JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 3, pp. 609-630, Warsaw 2005 STABILITY OF ACTIVELY CONTROLLED ROTATING SHAFT MADE OF FUNCTIONALLY GRADED MATERIAL Piotr M. Przybyłowicz Institute of Machine Design Fundamentals, Warsaw University of Technology e-mail: piotrp@ipbm.simr.pw.edu.pl In the paper, the problemof active stabilisation of a rotating shaftmade of a three-phaseFunctionallyGradedMaterial (FGM)with piezoelectric fraction is presented. Due to internal friction, at a certain critical rotation speed, the shaft loses its stability and starts to vibrate in a self-excited manner. In the paper, a method protecting the system from such a phenomenon by making use of a FGM controlled by electrodes bonded and embedded in the structure of the shaft is discussed in detail. The critical threshold is found by examination of the eigenvalues corresponding to linear formulas derived via uni-modal Galerkin’s discretisation of partial differential equations of motion. The main goal of the paper is determination of such a distribution of the volume fraction of the active phase within the shaft, which makes the system possibly most resistant to self-excitation on the one hand, and still attractive in terms of strength properties on the other. The results are presented in the form of diagrams depicting the critical rotation speed vs. exponents describing volume distribution of the active phase as well as gain factors applied in the control system. Key words: rotating shaft, stabilisation, functionally gradedmaterial, piezo- ceramic 1. Introduction Rotating shafts, even when perfectly balanced, exhibit self-excited vibra- tion brought about by internal dissipation due to damping in the material, structural friction in articulated joints, supports, etc. The instability occurs due to exceeding the critical rotation speed (over thefirst eigenfrequency corre- sponding to flexural vibration of the given shaft as a beam), and ismanifested by sudden growth in the amplitude of transverse vibration for a slight change 610 P.M.Przybyłowicz of the rotation speed. It is to be emphasised that thementioned critical speed is definitely different from that classically understood andbeing related to the resonance of rotors undergoing excitation by unbalanced inertia, see Kurnik (1992). The last several years have been characterized by animated interest of scientific researchers and engineers in the so-called smartmaterials and struc- tures which, in contradistinction to the classical ones, can adapt their proper- ties to varying operating conditions according to the given algorithm. Smart systems combinemechanical properties with non-mechanical ones, most often with electric, magnetic, thermal, or sometimes, optical fields of interaction. The most popular smart structures employ elements controllable by easy-to- transduce electric signals. Predominantly, piezoelectric elements made of lead zirconate titanate (PZT) or polyvinylidene fluoride (PVDF) are applied. Despite good electromechanical properties of piezoelectric ceramics, the surface-bonded actuators are brittle, heavy, non-flexible and non-conformable elements. Their alternatives, PVDF copolymers are, admittedly, flexible and conformable to any shape but they exhibit high electrical losses, are difficult to pole and their properties are sensitive to temperature variation. Instead of looking for an entirely new class of piezoelectric materials without the above- mentioned drawbacks, a concept of piezoelectric composites came into being (Newnham et al., 1980). First patterns of diphasic materials and several dif- ferent piezocomposites were fabricated at the Pennsylvania State University. Properties of ceramic/polymer composites can be tailored by changing con- nectivity of the phases, volume fraction of the active component and spatial distribution of the ceramic phase. Piezocomposites exhibit excellent electro- mechanical properties and limit the detrimental characteristics of monoliths at the same time. In literature, one can come across various techniques used to form a variety of novel piezoceramic/polymer composites. Among them, the Solid Freeform Fabrication (SFF) containing Fused Deposition of Ceramics (FDC) and Sanders Prototype (SP) are most popular. The FDC or the lost mould endows piezocomposites with particularly good electromechanical pro- perties. That way, Volume Fraction Gradients (VGFs), staggered rods, radial tubes, curved transducers andmany other composite structures are fabricated (Safari, 1999). The concept of piezoelectric stabilization of rotating shafts was described byPrzybyłowicz (2002a), who investigated the efficiency of using piezoelectric actuators glued around the perimeter of the shaft. Recent developments in the field of smart structures and the coming of active composites into being have opened new possibilities to the control of rotating shafts. Composite ro- Stability of actively controlled rotating shaft made... 611 tors, due to low specific weight, anisotropic properties and excellent torsional stiffness became competitive materials as compared to their traditional steel counterparts. The application of active piezoelectric fibersmade them compe- titive even more. Efficient use of PFCs in rotating structures was confirmed byKurnik and Przybyłowicz (2003). The latest advances inmaterial science have led to the emergence of a new class of smart materials, called functionally graded materials (FGMs). Their main feature, i.e. spatial variation of themicrostructure endows such structu- res with optimal mechanical, electrical and thermal properties. In particular, FGMs are an answer to the electro- elastic mismatch between a passive host structure andpiezoelectric actuator attached to its surface (Tylikowski, 2001). It is achieved through continuous gradation of volume fraction of constituent phases. FGMs are superior to the conventional smartmaterials because of lack of internal stress concentrations and interfacial debonding, whichmakes them have improved lifetime and reliability with respect to traditional structures, Liew et al. (2003). This paper is concernedwith active flutter suppression of a rotating shaft, i.e. stabilization and reduction of transverse vibration by making use of an active, functionally graded material containing three phases: two structural phases (traditional passive materials - carrying layer of aluminium and insu- lating layer of a polymer) and an active one (piezoceramic). The three consti- tuents are a mixture of continuously varying volume fraction, see Fig.1. The active phase is able to producemechanical stress and strain under an electric field, which stabilizes operation of the rotating shaft if properly controlled. It occurs that the application of only three actuating electrodes around the shaft circumference is enough to generate a constant counter-bendingmoment despite rotary motion of the entire structure, see Przybylowicz (2002b). Such a moment opposes the internal interactions in the shaft that lead directly to self-excitation while exceeding the critical angular velocity. A quite detailed discussion on rotating shafts was given by Librescu et al. (2005) who studied spinning thin-walled beams made of FGMs. They thoro- ughly examined the influence of a temperature field aswell as an axial force on stability and evolution of the first natural frequency of circular cylindrical sha- fts. Their systems remained passive, however. In this work, the author’s effort is focused on ”active” aspects of such systems if manufactured with ”smart” components integrally immersed in the entire FGM structure. In the paper, the effect of radial distribution of components constituting the internal structure of a rotating shaft on the critical rotation speed is in- vestigated. The shaft consists of three components, among which one of them 612 P.M.Przybyłowicz (the inner surface) comprises active piezoceramic (PZT) fraction. The stabi- lity is examined in terms of eigenvalues corresponding to equations of motion linearized around the trivial equilibrium position. The governing equations include terms describing internal friction in the shaft material as well as the control strategy based on the velocity feedback. Results of numerical simu- lations on the formulated model indicate considerable growth of the rotation speed at which the structure loses its stability. Fig. 1. A shaft made of a three-phase functionally gradedmaterial 2. Properties of the three-phase FGM structure Consider a slender shaft rotating with a constant angular velocity ω aro- und the vertical axis. The shaft is entirely made of a three-phase functionally graded material being a composition of an active constituent, which is a pie- zoceramic PZT, and two passive ones metallic (carrying) and polymeric (in- sulating). These components are mixed through the shaft thickness and their fraction in a certain point along the radius varies exponentially according to the following rules ξ1pol = ( r−r1 r2−r1 )n ξ2pol = ( r3−r r3−r2 )n (2.1) where n is the assumed exponent of the applied distribution, ξ1pol denotes volume fraction of the polymeric constituent within the inner part of the shaft cross-section, i.e. between pure piezoceramic and pure polymer: r ∈ (r1,r2), see alsoFig.1.Obviously, ξ2pol is the volume fraction in the outerpart, between Stability of actively controlled rotating shaft made... 613 the polymer and aluminium: r ∈ (r2,r3). Accordingly, the effective Young’s modulus, effective coefficient of internal friction (the dampingdescribedby the Kelvin-Voigt rheological model) as well as effective constant of electromecha- nical coupling in the piezoceramic-polymer phase are given by the following formulas and corresponding figures (Fig.2 and Fig.3) Yef =    YPZT(1− ξ1pol)+Ypolξ1pol for r1 (4.5) where the overbars are relative to the rotated co-ordinate system: d=TdT −1 (4.6) where T is the transformation matrix T=   cos2θ sin2θ sin2θ sin2θ cos2θ −sin2θ −1 2 sin2θ 1 2 sin2θ cos2θ   Moreover σ=Qε⇒D=dQε ε= [ εx,εy, 1 2 γxy ]> (4.7) where Q = TQT−1, which entails D = TdT−1TQT−1ε = TdQT−1ε. Since PVDFs are poled in the 3-rd direction, the electrodes must be attached to the normal surfaces. Therefore, the dielectric displacement (and the resulting charge) is of the greatest interest D3 =D3 =Dz = {TdQT−1}3ε= [∆1,∆2,∆3] · [ εx,εy, 1 2 γxy ]> (4.8) A single PVDF sensor patch is glued to the surface of each structure shown in Fig.7. The structure can be subject to a combination of the following vibration modes: transverse vibration due to radial motion (pure swelling in the radial direction), transverse vibration due to bending, and torsional vibration (twi- sting). Find now, how effective can be the PVDF sensor inmeasuring each of the indicated vibration types. Stability of actively controlled rotating shaft made... 619 Fig. 7. Polarisation of a PVDF film Let the sensor be a small rectangular patch bonded to the host structure as shown in Fig.8. This time (pure bending) the strain vector assumes the form: ε= [εx,0,0], and the dielectric displacement in the 3-rd (z) axis is D3 = [∆1,∆2,∆3]   εx 0 0  =∆1εx (4.9) Fig. 8. Model of the sensors for measuring bending modes The electric charge and the resulting voltage on the ith sensor patchwill be q3i = ∫ A D3 dAi =∆1r 2 xs+ ls 2∫ xs− ls 2 κ(x) dx ϕi+ χ 2∫ ϕi− χ 2 sinϕdϕ (4.10) USi = q3i C = ∆1hsr �sχls xs+ ls 2∫ xs− ls 2 κ(x) dx [ cos ( ϕi− χ 2 ) − cos ( ϕi+ χ 2 )] 620 P.M.Przybyłowicz where κdenotes the curvature: κ= ∂2w/∂x2, the substitutionofwhichentails USi = ∆1hsr �sχls [∂w ( xs+ ls 2 ) ∂x − ∂w ( xs− ls2 ) ∂x ] 2sinϕi sin χ 2 (4.11) As it canbe seen, theproducedvoltage isdirectlyproportional to thedifference between the slopes at the beginning and ending points of the sensor patch. For very small sensor elements (both in the radial and longitudinal directions), equation (4.11) can be rewritten as follows USi = lim ls→0 χ→0 US(ls,χ)= ∆1hsr �s ∂2w(xs) ∂x2 sinϕi (4.12) Again, the term ∆1 strongly depends on the geometric configuration of the attached sensor. This dependence is shown in Fig.9. Fig. 9. Efficiency of voltage generation for bending vibrationmodes vs. orientation of the PVDF sensing patch As it can be seen, themost favourable geometric orientation of the PVDF sensing patches is for θ=0◦. Then USi = ∆1(0 ◦)hsr �s ∂2w(xs) ∂x2 sinϕi (4.13) or briefly USi = γS ∂2w(xs) ∂x2 sinϕi (4.14) 5. Control of the rotating FGM structure Consider now the problem of control of transverse vibration of a rotating shaft made of a Functionally Graded Material comprising an active piezoce- Stability of actively controlled rotating shaft made... 621 ramic (PZT) fraction. Let us analyse a single electrode covering a part of the shaft circumference as shown in Fig.10. Fig. 10. Electrode patch controlling a sector of piezoceramic structure According to the converse law of piezoelectricity (Nye, 1985), the stress generated within a piezoelectric gradient structure is σ=Yef(ε−def31E3) (5.1) where Yef denotes the effective Young’s modulus, ε – strain, d ef 31 – effective electromechanical coupling constant, E3 – applied electric field. Integrating the terms of stress purely related to d ef 31 one can write down (see also Fig.5) MAi =− ∫ A PS ·Yefdef31E3i dA (5.2) It is a bending moment produced by the ith electrode, contributing to the entiremoment controlling the shaft.Putting it downmore explicitly, onefinds: MAi =−E3i r2∫ r1 Yef1(r)d ef 31(r)r 2 dr ϕi+ α 2∫ ϕi− α 2 sinϕdϕ=−E3iΞ sinϕi (5.3) where the introduced constant is Ξ =2sin α 2 r2∫ r1 Yef1(r)d ef 31(r)r 2 dr (5.4) and the electric field results from the applied voltage and the distance between the electrodes, see Fig.1 E3i = UAi r2−r1 (5.5) 622 P.M.Przybyłowicz It is here assumed that the electric field is constant through the shaft thick- ness, which is only a simplification of real electrical conditions. Admittedly, the electric field intensity does not behave as a constant quantity since the dielectric permittivity varies with radius due to the gradual change of contac- ting component (having, naturally, different values of �i), see Tylikowski and Przybyłowicz (2004). Obviously, the voltage itself follows the incorporated control law, which is here based on a differential regulator. Hence UAi = kd dUSi dt (5.6) where kd is the gain factor. Substituting the equation describing the voltage generated in the ith single sensor supplying signals to the ith electrode via the control unit, see Eqs (4.13) and (4.14), one obtains E3i = kd r2−r1 d dt ( γS ∂w(xS) ∂x2 sinϕi ) = (5.7) = kd r2−r1 γS (∂ẇ(xS) ∂x2 sinϕi+ω ∂w(xS) ∂x2 cosϕi ) And the ith bendingmoment MAi = kd r2−r1 γSΞ (∂ẇ(xS) ∂x2 sin2ϕi+ω ∂w(xS) ∂x2 sinϕi cosϕi ) (5.8) Decomposing the thus determined moment into two directions y and z, one reads   MzAi M y Ai  = γd(n)      ∂3y(xS) ∂x2∂t ∂3z(xS) ∂x2∂t   sin2ϕi+ ω 2   ∂2y(xS) ∂x2 ∂2z(xS) ∂x2   sin2ϕi    (5.9) where γd stands for the general coefficient shortly expressing the multiplier kdγSΞ/(r2−r1). It is here underlined that this factor depends on the applied function of volumedistribution of the active piezoceramic fraction in theFGM structure, described by the exponent n: γd = γd(n). The resultant controlling moment will be M (z,y) A = N∑ i=1 M (z,y) Ai (5.10) Stability of actively controlled rotating shaft made... 623 where N is the number of electrodes placed around the shaft perimeter. No- te, that the angular distance between the jth and (j + 1)th electrode is ϕj+1 −ϕj = 2π/n. Denoting the position of the first pair of electrodes by ϕ1 =ϕ, one finds locations of the subsequent electrodes ϕk =ϕ+ 2πk N (5.11) Now analyse the sums involving expressions of the angular position ϕj in (5.10). According to (5.9), there are two such sums N∑ k=1 sin2ϕk = N∑ k=1 sin2 ( ϕk+k 2π N ) =    sin2ϕ for N =1 2sin2ϕ for N =2 N 2 for N ­ 3 (5.12) N∑ k=1 sinϕk cosϕk = 1 2 N∑ k=1 sin2 ( ϕj +k 2π N ) =    1 2 sin2ϕ for N =1 sin2ϕ for N =2 0 for N ­ 3 It is clearly seen that application of 3 ormore electrode patches ensures gene- ration of a constant bendingmoment (non-oscillating), which is a very desired and advantageous effect making the control of the rotating structure quite convenient. Finally, the vector of actuating moment assumes the form MA = γd(n) N 2    ∂3y(xS) ∂x2∂t ∂3z(xS) ∂x2∂t    {H(x)−H(x− l)} (5.13) The presence of Heaviside’s step functions H(·) results from the fact that the electrodes may not necessarily cover the entire length of the shaft, but only a part of it, e.g. between x = x1 and x = x2. Here it is assumed that x1 = 0 and x2 = l. The second derivative of MA present in the equations of motion becomes ∂2MA ∂x2 = cd(n)   ∂3y(xS) ∂x2∂t ∂3z(xS) ∂x2∂t   {∂δ(x) ∂x − ∂δ(x− l) ∂x } (5.14) where cd = γdN/2 and δ(·) is the Dirac Delta function. 624 P.M.Przybyłowicz 6. Stability investigations Having derived the resultant bendingmoment generated within the active piezoelectric fraction, see (5.14), one substitutes it into equations of motion (3.10), which leads to (in a dimensionless form) ∂2ỹ ∂t̃2 + ∂4ỹ ∂x̃4 +Γ ( ∂5ỹ ∂x̃4∂t̃ +Ω ∂4z̃ ∂x̃4 ) − c̃d(n) ∂3ỹ(x̃S) ∂x̃2∂t̃ (∂δ(x̃) ∂x̃ − ∂δ(x̃−1) ∂x̃ ) =0 (6.1) ∂2z̃ ∂t̃2 + ∂4z̃ ∂x̃4 +Γ ( ∂5z̃ ∂x̃4∂t̃ −Ω∂ 4ỹ ∂x̃4 ) − c̃d(n) ∂3z̃(x̃S) ∂x̃2∂t̃ (∂δ(x̃) ∂x̃ − ∂δ(x̃−1) ∂x̃ ) =0 In order to examine dynamic stability of the analysed system, equations of motion (6.1) will be transformed into a set of ordinary differential equations bymaking use of a unimodalGalerkin’s discretisation based on the first eigen- form corresponding to a simply supported beam. Let (6.1) be represented in the form of differential operators =i[ỹ(x̃, t̃), z̃(x̃, t̃)], i=1,2. Let the solution to (6.1) be predicted as ỹ(x̃, t̃) = T1(t̃)F(x̃) and z̃(x̃, t̃) = T2(t̃)F(x̃) where F(x̃) = sin(x̃) and T1(t̃), T2(t̃) are arbitrary time functions. The Galerkin discretisation implies 1∫ 0 =i { ỹ[(F(x̃)T1(t̃)], z̃[(F(x̃)T2(t̃)] } F(x̃) dx̃=0 i=1,2 (6.2) which leads to a set of two second-order ordinary differential equations with respect to T1 and T2 T̈1 1∫ 0 F2 dx̃+[T1+Γ(Ṫ1+ΩT2)] 1∫ 0 FFIV dx̃+ −c̃dṪ1 1∫ 0 FII(x̃S)[δ I(x̃)−δI(x̃−1)]F dx̃=0 (6.3) T̈2 1∫ 0 F2 dx̃+[T2+Γ(Ṫ2−ΩT1)] 1∫ 0 FFIV dx̃+ −c̃dṪ2 1∫ 0 FII(x̃S)[δ I(x̃)−δI(x̃−1)]F dx̃=0 Stability of actively controlled rotating shaft made... 625 Taking into account fundamental properties of the eigenfunction F(x̃), i.e. knowing that 1∫ 0 F2(x̃) dx̃= 1 2 1∫ 0 F(x̃)FIV (x̃) dx= π4 2 (6.4) andmaking use of the fact that for any function f(x) the following holds 1∫ 0 f(x) dδ(x−xS) dx dx=− df(xS) dx if 0