# Damping of Beam Vibrations Using Tuned Particles Impact Damper
## 5 Modelling  
At the initial stage of TPID investigation the 2-DOF linear model (Figure 9), described by equations of motion (2) has been selected to capture the real behavior of the considered system. The authors want to emphasize the dynamics of the tested object in the resonant range for the TPID being in switched \"on\" and \"off\" states. \[M\ddot{X}+C_{1}\dot{X}+K_{1}X+C_{2}\dot{u}+K_{2}u=0. \tag{2}\]  
\[ \begin{array}{cc}M=\left(\begin{array}{cc}M&0\\ 0&M_{p}\end{array}\right)&K_{1}=\left(\begin{array}{cc}K+K_{p}&-K_{p}\\ -K_{p}&K_{p}\end{array}\right)&C_{1}=\left(\begin{array}{cc}C+C_{p}&-C_{p} \\ -C_{p}&C_{p}\end{array}\right)\\ C_{2}=\left(\begin{array}{cc}-C&0\\ 0&0\end{array}\right)&K_{2}=\left(\begin{array}{cc}-K&0\\ 0&0\end{array}\right)\end{array} \]  
where \(M\)--reduced mass of the system (mass of the container included), \(M_{p}\)--mass of particles, \(K\)--beam stiffness, \(C\)--beam damping, \(K_{p}\)--TPID stiffness, \(C_{p}\)--controllable TPID damping. Figure 8: Exemplary spectrum analysis. Figure 9: Mathematical model. \(K\), \(C\) and \(M\) parameters were identified for free vibrations of the investigated cantilever beam in separate experimental tests. It was assumed that the value of \(K_{p}\) parameter is constant during laboratory tests and results from the attachment of the container to the beam. It was calibrated basing on the experiments conducted for the fully compacted particles that is, for the lowest volume value (\(V_{min}\)) of the TPID container. The \(C_{p}\) parameter values depend on the type of inelastic collisions between single particles and the character of their mutual contact with the container. In the paper it is assumed that variations of this parameter are the function of the TPID operation volume and can be described as:  
\[C_{p}=C_{TPID}\cdot A(t)+C_{p_{0}}, \tag{3}\]  
where \(C_{TPID}\)-damping parameter for the fully opened TPID device (volume of the container is \(V_{max}\)), \(A(t)\) activation function (4), \(C_{p_{0}}\) -initial damping in the system (considered as the \"intergranular\" friction forces acting in the switched \"off\" state). The \(C_{TPID}\) parameter was determined for the test conducted in a fully turned \"on\" state. It involves the maximal number of collisions between loose material and container. On the contrary, the \(C_{p_{0}}\) parameter value was captured for the minimal value of the container volume. Although the mathematical model (Figure 2) seems to be simple, its complexity lies in the previously mentioned complex phenomenon related to not fully understood interactions between internal particles. The confirmation of this fact is the data depicted in Figure 10 where the logarithmic damping decrement for the investigated beam free vibrations is illustrated. In Figure 10a,b the damping characteristics for various particles masses and different container volumes are presented respectively. The characteristics have been obtained on the basis of supplementary experiments. They consisted of measurements of free vibrations of the considered beam initially deflected at the free right end by 30 mm. The release of the deflected beam was achieved by burning out a thin thread attached to the tip of the element. Basing on data presented in Figure 10 it can be stated that the higher mass of grains results in a more efficient energy dissipation process. Similarly, the higher volume of the TPID compartment is related to the more effective vibrations attenuation. The next problem is encountered during the volume changing of the compartment, that is, inflating and deflating of the internal balloon. These transition processes are not fully recognized yet, therefore the authors proposed exponential functions to reproduce them (Figure 11). The function (4) takes 0 value for the deflated balloon (the grains are compacted and can be treated as a concentrated mass). In the contrary, for the \(V_{max}\) volume of the container the activation function takes a value of 1. Figure 10: Logarithmic damping decrement.