*3.2. Nondestructive Testing Based on Vibration Signal*

In order to validate the damage identification of the algorithm proposed in this paper, a numerical model of the frame structure in Figure 2 was established as shown in Figure 4, in which the friction between the bottom plate and the track is neglected.

**Figure 4.** Mathematic dynamical model of the frame structure.

The motion equation of the numerical model in Figure 4 is established as follows:

$$\mathbf{M}\ddot{\mathbf{x}} + \mathbf{C}\dot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{f}(t) \tag{27}$$

where, **M**, **C** and **K** are the mass, damping and stiffness matrices, respectively; **f**(*t*) is the input excitation vector; **x** = [*x*0, *x*1, *x*2, *x*3] <sup>T</sup> is the displacement of each layer, and the 0th floor represents the bottom plate; **. <sup>x</sup>** and **.. x** denote the speed and acceleration, respectively. The expressions of **M** and **K** are as follows:

$$\mathbf{M} = \begin{bmatrix} m\_0 & 0 & 0 & 0 \\ 0 & m\_1 & 0 & 0 \\ 0 & 0 & m\_2 & 0 \\ 0 & 0 & 0 & m\_3 \end{bmatrix}, \mathbf{K} = \begin{bmatrix} k\_1 & -k\_1 \\ -k\_1 & k\_1 + k\_2 & -k\_2 \\ & -k\_2 & k\_1 + k\_2 & -k\_3 \\ & & -k\_3 & k\_3 \end{bmatrix} \tag{28}$$

where, *m*<sup>0</sup> ∼ *m*<sup>3</sup> denotes the mass of each layer; *k*<sup>0</sup> ∼ *k*<sup>3</sup> denotes the interlayer stiffness. Damping matrix **C** can be assumed to be the Rayleigh damping matrix, as shown in Equation (5).

The augmented state variables as shown in Equation (4) are established by selecting the relevant displacement, velocity, story stiffness and Rayleigh damping coefficient.

*Materials* **2020**, *13*, 3301

For the numerical model of Figure 4, the initial values of the parameters are set according to Figure 2 as follows: (1) it is considered that the mass matrix **M** remains unchanged and the density of aluminum is 2700 kg/m<sup>3</sup> in the processing; thus, *m*<sup>0</sup> = *m*<sup>1</sup> = *m*<sup>2</sup> = *m*<sup>3</sup> = 6.7 kg is calculated according to the structure size; (2) The modulus of the elasticity of aluminum is set 70 Gpa so that *<sup>k</sup>*<sup>0</sup> = *<sup>k</sup>*<sup>1</sup> = *<sup>k</sup>*<sup>2</sup> = *<sup>k</sup>*<sup>3</sup> = 4.2 <sup>×</sup> 105 N/m; (3) By analyzing the excitation response test of the beam element shown in Figure 2, it can be found that the measured mode damping ratio usually has less influence than the inertia and stiffness of the structure. We can determine the Rayleigh damping coefficient by using the orthogonality experiment between the damping matrix and the vibration response mode shape of the structure. Rayleigh damping coefficients are estimated as <sup>α</sup> = 3.12 <sup>×</sup> 10−3, <sup>β</sup> = 1.77 <sup>×</sup> 10−4; (4) **<sup>Q</sup>** = 1.2 <sup>×</sup> 10−7**I**, **<sup>R</sup>** = 1.15**I**, and these will be slightly and appropriately adjusted in the calculation. (5) Take the initial displacement and velocity as 0, so that **<sup>X</sup>**0|<sup>0</sup> <sup>=</sup> 0, 0, 0, 0, 0, 0, 0, 0, 4.2 <sup>×</sup> 105, 4.2 <sup>×</sup> <sup>10</sup>5, 4.2 <sup>×</sup> <sup>10</sup>5, 3.12 <sup>×</sup> <sup>10</sup><sup>−</sup>3, 1.77 <sup>×</sup> <sup>10</sup>−<sup>4</sup> **T** . By applying the magnitude balance technology, we can set <sup>κ</sup>amp = 4.2 <sup>×</sup> <sup>10</sup>5, <sup>α</sup>amp = 3.12 <sup>×</sup> <sup>10</sup><sup>−</sup>3, <sup>β</sup>amp = 1.77 <sup>×</sup> <sup>10</sup>−<sup>4</sup> and adjust **X**0|<sup>0</sup> as [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1] T; (6) **PX**,0|<sup>0</sup> is set as the identity matrix **<sup>I</sup>**.

By substituting the above parameters into the structural state equations and the observation equations that are expressed as Equations (6)~(10), the results of the NDT analysis can be obtained as described in the following section.
