JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 3, pp. 695-706, Warsaw 2005 STABILIZATION OF PLATE PARAMETRIC VIBRATION VIA DISTRIBUTED CONTROL Andrzej Tylikowski Institute of Machine Design Fundamentals Warsaw University of Technology e-mail: aty@simr.pw.edu.pl A theoretical investigation of vibration control for linear laminated pla- te due to uniform, harmonically or arbitrarily varying in-plane forces is presented.Adistributed controller in an active systemconsisting of elec- troded piezoelectric sensors/actuatorswith suitable polarization profiles is considered. To satisfy theMaxwell electrostatics equation in the actu- ator, a constant electrical potential distribution in the in-planedirections and linear distribution in transverse direction cannot be assumed but is rather obtained by solving the coupled governing equations by assuming a certain theoretically advisable distribution in the thickness direction. Coupled dynamics equations with respect to a plate displacement and an electric field are derived using the Hamilton principle. The rate velo- city feedback is applied to stabilize the plate parametric vibration. The almost sure stability of the trivial solution is analysed using the appro- priate Liapunov functional. Key words: piezoelectric layers, coupled dynamics equations, parametric excitation, stability analysis 1. Introduction Distributed piezoelectric layers can be used as distributed sensors and ac- tuators for structural monitoring and control of elastic structures. In most of the published literature on active systems consisting of continuousmechanical systems with piezoelectric sensors and actuators, the actuator equations were simplified by assuming that the generated electric fields depend only on the applied external voltages (Lee, 1990; Tylikowski, 2001). In addition, the di- stribution of the electric potential ϕ is assumed to be uniform in the in-plane directions of the piezoelectric actuator and linear in its thickness direction. 696 A.Tylikowski The approach was criticized since the Maxwell electrostatics equation is not satisfied for the actuator layers (Gopinathan et al., 2000; Krommer and Ir- schik, 1999, 2000; Tylikowski, 2005; Wang and Quek, 2000). In this paper, the coupled partial differential equations describing the transverse plate mo- tion w and the electric potential are derived. The two piezoelectric layers are placed symmetrically with the poling direction oriented in the transverse di- rection of the plate. When voltages of equal magnitude but opposite phase are applied to the upper and the lower piezoelectric layers of the plate, the induced strains are resulting in flexure action. The sensor layer is made of a thin piezoelectric foil PVDF with a negligible stiffness as compared with the plate and the piezoelectric actuators stiffnesses. The derived plate dynamics equationwhich contains the component dependent on the second derivative of electric field is derived. Similarly, theMaxwell equation contains an additional component dependent on the plate curvatures. The plate is supposed to be bi-axially loaded by time-dependent forces, which can excite parametric vi- bration and destabilize the equilibrium state. The present paper is devoted to formulating a control lawwithout the necessity ofmodeling the plate in terms of its vibrationmodes. The voltage applied to the actuators is calculated from the rate feedback. The derived coupled equations are applicable to the stabi- lity analysis of plate parametric vibration. The control law is derived using the Liapunov approach, with a functional as a sum of the modified mechani- cal plate energy and the energy of electric field. The results indicate that the transverse plate vibrations can be effectively stabilized using the distributed piezoelectric elements. 2. Derivation of dynamic equations Consider the symmetrically laminated rectangular plate of length a, width b and the total thickness 2h with two piezoelectric actuators of thick- ness h1, embedded symmetrically at the distance e from the plate middle plain, with poling direction oriented in the transverse direction of the plate. Whenvoltages of equalmagnitude but opposite phase are applied to the upper and the lower piezoelectric layers of the plate, the induced strains are resulting in flexure action. The sensor layer is made of a thin piezoelectric foil PVDF with a negligible stiffness as compared with the plate and the piezoelectric actuators stiffnesses andwith the appropriate polarization function. The ana- lysiswill use theKirchhoffkinematic assumptions to describe theplate strains. The plate is assumed to be simply supported on all edges. Stabilization of plate parametric vibration... 697 Strains and stresses in a laminated plate according to theKirchhoff theory are εx =−zw,xx εy =−zw,y γxy =−2zw,xy (2.1) σ2x = Yp 1−ν2P (εx+νpεy)+e31Ez σ 2 y = Yp 1−ν2P (εy +νpεx)+e32Ez σ2xy = 2Yp 1+νp γxy σ 1(k) x =Q (k) 11 εx+Q (k) 12 εy σ 1(k) y =Q (k) 12 εx+Q (k) 22 εy σ 1(k) xy =Q (k) 66 γxy (2.2) where the superscripts 1 and2 represent the laminatedplate and thepiezoelec- tric material, respectively, w is the plate transverse displacement, z denotes thedistance fromtheplatemiddleplain, Yp and νp denote theYoungmodulus and the Poisson ratio of the piezoelectric material, respectively. {Q(k)ij } deno- tes the stiffnessmatrix of the kth lamina, e31 and e32 denote the piezoelectric constants and Ez denotes the component of electric field in z-direction. The electric potential ϕ is assumed in the formproposedbyWang andQuek (2000) as a combination of an approximate half-cosine and linear variation ϕ=ϕ(x,y,z,t) =−cos (πzl h1 ) ϕ(x,y,t)+ 2zl h1 ϕa (2.3) where zl is a local co-ordinate measured from the center of the piezoelectric actuator in the global z-direction, ϕa is the value of external electric voltage applied to the actuator electrodes and ϕ(x,y,t) is the time and spatial varia- tion of the electric potential in x-direction. The components of electric field E are given as follows Ex =−cos (πzl h1 ) ϕ,x Ey =−cos (πzl h1 ) ϕ,y Ez =−sin (πzl h1 ) ϕ π h1 + 2 h1 ϕa (2.4) Components of the electric displacement in a piezoelectric layer are given by Dx =Ξ11Ex Dy =Ξ22Ey Dz =Ξ33Ez +e31εx+e32εy (2.5) The systemLagrangian is written as the volume integral of kinetic energy and electric enthalpy L= ∫ Ω (1 2 ρw2,T −U+DxEx+DyEy +DzEz ) dΩ (2.6) 698 A.Tylikowski Substituting the components of electric field and strains we have L= 1 2 ∫ Ω { ρ(z)w2,t−Q (k) 11 z 2w2,xx−2Q (k) 12 z 2w,xxw,yy−Q (k) 22 z 2w2,yy + −4Q(k)66 z 2w2,xy+ S0x+Sx(t) 2h w2,x+ S0y +Sy(t) 2h w2,y+ (2.7) +Ξ11 cos 2 (πzl h1 ) ϕ2,x+Ξ22 cos 2 (πzl h1 ) ϕ2,y+Ξ33 [ −sin (πzl h1 ) ϕ π h1 + 2 h1 ϕa ]2 + +2 [ sin (πzl h1 ) ϕ π h1 − 2 h1 ϕa ] (e31zw,xx+e32zw,yy) } dΩ Integrating with respect to z over the thickness 2h we obtain the Lagran- gian of the system as a function of the transverse displacement w(x,y) and electric potential ϕ(x,y) of the plate L= 1 2 a ∫ 0 b ∫ 0 { 2ρhw2,t−D11w2,xx−2D12w,xxw,yy−D22w2,yy −4D66w2,xy+ +Ξ11h1ϕ 2 ,x+Ξ22h1ϕ 2 ,y +[S0x+Sx(t)]w 2 ,x+[S0y +Sy(t)]w 2 ,y + (2.8) +Ξ33 (π2 h1 ϕ2+ 4ϕ2a h1 ) +4(e31w,xx+e32w,yy) [2h1 π ϕ− ( e+ h1 2 ) ϕa ]} dxdy Remembering that the variation of external potential ϕa is equal to zero we apply the Hamilton principle and finally we obtain t ∫ t0 a ∫ 0 b ∫ 0 {[( −2ρhw,tt−D11w,xxxx−2(D12+2D66)w,xxyy−D22w,yyyy + −[S0x+Sx(t)]w,xx− [S0y +Sy(t)]w,yy + 4h1 π (e31ϕ,xx+e32ϕ,yy)+ −2 ( e+ h1 2 ) (e31ϕa,xx+e32ϕa,yy) ) δw+ (2.9) + ( Ξ33h1 ( π h1 )2 ϕ−Ξ11h1ϕ,xx−Ξ22h1ϕ,yy+ 4h1 π (e31w,xx+e32w,yy) ) δϕ ] dxdy+ + b ∫ 0 (−D11w,xx−D12w,yy+ 4e31h1 π ϕ−2 ( e+ h1 2 ) e31ϕa ) ∣ ∣ ∣ a 0 δw,x dy+ Stabilization of plate parametric vibration... 699 + a ∫ 0 ( −D12w,xx−D22w,yy+ 4e32h1 π ϕ−2 ( e+ h1 2 ) e32ϕa ) ∣ ∣ ∣ b 0 δw,y dx+ − b ∫ 0 Ξ11h1ϕ,x ∣ ∣ ∣ a 0 δϕ dy− a ∫ 0 Ξ22h1ϕ,y ∣ ∣ ∣ b 0 δϕ dx } dt=0 TheEuler equationswhichareobtained fromthecondition that δw and δϕare independent of the transverse plate displacement w and the in-plane electric potential vp are as follows 2ρhw,tt+D11w,xxxx+2(D12+2D66)w,xxyy+D22w,yyyy + +[S0x+S(t)]w,xx+[S0y +Sy(t)]w,yy + 4h1 π (e31ϕ,xx+e32ϕ,yy)+ −2 ( e+ h1 2 ) (e31ϕa,xx+e32ϕa,yy)= 0 (2.10) Ξ33 π2 h1 ϕ−Ξ11h1ϕ,xx−Ξ22h1ϕ,yy + 4h1 π (e31w,xx+e32w,yy)= 0 (x,y)∈ (0,a)× (0,b) FromEq. (2.9) we also obtain the natural boundary conditions for x=0 and x= a w=0 D11w,xx+D12w,yy− 4h1 π e31ϕ+2 ( e+ h1 2 ) e31ϕa =0 ϕ=0 or ϕ,x =0 (2.11) and for y=0 and y= b w=0 D12w,xx+D22w,yy− 4h1 π e32ϕ+2 ( e+ h1 2 ) e32ϕa =0 ϕ=0 or ϕ,y =0 (2.12) Introducing the passive damping viscous term with coefficient α=4ρhβ, and the active damping feedback term (Tylikowski, 2005) ga(x,y) =G(e31χa,xx+e32χa,yy) a ∫ 0 b ∫ 0 χs(x,y)(e s 31w,xx+e s 32w,yy),t dxdy (2.13) where χa is the actuator polarization function and χs is the sensor sensitivity function (Gardonio andElliott, 2004) with piezoelectric constants es31, e s 32, we 700 A.Tylikowski obtain the following basic system of partial differential equations with respect to w, ϕ 2ρhw,tt+4ρhβw,t+D11w,xxxx+2(D12+2D66)w,xxyy +D22w,yyyy + +[S0x+Sx(t)]w,xx+[S0y +Sy(t)]w,yy + 4h1 π (e31ϕ,xx+e32ϕ,yy)+ +ga(x,y)= 0 (2.14) Ξ33 π2 h1 ϕ−Ξ11h1ϕ,xx−Ξ22h1ϕ,yy + 4h1 π (e31w,xx+e32w,yy)= 0 (x,y)∈ (0,a)× (0,b) The sensor voltage is calculated from an elementary formula relating the charge, the generated voltage and the sensor capacity. Distributed piezoelec- tric elements are implemented to suppress the motion caused by parametric disturbances. A proportional controller in an active system consisting of elec- troded piezoelectric sensors/actuators with a suitable polarization profile is considered. The active stabilizing effect with velocity feedback is described by term ga(x,y) with the gain G. 3. Energy extraction Vibration damping of the plate with parametric excitation can be exami- ned by means of the total energy considerations. The method can be applied without earliermodal or finite-dimensional approximations. The energy consi- sts of the kinetic energy, the bending energy, the elastic energy of compression due to constant components of the in-plane forces S0x and S0y, and the energy of electric field E = 1 2 a ∫ 0 b ∫ 0 ( 2ρhw2,t+D11w 2 ,xx+2D12w,xxw,yy+D22w 2 ,yy +4D66w 2 ,xy+ (3.1) −S0xw2,x−S0yw2,y+Ξ11h1ϕ2,x+Ξ22h1ϕ2,y +Ξ33 π2 h1 ϕ2 ) dx The energy is positive definite if the clasic buckling condition is fulfilled by the constant components of in-plane forces. By differentiating Eq. (3.1) with respect to time, the rate of energy extraction is given by Stabilization of plate parametric vibration... 701 dE dt = a ∫ 0 b ∫ 0 ( 2ρhw,ttw,t+D11w,xxtw,xx+D12w,xxtw,yy +D12w,xxw,yyt+ +D22w,yyw,yyt+4D66w,xyw,xyt−S0xw,xw,xt−S0yw,yw,yt+ (3.2) +Ξ11h1ϕ,xϕ,xt+Ξ22h1ϕ,yϕ,yt+Ξ33 π2 h1 ϕϕ,t ) dx Integrating by parts we have I1 = a ∫ 0 b ∫ 0 ( Ξ11h1ϕ,xϕ,xt+Ξ22h1ϕ,yϕ,yt+Ξ33 π2 h1 ϕϕ,t ) dxdy= (3.3) = a ∫ 0 b ∫ 0 ( −Ξ11h1ϕ,xxt−Ξ22h1ϕ,yyt+Ξ33 π2 h1 ϕ,t ) ϕdxdy Substituting the electrostatic equation (2.14)2 we have I1 =− 4h1 π a ∫ 0 b ∫ 0 ( e31w,xxt+e32w,yyt ) ϕdxdy a ∫ 0 b ∫ 0 [ D11w,xxxx+D12w,xxyy− 4h1 π ϕ,xx+2 ( b+ h1 2 ) e31ϕa,xx ) w,t dxdy= = a ∫ 0 b ∫ 0 ( D11w,xx+D12w,yy− 4h1 π ϕ+2 ( b+ h1 2 ) e31ϕa ] w,xxt dxdy (3.4) a ∫ 0 b ∫ 0 [ D12w,xxyy+D22w,yyyy − 4e31h1 π ϕ,yy +2 ( b+ h1 2 ) e31ϕa,yy ] w,t dxdy= = a ∫ 0 b ∫ 0 [ D12w,xx+D22w,yy − 4e32h1 π ϕ+2 ( b+ h1 2 ) e31ϕa ] w,yyt dxdy Eliminating the acceleration in the first part of integrand of Eq. (3.2) by means of dynamic equation (2.14)1 and using Eqs. (3.4) gives 702 A.Tylikowski dE dt =− a ∫ 0 b ∫ 0 4ρhβw2,t dxdy+ a ∫ 0 b ∫ 0 [Sx(t)w,xx+Sy(t)w,yy]w,t dxdy+ (3.5) −2 ( b+ h1 2 ) a ∫ 0 b ∫ 0 ϕa(e31w,xxt+e32w,yyt) dxdy The first negative component in Eq. (3.5) represents the rate at which the energy is extracted from the plate by the passive viscous damping. The se- cond component with an undefined sign is the power flow due to the para- metric excitation. The third term in Eq. (3.5) represents the active damping. Rewriting the active damping feedback term in the form (2.13) and assuming the same polarization functions of the sensor and actuator χa =χs, it can be shown that the third term is also negative dE dt =− a ∫ 0 b ∫ 0 4ρhβw2,t dxdy+ a ∫ 0 b ∫ 0 [Sx(t)w,xx+Sy(t)w,yy]w,t dxdy+ (3.6) −G [ a ∫ 0 b ∫ 0 χa(x,y)(e31w,xxt+e32w,yyt) dxdy ]2 Therefore, for a sufficiently large gain factor G it is possible to stabilize the parametric vibration excited by the time-dependent in-plane forces. 4. Stability analysis The derived Eq. (3.6) does not provide an effective quantitative estima- tion of the minimal active damping coefficient or the gain factor stabilizing the parametric vibration. In order to derive an analytical relation involving characteristics of the parametric excitation, and parameters of the passive damping and the active damping, it is necessary to define precisely the class of the parametric excitation. The derived equations are applicable to stability analysis of the parametric plate vibration due to the action of time-dependent in-plane forces. As the equations (2.14) are linear, it is sufficient to examine the asymptotic stability of the trivial solution.We look for conditions imposed on the plate geometry, the viscous damping, the gain factor and the in-plane force characteristics which imply tending to zero of the distance of the distur- bed solutions from the trivial one. If the in-plane forces are stochastic ergodic Stabilization of plate parametric vibration... 703 processes with physically realizable trajectories, we examine the almost sure stochastic stability (cf. Kozin, 1972) P { lim t→∞ ‖w(t),ϕ(t)‖=0 } =1 (4.1) where ‖·, ·‖ denotes the distance between solutions. The energy-like Liapunov functional containing among others the kinetic energy, the elastic energy and the energy of electric field, is introduced to examine the asymptotic stability V = 1 2 a ∫ 0 b ∫ 0 ( 2ρhw2,t+4ρhβww,t+4ρhβ 2w2+D11w 2 ,xx+2D12w,xxw,yy+ +D22w 2 ,yy+4D66w 2 ,xy−S0xw2,x−S0yw2,y+Ξ11h1ϕ2,x+ (4.2) +Ξ22h1ϕ 2 ,y +Ξ33 π2 h1 ϕ2 ) dx For sufficiently small in-plane forces S0x and S0y functional (3.1) is positive definite, and its square root can be chosen as the distance between the distur- bed solution and the trivial one ‖ · ‖ = √ V . If time-dependent components of the in-plane forces are continuous and physically realizable, the functional can be differentiated in a classical way dV dt = a ∫ 0 b ∫ 0 [ 2ρhw,tt(w,t+βw)+4ρhβ 2ww,t+2ρhβw 2 ,t+D11w,xxtw,xx+ +D12w,xxtw,yy +D12w,xxw,yyt+D22w,yyw,yyt+4D66w,xyw,xyt+ (4.3) −S0xw,xw,xt−S0yw,yw,yt+Ξ11h1ϕ,xϕ,xt+Ξ22h1ϕ,yϕ,yt+Ξ33 π2 h1 ϕϕ,t ] dx Eliminating the acceleration of transverse motion by means of Eq. (2.14)1, using Eqs. (3.3) and (3.4) and integrating by parts using the boundary con- ditions (2.11) and (2.12), we rewrite the time derivative of functional in the form dV dt =−2βV +2U (4.4) where U denotes the auxiliary functional as follows U = 1 2 a ∫ 0 b ∫ 0 { 4ρhβ2ww,t+4ρhβw 3+ (4.5) +[Sx(t)w,yy +Sy(t)w,yy](w,t+βw)−ga(x,y)(w,t+βw) } dx 704 A.Tylikowski Solving the variational inequality U ¬λV (4.6) in the set of functions satisfying boundary conditions (2.11)-(2.13), we obtain function λ as follows λ= max m,n=1,2,...          1 2 √ √ √ √ √ √ √ G2+ 4 [ (2ρhβ)2+ρhβG+1 2 Sx(t) ( mπ a )2 +1 2 Sy(t) ( nπ a )2]2 (2ρhβ)2(1+ω2mn)+κmn−S0x ( mπ a )2 −S0y ( nπ a )2 − G 2          (4.7) where ωmn is the frequency of plate free vibrations without in-plane forces ω2mn = D11 ( mπ a )4 +2(D12+2D66) ( mπ a )2( nπ b )2 +D22 ( nπ b )4 2ρh (4.8) κmn is a coefficient dependent on electric properties of the embedded actuator κmn = 16h21 π2 [ e31 ( mπ a )2 +e32 ( nπ b )2]2 Ξ11h1 ( mπ a )2 +Ξ22h1 ( nπ b )2 +Ξ33 π2 h1 (4.9) and G is the gain factor for the mnmode. The almost sure stability condition has the form 〈λ〉< 2ρhβ (4.10) where 〈·〉 denotes the mathematical expectation. Analyzing Eq. (4.7) it is evident that as the gain factor G increases, we observe a decrease of λmn resulting in a saturation effect. The presence of a positive coefficient κmn in the denominator of Eq. (4.7) increases the stability domain described by Eq. (4.10). Therefore, the results are more conservative when the Maxwell electrostatic equation is neglected. 5. Conclusions Coupled dynamics equations with respect to the plate displacement and the electric field are derived using Hamilton’s principle. The rate velocity fe- edback is applied to stabilize the parametric plate vibration. A new form of Stabilization of plate parametric vibration... 705 Liapunov functional suitable for the stability analysis of a coupled problem is proposed.The almost sure stability of the trivial solution is analyzed using the appropriate Liapunov functional. It is shown that stability domains are incre- ased when the stability problem is solved for coupled equations, i.e. when the electric potential is taken into account. Therefore, the results are more con- servative when the Maxwell electrostatic equation is neglected. A saturation effect is observed during increasing of the feedback gain factor. An unlimited increase of the gain factor does not lead to increasing of the critical variance of axial force. A further increase of the critical variance can be obtained applying multimode control. References 1. GardonioP., ElliottS.J., 2004, Smart panels for active structural acoustic control, Smart Materials and Structures, 13, 1314-1336 2. Gopinathan S.V., Varadan V.V., Varadan V.K., 2000, A review and critique of theories for piezoelectric laminates, SmartMaterials and Structures, 9, 24-48 3. Kozin F., 1972, Stability of the linear stochastic systems, Lecture Notes in Mathematics, 294, Springer Verlag, BerlinHeidelberg-NewYork, 186-229 4. Krommer M., Irschik H.A., 1999, 0n the influence of the electric field on free transverse vibrations of smart beams, Smart Materials and Structures, 8, 401-410 5. Krommer M., Irschik H.A., 2000, Reissner-Mindlin type plate theory in- cluding the direct piezoelectric and pyroelectric effect, Acta Mechanica, 141, 51-69 6. Lee C.K., 1990, Theory of laminated piezoelectric plates for the design of distributed sensors/actuators, Part I: Governing equations and reciprocal rela- tionships, J. Acoustical Society of America, 87, 1144-1158 7. Tylikowski A., 2001,Active damping of parametric vibrations ofmechanical distributed systems, In: Watanabe K., and Ziegler F., (edit.), Dynamics of Advances Materials and Structures, Dordrecht, Kluwer 8. Tylikowski A., 2005, Refined approach to active damping of thermally in- duced parametric vibrations, Proceedings of The Sixth International Congress on Thermal Stresses, Ziegler F., Heuer R., AdamC., (edit.), TUWien, Vol. 1, 347-350 706 A.Tylikowski 9. Wang Q., Quek S.T., 2000, Flexural vibration analysis of sandwich be- am coupled with piezoelectric actuators, Smart Materials and Structures, 9, 103-109 Stabilizacja drgań parametrycznych płyty za pomocą rozłożonego sterowania Streszczenie W dotychczasowym opisie układów aktywnych składających się z ciągłego ukła- du mechanicznego z warstwami piezoelektrycznymi upraszczano równanie aktuatora zakładając, że generowanew nim pole elektryczne zależy jedynie od przyłożonego ze- wnętrznego napięcia generowanego przez układ sterowania. Zakładano również arbi- tralnie liniowy rozkładnapięcia na grubościwarstwy.Nie spełnianow ten sposób rów- nania elektrostatykiwarstwywykonanej zmateriału piezoelektrycznego.Wniniejszej pracy wyprowadzono sprzężony układ równań opisujący dynamikę płyty prostokąt- nej.W płycie zanurzone są dwie symetryczne warstwy piezoelektryczne o polaryzacji prostopadłej do powierzchni płyty. Sensor wykonany jest z cienkiej folii piezoelek- trycznej PVDF o pomijalnie małej w porównaniu z belką i piezoceramicznymi aktu- atorami sztywności. Odkształcenie w płycie i warstwach piezoelektrycznych opisano zgodnie z teoriąKirchhoffa. Za pomocą zasadyHamiltona otrzymano zmodyfikowane równanie dynamiki płyty i równanie elektrostatyki zawierające składniki zależne od krzywizn i torsji oraz pochodnych potencjału elektrycznego.Wyprowadzono również zmodyfikowane warunki brzegowe odpowiadające swobodnemu podparciu. Napięcie działające na piezoceramiczne aktuatory wyznaczono przy założeniu prędkościowego sprzężenia zwrotnego na podstawie zmierzonego napięcia. Otrzymane równania posłużyły do analizy stateczności i stabilizacji drgań pa- rametrycznych płyty poddanej działaniu sił jawnie zależnych od czasu działających w płaszczyźnie środkowej. Wprowadzono nowy funkcjonał Lapunowa, zawierający obok składnikówmechanicznych składnikibędące energiąpola elektrycznego.Pozało- żeniu rozkładu gęstości prawdopodobieństwa sił błonowychmożliwe jest wyznaczenie obszaru stateczności w funkcji parametrów, to jest współczynnika tłumienia, współ- czynnika wzmocnienia sprzężenia zwrotnego, średnich wartości i wariancji sił. Z ana- lizy wzorówwynika, że obszar stateczności jest większy przy uwzględnieniu działania pola elektrycznego opisanego równaniem elektrostatyki podczas badania stateczno- ści. Pominięcie równania elektrostatyki prowadzi do zbyt konserwatywnychwyników stabilizacji.Występuje tu zjawiskonasycenia podczaswzrostuwspółczynnikawzmoc- nienia. Manuscript received April 25, 2005; accepted for print May 23, 2005