Analysis of a ceramic bimorph piezoelectric gyroscope

Int. J. of Applied Electromagnetics and Mechanics 0 (999) 459 47 459 IOS Press Analysis of a ceramic bimorph piezoelectric gyroscope J.S. Yang a,, H.Y. Fang a andq.jiang b a Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588, USA b Department of Mechanical Engineering, University of California, Riverside, CA 95, USA Received 5 June 999 Revised 6 October 999 Abstract. The mechanism of piezoelectric transformers for producing high voltage is incorporated into a ceramic beam bimorph piezoelectric gyroscope for generating high voltage sensing signal. For the analysis of the gyroscope, one-dimensional equations of beam piezoelectric bimorphs are derived from the three-dimensional equations of linear piezoelectricity. The equations are employed in the analysis of the bimorph gyroscope. Free and forced vibration solutions are obtained showing high voltage sensitivity.. Introduction Piezoelectric materials can be used to make gyroscopes (angular rate sensors) [,] to measure the angular velocity of a rotating body. These gyroscopes make use of two vibration modes of a vibrating piezoelectric body in which material particles move in perpendicular directions. When a piezoelectric gyroscope is excited into vibration of one of the two modes by an applied alternating voltage and attached to a rotating body, Coriolis force will excite the other mode through which the rotation rate of the body can be detected. The natural frequencies of the two modes must be very close so that the gyroscope works at resonant conditions with maximum sensitivity. Examples are flexural vibrations of beams or tuning forks [ 9], and thickness shear vibrations of plates []. Piezoelectric materials can also be used to make transformers to produce high voltage [0,]. A beamlike ceramic piezoelectric transformer for high voltage applications [0] is shown in Fig. (Rosen transformer). The driving electrodes at x = ±h are bounded by the thick lines. One of the two receiving electrodes is placed at x = l, the other is shared with a driving electrode. This transformer can be driven into vibrations in its lowest extensional mode by an applied voltage V at its driving portion x < 0 across the smaller, lateral dimension of the beam (thickness h), so that a low driving voltage can generate a reasonably strong lateral electric field E and the accompanying axial mechanical field through piezoelectric coupling. This axial mechanical field in the receiving portion x > 0 produces an axial electric field E and hence voltage there, again because the material is piezoelectric. This voltage accumulates spatially in the axial direction of the receiving portion and is picked up by electrodes across the larger, axial dimension of the beam (l in the length direction). The voltage transforming ratio V /V strongly depends on the ratio l /(h) of the length of the receiving portion to the thickness of the driving portion. * Corresponding author. Fax: + 40 47 89; E-mail: jyang@unl.edu 8-546/99/$8.00 999 IOS Press. All rights reserved

460 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope Fig.. A ceramic piezoelectric transformer working with the first extensional mode. It is important to observe that the origin of the high output voltage of the extensional transformers in [0,] is the axial electric field E in the receiving portion. All of the beam (or tuning fork) gyroscopes in [ 9] use lateral electric fields E or E in the beam thickness and width directions for both driving and receiving. If high output voltage signal is desired for a beam-like gyroscope, axial electric field along the beam length direction should be used in the receiving portion. This is equivalent to incorporating the transformer mechanism into a gyroscope for high voltage sensing signal, which is what the present paper will investigate. It will be shown that it is possible to design bimorph beam or tuning fork gyroscopes with high voltage sensitivity. A ceramic beam piezoelectric gyroscope with transformer mechanism will be studied. For the analysis of the ceramic gyroscope, one-dimensional equations for beam piezoelectric bimorphs are derived from the three-dimensional equations of piezoelectricity. The one-dimensional equations are employed in the analysis of the ceramic gyroscope. Free and forced vibration solutions are obtained showing high voltage sensing signal.. Structure of a gyroscope with the mechanism of transformers The structure of the ceramic gyroscope we will study is shown in Fig. in which 0 <x <a is the driving portion and a <x <lthe receiving or sensing portion. The gyroscope consists of twolayered piezoelectric beams (bimorphs). We note that the polarizations in the two layers are opposite. When a driving voltage V is applied across the electrodes at x = ±c, one layer tends to extend while the other tends to contract due to switched polarizations. This will allow the beam to be driven into the lowest flexural mode in the x direction with a proper driving frequency. If the beam is rotating about the x axis with a constant angular velocity Ω which is what we want to measure, Coriolis force which is perpendicular to the directions of the driving motion and the angular velocity will cause a flexural motion in the x direction. This Coriolis force generated flexure will produce a voltage V in Fig.. Structure of the ceramic bimorph piezoelectric gyroscope.

J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope 46 the receiving portion which can be picked up by the sensing electrodes. We note that this gyroscope is like two Rosen transformers bound together with the receiving portions rotated by 90 about the x axis. The two layers with switched polarizations change the extensional mode of a Rosen transformer into the flexural mode of the gyroscope. The rotated receiving portion is for picking up the flexure in the x direction due to Coriolis force. We note that flexure in the x direction does not cause any output voltage. Since the gyroscope in Fig. is effectively a rotating transformer in flexural modes, high output sensing voltage similar to Rosen transformers can be expected.. One-dimensional equations for a beam piezoelectric bimorph To obtain one-dimensional equations for the elementary (classical) flexural motion of the beam bimorph shown in Fig., we begin with the following expansions of the three-dimensional mechanical displacement and electric potential u (x, x, x, t) = x, (x, t) x, (x, t), u (x, x, x, t) = (x, t), u (x, x, x, t) = (x, t), φ(x, x, x, t) = φ (0,0) (x, t) + x φ (,0) (x, t) + x φ (0,) (x, t), (.) where and are the flexural displacements in the x and x directions, φ (0,0) will be shown later to be responsible for the axial electric field in the x direction, φ (,0) and φ (0,) are related to the lateral electric fields in the x and x directions. Substituting (.) into the variational formulation of threedimensional piezoelectricity [], with integration by parts and making use of the interface continuity conditions between the two layers of the bimorph, in a manner similar to the derivation of equations for piezoelectric beams [] and composite piezoelectric plates [4], we obtain the following equations of flexure and electrostatics for the bimorph T (,0), + F (,0), + F (0,0) = mü (0,0), T (0,), + F (0,), + F (0,0) = mü (0,0), (.) D (0,0), + D (0,0) = 0, (.) where the equations corresponding to φ (,0) and φ (0,) are not shown because in the problems to be studied in this paper φ (,0) and φ (0,) are known applied driving voltages and therefore equations for determining them are not needed. This will be clear in Section 4.. In (.) and (.), m = 4bcρ, T (,0) and T (0,) are the bending moments in the x and x directions, D (0,0) is the one-dimensional electric displacement, and m is mass per unit length. F (0,0), F (0,0) and D (0,0) are related to surface force and charge []. Equation (.) also implies the following beam strains and electric fields relevant to our gyroscope problem [] S (,0) =,, S(0,) =,, (.4) E (0,0) = φ (0,0),, E (0,0) = φ (,0), E (0,0) = φ (0,). (.5)

46 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope We note that S (,0) and S (0,) are bending curvatures in the x and x directions. For long and thin beams, since T = T = 0atx = ±b and T = T = 0atx = ±c, these stress components are also very small inside the beam and are approximately zero everywhere. When the beam is not in pure bending, there are shear stresses T and T related to bending which are responsible for the shear resultants T (0,0) and T (0,0). For polycrystalline ceramics, shear stresses are not coupled to extensional strains. For materials with general anisotropy, coupling between shear and extension exists but is usually an order of magnitude weaker than the coupling between extensions in different directions. Therefore, in calculating the bending strains, the only stress component needs to be considered is T, which is the usual assumption in the elementary theory of bending. With the compact notation [], we obtain the bending moment in the x direction as T (,0) = c (,0) S(,0) + c (,) S(0,) e (,0) k E(0,0) k, (.6) where c (,0) x = da, c (,) x x = da, e (,0) d k k = x da. (.7) A s A s A s Clearly, c (m,n) represent various cross-sectional properties related to moments of inertia and products of inertia. Similarly, the bending moment in the x direction can be written as T (0,) = c (,) S(,0) + c (0,) S(0,) e (0,) k E(0,0) k, (.8) where c (0,) x = da, e (0,) d k k = x da. (.9) A s A s The electric constitutive relation is found to be D (0,0) i = e (,0) i S (,0) + e (0,) i S (0,) + ε (0,0) ik E(0,0) k, (.0) where ε (0,0) ik = ε ik da, A ε ik = ε T ik d id k s. (.) The transverse shearing forces in the beam are related to bending moments by [] T (0,0) = T (,0), + F (,0), T (0,0) = T (0,), + F (0,), (.) which are needed to prescribe one-dimensional boundary or continuity conditions. In summary, we have obtained one-dimensional equations of motion (.) and electrostatics (.), strain displacement relation (.4), electric field potential relation (.5), constitutive relations (.6), (.8) and (.0). With successive substitutions, (.) and (.) can be written as three equations for, and φ (0,0).

4. Equations for the gyroscope J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope 46 We now specialize the above equations to the ceramic gyroscope in Fig.. 4.. The driving portion First we consider the driving portion of the gyroscope 0 <x <a. This portion is under given driving voltage V which implies, from (.) 4, (.5) and the fact that on an electrode the potential is constant, the following φ (0,0) = φ (,0) = 0, φ (0,) = V c, (4.) E (0,0) = 0, E (0,0) = 0, E (0,0) = V c. (4.) The electrostatic equation (.) is no longer needed because φ (0,0) is already known. The top layer of the driving portion is poled in the x direction for which the material matrices are given by [5] s s s 0 0 0 s s s 0 0 0 s s s 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 66, 0 0 d 0 0 d 0 0 d 0 d 5 0, d 5 0 0 0 0 0 where s 66 = (s s ). From (.7), (.9), (.) and (4.) we obtain c (,0) = 4b c s, c (,) = 0, c (0,) = 4bc s, ε T 0 0 0 ε T 0, (4.) 0 0 ε T e (0,) = c bd, e (,0) = 0, ε (0,0) = 4bcε, ε = ε T s d /s, (4.4) where we have made use of the fact that the bottom layer has a reversed polarization and the corresponding d has the opposite sign [6]. Then, from (.6) and (.8) we have T (,0) = c (,0) S(,0), T (0,) = c (0,) S(0,) e (0,) E(0,0). (4.5) We note that the second term on the right hand side of (4.5) effectively can be interpreted as the moment of two equal and opposite forces with magnitude bcd E (0,0) /s and at distance c apart, which represents the moment due to the axial forces in each layer of the bimorph under the applied voltage, one in extension and the other in compression. From (.0) we obtain D (0,0) = e (0,) S(0,) + ε (0,0) E(0,0), (4.6) which is needed to calculate the charge and current on the driving electrode at x = c b a Q = b 0 4bc D(0,0) dx dx, I = Q. (4.7)

464 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope The equations of motion (.) now take the following form T (,0), = m( ü (0,0) Ω Ω ), T (0,), = m( ü (0,0) Ω Ω ), (4.8) where we have included Coriolis and centrifugal forces [5] and neglected the surface and body force terms which are zero in our gyroscope problem. The coordinate system is assumed to be rotating with the beam. In this co-rotating frame the beam undergoes small amplitude vibrations under the driving voltage. 4.. The sensing portion Next we consider the sensing portion of the gyroscope a <x <l. For ceramics with the polarization vector in the x direction, we have the following materials matrices which are obtained from (4.) by rotating rows and columns properly s s s 0 0 0 s s s 0 0 0 s s s 0 0 0 0 0 0 s 66 0 0, 0 0 0 0 s 44 0 0 0 0 0 0 s 44 d 0 0 d 0 0 d 0 0 0 0 0, 0 0 d 5 0 d 5 0 ε T 0 0 0 ε T 0. (4.9) 0 0 ε T In the sensing portion, the lateral surfaces are unelectroded hence we have D = 0onx = ±b and D = 0onx = ±c, because the dielectric constants of ceramics are much greater than that of the surrounding air. Since the beam is thin, we have approximately D = D = 0 everywhere in the receiving portion. This implies, through (4.9), that E = E = 0 everywhere in the receiving portion. Equation (4.9) also implies that in the receiving portion the mechanical extensional field will generate an axial electric field. Therefore the major electric field in the receiving portion is the axial electric field E. Hence we have φ (0,0) = φ (0,0) (x, t), φ (,0) = φ (0,) = 0, (4.0) E (0,0) = φ (0,0),, E (0,0) = 0, E (0,0) = 0. (4.) The beam material constants are found to be c (,0) = 4b c =ĉ (,0) s, c(,) = 0, c (0,) = 4bc =ĉ (0,) s, e (0,) = 0, e (,0) = cb d s, ε (0,0) = 4bcε, ε = ε T d s. (4.) The beam constitutive relations take the form T (,0) =ĉ (,0) S(,0) e (,0) E(0,0), T (0,) =ĉ (0,) S(0,), (4.) D (0,0) = e (,0) S(,0) + ε (0,0) E(0,0). (4.4)

J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope 465 The equations of motion are the same as (4.8) and the equation of electrostatics (.) takes the following form D (0,0), = 0, (4.5) where the term for the electric charge on the lateral surfaces vanishes because D = D = 0. Equations (4.5) and (4.4) imply D (0,0) = e (,0) S(,0) + ε (0,0) E(0,0) = Q, (4.6) where Q is an integration constant which may still be a function of time t. Physically, Q represents the electric charge on the sensing electrode at x = l. The current flows out of this electrode is given by I = Q. (4.7) Integrating (4.6) once with respect to x, with (.4) and (.5), we obtain e (,0) u(0,0), + ε (0,0) φ(0,0) = Q x + L, (4.8) where L is another integration constant. Equation (4.8) allows us to obtain φ (0,0) once is known. Solving from (4.6) for E (0,0) and substituting the result into (4.) we obtain where T (,0) = c (,0) S(,0) c (,0) =ĉ (,0) + e(,0) ε (0,0) + e(,0) e(,0) ε (0,0) is a piezoelectrically stiffened bending stiffness. 4.. Boundary and continuity conditions Q, (4.9), (4.0) First we note that in gyroscope applications the driving voltage V is usually considered known and is time harmonic. The sensing electrodes are usually connected by an output circuit with impedance Z when the motion is time harmonic. In the special cases when Z = 0or we have short or open output circuit conditions with V = 0orI = 0. In general, neither V nor I is known and we have the following circuit condition where I = V /Z, (4.) V = φ (0,0) (l, t). (4.)

466 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope At the junction of the two portions we have φ (0,0) (a +, t) = 0, (4.) and the mechanical continuity conditions of the deflections and, slopes, and,, bending moments T (,0) and T (0,), as well as shear forces T (0,0) and T (0,0) which are given by (.). Various combinations of mechanical boundary conditions can be prescribed at the two ends of the gyroscope. 5. Analysis of the gyroscope With successive substitutions, from the equations in the previous section we obtain, for the driving portion 0 <x <a, the following equations in terms of and c (,0) u(0,0), = m( ü (0,0) Ω Ω ), c (0,) u(0,0), = m( ü (0,0) + Ω Ω ). (5.) Similarly, for the receiving portion a <x <l,wehave c (,0) u(0,0), = m( ü (0,0) Ω Ω ), ĉ (0,) u(0,0), = m( ü (0,0) + Ω Ω ), ε (0,0) φ(0,0) = e (,0) u(0,0), + Q x + L. (5.) We consider the case that the two ends of the beam are simply supported in both directions of x and x. Then at the left end x = 0, the boundary conditions are = 0, = 0, T (,0) = c (,0) u(0,0) (0,), = 0, T = c (0,) u(0,0), + e(0,0) V /c = 0. (5.) At the right end x = l, the boundary conditions are = 0, = 0, T (,0) = c (,0) u(0,0), + e(,0) ε (0,0) Q = 0, T (0,) = ĉ (0,) u(0,0), = 0, φ (0,0) (l) = V, Q = V /Z. (5.4) At the junction of the two portions x = a,wehave = 0, = 0,, = 0,, = 0, T (,0) = 0, T (0,) = 0, T (0,0) = 0, T (0,0) = 0, φ (0,0) (a + ) = 0, (5.5)

J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope 467 where = (a + ) u(0,0) (a ) is the jump of u(0,0) across x = a, and the others are similar. Equations (5.) and (5.5) 6 show that effectively the beam is driven by a pair of concentrated moments at x = 0andx = a.sincev is given with harmonic time dependence, for time-harmonic solutions all other quantities have the same time dependence and the time differentiation can be carried out. We then have two forth-order ordinary differential equations in each portion. This will lead to sixteen integration constants. With V and the two integration constants Q and L that are already in (5.), the total number of constants to be determined is nineteen. We have four boundary conditions in (5.) at the left end, six boundary conditions in (5.4) at the right end, and nine conditions in (5.5) at the junction of the two portions. The total number of conditions is also nineteen. Since the driving voltage is time harmonic, we use the complex notation {V, V, Q, L} = {V, V, Q, L}e iωt. (5.6) The general solution to (5.) for the driving portion is found to be = = 8 α (p) U (p) e k(p) x e iωt, p= 8 βu (p) e k(p) x e iωt, (5.7) p= where U (p) are undetermined constants, k (p) are the eight roots of the equation and c (,0) c(0,) k8 m(ω + Ω ) ( c (,0) + c (0,) ) k 4 + m (ω Ω ) = 0, (5.8) α (p) = α(k (p) ), α(k) = c (0,) k4 m(ω + Ω ), β = imωω. (5.9) Similarly, the general solution to (5.) for the sensing portion can be written as = = ε (0,0) φ(0,0) = 8 α (p) U (p) e k(p) (x l) e iωt, p= 8 βu (p) e k(p) (x l) e iωt, p= 8 p= e (,0) k(p) α (p) U (p) e k(p) (x l) e iωt + Q x e iωt + Le iωt, (5.0) where U (p) are undetermined constants, k (p) are the eight roots of (5.8) when c (,0) and c (0,) are replaced by c (,0) and ĉ (0,), α(p) = α(k (p) ) with c (0,) replaced by ĉ (0,). Corresponding to (5.) (5.5), with the substitution of (5.7) and (5.0), we obtain the nineteen liner equations for the nineteen undetermined

468 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope constants U (p), U (p), V, Q,andL. These equations are driven by V. In (5.4) 6, the impedance Z in general is a function of ω. The specific form of this function depends on the structure of the output circuit joining the sensing electrodes. Since we are mainly interested in the voltage sensing signal, in the following we will consider the case when the sensing electrodes are open with Z =. In this case the output voltage signal assumes maximum [7]. For open sensing electrodes, Eq. (5.4) 6 simply implies Q = 0. The dependence of the voltage sensitivity on Z and other parameters were discussed in [7] for a plate piezoelectric gyroscope. 5.. Free vibration analysis For free vibrations the driving voltage V = 0 which means that the driving electrodes are shorted. The equations and boundary conditions become homogeneous and we have an eignevalue problem., φ (0,0) and ω are unknowns and nontrivial solutions are to be obtained. For piezoelectric gyroscopes, it is more revealing to study the free vibration modes when the beam is not rotating. Separate flexural modes in the x and x directions can be obtained. Once these modes have been obtained, it is easy to see that flexures in the x and x directions will be coupled by Coriolis force when the beam is rotating, at the same time the frequencies of these modes shift a little due to rotation. This frequency dependence on Ω was examined in [7], which is not relevant to the main purpose of this paper. Therefore we will examine the modes of the beam in Fig. when it is not rotating. Since the main purpose of this paper is to demonstrate the high voltage sensing capability, it is sufficient for our purpose to study the sensing mode or flexure in the x direction. From (5.),whenΩ = 0, for the driving portion, we have c (,0) k4 mω = 0, (5.) which has four roots of k () = 4 mω /c (,0) = k, k () = ik (), k () = k (),andk (4) = k ().The solution for is = ( U () sin kx + U () sinh kx ) e iωt, (5.) which already satisfies (5.),. Similarly, for the sensing portion, the solution to (5.) can be written as = [ U () sin k(x l) + U () sinh k(x l) ] e iωt, (5.) where k = 4 mω /c (,0),and(5.4), have been satisfied (Q = 0 for open circuit). The continuity conditions of (5.5),,5,7 require sin ka sinh ka sin ka sinh ka k cos ka k cosh ka k cosh ka k cosh ka c (,0) k sin ka c (,0) k sinh ka c (,0) k sin ka c (,0) c (,0) k cos ka c (,0) k cosh ka c (,0) k cos ka c (,0) k cosh ka, U () U () k sinh ka U () = 0. (5.4) U () Vanishing of the determinant of the coefficient matrix of (5.4) yields the frequency equation for the sensing mode. Once the sensing mode is known, the corresponding electric potential distribution can be

J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope 469 Fig.. Normalized deflection and potential distribution of the sensing mode showing voltage rise (a = a = 0 cm, b = c = cm,ω = 0, U = / max, Φ = φ(0,0) /φ (0,0) max ). obtained from (5.) and (5.5) 9 as ε (0,0) φ(0,0) = e (,0) U () k [ cos ka cos k(x l) ] e iωt + e (,0) U () k [ cosh ka cosh k(x l) ] e iωt. (5.5) As an example, we consider PZT-5H with [5] s = 6.5, s = 0.7, s 44 = 4.5, s = 4.78, s = 8.45 0 m /N, d = 74, d 5 = 74, d = 59 0 C/N, (5.6) ε T = 0ε 0, ε T = 400ε 0, ε 0 = 8.854 0 F/m, ρ = 7500 kg/m. For geometric parameters we choose a = a = 0 cm, b = cm,andc = cm. The flexural deformation and electric potential distribution corresponding to the first flexural mode in the x direction are plotted in Fig. according to (5.) (5.5), normalized by their maximum values. When the beam is deflected into its lowest flexural mode in the x direction, it is seen that in the sensing portion the potential keeps rising which is typical for a piezoelectric transformer [8] and has now been incorporated into our gyroscope. This mode is obtained from the equations of a non-rotating beam. When the beam is rotating about the x axis and is electrically excited into flexure in the x direction, flexure in the x direction will be excited by Coriolis force and the above high voltage output will be generated between the sensing electrodes.

470 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope Fig. 4. Sensitivity versus the driving frequency ω near the first pair of flexural modes (a = a = 0 cm, b = c = cm). Fig. 5. Sensitivity versus the driving frequency ω near the first few pairs of flexural modes (a = a = 0 cm, b = c = cm). 5.. Forced vibration analysis For forced vibrations driven by V, Eqs. (5.) (5.5) are solved on a computer. Viscous damping is introduced by allowing the relevant elastic constants to assume complex values. The real elastic constants s pq in the above expressions are replaced by s pq ( iq), where the value of the real number Q for ceramics is usually in the order of 0 to 0 [9]. In the following, we will fix the value of Q to be 0 in our calculations. Another type of damping, electric hysteresis, is common in soft ceramics. Macroscopically, electric hysteresis can be modeled by, e.g., the internal variable theory which is beyond the scope of this paper. The analysis presented here is valid for hard ceramics for which electric hysteresis is weak. The relation of the voltage sensitivity V /V as a function of the driving frequency ω is plotted in Fig. 4 for two values of Ω, whereω 0 is the frequency of the first flexural mode in the x direction when

J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope 47 Fig. 6. Sensitivity versus the rotation rate Ω (a = a = 0 cm, b = c = cm). Fig. 7. Sensitivity versus the aspect ratio a /(c). the beam is not rotating. It is seen that near the first pair of flexural resonant frequencies the sensitivity assumes maximum values. The two frequencies split due to rotation, as also shown in [7]. V /V versus ω around the first few pairs of flexural modes is given in Fig. 5. Near the resonant frequencies of the second pair of flexural modes there also exist reasonably high voltage sensitivity. When the beam is deformed into the second pair of flexural modes, the only nodal point is at about the middle point of the beam and the sensing portion is bent into the same direction with voltage accumulated along the sensing portion. This, however, will change for the third pair of flexural modes with a nodal point inside the sensing portion which is bent into opposite directions with voltage cancellation inside the sensing portion and a small voltage output. For the fourth pair of flexural modes with a nodal point at about the middle point of the sensing portion the voltage is almost completely canceled with essentially no output voltage.

47 J.S. Yang et al. / Analysis of a ceramic bimorph piezoelectric gyroscope The dependence of the maximum sensitivity V /V on the rotation rate Ω is shown in Fig. 6. When Ω is much smaller than the first flexural frequency, the relation between the sensitivity and Ω is essentially linear. Therefore, in the analysis of piezoelectric gyroscopes, the centrifugal force which represents quadratic effects of Ω can be neglected and the contribution to sensitivity is essentially from the Coriolis force which is linear in Ω. We note that Figs 4 and 6 show the same qualitative behavior as the plate gyroscope analyzed in [7]. What is unique to the present gyroscope is the sensitivity-aspect ratio relation shown in Fig. 7. An increasing behavior of the sensitivity versus the aspect ratio is predicted. This shows that when the sensing portion is relatively long compared to the thickness of the driving portion, high output voltage can be produced. This is in agreement with the prediction of the mode in Fig., which is characteristic of a piezoelectric transformer and is now built into this gyroscope. 6. Conclusions The ceramic bimorph piezoelectric gyroscope with transformer mechanism is shown to be able to produce high voltage sensing signal. The idea of using axial electric field to include transformer mechanism for high voltage sensitivity can be applied to gyroscopes made from materials other than ceramics, and tuning fork gyroscopes. The one-dimensional equations for piezoelectric beam bimorphs can also be used to analyze other piezoelectric devices. Acknowledgement This work was supported by the Office of Naval Research under contract number ONR N0004-94-- 005. References [] W.D. Gates, Vibrating angular rate sensor may threaten the gyroscope, Electronics 4 (968), 0 4. [] J. Soderkvist, Micromachined gyroscopes, Sensors and Actuators A 4 (994), 65 7. [] M. Rodamaker and C.R. Newell, Finite element analysis of a quartz angular rate sensor, in: Proc. ANSYS Conference, 989, pp..5.48. [4] J. Soderkvist, Piezoelectric beams and vibrating angular rate sensors, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control 8 (99), 7 80. [5] C.S. Chou, J.W. Yang, Y.C. Huang and H.J. Yang, Analysis on vibrating piezoelectric beam gyroscope, Int. J. of Applied Electromagnetics in Materials (99), 7 4. [6] N. Wakatsuki, M. Ono, S. Yamada, Y. Takahashi and K. Kikuohi, LiTaO crystal fork vibratory gyroscope, in: Proc. IEEE Ultrasonics Symp., 994, pp. 58 584. [7] I.A. Ulitko, Mathematical theory of the fork-type wave gyroscope, in: Proc. IEEE Int. Frequency Control Symp., 995, pp. 786 79. [8] S. Kudo, S. Sugawara and N. Wakatuki, Finite element analysis of single crystal tuning forks for gyroscopes, in: Proc. IEEE Int. Frequency Control Symp., 996, pp. 640 647. [9] J.S. Yang, Some analytical results on piezoelectric gyroscopes, in: Proc. IEEE Int. Frequency Control Symp., 998, pp. 7 74. [0] C.A. Rosen, Ceramic transformers and filters. Ph.D. thesis, Syracuse University, 956. [] S. Kawashima, O. Ohnishi, H. Hakamata, S. Tagami, A. Fukuoka, T. Inoue and S. Hirose, Third order longitudinal mode piezoelectric ceramic transformer and its application to high-voltage power inverter, in: Proc. IEEE Ultrasonics Symp., 994, pp. 55 50. [] H.F. Tiersten, Linear Piezoelectric Plate Vibrations, Plenum, New York, 969.

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