# Point Deflection in Topological Interlocking Plates
## 5 Oscillations of TI Beams with Discontinuous Linear Damping  
Discontinuous damping is understood here as damping only working during a part of the oscillation period. Figure 12 illustrates the timeline of emerging discontinuities of damping produced by energy absorption due to air squeezing between the contacting surfaces of the neighboring blocks. It is seen that the oscillating cycle consists of two undamped-damped phases (Figure 12b-e). Therefore, the sequence of undamped-damped phases is antisymmetric, leading to equality in durations of the undamped and damped phases. The damping phases involving air squeezing are discontinuous (passing from low damping pumping-in to high damping pumping-out) and non-linear [45]. Here, we concentrate on the analysis of the effect of the discontinuity of the damping. To this end, we consider a simplified model whereby the damping, when operational, is linearly characterized by damping ratio \(\zeta\). We assume that the oscillations start with an impulse applied at time \(t=0\) without the presence of external force. Then, the equations of motion for discontinuous damping read as follows:  
\[ \begin{array}{l}\ddot{\mathbf{v}}+\mathbf{\omega_{0}^{2}}\mathbf{v}=\mathbf{0}\text{ for phases (b), (c)}\\ \ddot{v}+2\zeta\dot{v}+\omega_{0}^{2}v=0\text{ for phases (d), (e)}\end{array}  \tag{7}\]  
Here, \(v\) is the displacement associated with oscillations, and \(\omega_{0}\) is the angular natural frequency of non-damped oscillations. The initial conditions read as follows:  
\[v(0)=0,\ \ \ \dot{v}(0)=\mathbf{V_{0}} \tag{8}\]  
For the first undamped stage, the solution of Equation (7) with initial conditions (8) is obvious:  
\[v(t)=\frac{V_{0}}{\omega_{0}}\sin\omega_{0}t \tag{9}\]  
Subsequently, the maximum displacement \(v_{max}=\frac{V_{0}}{\omega_{0}}\) is reached at \(t=\frac{\pi}{2\omega_{0}}\). After reaching the maximum displacement, phase (c) starts, which is the damping phase. For this phase, the solution reads as follows:  
\[v(t)=\frac{V_{0}}{\omega_{0}}e^{-\zeta\omega_{0}\tau}\Bigg{[}\cos\omega_{D} \tau+\frac{\zeta}{\sqrt{1-\zeta^{2}}}\sin\omega_{D}\tau\Bigg{]},\ \tau=t-\frac{\pi}{2\omega_{0}} \tag{10}\]  
\[\dot{v}(t)=-V_{0}e^{-\zeta\omega_{0}\tau}\frac{\zeta}{\sqrt{1-\zeta^{2}}}\sin \omega_{D}\tau \tag{11}\]  
This trajectory lasts until the contact faces are closed (end of phase (c)), that is, until time \(\tau_{D}\), such that \(v(\tau_{D})=0\). Using Equation (10), one obtains the following:  
\[ \begin{array}{c}V_{1}=V_{0}\exp\left(-2\zeta\frac{sin^{-1}\sqrt{1-\zeta^{2} }}{\sqrt{1-\zeta^{2}}}\right)\\ \omega_{D}\tau_{D}=\frac{\sin^{-1}\sqrt{1-\zeta^{2}}}{\sqrt{1-\zeta^{2}}} \end{array}  \tag{12}\]  
The velocity at the end of this phase is as follows:  
\[V_{\frac{1}{2}}=\dot{v}(\tau_{D})=-V_{0}e^{-\zeta\omega_{0}\tau_{D}} \tag{13}\]  
Figure 12: Illustration of air squeezing damping discontinuity during an oscillation period: **(a)** the period starts with neighboring blocks in close contact; **(b)** the first oscillation phase consists of opening the contact to its maximum, during which the air is pumped in (the damping associated with friction between the air and the block surfaces is neglected). This is a conventional linear phase; its duration is a quarter of undamped period \(\frac{\pi}{2\omega_{0}}\), where \(\omega_{0}\) is the angular natural frequency of non-damped oscillations. Phase **(c)** corresponds to closing the contact and squeezing the air out. This phase lasts for time \(\tau_{D}\) and ends up with the closed contact. **(d)** is another opening phase without damping. It essentially repeats phase **(b)** but with a negative sign. Phase **(e)** corresponds to closing the contact and squeezing the air out, similarly to phase **(c)**, but with the opposite sign. Now, due to the antisymmetry, the following phases will give the same result as Equation (13), but with the opposite sign.