore the criterion of adjacency of the machined contour and the programmed one cannot be used. Thus at the design stage, we can only make demands on the transfer function of the drive. © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 224, 05005 (2019) https://doi.org/10.1051/epjconf/201922405005 MNPS-2019


Feed drives design optimization criterion
Increasing productivity and ensuring high precision machining of parts are priority areas for improving metalworking machines. Part profile is the result of coordinated motions of tool and workpiece. In the CNC machine tools mechatronic systems of surface generation, the feed drives are the crucial elements which ensure coordination of their motions. They receive their job via information network of the controller. The change from mechanical links to data communication allowed to increase the equipment flexibility and part profile variety significantly, while simultaneously increasing the machining accuracy, due to the exclusion of the influence of the error of the mechanical links of kinematic chains which provide mortions coordinations in the manually operated machine tools [1][2][3][4]. However, axis drives errors and the information network latency have direct impact on the part accuracy.
The relationship between movements of the machine components along generalized coordinates and paths of tool and workpiece relative motion is determined by dependencies: which are formalism for direct and inverse kinematics.
The analysis of expressions (1) to (3) allows us to identify the main directions of improving the accuracy of contouring. We can vary system of surface generation, transfer functions of drives and the control signal. However, all of these are by some means or other depend on the characteristics of the machine tool drives.
The biggest challenge in designing a machine drive is the search for the optimality criterion of its design, since as well as in servo drive, the law of variation of the master control is unknown in advance. The CNC program is loaded into the machine tool immediately prior to part machining, therefore the criterion of adjacency of the machined contour and the programmed one cannot be used. Thus at the design stage, we can only make demands on the transfer function of the drive. However, the time delays between output and input signals in the machine drive do not play such a significant role as in the classic servo. It is enough to ensure the unit values of the transfer function modules, the equality of the time delays of the drives which are the components of the machine system of surface generation, and the equality of the time delays of the components of the output signal spectrum in the range of frequencies of the control signal in order to reproduce it without distortion.
The condition of undistorted drive response to an arbitrary input signal along the i-th generalized coordinate is: where t i ≥ 0 is time delay (group-delay distortion) at a given coordinate. Drive specifications must match those of an ideal low-pass filter. If 0 0 2 f ω π ≤ ≤ then its frequency response and phase response are ( ) 1 Since it is impossible to implement an ideal filter, the transfer function of each drive is to be searched for among the filters whose frequency characteristics are the best approximations of the useful component of the input signal that is ideal in the frequency range [5,6].
A straightforward decision is to choose a single reference transfer function ( ) э W s , that satisfies the expression a function that will ensure compliance with the requirements specified above and the subsequent structural and parametric synthesis of the corresponding drives.

Modeling drive mechanisms
The elastic properties and inertial characteristics of the mechanism have a significant effect on the dynamic processes in the feed drive, since the resonance phenomena arising in the mechanism impede the effective adjustment of its contours. Despite its importance, this problem is under-examined. The analysis of the results of experiments conducted at the test bench ( Fig. 1), showed that the frequency response spectrum of its speed loop taken from the motor sensor has a complex composition with a large number of resonance peaks, most of which are outside the loop bandwidth in the frequency range of 50 to 2500 Hz (Fig. 2).
Here you can also see the dips corresponding to the speed loop transfer function zeros. An increase in the gain of the controller is accompanied by the speed loop bandwidth widening and increasing of the amplitudes of the oscillations at frequencies outside of the bandwidth (Fig. 3). A further gain increase leads to a loss of circuit stability.   Simulation of the rotating components of the drive was performed with the ANSYS finite element analysis software (see the model layout in Fig.4). Waveforms of the torsional oscillations of mechanical components of the drive with a free and fixed rotor are shown in Fig. 5. The finite element model of the rotating component of the feed drive mechanism cannot be used directly to model its control loops, because of its bulkiness, redundancy and closeness. The most convenient for these purposes are discrete models shown in Fig. 6, which provide a satisfactory description of dynamic processes in the mechanical part with minimal resources [7,8]. Using this approach, we build a system of n equations in operator form. Here J 1 … J n-1 are equivalent moments of inertia of the rotating components of the mechanism, J n is equivalent moment of inertia of the translational component (     Fig.7a , it was found that the zeros of its transfer function correspond to the poles of the fixed rotor system (Fig.  7b). Here, G n (s), G n-1 (s), G n-2 (s) and G n-3 (s) are characteristic polynomials of chain systems of dimensions n, n-1, n-2 and n-3.
To find a relation between the finite element models of torsion systems and the chain models shown in Fig. 7, let's consider the peculiarities of formulating the eigenvalue problem for them. Neglecting damping, let's represent the model of the chain system ( Fig. 7a), in matrix form: are i-th eigenvalue and vector.

Elements of the tridiagonal matrix
[ ] are squares of partial frequencies where where j=1,…,m (m=2n-2). The partial frequency squared matrix of a system with a fixed rotor [ ] 1 n α − is obtained from it by deleting the last row and column. In solving the eigenvalue problem, an important advantage of symmetric tridiagonal matrices is the relative simplicity of finding the coefficients of the characteristic polynomial, which is used in some algorithms in the final stage of finding eigenvalues [9,10]. A characteristic polynomial of a system (9) with a symmetric general tridiagonal matrix is recursively determined by the sequence of polynomials where a r and b r where ar and br are the corresponding diagonal and off-diagonal elements of this matrix, 2,..., r n = . Assuming ( ) obtain the sequence of polynomials known as the Sturm sequence [9]. Each of these polynomials is a characteristic polynomial of the matrix , which can be written as The coefficients of the characteristic polynomial ( ) Here ( )  From the first two dependencies of the newly formed scheme, we can determine the squares of the partial frequencies α 1 and α 2 . From the last (n-1) of its dependencies we obtain the coefficents ( ) It is more convenient to divide the computational procedure into (n-2) cycles; within each of them we will perform the same steps. The computational algorithm in the i-th cycle, where i=1,…,n-2, can be represented as: Step 1.
Given that the number of equations (17) is one less than the number of variables, we choose one of them as a Table 1. Relationship between the indices of the squares of partial frequencies, moments of inertia and stiffness.
Using expressions (18), we build the correspondence table of indices 1. It follows from it that in equations (17), two different partial frequencies correspond to one moment of inertia or rigidity of the model. This correspondence is violated only for the first and last mass. Using it, we get from the relationship between α 2i-3 and α 2i-2 we obtain Since we can easily obtain information about the total moment of inertia of the rotating components of the drive using a solid modeling system, we will use this dependence to find the free parameter. The expression for calculating the moment of inertia of the first body is: The total rigidity of the chain system can also be expressed in terms of the moment of inertia of the first body Step one. Considering the obtained results, the method for determining unknown values of the moments of inertia of lumped masses and stiffnesses of elastic bonds of system (9) can be divided into 4 steps.
Nonzero eigenvalues of the rotating components of the drive with a free (poles pi) and a fixed rotor (zeros z i ).
Step two. Using nonzero eigenvalues of the poles and zeros, using the recurrence dependences (16) obtained from Newton's formulas for determining power sums of the roots of the polynomial, the coefficients of the characteristic polynomials (12) of the tridiagonal square matrixes of partial frequencies of systems with a free and fixed rotor are determined.
Step three. From the characteristic polynomials of these tridiagonal matrices coefficients obtained at Step two, the squares of partial frequencies are found using the computational procedure consisting of (n-2) cycles, within the framework of each of which the same steps described by the recurrence dependences (14) are performed.
Step four. Unknown moments of inertia of concentrated masses and stiffness of elastic bonds are determined by formulas (20) based on information about the total moment of inertia of the rotating elements of the drive hardware.
As an example of the application of the developed method, the model parameters of the rotating part of the mechanism of the experimental stand using the poles and zeros obtained both using finite element modeling and experimentally determined. The calculation results are shown in tables 2 and 3.     To obtain the frequency characteristics of the drive, a model of its hardware shown in Fig. 8 was used, which in combination with a rather simple model of drive control loops forms a mathematical model of a dynamic system of a multiloop feed drive. Fig. 9 shows a comparison of the simulation results obtained in the MATLAB environment with experimental frequency characteristics of the drive.

Parametric synthesis
Unlike torsional vibrations, the natural frequencies of the axial vibrations of the translational component are either within the bandwidth of the position loop or in close proximity to it and have a direct effect on the quality of machining. If the torsional stiffness of the rotating elements of the drive can be neglected, the entire drive hardware can be represented as two rotating masses that simulate a translational component and drive mechanisms brought to the motor shaft, and an elasticdissipative coupling between them. For this case, the block diagram of the closed-loop feed drive with current, speed and position loops, is shown in Fig. 10 The dependence of the reduced angle of rotation of the moving unit on the reference angle and the moment of resistance corresponding to this scheme has the form: where ( ) ( ) The characteristic polynomial of the transfer functions of system (21) is of the form: If we design the drive mechanism, power unit and control system simultaneously, it is possible to vary the entire design considering the limitations caused by energy feasibility, which provides an intangible advantage in finding solution to the problem.
The parametric drive synthesis is based on equating the coefficients of the characteristic polynomial (22) of system (21) to the coefficients of the reference polynomial ( ) . After changing the variables in (22) and equating the coefficients for the corresponding degrees of the polynomials, we obtain the following system of equations:

T T T T T T a T T T T k T T T a T T T k T T k T T T a T T k T T k T T T a T k T T T T T a T T T a T T a
Converting the variables T m , T min , T J , T рс , T c , T b and T рп of system (24) into a dimensionless form by multiplying them by ω ν , we obtain the following system: Given the value of the base constant T m , which determines the speed of the drive control system, we can go to the initial set of variables of system (24), and then calculate the design parameters of the drive.
An advanced methodology for designing a feed drive with a rolling screw-nut transmission includes 4 stages as follows.
At the first stage, based on information about the mass, speed of movement of the actuating unit and the external loads acting on it, the reduction coefficient of the traction device and the complete drive are selected using traditional methods. The components of the drive have the required speed and power characteristics.
At the second stage, based on a requirements analysis of accuracy and dynamic quality of the equipment, a reference polynomial is selected and a verification of its correspondence with the drive speed is performed. Preliminary estimates of stiffness, moment of inertia of the rotating components and damping of the drive mechanism are calculated.
At the third stage, using the information on the maximum travel, speed, acceleration and dimensions of the actuating component, the design of the following units is selected: rolling screw-nut transmission, bearings, clutch or additional gearbox. Then, the traction device is designed with evaluation of its rigidity and the moment of inertia of the drive mechanism rotating components. The results are compared with the calculated values. If the difference is significant, changes in the mechanical design are made. For this purpose, a specialized software package is used, which allows performing the calculation of dynamic characteristics and selecting the main structural elements of the designed feed drive in interactive mode.
At the final stage, the Simulink environment simulates the designed drive. The correspondence between its frequency characteristics and the characteristics of the reference polynomial is checked. Performance and overshoot are estimated. The influence of torsional vibrations in the mechanism on the dynamic characteristics of the drive is analyzed.
The design workflow is iterative. If problems arise, a transition to earlier stages of design is possible.  The results of the parametric synthesis of the test bench drive corresponding to the second stage of the design process for two types of motors (1FT7046-* and 1FK7061-*) and reference polynomials from Table 4 are presented in Table 5. The type and bandwidth of the frequency characteristics of the position loop shown in Fig. 11 depend only on the choice of the reference polynomial and the base constant T m . Consequently, choosing the same reference polynomials and complete drives with the necessary speed for all axes, we get the same transfer functions for their position contours. However, it should be borne in mind that the distribution of the moments of inertia of the rotating drive elements forms an unpredictable resonant picture of torsional vibrations. To control this situation, it is necessary to use a full-fledged multi-mass drive model.  A comparison of the experimental frequency characteristics obtained after suppressing resonance phenomena in the test bench drive mechanism with the simulation results (see Fig. 12) allows us to conclude that, thanks to a rational choice of the design parameters, it was possible to expand the drive bandwidth from 52 to 142 Hz.

Conclusions
1. The feed drive mechanism with rolling screw-nut transmission is a complex dynamic object. The resonance phenomena in this unit have a significant impact on the operational characteristics of the entire machine tool. In most cases, the resonance is caused by axial and torsional vibrations of the structural elements of feed drive. The natural frequencies of axial vibrations of the translational unit located in the bandwidth of the position loop from (30-40 Hz to 200 Hz) or in close proximity of it have a direct imact on the quality of machining process. Resonance phenomena during torsional vibrations of rotating structural elements of the traction device (screw, clutch, engine rotor, etc.) occur at frequencies from several hundred to several thousand Hz. They practically do not appear on the moving unit, but they limit the possibility of optimizing the dynamic characteristics of the drive, leading to a loss of stability of its speed loop during tuning.
2. A peculiarity of the dynamic system of the feed drive mechanism is the alternation of zeros and poles of the transfer function of the speed loop. The characteristic polynomials in the numerator and denominator of this transfer function form the Sturm sequence. The numerator of the transfer function of a mechanism with a free motor rotor is a characteristic polynomial of a system with a fixed rotor. The series of natural frequencies corresponding to the poles and zeros of the transfer function converge quickly and, starting from a certain frequency, practically coincide. This peculiarity explains the limited spectra of the resonant frequencies of real drives by the effect of mutual filtering, coincident poles and zeros of the transfer function.
3. The developed method of parametric approximation allows obtaining models of replacing chain systems with properties close to the original model. However, the finite element model of the rotating part of the drive mechanism, which does not take into account the joints between its elements, does not provide a complete quantitative correspondence between the calculation results and experimental data. Significantly better correlation with experimental data is provided by models of substitute chain systems obtained on the basis of experimental information. In this case, we can talk about parametric identification of the considered dynamic system. 4. The parametric synthesis of a two-mass model with an elastic-dissipative coupling between equivalent lumped masses which simulate a translational component and rotating elements of the drive mechanism consists in choosing its design parameters and making the regulators adjustments wchich provide a