## 2 Mechanical Model
### Free Body Diagram of SA-TLGD  
The single tube SA-TLGD is considered in a deflected position separated from the floor of the SDOF-main system under horizontal floor excitations \(w_{t}=w_{g}+w\) (Figure 2). The horizontal orientated tube with diameter \(A_{T}\) is sealed at both ends and filled with a fluid (density \(\rho_{f}\)) over the length \(L_{f}\), i.e., the total fluid mass yields to \(m_{f}=\rho_{f}\,A_{T}\,L_{f}\). The relative motion of the fluid interface to gas is described by the displacement \(u_{1}=u_{2}=u(t)\). A large deformable elastomeric membrane with neglectable stiffness is used as the interface. To formulate the SA-TLGDs dynamic, the shape of the membrane in static equilibrium is approximated as a pure vertically orientated surface. During the vibration process, the gas inside the air chambers \(V_{0i}\) is quasi-statically compressed and released by the moving fluid interface in relatively slow motion. A pressure difference \(\Delta p=p_{2}\,-p_{1}\) is created between the left and right gas chambers and it changes the undamped circular natural frequency of the SA-TLGD defined in Equation (8). The absolute acceleration of the chosen reference Point \(P\) in the prescribed rigid body motion is given by  
\[\bar{w}_{t}=\bar{w}_{g}+\bar{w}. \tag{1}\]  
To derive the nonlinear equation of motion, the floor excited SA-TLGD is transformed into an equivalent single-mass oscillator [39]. Thereby, the fluid mass \(m_{f}=\rho_{f}\,A_{T}\)\(L_{f}\) represents the equivalent mass \(M^{*}\) of the harmonic oscillator. Neglecting the stiffness contribution of the elastic membranes, the equivalent spring stiffness \(K^{*}\) results solely from the gas spring effect. It is assumed that the gas pressure \(p_{0}\) is present in both gas chambers in the static equilibrium position of the fluid. \(p_{0}\) can either be chosen equal to the atmospheric pressure (\(p_{0}\cong 10^{5}\) Pa) or to any other arbitrary desired value (negative or positive pressure). The initial size of the gas volume \(V_{0}\) enclosed on the left and right sides of the fluid in the static equilibrium position is exposed to a constant change in compression and expansion, i.e., the size of \(V_{0}\) fluctuates as a function of the horizontal dynamic deformation \(u(t)\) of the fluid mass. Hence, the time varying size of \(V_{0}\) leads to a change in the pressure state of the gas volume and in consequence of a restoring effect of the deflected fluid mass. The resulting gas pressure difference \(\Delta p=p_{2}\,-p_{1}\) approximately follows the quasistatic polytropic law [39] (\(\rho_{0}\) is the gas density in the static equilibrium position of the fluid mass),  
\[\Delta p\ =\ p_{0}\,\left(\frac{\Delta\rho}{\rho_{0}}\right)^{n}\ =\ p_{0}\,\left(\frac{V_{0}}{\Delta V}\right)^{n},\, \Delta V=V_{2}-V_{1},\ V_{0}=\sum_{i=1}^{k}V_{i}. \tag{2}\]  
where \(n\) denotes the polytropic coefficient and \(k\) is the number of conducted separated gas chambers \(V_{i}\). In the case of very slow fluid velocities (low eigenfrequency) the gas spring acts approximately under isothermal conditions \(n=1.0\), while in the case of higher velocities an adiabatic condition occurs and the coefficient changes to a value of \(n=1.4\). In any other circumstances, \(n\) takes a value in the range between \(1.0\) and \(1.4\)[25].