## 2 Equation of Motion and Its Solution Considering Structural Nonlinearity
### Determination of the Order for the Orthogonal Decomposition
It can also be deduced that there is a relation between the number of periods in signal and the number of terms in the Chebyshev polynomial decomposition. A brief conclusion can also be drawn that the combined bandwidth of the polynomials should cover the bandwidth of the original signal. In other words, the frequency of the higher order of Chebyshev polynomial decomposition should at least exceed the concerned upper limit of the bandwidth of the signal. In this way, the Fourier basis used for the decomposition compared to the polynomials will result in a large number of unknowns to cover the bandwidth of the excitation signal. This is also the reason the Chebyshev polynomial was selected to conduct the decomposition in this study. Multi-frequency signal can also be decomposed with Chebyshev polynomials. The highest order of the polynomial used in the decomposition can be approximately determined by the highest frequency of the signal, which is similar to decomposition of the single frequency signal. The Chebyshev decomposition can be expanded to the random signals. Generally, a random signal can be taken as the summation of the periodic signals which can also be decomposed with Chebyshev polynomials. The order of the Chebyshev polynomial will depend on the highest frequency concerned and the length of the signal. Another example is provided to demonstrate the decomposition of multi-frequency signals. A four-second signal defined as a summation of sine waves **F** = 0.2 \(\times\) sin(2\(\pi t\)) + 0.05 \(\times\) sin(6\(\pi t\)) + 0.03 \(\times\) sin(9\(\pi t\)) + 4. The frequency analysis based on fast Fourier transform is shown in Figure 3. The frequency analysis result is shown in Figure 3a. The highest frequency is 4.5 Hz and other two frequencies are approximately 2.93 Hz and 0.98 Hz. Therefore, the order of the highest Chebyshev polynomial can be determined as \(\alpha\) \(\times\) 4.5 \(\times\) 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Signal** & **Order of the Chebyshev Polynomial** & **Error (\%)** \\ \hline \multirow{3}{*}{One-period of sine wave} & 4 & 9.43 \\  & 5 & 9.39 \\  & 6 & 0.61 \\ \hline \multirow{3}{*}{Two-period of sine wave} & 7 & 32.24 \\  & 8 & 5.42 \\  & 9 & 5.40 \\  & 10 & 0.57 \\ \hline \multirow{3}{*}{Ten-period of sine wave} & 37 & 1.99 \\  & 40 & 0.11 \\ \cline{1-1}  & 41 & 0.11 \\ \cline{1-1}  & 42 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Error of the Re-construction with different order Chebyshev polynomial. The value of \(\alpha\) can be determined through the inverse study as shown in Equation (22) and varies with the number of the periods '\(np\)' in the signal. The formulation for \(\alpha\) is shown as follows. \[\alpha=\left\{ \begin{array}{ll}6&\quad np=1\\ 4&\quad np=2,3\ldots 8\\ 3.37&\quad np\geq 9\end{array} \right. \tag{22}\]  
where \(\alpha\) is obtained with the general curve fitting method based on the first order approximation. With the determined highest order of Chebyshev polynomial, the signal is re-constructed as shown in Figure 3b. The Chebyshev polynomial can accurately re-construct the original signal. The weighing coefficients of the Chebyshev decomposition exist as long as the highest order of the polynomial can be stably determined. It is also expected that the coefficients of the Chebyshev polynomial can be identified with a suitable identification tool in the force identification problem. The random excitation is a special kind of multi-frequency load which may be applied on structures. Although it is difficult for the random signal to be represented by Chebyshev polynomials with only very few terms, this paper will provide an approximation method to evaluate the crucial component in the random excitation for the structure as introduced in the numerical study. Figure 3: Re-construction of multi-frequency signals with Chebyshev polynomial. **(a)** Frequency analysis of the signal. **(b)** Re-construction based on the frequency analysis.