Research Article Volume 7 Issue 1
National Research University of Electronic Technology MIET, Russia
Correspondence: Afonin SM. National Research University of Electronic Technology MIET, Moscow, Russia
Received: October 10, 2023 | Published: November 14, 2023
Citation: Afonin SM. Absolute stability of system with nano piezoengine for biomechanics. MOJ App Bio Biomech. 2023;7(1):211-213. DOI: 10.15406/mojabb.2023.07.00197
The nano piezoengine is used for biomechanics and nano sciences in dosing device, scanning microscopy, nano manipulator, nano pump. For the nano piezoengine with hysteresis in control system its set of equilibrium positions is the segment of line. The frequency method for studying the stability of system is used. By applying Yakubovich criterion for system with the nano piezoengine the absolute stability of system is calculated for biomechanics. The ratio of the piezomodules of the nano piezoengine with transverse, longitudinal, shear piezoelectric effects is proportional the ratio of its tangents of the angle of inclination to the hysteresis.
Keywords: absolute stability system, nano piezoengine, hysteresis, set equilibrium positions, biomechanics
Many equilibrium positions are found in system with nano piezoengine for biomechanics and nano science. For calculation absolute stability of system with the nano piezoengine is using Yakubovich criterion, which is the development of the Lyapunov and Popov criterions.1–20 The nano piezoengine is used for biomechanics and nano sciences in dosing device, scanning microscopy, nano manipulator, nano pump.3–30
The frequency method for studying the stability of system is used to study the absolute stability of control system with the nano piezoengine.
In this work for discussions stability of system with the nano piezoengine are used three main problems: the set of equilibrium positions, the Yakubovich criterion for the absolute stability of system, the maximum of the tangent the angle of inclination to the hysteresis loop of the nano piezoengine.
For written the hysteresis of the nano piezoengine for biomechanics and nano science the Preisach model is used for its hysteresis deformation. The hysteresis Preisach function of the relative deformation the nano piezoengine on Figure 1 is determined.3–28
Si=F[Em|t0,t,Si(0),sign˙Em]Si=F[Em|t0,t,Si(0),sign˙Em]
here SiSi - the hysteresis deformation, t - time, Si(0)Si(0) - the initial condition, EmEm - the strength of electric field, sign ˙Emsign˙Em - the sign for velocity of change strength of electric field.
In control system the set of equilibrium positions is the set of points M of intersection of the line L with the hysteresis characteristic of nano piezo engine on Figure 1 in the form of the selected line segment.3 Respectively, the equation of the line L is evaluated
Em+kSi=0Em+kSi=0
here kk - the transfer coefficient for the linear part of control system.
The expression for the symmetric main hysteresis loop28 of the characteristic of nano piezoengine on Figure 1 is determined in the form
Si=dmiEm−γmiEm max(1−E2mE2m max)nmisign˙EmSi=dmiEm−γmiEmmax(1−E2mE2mmax)nmisign˙Em
here dmidmi - the piezo module, γmi=S0i/Em maxγmi=S0i/Emmax - the coefficient of hysteresis, S0iS0i - the relative deformation at Em=0Em=0 , nminmi - the coefficient for the nano piezoengine from PZT nminmi = 1.
The width of the resting zone at ΔEm maxΔEmmax is obtained
ΔEm max+kS+i(ΔEm max)=0ΔEmmax+kS+i(ΔEmmax)=0
here ΔΔ - the relative value of electric field strength; S+i(ΔEm max)S+i(ΔEmmax) - the value of the relative deformation on the ascending branch for ˙Em>0˙Em>0 , S−i(−ΔEm max)S−i(−ΔEmmax) - the value of the relative deformation on the descending branch for ˙Em<0˙Em<0 on Figure 1.
At the symmetric main hysteresis loop characteristic of the nano piezo engine the equation is evaluated
S+i(ΔEm max)=dmiΔEm max−γmiEm max(1−(ΔEm max)2E2m max)S+i(ΔEmmax)=dmiΔEmmax−γmiEmmax(1−(ΔEmmax)2E2mmax)
After transformation this expression is determined
S+i(ΔEm max)=dmiΔEm max−γmiEm max(1−Δ2)S+i(ΔEmmax)=dmiΔEmmax−γmiEmmax(1−Δ2)
From equation for the width of the resting zone the expression is calculated
ΔEm max+kEm max[dmiΔ−γmi(1−Δ2)]=0ΔEmmax+kEmmax[dmiΔ−γmi(1−Δ2)]=0
Then the equation is determined
Δ+k[dmiΔ−γmi(1−Δ2)]=0Δ+k[dmiΔ−γmi(1−Δ2)]=0
The quadratic equation is calculated
Δ2+(1+kdmi)kγmiΔ−1=0Δ2+(1+kdmi)kγmiΔ−1=0
The relative width of the rest zone of system with the nano piezoengine for biomechanics and nano sciences is obtained from this quadratic equation for the symmetric loop characteristic in the form
2Δ=−(1+kdmi)kγmi+√(1+kdmi)2k2γ2mi+42Δ=−(1+kdmi)kγmi+√(1+kdmi)2k2γ2mi+4
and for the asymmetric loop characteristic its relative width of the rest zone of system is evaluated in the form
Δ++Δ−=−(1+kdmi)2k(1γ+mi+1γ−mi)+12√(1+kdmi)2k2(γ+mi)2+4+12√(1+kdmi)2k2(γ−mi)2+4
From the Yakubovich criterion,1–4 which is the development of the Lyapunov and Popov criterions, the absolute stability of system with the nano piezoengine for biomechanics is obtained. The condition for the absolute stability of system with nano piezoactuator from PZT for biomechanics on Figure 2 is evaluated in the form
ReνmiW(jω)≥−1
here ω - the frequency, j - the imaginary unit, νmi - maximum of the tangent the angle of inclination to the hysteresis loop. The amplitude-phase frequency characteristic on Figure 2 shows the frequency transfer function W(jω) with boundary vertical line B, passing point -1 on real axis.
At the maximum strength of electric field in the nano piezoengine the minimum for the tangent the angle of inclination has the form min[dSi/dEm]=0 and maximum has the form max[dSi/dEm]=νmi. For the nano piezoengine from PZT for biomechanics we have its maximum tangents ν31 = 0.55 nm/V for transverse piezoeffect, ν33 = 1 nm/V for longitudinal piezoeffect, and ν15 = 1.25 nm/V for shear piezoeffect at error 10%.
Therefore, the ratio of the piezomodules of the nano piezoengine from PZT with transverse, longitudinal, shear piezoelectric effects is proportional the ratio of its tangents of the angle of inclination to the hysteresis in the form: d31:d33:d15=ν31:ν33:ν15
For the nano piezoengine with hysteresis in the control system for biomechanics its set of equilibrium positions of the control system is the segment of line. The frequency method for studying the stability of system is used. By using Yakubovich criterion for system with the nano piezoengine the absolute stability of control system is obtained for biomechanics. The ratio of the piezomodules of the nano piezoengine with transverse, longitudinal, shear piezoelectric effects is proportional the ratio of its tangents of the angle of inclination to its hysteresis deformation.
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The authors declare that there are no conflicts of interest.
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