# Vibration Analysis

Given the well-know mathematical similarity between the stability (bifurcation) and vibration eigen­value problems, the derivation of a GBT vibration formulation constitutes a relatively easy task: it suffices to replace the geometric effects by their dynamic counterparts, as done by Schardt and Heinz (1991) for isotropic linear elastic members, and by Silvestre and Camotim (2004d), for orthotropic
linear elastic laminated plate members. Very recently, these authors proposed a novel combined formulation that makes it possible to analyse the vibration behaviour of loaded orthotropic members (Silvestre and Camotim, 2005a, b). In this case, the GBT system of equilibrium equations reads

(Kik + WmGik – шгЖ[к)фк + (Kij – a>2Mij)vj = 0, (6)

(Khk – o/Mhk^k + (Khj – a>2Mhj)Vj = 0, (7)

and corresponds to the assembly of two subsystems, both including coupling components – while the first is associated with the conventional modes (see Figure 2 – amplitudes фk), the second is related to the shear modes (see Figure 9 – amplitudes y). The system is expressed in terms of dif­ferential operators concerning linear stiffness (Kik, Kij, Khk, Khj), geometric stiffness (fyik) and mass (Mik, Mij, Mhk, Mhj) effects. The boundary conditions include generalised normal and shear stress resultants, involving terms that stem from (i) normal stress equilibrium and (ii) the variation of the shear stresses along the cross-section wall thickness. For composite members displaying cross – ply orthotropy (the case of the illustrative example presented in Section 6.2) the system (6)-(7) may be rewritten in matrix form as (Silvestre and Camotim, 2005a, b)

(Cc)f, xxxx – Dсф, хх + Всф + Ccs Ф, xxx) – fBWm(Xc. m,f, xx)

– &Ш2(Ксф – Qcф, xx + Qcs V, x) = 0, (8)

(Csc^xxx + Cs V, xx – DS Ф) – fyOX (Qs^,x + Qs V) = 0 (9)

where (i) the subscripts (-)c, (-)s and (-)cs stand conventional, shear and coupling conventional- shear mode quantities, and (ii) the various dynamic stiffness matrices Rik and Qik account for the effects of mass forces related to the in and out-of-plane cross-section translations, rotations and translation-rotation couplings. By making (i) fB = 1 and fу = 0, (ii) fB = 0 and fу = 1 or (iii) fB = ф(0 < ф < 1) and fy = 1, this system defines linear eigenvalue problems associated with (i) buckling analyses, (ii) a free vibration analyses of load-free members and (iii) free vibration analyses of loaded members – in the last case, the loads Wm are known a priori and ш2 are the problem eigenvalues.