## 3 Results and Discussions
### Effect of Damping  
In the above parameter analysis process, damping is not considered, but damping exists in actual structures. Therefore, this section will focus on the vibration characteristics of damped structures. When damping is taken into account, the governing equation of the forced vibration of the lattice sandwich beam can be expressed as follows:  
\[\mathbf{M\ddot{X}}+\mathbf{C\dot{X}}+\mathbf{K}\mathbf{X}=\mathbf{B}_{0}f(t) \tag{18}\]  
where \(\mathbf{M}\) and \(\mathbf{K}\) are the mass and rigidity matrices of lattice sandwich beams, respectively, which can be calculated by the expression in Appendix A. \(\mathbf{C}\) is the viscous damping matrix, \(f\)(t) is an external excitation and B0 is the position vector of the external excitation.  
\[\mathbf{C}=\left(\phi^{T}\right)^{-1}\mathbf{C}_{0}(\phi)^{-1} \tag{19}\]  
\[M_{\mathrm{m}}=\phi_{m}^{T}\mathbf{M}\phi_{m} \tag{20}\]  
\[\mathbf{C}_{0}=\left[\begin{array}{cccc}2\omega_{1}\zeta_{1}M_{1}&\\ &2\omega_{2}\zeta_{2}M_{2}&\\ \vdots&\ddots&\vdots\\ 0&\cdots&2\omega_{m}\zeta_{m}M_{m}\end{array}\right] \tag{21}\]  
Figure 21: The relationship between the inclination angle and the equivalent transverse shear modulus of lattice sandwich beams under different truss cores.  
Figure 20: The relationship between the inclination angle and the equivalent density of lattice sandwich beams under different truss cores.  
where \(C_{0}\) is the generalized damping matrix, \(\omega_{m}\) is the \(m\)th order natural frequency of lattice sandwich beams, \(M_{m}\) is the \(m\)th order generalized mass, \(\phi\) is the vibration mode matrix, \(\phi_{m}\) is the vibration mode vector corresponding to the \(m\)th order natural frequency and \(\zeta_{m}\) presents the \(m\)th order modal damping ratio. Generally, the modal damping ratio needs to be measured according to experimental data. For the convenience of research, \(\zeta_{m}\) will be assumed to be 0.005 and 0.01 for all generalized coordinates in this section.  
In order to solve the vibration response of lattice sandwich beams subjected to an initial external excitation, it is first necessary to transform Equation (18) into the state space equation form:  
\[\mathbf{Z}=\left[\dot{\mathbf{X}},\mathbf{X}\right] \tag{22}\]  
\[\dot{\mathbf{Z}}=\left[\begin{array}{cc}0&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array}\right] \mathbf{Z}+\left[\begin{array}{cc}0\\ -\mathbf{M}^{-1}\mathbf{B}_{0}f(t)\end{array}\right] \tag{23}\]  
\[\mathbf{B}_{0}=\left[\left(\cos\lambda_{r}x_{a}\right)^{T}\left(\sin\lambda_ {s}x_{a}\right)^{T}0\ 0\right]^{T} \tag{24}\]  
where \(x=\left[w^{T},\theta^{T}\right]^{T}\) is the generalized coordinate vector, \(x_{a}\) represents the external excitation position on the lattice sandwich beam and \(r=0,\cdots,M,s=1,2,3,4\).  
The natural frequency of the structure remains constant when damping is considered. Therefore, the viscous damping has no effect on the mass and rigidity of the structure. In this section, the transient exciting force of 10N lasting for only 0.001s is applied on the mid-point of the lattice sandwich beam. Since the force applied is small in magnitude and time, the energy generated is small. Therefore, after the excitation force disappears, the structure will be in a state of free vibration. On this basis, the \(lsim\) function in Matlab is used to calculate the time history curve at the mid-point of the lattice sandwich beam. As can be seen in Figures 22 and 23, when damping is not considered, the displacement amplitude of the lattice sandwich beam does not change with time. However, when the influence of damping is considered, the displacement amplitude of the lattice sandwich beam will continue to decrease with time and, the greater the damping, the more the attenuation.  
Figure 22: Time-response history at the middle point of the lattice sandwich beam, when \(\zeta=0\) and \(\zeta=0.005\).