## 6 Results and Discussion
### Free Rocking on a Rigid Base  
Based on the previously performed convergence tests, herein, the mesh of six elements and the time-step of \(10^{-5}\) s are used. In each test, an arbitrary inclination angle is chosen since it has no significant influence on the rocking behaviour, and the reported value was measured from the experiment's initial conditions. Obviously, it was considered that the initial angle does not exceed the value \(\alpha\), in which case immediate overturning would occur. First, we consider the example involving block B6M with a slenderness ratio of 4.5, which corresponds to slender rocking bodies. In the initial configuration, the block is inclined around one of its corners touching the base at an angle of \(-11.268^{\circ}\) with respect to the stable vertical position and is released from rest and left to move under the action of gravitational force. Rotation time histories obtained experimentally and numerically are given in Figure 12. It can be seen that after the first impact, a delay is present in the numerical result with respect to the experimentally obtained ones. The experimentally measured block angle after the first impact was \(-6.92^{\circ}\), while the simulation predicted an angle of \(-7.76^{\circ}\). Nonetheless, good overall behaviour is predicted by the simulation, in terms of both tilt amplitude and decay. Both time histories reach a rest state around 3.5 s. The corresponding angular velocity time histories for this block are shown in Figure 13, and observations similar to the ones made for the block angle can also be noticed when analyzing angular velocities. Since an \(h/b\) slenderness ratio equal to 4 is usually taken as the limit to considering slender rocking bodies, the analysis of block B6M (\(h/b=4.5\)) is complemented by an analysis of two additional blocks: one considered bulky (block B3M with \(h/b=3\)), and one considered significantly slender (block B8M with \(h/b=8\)). Figure 14 gives a comparison between numerical and experimental results for the block with the smallest slenderness ratio \(-b/h=3.0\). The initial rotation angle is \(-14.839^{\circ}\). This block is considered bulky, and inverted-pendulum-based models (e.g., Housner's Housner (1970)) are considered unable to simulate its behaviour accurately enough. From Figure 14, we can notice that the overall time history of the block rotation angle is qualitatively well predicted by the numerical analysis. The smaller number of vertex impacts is successfully predicted. In the simulation, a faster decrease in the rocking _period_ (the time between two impacts at each rocking _half-cycle_) is obtained with respect to the experiment. The numerically predicted block rotation angle is similar to the one measured in the experiment; they are, respectively, \(-8.28^{\circ}\) and \(-8.35^{\circ}\). Figure 11: Analysis of the influence of FEM mesh density on the rocking response. Figure 14: Block B4M—experimental and numerical results for the block rotation angle in time. Figure 12: Block B6M—experimental and numerical results for the block rotation angle in time. Figure 13: Block B6M—experimental and numerical results for the block angular velocity in time. The last block from the experimental campaign is the B8M block. This is the most slender of the three, and therefore, it exhibits shorter rocking _periods_, as can be seen in Figure 15. The initial rotation angle was \(-7.731^{\circ}\). For this example, differences start to appear already after the first impact (somewhere around 0.5 s), with the simulation predicting a maximum angle of \(-6.18^{\circ}\) after the first impact, while in the experiment, an angle of \(-5.16^{\circ}\) was measured. The delay of the numerical results with respect to the experiment increases with time. However, we can note that the overall behaviour is predicted well, particularly the decay rate of the inclination angle when a larger number of impacts occur.