Introduction to linear and nonlinear programming

The concept of criticality in multi-degree-of-freedom systems is discussed. Sufficient conditions for overdamping, critical damping, and underdamping are derived in terms of the matrices appearing in the modal coordinates. It is noted that results available in the literature for the case of overdamping and mixed damping are erroneous. This has been pointed out by Bhaskar (1991, 1992) for the cases of overdamping and mixed damping, and by Barkwell et al. (1992) for the case of overdamping. The error in the proof of the conditions for overdamping is brought out. A sufficient condition for overdamping is presented. Results obtained for the symmetric systems are then generalized to the symmetrizable systems. Theorems on eigenvalue bounds are applied to establish criticality.

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Criticality of Damping in Multi-Degree-of-Freedom Systems

Oscillatory damped linear systems

Eigenvalue bounds for damped linear systems

In active magnetic bearing (AMB), the beat vibration will affect the flywheel rotary precision. A vibration model, which includes general dynamic stiffness and unbalance response different from the existing vibration research, is proposed to analyze the vibration characteristics, and the vibration model can express the actual system vibration features from the mechanism. In order to solve the vibration model, it is considered as a multi-degree-of-freedom vibration system. Both the free vibration and the forced vibration are analyzed according to the proposed model's solution, and therefore the beating vibration is discovered when the two vibration frequencies are similar. The weakening measures of the unbalance vibration and the beat vibration are also discussed by analyzing the general dynamic stiffness changing. The simulation and experimental results demonstrate the validity of the theoretical analysis. The vibration analysis results can provide effective mechanism support for active magnetic bearings vibration characteristics research, and the vibration model can provide simulation platform for others applied Research.

D.W. Nicholson

Papers

Eigenvalue bounds for damped linear systems

A note on vibrations of damped linear systems

On the forced vibration of a damped linear system

On preconditioning a nonsingular matrix