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The schematic of a piezoelectric nano-beam bogus from a adaptable ellipsoidal arm and a anchored bowl as the substrate is represented in Fig. 1. The deformable electrode deflects against the substrate due to the attractions of Casimir and electric forces. Furthermore, the piezobeam is actuated by the absolute accepted VP as the animosity voltage, which is activated forth the arrangement of the nano-beam and leads to a amount in the longitudinal direction33. Such a agreement as an adjustable arrangement is advantageous in several nano-bridges, tunable filters, thermal gates, cooling devices, and capricious switches31,35,36. In this case, the abuttals altitude (BCs) of the anatomy are such that the axle does not abide any absorption forth its aloof axis. There is not any circling or vertical displacement at the appropriate end.

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Schematic of a piezoelectric controllable nano-system with fixed-sliding abuttals altitude beneath Casimir and electrostatic effects.

Based on the thermo-elasticity mechanics, the ache activity of a thermo-sensitive axle Ut is

$${U}_{t}=-,frac{EADelta T}{2}{alpha }_{t}{int }_{0}^{L}{(frac{partial w(x,t)}{partial x})}^{2}dx,$$

(1)

where A, E, ΔΤ, and αt are the ellipsoidal cross-sectional area, aggregate Young’s modulus of the axle or modulus of elasticity, temperature variation, and thermal amplification coefficient, respectively. Furthermore, w is the axle displacement forth z alike axis.

It should be mentioned that all the symbols acclimated in this assignment are authentic in the Nomenclature. The nonlinear curvature aftereffect should be advised back the deformable axle undergoes baloney due to the alien force. Therefore, we access the ache as

$${xi }_{xx}(x,t)=frac{ds-dx}{dx},$$

(2)

where s and ξxx are the absolute breadth angled axle during bend and axle axial ache at its aloof axis, respectively.

It should be mentioned that the activity of the beeline bounce can be accustomed as

$${U}_{k}=frac{1}{2}{K}_{s}{u}^{2}(x,t),$$

(3)

where Uk, Ks, and u are the bounce abeyant energy, axle displacement forth x alike axis, and affiliated bounce stiffness, respectively.

The nonlinear curvature ζ is declared as37

$$zeta (x,t)=frac{dtheta }{ds},$$

(4)

where θ is the bend of angled element.

By demography the nonlinear curvature of the piezobeam into account, both of axial ache εxx and accent σxx can be bidding as19,38

$${varepsilon }_{xx}(x,t)={xi }_{xx}(x,t)-zzeta (x,t);,{rm{all}},{rm{other}},{varepsilon }_{ij}=0,$$

(5)

$${sigma }_{xx}(x,t)=E{varepsilon }_{xx}(x,t)-{e}_{31}{E}_{z};,{rm{all}},{rm{other}},{sigma }_{ij}=0,$$

(6)

where e31 is the angular piezoelectric accessory (−6.4 C.m−2).

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To archetypal the electrostatic acknowledgment of the piezobeam, the activity of the piezobeam Up is declared as38

$${U}_{P}=-,{e}_{31}frac{{V}_{P}}{h}frac{A}{2}{int }_{0}^{L}{(frac{partial w}{partial x})}^{2}dx,$$

(7)

where h, L, and VP are the axle thickness, axle length, and the piezovoltage amid top and basal surfaces, respectively.

Moreover, the moment of the nano-beam by because aggregate and apparent band is bidding as

$$M={int }_{A}{sigma }_{xx}zdz {int }_{S}({tau }_{0} {E}^{s}{varepsilon }_{xx}^{s})zdz=-,zeta (x,t)(frac{Eb{h}^{3}}{12} frac{{E}^{s}b{h}^{2}}{2} frac{{E}^{s}{h}^{3}}{6}),$$

(8)

where b, Es, τ0, and εxxs are the axle width, apparent Young’s modulus about the beam, apparent ache about the axle (beam apparent band strain), and balance accent of the axle apparent area, respectively.

Therefore, the activity of the nano-beam is acquired as

$${U}_{m}=frac{{E}_{eff}{I}_{eff}}{2}{int }_{0}^{L}{zeta }^{2}(x,t)dx,$$

(9)

where Um and EeffIeff are the automated ache activity and able Young’s modulus and added moment of breadth of axle (bulk and apparent layer), respectively.

The activity due to the apparent band astriction Us is acquired as

$${U}_{s}={tau }_{0}b{int }_{0}^{L}zeta (x,t)w(x,t),dx$$

(10)

Due to the nonclassical adapted brace accent access (MCST), the activity of a axle in the submicron-scale is accustomed by17

$${U}_{l,{rm{MCS}}}=frac{bhE{ell }^{2}}{4(1 nu )}{int }_{0}^{L}{(frac{{partial }^{2}w(x,t)}{{partial }^{2}x})}^{2}dx,$$

(11)

where Ul, ℓ, and ν are the ache activity due to actual size, actual nano-scale agency or size-dependent connected according to MCST, and Poisson’s ratio, respectively.

According to accession nonclassical SGT17, the activity of a nano-beam is39

$${U}_{l,{rm{SG}}}={int }_{0}^{L}[begin{array}{c}frac{bhE}{4(1 nu )}(2{l}_{0}^{2} frac{8{l}_{1}^{2}}{15} {l}_{2}^{2}){(frac{{partial }^{2}w(x,t)}{partial {x}^{2}})}^{2}\ , ,frac{b{h}^{3}E}{48(1 nu )}(2{l}_{0}^{2} 0.8{l}_{1}^{2}){(frac{{partial }^{3}w(x,t)}{partial {x}^{3}})}^{2}end{array}]dx,$$

(12)

where ℓi (i = 0, 1, 2) are the added breadth ambit (dilatation, deviatoric stretch, and circling gradients).

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Furthermore, the active activity of the axle is

$${E}_{k}=frac{1}{2}{int }_{0}^{L}rho A{(frac{partial w(x,t)}{partial t})}^{2}dx,$$

(13)

where ρ is the axle density.

In general, the assignment agitated out by accessible attractions can be accustomed by

$${W}_{ext}={int }_{0}^{L}{F}_{ext}w(x,t)dx,$$

(14)

where Wext and Fext are the alien assignment and broadcast alien force (per assemblage length), respectively.

Considering two alongside plates (moveable electrode and substrate) after because FFC, the electric allure can be estimated as12,25

$${F}_{els}=frac{{varepsilon }_{0}b{V}^{2}}{2{(G-w(x,t))}^{2}},$$

(15)

where G, V, and ε0 are the antecedent break or gap, alien activated DC voltage or electric abeyant difference, and electrical permittivity connected of chargeless amplitude exhaustion (8.854 × 10−12 F.m−1), respectively.

Although, Parallel-Plates (PP) archetypal for electric force is the best accepted agreement in abundant applications; NEMS designers are axis the focus to the electric FFC. In adjustment to accompaniment this effect, altered corrections can be appropriate to the authentic electric expression. Demography into annual the simplest as able-bodied as the best abundantly acclimated FFC in the literature, which is accepted as Palmer’s (PM) model, the electrostatic allure can be accustomed by12

$${F}_{els,PM}=frac{{varepsilon }_{0}b{V}^{2}}{2{(G-w(x,t))}^{2}} frac{0.65{varepsilon }_{0}{V}^{2}}{2(G-w(x,t))}.$$

(16)

In addition, Mejis-Fokkema (MF) archetypal is accession acclaimed one, which can acclimatize the electrostatic force as follows13:

$${F}_{els,MF}=frac{{varepsilon }_{0}b{V}^{2}}{2{(G-w(x,t))}^{2}}(1 0.265{(frac{G-w(x,t)}{b})}^{0.75} 0.53frac{h}{b}{(frac{G-w(x,t)}{b})}^{0.5}),$$

(17)

where the furnishings of both axle amplitude and arrangement accept been reflected.

On the added hand, Casimir administration is important in nano-devices, which can be abandoned above this scale. Casimir force is accustomed by40

$${F}_{cas}=frac{{pi }^{2}bc{h}_{p}}{240{(G-w(x,t))}^{4}},$$

(18)

where c and hp are the ablaze acceleration (2.9979 × 108 m.s−1) and bargain Planck’s connected (1.055 × 10−34 J.s), respectively.

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The advised nano-manipulator is beneath both Casimir Fcas and electric Fels armament by because FFC, so the performed assignment by the mentioned armament is accustomed by

$${W}_{ext}={int }_{0}^{L}({int }_{0}^{w}({begin{array}{c}{F}_{els}\ {F}_{els,PP}\ {F}_{els,MF}end{array} {F}_{cas})dw(x,t))dx.$$

(19)

To acquire the nonlinear EOM of the presented nano-manipulator, the acclaimed continued Hamilton’s principle41 is activated as

$$delta {int }_{0}^{t}({U}_{t} {U}_{k} {U}_{p} {U}_{m} {U}_{s} {U}_{l}-{E}_{k}-{W}_{ext})dt=0.$$

(20)

By replacing the potential, kinetic, and ache activity additional the performed assignment by added (corrected electrostatic as able-bodied as Casimir) armament because the accompanying added relations into Eq. (20), application the variational approach, the afterward PDE for the piezoelectric size-dependent nano-beam can be obtained

$$begin{array}{c}{begin{array}{c}frac{bhE}{2(1 nu )}{ell }^{2}frac{{{rm{partial }}}^{4}w}{{rm{partial }}{x}^{4}}(forMCST)\ frac{b{h}^{3}E}{24(1 nu )}(2{l}_{0}^{2} frac{8{l}_{1}^{2}}{10})frac{{d}^{6}w}{d{x}^{6}} frac{bhE}{2(1 nu )}(2{l}_{0}^{2} 8{l}_{1}^{2}/15 {l}_{2}^{2})frac{{{rm{partial }}}^{4}w}{{rm{partial }}{x}^{4}}(forSGT)end{array}\ ,(Efrac{b{h}^{3}}{12} {E}_{s}(frac{b{h}^{2}}{2} frac{{h}^{3}}{6}))[frac{{{rm{partial }}}^{4}w}{{rm{partial }}{x}^{4}} frac{{rm{partial }}}{{rm{partial }}x}(frac{{rm{partial }}w}{{rm{partial }}x}frac{{rm{partial }}}{{rm{partial }}x}(frac{{{rm{partial }}}^{2}w}{{rm{partial }}{x}^{2}}frac{{rm{partial }}w}{{rm{partial }}x}))]-b{tau }_{0}frac{{{rm{partial }}}^{2}w}{{rm{partial }}{x}^{2}}(4 {(frac{{rm{partial }}w}{{rm{partial }}x})}^{2})\ ,(EbhvarDelta T{alpha }_{T} {e}_{31}{V}_{{rm{P}}}b-sign(theta )frac{{K}_{s}}{2}{int }_{0}^{L}{(frac{{rm{partial }}w}{{rm{partial }}x})}^{2}dx)frac{{{rm{partial }}}^{2}w}{{rm{partial }}{x}^{2}} rho bhfrac{{{rm{partial }}}^{2}w}{{rm{partial }}{t}^{2}}={F}_{ext}.end{array}$$

(21)

The abuttals altitude (BC) of the advised arrangement are

$$w(0)=w(L)=0,dw(0)/dx=dw(L)/dx=0.$$

(22)

The mentioned BC are not acceptable for the ache acclivity model. Accordingly, they are accounting as

$$begin{array}{c}w(0)=w(L)=0,dw(0)/dx=dw(L)/dx=0,\ {d}^{2}w(0)/d{x}^{2}={d}^{2}w(L)/d{x}^{2}=0,OR,{d}^{3}w(0)/d{x}^{3}={d}^{3}w(L)/d{x}^{3}=0.end{array}$$

(23)

For the account of the scalability of the problem, the nonlinear equations of motion are advised in the dimensionless anatomy as

$$begin{array}{c}chi =frac{x}{L},,varpi =frac{w}{G},,tau =frac{ht}{2{L}^{2}}sqrt{frac{E}{3rho }},,phi =frac{G}{b},,xi =frac{{G}^{2}}{{L}^{2}},,beta =frac{{L}^{2}}{{h}^{2}},,eta =frac{2{E}^{s}}{E}(frac{3}{h} frac{1}{b}),\ iota =frac{6{ell }^{2}}{(1 nu ){h}^{2}},,{iota }_{0}=frac{{l}_{0}^{2}}{(1 nu ){L}^{2}},,{iota }_{1}=frac{0.4{l}_{1}^{2}}{(1 nu ){L}^{2}},,{iota }_{2}=frac{6{l}_{2}^{2}}{(1 nu ){h}^{2}},,lambda =frac{48{tau }_{0}{L}^{2}}{E{h}^{3}},\ vartheta =frac{12{L}^{2}varDelta T{alpha }_{T}}{{h}^{2}},,varUpsilon =frac{12{L}^{2}{e}_{31}{V}_{{rm{P}}}}{E{h}^{3}},,{c}_{cas}=frac{{pi }^{2}{h}_{p}c{L}^{4}}{20{h}^{3}{G}^{5}E},\ kappa =sign(varDelta T)frac{6L{G}^{2}{K}_{s}}{Eb{h}^{3}}{int }_{0}^{L}{(frac{partial varpi }{partial chi })}^{2}dchi ,,upsilon =frac{V{L}^{2}}{hG}sqrt{frac{6{varepsilon }_{0}}{hGE}},end{array}$$

(24)

χ: dimensionless breadth according to axle breadth (x/L); ϖ: dimensionless axle mean displacement (w/G); τ: dimensionless time; ϕ: arrangement of antecedent gap to axle amplitude (G/b); ξ: aboveboard arrangement of antecedent gap to axle length; β: aboveboard arrangement of axle breadth to thickness; η: apparent animation dimensionless parameter; ι: actual admeasurement dimensionless connected according to MCST; ιi (i = 0, 1, 2): breadth dimensionless ambit according to SGT; λ: balance surface-induces accustomed accent dimensionless parameter; ϑ: thermal amplification dimensionless parameter; ϒ: piezoelectric voltage dimensionless parameter; ccas: Casimir dimensionless coefficient; κ: dimensionless bounce coefficient; υ: dimensionless voltage.

Substituting the dimensionless agreement Eq. (24) into Eq. (21), the equations of motion are acquired from the calm blueprint as

$$begin{array}{c}{begin{array}{c}iota frac{{partial }^{4}varpi }{partial {chi }^{4}}\ ({iota }_{0} {iota }_{1})frac{{partial }^{6}varpi }{partial {chi }^{6}} (12{iota }_{0}beta 8{iota }_{1}beta {iota }_{2})frac{{partial }^{4}varpi }{partial {chi }^{4}}end{array} (1 eta )frac{{partial }^{4}varpi }{partial {chi }^{4}}\ ,(vartheta varUpsilon -kappa -4lambda )frac{{partial }^{2}varpi }{partial {chi }^{2}} xi ((1 eta )[frac{partial }{partial chi }(frac{partial varpi }{partial chi }frac{partial }{partial chi }(frac{{partial }^{2}varpi }{partial {chi }^{2}}frac{partial varpi }{partial chi }))]-lambda {(frac{partial varpi }{partial chi })}^{2}frac{{partial }^{2}varpi }{partial {chi }^{2}})\ ,frac{{partial }^{2}varpi }{partial {tau }^{2}}=frac{{upsilon }^{2}}{{(1-varpi )}^{2}}{begin{array}{c}1\ 1 0.65phi (1-varpi )\ 1 0.265{(phi (1-varpi ))}^{0.75} 0.53sqrt{xi /beta }{phi }^{1.5}{(1-varpi )}^{0.5}end{array} frac{{c}_{cas}}{{(1-varpi )}^{4}}.end{array}$$

(25)

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