## 4 Our Current Work and Contributions  
Based on the literature review, there are two vibration isolation systems proposed by [PERSON] et al. [46] and by [PERSON] et al. [50] where a scissors-like structure (SLS) is used to provide the QZS characteristics. The designs have met the vibration isolation performance criteria well because the passive vibration isolators are able to attenuate the vibrations from multiple directions. The scissor-like structure as a support leg introduces nonlinearity, which helps to achieve a trade-off between the load capacity and vibration isolation performance. The most important feature of the scissor-like structure is that it can attenuate the vibrations of the 6 DOFs, which has been verified by the results in [46]. Our research aims to 3D-print a seat cushion mattress/structure. In order to investigate the vibration transmissibility of the seat cushion mattress/structure, the cell/unit/element structure of the 3D-printed seat cushion mattress has to be studied. A double-diamond structure (SLS supporting leg) is selected as the cell/unit/element structure of the 3D-printed seat cushion mattress. The double-diamondisolator structure (SLS supporting leg) supporting a mass is compared to a linear spring isolator structure supporting a mass for their vibration transmissibility using the analytical method and Adams simulation software model where the SLS supporting leg has the number of diamond structure elements of \(n=2\), the diamond sidebar length of \(L=80\) mm, and the assembly angle of \(\theta=30^{\circ}\), as shown in Figure 14. The double-diamond isolator structure is selected as one of the cells or units of the seat cushion mattress/structure, which does not represent the whole seating suspension system. In order to further study the transmissibility ratio of the double-diamond isolator element structure, the analytical model will have to be developed. First, the kinematic energy of the system, the elastic potential energy of the spring deformation, and the friction dissipation energy of the joints are established. Second, the Lagrange principle is applied to develop the dynamic equation of the single-degree-of-freedom system where three complex nonlinear functions of the oscillator displacement related to the coefficients of the rotational inertia of the sidebars, the damping coefficients of the joints, and the stiffness coefficient of the spring have been established. Third, the three complex nonlinear functions of the oscillator displacement are calculated from the lateral displacement of the left- and right-hand-side joints and the relative rotational angle of the sidebars. The lateral displacement of the left- and right-hand-side joints and the relative rotational angle of the sidebars are functions of the oscillator displacement and are then expanded as a Taylor series of the first and second-order terms of the oscillator displacement where their coefficients are calculated from differentiating the geometric relationship of the double-diamond structure twice and the initial conditions of the structure. Fourth, the equation of motion is then simplified as a non-linear dynamic equation with the linear, squared, and cubic displacement terms. The non-linear dynamic equation can be solved and simulated through the Range-Kutta numerical integration method to obtain the vibration transmissibility ratio of the double-diamond isolator structure. The linear, squared, and cubic displacement terms can also be linearized to an equivalent linear displacement term. The vibration transmissibility ratio through the double-diamond isolator structure can be calculated from the Fourier transform and displacement frequency response function. The coefficient of the equivalent linear displacement term can be identified through matching the peak values of the vibration transmissibility ratios calculated from the Range-Kutta numerical integration method of the non-linear dynamic equation with those calculated from the Fourier transform and displacement frequency response function methods.