## 1 Introduction
### Wave-Based Modeling of Joints: Scattering Relations
#### 2.2.2 Periodic Conditions between the Unit Cells and Scattering Relations at Exterior Joints  
At the exterior joint pair \(1\) and \(N\) where the current unit cell is rigidly connected to the left and right adjacent unit cells, periodic conditions exist between the generalized displacements (forces) of joint pairs due to the Floquet-Bloch principle [1; 2; 3]. Further considering the coupling conditions between the generalized displacements (forces) of the exterior joint and the corresponding generalized displacements (forces) of the correlated end member at the section of the joint, the spectral periodic conditions between the unit cells [77; 78] are written as:  
\[\begin{array}{c}-\phi^{N(N-1)}(0)=\hat{u}_{YR}=\mathrm{e}^{\mathrm{i}qI}\hat{u }_{YL}=\mathrm{e}^{\mathrm{i}qI}\hat{\varrho}^{12}(0),\,\hat{\varphi}^{N(N-1)}( 0)=\hat{\theta}_{ZR}=\mathrm{e}^{\mathrm{i}qI}\hat{\varrho}_{ZL}=\mathrm{e}^{ \mathrm{i}qI}\hat{\varrho}^{12}(0);\\ \hat{Q}^{N(N-1)}(0)=\hat{p}_{YR}=-\mathrm{e}^{\mathrm{i}qI}\hat{p}_{YL}= \mathrm{e}^{\mathrm{i}qI}\hat{Q}^{12}(0),\,-\hat{M}^{N(N-1)}(0)=\hat{m}_{ZR}=- \mathrm{e}^{\mathrm{i}qI}\hat{n}_{ZL}=\mathrm{e}^{\mathrm{i}qI}\hat{M}^{12}(0) ;\end{array} \tag{9}\]  
where \(q\) is the wavenumber of the characteristic wave in the periodic beam, and \(l\) is the length of the unit cell. \(\mathrm{i}ql=\mu\) is known as the complex propagation constant, with its imaginary part \(q_{R}l=\mu_{I}\) and real part \(-q_{l}l=\mu_{R}\) are the well-known phase constant and attenuation constant, respectively.  
Introduction of the solutions to the generalized displacements and forces of members provided respectively in Equations (3) and (5) into Equation (9) leads to the local scattering relations of joint pairs \(1\) and \(N\). These scattering relations are written as:  
\[\left[\begin{array}{cc}\mathbf{A}^{1}(\omega,q)&\mathbf{A}^{1N}(\omega,q)\\ \mathbf{A}^{N1}(\omega,q)&\mathbf{A}^{N}(\omega,q)\end{array}\right]\left\{ \begin{array}{c}\mathbf{a}^{1}\\ \mathbf{a}^{N}\end{array}\right\}+\left[\begin{array}{cc}\mathbf{D}^{1}( \omega,q)&\mathbf{D}^{1N}(\omega,q)\\ \mathbf{D}^{N1}(\omega,q)&\mathbf{D}^{N}(\omega,q)\end{array}\right]\left\{ \begin{array}{c}\mathbf{d}^{1}\\ \mathbf{d}^{N}\end{array}\right\}=\left\{\begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array}\right\}, \tag{10}\]  
where \(\mathbf{a}^{1}=[a_{2}^{12},a_{3}^{12}]^{\mathrm{T}}\) (\(\mathbf{a}^{N}=[a_{2}^{N(N-1)},a_{3}^{N(N-1)}]^{\mathrm{T}}\)) and \(\mathbf{d}^{1}=[d_{2}^{12},d_{3}^{12}]^{\mathrm{T}}\) (\(\mathbf{d}^{N}=[d_{2}^{N(N-1)},d_{3}^{N(N-1)}]^{\mathrm{T}}\)) are the arriving and departing wave vectors of exterior joint \(1\) (\(N\)), respectively, and the corresponding partitions of the coefficient matrices are:  
\[\begin{array}{c}\mathbf{A}^{1}=\mathrm{e}^{\mathrm{i}qI}\left[\begin{array} []{cc}1&1\\ \beta_{2(1)}&\beta_{3(1)}\end{array}\right],\mathbf{A}^{N1}=\mathrm{e}^{ \mathrm{i}qI}\left[\begin{array}{cc}\zeta_{2(1)}&\zeta_{3(1)}\\ \mu_{2(1)}&\mu_{3(1)}\end{array}\right],\,\mathbf{A}^{N}=\left[\begin{array} []{cc}1&1\\ -\beta_{2(m)}&-\beta_{3(m)}\end{array}\right],\,\mathbf{A}^{N}=\left[\begin{array} []{cc}-\zeta_{2(m)}&-\zeta_{3(m)}\\ \mu_{2(m)}&\mu_{3(m)}\end{array}\right];\\ \mathbf{D}^{1}=\mathrm{e}^{\mathrm{i}qI}\left[\begin{array}{cc}1&1\\ -\beta_{2(1)}&-\beta_{3(1)}\end{array}\right],\,\mathbf{D}^{N1}=\mathrm{e}^{ \mathrm{i}qI}\left[\begin{array}{cc}-\zeta_{2(1)}&-\zeta_{3(1)}\\ \mu_{2(1)}&\mu_{3(1)}\end{array}\right],\,\mathbf{D}^{1N}=\left[\begin{array} []{cc}1&1\\ \beta_{2(m)}&\beta_{3(m)}\end{array}\right],\,\mathbf{D}^{N}=\left[\begin{array} []{cc}\zeta_{2(m)}&\zeta_{3(m)}\\ \mu_{2(m)}&\mu_{3(m)}\end{array}\right]\end{array} \tag{11}\]  
with \(\Gamma_{r(1)}\) and \(\Gamma_{r(m)}\) (\(\Gamma=\beta,\zeta,\mu,\,r=2,3\)) denoting the influence coefficients of rotation, shear force and bending moment for members (1) and \((m)\) associated with \(r\) wave mode, respectively.