## 5 Experimental and Simulation Studies of the Harmonic Motion  
Tests were carried out using harmonic forcing, which is the simplest form of analysing the behaviour of a system subjected to external forcing. The tests carried out are an appropriate form of confirmation of the correct choice of model and its parameterisation. That is, it is a type of test to verify the correctness of the research method used. Figure 16 shows the plots of the actual sinusoidal waveforms for vertical accelerations with a maximum amplitude of \(y_{0}=63\) mm. Oscillatory motion is defined by the position relative to the equilibrium point. The positive and negative values of the oscillation represent the positions on either side of the equilibrium point.  
For comparison, Figure 17 shows the results of the simulation studies of the model's response in terms of displacements \(y_{f1}\) to a sinusoidal forcing \(y_{f0}\). The parameters of the forcing waveform are analogous to the experimental studies of Figure 16.  
Figure 16: _Cont._  
Figure 15: The variation curves, as a function of time, were obtained from the simulation for \(k_{\mathrm{s}}\) and the experimental tests of \(p\) and \(h\) for an initial pressure value of \(p=4.5\) bar.  
Figure 16: Displacement curves \(y\)(seat) and \(y_{0}\) (base) for suspensions: (**a**) passive optimum fixed \(p\) = 4 bar, (**b**) passive optimum fixed \(p\) = 4.5 bar, (**c**) passive soft \(p\) = 3.75 bar, and (**d**) passive hard \(p\) = 5.5 bar.  
Figure 17: Simulation displacement diagrams \(y_{\text{f1}}\) (seat) and \(y_{\text{f0}}\) (base) for suspensions: (**a**) passive optimal \(p\) = 4 bar, (**b**) passive optimal \(p\) = 4.5 bar, (**c**) passive soft \(p\) = 3.75 bar, and (**d**) passive hard \(p\) = 5.5 bar.  
Table 3 shows the values of the vibration amplitude amplification coefficients for the passive suspension system of the operator seat. Tests were carried out for different air pressures in the bellows actuator and a constant mass load (\(m=69.6\) kg).  
For a hypothetical improvement in operator comfort, a range of 0.3 to 0.7 was assumed for the amplitude gain (\(K_{u}=y/y_{0}\) and \(K_{fu}=y_{f1}/y_{0}\)) of the excitation signal.  
The results of the bench experiments (Figure 2) for a modulated displacement (forcing) frequency \(y_{0}\) ranging from 1 to 6 Hz and \(y\) displacement are shown in Figure 18. A frequency study for the \(K_{u}(y/y_{0})\) amplification at different modulation frequencies produced the frequency response of the pneumatic vibration isolation system (Figure 18b). From this diagram, it is possible to estimate the points at which the gain is maximum or minimum. Naturally, for low frequencies of the input signal \(y_{0}\), the amplitude of the output signal \(y\) is close to that of the input signal. However, as the frequency increases, the amplitude of the output signal becomes smaller than the amplitude of the input signal. This is due to the properties of the circuit, which attenuates the higher frequencies. In this case (\(p=4.0\) bar), there is a local amplitude amplification for the frequency \(f=2.12\) Hz. For this frequency, the instantaneous effectiveness of the pneumatic vibration isolation is limited. However, in the set range (up to 6 Hz), vibration isolation effectiveness can be demonstrated for the system.