# Mechanical Performance Analysis and Experimental Study of Four-Star-Type Crank-Linkage Mechanism
## 2 Numerical Calculation of Crank Linkage
### Modal Analysis of Four-Star-Type Crank Linkage
#### 2.4.2 Modal Analysis  
Modal analysis includes free modal analysis, installed modal analysis, and operational modal analysis. Free modal analysis is usually used to verify the reasonableness of the finite element model. It is the basis for further finite element analysis. In this paper, the free modal analysis of the crankshaft is carried out. The free mode of the crankshaft is measured by multi-point knocking and single-point response. Figure 9 shows the crankshaft free-suspension state. There are two hooking bolt mounting holes on the end face of the crankshaft, and the motorcycle elastic band is used to lift it with the hooking bolt. Figure 10 shows the knock point and acceleration sensor arrangement. The arrows show the sensor mounting direction. The black dot hammer indicates the hammering point and direction, and a total of 27 points are hammered. Each point is struck three times. The number of strikes must be recalculated in case of consecutive hits. Therefore, the hammering method requires some operational experience. In the finite element free modal analysis, the material properties are defined as follows: density is 7820 kg/m\({}^{3}\), modulus of elasticity is 2.1 \(\times\) 10\({}^{11}\) N/m\({}^{3}\), and Poisson's ratio is \(\delta=0.266\). The mesh is divided into tetrahedral elements. The mesh is refined at the rounded corners and round holes. The number of mesh nodes is 195,652, and the number of divided cells is 136,554. In addition, 10 Hz-2000 Hz is chosen as the analysis bandwidth. The comparison of the modal vibration patterns is shown in Figure 11. In the frequency band of 10-2000 Hz, the crankshaft has only two bending modes. The bending directions of the modes can be aligned. The first-order resonance frequency is 1.5% different. The second-order resonance frequency difference is 1.9%. The error is controlled within 2% due to the influence of casting process and the lubricating oil left in the oil hole, which can meet the calculation requirements. Figure 10: Arrangement of knocking points and acceleration sensors. Figure 9: Diagram of crankshaft free suspension. In the process of finite element analysis, it is generally necessary to make appropriate simplifications to the model. One is to facilitate meshing, and the other is to save computer resources. In order to discuss the effect of the degree of simplification on the modal frequency, the simplification process of the model is shown in Figure 12. Finite element modal analysis is carried out for the original 1:1 model, then the model with rounded holes and chamfers is removed, and the model with peripheral details is removed for the three degrees of simplification, respectively. The analysis results are shown in Table 3. The errors of the first two orders for the de-rounded hole and chamfered model are 2.6% and 1.9%, respectively. The number of finite element units of the final simplified model differs from the original 1:1 model by a factor of nearly 10. The difference between the first-order inherent frequency of the simplified model and the experiment is 6.6%, and the second-order inherent frequency is 5.6%. The error is larger than that of the 1:1 model, but it can basically meet the general calculation requirements. The linkage is freely suspended, as shown in Figure 13. First, we hang the small end of the linkage freely by the elastic rope, then tighten the linkage bolt at the large end of the linkage. The test bandwidth is 10-2000 Hz. It also adopts the method of multi-point knocking and single-point response, and the schematic diagram of hammering and response points is shown in Figure 14.  
Figure 11: Comparison of modal vibration shapes of crankshaft. (**a**) Finite element first-order vibration shapes (1336 Hz). (**b**) Experimental first-order vibration shapes (1356 Hz). (**c**) Finite element second-order vibration shapes (1461 Hz). (**d**) Experimental second-order vibration shapes (1489 Hz).