**3.3 Optimization method and parameters**

In the present research, Sequential Quadratic Programming (SQP), also known as Quadratic Approximation, is applied. It has arguably become the most successful method for solving nonlinearly constrained optimization problems. Its outstanding strongpoint is the less number of function and gradient evaluation, and the higher computational efficiency, especially for the rudder structural optimization objective function being a linear or nonlinear function of the design variables, and constraints such as frequencies for a function of the design variables. Using the method to optimize its stiffness, applying the mode shape error derivative concept to calculate


**Table 3.** *Structural material parameters.* *Verification and Validation of Supersonic Flutter of Rudder Model for Experiment DOI: http://dx.doi.org/10.5772/intechopen.98384*

the sensitivity of the stiffness, and employing this Method in the iterative process to make rapid the optimization convergence, and reliable the computational results.

When the optimal model was established, it was possible to define the optimization conditions and perform mode analysis.

1.The objective function:

From **Table 1**, the vibration amplitude distribution of the first frequency of the rudder was nonlinear in Case 1, and therefore, the eigenvector error equation was written by.

$$error1 = \sqrt{\sum\_{i=1}^{n} \left(\chi\_i - \chi\_i\right)^2 / n} \tag{1}$$

N sample nodes were located at the same location of the test points from the upper rudder surface in **Figure 6**.

For Case 2, the eigenvector error equation was defined by linear, due to the linear changing of the vibration displacements.

$$error2 = \sum\_{i=1}^{n} |\mathbf{x}\_i - \mathbf{y}\_i| / n \tag{2}$$

The two errors in modal analysis of the rudder were the objectives to be minimized in this research, respectively.

2.Constraints:

The first and second frequencies were taken as constraints. At the same time, the stiffness of the rotation shaft was also restricted to some limits.

#### 3.Design variables:

The stiffness of the rudder axle was defined as design variables.

**Figure 6.** *Monitoring nodes of FEM.*

### **3.4 Optimization results**

The optimization typically took 9 and 26 iterations, for Case 1 and Case 2 respectively to converge to the precision required for the gradient optimization in **Figures 7** and **8**.

In Case 1, the first mode surface was fitted in the test data and the optimization monitoring points in **Figure 9**.

The **Figure 9** above shows that the experimental points change dramatically and disorderedly, while the optimized ones transit gradually and softly. It also demonstrates the 2nd-order error for the first mode shape in Case 1 is selected very right.

In Case 2, the first mode surface is fitted in the test data and the optimization monitoring points in **Figure 10**.

It is indicated in **Figure 10** that more test data deviate from the fit left surface, but almost all of optimization points lie in the right surface.

In cases 1 and 2, the natural frequencies and mode shapes are shown in **Figures 11**–**14**.

Compared with the experimental frequencies, the errors are shown in **Table 4**.

**Figure 7.** *History of case 1 (vertical axis shows percent error in Eq. (1)).*

*Verification and Validation of Supersonic Flutter of Rudder Model for Experiment DOI: http://dx.doi.org/10.5772/intechopen.98384*

**Figure 8.** *History of case 2 (vertical axis shows percent error in Eq. (2)).*

#### **Figure 9.**

*First mode surface of the test data(left) VS. first mode surface of the optimization monitoring points (right).*

#### **Figure 10.**

*First mode surface of the test data(left) VS. first mode surface of the optimization monitoring points(right).*

**Figure 11.** *First bending mode (case 1).*
