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Dynamic basic displacement functions in free vibration analysis of centrifugally stiffened tapered beams; a mechanical solution

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Abstract

This paper deals with enhancing the existing Finite Element formulations through employing basic principles of structural mechanics accompanied with mathematical techniques. Introducing the concept of Basic Displacement Functions (BDFs), the free vibration analysis of rotating tapered beams is studied from a mechanical point of view. It is shown that exact shape functions could be derived in terms of BDFs. The new shape functions turn out to be dependent on the rotational speed, circular frequency, the position of element along the beam and variation of cross-sectional dimensions along the element. Dynamic BDFs are obtained by applying Adomian Modified Decomposition Method (AMDM) to the governing differential equation of motion. Carrying out numerical examples, the competency of the method is verified.

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Abbreviations

A(x):

cross-sectional area at distance x

A 0 :

cross-sectional area at x=0

b :

vector of BDFs

E :

modulus of elasticity

F :

vector of nodal forces

F II ,F JJ :

nodal flexibility matrices of the left and right nodes respectively

G :

matrix containing nodal stiffness matrices

I(x):

moment of inertia at distance x

I 0 :

moment of inertia at x=0

K :

flexural stiffness matrix

K G :

geometric stiffness matrix

K II ,K JJ :

nodal stiffness matrices of the left and right nodes respectively

L :

length of the beam

l e :

length of the element

M :

bending moment

M :

consistent mass matrix

N :

vector of shape functions

q :

external transverse load

R :

hub radius

t :

time

T :

centrifugal force

V :

shear force

w :

transverse displacement

x :

longitudinal coordinate along whole beam

\(\bar {x}\) :

longitudinal coordinate along beam element

δ :

hub radius parameter

η :

non-dimensional rotational speed

θ :

angle of rotation

μ :

non-dimensional natural frequency

ρ :

mass density

ω :

circular frequency

Ω:

rotational speed

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Attarnejad, R., Shahba, A. Dynamic basic displacement functions in free vibration analysis of centrifugally stiffened tapered beams; a mechanical solution. Meccanica 46, 1267–1281 (2011). https://doi.org/10.1007/s11012-010-9383-z

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  • DOI: https://doi.org/10.1007/s11012-010-9383-z

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