# Design Optimization of a Hydrodynamic Brake with an Electrorheological Fluid
## 3 The Mathematical Model of the Hydrodynamic Brake
### Coefficient Selection for the Mathematical Model of the Hydrodynamic Brake  
Before using the developed HB model in optimization calculations, it is necessary to determine the values of the model's coefficients: \(\alpha\), \(\xi\), \(c\), \(a\), b, \(\beta\), \(\varepsilon_{\rm r}\). The values of the coefficients are determined on the basis of the design analysis and PHB experimental research. The coefficient \(\alpha\) is calculated on the basis of the PHB weight, assuming that the modelled HB has an identical structure and proportional dimensions. Calculating the coefficient \(\xi\), demanded PHB bench tests without connecting the blades to the high voltage power supply. The test bench scheme is shown in Figure 5.  
The main elements of the test bench are: an electric drive motor and a torque gauge. The computer control system of the test bench rendered it possible to obtain the measurement values specified in the test program, as well as to collect and process measurement data. The values recorded during the test are: the angular velocity of the pump shaft \(\omega_{1}\), the braking torque M, and the temperature of the working fluid \(\Theta\). The temperature \(\Theta\) is maintained at 35 \({}^{\circ}\)C by cooling the PHB with compressed air. Subsequently, the braking torque M is measured at a predetermined angular velocity of the PHB input shaft. The coefficient \(\xi\)is determined on the basis of Formulas (14) and (23). During the measurement, apart from the braking torque M, the rotational torques are of resistance to motion of the bearings and sealing. In further considerations, the torques of resistance are omitted due to their low values [42]. The value of the a coefficient is determined on the basis of the data from the manufacturer of the fluid LID3354S [46].  
In order to determine the coefficient c for the immobile turbine shaft and for a specified angular velocity of the pump shaft \(\omega_{1}\), the braking torque M of the PHB is measured for a few selected voltages U. To generate the high voltage U, an electric power supply with a rated power of 18W is used. The maximal voltage U obtained during these measurements is U\({}_{\rm max}\) = 3 kV. The coefficient c is selected, so that for the voltage U\({}_{\rm max}\) = 3 kV, the torque calculated with the use of the dependence (22) is equal to the torque measured with the use of the test bench, i.e., M = 1.733 Nm, Figure 6.  
Figure 5: Test bench scheme: 1—electric motor, 2—speed sensor, 3—slip rings used to supply high voltage, 4—the tested brake with ER fluid, 5—shaft, 6—connecting clutch, 7—torque gauge.  
Figure 6: Method of determining the coefficient \(c\): solid line—model, points—test.  
The coefficient b is selected in order to obtain the measured total leakage current i\({}_{\rm c}\) = 6 mA for the voltage U = 3 kV. The value of the relative electric permittivity \(\varepsilon_{\rm r}\) of the ER fluid is determined on the basis of [47; 48]. The coefficient values of the HB mathematical model determined for \(\omega_{1}\)= 200 rad/s and for the temperature \(\Theta\) = 35 \({}^{\circ}\)C are presented in Table 2.  
Due to the chosen method of selecting the coefficients shown in Table 2, only the modelling error of the dependence of the torque M on the high voltage U can be assessed. The solid line shown in Figure 6, obtained on the basis of the mathematical model, according to the assumptions, passes through the points for U = 0 kV and U = 3 kV. The average relative error calculated for U = 1 kV and U = 2 kV is under 2%.