## 3 Sabot-Stopping System Conceptual Design
### Energy Damping System Design  
This section will describe a set of simulations carried out on the most critical energy damping component, i.e., the rubber cylinder. In an effort to understand more about the deformation and the stresses in the energy damper rubber cylinder, a non-linear dynamic hyper-elastic FEM of a 20 mm long partial section was created and shown in Figure 9. In these preliminary studies, only a section of the rubber cylinder was analysed to reduce computational resources via high element count. The main goal of these studies is to help specify the appropriate rubber material that can withstand the impact of the sabot with projectile for multiple consecutive impacts. The predicted deformation can also help with designing a key component that will not deform such that it will be placed in the path of the projectile and cause contamination of the target impact. The dimensions chosen are a first iteration to study the behaviour of the component; the results will be used to guide the final design. For these initial studies, 5-parameter Mooney Rivlin constants (C10 = \(-\)0.55 MPa, C01 = 0.7 MPa, C20 = 1.7 MPa, C11 = 2.5 MPa, C02 = \(-\)0.9 MPa) were adopted and taken from the ANSYS materials library. A mass density of 1000 kg/m\({}^{3}\) and Poisson's ratio of 0.49 was assigned and amounted to a rubber cylinder mass of 145 g. Rayleigh damping constants for the mass (\(\alpha\)) and stiffness (\(\beta\)) matrix were taken as 1.26 and 8.69 \(\times\) 10\({}^{-6}\), respectively. The sabot and projectile, having a mass of 40 g, will make contact with the impact tube, transferring the full momentum to the energy damping system (rubber cylinder). This is a conservative assumption. It is also assumed that the impact tube and the rubber cylinder will share the same impact surface dimensions. The velocity of this assembly is taken as 250 m/s and amounts to 1250 J. Equating strain energy to work and using an estimated travel of 0.003 m, the impact force is taken as 416.6 kN. The rubber disk is held in a cylindrical metal tube holder and is pressed stationary against the exiting wall. The transfer of momentum upon impact between the sabot and rubber disk causes the rubber disk to compress along its longitudinal axis; but also, through conservation of volume, a reduction of the internal rubber tube diameter occurs. The full surface contact is assumed between the sabot and the rubber disk. In the FEM, the impact force is applied to all nodes on face 1 and acting in the direction along the longitudinal axis, while the opposite end of the disk (face 2) is fixed in all D.O.F., representing the fixed exiting wall surface. Nodes on the surface of the outer cylinder wall are fixed in all D.O.F. of the cylinders longitudinal axis, i.e., the external diameter is fixed, representing the metal tube wall holder. Figure 10 shows the model boundary conditions. The duration of the solution time is based on the period, i.e., 1/eigenfrequency. This period was then scaled by a factor of three to ensure capture of three full waves. A modal study predicted the first significant mode  
Figure 9: Rubber cylinder (20 mm section). (70% mass participation) at 533 Hz, resulting in a solution time of 5.62 ms. The time-step was then estimated by dividing the maximum element dimension (4 mm) in the direction of the wave propagation divided by the speed of sound in rubber, taken as 39 m/s, and this amounted to a time-step of 0.1 ms. This technique is taken from the literature [49]. Mesh density sensitivity was considered for each model. Element size, element growth ratio, and model discretization of the curved surfaces were adjusted and inspected for the maximum element aspect ratio. The meshing of the model employed blended curvature-based solid tetrahedral 3D elements with 16 Jacobian points, resulting in approximately 160 k elements and 227 k nodes, with a maximum element size of 2 mm. An element growth ratio of 1.4 was utilized, and the maximum aspect ratio was 4; 99.8% of elements remained below 3.