# Design Optimization of a Hydrodynamic Brake with an Electrorheological Fluid
## 1 Introduction  
Hydrodynamic brakes (HB) belong to the group of hydraulic components of machines, as do clutches and hydrodynamic torque converters. They are used in drive systems of machines, mainly due to their favorable weight-to-torque ratio on the input shaft and the long durability [1]. The basic way of controlling the HBs is by changing their working fluid filling degree. The lower the filling degree, the smaller the torque on the input shaft of the brake. However, the dependence of the torque on the filling degree is not linear. Moreover, this manner of control has a significant disadvantage, which is the unstable operation of HB at low filling degrees. The analysis of the operation of hydrodynamic components leads to the conclusion that controlling them is also possible by changing properties of the working fluid [2]. Thus, in this case, it is possible to use smart fluids, which react with a change in shear stress when exposed to changes in the electrical field (electrorheological fluids) or in the magnetic field (magnorheological fluids) [3, 4]. Currently, smart fluids are used in numerous devices and components of machines, e.g., in viscous clutches and brakes [5, 6, 7], shock absorbers [8, 9, 10], valves [11, 12, 13], sealings [14], cantilever sandwich beams [15, 16], landing gear systems in planes [17, 18], and industrial robots [19, 20, 21]. In addition, prototypes of hydrodynamic clutches with electrorheological fluids (ER) and magnetorheological fluids (MR) are constructed [22, 23, 24, 25]. A frequent method of perfecting the design of machine components is the use of construction optimization. The construction optimization is performed on the basis of the component's mathematical model. The model parameters whose values are changed during the optimization process are called the design variables. The objective function is determined, and then its global extreme is sought on the basis of the requirements analysis assigned to the machine component (i.e., criteria) and to the mathematical model. To search for the global extreme of the objective function, multiple methods have been developed. Among the most frequently used methods are randomization methods, including the Monte Carlo Method and the Genetic Algorithm method [26; 27; 28]. It is advantageous to use structure optimization methods in the design process due to the complex design of hydraulic clutches with ER or MR fluids, which results from the necessity to generate an electric field inside. For mathematical modeling of clutches, brakes and torque converters, and one-, two-, and three-dimensional flows within the rotors are taken into consideration [29; 30; 31]. In the model based on the one-dimensional flow, it is assumed that the working fluid moves with the mean flow path with a uniform through-flow velocity [32]. The high accuracy of this model, comparable to the accuracy of two- and three-dimensional models, is obtained as a result of parameter selection based on experimental research. Due to their simplicity, these models are particularly suitable for use in design optimization [33]. Publications concerning the optimization of brakes and hydraulic clutches with smart fluids most frequently concern viscous clutches and brakes with MR fluids [34; 35]. The article [36] describes the design optimization of a disc viscous clutch with ER fluids. The optimization is performed with the use of two optimization methods: the Monte Carlo method and the Genetic Algorithm method. The purpose of the design optimization is to seek such values of design variables, for which the viscous clutch with the ER fluid transmits a large torque, while the clutch's dimensions are small and its determined operating temperature is low. The optimization calculations are performed for two models including three and eight design variables. As a result of the design optimization, a disc viscous clutch with the ER fluid is constructed. In the publication [37], the object of the optimization process is a disc brake with a MR fluid.