## 2 Problem Description
For the block, the contact forces have been reduced to the centre of gravity; therefore, a moment of \(r_{x}\) and \(r_{y}\) with respect to the centre of gravity is added in (10) and the index \(k\) in (8) and (9) indicates the force is acting at \(k\) (Figure 3a). In the contact formulation, it is convenient to express the contact forces in a local reference frame with unit vectors \(\vec{n}\) and \(\vec{t}\) as depicted in Figure 5, in which points \(b\) and \(d\) form a contact pair. The origin of the local frame of reference is at the beam's contact point \(b\). The unit normal vector \(\vec{n}\) is normal to the beam axis at \(b\) and is directed toward the block, while the tangential axis (unit vector \(\vec{t}\)) is rotated counter-clockwise with respect to \(\vec{n}\) and is orthogonal to \(\vec{n}\). In this work, a frictional contact model is used, and in order to formulate it, we need the following quantities:  
* \(g\)--gap function between the contacting bodies defined as \(g=\overline{bd}\), i.e., the distance between \(b\) and \(d\);
* \(\mathbf{R}=[R_{n},R_{t}]^{T}\)--contact force exerted on the block by the beam expressed in the local frame, where \(R_{n}\) is the normal component and \(R_{t}\) is the tangential component;
* \(\mathbf{U}=[U_{n},U_{t}]^{T}\)--relative velocity of point \(d\) with respect to point \(b\) expressed in the local frame, with \(U_{n}\) being the normal component and \(U_{t}\) the tangential one. In the non-smooth contact dynamics method [32] used in this work, it is assumed that there is no penetration between the block and the beam, the contact force is repulsive, and when the bodies are not in contact, the contact force equals zero. All this is encompassed with the following set of unilateral constraints, known as the Signorini condition [32]:  
\[m\frac{\partial^{2}v(x,t)}{\partial t^{2}}+c\frac{\partial v(x,t)}{\partial t }+EI\frac{\partial^{4}v(x,t)}{\partial x^{4}}=p_{y}(x,t), \tag{1}\]5  
When contact with deformable bodies is concerned, the contact constraint can be formulated at the velocity level [41]:  
\[m\frac{\partial^{2}v(x,t)}{\partial t^{2}}+c\frac{\partial v(x,t)}{\partial t }+EI\frac{\partial^{4}v(x,t)}{\partial x^{4}}=p_{y}(x,t), \tag{1}\]6  
where \(U_{n}^{+}(t)\) is the normal component of the velocity immediately after impact. Condition (12) is referred to as the velocity Signorini condition. In order to complete the model, an impact law is necessary. [PERSON]'s impact law can be used, which relates the normal velocity just before the impact to the velocity just after the impact as \(U_{n}^{+}(t)=-eU_{n}^{-}(t)\), where \(e\) is the coefficient of restitution ranging from 0 to 1, and \(U_{n}^{-}(t)\) is the velocity immediately before impact. In this work, the impact is assumed to be plastic, which implies a value of \(e=0\), and the velocity Signorini condition is used. To account for frictional forces, the Coulomb law is used, which states the following:  
\[m\frac{\partial^{2}v(x,t)}{\partial t^{2}}+c\frac{\partial v(x,t)}{\partial t }+EI\frac{\partial^{4}v(x,t)}{\partial x^{4}}=p_{y}(x,t), \tag{1}\]7 Arbitrary contact pair \((bd)\) and its local frame of reference. \[m\frac{\partial^{2}v(x,t)}{\partial t^{2}}+c\frac{\partial v(x,t)}{\partial t }+EI\frac{\partial^{4}v(x,t)}{\partial x^{4}}=p_{y}(x,t), \tag{1}\]8 \[m\frac{\partial^{2}v(x,t)}{\partial t^{2}}+c\frac{\partial v(x,t)}{\partial t }+EI\frac{\partial^{4}v(x,t)}{\partial x^{4}}=p_{y}(x,t), \tag{1}\]9  
where \(\mu\) is the coefficient of friction and \(U_{t}^{+}\) is the relative tangential velocity of \(d\) with respect to \(b\) during contact. From the last two equations, we see that \(R_{t}\) has a limiting value and its direction is opposite to the sliding velocity.