Hard loss of stability of Ziegler’s column with nonlinear damping

Piezoelectric control of the Hopf bifurcation of Ziegler’s column with nonlinear damping

“…Each of the two viscoelastic devices consists of a linear spring of stiffness k j = k ( j = 1, 2) and a nonlinear dashpot, of linear viscosity coefficients c 1 j = c ( j = 1, 2), and cubic viscosity coefficients c 3 j ( j = 1, 2), these latter accounting for a nonlinear dissipation here assumed to be ruled by Van der Pol-type law (see, e.g., [14,16,17]); the piezoelectric stiffness is k p (assumed negligible throughout the paper), its capacitance C p and its coupling coefficient g. The column is loaded at the point C by a follower force of intensity F.…”

“…Moreover, when finite kinematics, in addition to the nonlinear damping, is considered for the uncontrolled system, the relevant equations of motion are those derived in [14,16,17]; finally, the relevant equations of the controlled and linearly damped column, in finite kinematics, are those discussed in [55,56].…”

“…Relevant results concerning the nonlinear dynamical response of the uncontrolled Ziegler's column, with or without nonlinear damping, are presented in [15][16][17] and are briefly recalled here. Mainly, they can be summarized in the following points: (i) if the column is linearly damped, supercritical limit-cycles manifest themselves, whose large amplitude increases as the destabilizing effect of the damping increases; (ii) if a proper nonlinear damping form, such as the Van der Pol-like damping, is introduced, the amplitude of the supercritical limit-cycles can be reduced; (iii) the same nonlinear damping can have a detrimental effect on the nonlinear dynamics of the column, by giving rise to subcritical limit-cycles and, in particular, to the hard loss of stability bifurcation, according to which a regain of stability is experienced by a subcritical branch of a periodic solution through a turning point.…”

“…The analysis on the effects of the paradox extended to the nonlinear behavior of the Ziegler's column, i.e., when finite kinematics is assumed and nonlinear damping, in addition to the linear one, is introduced, is carried out in relatively few papers in the literature, e.g., in [13][14][15][16][17], with different aims. In [13,14], the nonlinear analysis is performed to investigate the stability of the trivial equilibrium via the Normal Form Theory and the second method of Lyapunov, respectively, and for different damping forms, namely: linear in [13], linear and nonlinear of Van der Pol-type, also called 'hysteretic,' in [14].…”

“…In [15] a Multiple Time Scale analysis [18][19][20] is developed to analyze the nonlinear dynamics of the linearly and uniformly damped Ziegler's column; the supercritical nature of the limitcycle, occurring when the critical load is exceeded, is shown. In [16,17] the Multiple Scale approach is followed, along the lines illustrated in [21][22][23][24][25][26] for bifurcation analysis, with the aim to detect the nonlinear behavior of the Ziegler's column endowed with nonlinear damping, ruled by a Van der Pol-like law; it is seen that large-amplitude limit-cycles manifest themselves, in the absence of nonlinear damping, and that, if this kind of damping is added to the system, the amplitude of the limit-cycle can be reduced. However, it is also shown that nonlinear damping can have a detrimental effect on the nonlinear dynamics, due to the possible occurrence of the 'hard loss of stability bifurcation' (see, e.g., [27][28][29]), according to which a regain of stability is experienced after a subcritical branch of the periodic solution manifests itself.…”