Non-linear Frequency Response and Stability Analysis of Piezoelectric Nanoresonator Subjected to Electrostatic Excitation

Sayyid H. Hashemi Kachapi; Morteza Dardel; Hamidreza Mohamadi daniali; Alireza Fathi

doi:10.1515/ijnsns-2018-0269

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Non-linear Frequency Response and Stability Analysis of Piezoelectric Nanoresonator Subjected to Electrostatic Excitation

Sayyid H. Hashemi Kachapi , Morteza Dardel , Hamidreza Mohamadi daniali and Alireza Fathi

Published/Copyright: April 11, 2019

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 20 Issue 5

Abstract

The effects of surface energy on the non-linear frequency response and stability analysis of piezoelectric cylindrical nano-shell as piezoelectric nanoresonator are investigated in the current paper using Gurtin–Murdoch surface elasticity and von Karman–Donnell’s theory. The nanoresonator is embedded in visco-Pasternak medium and electrostatic excitation. The governing equations and boundary conditions are derived using Hamilton’s principle and also the assumed mode method is used for changing the partial differential equations into ordinary differential equations. Complex averaging method combined with arc-length continuation is used to achieve an approximate solution for the steady-state vibrations of the system. The validation of the mentioned system is achieved with excellent agreements by comparison with numerical results. The parametric studies such as the effects of geometrical and material properties, different boundary conditions, the ratio of length to radius L/R for different values of the voltages VDC and VAC, the gap width of the nanoresonator b/L, the effect of the voltages VDC and VAC and also the effect of piezoelectric voltage Vp are conducted on the non-linear frequency response and stability analysis of the piezoelectric nanoresonator.

Keywords: Applied Mathematics; Nonlinear and Complex Systems; surface elasticity; piezoelectric nanoresonator; non-linear frequency response; stability analysis; electrostatic excitation; complex averaging method; arc-length continuation

MSC 2010: 65P40; 65P30; 37M10; 37M20; 34C28; 34C15

Conflict of interest: The authors report no conflict of interest.

Appendix A

(54)AijN,BijN,DijN=∫−hNhNCijN1,z,z2dz,Aijp,Bijp,Dijp=∫hNhN+hpCijp1,z,z2dz,

(55)Nxp,Nθp=∫hNhN+hpe31p,e32pEˉzpdz+e31ps,e32psEˉzp,

(56)Mxp,Mθp=∫hNhN+hpe31p,e32pEˉzpzdz+e31ps,e32psEˉzphN+hp,

(57)A11∗=A22∗=λs1+2μs1+λs2+2μs2,A12∗=A21∗=τ0s1+λs1+τ0s2+λs2,A66∗=μs1−τ0s12+μs2−τ0s22,

(58)B11∗=B22∗=λs2+2μs2hN+hp−λs1+2μs1hN,B12∗=B21∗=τ0s2+λs2hN+hp−τ0s1+λs1hN,B66∗=μs2−τ0s22hN+hp−μs1−τ0s12hN,

(59)D11∗=D22∗=λs2+2μs2hN+hp2+λs1+2μs1hN2,D12∗=D21∗=τ0s2+λs2hN+hp2+τ0s1+λs1hN2,D66∗=μs2−τ0s22hN+hp2+μs1−τ0s12hN2,

(60)F11N∗=∫−hNhNυN1−υNτ0s2−τ0s12+τ0s2+τ0s1z2hN+hpdz,F11p∗=∫hNhN+hpυp1−υpτ0s2−τ0s12+τ0s2+τ0s1z2hN+hpdz,

(61)J11N∗=∫−hNhNυN1−υNρs1−ρs22−ρs1+ρs2z2hN+hpdz,J11p∗=∫hNhN+hpυp1−υpρs1−ρs22−ρs1+ρs2z2hN+hpdz,

(62)E11N∗=∫−hNhNυN1−υNτ0s2−τ0s1z2+τ0s2+τ0s1z22hN+hpdz,E11p∗=∫hNhN+hpυp1−υpτ0s2−τ0s1z2+τ0s2+τ0s1z22hN+hpdz,

(63)G11N∗=∫−hNhNυN1−υNρs1−ρs2z2−ρs1+ρs2z22hN+hpdz,G11p∗=∫hNhN+hpυp1−υpρs1−ρs2z2−ρs1+ρs2z22hN+hpdz,

Appendix B

α1=1m3Aˉ11, α2=m02m3Aˉ66, α3=m0m3Aˉ12+Aˉ21, α4=2m0m3Aˉ66, α5=m0m3Aˉ12+Aˉ21,

α6=12m1m32Aˉ11−τˉ0s1−τˉ0s2, α7=m0m22m3Aˉ12+Aˉ21, α8=2m0m2m3Aˉ66, α9=m02m3Aˉ22,

α10=m02m22m32Aˉ22−τˉ0s1−τˉ0s2,α11=m22m3Aˉ12+Aˉ21, α12=m02m32Aˉ22−τˉ0s1−τˉ0s2,

α13=1m3Aˉ66, α14=2m2m3Aˉ66,α15=m02m3Aˉ22−τˉ0s1−τˉ0s2, α16=14m12m3Aˉ11−τˉ0s1−τˉ0s2,

α17=m02m224m3Aˉ22−τˉ0s1−τˉ0s2, α18=m224m34Aˉ66+Aˉ12+Aˉ21, α19=m22m3Aˉ12+Aˉ21,

α20=m02m2m3Aˉ22−τˉ0s1−τˉ0s2, α21=−4m0m2m3Bˉ66,α22=m0m2m3Fˉ11−Bˉ12−Bˉ21,

α23=1m1m3Fˉ11−2Bˉ11, α24=m2m3Fˉ11−Bˉ12−Bˉ21,α25=−4m2m3Bˉ66,

α26=m02m2m3Fˉ11−2Bˉ22,α27=m222m3Fˉ11−Bˉ12−Bˉ21,α28=m2m3Fˉ11−Bˉ12−Bˉ21,

α29=m222m3Fˉ11−Bˉ12−Bˉ21, α30=12m12m3Fˉ11−2Bˉ11+τˉ0s21+m4−τˉ0s1,

α31=m02m222m3Fˉ11−2Bˉ22+τˉ0s21+m4−τˉ0s1,α32=−4m22m3Bˉ66,

α33=m02m2m3Fˉ11−2Bˉ22+τˉ0s21+m4−τˉ0s1,α34=1m12m3Dˉ11−Eˉ11,

α35=m02m22m3Dˉ22−Eˉ11,α36=4m22m3Dˉ66,α37=m22m3Dˉ12+Dˉ21−2Eˉ11,

α38=m0m1m3τˉ0s1+τˉ0s2−Nˉθp,α39=m1m3τˉ0s1+τˉ0s2−Nˉxp,α40=m0m1m3τˉ0s1+τˉ0s2−Nˉθp,

α41=12m3τˉ0s1+τˉ0s2−Nˉxp,α42=m022m3τˉ0s1+τˉ0s2−Nˉθp, α43=1m3Mˉxp+τˉ0s1−τˉ0s21+m4,

α44=m02m3Mˉθp+τˉ0s1−τˉ0s21+m4,α45=−12m12m3Gˉ11∗,α46=−m222m3Gˉ11∗,α47=12m1m3Jˉ11∗,

α48=m22m3Jˉ11∗,α49=m22m3Jˉ11∗,α50=14m12m3Jˉ11∗,α51=m224m3Jˉ11∗,

Appendix C

Muu=∫χeχidξ∫ϑfϑjdθ

Muw=12α47∫χe′βodξ∫ϑfψldθ

Kuu=α1∫χe′χi′dξ∫ϑfϑjdθ+α2∫χeχidξ∫ϑf′ϑj′dθ

Kuv=12α3∫χe′ϕkdξ∫ϑfαl′dθ+12α4∫χeϕk′dξ∫ϑf′αldθ

Kuw=12α5∫χe′βodξ∫ϑfψldθ+12α21∫χeβ0′dξ∫ϑf′ψl′dθ+12α22∫χe′βodξ∫ϑfψl′′dθ+12α23∫χe′β0′′dξ∫ϑfψldθ

NLuw=12α6∫χe′β0′βt′dξ∫ϑfψpψvdθ+12α7∫χe′βoβtdξ∫ϑfψp′ψv′dθ+12α8∫χeβ0′βtdξ∫ϑf′ψpψv′dθ

Fˉup=12α39∫χe′dξ∫ϑidθ

Mvv=∫ϕqϕkdξ∫afaldθ

Mvw=12α48∫ϕqβodξ∫αf′ψldθ

Kvu=12α3∫ϕqχi′dξ∫αf′ϑldθ+12α4∫ϕq′χidξ∫αfϑl′dθ

Kvv=α9∫ϕqϕkdξ∫αf′αl′dθ+α13∫ϕq′ϕk′dξ∫αfαldθ

Kvw=12α12∫ϕqβodξ∫αf′ψldθ+12α24∫ϕqβ0′′dξ∫αf′ψldθ+12α25∫ϕq′β0′dξ∫αfψl′dθ+12α26∫ϕqβodξ∫αf′ψl′′dθ

NLvw=12α10∫ϕqβoβtdξ∫αg′ψp′ψv′dθ+12α11∫ϕqβ0′βt′dξ∫αg′ψpψvdθ+12α14∫ϕq′β0′βtdξ∫αgψpψv′dθ

Fˉvp=12α40∫ϕqdξ∫αf′dθ

Mww=∫βrβodξ∫ψsψpdθ+12α45∫βr′′βodξ∫ψsψpdθ+12α46∫βrβodξ∫ψs′′ψpdθ+12α49∫βrβodξ∫ψsψpdθ

Cww=Cˉw∫βrβodξ∫ψsψpdθ

Kwu=12α5∫βrχi′dξ∫ψsϑjdθ+12α21∫βr′χidξ∫ψp′ϑj′dθ+12α22∫βrχi′dξ∫ψs′′ϑjdθ+12α23∫βr′′χi′dξ∫ψsϑjdθ

Kwv=12α12∫βrϕkdξ∫ψsαl′dθ+12α24∫βr′′ϕkdξ∫ψsαl′dθ+12α25∫βr′ϕk′dξ∫ψs′αldθ+12α26∫βrϕkdξ∫ψs′′αl′dθ

Kww=α15∫βrβodξ∫ψsψpdθ+12a28∫βrβ0′′dξ∫ψsψpdθ+12a28∫βr′′βodξ∫ψsψpdθ+12a33∫βrβodξ∫ψsψp′′dθ+12a33∫βrβodξ∫ψs′′ψpdθ+a34∫βr′′β0′′dξ∫ψsψpdθ+α35∫βoβrdξ∫ψs′′ψp′′dθ+α36∫βr′βo′dξ∫ψs′ψp′dθ+12α37∫βrβo′′dξ∫ψs′′ψpdθ+12α37∫βr′′βodξ∫ψsψp′′dθ+α41∫βr′βo′dξ∫ψsψpdθ+α42∫βrβodξ∫ψs′ψp′dθ−kˉw∫βrβodξ∫ψsψpdθ+kˉp∫βrβ0′′dξ∫ψsψpdθ+kˉpm02∫βrβodξ∫ψsψp′′dθ−Fˉe2Keww

Keww=∫βrβodξ∫ψsψpdθ

NLwu=α6∫βr′βo′χi′dξ∫ψsψpϑjdθ+α7∫βrβoχi′dξ∫ψs′ψp′ϑjdθ+12α8∫βr′βoχidξ∫ψsψp′ϑj′dθ+12α8∫βrβo′χidξ∫ψs′ψpϑj′dθ

NLwv=α10∫βrβoϕkdξ∫ψs′ψp′αl′dθ+α11∫βr′βo′ϕkdξ∫ψsψpαl′dθ+12α14∫βr′βoϕk′dξ∫ψsψp′αldθ+12α14∫βrβo′ϕk′dξ∫ψs′ψpαldθ

(NL)w2w=12α19∫βrβ′oβ′tdξ∫ψsψpψvdθ+α19∫β′rβ′oβtdξ∫ψsψpψvdθ+12α20∫βrβoβtdξ∫ψsψ′pψ′vdθ+α20∫βrβoβtdξ∫ψ′sψ′pψvdθ+12α27∫β″rβoβtdξ∫ψsψ′pψ′vdθ+α27∫βrβoβ″tdξ∫ψ′sψ′pψvdθ+12α29∫βrβ′β′tdξ∫ψ′sψpψvdθ+α29∫β′rβ′oβtdξ∫ψsψpψ′vdθ+12α30∫β″rβ′oβ′tdξ∫ψsψpψvdθ+α30∫β′rβ′oβ″tdξ∫ψsψpψvdθ+12α31∫βrβoβtdξ∫ψ″sψ′pψ′vdθ+α31∫βrβoβtdξ∫ψ′sψ′pψ″vdθ+12α32∫β′rβ′oβtdξ∫ψ′sψpψ′vdθ+12α32∫β′β′oβtdξ∫ψsψ′pψ′vdθ+12α32∫βrβ′oβ′tdξ∫ψ′sψpψ′vdθ−F¯e3(NL2e)ww

NL2eww=∫βrβoβtdξ∫ψsψpψvdθ

NLw3w=2α16∫βr′βo′βt′βa′dξ∫ψsψpψvψbdθ+2α17∫βrβoβtβadξ∫ψs′ψp′ψv′ψb′dθ+α18∫βr′βo′βtβadξ∫ψsψpψv′ψb′dθ+α18∫βrβoβt′βa′dξ∫ψs′ψp′ψvψbdθ−Fˉe4NL3eww

NL3eww=∫βrβoβtβadξ∫ψsψpψvψbdθ

Fˉwp=12α38∫βrdξ∫ψsdθ+12α43∫βr′′dξ∫ψsdθ+12α44∫βrdξ∫ψs′′dθ

Fˉ1=∫βrdξ∫ψsdθ, Fˉe1=FˉeFˉ1, FˉeDC=FˉeVˉDC2, Fˉwe=Cˉ1FˉeDCFˉ1,

Fˉe2=Cˉ2FˉeDC, Fˉe3=Cˉ3FˉeDC, Fˉe4=Cˉ4FˉeDC,

Appendix D

MˆuwMˆvw=KuuKuvKvuKvv−1MuwMvwKˆuwKˆvw=KuuKuvKvuKvv−1KuwKvw,NLˆuwNLˆvw=KuuKuvKvuKvv−1NLuwNLvwFˆupFˆvp=KuuKuvKvuKvv−1FˉupFˉvp,

FˆwupFˆwvp=KwuKwvFˆupFˆvpMˆuwuMˆvwv=−KwuKwvMˆuwMˆvw,KˆuwuKˆvwv=−KwuKwvKˆuwKˆvwNLˆuwuNLˆvwv=−KwuKwvNLˆuwNLˆvw,

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Received: 2018-09-10

Accepted: 2019-03-11

Published Online: 2019-04-11

Published in Print: 2019-08-27

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2018-0269

Keywords for this article

Applied Mathematics; Nonlinear and Complex Systems; surface elasticity; piezoelectric nanoresonator; non-linear frequency response; stability analysis; electrostatic excitation; complex averaging method; arc-length continuation