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Research Articles

Stochastic Evaluation and Analysis of Free Vibrations in Simply Supported Piezoelectric Bimorphs

[+] Author and Article Information
Alberto Borboni

e-mail: borbonia2@asme.org

Rodolfo Faglia

Dipartimento di Ingegneria Meccanica,
Università degli Studi di Brescia,
Via Branze 38,
25123 Brescia, Italy

1Corresponding author.

Manuscript received February 25, 2011; final manuscript received September 7, 2012; accepted manuscript posted September 29, 2012; published online January 22, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(2), 021003 (Jan 22, 2013) (7 pages) Paper No: JAM-11-1064; doi: 10.1115/1.4007721 History: Received February 25, 2011; Revised September 07, 2012; Accepted September 29, 2012

Piezoelectric bimorph benders are a particular class of piezoelectric devices, which are characterized by the ability to produce flexural deformation greatly larger than the length or thickness deformation of a single piezoelectric layer. Due to extensive dimensional reduction of devices and to the high accuracy and repeatability requested, the effect of erroneous parameter estimation and the fluctuation of parameters due to external reasons, sometimes, cannot be omitted. As such, we consider mechanical, electrical and piezoelectric parameters as uniformly distributed around a nominal value and we calculate the distribution of natural frequencies of the device. We consider an analytical model for the piezoelectric bimorph proposed in literature. The results show how the parameters errors are reflected on the natural frequencies and how an increment of the error is able to change the shape of the frequencies distribution.

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Figures

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Fig. 1

Simply supported piezoelectric bimorph

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Fig. 2

Mechanical equilibrium

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Fig. 3

Probability density function of a rectangular distributed parameter

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Fig. 4

Δω versus λ for mechanical (dC), piezoelectric (de), electric (deps), and mixed mechanical, piezoelectric and electric (dtot) parametric errors with ΔInput = 0.01 and shear coefficient k = 5/6

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Fig. 5

Δω versus λ for different values of mechanical relative error ΔC with shear coefficient k = 5/6

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Fig. 6

γ1 versus λ for different values of mechanical relative error ΔC with shear coefficient k = 5/6

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Fig. 7

γ2 versus λ for different values of mechanical relative error ΔC with shear coefficient k = 5/6

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Fig. 8

Δω versus λ for different values of mixed relative error ΔC,e,eps with shear coefficient k = 5/6

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Fig. 9

γ1 versus λ for different values of mixed relative error ΔC,e,eps with shear coefficient k = 5/6

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Fig. 10

γ2 versus λ for different values of mixed relative error ΔC,e,eps with shear coefficient k = 5/6

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Fig. 11

Δω versus λ for different values of mixed relative error ΔC,e,eps with shear coefficient k = 8/9

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Fig. 12

γ1 versus λ for different values of mixed relative error ΔC,e,eps with shear coefficient k = 8/9

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Fig. 13

γ2 versus λ for different values of mixed relative error ΔC,e,eps with shear coefficient k = 8/9

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Fig. 14

Δω versus λ for different values of shear coefficient k with mechanical relative error ΔC = 0.01

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Fig. 15

Average Δω versus ΔC when mechanical errors are present (dC), when mixed errors are present (dtot) and empirically forecasted values (expr17) with shear coefficient k = 5/6

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Fig. 16

Comparing Δω (continuous lines) with forecasted average Δω (dot-point lines)

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Fig. 17

Higher order forecasted average Δω

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Fig. 18

Difference between average Δω and higher order forecasted average Δω

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Fig. 19

Sum of modulus of errors in every considered geometrical ratio between average Δω and higher order forecasted average Δω, for every considered fitting function

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Fig. 20

Comparing Δω (continuous lines) with exponentially forecasted average Δω (dot-point lines)

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