## 3 The Contact Model and Its Piecewise Linearity  
The contact forces at a particular instant in time and at a given contact node pair \(\mathbf{f_{i}^{*}}\) are computed starting from the relative displacements at the contact \(\mathbf{q_{iIP}^{*}}-\mathbf{q_{iID}^{*}}\) using a friction contact model with normal load variation that is widely used in the nonlinear forced response analysis of frictional constraint structures [18; 22], see also Figure 3. The time-discrete implementation facilitates the contact model application in time-marching solution methods. The main drawback is that contact state transitions are captured with finite accuracy, i.e., the length of the chosen time step \(\Delta\tau\). This has, however, negligible effects on the solution for the range of investigated \(\Delta\tau\). Figure 3 shows the contact element in case of 2D motion. In case of 3D motion, e.g., see Figure 2a, it is assumed that the in-plane tangential components of the contact force are independent from each other (i.e., quasi-3D). In other words, an uncoupled in-plane motion/force coordinate, here termed \(z-Z\), orthogonal to \(t-T\), is added. This assumption is commonly used in the calculation of the nonlinear forced response of structures with frictional constraints, see [20; 41; 42; 43]  
Each contact element can undergo, alternatively, three different contact states. * Lift-off: There is a gap between the platform and damper contact points (i.e., \(\mathrm{n_{iP}}-\mathrm{n_{iD}}\leq 0\)), and contact forces are null \(\mathrm{T}=\mathrm{N}=0\). * Stick: There is interference between the platform and damper contact points (i.e., \(\mathrm{n_{iP}}-\mathrm{n_{iD}}>0\)). Furthermore, tangential relative displacement is limited, and a proportional relation between tangential forces and displacements exists: \(\mathrm{T_{i}}=\mathbf{k_{t}}\cdot(\mathrm{t_{IP}}-\mathrm{t_{iD}}-\mathrm{s_ {i}})\), where \(\mathrm{s_{i}}\) is the relative displacement of the tip of the slider with respect to the contact point on the damper. * Slip: There is interference between the platform and damper contact points (i.e., \(\mathrm{n_{iP}}-\mathrm{n_{iD}}>0\)). Furthermore, the tangential relative displacement exceeds a threshold value, and a proportional relation between tangential and normal forces exists: \(T_{i}=\mu\cdot N_{i}\). The full set of conditions for the three contact states described above can be found in [11]. The standard implementation of this contact model sees a \"black-box-like\" subroutine, which receives as input the displacement vectors \(\mathbf{q}_{\mathbf{i}\mathbf{P}}^{*}\), \(\mathbf{q}_{\mathbf{i}\mathbf{D}}^{*}\) at the time step \(\tau\) and the slider displacement \(\mathbf{s}_{\mathbf{i}}\) at time \(\tau-\Delta\tau\) and gives, as output, the contact forces \(\mathbf{f}_{\mathbf{i}}^{*}=\left(\mathrm{T}_{\mathbf{i}},\mathrm{N},\mathrm{ i}\right)^{\prime}\). Inside the sub-routine a predictor-corrector scheme (see Figure 3) checks the inputs against a series of \"conditions\" in order to assign the correct contact state and the corresponding values of contact forces. This paper proposes a novel alternative to implementing the contact model constitutive equations. The contact state transition conditions and contact forces expressions are unchanged, so the ultimate result is the same. However, this novel formulation brings out an important feature of the chosen kind of contact model: **nonlinearity is obtained through a time sequence of linear systems**. The nonlinear system is actually a piecewise-linear system. With reference to Equation (3), let us consider a specific instant in time. The local vector of contact forces \(\mathbf{f}_{\mathbf{i}}^{*}(\tau)\) is substituted by the product of the contact stiffness matrix \(\mathbf{K}_{\mathbf{i}}(\tau)\) and the vector of local relative displacement at the contact \(\mathbf{q}_{\mathbf{i}\mathbf{P}}^{*}(\tau)-\mathbf{q}_{\mathbf{i}\mathbf{D}} ^{*}(\tau)\), plus a contribution from the slider.