Keywords

1 Introduction

Changes of requirements for asphalt pavement quality create conditions for the implementation of intelligent and “smart” technologies in the road building industry. The main stages of the asphalt pavements construction process affect coating performance [1]: manufacture, transportation from the plant to the facility, spreading, mixture placing, and compaction. Mixture compaction by pavers and road rollers eliminates up to 50% of the disadvantages of asphalt pavements and increases their service life [2].

Systems of continuous non-destructive compaction control of hot mix asphalt with intelligent compaction technologies, continuous compaction control [3,4,5,6,7,8,9], installed on vibratory rollers, and a neural network automatic control system [10, 11] are modern methods of compaction control of road surfaces.

Known methods and control means have a significant error, do not allow to evaluate the efficiency of the rolling process, and are not able to record the compaction characteristics [12]. These drawbacks are confirmed in many scientific papers [12,13,14].

Over the 40 years of the existence of the idea of continuous compaction control, the developers of automatic systems have not come up with effective means of reducing drawbacks in measuring the compaction degree of hot mix asphalt. The main variable of control systems is the acceleration of the vibratory roller. However, the material characteristics during compaction are not constant. The structure of the road-building mixture, humidity, temperature of the layer, stiffness of the base, layer thickness are changing. In well-known developments of continuous compaction control systems of leading companies such as Trimble, MOBA, Topcon, Hydac, etc., there are no data on the consideration of several significant factors in predicting the compaction effectiveness. Thus, the problem of increasing the efficiency of continuous non-destructive compaction control, taking into account significant factors of the changing properties of the material, strain–stress state, requires a solution.

The relevance of solving this problem increases in connection with the development of the scientific field—cybernetic physics [15]. The automation level of road-building machines—objects of cyber-physical systems (CPS), should be high. The performance of the entire CPS depends on the quality of measurements and forecasts of the parameters of the automated control system. We can state the presence of developed mathematical models of the compaction process by vibratory rollers [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], and oscillating rollers, which allows solving current issues of non-destructive technologies in the road construction field.

The set of road construction machines providing compaction of hot mix asphalt, in addition to road rollers, includes pavers that can compact the material to standard values when equipped with a highly effective working body. Depending on the achieved compaction factor, we can reduce the number and types of road rollers, the number of passes after the stacker. These technological measures boost productivity and cost reduction of road construction. But it is impossible to completely exclude the use of existing road rollers with existing technologies isn’t possible, because they are necessary to confirm the achieved compaction coefficient of the material and increase the structural strength of asphalt concrete. To increase the efficiency and productivity of the compaction processes of pavements, it is necessary to carry out continuous non-destructive compaction control and regulation of operating modes—tamping frequency, frequency of plate vibration, impulse movements frequency of the pressing bars, taking into account the technological capabilities of modern pavers. Currently, such automatic control and management systems for pavers are not available. Having such systems, it is possible to create a “smart” paver with a significant increase in productivity by optimizing the compaction process and reducing the workload of the driver. To increase the effectiveness of scientific research and the design of objects of cyber-physical systems, mathematical models of real processes are needed.

This work is devoted to the theoretical description of the object of continuous compaction control of a cyber-physical road-building system—an asphalt paver with a highly effective working body. Theoretical studies on the compaction of various materials (soils, asphalt mixtures) have been and are being carried out by many Russian [16] and foreign scientists [17,18,19,20]. The disadvantages of previously developed mathematical models are the problems of algorithmization of tasks, computational difficulties in the study of dynamic systems, the design of automatic control and management systems.

To eliminate these drawbacks and to increase the efficiency of theoretical research of the control object using the modern MATLAB/Simulink software, it is recommended to use the state space method, which allows making clear characterization and automation of computational procedures [21,22,23].

2 The Mathematical Formulation of the Problem

For research, a highly effective working body was taken in the following composition of the compacting units: tamper—vibration plate—two pressing bars. The compaction process of the mixture is carried out with constant contact of the vibrating plate with the mixture. The main compaction is performed by compressing the mixture with a horizontal platform of a tamping beam when moving in a vertical plane for 4–6 effects. A vibrating plate and pressing bars fix the achieved compaction result and improve the structure of the material. Pressing bars creates periodic impulse loadings with a frequency of 50–70 Hz and pressure in the hydraulic system from 5 to 15 MPa [24].

The dynamic parameters of compacting units and particles of material are frequency, amplitude, speed, and acceleration.

A simulation model of a dynamic model of the compaction process of the mixture with a highly effective working body of the paver is obtained based on the technical documentation [24], Fig. 1.

Fig. 1
figure 1

The dynamic model of the mixture compaction process by the working body (tamper, vibration plate, two pressure bars) of the paver

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On the Fig. 1 the following notation is used: \(m_{1}\)—mass of the smoothing plate frame, kg; \(m_{2}\)—weight of vibrator, kg; \(m_{3}\)—weight of tamper, kg; \(m_{4}\)—weight of the first pressure bar, kg; \(m_{5}\)—weight of the second pressure bar, kg; \(m_{6}\)—weight of the mix under the tamper, kg; \(m_{7}\)—weight of the mix under the vibration plate, kg; \(m_{8}\)—weight of mix under the first pressure bar, kg; \(m_{9}\)—weight of the mix under the second pressure bur, kg; \(k_{1}\)—coefficient of elastic resistance of the compacted mix under the plate, N/m; \(c_{1}\)—damping coefficient of the compacted mix under the plate, N s/m; \(k_{2}\)—coefficient of elastic resistance of the shock absorbers of the vibrator, N/m; \(c_{2}\)—damping coefficient of shock absorbers of the vibrator, N s/m; \(k_{3}\)—coefficient of elastic resistance of the compacted mix under the tamper, N/m; \(c_{3}\)—damping coefficient of the compacted mix under the tamper, N s/m; \(k_{4}\)—coefficient of elastic resistance of the compacted mix under the first pressure bar, N/m; \(k_{5}\)—coefficient of elastic resistance of the compacted mix under the second pressure bar, N/m; \(c_{4}\)—damping coefficient of the compacted mixture under the second pressure bar, N s/m; \(k_{14}\) and \(k_{15}\)—coefficient of elastic resistance of the first and second pressure bars, N/m; \(c_{14}\) and \(c_{15}\)—damping coefficient of the first and second pressure bars, N s/m; \(y_{1} ,\,y_{2} ,\,y_{3} ,\,y_{4} \,\) and \(y_{5}\)—the moving system elements, respectively, m.

We developed a mathematical model that presents both the vibration dynamics of structural elements and the rheological properties of the compacted material. Differential equations describing the movement of the first and second pressure bar

$$\begin{aligned} \left( {{m_4} + {m_8}} \right) \cdot {{\ddot y}_4} &\quad - \,{c_{14}} \cdot {{\dot y}_1} + ({c_4} + {c_{14}}) \cdot {{\dot y}_4} - {k_{14}} \cdot {y_1} \\ + \,\left( {{k_4} + {k_{14}}} \right) \cdot {y_4}{\mkern 1mu} = {F_4} + {m_4} \cdot g + {m_8} \cdot g{\mkern 1mu} ,\\ \end{aligned}$$
(1)
$$\begin{aligned} \left( {{m_5} + {m_9}} \right) \cdot {{\ddot y}_5} &\quad - \,{c_{15}} \cdot {{\dot y}_1} + \left( {{c_5} + {c_{15}}} \right) \cdot {{\dot y}_5} - {k_{15}} \cdot {y_1} \\& + \,\left( {{k_5} + {k_{15}}} \right) \cdot {y_5} = {F_5} + {m_5} \cdot g + {m_9} \cdot g{\mkern 1mu} , \\ \end{aligned}$$
(2)

where \(F_{4}\), \(F_{5}\) is the force of the first and second pressure bar, respectively, N.

The differential equation of the smoothing plate vibrator

$$m_{2} \cdot \ddot{y}_{2} - c_{2} \cdot \dot{y}_{1} + c_{2} \cdot \dot{y}_{2} - k_{2} \cdot y_{1} + k_{2} \cdot y_{2} = m \cdot \omega_{2}^{2} \cdot r \cdot \sin \left( {\omega_{2} \cdot t\,} \right)\,,$$
(3)

where \(F_{3}\) is the force of tamping beam ram, N.

With allowance for the principle of relative motion, we obtain an additional equation

$$y_{3} = y_{1} + e \cdot \sin \left( {\omega_{3} \cdot t} \right)\,,$$
(4)

where \(e\)—tamper eccentricity size, m; \(\omega_{3}\)—the rotational speed of the tamper gear, rad/s.

Substituting Eq. (4) into (3) with transformations, we obtain the following equation

$$\begin{aligned} F_{3} &= \left( {m_{3} + m_{6} } \right) \cdot \ddot{y}_{1} + c_{3} \cdot \dot{y}_{1} + k_{3} \cdot y_{1} - \left( {m_{3} + m_{6} } \right) \cdot e \cdot \omega_{3}^{2} \cdot \sin \left( {\omega_{3} \cdot t} \right) \hfill \\& \quad + \,\,k_{3} \cdot e \cdot \sin \left( {\omega_{3} \cdot t} \right) + c_{3} \cdot e \cdot \omega_{3} \cdot \sin \left( {\omega_{3} \cdot t + \pi {/}2\,} \right)\, - m_{6} \cdot g. \hfill \\ \end{aligned}$$
(5)

The differential equation of smoothing plate motion

$$\begin{gathered} \left( {m_{1} + m_{7} } \right) \cdot \ddot{y}_{1} + \left( {c_{1} + c_{2} + c_{14} + c_{15} } \right) \cdot \dot{y}_{1} - c_{2} \cdot \dot{y}_{2} \, - c_{14}^{{}} \cdot \dot{y}_{4} - c_{15}^{{}} \cdot \dot{y}_{5} \hfill \\ \,\,\,\,\,\,\,\,\, + \,\,\left( {k_{1} + k_{2} + k_{14} + k_{15} } \right) \cdot y_{1} - k_{2} \cdot y_{2} - k_{14}^{{}} \cdot y_{4} - k_{15}^{{}} \cdot y_{5} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - F_{3} - F_{4} - F_{5} + \left( {m_{1} + m_{7} } \right) \cdot g\,. \hfill \\ \end{gathered}$$
(6)

Substituting Eq. (5) into (6), we obtain the following differential expression

$$\begin{gathered} \left( {m_{1} + m_{7} + m_{3} + m_{6} } \right) \cdot \ddot{y}_{1} + \left( {c_{1} + c_{2} + c_{3} + c_{14} + c_{15} } \right) \cdot \dot{y}_{1} - c_{2} \cdot \dot{y}_{2} - c_{14} \cdot \dot{y}_{4} - c_{15} \cdot \dot{y}_{5} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\left( {k_{1} + k_{2} + k_{3} + k_{14} + k_{15} } \right) \cdot y_{1} - k_{2} \cdot y_{2} - k_{14} \cdot y_{4} - k_{15} \cdot y_{5} \, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - F_{4} - F_{5} + \left( {\left[ {m_{3} + m_{6} } \right] \cdot e \cdot \omega_{3}^{2} - k_{3} \cdot e} \right) \cdot \sin \left( {\omega_{3} \cdot t} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - c_{3} \cdot e \cdot \omega_{3} \cdot \sin \left( {\omega_{3} \cdot t + \pi {/}2} \right)\, + \left( {m_{1} + m_{6} + m_{7} } \right) \cdot g. \hfill \\ \end{gathered}$$
(7)

3 Methods and Materials

The state-space method allows you to represent the control system in the form of an equations system [21, 22]:

$$\dot{x}(t) = A(t) \cdot x(t) + B(t) \cdot u(t),$$
(8)
$$y(t) = C(t) \cdot x(t) + D(t) \cdot u(t),$$
(9)

where \(x(t)\) is the state vector of dimension \((n \times 1)\), whose components are state variables of the n-th order system, \(x(t) = \left[ {x_{1} (t),\,\,x_{2} (t),...,\,\,x_{n} (t)} \right]^{T}\), \(y(t)\) is the dimension output vector \((p \times 1)\), whose components are the output variables of the system, \(y(t) = \left[ {y_{1} (t),\,\,y_{2} (t),...,\,\,y_{p} (t)} \right]^{T}\), \(u(t)\) is the dimension input vector \((r \times 1)\), whose components are the output variables of the system \(u(t) = \left[ {u_{1} (t),\,\,u_{2} (t),...,\,\,u_{r} (t)} \right]^{T}\), \(A(t)\) is the matrix of the system \((n \times n)\), \(B(t)\) is the input matrix \((n \times m)\), \(C(t)\) is the output matrix \((p \times n)\), \(D(t)\) is the detour matrix \((p \times m)\), determining the direct dependence of the output on the input,\(p\) is the output quantity number.

The state parameters of a dynamic system are defined as follows: \(x_{1}\) is the vertical movement of the vibrating plate, \(x_{1} = y_{1}\), \(x_{2}\) is the vertical velocity of the vibrating plate, \(x_{2} = \dot{y}_{1}\), \(x_{3}\) is the vibrator vertical movement, \(x_{3} = y_{2}\), \(x_{4}\) is the vibrator vertical velocity \(x_{4} = \dot{y}_{2}\), \(x_{5}\) is the first pressing bar vertical movement, \(x_{5} = y_{4}\), \(x_{6}\) is the first pressing bar vertical velocity, \(x_{6} = \dot{y}_{4}\), \(x_{7}\) is the second pressing bar vertical movement, \(x_{7} = y_{5}\), \(x_{8}\) is the second pressing bar vertical velocity, \(x_{8} = \dot{y}_{5}\).

We get a mathematical formulation of the object dynamics in the form of Cauchy on the rearrangement of the equations system of (1)–(7):

$$.\begin{aligned} x_{2} & = \dot{y}_{1} ;\hfill \\\dot{x}_{2} &= \frac{{\left[ {\begin{array}{*{20}l} { - \left( {c_{1} + c_{2} + c_{3} + c_{{14}} + c_{{15}} } \right) \cdot x_{2} + c_{2} \cdot x_{4} + c_{{14}} \cdot x_{6} + c_{{15}} \cdot x_{8} } \hfill \\ { + \left( {k_{1} + k_{2} + k_{3} + k_{{14}} + k_{{15}} } \right) \cdot x_{1} \, + k_{2} \cdot x_{3} + k_{{14}} \cdot x_{5} + k_{{15}} \cdot x_{7} - F_{4} - F_{5} } \hfill \\ { + \left( {\left[ {m_{3} + m_{6} } \right] \cdot e \cdot \omega_{3}^{2} - k_{3} \cdot e} \right) \cdot \sin \left( {\omega_{3} \cdot t} \right)} \hfill \\ { + \,c_{3} \cdot e \cdot \omega_{3} \cdot \sin \left( {\omega_{3} \cdot t + {{\pi /2}}} \right) + \left( {m_{1} + m_{6} + m_{7} } \right) \cdot g} \hfill \\ \end{array} } \right]}}{{m_{1} + m_{3} + m_{6} + m_{7} }}, \\ x_{4} & = \dot{y}_{2} ;\,\,\dot{x}_{4} = \frac{1}{{m_{2} }} \cdot \left( {c_{2} \cdot x_{2} - c_{2} \cdot x_{4} + k_{2} \cdot x_{1} - k_{2} \cdot x_{3} + m \cdot r \cdot \omega_{2}^{2} \cdot \sin \left( {\omega_{2} \cdot t} \right)} \right), \\ x_{6} & = \dot{y}_{4} ;\,\,\dot{x}_{6} = \frac{1}{{m_{4} + m_{8} }} \cdot \left( {c_{{14}} \cdot x_{2} - \left[ {c_{4} + c_{{14}} } \right] \cdot x_{6} + k_{{14}} \cdot x_{1} - \left[ {k_{4} + k_{{14}} } \right] \cdot x_{5} + F_{4} + \left[ {m_{4} + m_{8} } \right] \cdot g} \right), \\ x_{8} & = \dot{y}_{5} ;\,\,\dot{x}_{8} = \frac{1}{{m_{5} + m_{9} }} \cdot \left( {c_{{15}} \cdot x_{2} - \left[ {c_{5} + c_{{15}} } \right] \cdot x_{8} + k_{{15}} \cdot x_{1} - \left[ {k_{5} + k_{{15}} } \right] \cdot x_{7} + F_{5} + \left[ {m_{5} + m_{9} } \right] \cdot g} \right). \\ \end{aligned} . $$

We can obtain the following results by moving these parameters into the corresponding vector and matrix.

Enter parameter value

$$\begin{gathered} M = m_{1} + m_{3} + m_{6} + m_{7} \,,\, \hfill \\ K = k_{1} + k_{2} + k_{3} + k_{14} + k_{15} \,,\,\, \hfill \\ Q = c_{1} + c_{2} + c_{3} + c_{14} + c_{15} \,. \hfill \\ \end{gathered}$$

Then the model of the process understudy in the state space, in the vector–matrix form

$$\begin{aligned}\,&y(t) = C \cdot x(t) + D \cdot u(t)\,,\\&\left[ {\begin{array}{*{20}c} {y_{1} (t)} \\ {y_{2} (t)} \\ {y_{3} (t)} \\ {y_{4} (t)} \\ {y_{5} (t)} \\ {y_{6} (t)} \\ {y_{7} (t)} \\ {y_{8} (t)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {x_{1} (t)} \\ {x_{2} (t)} \\ {x_{3} (t)} \\ {x_{4} (t)} \\ {x_{5} (t)} \\ {x_{6} (t)} \\ {x_{7} (t)} \\ {x_{8} (t)} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\, \cdot u(t)\,, \end{aligned}$$
$$\dot{x}(t) = A \cdot x(t) + B \cdot u(t)\,,$$
$$\begin{gathered} \left[ {\begin{array}{*{20}c} {\dot{x}_{1} (t)} \\ {\dot{x}_{2} (t)} \\ {\dot{x}_{3} (t)} \\ {\dot{x}_{4} (t)} \\ {\dot{x}_{5} (t)} \\ {\dot{x}_{6} (t)} \\ {\dot{x}_{7} (t)} \\ {\dot{x}_{8} (t)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - \frac{K}{M}} & { - \frac{Q}{M}} & {\frac{{k_{2} }}{M}} & {\frac{{c_{2} }}{M}} & {\frac{{k_{14} }}{M}} & {\frac{{c_{14} }}{M}} & {\frac{{k_{15} }}{M}} & {\frac{{c_{15} }}{M}} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ {\frac{{k_{2} }}{{m_{2} }}} & {\frac{{c_{2} }}{{m_{2} }}} & { - \frac{{k_{2} }}{{m_{2} }}} & { - \frac{{c_{2} }}{{m_{2} }}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ {\frac{{k_{14} }}{{m_{4} + m_{8} }}} & {\frac{{c_{14} }}{{m_{4} + m_{8} }}} & 0 & 0 & { - \frac{{k_{4} + k_{14} }}{{m_{4} + m_{8} }}} & { - \frac{{c_{4} + c_{14} }}{{m_{4} + m_{8} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ {\frac{{k_{15} }}{{m_{5} + m_{9} }}} & {\frac{{c_{15} }}{{m_{5} + m_{9} }}} & 0 & 0 & 0 & 0 & { - \frac{{k_{5} + k_{15} }}{{m_{5} + m_{9} }}} & { - \frac{{c_{5} + c_{15} }}{{m_{5} + m_{9} }}} \\ \end{array} } \right] \hfill \\ \cdot \left[ {\begin{array}{*{20}c} {x_{1} (t)} \\ {x_{2} (t)} \\ {x_{3} (t)} \\ {x_{4} (t)} \\ {x_{5} (t)} \\ {x_{6} (t)} \\ {x_{7} (t)} \\ {x_{8} (t)} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \frac{1}{M} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & {\frac{1}{{m_{2} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{1}{{m_{4} + m_{8} }}} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{{m_{5} + m_{9} }}} \\ \end{array} } \right]\, \cdot u(t)\,\,, \hfill \\ \end{gathered}$$

where

$$u(t) = \left[ {\begin{array}{*{20}c} \begin{gathered} - F_{4} - F_{5} + \left( {\left[ {m_{3} + m_{6} } \right] \cdot e \cdot \omega_{3}^{2} - k_{3} \cdot e} \right) \cdot \sin \left( {\omega_{3} \cdot t} \right) \hfill \\ - c_{3} \cdot e \cdot \omega_{3} \cdot \sin \left( {\omega_{3} \cdot t + \pi {/}2} \right) + \left( {m_{1} + m_{6} + m_{7} } \right) \cdot g \hfill \\ \end{gathered} \\ {m \cdot \omega_{2}^{2} \cdot r \cdot \sin \left( {\omega_{2} \cdot t} \right)} \\ {F_{4} + \left( {m_{4} + m_{8} } \right) \cdot g} \\ {F_{5} + \left( {m_{5} + m_{9} } \right) \cdot g} \\ \end{array} } \right]\,.$$

4 Results and Analysis

A simulation model is obtained in the environment of the MATLAB/Simulink program. The initial data from scientific work [25] were used to simulate the process.

$$\begin{aligned} &k_{1} = 4.2 \cdot 10^{6} \,{\text{N/m , }}k_{2} = 10.3 \cdot 10^{7} \,{\text{N/m, }}k_{3} = 1.68 \cdot 10^{6}\,{\text{N/m,}}\,\hfill \\&k_{4} = 4.2 \cdot 10^{6} \,{\text{N/m,}}\,k_{5} = 5 \cdot 10^{6} \,{\text{N/m,}}\,\,k_{14} = 1.1 \cdot 10^{4} \,{\text{N/m, }}\,\hfill \\&k_{15} = 1.1 \cdot 10^{4} \,{\text{N/m, }}\,c_{1} = 1200\,{\text{N}} \cdot {\text{s/m, }}\,c_{2} = 17600\,{\text{N}} \cdot {\text{s/m, }}\,\hfill\\& c_{3} = 3160\,{\text{N}} \cdot {\text{s/m, }}c_{4} = 1200\,{\text{N}} \cdot {\text{s/m,}}\,c_{5} = 1200\,{\text{N}} \cdot {\text{s/m,}}\,\hfill\\&c_{14} = 200\,{\text{N}} \cdot {\text{s/m,}}\, c_{15} = 200\,{\text{N}} \cdot {\text{s/m,}}\,\, \hfill \\& m = 9.6\,{\text{kg,}}\,m_{1} = 3000\,\,{\text{kg,}}\,m_{2} = 80\,\,{\text{kg,}}\,\,m_{3} = 260\,{\text{kg,}}\,\,m_{4} = 260\,\,{\text{kg,}} \hfill \\& m_{5} = 260\,\,{\text{kg, }}m_{6} = 0.2 \cdot \,m_{3} {,}\,\,m_{7} = 0.2 \cdot \,m_{1} {,}\,\,m_{8} = 0.2 \cdot \,m_{4} {,}\,\hfill\\&m_{9} = 0.2 \cdot \,m_{5} {, }\,r = 0,035\,{\text{m,}}\,e = 0,007\,{\text{m, }}f_{2} = 20\,{\text{Hz, }}f_{3} = 15\,{\text{Hz, }}\,\hfill\\&f_{4} = 25\,{\text{Hz}}\,f_{5} = 25\,{\text{Hz,}}\,\,\,F_{4} = 9000\,{\text{N,}}\,F_{5} = 9000\,{\text{N}}{.} \hfill \\ \end{aligned}$$

A simulation model of the studied process is presented in Fig. 2.

Fig. 2
figure 2

Simulation model in the language of the program MATLAB/Simulink

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The timing diagram of work process parameters was obtained by computer simulation: displacement, velocity, acceleration. Figure 3 shows the graphical dependencies for the vibration plate.

Fig. 3
figure 3

The dependences of the oscillatory process parameters of the smoothing plate of the paver

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The obtained dependencies correspond to the oscillatory process, and stability features of a dynamic system are observed. The vibration plate acceleration with peak values up to 25 m/s2 indicates a good correlation with the research results [1, 25].

The simulation results of the compaction process of the material by the second pressing bar are shown in Fig. 4.

Fig. 4
figure 4

The dependences of the oscillatory process parameters of the second pressing bar of the paver working body

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The graphs in Fig. 4 show a significant decrease in the amount of the bar movement, a slight increase in acceleration due to gear vibration. Dependencies correspond to the nature of the physics of the workflow.

5 Determination of the Transfer Function of the Control Object

The transfer function was determined to conduct a study of a mathematical model using block simulation software. As a result of the MATLAB program tools implementation, we obtained advanced functions with the regulated variable—the angular frequency of vibration.

Transfer function regulated variable—vibrating plate displacement

$$W(s) = \frac{{\left[ \begin{gathered} 0,0002556 \cdot s^{6} + 0,05853 \cdot s^{5} + 338,2 \cdot s^{4} + 4890 \cdot s^{3} \hfill \\ + 1,119 \cdot 10^{7} \cdot s^{2} + 6,576 \cdot 10^{7} \cdot s + 9,337 \cdot 10^{10} \hfill \\ \end{gathered} \right]}}{{\left[ \begin{gathered} s^{8} + 234,7 \cdot s^{7} + 1,351 \cdot 10^{6} \cdot s^{6} + 2,147 \cdot 10^{7} \cdot s^{5} + \,4,667 \cdot 10^{10} \cdot s^{4} \hfill \\ + 3,446 \cdot 10^{11} \cdot s^{3} + 4,391 \cdot 10^{14} \cdot s^{2} + \,8,324 \cdot 10^{14} \cdot s + 5,51 \cdot 10^{17} \hfill \\ \end{gathered} \right]}}\,\,,$$

transfer function regulated variable—vibrating plate velocity

$$W(s) = \frac{{\left[ \begin{gathered} 0,0002556 \cdot s^{7} + 0,05853 \cdot s^{6} + 338,2 \cdot s^{5} + 4890 \cdot s^{4} \hfill \\ + 1,119 \cdot 10^{7} \cdot s^{3} + 6,576 \cdot 10^{7} \cdot s^{2} + 9,337 \cdot 10^{10} \cdot s \hfill \\ \end{gathered} \right]}}{{\left[ \begin{gathered} s^{8} + 234,7 \cdot s^{7} + 1,351 \cdot 10^{6} \cdot s^{6} + 2,147 \cdot 10^{7} \cdot s^{5} + 4,667 \cdot 10^{10} \cdot s^{4} \hfill \\ + 3,446 \cdot 10^{11} \cdot s^{3} + 4,391 \cdot 10^{14} \cdot s^{2} + 8,324 \cdot 10^{14} \cdot s + 5,51 \cdot 10^{17} \hfill \\ \end{gathered} \right]}}\,\,.$$

The obtained transfer functions correspond to the correct form since the degree of the numerator is less than the degree of the denominator. In further studies, when designing an automatic controller, it is necessary to take into account the high order of the mathematical model of the control object.

6 Conclusion

The chapter discusses the task of simulation as a highly efficient object of continuous compaction control of the cyber-physical system. A mathematical model of the mixture compaction process is obtained by a highly effective working body of the paver using the state space method. The obtained model was verified by simulation in the environment of the MATLAB/Simulink program. The results of the work are a stage of scientific research in the field of designing intelligent control and management systems for cyber-physical road-building systems.