## 5 Calculation of the Tail Boom Vibrations
Using the basic functions \(X_{n}(\overline{x})\), one can determine the function \(\overline{v}_{1}\left(\overline{t},\overline{x}\right)\) for the forced tail boom vibrations (analytical transformations and calculations were performed using Maple \(17^{\text{TM}}\) software). One can note here that the behavior of the \(\overline{v}_{1}\left(\overline{t},1\right)\) function is determined primarily by the first two components \(S_{1}\left(\overline{t}\right)X_{1}(\overline{x})\) and \(S_{2}\left(\overline{t}\right)X_{2}(\overline{x})\). The eigenfrequencies \(f_{1}\) and \(f_{2}\) depend on the geometry parameter \(\gamma\). Nevertheless, in this work, all components \(S_{n}\left(\overline{t}\right)X_{n}(\overline{x})\), \(n=1,\ldots,4\) were used to determine \(\overline{v}_{1}\left(\overline{t},\overline{x}\right)\). Thus, a general behavior of the \(\overline{v}_{1}\left(\overline{t},1\right)\) function depends on the interaction of the eigen and forced vibrations. Figure 13 shows the functions \(S_{1}\left(\overline{t}\right)\) and \(S_{2}\left(\overline{t}\right)\) for the different values of parameter \(\gamma\). The time behavior of the functions \(S_{1}\left(\overline{t}\right)\) and \(S_{2}\left(\overline{t}\right)\) is predominantly determined by the frequencies \(\overline{\omega}_{1}\) and \(\overline{\omega}_{2}\), respectively, and the amplitude of the total function \(\overline{v}_{1}\left(\overline{t},\overline{x}\right)\) depends on the mass and geometry data. For variant 4, the function \(S_{2}\left(\overline{t}\right)\) demonstrates resonance due to the second blade passing harmonic \(2N\!f\) being equal to the second eigenfrequency \(f_{2}\) and \(2N\overline{\omega}=\overline{\omega}_{2}=22.0336\). Figure 13: _Cont._  
From Figure 13, it follows that, for variants 1 and 7, the amplitude of the forced tail boom oscillations is determined by the \(S_{1}(\overline{t})X_{1}(\overline{x})\) term and one can expect that the dominant frequency of oscillations corresponds to the \(f_{1}\) eigenfrequency (\(\overline{\omega}_{1}=\overline{a}_{1}^{2}\)). On the contrary, for variants 2-6, the amplitude of the forced oscillations is determined by the \(S_{2}(\overline{t})X_{2}(\overline{x})\) term and the dominant frequency of oscillations corresponds to the \(f_{2}\) eigenfrequency (\(\overline{\omega}_{2}=\overline{a}_{2}^{2}\)). Figure 14 shows the function \(v_{1}\big{(}\overline{t},L\big{)}=L\overline{v}_{1}\big{(}\overline{t},1\big{)}\) for the forced oscillations of the free tail boom end (data are presented in mm) for different values of the parameter \(\gamma\) except of the resonance variant 4). In general, one can note that the interaction between eigen and forced vibrations leads to two oscillation types having short (\(\Delta\overline{T}_{s}\)) and long (\(\Delta\overline{T}_{l}\)) dimensionless periods. Thus, all considered variants can be divided into two cases. Case I includes variants 1 and 6, for which \(2N\overline{\omega}\
eq\overline{\omega}_{2}\), and case II includes variants 2-5, for which \(2N\overline{\omega}\approx\overline{\omega}_{2}\). For case I, the short period \(\Delta\overline{T}_{s}\) of oscillations is determined by the eigenfrequency \(f_{1}\) (\(\overline{\omega}_{1}=3.515625\)) and oscillations of the free tail boom end have approximately constant amplitudes of about 0.1 mm (peak to peak relative deviation is approximately 50 microstrains). In general,this value of the oscillation amplitudes corresponds to estimations of paper [11] for static deformations of a light helicopter tail boom. Figure 14: Forced free end tail boom oscillations \(v_{1}(0,L)\) (in mm) at different \(\gamma\) values (variant numbers). For all variants of case II, the short period \(\Delta\overline{T}_{s}\) of the oscillations is determined by the eigenfrequency \(f_{2}\) (\(\overline{\omega}_{2}=22.033636\)) and the long (interactional) oscillations period \(\Delta\overline{T}_{l}\) depends on the \(\gamma\) parameter value.