Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

Abstract

To cite this version:M.R. Brake, J.A. Wickert. Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints. Journal of Sound and Vibration, Elsevier, 2010, 329 (7) A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed. This method assumes that the nonlinear constraint can be expressed as a piecewise linear force-deflection profile located at an arbitrary position within the domain. Using this assumption, the mode shapes and natural frequencies are first found for each state, then a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between the in-contact and out-of-contact states. To illustrate this method, a model for the vibration of a traveling string in contact with a piecewise-linear constraint is developed as an analog of the interaction between magnetic tape and a guide in data storage systems. Five design parameters of the guide are considered: flange clearance, flange stiffness, symmetry of the force-deflection profile in terms of flange stiffness and offset, and the guide's position along the length of the string. There are critical bifurcation thresholds, below which the system exhibits no chaotic behavior and is dominated by period one, symmetric behavior, and above which the system contains asymmetric, higher periodic motion with windows of chaotic behavior. These bifurcation thresholds are particularly pronounced for the transport speed, flange clearance, symmetry of the force deflection profile, and guide position. The stability of the system is sensitive to the system's velocity, and, compared to stationary systems, more mode shapes are needed to accurately model the dynamics of the system.

Model Reduction Methods for Rotor Dynamic Analysis: A Survey and Review

“…From an extensive survey of the literature, the most common methods used for structural reduction of rotor systems appear to be Guyan Reduction [38,58], Modal Analysis [39,[59][60][61][62][63] and Component Mode Synthesis [20,40,43,[64][65][66]. More specifically research on reduction methods, not necessarily applied to rotor systems, looks to handle damping [59,65,[67][68][69] and gyroscopics [63,[70][71][72] within the systems but rarely together. Other methods such as Krylov-based methods have been applied to structural systems for FEA analysis [73] but are not commonly found in the literature nor are they normally employed in rotor systems.…”

On Travelling Wave Modes of Axially Moving String and Beam

“…Park et al analyzed the deploying and retracting axially moving beam and derived the dynamic responses of the longitudinal and transverse vibrations [6]. Modal analysis is one of the classical methods to derive the response of gyroscopic systems [7][8][9][10]. Using the Galerkin truncation method to discretize the gyroscopic system is an effective method to truncate the partial differential equations into a set of ordinary differential equations [11][12][13].…”

Dynamics of Structures with Distributed Gyroscopes: Modal Discretization Versus Spatial Discretization

“…The dynamics of flexible structures with distributed appendages can be investigated by modal discretization techniques such as the Galerkin method by introducing a set of trial mode functions, which are usually the modal functions of the corresponding structure without appendages [7][8][9]. Modal discretization techniques have shown powerful applications to structures with regular shapes (explicit modal functions) [10][11][12][13][14]. However, modal discretization becomes unpractical when treating structures with irregular or complicated contours.…”