4 S. Wang et al. Fig. 1.4 Damped multi degree of freedoms system matrix method are introduced as general discretization methods, and the discretization process of the vibration system is explained. The equation of motion in matrix form is derived as shown in Eq. (1.7). MRx N C CPx N C Kx N DF N .t/ (1.7) The MDOF undamped free-vibration problem is switched to an eigenvalue problem, and the result shows that eigenvalues and eigenvectors can be equally considered as natural frequencies and mode shapes respectively. Also, the relationship between the number of the degree of freedom and the number of the natural frequencies and mode shapes can be explained. The eigenvectors of the system are shown to be orthogonal with respect to both mass and stiffness matrices. Modal matrix which assembles eigenvectors into a square matrix is introduced. By using the modal matrix, decoupling of the forced vibration terms and modal damping concept can be explained. In forced vibration case, FRF as shown in Eq. (1.8) can be obtained using the orthogonality of eigenvectors. Additionally, Maxwell’s reciprocity theorem states that Hik DHki for the linear system, Hik .!/ D N X rD1 i r k r .kr !2mr/ Cj .!cr/ (1.8) When the system becomes larger and more complex, DOF is increased. This leads to the difficulty of calculating the exact solution. To solve this kind of problem, an approximate solution is introduced. Mainly, superposition methods such as mode displacement method (MDM), mode acceleration method (MAM), load dependent Ritz vectors (LDRV) method, Krylov sequence, and Lanczos algorithm are explained, and the advantages and disadvantages of the each method are explained as well. Newton’s second law, energy method, and virtual work method are compared with each other so that the equation of motion is formulated. Consequently, Lagrange’s equation as shown in Eq. (1.9) is introduced to formulate the large and complex system. d dt @T @Pqi @T @qi C @U @qi DQi (1.9) 1.2.3 Experimental Modal Analysis Experimental modal analysis (EMA) contains experimental measurement process for the FRF of the system; signal processing, and extracting the modal parameters (natural frequencies, mode shapes, and damping ratios) from measured FRF. Verifying a numerical model, determining dynamic durability by experiments, and machinery diagnostics for maintenance are possible by using the modal parameters. The techniques needed to experimentally determine the FRF showing the relationship of response and excitation force are introduced. There are two kinds of methods to excite a structure. First case is supplying excitation force by attaching a vibration exciter as shown in Fig. 1.5a. In this case, in order to reduce the mass loading due to an attached vibration exciter, a stinger should be used. The type of signals that is applied to the vibration exciter is introduced,and the characteristics in the
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