## 2 Aeroelastic Model and Analysis Method
### Analysis Methods for Aeroelastic Model with Freeplay
#### 2.3.3 Harmonic Initial Conditions Derived from Frequency-Domain LCO Solutions  
The relationship between the free-stream velocity and the nondimensional LCO amplitude can be finally obtained by using the DF method. However, in this process, most studies ignore the information of the corresponding flutter eigenvector, which contains the amplitude and phase information of LCOs. According to Equation (13), here, we consider an \(n\)-dimensional linear system \(\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)\). The system response can be expressed as follows [41]:  
\[\mathbf{x}(t)=\mathbf{V}e^{\mathbf{L}t}\mathbf{V}^{-1}\mathbf{x}(0) \tag{15}\]  
where: \(\mathbf{L}\) is the diagonal matrix of eigenvalues, the \(j\)th element of which is the \(j\)th eigenvalue \(\lambda_{j}\) of \(\mathbf{A}\); and \(\mathbf{V}\) is the matrix of the eigenvectors, and its \(j\)th column is the \(j\)th eigenvector \(\boldsymbol{\varphi}_{j}\), which satisfies \(\mathbf{A}\boldsymbol{\varphi}_{j}=\lambda_{j}\boldsymbol{\varphi}_{j}\). The response of the \(k\)th state variable \(\mathbf{x}^{(k)}(t)\) can be written as a superposition of all eigenvalue exponential responses:  
\[\mathbf{x}^{(k)}(t)=\sum\
olimits_{j=1}^{n}\left(c_{j}\text{e}^{\lambda_{j}t}\right) \tag{16}\]  
where: the weight \(c_{j}=\boldsymbol{\varphi}_{j}^{(k)}\sum\
olimits_{i=1}^{n}\mathbf{\tilde{x}} _{i}^{(j)}\mathbf{x}^{(i)}(0);\boldsymbol{\varphi}_{j}^{(k)}\) represents the \(k\)th element of the column vector \(\boldsymbol{\varphi}_{j}\); \(\mathbf{\tilde{x}}_{i}^{(j)}\) represents the \(j\)th element of \(\mathbf{\tilde{x}}_{i}\), which can be obtained by inverting the matrix \(\mathbf{V}\), i.e., \(\mathbf{V}^{-1}=[\mathbf{\tilde{x}}_{1},\dots,\mathbf{\tilde{x}}_{n}]\); and \(\mathbf{x}^{(i)}(0)\) is the \(i\)th element of \(\mathbf{x}(0)\). For a real matrix \(\mathbf{A}\), its eigenvalues \(\lambda_{j}\) are real or complex. If \(\lambda_{j}\) is complex, its complex conjugate \(\tilde{\lambda}_{j}\) is also an eigenvalue of \(\mathbf{A}\), and their eigenvectors are conjugate. Here, we denote the real and imaginary parts of \(\lambda_{j}\) as \(\Re(\lambda_{j})\) and \(\Im(\lambda_{j})\), respectively. If \(\Im(\lambda_{j})\
eq 0\), the superimposed exponential response of \(\lambda_{j}\) and \(\tilde{\lambda}_{j}\) can be written as follows:  
\[c_{j}\mathrm{e}^{\lambda_{j}t}+\tilde{c_{j}}\mathrm{e}^{\tilde{\lambda}_{j}t}= 2\big{|}c_{j}\big{|}\mathrm{e}^{\Re(\lambda_{j})t}\cos\big{(}\Im(\lambda_{j})t +\angle c_{j}\big{)} \tag{17}\]  
where \(\big{|}c_{j}\big{|}\) is the magnitude of \(c_{j}\), and \(\angle c_{j}\) is the phase of \(c_{j}\). Equation (17) represents an oscillatory response with angular frequency \(\Im(\lambda_{j})\). If \(\Re(\lambda_{j})<0\), the oscillatory amplitude decreases; if \(\Re(\lambda_{j})>0\), it will increase. If \(\Im(\lambda_{j})=0\), the exponential response of \(\lambda_{j}\) is \(c_{j}\mathrm{e}^{\Re(\lambda_{j})t}\), which diverges or decays exponentially over time. For a linear aeroelastic system, when the free-stream velocity does not reach the flutter boundary, the real parts of the eigenvalues of the system matrix are all negative, and the system response decays. When the flutter boundary is reached, the real parts of one or two eigenvalues will change from negative to zero, and the exponential responses of the other eigenvalues with negative real parts will decay over time, eventually leaving only the response of the eigenvalues with zero real parts. If the imaginary parts of the remaining eigenvalues are zero, this indicates a static divergence; otherwise, it is a typical dynamic aeroelastic flutter. Here, we are mainly concerned with the latter. Linear aeroelastic systems tend to be in two states: convergence or divergence, and there is almost no case of a critical constant-amplitude vibration; there is only an approximate constant-amplitude vibration that can be maintained for a longer time near the flutter boundary. However, for nonlinear systems, amplitude-stabilized LCO is relatively common.