# Optimization Design of Lattice Structures in Internal Cooling Channel of Turbine Blade
## 2 Functions of Heat Transfer and Mechanical Performances
### The Function of Vibration Performance  
Low-frequency vibration and vibration causing structural resonance of turbine blades should be avoided. The natural frequency of turbine blades should be increased. To achieve these objectives, LS cores were set at internal cooling channel of turbine blades, and the finite element analysis method (FEA) was used to set up the relevance between the first-order natural frequency (_freq_1) and geometric parameters of LS channel. This relevance was determined by an implicit function box by numerical calculation using ABAQUS software. Therefore, the function of _freq_1 can be expressed as:  
\[f_{2}(x)\ =freq1=f(D,\omega) \tag{16}\]  
This value was given by ABAQUS directly by numerical calculation. The boundary condition of LS channel was fixed support on one side and free on the other side. Herein, vibration of PLS channel and XLS channel were mainly investigated. However, the XLS had symmetrical characteristic in topology. Therefore, the value of Equation (16) was based on the XLS channel as well as this value validation. \(f_{2}(x)\) was the second objective, and this function was validated as follows. The XLS channel is shown in Figure 4a. Material used for the printed XLS channel was PLA, as shown in Table 1. The modified quadric tetrahedron element (C3D10M) was used in the process of mesh generation. After the verification of grid independence, the first six-order natural frequencies (_freq_1-6) tended to be stable when the grid number was approximately 100,000. Therefore, the computational grid number of the XLS channel was approximately 100,000. Relevant experiments were carried out to further verify the reliability of the finite element results. In the experiment, a SIEMENS SCM205 dynamic signal analyzer was used; the model of force hammer was PCB086C03 with a sensitivity of 2.25 mV/N; a PCB356A66 piezoelectric accelerometer sensor was used with a sensitivity of 1.034 mV/(m-s\({}^{2}\)), and a vice was used to fix the test sample so as to reduce its impact on the accuracy of the experimental results. Figure 4a shows the sample of XLS channel and the layout of the relevant measuring points. The sample was divided into five parts in the length direction and two parts in the width direction with equal proportion. The nodes were numbered in sequence, and the geometric model, established in LMS TestLab, was consistent with those settings. Acceleration sensors were set at points 5, 11, and 14 of the sample to pick up vibration signals, and a hammer was used to knock at point 17. The average value was obtained after striking the point five times. The signals were captured by acceleration sensor and force hammer sensor and were transmitted to multi-channel  
Figure 3: Comparison of the experimental value of Kim [38] and calculated value with the fin method. data acquisition apparatus for processing. Then, the data were transmitted to PC end for relevant data processing, and the relevant parameters were obtained. Figure 4b shows that in _freq_1-3, the experimental value and calculated value showed good consistency, of which the maximum error was within 10.5%. However, when it came to _freq_4-6, the error between them was relatively large, and the maximum error reached 24.9%. The reason for the analysis error lay in two aspects. On the one hand, due to the limitation of printing accuracy of the printer, the printed sample had certain defects. Particularly, the printing mode was adopted without internal support at the top of the channel, which made the resin material on the top of the sample not fit tightly enough. On the other hand, because the sample was a thin-walled structure, it was inevitable that some deformation would occur in the sample fixing with the vice. However, in the simulation process using FEA, there were no defects in the three-dimensional model, and its deformation was not considered when the constraints were applied.