# Damping of Beam Vibrations Using Tuned Particles Impact Damper
## 5 Modelling
(**a**) Variable particles mass; (**b**) Variable container volume. \[A(t)=\left\{ \begin{array}{ll}0&\Leftrightarrow 0<t<t_{1}\\ (e^{a\frac{(t-t_{1})^{k}}{\Delta t_{1}^{2}}}-1)\cdot(H(t-t_{1})-H(t-t_{2}))& \Leftrightarrow t_{1}<t<t_{2}\\ 1&\Leftrightarrow t_{2}<t<t_{3}\\ (e^{b\frac{(t_{2}-t)^{k}}{\Delta t_{1}^{2}}}-1)\cdot(H(t-t_{3})-H(t-t_{4}))& \Leftrightarrow t_{3}<t<t_{4}\\ 0&\Leftrightarrow t_{4}<t<\infty\\,&\end{array} \right. \tag{4}\]  
where \(a\)--arbitrary coefficient (\(\alpha=3\) was assumed during numerical simulations), \(\Delta t=t_{2}-t_{1}=t_{4}-t_{3}\)--the activation and deactivation time (at this stage of investigations it was assumed that inflating and deflating times of the balloon were the same) and coefficients \(a=b=0.69\) because simplifying the formula (4) for cases \(t_{1}<t<t_{2}\) or \(t_{3}<t<t_{4},e^{a}-1=1\). In the experiments, two various forms of activation functions were taken into consideration (Figure 12a,b). They corresponded to changes in the compartment volumes from \(V_{min}\) to \(V_{max}\) in \(\Delta_{t_{1}}=4.5\) s and \(\Delta_{t_{2}}=1.5\) s respectively. The investigated beam tip waveforms for the resonant range of excitations are depicted in Figure 13. For all described cases the plastic particles were involved. Two various numbers of grains corresponding to 10% and 5% of the total mass of the beam were taken into account. Additionally, different activation times, according to the data depicted in Figure 12, were applied in the experiments. Moreover, in Figure 13 the numerical simulation results are verified with the real response of the system. The data depicted in Figure 13 reveal interesting information. It is evident that the activation of the TPID device significantly reduced the resonant vibration amplitudes. The efficiency of granular damper is strongly dependent on the number (mass) of particles being involved in the damping process. On the basis of Figure 12 it can be observed that the larger amount of granular media increased the dissipative properties of the system. Changing the mass of the grains from 5% of the total mass into 10% increased the damping properties of considered device by 35%. This phenomenon is apparent for both damper activation times. The detailed analysis of data depicted in Figure 13 also revealed some damping inertia of the system. For the case of larger amount of particles and applied activation time \(\Delta_{t_{1}}=4.5\) s (resulted from inflating the balloon to the maximal volume) the \(\Delta_{t_{11}}\) time necessary for the recorded waveforms stabilization was 5.4 s (Figure 13d). Similarly to stabilize the vibrations of the beam dampened by lower number of particles (5% of the total mass) the same activation function \(\Delta_{t_{12}}=6.4\) s was required. Such observations were also valid for the shorter activation time \(\Delta_{t_{2}}=1.5\) s. The damping inertia for this case was \(\Delta_{t_{21}}=1.9\) s and \(\Delta_{t_{22}}=3.5\) s for larger and smaller amount of  
Figure 11: Activation function \(A(t)\). particles respectively. It is worth emphasizing that the transition from the TPID deactivated into fully activated state was relatively smooth. There were no uncontrolled increases in vibration amplitudes during the balloon pumping process. Figure 14, which depicts the envelopes of registered waveforms for various parameters of the investigated TPID prototype, can be a transparent summary of the experimental work carried out in the paper. It is worth emphasizing that the proposed mathematical model of the system is quite reliable. The verification between real experimental responses of the vibrated beam and numerical simulations being carried out on the basis of Equations (2)-(4) revealed a satisfactory convergence (Figure 13). The differences between real and theoretical waveforms for the \"turned on\" and \"turned off\" states are negligible.