**2. Mathematical Model for the Serial Throttle Control Method**

A mathematical description of the start-up of a hydrostatic transmission with serial throttle control using a proportional valve was obtained based on a set of ordinary differential equations (a model with focused parameters). One of the equations for the model is the flow continuity equation at particular points of the hydraulic circuit, and the other is the equation of the equilibrium of torque values on the shaft of the hydrostatic motor [22].

In order to solve this set of equations, it is also necessary to formulate the initial conditions.

In the presented mathematical model, the following simplifying conditions were adopted (among others):


The flow continuity equation can be formulated as follows:

$$Q\_{pt} = Q\_{vp} + Q\_{cp} + Q\_{RD} + Q\_z \tag{1}$$

The throttling valve flow rate is determined as follows:

$$Q\_{RD} = Q\_s + Q\_{vs} + Q\_{cs} \tag{2}$$

$$Q\_{RD} = G\_{RD}\sqrt{p\_p - p\_s - p\_d} \tag{3}$$

In a system with a proportional flow valve, it is usually assumed that there is a linear dependency between the area of the surface through which the liquid flows and the displacement of the spool. Unfortunately, this simplification does not work in simulations of systems where even the slightest displacement of the spool is a significant factor.

In the model, it was assumed that the aforementioned relation is a polynomial function (4), whose degree and values of coefficients were selected based on the characteristics

listed in the catalogue specification provided by the manufacturer of the spool valve. The result of model matching has been presented in Figure 3.

$$s\_{\rm ll} = A\_3 \mathbf{s}^3 + A\_2 \mathbf{s}^2 + A\_1 \mathbf{s} + A\_0 \tag{4}$$

**Figure 3.** Opening characteristic curve of the spool valve from the data sheet with a fitted curve of model based on Equation (4). Curves 1–5 correspond to different valve pressure differential, the red dashed line is the fitted curve.

In the system with a proportional flow valve, it was assumed that the dynamics of the proportional flow valve are characterised by the following first-order differential equation, identical to the first-order inertial element:

$$s\_m \cdot G\_{RD\text{max}} = G\_{RD} + T\_{RD} \frac{dG\_{RD}}{dt} \tag{5}$$

The flow through the hydraulic motor is described by the following equation:

$$Q\_s = q\_s \omega\_s \tag{6}$$

The flow caused by the compressibility of the working medium [23] and the deformation of the elements of the system was assumed to be as follows:

• On the section from the pump to the proportional spool valve:

$$Q\_{cp} = c\_p \frac{dp\_p}{dt} \tag{7}$$

• On the section from the proportional spool valve to the motor:

$$Q\_{\rm cs} = c\_{\rm s} \frac{dp\_{\rm s}}{dt} \tag{8}$$

Losses caused by leakages in the pump and in the motor can be described linearly with the following equations:

$$Q\_{vp} = a\_{vp} p\_p \tag{9}$$

$$Q\_{\rm vs} = a\_{\rm vs} p\_{\rm s} \tag{10}$$

The pressure drop caused by the total loss of pressure resulting from viscous drag values (Hagen–Poiseuille equation) as well as the turbulent flow (Bernoulli's equation) were modelled using the following relation:

$$p\_d = \frac{8\mu L Q\_{RD}}{\pi R^4} + \sum\_j \zeta\_j \frac{\rho}{2} \left(\frac{Q\_{RD}}{\pi R^2}\right)^2 \tag{11}$$

The equation of the flow through the safety valve can be presented in the following form [24]:

$$Q\_z(t) = \begin{cases} \begin{array}{l} h\_z(p\_p - p\_0) - T\_z \frac{dQ\_z}{dt} \end{array} \text{for} \quad p\_p > p\_0\\ 0 - T\_z \frac{dQ\_z}{dt} \qquad \text{for} \quad p\_p \le p\_0 \end{cases} \tag{12}$$

In the analysed case, the motor load torque consists of three components: the constant one coming from static friction, the one coming from viscous friction in the motor and gearbox, and the moment of inertia. The torque value equilibrium condition on the hydrostatic motor shaft is described by the following relation:

$$q\_s p\_s = M\_b + f\omega\_s + I\_{zr}\frac{d\omega\_s}{dt} \tag{13}$$

To solve the above equations, the following initial conditions were assumed (slightly different from those found in the literature):

$$p\_p(0) = p\_0 + \frac{Q\_{pt}}{h\_z} \tag{14}$$

$$p\_s(0) = 0\tag{15}$$

$$Q\_z(0) = Q\_{pt} - a\_{\upsilon p} p\_0 \tag{16}$$

$$
\omega\_s(0) = 0\tag{17}
$$

The limit condition for the hydrostatic motor was defined as follows:

$$\text{if } q\_s p\_s \le M\_{b\prime} \text{ then } \omega\_s = 0, \,\frac{d\omega\_s}{dt} = 0 \tag{18}$$

Solving the above equations numerically requires their parametrization; this was carried out based on the catalogue data and the information found in the literature. However, the available literature does not specify the value of some of the coefficients for the equations; therefore, experiments were conducted in order to determine the friction coefficient of the hydraulic motor and of the coupled planetary gear.

In hydrostatic drive units, damping is caused predominantly by internal leakage, resistance related to the flow of the working medium, and friction forces caused by the movement of the hydraulic motor and the driven mechanism. In the dynamic model of hydrostatic transmission, leakages are taken into account in the flow balance equation, while the resistance of the movement of the hydraulic motor and of the coupled mechanism (independent of the speed in the case of Coulomb friction and linearly dependent on the speed in the case of viscous friction) are described via the equation of the equilibrium of the torque values acting on the shaft of the motor. The viscous friction coefficients of the hydraulic motor and of the coupled planetary gear were determined by measuring the resistance to idle running motion as a function of the angular velocity of the shaft of the motor. The pressure differences, *ps*, in the connection pipes of the hydraulic motor were taken as the measure of the aforesaid resistance values. Figure 4 shows the relation *ps* as a function of the angular velocity, *ωs*, of idle running. As the graph of this function shows, it is a linear relation, which—with accuracy sufficient for practical purposes—can be approximated with a straight line, which confirms the assumption of viscous friction.

**Figure 4.** Resistance to motion of the gear motor M2C1613 and of the coupled planetary gear OH-500 as a function of angular velocity, *ωs*, of idle running.

The findings presented here pertain to an M2C1613 gear motor coupled with an OH-500 planetary gear. The planetary gear was filled with SAE 85W90 transmission oil, whose temperature was kept within the range *t*<sup>1</sup> = 25 − 28 ◦C during the measurements, and the hydraulic motor was supplied with ISO-VG 46 hydraulic oil at a temperature of *t*<sup>2</sup> = 40 ◦C ± 2 ◦C.

Based on the presented measurement results, one can determine the value of the viscous friction torque, and, in consequence, also the coefficient *f* of that friction, according to the following equation:

$$f = \frac{(p\_s - p\_c)q\_s}{\omega\_s} \tag{19}$$

Considering the specific absorptivity of the motor (*qs* <sup>=</sup> 5.03 <sup>×</sup> <sup>10</sup>−<sup>6</sup> [m<sup>3</sup> /rad]), the viscous friction coefficient—determined based on the data presented in Figure 4, in accordance with Equation (19)—was *<sup>f</sup>* = 6.3 × <sup>10</sup>−<sup>2</sup> <sup>N</sup> · <sup>m</sup> · s/rad<sup>2</sup> .

Due to the specific nature of the rotational speed measurement, it was assumed in the model that the speed measurement system can be described using a first-order inertial element with the following transmittance:

$$G\_{\eta}(s) = \frac{1}{T\_{\eta}s + 1} \tag{20}$$

Once the equations of the mathematical model had been parametrized and the initial conditions adopted, it was possible to solve the model numerically, and subsequently to present in graphical form the pumping pressure of the pump, *pp*, the pressure on the motor, *ps*, and the angular velocity of the motor's shaft, *ns*, over time, for various waveforms of the control signal, *s*, supplied to the coils of the proportional electromagnet, as described by Equation (21) and presented in graphical form in Figure 5:

$$s = \begin{cases} s\_0 + \frac{s\_{\text{max}} - s\_0}{t\_0} t & \text{for } 0 < t < t\_0 \\ s\_{\text{max}} & \text{for } t > t\_0 \end{cases} \tag{21}$$

**Figure 5.** Waveform of the control signal, *s*, for the proportional valve.

The coefficient *s*<sup>0</sup> was taken in such a form as to compensate for the idle stroke of the spool of the proportional spool valve resulting from the stationary overlap, *s*max was assumed to be the maximum value of the signal as specified in the specification sheet of the spool valve, and *t*<sup>0</sup> is the signal rise time from *s*<sup>0</sup> to *s*max.

Figures 6–8 show the waveforms obtained by solving the mathematical models for, respectively, the pressure in the discharge flange of the pump *pp*, the pressure at the inlet port of the motor *ps*, and the rotational speed of the motor shaft *ns*.

**Figure 6.** Result of simulation of the waveform of pressure on the motor *ps* for various signal rise times, *t*0.

**Figure 7.** Result of simulation of the waveform of rotational speed *ns* for various signal rise times, *t*0.

**Figure 8.** Result of simulation of the waveform of pressure on the pump *pp* for various signal rise times, *t*0.

The simulations were carried out for various values of the ramp time (edge rise time) *t*<sup>0</sup> = { 0.5 s, 1 s, 2.5 s, 5 s, 8 s, 10 s}.

For the obtained results, the following parameters enabling assessment of the transitional state of the transmission were determined:


Table 1 presents a comparison of the parameters of the control signal along with the above-described values of the parameters for assessment of the transition state.

**Table 1.** Comparison of the control signal parameters and dynamic surplus pressure values at the inlet port of the motor, and the transmission start-up duration and reaction time. Transmission controlled with the serial throttle method without feedback.


Analysing the simulation results obtained for the open control system (Figures 6–8 and Table 1), we can observe that for a signal rise time below *t*<sup>0</sup> < 5 s the start-up time, *ts*, varies in a small range. This is directly due to the performance of the system, specifically the large moment of inertia and the maximum pressure set on the relief valve. Additionally, Table 1 shows the energy generated by the pump during start-up process. It can be observed that increasing the rise time of the signal increases the energy used for start-up. This is due to the fact that the pressure on the pump remains high for a longer period of time when the signal rise time is increased.

As the next step, the control system was equipped with a feedback loop from the angular velocity on the shaft of the hydraulic motor of the tested transmission. The following relation describes the transmittance of the PI controller used:

$$G\_{PI}(s) = K\_P \left( 1 + \frac{1}{T\_{IS}} \right) \tag{22}$$

The linear rise from *n*<sup>0</sup> = 0 rpm to the set value of *n*max = 200 rpm was taken to be the signal of the set value to the controller, and *t*<sup>0</sup> = 10 s was taken to be the signal rise time. The following is the mathematical description of the signal of the set value:

$$n\_r = \begin{cases} \begin{array}{c} n\_0 + \frac{\eta\_{\text{max}} - \eta\_0}{t\_0} t \end{array} \text{for} \quad 0 < t < t\_0\\ \begin{array}{c} n\_{\text{max}} \end{array} \text{for} \quad \begin{array}{c} 0 < t < t\_0\\ t > t\_0 \end{array} \end{cases} \tag{23}$$

A series of simulations for various values of parameters of the PI controller was conducted for the assumed control signal. To assess the state of adjustment, apart from the above-mentioned parameters, two additional parameters were introduced:

• Overshoot parameter described by the following relation:

$$\kappa\_{\rm ll} = \frac{\max\{n\}}{n\_{\rm max}} \tag{24}$$

• Steady-state error for the ramp input measured at the end of the ramp signal:

$$e\_r = n\_r - n\_s \tag{25}$$

Table 2 presents a comparison of the parameters of the PI controller for which the simulation results are presented, as well as the determined values of the assessment parameters.


**Table 2.** Comparison of the parameters of the PI controller and the obtained values of the dynamic surplus pressure on the motor, the start-up time, the reaction time, and the overshoot.

The diagrams in Figures 9 and 10 illustrate the significant impact of the control and adjustment parameters on the waveforms of the pressure on the pump and on the motor, as well as the speed on the hydrostatic motor shaft.

Figures 11 and 12 show simulation results for different moments of inertia of the system in the range *I* = 0.5 − 2.5*Izr*. The presented model corresponds to other rotating systems of machines in which the moment of inertia changes significantly by changing the value and position of the load. In addition, the changes of the static load are insignificant, so it is assumed in the simulation that they are constant.

**Figure 9.** Result of simulation of the waveform of rotational speed, *ns*, for various parameters of the PI controller.

**Figure 10.** Result of simulation of the waveform of pressure on the motor *ps* for various parameters of the PI controller.

**Figure 11.** Result of simulation of the waveform of rotational speed, *ns*, for various moments of inertia of the system (*KP* = 0.015, *TI* = 0.75 s).

**Figure 12.** Result of simulation of the waveform of pressure on the motor *ps* for various moments of inertia of the system (*KP* = 0.015, *TI* = 0.75 s).
