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Dynamic instability of axially loaded shafts in the Mathieu map

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Abstract

It is shown that the Mathieu eigenvalues can be straightforwardly utilized to study the instability regions of an axially loaded simply supported shaft, the shaft being modeled as a continuous rotating beam. When a harmonic axial load is taken into account, the equation of motion of the system, here written according to the Lagrangian formulation of continuous systems, proves to be the Mathieu equation. It follows that the conditions for the stable or unstable motion of the shaft can be graphically investigated once the operation line corresponding to an actual rotating shaft is drawn in the Mathieu map.

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Abbreviations

A, l :

Area, length

E, G :

Young’s modulus, shear modulus

ρ :

Mass density

χ :

Shear factor

J :

Diametral area moment of inertia

J m , I m :

Diametral, polar mass moment of inertia per unit length

N s , N d :

Amplitude of the axial load static, dynamic component

\(\mathcal{T}\) :

Kinetic energy density

\(\mathcal{V}\) :

Potential energy density

\(\mathcal{W}\) :

Work density

ℒ:

Lagrangian density

x,y,s :

Fixed coordinates

t :

Time

u,v :

Displacements in the x, y axes

ϑ x ,ϑ y :

Bending rotations about the x, y axes

w :

Complex displacement

λ k , Λ k :

k-th natural frequencies

ω,Ω :

Rotational speed, axial excitation frequency

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Authors and Affiliations

  1. Dipartimento di Meccanica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Francesco A. Raffa & Furio Vatta

Authors
  1. Francesco A. Raffa
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  2. Furio Vatta
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Corresponding author

Correspondence to Francesco A. Raffa.

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Raffa, F.A., Vatta, F. Dynamic instability of axially loaded shafts in the Mathieu map. Meccanica 42, 547–553 (2007). https://doi.org/10.1007/s11012-007-9079-1

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  • DOI: https://doi.org/10.1007/s11012-007-9079-1

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