## 5 Results and Discussion  
Using governing Equation (16) of the system shown in Figure 5, the damping effectiveness of the system with the slit membrane damper was analyzed. The drag force derived in Section 4.2 (Equation (15)) is used in the model. The typical system parameters for the quarter car model were used: mass \(m=250\) kg; \(\varepsilon=0.0000001\); \(a=0.01\) m; \(h=0.02\); \(\Theta=0.04\); \(k=25000\) N/mv. Two cases were analyzed where harmonic excitation and impulse excitation were applied. The results of harmonic excitation in the form of \(f=\sin wt\) (\(f=100\times\sin(2\pi 0.5t)\)) are presented in Figure 6.  
Figure 4: Relationship between drag force and \(x^{\prime}\) at \(h=0.02\), \(\Theta=0.04\), \(\varepsilon=0.0000001\) and \(a=0.01\). Red line the slit membrane damper, blue line for linear damper. Figure 5: Single DOF mass–spring–damper dynamical model. The nonlinear effect can be observed. We can see clear hysteresis in the graph of damping force displacement (Figure 6g). This can be explained by motion of the membrane due to external excitation. The motion of the excited membrane is described in Sections 3.1 and 3.2. This leads to the slit opening area given by Equation (12). The flow of viscous fluid through the variable-area openings of the slits, associated with the drag force expressed by Equation (15), gives a nonlinear effect in the form of hysteresis. Then, the resultant force acting on mass m is shown in Figure 6e,f. Also, it is worth noting that at lower displacements an increased damping force is obtained as the opening area of the slits is smaller. The increase in damping force at lower velocities (Figure 6h) can also be explained by the smaller opening area of the slits. In the second case, the effectiveness of the system with the slit damper was analyzed when shock (impulse) excitation was applied. The results are presented in Figure 7. Here, the system response with a linear damper and the system response with the nonlinear slit membrane damper are compared. The results are consistent with the obtained relationship of drag force versus velocity (Figure 4). At lower velocities, the drag force is higher compared to the linear damper; therefore, we see a lower velocity amplitude (Figure 7b) and an increased damping force (Figure 7c). Thus, conditions are created for more effective shock suppression. A shorter time is necessary for the oscillations to decay. Or, we can say that the nonlinearity of the damping characteristic allows a shorter transient process to be achieved. Figure 6: Motion characteristics of the mass–spring–damper system: (**a**) displacement of mass x; (**b**) velocity of mass x; (**c**) net force acting on mass m; (**d**) damping force; (**e**) net force acting on mass m versus displacement; (**f**) net force acting on mass m versus velocity; (**g**) damping force versus displacement; (**h**) damping force versus velocity. An innovative side of the research is that the nonlinear damping characteristic was achieved with a damper with a simple structure. It is much simpler than the passive dampers presented in [4; 5; 6; 8], as no parts with complicated geometrical form are necessary. It is simpler than active dampers [9; 10; 13; 15; 17] because no sophisticated control system is necessary. But on the other hand, the dynamics of its main component, the slit membrane, have a great influence on damping effectiveness. Therefore, the modelling and simulation of membrane behavior is an important step in developing a slit membrane damper for particular dynamic loads and system parameters. In further research, this property can be used to control the reaction of the mechanical system to dynamic effects, as shown by the principal diagram in Figure 8.  
In a mechanical system, an input dynamical load can be transmitted directly without suppression (without a damper), using linear suppression (linear damper) and using nonlinear suppression (nonlinearity damper), which is the fastest.