## 2 Dynamic Response of the Vibrating Compactor Roller  
For the Voigt-Kelvin linear model, the equivalent stiffness \(k\) of the layer of soil at the vibrating roller-soil contact, the equivalent viscous damping \(c\) and the harmonic disturbing force \(F=F(t)=F_{0}\)_s_in_ have been considered. The force amplitude is \(F_{0}=m_{0}r\omega^{2}\), where \(m_{0}r\) is the static moment of the dynamic mass imbalance in the vibrating compactor roller. The mass of the compactor roller in motion is \(m\)[2; 3; 5; 14; 15]. The linear dynamic system (\(m\),\(c\),\(k\)) has a single degree of freedom and is driven by the harmonic disturbing force \(F(t)\). For each discrete variable parameter \(k\), \(c\) or (\(k\),\(c\)), the excitation frequency (pulse) is swept continuously by using the command and control system of the vibrator's hydrostatic actuation. Thus, the dynamic response curves can be plotted by numerical simulation with the parameters (\(m\), \(c\), \(k\)) of the soil and (\(m_{0}\), \(\omega\)) of the vibrating roller, as well as the experimental response curves, by measuring the amplitude of the vibrations \(A(\omega\), \(k)\), \(A(\omega\), \(c)\), \(A(\omega\), \(k\), \(c)\) \"in every detail\" for each significant value of \(\omega\)[2; 6; 8; 12; 16; 19]. The dynamic response functions parameterized by \(k\), \(c\) and with the continuous variation of the excitation frequency are for the amplitude \(A(\omega\), \(k\), \(c)=A(\omega)\) and for the maximum force transmitted to the soil \(Q_{0}(\omega\), \(k\), \(c)=Q_{0}(\omega)\), signifying the efficiency of the dynamic compaction process. The calculation relations for the response functions are in the form:  
\[A(\omega,k,c)=\frac{m_{0}\
u a^{2}}{\sqrt{\left(k-m\omega^{2}\right)^{2}+c^{2 }\omega^{2}}} \tag{1}\]  
\[Q_{0}(\omega,k,c)=m_{0}\
u a^{2}\sqrt{\frac{k^{2}+c^{2}\omega^{2}}{\left(k-m \omega^{2}\right)^{2}+c^{2}\omega^{2}}} \tag{2}\]  
In compaction, the significant dynamic regimes are the following:  
* The resonance regime for \(\omega=\omega_{n}\), where \(\omega_{n}=\sqrt{\frac{k}{m}}\), meaning that the condition \(k=m\omega^{2}\) needs to be fulfilled. * The post-resonance regime for \(\omega\gg\omega_{n}\), or at the limit \(\omega\rightarrow\infty\). Practically, \(\omega=(3\ldots 6)\omega_{n}\) is adopted. **Resonance regime**. The parameters \(A(k,c,\ \omega_{n})\equiv A_{rez}\) and \(Q(k,c,\ \omega_{n})\equiv AQ_{rez}\) for \(\omega=\omega_{n}\) have the following calculation relations:  
* The amplitude of the vibration _at resonance_ is expressed as \[A_{rez}=\frac{m_{0}\tau}{c}\omega_{n}\] where we take into account \(\omega_{n}=\sqrt{\frac{k}{m}}\) and \(\zeta=\frac{c}{2\sqrt{m}}\), obtaining \[A_{rez}=\frac{m_{0}\tau}{c}\sqrt{\frac{k}{m}}\] (3) or \[A_{rez}=\frac{m_{0}\tau}{m}\frac{1}{2\zeta}\] (4)
* The amplitude (maximum value) of the force transmitted to the soil _at resonance_ may be calculated as follows: \[Q_{0}^{res}=m_{0}\tau\omega^{2}\sqrt{\frac{k^{2}+c^{2}\omega^{2}}{c^{2}\omega ^{2}}}\] (5) where we introduce \(\eta=\frac{c\omega_{n}}{k}\) the hysteretic damping and \(\omega_{n}^{2}=\frac{k}{m}\), so that we obtain \[Q_{0}^{res}=m_{0}\tau\frac{k}{m}\sqrt{1+\frac{1}{\eta^{2}}}\] (6)Considering that \(\eta=2\zeta\), where \(\zeta=\frac{c}{c_{\sigma}}\) is the fraction of the critical damping, for linear viscoelastic systems, we have:  
\[Q_{0}^{\text{res}}=m_{0}r\frac{k}{m}\sqrt{1+\frac{1}{4\zeta^{2}}}Q_{0}^{\prime \text{res}}=m_{0}r\frac{k}{m}\sqrt{1+\frac{1}{\eta^{2}}} \tag{7}\]  
Thus, \(Q_{0}^{\text{res}}\) can be calculated depending on the stiffness \(k\) and on the damping expressed by \(c\), \(\eta\) or \(\zeta\). **Post-resonance regime**. For the limit condition \(\omega\rightarrow\infty\), the relations for \(A(\omega,k,c)\) and \(Q_{0}(\omega,k,c)\) will be expressed as follows:  
(a) Amplitude in post-resonance is given by the relation  
\[A_{postres}=A_{\infty}=\frac{m_{0}r}{m} \tag{8}\]  
i.e., in post-resonance conditions, the vibration amplitude is constant, regardless of the variation of stiffness or damping.