Research Papers

Effect of Functionally Graded Materials on Resonances of Bending Shafts Under Time-Dependent Axial Loading

[+] Author and Article Information
Arnaldo J. Mazzei

Department of Mechanical Engineering, C. S. Mott Engineering and Science Center,  Kettering University, 1700 University Avenue, Flint, MI 48504amazzei@kettering.edu

Richard A. Scott

Department of Mechanical Engineering,  University of Michigan, G044 W. E. Lay Automotive Laboratory, 1231 Beal Avenue, Ann Arbor, MI 48109car@umich.edu

Note that the second mode choice could be problematical. For the homogeneous case, there is an interior node at ξ=1/2. This is not true for the nonhomogeneous case and use of sin(2πξ)η2(τ) could lead to inaccuracies, but, here, that turns out not to be the case.

J. Vib. Acoust 133(6), 061005 (Oct 12, 2011) (11 pages) doi:10.1115/1.4004605 History: Received November 24, 2009; Revised March 08, 2011; Published October 12, 2011; Online October 12, 2011

The effect of functionally graded materials (FGMs) on resonances of bending shafts under time-dependent axial loading is investigated. The axial load is taken to be a sinusoidal function of time and the shaft is modeled via an Euler–Bernoulli beam approach (pin-pin boundary conditions). The axial load enters the formulation via a “buckling load type” approach. For generality, two distinct particulate models for the FGM are considered, namely, one involving power law variations and another based on a volume fraction approach, for both Young’s modulus and material density. The spatial dependence in the partial differential equation of motion is suppressed by utilizing Galerkin’s method with homogeneous beam mode shapes. To check the accuracy of this approximation, numerical solutions for the boundary value problem represented by the original partial differential equation are obtained using MAPLE® ’s PDE solver. Good agreement (within 5%) was found between the PDE results and the one-mode approximation. The approximation leads to ordinary differential equations that have time-dependent coefficients and are prone to parametric and forced motions instabilities. Hill’s infinite determinant approach is used to study stability. The main focus is on the primary parametric resonance. It was found that in most cases the FGM shafts increase the parametric resonance frequencies substantially, while decreasing the zone thicknesses, both desirable trends. For instance, for an axial load about one-third of the buckling value, an aluminum/silicon carbide shaft, when compared to a pure aluminum shaft, increases the primary parametric resonance by 21% and decreases instabilities by 23%. For one model of FGM, the sensitivity of the results to volume fraction variations is examined and it was found that increasing the volume fraction is not uniformly beneficial. Results for other parametric zones are also presented. Forced resonances are also briefly treated.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 16

Comparison between second-order zones of instability: model 1 versus model 3 and model 2 versus model 3

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Figure 17

Forced response of aluminum shaft for vf=0.9,1.0 and 1.1, respectively

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Figure 18

Forced response of aluminum/silicon carbide shaft for vf=1.1,1.2, and 1.3, respectively

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Figure 19

Forced response of aluminum/steel shaft for vf=1.00,1.01, and 1.2, respectively

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Figure 1

Beam differential element

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Figure 2

Primary zone of instability for homogeneous material

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Figure 3

Primary zone of instability for aluminum/silicon carbide

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Figure 4

Primary zone of instability for aluminum/steel

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Figure 5

Deflection of homogeneous aluminum shaft at nonresonant and resonant conditions

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Figure 6

Deflection of aluminum/silicon carbide shaft at nonresonant and resonant conditions

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Figure 7

Deflection of aluminum/steel shaft at nonresonant and resonant conditions

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Figure 8

Primary instability zones for model 1: Approximation versus PDE solution

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Figure 9

Primary instability zones for models 3, 1, and 2, respectively, using a two-mode approximation

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Figure 10

Comparison between zones of instability—model 3 versus model 1

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Figure 11

Angular measure of instability zone spread

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Figure 12

Comparison between zones of instability—model 3 versus model 2

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Figure 13

Normalized primary zone thickness for model 2 as a function of λ for γ2=0.03

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Figure 14

Secondary instability zones for models 3, 1, and 2

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Figure 15

Second-order instability zones for models 3, 1, and model 2




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