## 2 Materials and Methods
### Formulation of the Optimization Problem  
The aim of the optimization process is to avoid crossing the natural frequency of the analyzed system in the operating range with the forcing function. Figure 5 shows a notional Campbell diagram that illustrates the natural frequency of the system with negative slope caused by the increased temperature that reduces the stiffness and thereby the frequency. The engine order is the function of the rotational speed. The intersection point of two straight lines indicates the resonance condition (natural frequency is aligning with engine order excitation frequency) or the risk situation of high dynamic response and so wears on mating surfaces (contact interfaces between the nozzle hook and turbine casing rails).  
The desired scenario from a design perspective is to have natural frequency crossing with an excitation harmonic force outside the operating range as presented in Figure 6 and fulfilled in Equation (11).  
\[f_{1}>f\omega_{MAX} \tag{11}\]  
The characteristic of the first system mode is presented in Figure 7. The nozzle is moving as a pendulum (the inner band is traveling back and forth) being bonded to the turbine casing. The second and the following system modes of the structure are not in the scope of interest since they are outside of the operating range and are not in interaction with excitation from shaft imbalance.  
Figure 5: Notional Campbell diagram.  
Figure 6: Notional Campbell diagram with optimal location of the natural frequency mode.  
The general optimization problem is to minimize or maximize the objective function considering the assumed constraints in the design domain. The set of design variables for which the function of the optimization problem reaches the global minimum can be described as follows:  
\[x=[x_{1},\ x_{2},\ldots\ x_{i},\ldots x_{n}] \tag{12}\]  
where:  
\(n\)--number of design variables (geometrical parameters)  
\(x_{i}\)--\(i\)th design variable from design domain \([x_{i}]_{\text{MIN}}<x_{i}<[x_{i}]_{\text{MAX}}\)  
\[\min_{x}J_{0}(x) \tag{13}\]  
where:  
\(J_{0}\)--objective function  
\(x\)--design variable vector  
In the optimization process, two different objective functions were taken into account. The first optimization process is based on minimalization of the model area (Equation (14)) that is a derivative of the system weight (including all the defined further geometrical parameters) by the constraint imposed on the first natural frequency which should exceed 140 Hz (\(f_{1}\)). The frequency threshold is defined as the expected response in the defined domain.  
\[J_{1}=\int\limits_{\Omega}\rho d\Omega\ \rightarrow\ \min_{x}J_{1}(x) \tag{14}\]  
\[f_{1}=\frac{\sqrt{\lambda_{1}(K,\ M)}}{2\ \pi}\geq 140\ Hz \tag{15}\]  
where:  
\(\Omega\)--model area  
\(\rho\)--model density  
\(f_{1}\)--first natural frequency of the system  
\(\lambda_{1}\)--eigenvalue obtained from Equation (7)  
The second criterion used in the optimization process is to maximize the natural frequency of the first system mode using a reduced set of geometrical parameters (Equation (16)).  
\[J_{\text{ZA}}=\frac{\sqrt{\lambda_{1}(K,\ M)}}{2\ \pi}\ \rightarrow\ \max_{x}J_{\text{ZA}}(x_{A})\ \ \ \text{OR}\ J_{\text{ZB}}=\frac{\sqrt{\lambda_{1}(K,\ M)}}{2\ \pi}\ \rightarrow\ \max_{x}J_{\text{ZA}}(x_{B}) \tag{16}\]  
Figure 7: First system mode of the nozzle-casing system.  
where:  
\(x_{A}\)--reduced set of geometrical parameters on nozzle only  
\(x_{B}\)--reduced set of geometrical parameters on nozzle only (excluding thicknesses)  
Obtained results compared to study with full set of geometrical characteristics for what concern model area impact and frequency. The aim of the comparison was to demonstrate the magnitude of the change between the baseline reference configuration and the proposed one.