Experimental and Theoretical Nonlinear Dynamic Response of Intact and Cracked Bone-Like Specimens With Various Boundary Conditions

[+] Author and Article Information
Erick Ogam, Armand Wirgin, Z. E. A. Fellah

 CNRS Laboratoire de Mécanique et d’Acoustique UPR7051, 31 Chemin Joseph Aiguier, 13402 Marseille, France

Yongzhi Xu

Department of Mathematics, University of Louisville, Louisville, KY 40292

J. Vib. Acoust 129(5), 541-549 (Feb 06, 2007) (9 pages) doi:10.1115/1.2731413 History: Received July 28, 2006; Revised February 06, 2007

The potentiality of employing nonlinear vibrations as a method for the detection of osteoporosis in human bones is assessed. We show that if the boundary conditions (BC), relative to the connection of the specimen to its surroundings, are not taken into account, the method is apparently unable to differentiate between defects (whose detection is the purpose of the method) and nonrelevant features (related to the boundary conditions). A simple nonlinear vibration experiment is described which employs piezoelectric transducers (PZT) and two idealized long bones in the form of nominally-identical drinking glasses, one intact, but in friction contact with a support, and the second cracked, but freely-suspended in air. The nonlinear dynamics of these specimens is described by the Duffing oscillator model. The nonlinear parameters recovered from vibration data coupled to the linear phenomena of mode splitting and shifting of resonance frequencies, show that, despite the similar soft spring behavior of the two dynamic systems, a crack is distinguishable from a contact friction BC. The frequency response of the intact glass with contact friction BC is modeled using a direct steady state finite element simulation with contact friction.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

(a) A bovine-bone femoral shaft and its idealization, a drinking glass. (b) A glass vibrating with stress-free boundary conditions (suspended on nylon threads). The glass is then turned upside down and a contact friction boundary condition (CFBC) prevails at the interface between the glass specimen and (c) a fairly-smooth pine table. (d) A thick aluminium plate (dimensions 22.5×20.5×1.5cm).

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Figure 2

(1) Scheme of the experimental setup with one of the configurations; the glass placed on an aluminium plate. The excitation force and response are applied and measured, respectively, using piezoelectric (PZT) devices. (2) An extra input force is applied to the base (at the top in the figure) of the glass using two small mutually-repulsive magnets.

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Figure 3

(a) The experimental response of the suspended (stress-free) bovine-bone. (b) The experimental response of the same bone with a contact friction boundary condition. The response is nonlinear.

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Figure 4

The mode shapes and vector plots of their displacements for the modes used in the inversion algorithm to recover Young’s modulus and the Poisson ratio (a)1111Hz; (b)2685Hz; (c)4489Hz. (d) The mesh distribution, the deformed shape, and the displacement vectors of the glass in CFBC with the support.

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Figure 5

The computed acceleration (acc) (dotted lines), and the experimental (continuous line) response of the upside down glass. The contact friction coefficient on the interface between the glass and the supporting aluminium block is μf=0.17.

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Figure 6

(a) The FES model (third mode) showing the arbitrarily chosen slit mimicking an open crack. (b) The computed response depicting the third and fourth resonance frequencies (second in-plane and out-of-plane modes) for the intact and the cracked (two sizes) case.

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Figure 7

(a) The wide band experimental response spectrum of the intact and cracked glasses with SFBC. (b) The resonant responses in the vicinity of the third mode of the glasses and their peaks. (c) The resonant peaks after peak-picking. The peaks of the sound glass with SFBC do not change as the excitation amplitude is being increased (linear behavior). Those of the cracked glass are split (the in-plane and out-of-plane modes are different) and have shifted downwards. The lowest one shifts downwards as the excitation amplitude increases.

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Figure 8

(a) The experimental spectra of intact and cracked glasses turned upside down on a plate (a contact friction boundary condition (CFBC) prevails at their interface). Note the multiplication of peaks (mode splitting) in the cracked case. (b) The glasses are on a massive pine table with CFBC. (c) The peak resonant responses in the vicinity of a nonlinear mode for the case in (a). (d) The peak resonant responses in the vicinity of a nonlinear mode for the case in (b). In (a) and (b) both glasses behave in a nonlinear manner that is, the resonance frequency shifts downwards with the increase in the amplitude of the exciting force.

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Figure 9

The measured response (continuous line), the computed response (open circles) using Duffing’s equation. (a) The cracked glass with stress-free boundary conditions, αs=−150 and μm=0.0048. (b) The intact glass on a plate with contact friction boundary conditions, αs=−600 and μm=0.019. (c) The cracked glass in CFBC with a plate, αs=−1100 and μm=0.0095. (d) The intact glass in CFBC with the table, αs=−65 and μm=0.004. (e) The cracked glass in CFBC with the table, αs=−100 and μm=0.0095.

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Figure 10

Different normal loads (force variation by use of two repelling magnets, one fixed on the glass) are applied by approaching the second magnet while the excitation amplitude remains constant and the frequency is swept. The distances between the two magnets are indicated on the curves. The system exhibits a nonlinear hard spring behavior with an increase in the frequency and amplitude of the force.



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