Generalization of a model of hysteresis for dynamical systemsa) Jean C. Piquetteb) and Elizabeth A. McLaughlin Naval Undersea Warfare Center, Division Newport, 1176 Howell Street, Newport, Rhode Island 02841
Wei Ren and Binu K. Mukherjee Department of Physics, Royal Military College of Canada, Kingston, Ontario K7K 7B4, Canada
共Received 24 September 2001; revised 15 March 2002; accepted 4 April 2002兲 A previously described model of hysteresis 关J. C. Piquette and S. E. Forsythe, J. Acoust. Soc. Am. 106, 3317–3327 共1999兲; 106, 3328 –3334 共1999兲兴 is generalized to apply to a dynamical system. The original model produces theoretical hysteresis loops that agree well with laboratory measurements acquired under quasi-static conditions. The loops are produced using three-dimensional rotation matrices. An iterative procedure, which allows the model to be applied to a dynamical system, is introduced here. It is shown that, unlike the quasi-static case, self-crossing of the loops is a realistic possibility when inertia and viscous friction are taken into account. 关DOI: 10.1121/1.1481061兴 PACS numbers: 43.38.Ar, 43.20.Px, 43.30.Yj 关SLE兴
I. INTRODUCTION
A model of hysteresis is described in Refs. 1 and 2. That model accommodates hysteresis by first introducing a data transformation that collapses the hysteresis loops into onedimensional curves. Since this transformation effectively eliminates hysteresis, an anhysteretic model3 can be least squares fitted to the result. While this model of hysteresis has been well verified by application to laboratory data, the use of three-dimensional rotation matrices, which the model incorporates, makes it unclear how it might be applied to a dynamical system, such as a transducer. A generalization of the model that makes such application straightforward is described here. 关Readers unfamiliar with other work done in hysteresis modeling may wish to consult Refs. 4 –7, and references within. Those who would like to see a study of the transducer properties of lead magnesium niobate 共PMN兲, the material of primary applications interest here, may wish to consult Ref. 8.兴 In Sec. II an iterative procedure is described for effecting the model generalization. By replacing the stress term of the original model3 with suitable dynamic 共and static兲 terms, an ordinary differential equation, solvable by the usual methods, is produced. The iterative process renders the solutions of the differential equation consistent with the rotations of the original model of hysteresis.1,2 A specific numerical example of the process is given in Sec. III. It is shown here that, unlike the quasi-static case, self-crossing of the loops is a realistic possibility when inertia and viscous friction are taken into account. A summary and the conclusion are given in Sec. IV. II. THEORY MODIFICATIONS FOR DYNAMIC EFFECTS
We first consider what must be done to render the model of Refs. 1 and 2 applicable to a dynamic system. It is ema兲
A preliminary version of this work was reported at the 2001 ONR Workshop on Transducers and Transduction Materials, Baltimore, MD, May 2001. b兲 Electronic mail:
[email protected] J. Acoust. Soc. Am. 111 (6), June 2002
phasized at the outset that, while the drive frequency is introduced here as a specific parameter that must be taken into account, any explicit frequency dependence of the model parameters is ignored. That is, the effects of frequency are accounted for only through their influences upon the inertia and viscous loss mechanisms contained in the system. If other parameters, such as the permittivity, are frequency dependent, it is assumed that the values of these parameters are evaluated at the frequency of interest. That is, no specific model of the variation of these parameters with frequency is advanced here. The dynamic system of interest is shown in Fig. 1. Here, a PMN rod of length L 0 , operating in the length-expander mode, drives a mass M. It is assumed that the mass of the driver is insignificant compared with that of the driven mass. The driven mass is also subject to a viscous frictional force, associated with which is the viscous coefficient b. It is also assumed that the driver is subjected to a constant stress T 0 , such as might be supplied by a prestress bolt in a transducer. It is first necessary to apply the model of Refs. 1 and 2 共in its original form兲 to quasi-static laboratory polarization and strain data acquired from the active material 共herein assumed to be PMN兲. A data transformation that collapses the hysteresis loops from the measured data into onedimensional curves, through the use of three-dimensional rotation matrices, was introduced in Refs. 1 and 2. Quasi-static laboratory data are thus still required for implementation of the present dynamical version of the theory, in order that the required rotation angles can be ascertained. An anhysteretic theory3 is then applied to the resulting one-dimensional curves to determine the required model parameters via least squares fitting. The inverse rotations are applied to the anhysteretic theory to yield a theoretical hysteretic 共but still quasi-static兲 polarization loop. The interested reader is directed to Refs. 1 and 2 for more details of the original theory of hysteresis. The strain equation of the original anhysteretic model,3 viz.,
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TABLE I. Parameter values used in an example calculation for the system of Fig. 1. M ⫽2 kg L 0 ⫽0.0625 m A⫽5⫻10⫺3 m2 b⫽1.0⫻104 N•s/m T 0 ⫽⫺4.13⫻107 N/m2 s D33⫽8.7⫻10⫺12 m2 /N E drive⫽E bias⫽6.7⫻105 V/m
res⫽
FIG. 1. Dynamical system to which the theory of Refs. 1 and 2 is herein generalized to apply. PMN—lead magnesium niobate driver. L 0 —unstressed length of driver. T 0 —constant prestress. A—face area of driver. M—driven mass 共assumed much larger than driver mass兲. b—coefficient of viscous friction. x—position coordinate.
D S 3 ⫽s 33 T 3 ⫹Q 33D 23 ,
共1兲
first is transformed into a differential equation by replacing the external stress T 3 with suitable dynamic and static terms 共cf. Fig. 1兲. This gives D D M x¨ s 33 bx˙ s 33 x D ⫽⫺ T 0 ⫹Q 33D 23 . ⫺ ⫹s 33 L0 A A
共2兲
关Equation 共2兲 is derived by separately isolating the PMN rod and the mass M driven by it, and applying standard methods of dynamics, using Eq. 共1兲 to describe the dynamics of PMN. In this way the stress term T 3 of Eq. 共1兲 is evaluated as the reaction supplied by the external mass, including the assumed constant stress T 0 . In Eqs. 共1兲 and 共2兲 we have replaced the notations  1 and  2 originally used in Ref. 3 with D and Q 33 , respecthe more commonly used notations s 33 tively.兴 In Eqs. 共1兲 and 共2兲, Q 33 is the electrostriction constant, b is the coefficient of viscous friction 共which acts on the driven mass兲, L 0 is the original length of the active maD is the terial, A is the end-face area of the active material, s 33 elastic compliance at constant D, S 3 is the strain, and T 3 is the stress. 共T 0 in Fig. 1 is a constant stress, such as a prestress. It is only a portion of the total external stress T 3 ‘‘seen’’ by the active material, owing to the dynamic effects of the reaction force arising from the contact between the active material and the driven mass M.兲 Over-dots denote time derivatives. Each time Eq. 共2兲 is solved for x(t), the 共current兲 hysteretic version of D 3 is substituted, and thus acts as a source 共or driving兲 term in this differential equation. The initial version of the hysteretic D 3 source function required in Eq. 共2兲 to start the iterative procedure is obtained by carrying out the theoretical solution for D 3 as detailed in Refs. 1 and 2, applied to the quasi-static laboratory data, replacing the external stress T 3 with the constant value T 0 共i.e., inertial effects are initially ignored兲. That is, the initial form of the 2672
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1 2
冑
A ⇒ res⫽10 800 Hz s D33L 0 M
D 3 source function simply results from applying the original theory of Refs. 1 and 2 to the quasi-static laboratory polarization and strain data acquired from the active material. The dynamic solution for the system of Fig. 1 is then computed by the following procedure: 共1兲 Substitute the current hysteretic D 3 共initially quasistatic兲 into Eq. 共2兲, and solve for x(t). 共2兲 Use Eq. 共1兲 to determine the new ‘‘current’’ stress T 3 , evaluating the required strain S 3 using the solution of Eq. 共2兲, together with the ‘‘current’’ 共hysteretic兲 D 3 . 共3兲 A new 共approximately兲 anhysteretic strain versus electric field hysteresis loop is now computed by applying the original rotation matrices 共as originally derived from the quasi-static laboratory data兲 to the new strain versus E-field loop obtained from combining the solution of Eq. 共2兲 with the given E. A corresponding approximately anhysteretic D vs E hysteresis loop is computed by applying the rotation matrices to the current hysteretic D 3 . 共4兲 Solve the E-field equation from the original anhysteretic model,3 viz., E 3⫽
D 3⫺ P 0 ⫺2Q 33T 3 D 3 , T 2 共 0 33兲 ⫺a 共 D 3 ⫺ P 0 兲 2
冑
共3兲
for a new 共approximately兲 anhysteretic D 3 . Here the approximately anhysteretic value of dynamic stress T 3 , as determined by substituting the approximately anhysteretic S 3 and D 3 from step 3 into Eq. 共1兲, is used. 共5兲 Apply the inverse rotations of Refs. 1 and 2 to the current 共approximately anhysteretic兲 D 3 to produce a new hysteretic D 3 . Steps 1–5 are then iterated until the solution does not change. This typically requires three iterations.
III. EXAMPLE
As a specific example, consider the parameter values shown in Table I. The results of applying the iterative process using these numerical values are shown in Figs. 2– 4. 共In these figures, the term ‘‘polarization’’ is used to denote the D-field within the sample. Owing to the very large permittivity of PMN, the polarization and D-field are essentially equivalent. The term ‘‘strain’’ refers to the relative change in length of the PMN rod, compared with its initial length.兲 The original quasi-static data required for initializing the iterative procedure are also shown for reference in each of these figures, so that the changes caused by the system dynamics may be more easily seen. 共It is worthwhile noting that the theory described here closely matches the shown quasi-static data if Piquette et al.: Hysteresis model for dynamical systems
FIG. 2. Computed response of the system of Fig. 1 when driven at a frequency of 1500 Hz 共solid lines兲. Quasi-static data 共dots兲 are shown for reference. 共a兲 Polarization. 共b兲 Strain.
FIG. 4. Computed response of the system of Fig. 1 when driven at a frequency of 10 000 Hz 共solid lines兲. Quasi-static data 共dots兲 are shown for reference. 共a兲 Polarization. 共b兲 Strain.
the drive frequency is sufficiently low that inertia and viscous friction become negligible.兲 At the 1500-Hz frequency considered in Figs. 2共a兲 and 共b兲, the effects of inertia are just starting to become visible, as can be seen from the slight departure of the solid-line curves from the dots. As can be seen in Fig. 3共b兲, where the drive frequency is now 2500 Hz, self-crossing of the hysteresis loop is a realistic possibility in a dynamic system, whereas this should never occur in a quasi-static system.9 The self-crossing seen here arises from the interaction of hysteresis with the effects of both viscous friction and inertia. Finally, in Figs. 4共a兲 and 共b兲 is shown the system response as resonance 共10 800 Hz兲 is approached. The drive considered here is 10 000 Hz, or just slightly below resonance. At this point, the effects of viscous friction overwhelm the effects of hysteresis, and the elliptical shape of the loop seen in Fig. 4共b兲 is almost entirely due to viscous friction. The influence of the inertial effects are also significant, as is reflected by the modified polarization response seen in Fig. 4共a兲.
IV. SUMMARY, DISCUSSION, AND CONCLUSION
FIG. 3. Computed response of the system of Fig. 1 when driven at a frequency of 2500 Hz 共solid lines兲. Quasi-static data 共dots兲 are shown for reference. 共a兲 Polarization. 共b兲 Strain. J. Acoust. Soc. Am., Vol. 111, No. 6, June 2002
A theory of hysteresis1,2 has been generalized to apply to a dynamical system by introducing an iterative procedure. A standard kind of ordinary differential equation is also introduced. 共The driving source term in this equation incorporates Piquette et al.: Hysteresis model for dynamical systems
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the effects of hysteresis into the solution.兲 It was shown that self-crossing of a hysteresis loop is a realistic possibility in a dynamic system. It should be understood that the presence of hysteresis in a transducer drive material is very undesirable, especially from the point of view of ‘‘wasted’’ energy that must be supplied to the material by the power amplifier. Significant also is the undesirable generation of output harmonics owing to hysteresis. However, by suitable preforming the driving voltage waveform,10 it is possible to linearize the output response. ACKNOWLEDGMENTS
This work was supported by the In-House Laboratory Independent Research Program 共ILIR兲 of the Naval Undersea Warfare Center, Division Newport, and by the Office of Naval Research, Code 321.
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J. C. Piquette and S. E. Forsythe, ‘‘One-dimensional phenomenological model of hysteresis I. Development of theory,’’ J. Acoust. Soc. Am. 106, 3317–3327 共1999兲. 2 J. C. Piquette and S. E. Forsythe, ‘‘One-dimensional phenomenological model of hysteresis II. Applications,’’ J. Acoust. Soc. Am. 106, 3328 – 3334 共1999兲. 3 J. C. Piquette and S. E. Forsythe, ‘‘A nonlinear material model for lead magnesium niobate 共PMN兲,’’ J. Acoust. Soc. Am. 101, 289–296 共1997兲. 4 R. C. Smith and C. L. Hom, ‘‘Domain wall theory for ferroelectric hysteresis,’’ J. Intell. Mater. Syst. Struct. 10, 195–213 共1999兲. 5 A. N. Soukhojak and Y-M. Chiang, ‘‘Generalized rheology of active materials,’’ J. Appl. Phys. 88, 6902– 6909 共2000兲. 6 K. Mayergoyz, Mathematical Models of Hysteresis 共Springer-Verlag, New York, 1991兲. 7 D. C. Jiles and D. L. Atherton, ‘‘Theory of ferromagnetic hysteresis,’’ J. Magn. Magn. Mater. 61, 48 – 60 共1986兲. 8 K. M. Rittenmeyer, ‘‘Electrostrictive ceramics for underwater transducer applications,’’ J. Acoust. Soc. Am. 95, 849– 856 共1994兲. 9 B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics 共Academic, New York, 1971兲, p. 40. 10 J. C. Piquette, E. A. Mclaughlin, G. Yang, and B. K. Mukherjee, ‘‘Nonlinear output control in hysteretic, saturating materials,’’ J. Acoust. Soc. Am. 110, 865– 876 共2001兲.
Piquette et al.: Hysteresis model for dynamical systems
