Determination of elastic-damping characteristics of the adaptive sprung suspension system to develop the optimal control of the simulation stand

Full Text

Introduction The springing system, realizing the connection of the propellers with the body and other structural elements of the mobile engineering device (MES), transmitting disturbances from the support surface to the body, ensures smooth running, affects the operator's condition, including comfort during operation both in transport mode and technological. Springing systems for the most part consist of the following main structural elements: damping devices – shock absorbers of various types (providing dissipation of vibrations caused by overcoming MES irregularities); elastic elements that provide a rigid connection between the propellers and the frame (pneumatic, rubber, hydropneumatic devices, as well as springs, torsion bars are usually used); suspension travel limiters and stabilizers. The existing various types of suspensions, such as: dependent and independent, active, adaptive, etc. have a number of advantages and disadvantages. To control and evaluate the quality of any automatic control systems during development, factory testing, commissioning, diagnostics and repair during operation by simulating the physical processes of the object, theoretical calculations are carried out in this work in order to find the optimal control range of the simulation stand for studying the parameters of the active springing system. Optimization of the springing system, its operation parameters, as well as active regulation depending on the support surface – can significantly affect fatigue resistance, increase operating comfort by ensuring smooth running, reduce soil pressure, and increase the overall energy efficiency of the MEA. The aim of the work is to develop optimal operating conditions for the simulation stand control system, minimize vibrations and vibrations, and predict the behavior of the system in various operating conditions and modes. Materials and methods The development of optimal control is based on the analysis of scientific papers, including publications, scientific articles and other sources of information on approaches to developing optimal control of the actuator in the active suspension system in order to most effectively dampen vibrations arising from road irregularities. The method of theoretical calculation of the three-factor experiment was used, the main dependencies were obtained. A large number of scientific papers [1-3] devoted to the mathematical description of the MES as a dynamic system in terms of the springing system confirms the special relevance, expediency and high efficiency of improvement and optimization, including the installation of adaptive control systems for springing systems of mobile power facilities. According to the conducted research regarding the use of adaptive springing systems, the work [4] presents the high efficiency of the adaptive cabin springing system. In this work, based on the presented oscillograms, it can be concluded that the developed system is highly efficient, which allows reducing the oscillation amplitudes by up to 50% at various frequencies. The use of this system significantly affects the working conditions of the MES operator. In the work [5] devoted to the adaptive springing system, special attention is paid to the operation of various controllers that ensure effective damping of vibrations due to the impact of irregularities of the support surface on the MES propellers. The above calculations and graphs demonstrate the possibility of effective damping of vibrations by means of various adaptive control controllers. Damping of vibrations in suspension systems of attachments is also an important aspect in improving the dynamic properties of complete MEAS, which is confirmed in [6]. When solving problems of optimizing the springing system, in order to solve the optimization problem of developing optimal control in the springing system, the particle swarm optimization (PSO) method was used in [7,8]. In this method, each particle of the system performs the process of searching in the solution space, bringing the entire swarm closer to the optimum, depending on the PSO variant and parameter values [9, 10]. In this paper, the vector differential equation with the dynamic feedback control equation is reduced to a closed system. In [11-13], the authors propose to form a vector for optimal control of damping in the suspension, taking into account transients. The disadvantage of the proposed algorithms is characterized by an increased probability of self-oscillation and sliding modes in the controlled suspension. Nevertheless, the method used in this work has shown high efficiency in terms of damping vibrations compared to an uncontrolled suspension. A large number of works are devoted to the development and finding of the optimal control law for the adaptive springing system [14-16], based on various approaches, high efficiency is confirmed by modeling. In [17], a quasi-optimal function is modeled, which compares the optimal control law. After analyzing the above materials, it can be concluded that the expediency and high efficiency of adaptive control systems of the springing system. The use of these systems in various structural elements: the cabin springing system, attachments (rod sprayer), as well as in the suspension system of mobile power equipment demonstrates a significant decrease in oscillation amplitudes both under the direct influence of road roughness or rocking, and with subsequent attenuation of vibrations. The use of adaptively adjustable PE has not been implemented in the designs of springing systems for both multi-main vehicles and wheeled vehicles with the 4x4 formula, however, it is a promising direction for improving both the springing system separately and the parameters of the entire MEA when performing technological or transport operations. The developed optimal control approaches allow the actuators to provide the most correct and effective control effect. According to the totality of the optimal control models being developed, the formation of the objective function, these approaches implementing optimal control are time-consuming, which opens up the field for solving the task with an automated approach based on input parameters. For optimal control of the simulation stand, it is necessary to determine the dependence of the elastic-damping characteristics of the adaptive springing system of the MES on the speed of movement and the height of the irregularities of the support surface along which the MES moves. To conduct research, we have developed a mathematical model of the oscillatory system of an agricultural mobile energy vehicle with attachments for the study of its adaptive springing system, described in [18]. The substantiation of the range of changes in the elastic-damping and inertial characteristics of the oscillatory system of agricultural MEAS with mounted technological equipment for conducting research is presented in [19]. The study of the dynamic characteristics of agricultural mobile energy facilities made it possible to determine the values of the factors [20]. As a result, accelerations of individual sprung masses of the MES (the center of mass of the skeleton, the center of mass of the mounted machine, the center of mass and the cabin) were obtained using a mathematical model. Based on the simulation results, the obtained acceleration values made it possible to calculate the coefficients of acceleration and intensity changes. The acceleration coefficients K1din and K2din, respectively, are calculated for the mounted vehicle and the frame, and the intensity coefficient Kint is calculated for the cabin, The coefficient of acceleration change Kdin was calculated using the formula: where Au is the steady–state acceleration amplitude, m/s2; Am is the maximum acceleration amplitude m/s2. The Kint intensity coefficient was calculated using the formula: [21], [22]. where σz is the mean square deviation of vertical accelerations with harmonic oscillations, ω0 = 62.8 s-1 is the frequency of reduction, ω is the frequency of oscillations. Of all possible factors, three were identified: the stiffness of the PE suspension of the front axle of the MES, the height of the irregularity of the microprofile and the speed of the tractor. The levels and intervals of variation of the factors are shown in Table 1. Table 1 –Levels and intervals of variation of factors Name of factors Designation of Factors Code designation Range of variation Natural values corresponding to the levels of coded factors Upper (+1) Main (0) Lower (-1) Front axle suspension PE stiffness Cp1, N/m X1 190·103 460·103 270·103 80·103 The height of the unevenness of the microprofile is H1, m X2 1,5·10-2 4,0·10-3 2,5·10-2 1,0·10-2 Speed of movement Vtr , m/s X3 2.0 5.0 3.0 1.0 The transition from encoded values of x to natural X is performed according to the formulas: where X is the natural value of the i-th factor; f0 is the natural value of the main level of the i-th factor (at the zero level); iint is the interval of variation of the i-th factor. As a plan for a three-factor experiment (plan type 33), a second-order non-positional plan was selected, the matrix of which is presented in Table 2. Table 2 – Matrix of a non-positional three-factor plan of the second order Experience no. To obtain the matrix of results Y, data obtained using equations (1) and (2) were used. The coefficient of change in accelerations Kdin was calculated using the formula (1). The intensity coefficient Kint was calculated using the formula (2).

Results and discussion

For each experiment, three observations were obtained (y1=K1din, y2=Kint, y3=K2din), the values of which are presented in Table 3. Table 3 – Values of observations for all experiments Experience no. y1 y2 y3 1 0,294 0,055 -0,754 2 0,286 0,576 -0,707 3 0,224 0,636 -0,196 4 0,436 0,025 0,111 5 0,353 0,591 0,039 6 0,239 0,068 -0,904 7 0,482 0,258 0,149 8 0,414 0,032 0,050 9 0,972 0,283 0,887 10 0,353 0,591 0,039 11 0,484 0,035 0,031 12 0,678 0,273 0,532 13 0,366 0,064 -0,038 14 0,681 0,275 0,558 15 0,353 0,591 0,039 Due to the fact that there are three observations for each experiment, it will be necessary to obtain three regression equations. The general form of these equations according to the experimental plan is as follows: The coefficients b will change for each equation. All calculations are performed in the Mathcad software environment. To calculate the coefficients of the regression equation, the experimental and observation plan must be entered into the matrices X and Y. Thus, the formulas of the regression coefficients have the form: The matrix of coefficients in (Table 4) is obtained from formula (5). Table 4 – Coefficients b for each equation Coef no. ba bb bc 0 0,353 0,591 0,039 1 -0,093 -0,003 -0,383 2 -0,011 0,007 -0,039 3 -0,164 -0,111 -0,373 4 0,055 -0,283 0,065 5 0,079 0,015 -0,054 6 0,030 -0,007 0,024 7 -0,034 -0,135 -0,325 8 -0,009 -0,133 -0,100 9 0,208 -0,296 0,332 The following notation is used in Table 4: where ba are the coefficients of the regression equation for observations y1, bb are the coefficients of the regression equation for observations y2, bc are the coefficients of the regression equation for observations y3. In the future, the indexes a, b, and c will also indicate belonging to one or another equation. Next, it is necessary to check the adequacy of the regression equation. The Fisher criterion is used for this [23]: where Da is the variance of adequacy, Dy is the residual variance. The residual variance of Dy can be assumed conditionally, rather than calculated. This is possible because the sample of observations is not random, but calculated. It is assumed that Dy = 0.05. We will determine the difference of the coefficients in Mathcad and write it into the MD matrix using the formula MD=Y–Yr. The obtained values of yr and MD are presented in Table 5. To visually assess the adequacy of the equations, graphs with y and yr can be constructed (Figure 1). Graphs clearly showing the discrepancy between the observation curves - y and the curves obtained using the polynomial (calculated curves – yr) are shown in Figure 1. Formula (6) in the Mathcad system has the form: Table 5 – Values of the yr and MD matrices Yra Yrb Yrc Mda MDb MDc 1 0,261 0,045 -0,744 0,033 0,010 -0,010 2 0,173 0,596 -0,796 0,113 -0,020 0,089 3 0,337 0,616 -0,107 -0,113 0,020 -0,089 4 0,470 0,035 0,101 -0,033 -0,010 0,010 5 0,353 0,591 0,039 0,000 0,000 0,000 6 0,349 0,062 -0,765 -0,110 0,006 -0,139 7 0,519 0,254 0,089 -0,037 0,004 0,060 8 0,378 0,037 0,110 0,037 -0,004 -0,060 9 0,862 0,289 0,748 0,110 -0,006 0,139 10 0,353 0,591 0,039 0,000 0,000 0,000 11 0,408 0,052 -0,118 0,076 -0,016 0,149 12 0,675 0,287 0,582 0,003 -0,014 -0,050 13 0,370 0,050 -0,087 -0,003 0,014 0,050 14 0,757 0,259 0,706 -0,076 0,016 -0,149 15 0,353 0,591 0,039 0,000 0,000 0,000 The final values of the adequacy variance and the Fisher criterion are presented in Table 6. c) Figure 1 – Graphs of the accuracy (adequacy) of the equations: a) – for curve Y1, b) – for curve Y2, c) – for curve Y3 Table 6 – Values of Da and F for all observations Adequacy variance Daa Dab Dac 0.022 0.001 0.037 Fischer's Criterion Fa Fb Fc 0.441 0.014 0.739 To assess the adequacy of the equations, we define the Fisher criterion. The tabular value of the criterion is Ftab = 3.59. In our case, Fa< Ftab, Fb < Ftab, Fc < Ftab – therefore all three equations are adequate. Let's find the critical points for each equation – these are the values x1, x2, x3. To do this, take the partial derivatives of these variables and equate them to zero. In general, partial derivatives are represented by a system of equations (10). To calculate the roots of x in Mathcad, it is necessary to fill in two matrices: the matrix of coefficients for variables x (Ax) and the matrix of free coefficients (Bs). Since the initial formula for this calculation looks like Ax×X=Bs, and the available free coefficients are on the left side of the equation, to start the calculation, it is necessary to transfer the values of Bs to the right side, i.e. take these coefficients with a minus. Then, the matrices Ax and Bs will look like: Using the inverse matrix method, we obtain the equation X=Ax-1*Bs, the values of X for each equation are obtained (Table 7). Table 7 – X values for all equations № xa xb xc 1 1,417 0,460 -0,667 2 3,642 -0,457 -0,349 3 -0,139 -0,171 0,520 The analysis of Table 7 shows that the change in factor Xa and Xc has a strong effect on the indicator Y. Substituting the values of X into equation (4), we obtain the values of Y. Under these conditions, we obtain the values ya = 0.278, yb = 0.599, yc = 0.077. Using the data in Table 1, we can obtain the natural values of x for all observations, the values of which are listed in Table 8. Table 8 – Natural values of X for all equations № xa xb xc 1 539324 0,032 1,665 2 961899 0,018 2,303 3 243573 0,022 4,041 Further, analyzing the obtained regression equations, we construct graphs of the response surface and graphs of the level lines for each of the equations. In this case, the values of x are set, at which two variables change from -1 to 1, and one is equal to 0. All combinations are shown in Figures 2-10. Figure 2 – Graphs of the response surface for Y1 (a) and the level line (b) at x1=-1..1, x2=-1..1, x3=0. Figure 3 – Graphs of the response surface Y1 (a) and the level line (b) at x1=0, x2=-1..1, x3=-1..1. Figure 4 – Graphs of the response surface Y1 (a) and the level line (b) at x1=-1..1, x2=0, x3=-1..1. Figure 5 – Graphs of the response surface Y2 (a) and the level line (b) at x1=-1..1, x2=-1..1, x3=0. Figure 6 – Graphs of the response surface Y2 (a) and the level line (b) at x1=0, x2=-1..1, x3=-1..1. Figure 7 – Graphs of the response surface Y2 (a) and the level line (b) at x1=-1..1, x2=0, x3=-1..1. Figure 8 – Graphs of the response surface Y3 (a) and the level line (b) at x1=-1..1, x2=-1..1, x3=0. Figure 9 – Graphs of the response surface Y3 (a) and the level line (b) at x1=0, x2=-1..1, x3=-1..1. Figure 10 – Graphs of the response surface Y3 (a) and the level line (b) at x1=-1..1, x2=0, x3=-1..1. The resulting equations have a high level of significance. The graphical representation allows for a qualitative assessment of the accuracy of the equations obtained. The equations allow you to control the parameters of the adaptive springing system according to several criteria. Each criterion included in the equation has an impact on the function itself. With substituted coefficients, equation (4) has the form (13)-(14). For the backbone of the MES To develop optimal control of the simulation stand, the obtained equations should be used in software development. These equations show not only the optimal parameters, but also the patterns of changes in the elastic-damping characteristics in the springing system of agricultural wheeled MES of traction class 2.0-3.0. Knowledge of the operating parameters allows us to develop laws of adaptive regulation depending on changes in the support surface. According to the totality of the obtained models (equations (13)-(15)), optimal control of the stand will be formed to solve the task – minimizing the level of vibration (level of displacement, speed and acceleration) transmitted through the springing system to the machine frame, by choosing the optimal control of the adaptive springing system. Conclusions: For optimal control of the simulation stand, the functional dependences of the elastic-damping characteristics of the adaptive suspension system of the wheeled MES on the speed of movement and the height of the irregularities of the support surface were determined. The conducted research on the development of optimal operating conditions for the control system of the simulation stand, minimizing fluctuations, predicting the behavior of the system in various conditions and operating modes allows us to draw the following conclusions: 1. The regression equations have a 95% confidence probability, and show relationships between independent variables such as speed, microprofile height, and suspension stiffness. 2. The parameters obtained as a result of calculations have the following values: a) the coefficient of acceleration change for a mounted agricultural machine is the parameter K2din = 0.077, while the stiffness of the pneumatic suspension element of the front axle of the tractor is Cp1, = 243573 N/m, at a speed of Vtr = 4.04 m/s, while the height of the unevenness of the microprofile H1, = 0.022 m, b) acceleration coefficients for the center of mass of the frame - parameter K1din = 0.278, while the stiffness of the pneumatic suspension element of the front axle of the tractor is Cp1, = 539324 N/m, at a speed of Vtr = 1.67 m /s, while the height of the irregularity of the microprofile is H1, = 0.032 m, c) the intensity coefficient for the cabin – Kint, = 0.599, while the stiffness of the pneumatic suspension element of the front axle of the tractor is Cp1, = 961899 N/m, at a speed of Vtr, = 2.30 m/s, while the height of the irregularity of the microprofile is H1, = 0.018 m, 3. The obtained equations and results are necessary for the development of software and the selection of characteristics of the simulation test bench for adaptive

About the authors

Zakhid Adygezalovich Godzhaev

FSAC VIM
Federal Scientific Agroengineering Center VIM

Email: fic51@mail.ru
ORCID iD: 0000-0002-1665-3730
SPIN-code: 1892-8405

Doctor of Technical Sciences, Professor, Corresponding Member of the Russian Academy of Sciences, Head of the Mobile Energy Vehicles Department;

Russian Federation, 1-st Institute driveway, b 5, Moscow, 109428

Sergey Evgenevich Senkevich

FSAC VIM
Federal Scientific Agroengineering Center VIM

Email: sergej_senkevich@mail.ru
ORCID iD: 0000-0001-6354-7220
SPIN-code: 7766-6626

Candidate of Technical Sciences, Associate Professor, Head of the Laboratory "Automated Drive of Agricultural Machinery", Senior Researcher;

1-st Institute driveway, b 5, Moscow, 109428

Ivan Sergeevich Malakhov

Federal Scientific Agroengineering Center VIM

Author for correspondence.
Email: malahovivan2008@mail.ru
ORCID iD: 0000-0001-8162-7718
Russian Federation

Ekaterina Nikolaevna Ilchenko

FSAC VIM
Federal Scientific Agroengineering Center VIM

Email: kat-sama@mail.ru
SPIN-code: 5672-1313

Engineer of the laboratory "Automated drive of agricultural machinery"

Russian Federation, 1-st Institute driveway, b 5, Moscow, 109428

Sergey Yurievich Uyutov

FSAC VIM
Federal Scientific Agroengineering Center VIM

Email: s_uyutov@mail.ru
SPIN-code: 7350-1489

Junior Researcher of the laboratory "Automated drive of agricultural machinery"

1-st Institute driveway, b 5, Moscow, 109428

References

Supplementary files