1. Introduction In the aerospace and automotive applications driveshafts are manufactured using fiber reinforced composite materials. During the design stage of the driveshaft, it is essential to determine the natural frequencies and the critical speeds of the driveshaft such that the resonance phenomena can be avoided. Compared to a conventional metallic driveshaft, a composite driveshaft gives higher natural frequencies and critical speeds, and lower vibration. In addition, they are also lightweight structures, especially when they are tapered. The design of the driveshaft is dependent on its fundamental natural frequency, and tapering the driveshaft can substantially improve the value of this natural frequency. Zinberg and Symonds [1] experimentally performed rotordynamic analysis of advanced composite shaft, and they determined the critical speeds of the composite shaft; the results showed superiority of the composite shafts over the aluminum alloy shafts. Ingle and Ahuja [2] designed an experimental set-up to investigate the vibration of a high speed composite shaft in aerostatic conical journal. Based on a first-order shear deformable beam theory, Chang et al. [3] developed equations of motion of uniform composite shaft using Hamilton’s principle. They performed rotordynamic analysis to find out the natural frequencies and critical speeds. Moreover, Chang et al. [4] studied the vibration of the rotating composite shafts containing randomly oriented reinforcements. Librescu et al. [5] studied the stability of rotating tapered composite shaft subjected to an axial compressive force; the results showed that tapering the composite shaft shifts the domain of divergence and flutter instability to larger rotating speeds. Moreover, Boukhalfa and Hadjoui [6] analyzed the free vibration of uniform composite shaft using the hierarchical finite element method. Al Muslmani and Ganesan [7] developed a finite element model for rotordynamic analysis of uniform composite shaft using Hermitian – conventional finite element. Qatu and Iqbal [8] provided the exact solution for a two-segmented composite driveshaft joined by a hinge. Kim et al. [9] studied the effect of steel core on the bending natural frequency of steel / CFRP hybrid shafts. Kim et al. [10] developed a mechanical model for a tapered composite Timoshenko shaft. The model represented an extended length tool holder at high speed in end milling or boring operation. The structure of the shaft had clamped – free supports. They used the general Galerkin method to obtain the spatial solutions of the equations of motion. They studied forced torsional vibrations, dynamic instability, forced vibration response, and static strength of a tapered composite shaft subjected to deflection- dependent cutting forces. In the present paper, a finite element model for tapered composite driveshaft using Lagrangian finite element formulation is developed for rotordynamic analysis. The strain and kinetic energy expressions for tapered composite driveshaft are obtained and then the Lagrange’s equation is applied to develop the governing equations. Timoshenko beam theory is adopted, so that the effect of shear deformation is included in the model in addition to the effects of the rotary inertia, the gyroscopic forces, the taper angle, the axial load, and the coupling effects due to the lamination of composite layers. 2. Rotordynamic Analysis Figure 1 shows a single lamina deformed into a conical tube with taper angle α that can change functionally in x direction. The principal material directions are denoted by 1, 2, and 3. The axis 1’ extends along the tapered tube surface while 3’- axis ROTORDYNAMICS OF TAPERED COMPOSITE DRIVESHAFT BASED ON A LAGRANGIAN FINITE ELEMENT M. Al Muslmani 1 , R. Ganesan 2 * Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada * Corresponding author ([email protected]) Keywords: (Tapered composite shaft, Rotordynamics, Finite element method)
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1. Introduction In the aerospace and automotive applications
driveshafts are manufactured using fiber reinforced
composite materials. During the design stage of the
driveshaft, it is essential to determine the natural
frequencies and the critical speeds of the driveshaft
such that the resonance phenomena can be avoided.
Compared to a conventional metallic driveshaft, a
composite driveshaft gives higher natural
frequencies and critical speeds, and lower vibration.
In addition, they are also lightweight structures,
especially when they are tapered. The design of the
driveshaft is dependent on its fundamental natural
frequency, and tapering the driveshaft can
substantially improve the value of this natural
frequency.
Zinberg and Symonds [1] experimentally performed
rotordynamic analysis of advanced composite shaft,
and they determined the critical speeds of the
composite shaft; the results showed superiority of
the composite shafts over the aluminum alloy shafts.
Ingle and Ahuja [2] designed an experimental set-up
to investigate the vibration of a high speed
composite shaft in aerostatic conical journal. Based
on a first-order shear deformable beam theory,
Chang et al. [3] developed equations of motion of
uniform composite shaft using Hamilton’s principle.
They performed rotordynamic analysis to find out
the natural frequencies and critical speeds. Moreover,
Chang et al. [4] studied the vibration of the rotating
composite shafts containing randomly oriented
reinforcements. Librescu et al. [5] studied the
stability of rotating tapered composite shaft
subjected to an axial compressive force; the results
showed that tapering the composite shaft shifts the
domain of divergence and flutter instability to larger
rotating speeds. Moreover, Boukhalfa and Hadjoui
[6] analyzed the free vibration of uniform composite
shaft using the hierarchical finite element method.
Al Muslmani and Ganesan [7] developed a finite
element model for rotordynamic analysis of uniform
composite shaft using Hermitian – conventional
finite element. Qatu and Iqbal [8] provided the exact
solution for a two-segmented composite driveshaft
joined by a hinge. Kim et al. [9] studied the effect of
steel core on the bending natural frequency of steel /
CFRP hybrid shafts. Kim et al. [10] developed a
mechanical model for a tapered composite
Timoshenko shaft. The model represented an
extended length tool holder at high speed in end
milling or boring operation. The structure of the
shaft had clamped – free supports. They used the
general Galerkin method to obtain the spatial
solutions of the equations of motion. They studied
forced torsional vibrations, dynamic instability,
forced vibration response, and static strength of a
tapered composite shaft subjected to deflection-
dependent cutting forces.
In the present paper, a finite element model for
tapered composite driveshaft using Lagrangian finite
element formulation is developed for rotordynamic
analysis. The strain and kinetic energy expressions
for tapered composite driveshaft are obtained and
then the Lagrange’s equation is applied to develop
the governing equations. Timoshenko beam theory is
adopted, so that the effect of shear deformation is
included in the model in addition to the effects of the
rotary inertia, the gyroscopic forces, the taper angle,
the axial load, and the coupling effects due to the
lamination of composite layers.
2. Rotordynamic Analysis
Figure 1 shows a single lamina deformed into a
conical tube with taper angle α that can change
functionally in x direction. The principal material
directions are denoted by 1, 2, and 3. The axis 1’
extends along the tapered tube surface while 3’- axis
ROTORDYNAMICS OF TAPERED COMPOSITE DRIVESHAFT
BASED ON A LAGRANGIAN FINITE ELEMENT
M. Al Muslmani1, R. Ganesan
2*
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada