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According to de Moivre’s formulawhere( cosϕ/ 2 isin/ 2)* *a 1 + ib1= ρ + ϕ , (2.27)*2*( a ) + ( ) 2= 1 b1so it is possible to represent as followswhereρ ,* * * *a 1 ib1= a 2 + ib2*1*1tanφ = b a =ϕ (2.28)+ , (2.29)*2*( b1) a 1*1 * 2a 2 = cos ϕ / 2 = ( a1) + +2ρ ,* 2 *( b1) − a 1*1 * 2b2 = ρ sin ϕ / 2 = ( a1) +2(2.30)Then the roots of the characteristic equation (2.23)s 1 = −n1+ iω1(31),s2 = −n2− iω , (2.31)where**ω 1 = ( b2− Aωb )/2 J o , ω 2 = ( b2+ Aωb )/2 J o , (2.32)2 *2 *n1 = ( β l − a 2 )/2 J o , n2= ( βl+ a 2 )/2 J o , (2.33)2 *while it is always β l > a2.The general solution of the system of differential equations (2.20)λ = C1 exp ( s1t) + C 2 exp ( s 2t), (2.34)Finally it was obtainedλ+= ( D1+ i D3) exp ( − n1t)( cos ω1t+ i sin ω1t)( D + i D ) exp ( − n t )( cos ω t + i sin ω t )24222+(2.35)After separating the imaginary part from the real in Eq. (2.35) according to theintroduced complex variable λ (Eq. (2.21)), the displacement of the blank endz+y+11=exp=exp( − n1t)( D1cos ω1t− D3sin ω1t)( − n2t )( D2cos ω2t + D4sin ω2t )( − n1t)( D1sin ω1t+ D3cos ω1t)( − n t )( − D sin ω t + D cos ω t )expexp22242++(2.36)A particular solution of Eqs (2.20) represents the forced vibrations of the system:**y 2 = a1cos pt + b1sin pt + Frl1/ k y l + c1cos ω t + d 1 sin ω t*z 2 = a 2 cos pt + b 2 sin pt + c 2 cos ω t (2.37)where29