IBC. 21(u¨2) + LBC21(u2) + iBC21,1(u˙, u˙) + iBC21,2(u, u¨) + iBC21,1(u, u˙, u˙) + iBC21,2(u, u, u¨) BC21(2) (3)(2)+ nBC21(u, u) + n(3)(u, u, u)} |s=1= 0, (8){LBC22(u2) + nBC22(u, u) + nBC22(u, u, u)} |s=1= 0,(2) (3){LBC22(u3) + nBC3(u, u) + nBC3(u, u, u)} |s=1= 0,0where u represents the 2 by 1 operator including the axial and the laggingcomponents of the beammotion, i.e., u = [u2(s, t) u3(s, t)]𝖳, while I, G and L are the linear inertia and the gyroscopic and elastic stiffness operators, respectively, whose expressions are given in Appendix I. Moreover, i(2), i(2), i(2) and i(3), i(3), i(3) are the quadratic and the cubic inertial terms, respectively, and n(2),1 2 0 1 2and n(3) represent the quadratic and the cubic stiffness, respectively. The quadratic operators are defined in Appendix II, while, for the sake of brevity, the third order operators are omitted.ω0Further nondimensional parameters adopted are: λ = ωR , α12 = EAL2 , α22 = EAL2 , α32 = EAL2EJEJEJS S S= EJ11 22 33and GJ3S3S332(1+ν¯), where ν¯ is the Poisson coefficient. Accordingly, for the case of symmetricbeams the only relations which provide the prestressed configuration and the associated bound-ary conditions, respectively, are simplified to α22u0′′(s) + λ2(r + s + u0(s)) = 0 and u0(0) =3 3 33