# Nonlinear Dynamics of Vibro-Impacting Indenter
## 3 Experimentally Calibrated Mathematical Modelling
Equation (1) describes the dynamics of the armature mass and is given as follows:  
\[m_{1}\ddot{x}_{1}+c_{2}\dot{\phi}_{1}+k_{2}\phi_{1}=F_{em}-F_{r} \tag{1}\]  
where \(m_{1}\) is the mass of the armature; \(x_{1}\) is its displacement; \(\phi_{1}\) is the displacement of the armature mass with respect to the displacement of the electrodynamic shaker body; \(k_{2}\) and \(c_{2}\) are the stiffness and damping between the armature and electrodynamic shaker body, respectively; and \(F_{em}\) is the excitation force, which is modelled by the sinusoidal function of Equation (2):  
\[F_{em}=F_{a}\sin(2\pi ft) \tag{2}\]  
where \(F_{a}\) and \(f\) are the amplitude and frequency of the excitation force, respectively. Equation (3) describes the motion of the electrodynamic shaker body, which is affected by the electromagnetic force \(F_{em}\):  
\[m_{2}\ddot{x}_{2}-c_{2}\dot{\phi}_{1}+c_{3}\dot{\phi}_{2}-k_{2}\phi_{1}+k_{3} \phi_{2}=-F_{em} \tag{3}\]  
where \(m_{2}\) is its mass, \(x_{2}\) is its displacement, and \(\phi_{2}\) is the displacement of the electrodynamic shaker body with respect to the displacement of the pneumatic cylinder piston-head. \(k_{3}\) and \(c_{3}\) are the equivalent stiffness and damping of air inside the pneumatic cylinder, respectively. Equation (4) is an expression of the translational motion pertaining to the mass of the pneumatic cylinder piston-head. The force arising from pneumatic pressure is denoted by \(F_{p}\), and a frictional force dependent on the direction of the velocity of the electrodynamic shaker body has an amplitude of \(p_{m}\). \[m_{3}\ddot{x}_{3}-c_{3}\dot{\phi}_{2}-k_{3}\phi_{2}-p_{m}\text{sgn}\left(\dot{ x}_{2}\right)=F_{p} \tag{4}\]  
where \(m_{3}\) is its mass, and \(x_{3}\) is its displacement. The pneumatic force, \(F_{p}\), can be related to the pressure of the supplied air in the pneumatic cylinder in bar, \(P\), using the following equation:  
\[F_{p}=25331P\pi D^{2} \tag{5}\]  
where \(D\) is the diameter of the piston-head. In addition, the equivalent damping of air inside the pneumatic cylinder is a function of both the excitation force and frequency, and it can be expressed as  
\[c_{3}=\frac{78.6}{\pi^{2}F_{af}}. \tag{6}\]  
For the specific case of a metal sample, the resistive force, \(F_{r}\), from the penetrated medium, which is equal in magnitude but opposite in direction to the force transmitted to the medium, is expressed as a discontinuous function in Equation (7):  
\[F_{r,metal}= \begin{cases}0&\text{if }x_{1}<G,\\ c_{1}\dot{x}_{1}+k_{1}x_{1}&\text{if }x_{1}\geq G,\end{cases}  \tag{7}\]  
where \(G\) is the gap between the indenter and the restraint, and \(k_{1}\) and \(c_{1}\) are the equivalent stiffness and damping between the armature and the metal sample upon penetration. The two conditions in Equation (7) dictate \(F_{r}\) when the indenter is not in contact and is in contact with the metal sample. However, for the specific case of a sandstone sample, the resistive force from the penetrated medium is discontinuous for three sets of conditions:  
\[F_{r,ss}= \begin{cases}0&\text{if }x_{1}<G+x_{p},\\ k_{l}\phi_{3}^{nl}&\text{if }x_{1}\geq G+x_{p},\dot{x}_{1}\geq 0,\\ k_{u}\phi_{4}^{nua}&\text{if }x_{1}\geq G+x_{p},\;x_{1}\geq G+x_{f},\;\dot{x}_{1}<0, \end{cases}  \tag{8}\]where \(k_{l}\) is the sandstone stiffness when it undergoes loading, \(\phi_{3}=x_{1}-G-x_{p}\), \(\phi_{4}=x_{1}-G-x_{f}\), and \(k_{u}\) is a different value of stiffness during unloading, and it was experimentally determined via quasi-static indentation tests in [26] that the stiffness value is greater during unloading than loading for sandstone samples. When the indenter is not in contact with the sandstone sample (i.e., the first condition), \(F_{r}\) is 0 N. When the indenter is in contact with the sandstone sample and it is in the loading phase, it meets the second condition, and the corresponding expression is used to calculate \(F_{r}\).