## 2 Equation of Motion and Its Solution Considering Structural Nonlinearity
### Determination of the Order for the Orthogonal Decomposition  
Before the identification of the coefficients of the Chebyshev basis, the determination of the Chebyshev polynomial order is of great importance. To discuss the order of the Chebyshev polynomial, the first two, six, ten, fifteen and twenty terms of Chebyshev polynomial for 3 s are first selected and shown in Figure 1. Here, it can be seen that the characteristics of the Chebyshev polynomials, including peaks and valleys, are a little similar to Fourier series, so this can be used to represent the periodic signal or the signal consisting of periodic components. It is also helpful to conduct a frequency analysis of the measured structural response which depends on the property of the external load. An example is first provided to demonstrate that the periodic signal can be represented by the Chebyshev polynomials as follows. A one second sine wave with the frequency at 1 Hz is shown in Figure 2. To solve the coefficients for the re-construction of the signal, Equation (15) can be rewritten as follows:  
\[ \begin{bmatrix}\mathbf{T}^{1}_{m}\\ \ldots\\ \mathbf{T}^{N_{f}}_{m}\end{bmatrix} \mathbf{W}=\mathbf{T}\mathbf{W}=\mathbf{F} \tag{19}\]  
Figure 1: Re-construction of single frequency signals with Chebyshev polynomial. where \(\mathbf{T}\) represents the determined Chebyshev polynomial basis \(\left[\mathbf{T}^{1}{}_{m}\quad\mathbf{T}^{2}{}_{m}\quad\ldots\quad\mathbf{T}^{ N}_{m}\right]^{\prime}\) and \(\mathbf{W}\) denotes the unknown vector of the Chebyshev polynomials. The coefficients vector can be solved with the following bounds to this re-construction problem as  
\[(\mathbf{T}^{\prime}\mathbf{T+}\lambda\mathbf{I})\mathbf{W}=\mathbf{T}^{ \prime}\mathbf{F} \tag{20}\]  
where \(\lambda\) is the non-negative damping coefficient governing the participation of the least-squares error in the solution. Solving Equation (20) is equivalent to minimizing the following function  
\[J(\mathbf{W},\lambda)=\|\mathbf{TW}-\mathbf{F}\|^{2}+\lambda\|\mathbf{W}\|^{2} \tag{21}\]  
Figure 2: _Cont._  
Figure 2: Re-construction of single frequency signals with Chebyshev polynomial. (**a**) Re-construction with first five terms of Chebyshev polynomial. (**b**) Re-construction with first six terms of Chebyshev polynomial. (**c**) Re-construction with first nine terms of Chebyshev polynomial. (**d**) Re-construction with first ten terms of Chebyshev polynomial. (**e**) Re-construction with first forty terms of Chebyshev polynomial. With the solution to Equation (21), the re-constructed signals are compared with the original signals as shown in Figure 2. First, the single period sine wave is constructed with the first five terms and six terms of Chebyshev polynomials. The re-construction results are shown in Figure 2a,b, where a one-period sine wave can be represented by the first sixth-term of Chebyshev polynomials accurately while the error is a little larger based on the five-term re-construction. Then, the re-construction of another sine wave is conducted the results of which are shown in Figure 2c,d. Here, we can see in the figure that ten terms of polynomials perform better than the nine-term Chebyshev polynomial re-construction. The conclusion is similar in that the bandwidth of the summation of the polynomials should cover the bandwidth of the original signal. Lastly, the investigation of a sine wave signal with 10 periods is conducted and decomposed by Chebyshev polynomials. The re-construction results are shown in Figure 2e, which clearly shows that the signal can be decomposed with 42-term Chebyshev polynomials accurately. The re-construction errors of the sine waves for the results in Figure 2 are listed in Table 1. From the results shown in Figure 2 and Table 1, it can be concluded that the order of Chebyshev polynomials is only related to the number of the period for the single frequency signal and the required number of order is nearly four times of the number of the period.