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Consistent matrices in rotor dynamic

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Sommario

Nel presente lavoro vengono ricavate le espressioni della matrice congruente delle masse e giroscopica per un elemento di albero a sezione costante. Tali espressioni tengono conto sia della deformazione a taglio che dell'inerzia trasversale. I risultati, in termini di velocità critiche e di diagramma di Campbell, per un albero a sezione costante vengono confrontati con soluzioni in forma chiusa di varia complessità. Vengono discussi gli effetti della deformazione a taglio, dell'inerzia trasversale, dello smorzamento e dello sforzo assiale sul comportamento dinamico. Viene inoltre presa in considerazione una distribuzione lineare nota di squilibrio. Il modello cosi ottenuto può venire usato anche per lo studio di sistemi smorzati, purchè lo smorzamento possa essere considerato di tipo «viscoso» o «strutturale». Viene infine mostrata un'applicazione ad un modello in cui si tiene conto anche dello smorzamento e che fa uso della condensazione delle matrici per ridurre l'ordine del problema dinamico.

Summary

The expressions for the consistent mass and gyroscopic matrices for a constant section shaft element are obtained taking into account both shear deformation and transversal inertia. The results are compared with closed form solutions, which are available in simple cases. The results obtained show that the study of the dynamic behaviour of the rotor with a model which includes rotational inertia but not shear deformation is, at least in the case examined, misleading. Formulae for matrix condensation and for taking into account the effects of axial load and of a linear unbalance distribution are given. Damped systems can be studied using the same model, provided that damping can be assumed to be of either viscous or hysteretic type. Some formulations found in the literature are however not considered correct. An application of consistent matrices to a model which includes damping and uses matrix condensation is shown.

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Abbreviations

{f}:

vector of the forces due to the unbalance of the element

[g]:

gyroscopic matrix of the element

i :

\(\sqrt { - 1} \)

[k]:

stiffness matrix of the element

l :

shaft length

l e :

length of the element

[m]:

mass matrix of the element

n :

number of elements

r i :

inner radius

r 0 :

outer radius

{q}:

generalized displacements at the nodes

x y z :

system of reference

t :

time

A :

area of the cross section

[C]:

viscous damping matrix

E :

Young's modulus

{F}:

force vector due to the unbalance

F a :

axial force

G :

shear modulus

[G]:

gyroscopic matrix

I :

area moment of inertia of the cross section

[I]:

identity matrix

[K′]:

real part of the stiffness matrix

[K″]:

imaginary part of the stiffness matrix

[M]:

mass matrix

T :

kinetic energy

U :

potential energy

α:

slenderness\(\left( {\alpha = 1\sqrt {A/I} } \right)\)

γ:

shear strain

∈:

«static» unbalance; strain

ζ:

non dimensional coordinate (ζ=z/l e )

η:

loss factor

ϑ:

«dynamic» unbalance

λ:

whirl speed

ν:

Poisson's ratio

ξ η ζ:

rotating system of reference (fixed to the shaft element)

ρ:

density

\(\phi _x ,\phi _y\) :

rotations aboutx andy axes

χ:

shear factor

ω:

spin speed

n :

non rotating element

r :

rotating element

I :

inertial section characteristics, imaginary part

R :

real part

S :

structural section characteristics

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Authors and Affiliations

  1. Dipartimento di Meccanica, Politecnico di Torino, Italy

    Giancarlo Genta (Associate Professor)

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  1. Giancarlo Genta
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Genta, G. Consistent matrices in rotor dynamic. Meccanica 20, 235–248 (1985). https://doi.org/10.1007/BF02336935

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  • DOI: https://doi.org/10.1007/BF02336935

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