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Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam

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Abstract

In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.

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Abbreviations

A :

cross-sectional area

b 0 :

beam breadth at the root section

c b :

breadth taper ratio

c h :

height taper ratio

E :

Young’s modulus

EA :

axial rigidity of the beam cross section

EI :

bending rigidity of the beam cross section

G :

shear modulus

h 0 :

beam height at the root section

\(\vec{i}, \vec{j}, \vec{k}\) :

unit vectors in the x, y, and z directions

I y :

second moment of inertia about the y axis

k :

shear correction factor

kAG :

shear rigidity

L :

beam length

P :

reference point after deformation

P 0 :

reference point before deformation

r :

inverse of the slenderness ratio S

\(\vec{r}_{0}\) :

position vector of P 0

\(\vec{r}_{1}\) :

position vector of P

R :

hub radius

S :

slenderness ratio

t :

time

T :

centrifugal force

u 0 :

axial displacement due to the centrifugal force

U b :

potential energy due to bending

U s :

potential energy due to shear

V x , V y , V z :

velocity components of point P

W[k], θ[k ]:

transformed functions

w :

flapwise bending displacement

w′:

flapwise bending slope

x :

spanwise coordinate

\(\bar{x}\) :

spanwise coordinate parameter

x0, y0, z0:

coordinates of P 0

x1, y1, z1:

coordinates of P

δ :

hub radius parameter

γ :

shear angle

\(\varepsilon_{0}\) :

uniform strain due to the centrifugal force

\(\varepsilon_{{ij}}\) :

classical strain tensor

\(\varepsilon_{{xx}}\) :

axial strain

\(\varepsilon_{{\eta \eta}}, \varepsilon_{{\xi \xi}}\) :

transverse normal strains

η :

sectional coordinate corresponding to major principal axis for P 0 on the elastic axis

μ :

natural frequency parameter

ξ :

sectional coordinate for P 0 normal to η axis at the elastic axis

ρ :

density of the blade material

ρA :

mass per unit length

θ :

rotation angle due to bending

ω :

circular natural frequency

Ω:

constant rotational speed

\(\bar{\Omega}\) :

rotational speed parameter

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Authors and Affiliations

  1. Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak, Istanbul, Turkey

    O. Ozdemir Ozgumus & M. O. Kaya

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  1. O. Ozdemir Ozgumus
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  2. M. O. Kaya
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Correspondence to M. O. Kaya.

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Ozdemir Ozgumus, O., Kaya, M.O. Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam. Arch Appl Mech 78, 379–392 (2008). https://doi.org/10.1007/s00419-007-0158-5

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  • DOI: https://doi.org/10.1007/s00419-007-0158-5

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