## 2 Mechanical Model
### Free Body Diagram of SA-TLGD
During the state of fluid mass vibration, the actual size of gas volume at the left and right gas chambers is defined as (Figure 2a),  
\[V_{1}\ =\ V_{0}+A_{T}\,u,\ V_{2}\ =\ V_{0}-A_{T}\,u. \tag{3}\]  
Inserting the expressions for the gas volumes \(V_{1}\) and \(V_{2}\) into Equation (2) yields the nonlinear gas pressure difference,  
\[\Delta p(u)\ =\ p_{0}\,\left[\left(\frac{V_{0}}{V_{0}-A_{T}\,u}\right)^{n}\ -\ \left(\frac{V_{0}}{V_{0}+A_{T}\,u}\right)^{n}\right]. \tag{4}\]A linearization of Equation (4) is obtained using a Taylor series expansion of the nonlinear function with respect to the equilibrium pressure \(p_{0}\) while neglecting the higher order terms,  
\[\Delta p(u)=\frac{2\;n\;p_{0}\;A_{T}}{V_{0}}\;u+O\;\Big{(}u^{3}\Big{)}\;\approx\; \frac{2\;n\;p_{0}}{L_{g}}\;u, \tag{5}\]  
where \(L_{g}=V_{0}/A_{T}\) defines the horizontal length of the gas spring at the left and right gas chambers of the SA-TLGD (Figure 2a). From Equation (5) and the relation \(K^{*}\;u=\Delta p\;A_{T}\) the single-mass oscillator equivalent spring stiffness \(K^{*}\) results as  
\[K^{*}=\;\frac{2\;n\;p_{0}\;A_{T}}{L_{g}}. \tag{6}\]  
The deviation of the exact solution for the nonlinear relative pressure difference \(\Delta p/p_{0}\) (Equation (4)) from the linearized solution (Equation (5)) is illustrated in Figure 3 as a function of the dimensionless relative displacement \(u(t)/L_{g}\). The polytropic coefficient was chosen with \(n=1.2\), i.e., mean value of isotherm and adiabatic state change. It is indicated that up to \(u(t)/L_{g}\leq 0.30\) the deviation to the exact nonlinear solution is insignificantly small, i.e., linearization is permissible in this range of the vibration amplitude. In the case of very large vibration amplitudes \(u(t)/L_{g}>0.50\), the restoring force of the nonlinear gas spring differs significantly from the linear solution and, therefore, leads to a disadvantageous detuning of the SA-TLGD eigenfrequency. Hence, when designing the SA-TLGD the relation of the maximum fluid mass displacement \(u(t)\) to the chosen gas spring length \(L_{g}\) is an important tuning parameter to achieve the optimum vibration damping effect. For this reason, the maximum amplitude of gas compression is limited to \(max\;|u(t)|=U_{max}<0.30\;L_{g}\) to keep the eigenfrequency of the SA-TLGD approximately constant. After derivation of the equivalent mass \(M^{*}=m_{f}=\rho_{f}\;A_{T}\;L_{f}\) and stiffness \(K^{*}\) given in Equation (6) of a single-mass oscillator, the nonlinear equation of motion for the floor excited SA-TLGD yields to,  
\[\ddot{u}+\delta_{L}|\dot{u}|\;\dot{u}+\omega_{A}^{2}\;u\;=\;-\ddot{w}_{t}, \tag{7}\]  
where an averaged turbulent damping term, that must be experimentally verified, has been added. The head loss coefficient \(\delta_{L}\) can be increased by properly selecting the diameter of the built-in orifice plate (Figure 2a). In the case of stationary flow, \(\delta_{L}\) is tabulated for  
Figure 3: Comparison of linear and nonlinear gas spring stiffness function \(\Delta p/p_{0}\). relevant pipe elements and cross-sectional areas in [40]. In Equation (7), the undamped circular natural frequency \(\omega_{A}\) of the both side sealed SA-TLGD is defined as,  
\[\omega_{A}=\sqrt{\frac{2\ n\ p_{0}\ A_{T}}{m_{f}\ L_{g}}}. \tag{8}\]  
In Equation (8) \(n\), \(p_{0}\), \(A_{T}\), \(m_{f}\), and \(L_{g}\) denote the polytropic coefficient (\(1.0\leq n\leq 1.4\)), the initial equilibrium pressure in the gas chambers, the cross-sectional area of the tube, the fluid mass, and the total gas spring length. \(L_{g}=V_{0}/A_{T}\) is an important design variable of the SA-TLGD, defining the volume of the gas chamber in terms of the cross-sectional area. The re-optimization of the damper eigenfrequency is performed during the state of operation by semi-actively controlling the appropriate number of either closed or open valves, i.e., adjusting the size of the gas volume \(V_{0}\).