Dynamic stability of viscoelastic columns loaded by a follower force

The nonlinear equations of motion for a cantilevered viscoelastic column subjected to a follower force at its free end are derived. The exact linear equations are obtained from the nonlinear form and studied. The motion of a column made of a linear viscoelastic material is analyzed by the dynamic approach. The models used in this study are many; in particular a Maxwell model, a Kelvin model, a three element elastic solid and viscous model, and a four element model. The last model is considered as the general model because the results for other models can be deduced from the results of this model. The boundary value problem for our column for each viscoelastic model described above is obtained. They are found to be non-self-adjoint boundary value problems.; The stability analyses of our systems are investigated by studying the eigenvalues of the characteristic equation for the solution to the boundary value problem of each viscoelastic model. More specifically, our stability analysis determined the smallest value of the follower load P, called P_cr, beyond which the system, under a suitable disturbance, will perform oscillations with increasing amplitudes (become unstable). The dynamic instability is found to occur in the form of flutter.; The effect of damping on the critical follower load is also examined; that is the damping effect on the stability analysis. Three damping parameters are found affecting our results. They are the damping coefficient related to the Maxwell unit, δ₁, the damping coefficient related to the Kelvin unit, δ₂, and the damping ratio γ. Also, the destabilization effects of viscosity is confirmed.; Finally, the results are compared with those from other models, and with the classic solution for the elastic solid column; Beck's problem.