Dynamic response of a guided circular saw

Sensitivity of Sub-critical Mode-coupling Instabilities in Non-conservative Rotating Continua to Stiffness and Damping Modifications

“…Moreover, all the roots of the polynomial (35) under restraints (36) and sufficiently small perturbations are pure imaginary if, in addition…”

“…Therefore, in the space of the shaft parameters there exists a non-trivial set given by expressions (36), (37) where all the eigenvalues of the system are pure imaginary (marginal stability) in the presence of dissipative and non-conservative forces. To study the connection of this set to the domains of instability and asymptotic stability in the parameter space, it is more convenient to use the perturbation formulas of the previous section than to analyse directly the characteristic polynomial (35) (34) into Eqns.…”

“…This is not the case in the subcritical speed range. Here the doublets in the Campbell diagrams that are sources of the combination resonance [49,50] in the supercritical range just veer away into avoided crossings [36,37,[39][40][41][42][44][45][46]. The instabilities in this speed range are provoked by the non-conservative positional forces and by the indefinite damping [8,22].…”

“…Typical rotor dynamical applications are related to stability of high-speed machinery such as turbine shafts and wheels, circular saws, disks of computer data storage devices, to name a few [34][35][36][37][38][39][40][41]. In such applications gyroscopic forces are significant and it is natural to consider non-conservative and dissipative forces as a perturbation of a conservative gyroscopic system.…”

“…It is known that variation in the mass and stiffness distribution may cause the so-called mass-and stiffness-instabilities for such speeds [37][38][39][40][41][42]. It is remarkable that these instabilities manifest themselves as instability bubbles [43] of complex eigenvalues that originate due to unfolding of double eigenvalues at some crossings of eigencurves of the Campbell diagram of the non-modified system [30][31][32][33][34][35][36][37][38][39][40][41][42][44][45][46][47][48]. The mass and stiffness instabilities can easily be identified with the interaction of the so-called waves of positive and negative energy -the instability mechanism that is well-known in hydrodynamics and that is typical for Hamiltonian systems such as conservative gyroscopic ones [42,43].…”