## 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
This can be approximated as the critical flutter state of the equivalent linear system from the point of view of the DF method. It should be noted that the state-space equation Equation (13) is only applicable to the stage of amplitude-stabilized LCO, and it cannot be applied to the previous transition stage. Here, we assume that the nonlinear system already has amplitude-stabilized LCO at time \(t=0\), and we refer to \(\mathbf{x}(0)\) as a harmonic initial condition. Therefore, the response of the \(k\)th state variable \(\mathbf{x}^{(k)}(t)\) can be written as:  
\[\mathbf{x}^{(k)}(t)=c_{s}\mathrm{e}^{\lambda_{s}t}+c_{s+1}\mathrm{e}^{\lambda_{s +1}t} \tag{18}\]  
where: \(c_{s}=\mathbf{\varphi}_{s}^{(k)}\sum_{i=1}^{n}\mathbf{\xi}_{i}^{(s)}\mathbf{x}^{(i)}(0)\); \(c_{s+1}=\mathbf{\varphi}_{s+1}^{(k)}\sum_{i=1}^{n}\mathbf{\xi}_{i}^{(s+1)}\mathbf{x}^{(i) }(0)\); \(\lambda_{s}=\mathrm{i}\omega\) and \(\lambda_{s+1}=-\mathrm{i}\omega\) are the eigenvalues with zero real part in the equivalent linear system; \(\omega\) is the angular frequency of the LCO; is the imaginary unit. Equation (18) indicates that the output response does not contain the exponential responses of the eigenvalue whose real part is not zero. Referring to Equation (16), \(\mathbf{x}(0)\) should satisfy the following conditions:  
1. \(\sum_{i=1}^{n}\mathbf{\xi}_{i}^{(j)}\mathbf{x}^{(i)}(0)=0\) when \(j\
eq s\) and \(j\
eq s+1\);
2. \(\mathbf{\varphi}_{s}\sum_{i=1}^{n}\mathbf{\xi}_{i}^{(s)}\mathbf{x}^{(i)}(0)+\mathbf{\varphi}_ {s+1}\sum_{i=1}^{n}\mathbf{\xi}_{i}^{(s+1)}\mathbf{x}^{(i)}(0)=\mathbf{x}(0)\). A general solution of the harmonic initial condition \(\mathbf{x}(0)\) is as follows:  
\[ \begin{split}\mathbf{x}(0)&=a\Big{(}\mathbf{\varphi}_{s} \mathrm{e}^{\mathrm{i}\phi_{c}}+\mathbf{\varphi}_{s+1}\mathrm{e}^{-\mathrm{i}\phi_ {c}}\Big{)}\\ &=2a\big{[}\Re(\mathbf{\varphi}_{s})\mathrm{cos}(\phi_{c})+\Im(\mathbf{ \varphi}_{s})\mathrm{sin}(\phi_{c})\big{]}\end{split}  \tag{19}\]  
where \(a\in\mathbf{R}^{+}\) and \(\phi_{c}\in[0,2\pi)\). The calculation of \(a\) is determined by the given LCO amplitude. For example, the given LCO amplitude in the bending DOF is \(A_{\mathrm{B}}\), which means that \(A_{\mathrm{B}}=2a\Big{|}\mathbf{\varphi}_{s}^{(0)}\Big{|}\), and we can calculate \(a=A_{\mathrm{B}}\big{/}\big{(}2\Big{|}\mathbf{\varphi}_{s}^{(0)}\Big{|}\Big{)}\). Here we call \(\phi_{c}\) the phase parameter of the harmonic initial condition. Its value is optional and different values correspond to different phases of the system at time \(t=0\). In this paper, the value of \(\phi_{c}\) is set to \(0\) or \(\pi/2\). Substituting Equation (19) into Equation (18) yields  
\[\mathbf{x}^{(k)}(t)=2a\Big{|}\mathbf{\varphi}_{\rm s}^{(k)}\Big{|}\cos\Bigl{(}\omega t+ \mathbf{\phi}_{\rm c}+\angle\mathbf{\varphi}_{\rm s}^{(k)}\Bigr{)} \tag{20}\]  
Therefore, for the all-movable fin in this paper, \(\mathbf{x}=[q_{\rm B},q_{\rm T},\dot{q}_{\rm B},\dot{q}_{\rm T}]^{T}\), and the LCO amplitude ratio \(R\) in the bending and torsional DOFs can be written as:  
\[R=\frac{2a\Big{|}\mathbf{\varphi}_{\rm s}^{(0)}\Big{|}}{2a\Big{|}\mathbf{\varphi}_{\rm s }^{(1)}\Big{|}}=\frac{\Big{|}\mathbf{\varphi}_{\rm s}^{(0)}\Big{|}}{\mathbf{\varphi}_{ \rm s}^{(1)}\Big{|}} \tag{21}\]  
which also represents the modal amplitude ratio of the equivalent linear system in the flutter state. The LCO phase difference \(\Delta\mathbf{\phi}\) between the bending and torsional DOFs is as follows. \[\Delta\mathbf{\phi}=\mathbf{\phi}_{\rm B}-\mathbf{\phi}_{\rm T}=\angle\mathbf{\varphi}_{\rm s }^{(0)}-\angle\mathbf{\varphi}_{\rm s}^{(1)} \tag{22}\]  
The above analysis is based on the harmonic oscillation assumption of the DF method. The essence of the harmonic initial condition is to select a state point on the LCO trajectory as the initial condition of time-domain integration according to the amplitude and phase information of the generalized coordinates of the system under the assumed harmonic oscillation. However, this harmonic oscillation assumption is approximate for actual nonlinear systems.