Analysis of the transient response of an automatic dynamic balancer for eccentric rotors

Abstract

This paper investigates the transient response of a dynamical system modelling an automatic dynamic balancing mechanism for eccentric rotors. By using recently developed computational techniques, pseudospectra of the linearisation of the system about an equilibrium are computed. This approach allows one to quantify which eigenvalues are most sensitive to perturbation. It is shown how the sensitivity of the eigenvalues directly influences the transient response. Furthermore, the effect which a variation of the damping coefficients has on the pseudospectra structure is considered. A transient growth due to the non-normality of the linearised system is shown to lead to an exponential decay or to a collapse back to the stable equilibrium; these effects are identified with the changes in the sensitivities of the eigenvalues under variation of the damping parameters. This provides a new insight into the full nonlinear system, in which qualitatively similar transient responses are shown to occur.

Automatic two-plane balancing for rigid rotors

“…For this special case, we are able to recover all the bifurcations and steady state solutions that have been found previously in the studies of a planar automatic balancer [3,4]. However, because our model includes the extra inclinational degrees of freedom, the rotor is now able to destabilise via out of plane motions, and so, the steady states will necessarily have smaller regions of stability.…”

“…Similarly, by setting the tilt angles φ x = φ y ≡ 0, the system reduces to the equations of motion for the planar automatic balancer [4]. It is also worth remarking that a careful inspection of equations (11)(12) reveals certain asymmetries in the moment terms corresponding to the balancing balls.…”

“…The first study of an ABB was carried out by Thearle in 1932 [1], and the existence of a stable steady state at rotation speeds above the first critical frequency was demonstrated. More recently, the equations of motion for a planar Jeffcott rotor with an ABB have been derived using Lagrange's method [2,3,4]. In particular Green et al [4] use rotating coordinates to forge an autonomous system of governing equations.…”

“…More recently, the equations of motion for a planar Jeffcott rotor with an ABB have been derived using Lagrange's method [2,3,4]. In particular Green et al [4] use rotating coordinates to forge an autonomous system of governing equations. The stability boundaries for the fully nonlinear system are then computed using numerical continuation techniques.…”

“…These out-of-plane motions are considered in the studies of Chung and Jang [6], Chao et al [7] and Sperling et al [8], but only linear stability analyses are provided. Here we extend the planar analysis of [4] so that we may model the tilting motion of the ABB and analyse the two-plane automatic balancer for rigid rotors. The rest of this paper is organised as follows.…”