## 2 Research on a Hydraulic Gear Pump with Undercut Tooth Roots  
In order to optimize a gear pump, its efficiencies must be calculated: hydraulic-mechanical (\(\eta_{hm}\)), volumetric (\(\eta_{l\mathcal{D}}\)) and total (\(\eta_{l\mathcal{D}}\)). Taking into account the conflict in the existing model, the total efficiency is often calculated using the direct product, assuming that the remaining efficiencies are met; this leads to a large possibility for saving energy. The total efficiency of a pump is defined by the ratio of the power output \(\left(N_{\textit{avg}}\right)\) to the power input \(\left(N_{\textit{ave}}\right)\) which can be written as [4; 5; 6; 7; 8]:  
\[\eta_{c}=\frac{N_{\textit{avg}}}{N_{\textit{ave}}}\cong\eta_{\
u}\times\eta_{ \textit{lim}} \tag{1}\]  
The volumetric efficiency of a gear pump (\(\eta_{\
u}\)) is determined by the ratio of the actual capacity \(Q_{\textit{rz}}\) to the theoretical capacity \(Q_{\textit{t}}\):  
\[\eta_{\
u}=\frac{Q_{\textit{rz}}}{Q_{\textit{t}}} \tag{2}\]  
The total volumetric losses in the pump are determined by the following parameters: compressibility of the liquid, deformation of the pump's components, internal leaks, the liquid's viscosity and density, and incomplete filling of the working chambers during the suction phase. Taking into account all of the coefficients and the interactions between them, the formula for the volumetric efficiency can be obtained as follows:  
\[\eta_{\
u}=1-c_{\mu}\times\frac{p}{2\pi\times\mu\times n}-c_{r}\times\frac{1}{ n}\times\sqrt{\frac{2p}{\rho}}\times\sqrt{q^{-1}} \tag{3}\]  
where: \(c_{\mu}\)--coefficient, which is a function of the size and number of slots in the pump, depending on the actual capacity of the pump,  
\(p\)--working pressure,  
\(q\)--actual performance,  
\(\rho\)--density of the liquid,  
\(n\)--the pump's rotational speed, \(\mu\)--dynamic viscosity of the liquid,  
\(c_{r}\)--coefficient that depends on the type and size of the slots and the actual capacity of the pump. The hydraulic-mechanical efficiency of a pump (\(\eta_{\textit{lim}}\)) is defined by the theoretical moment ratio \(M_{t}\) to the sum of the theoretical moment \(M_{t}\) and the hydraulic-mechanical loss moment \(\Delta M\) as follows:  
\[\eta_{\textit{lim}=\frac{M_{t}}{\Delta M+M_{t}}} \tag{4}\]  
Ultimately, the following formula can be obtained:  
\[\eta_{\textit{lim}}=\frac{1}{1+c_{\
u}\times 2\pi\times\frac{\mu\
u\alpha}{p}+c _{\rho}\times\frac{\rho\
u\alpha^{2}}{2p}\times\sqrt[3]{q^{2}+c_{p}}} \tag{5}\]  
where: \(c_{p}\)--coefficient that depends on the type of pump,  
\(c_{\rho}\)--coefficient that mainly depends on the actual capacity of the pump,  
\(c_{\
u}\)--coefficient that depends on the type of pump,  
\(p\)--working pressure,  
\(q\)--actual performance,  
\(\rho\)--density of the liquid,  
\(n\)--the pump's rotational speed,  
\(\mu\)--dynamic viscosity of the liquid. Using Equation (1), (3) and (5), the equation describing the total efficiency can be obtained as follows:  
\[\eta_{c}=\frac{1-c_{\mu}\times\frac{p}{2\pi\times\mu\
u\alpha}-c_{r}\times \frac{1}{n}\times\sqrt{\frac{2p}{\rho}}\times\sqrt[3]{q^{-1}}}{1+c_{\
u}\times 2 \pi\times\frac{\mu\
u\alpha}{p}+c_{\rho}\times\frac{\rho\
u\alpha^{2}}{2p} \times\sqrt[3]{q^{2}+c_{p}}} \tag{6}\]In this study, (\(\eta_{p}\)), (\(\eta_{lmm}\)) and (\(\eta_{c}\)) were considered as functions, while the parameters were assumed to be decision variables: \(M\), \(n\), \(p_{t}\), \(Q_{r2}\). The above procedure serves the purpose of using a gear pump, after tooth root undercutting, in various systems in order to demonstrate the accuracy and correctness of the mathematical calculations, and determine the discrepancies in the calculation that result from the different algorithms that were used during the design of the gear pump:  
* Determination of the maximum volumetric efficiency, assuming the permissible hydraulic-mechanical efficiency;
* Determination of the maximum overall efficiency [4; 5; 6; 7; 8]. * More detailed descriptions of the analyzed parameters can be given.