## 4 Energy Distribution  
In this section, we detail the energy distribution between different units for some configurations in series and in parallel described previously. We compute the total energy \(e(t)\) starting from the balance of mechanical energy (or theorem of power expended):  
\[\frac{d}{dt}K(t)+P_{int}(t)=P_{ext}(t), \tag{28}\]  
where \(K(t)\) is the kinetic energy, \(P_{int}(t)\) the stress-power and \(P_{ext}(t)\) the external mechanical power. The balance Equation (28) can be applied to every part of the considered system, in particular, to the single phases or the unit cell. Since a conservative mechanical system is considered, the external mechanical power \(P_{ext}(t)\) can be expressed by virtue of the rate of external potential energy \(U_{ext}(t)\), namely \(P_{ext}(t)=-dU_{ext}(t)/dt\), and the stress power \(P_{int}(t)\) can be expressed by virtue of the rate of internal potential energy (or total strain energy) \(U_{int}(t)\), namely \(P_{int}(t)=dU_{int}(t)/dt\). By means of these assumptions, the balance of mechanical energy implies that the total energy \(e(t)\), sum of the potential energy \(U(t)=U_{int}(t)+U_{ext}(t)\) and the kinetic energy \(K(t)\) is conserved (constant) during a dynamical process:  
\[e(t)=K(t)+U(t)=\text{const}. \tag{29}\]  
Figure 12: Oscillator in series. Distribution of the propagation band frequency \(\Omega\) as a function of the ratios \(\eta_{m}\) and \(\xi_{m}\). Results are given for \(\xi_{c}=0.22\) and \(\eta_{c}=0.5\) with \(\xi_{m}=8.71\) in (**a**) and \(\eta_{m}=1.265\) in (**b**). The grey dashed lines indicate the characteristic frequency of the oscillator. Then, in a continuous phase \(\alpha\), the total energy \(E_{\alpha}\) is obtained averaging \(e(t)\) over a period \(2\pi/\omega\) as:  
\[E_{\alpha}=\frac{1}{2}\mu_{\alpha}k_{\alpha}^{2}(A_{\alpha}\bar{A}_{\alpha}+B_{ \alpha}\bar{B}_{\alpha})d_{\alpha}\,, \tag{30}\]  
where \(A_{\alpha}\) and \(B_{\alpha}\) are the complex amplitudes of the displacement \(U_{\alpha}(x)\), as in Equation (2), and \(\bar{A}_{\alpha}\) and \(\bar{B}_{\alpha}\) their complex conjugates. The complex amplitudes may be computed from the transfer matrices of the single phases or the unit cells; in particular, referring for simplicity to a single continuous phase \(\alpha\) disposed between \(x_{0}\) and \(x_{1}=x_{0}+d_{\alpha}\):  
\[\mathbf{v}_{\alpha}(x_{1})=\mathbf{M}_{\alpha}\mathbf{v}_{\alpha}(x_{0})=e^{ \pm i\phi_{\alpha}}\mathbf{v}_{\alpha}(x_{0}), \tag{31}\]  
where \(\mathbf{v}_{\alpha}(x)=\left[U_{\alpha}(x)\ \tau_{\alpha}(x)\right]^{T}\). Therefore, the transfer matrix \(\mathbf{M}_{\alpha}\) has eigenvalues \(e^{\pm i\phi_{\alpha}}\) corresponding to propagation towards the positive and negative \(x\)-direction, respectively, and eigenvectors \(\mathbf{\tilde{v}}_{\alpha}^{\pm}\). Then, following the representation (2):  
\[\mathbf{\tilde{v}}_{\alpha}^{\pm}=\mathbf{H}_{\alpha}\mathbf{a}_{\alpha}^{\pm },\quad\text{with}\quad\mathbf{H}_{\alpha}=\left[ \begin{array}{cc}e^{ik_{ \alpha}x_{0}}&e^{-ik_{\alpha}x_{0}}\\ ik_{\alpha}\mu_{\alpha}e^{ik_{\alpha}x_{0}}&-ik_{\alpha}\mu_{\alpha}e^{-ik_{ \alpha}x_{0}}\end{array} \right]\,,\,\mathbf{a}_{\alpha}^{\pm}=\left\{ \begin{array} []{cc}A_{\alpha}^{\pm}\\ B_{\alpha}^{\pm}\end{array} \right\}. \tag{32}\]  
Relation (32) can be inverted to obtain the two sets of complex amplitudes \(\mathbf{a}_{\alpha}^{\pm}\) and \(\mathbf{a}_{\alpha}^{-}\). The energy \(E_{\chi_{d}}\) of the single oscillator arranged in parallel can be expressed in term of the amplitudes of the continuous phase 1 of Figure 13, identified with the label \(\alpha\):  
\[E_{\chi_{d}}=\frac{1}{2}\frac{\gamma\,m\omega^{2}(\gamma+m\omega^{2})}{( \gamma-m\omega^{2})^{2}}\bigg{[}\frac{A_{\alpha}\bar{A}_{\alpha}+B_{\alpha} \bar{B}_{\alpha}}{2}+\mathcal{R}e(A_{\alpha}\bar{B}_{\alpha}e^{i2\phi_{\alpha }})\bigg{]}. \tag{33}\]  
Figure 13: Energy distribution between different phases as a function of the radian frequency \(\omega\).