The theory of a trapped degenerate mode resonator

Resonance based biosensors are used in the detection of biological molecules for medical diagnostics. Sensing in a liquid environment is very desirable for this application, but presents a significant challenge for resonators based upon conventional technologies. In this paper, the major originality lies in the development and exposition of a fundamental theory enabling design of an original elastic resonant sensor whose modes are engineered to simultaneously possess three separate but complementary dynamical properties: namely, (1) in-plane displacement of the free interface whereby the SH waves are uncoupled from the SV and P waves; (2) intrinsic modal trapping; and finally, (3) cyclic symmetry and modal degeneracy. The modal trapping is due to the physical configuration of the resonator resulting in an imaginary wavenumber for one region of the resonator. The wave will be evanescent in this region and propagating elsewhere. The fundamental principles are elucidated, and analytical techniques are presented that facilitate the efficient design of this unique class of device.


I. INTRODUCTION
The purpose of this paper is to establish the theory of a trapped, degenerate mode, plate resonator which will permit only in-surface shear vibrations at its surface.The long term aim of this study is to design a resonant mass sensor which will perform under liquid.Since it has been reported 1 that out of surface displacements contribute significantly to the damping of the resonator-to such an extent that the resonant motion can be extinguished-it is important to understand how an in-surface resonant response can be realised.
A circular form is proposed and this is different from other shear mode resonators. 2 The resonator response is not based upon a plane strain wave travelling within a one dimensional rectilinear cavity.The circular design form has a closer relationship to the magnetic acoustic resonator (MARS) proposed in Refs. 3 and 4. The fundamental difference between the proposed resonator and the magnetic acoustic resonator is the modal characteristics of the resonator.For mass sensing, the magnetic acoustic resonator relies on the absolute frequency shift in the resonant frequency of a single mode of vibration.It is, therefore, susceptible to influences that would cause an unwanted change in the resonant frequency-such as temperature and added liquid mass.
It is known that resonators which have cyclic symmetry support pairs of independent modes which vary spatially as sinðnhÞ and cosðnhÞ, n 6 ¼ 0, and these share a common resonant frequency. 5Such resonators are called degenerate and will be investigated here.To first order it can be shown that structures with cyclic symmetries of order p, which break this degeneracy are determined by the condition 2n=p ¼ 1; 2; 3; :::. 6When mass or stiffness is added to p identical sectors of the resonator, then the degeneracy is broken and the single resonant frequency "splits" to yield two, close, resonant frequencies.This frequency split is used to record the added mass.Any distributions in mass or stiffness where 2n=p 6 ¼ 1; 2; 3; ::: will not affect the frequency split between the nth order cyclic modes.
When immersed in a fluid or temperature field the resulting loading is will be approximately uniform over of the resonator and will not affect the frequency split.Harmonic distributions from such environmental effects are unlikely to satisfy the condition 2n=p ¼ 1; 2; 3; ::: necessary for first order symmetry breaking.This makes the degenerate design very robust to temperature changes and liquid effects. 7There are several important examples of resonant sensors where the resonating element is formed from isotropic material and actuation is performed electromagnetically. 3,4 The highly successful Hemispherical Resonator Gyroscope (HRG) manufactured over several decades by Northrop Grumman and SAGEM uses isotropic fused quartz and electrostatic actuation.Piezoelectric actuation, although common in planar form resonators, are anisotropic and their use would compromise the high degree of symmetry required for exploiting modal degeneracy.Therefore, the focus of analysis will be on design form that can be realised in isotropic material in order to allow for the possible exploitation of modal degeneracy.

B. Energy trapped modes
The possibility of trapping a vibration within a region is well known and the basic phenomenon has been described by Auld for the case of shear vibrations in a long thin rectangular a) Electronic mail: b.j.gallacher@ncl.ac.uk strip on a plate. 8Displacement shapes consistent with trapping are proposed for inside and outside the strip and by matching displacements and stress at the edge of the strip a frequency equation is derived.

II. DESCRIPTION OF THE SYSTEM
This paper considers an infinite flat isotropic plate, of thickness 2h, and material characteristics, ðk; l; qÞ.Here ðk; lÞ are Lame's elastic constants and q is the density.Figure 1 shows a point O located on the mid-plane of the plate.This point is chosen to be the origin of a polar co-ordinate system ðr; h; zÞ; where z is parallel to the unit normal of the plate.Identical circular layers of a different material, specified by ðk l ; l l ; q l Þ; thickness dh and radius a, are deposited on the top and bottom surfaces of the plate and are positioned such that the top and bottom surfaces are mirror images of each other.The motion of the system, at a typical point PðR; h; ZÞ, is defined by the displacement vector Dimensionless variables are now introduced by setting: In this section the equations of motion of the unloaded plate in cylindrical coordinates will be obtained using a Helmholtz type decomposition of the displacement field. 9he Hankel transform is conventionally used in wave problems in cylindrical coordinates 9,10 and provides a convenient method of developing a solution.Standard relations pertaining to the manipulation of Bessel function, e.g., recursive relations, will be required as a consequence of applying the Hankel transform. 15,16 Equation of motion of the plate in terms of vector and scalar potentials If it assumed that the motion of the plate is periodic in time, such that the displacements are u ¼ u o e ixt , then it is well known from Ref. 9 that the equations of motion of an isotropic solid can be written as where In this equation is the dilatation and is the rotation vector.Following the conventional methodology as described in Refs.9 and 10, a solution in cylindrical co-ordinates can be obtained when it is assumed that u o , /; andcan be written in the form where n is the "modal" order.When the vector operations div and curl are performed, respectively, on Eq. ( 1) and the new variables introduced, it may be shown that the vector equations of motion can be decoupled and rewritten in "component" form as Here þ aÞ are interpreted as the non-dimensional shear and dilatation wave numbers and the modal operator r 2 n is B. Hankel transform of the vector and scalar equations of motion The radial dependency of Laplacian operator r 2 n in the vector and scalar equations of motion (6) can be transformed using the Hankel transform. 9To proceed with the analysis the variables are introduced and Eqs. ( 2), (3), (6), and (7) If a further rearrangement is made by setting it follows that Eqs. ( 2), (3), and (6) can be reduced to The procedure also shows that the additional variables R and D can be calculated directly from A as R ¼ ÀnA=2 and D ¼ ÀdA=dz.These quantities are not needed in what follows.Equations (11c) to (11e) can be integrated directly to provide expressions for (A,C,S) which are consistent with fields within the plate representing symmetric or skew symmetric displacements across the thickness. 9,10For example, for symmetric and skew symmetric responses we would set A ¼ A 1 cos c 1 z or A ¼ A 1 sin c 1 z; respectively.We will suppose that this choice has been made and that the appropriate forms have been chosen.

C. Stress boundary conditions
To complete the definition of the problem it is necessary to consider the stresses at the top and bottom surfaces of the plate and, for the symmetric and skew-symmetric cases, these are given by 10 By using the inverse transforms of the quantities defined in Eqs. ( 7)-( 11), these stresses and the surface displacements can be rewritten in the integral form Equations ( 11) and ( 13) define the dynamics of the plate and these will be used to construct a set of integral equations, the solution of which will complete the problem.

IV. EFFECT OF THE LAYER
It shall be assumed that the material layer deposited over the inner part of the plate is thin, such that D ( 1.This assumption allows the in-plane and out of plane motions of the layer to be considered separately by treating the layer as thin plate in plane stress and then as a thin plate in flexure.Since the flexural stiffness of a thin plate is proportional to dh 3 the elastic effects of bending will be neglected.It is further assumed that the displacement of the layer at its point of connection with the plate completely determines the motion of the layer.This displacement is taken to be constant through the thickness dh:

A. In-plane behaviour
The forces acting on the layer due to its connection with the plate will be regarded as body forces and these will have radial and tangential components given by It is shown in Ref. 10 that the in-plane displacements ðu lr ; u lh Þ of a thin disk can be written in terms of two functions u and w such that and When Eq. ( 15) is substituted into the equations of motion of the disk, 11 it follows that the functions u and w must satisfy the equations where 13) and ( 15) can be used to determine u and w.For example, the elimination of w from Eq. ( 15) shows that u can be calculated as the solution to Likewise w is given by When Eq. ( 13) is substituted into Eq.( 17) it follows that which then gives u as A similar procedure gives and The contribution made by the body forces is included in Eq. ( 16) by the terms u b and w b which are determined from Eq. ( 14) by defining When the results given by Eq. ( 13) are placed into Eq.( 22) it can be shown that and The in-plane inertia and elastic effects of the layer can now be accounted for by substituting Eqs. ( 18)-( 21) and Eqs. ( 23)-(24) into Eq.( 16).This substitution yields a pair of integral equations which can be written as and where Since the flexural stiffness of the layer is neglected, the normal stress ð r zz Þ z¼1 is determined by the inertia of the layer and is given by If this result is placed into Eq.( 13) it follows that and It is important to observe that the equations determining the response A, i.e., Eqs. ( 25) and (29), are decoupled from the equations which determine B and W. The responses B and W are coupled and are given by the simultaneous solutions of Eqs. ( 26), ( 27), (31), and (32).The interpretation of this result can be found in Ref. 10, where it is shown that a plate can support three types of wave: (i) the SH-horizontal shear-wave, where the motion is parallel to the surface, (ii) the SV-vertical shear-wave, where the motion is perpendicular to the wave vector, and (iii) the P-dilatation-wave, where the motion is along the wave vector.
It is also shown that the SH wave is not scattered at the surface, where as the SV and P waves are coupled at the surface and are scattered into a new set of SV and P waves.
The responses A and ðB; WÞ are thus interpreted as representing the SH and the (SV þ P) waves, respectively.These equations further show, at least for a thin layer, that the respective responses are not coupled at the edge r ¼ r o .For the design of a resonator to be used as a mass sensor these interpretations have practical significance.They suggest that a degenerate mode resonator can be designed which has surface motions that are always parallel to the sensing surface.The argument presented in Ref. 1 indicates that a resonator devoid of out of plane surface motion avoids the generation of compression waves which would radiate into the liquid and cause severe attenuation.As in the case of the quartz crystal microbalance (QCM) operating in a thickness shear mode, the resonator will be minimally effected by the damping caused by contact with a surrounding fluid.

VI. DESIGN OF A DEGNERATE MODE SHEAR RESONATOR
It will now be assumed that motion of the plate will be determined by A and that B ¼ W ¼ 0. In these circumstances, Eq. ( 9) gives A ¼ U ¼ V and the radial and tangential motions can be calculated from Eq. ( 13) as If it is also assumed, for example, that the motion is skewsymmetric in z then Equations ( 25) and (29) can thus be written as and where and A. Dispersion relations infinite un-layered and the infinite thin-layered plate Before the solution to the dual integral Eqs. ( 34) and ( 35) is determined, it is worthwhile to examine the functions d i ðnÞ and d o ðnÞ.It may be shown that the equations d o ðkÞ ¼ 0 and d i ðkÞ ¼ 0 are the dispersion equations for the infinite un-layered and the infinite thin-layered plates, respectively, and these have been the subject of much investigation. 12The roots of these equations determine the wave number parameter, k; as a function of the frequency parameter k 1 and give the conditions for which the motion is described by a travelling wave (k real) or an evanescent (trapped) wave (k imaginary).Figure 2 shows the first set of roots for the particular case when a layer of gold, of thickness D ¼ 0:02; is deposited on fused quartz.The dispersion curve for the layered arrangement is seen to be shifted along the frequency axis such that corresponding points occur at a lower frequency.The Bechmann number defines the value of non-dimensional frequency k 1 at which the wavenumber k becomes zero and thus establishes the cutoff frequencies.The curves show that if the frequency parameter is selected somewhere between the values corresponding to the Bechmann numbers B kA ¼ p=2 and B kB ¼ 1:33; then the wave in a layered region will be a travelling wave whilst the wave in the un-layered region will be evanescent.For frequencies within this range we may speculate that the arrangement shown in Fig. 1 will support travelling waves in the inner region, r < r o but that these waves will not be transmitted much beyond r > r o : If a resonant state can be produced then the arrangement will function as a trapped resonator.

B. Solution to the dual integral equations
The dual integral equations defined by Eqs. ( 34) and ( 35) relate to the un-layered and layered regions of the plate, respectively.The in-plane displacements are represented by the term A 1 which must be determined.Dual integral equations can solved by converting the pair of equations to a single Fredholm integral equation of the second kind, as described in Refs.13 and 14.The solution of Eqs.(34) and ( 35) is facilitated by making use of the discontinuous integral, 15 and from Ref. 13 it is proposed that where vðgÞ is some function to be found.The integral Eq. (38) shows that the solution form Eq. (39) satisfies Eq. ( 35).This result now allows Eq. (34) to be rewritten as Using Eq. ( 38), for the case r < r o , Eq. ( 40) can be further reduced to Thereafter, differentiation of Eq. ( 41) with respect to r yields Eq. ( 42), which can be recognised as a Fredholm integral equation of the second kind: 14 1 The kernel, Kðr; gÞ is given by the integral Kðr; gÞ ¼ and this can be evaluated once the frequency parameter k 1 is specified.If the factor 1=D is treated as an eigenvalue then the function vðrÞ can be interpreted as the eigenfunction of the kernel.Thus the problem becomes one of determining both 1=D and v(r) for a given value of k 1 .When used as a mass sensor mass loading from the fluid can be accommodated by shifting the specified value of k 1 from the unloaded case.
To construct a solution, the method of Galerkin and Bobov 14 will be used and this requires the function v(r) to be written as a Fourier-Bessel series over the interval ð0; r o Þ: Thus, where b j is chosen to satisfy the condition, and the factors q j have to be found through Eq. ( 42).It will be noted that a consequence of assuming the form Eqs. ( 44) and ( 45) is that the condition vðaÞ ¼ 0 is imposed on v.It is shown, in Appendix A, that this condition satisfies the physics of the arrangement by assuring the continuity of the stresses r zr and r zh at r ¼ r 0 .When Eqs. ( 44) and ( 45) are substituted into Eqs.( 42) and (45) and use is made of the orthogonal properties of the function J nþ1 ðb j rÞ over the interval ð0; aÞ it may be shown that Eq. ( 42) can be reduced to an infinite set of linear simultaneous equation given by where and According to the method of Galerkin and Bobov, Eq. ( 46) is solved by limiting the dimension of C to a finite value N. In this case, the problem reduces to a standard eigenvalue problem where the eigenvalue (1=D) and eigenvector ð xÞ can be found as a function of k 1 .In this problem the eigenvalues are always real because the symmetry of the matrix C.However, some caution must be exercised since the "thin layer" assumption imposes the requirement that only those values of D which satisfy D ( 1 can be selected.Negative values of D are also possible solutions since C is not positive definite.These must be rejected.

A. Determination of frequency as a function of plate geometry and material properties
To examine the in-plane behaviour of the construction show in Fig. 1, the case of a gold layer deposited on fused quartz has been considered with the numerical calculations performed using MathCad.The matrix coefficients C j;k have been evaluated for different values of k 1 in the range B kB to B kA and the eigenvalues and vectors of the matrix found for different matrix dimensions N.
It is possible to evaluate the integral Eq. ( 48) by applying the method contour integration if use is made of the results given in Ref. 17 pertaining to the integral representations of Bessel functions and their products.The integrals relevant to this evaluation are given in Appendix B. It is first necessary to identify the poles of the integrand.There are poles on the real axis at n ¼ 6b j and n ¼ 6b k .andat locations of n corresponding to cos ðb 1 Þ ¼ 0. These latter poles are imaginary and are positioned at It is shown in Appendix B that Eq. ( 48) can be rewritten as To provide an insight on how the principal geometrical parameters of the arrangement, D and r o relate to the value of the "natural" frequency parameter, k 1 , i.e., those values which give a solution to Eq. ( 46), the largest eigenvalue of Eq. ( 46) was calculated for different values of r o and for values of k 1 in the range B kB < k 1 < B kA .To assure good convergence of the result, it was sufficient to set N ¼ 20 The results are shown in Fig. 3 for cyclic order n ¼ 2. For each value of k 1 there is a value of D which corresponds to the Beckmann number of the fully layered plate and this is shown by the lowest curve of Fig. 3.The remaining curves, which relate to values of r o ¼ 5; 10; 20; 30, all lie above this Beckmann curve and show, for a given k 1 that the value of D increases disproportionally as r o is reduced from r o ¼ 30 to r o ¼ 5.For a given r o ; the curve for D decreases with increasing k 1 and is approximately parallel to the Beckmann curve.

B. Radial and tangential displacements
The surface displacements U r and U h can now be determined and since the integrals involved can be evaluated analytically the calculation is illustrated by considering U r .Substitution of Eqs. ( 39) and (44) into Eq.(32) gives The inner integral is a standard integral, 16 and, on applying Eq. ( 45), is given by For the values of k 1 under consideration, it is shown in Ref. 17, (1.421) that the function sin ðc 1 Þ=c 1 cos ðc 1 Þ may be expressed in terms of the poles n p and n as follows: When these are substituted into Eq.(51) U r can be rewritten as The integrals contained in Eq. ( 53) are special cases of a more general integral, 18 the results of which give, and for r > r o ) The corresponding expressions for and for r > r o ) Furthermore, it may be shown, by using the respective Bessel recurrent relationships and Wronskians, that U r and U h are continuous at r ¼ r o .

C. Interpretation of the expressions for the radial and tangential displacements
Nothing special can be gained from the relationships given by Eqs. ( 54) and (56), which are for the motion within the plated region, other than they made up of two parts.The first two functions within the second summation are most likely attributed to the "travelling" wave that is allowed in this region.The third term, which is exponential in character, relates to a developing motion which ultimately is terminated at the junction between the plated and un-plated region.Outside the plated region the character of the motion is clear and is determined by the functions K nÀ1 ðn p rÞ and K nþ1 ðn p rÞ.These functions show, for increasing values of r, that the displacements decay in an exponential fashion and that the motion is that of a trapped response.The calculation thus confirms the prediction made from an examination of the dispersion curves shown in Fig. 2.

D. Line plot of the mode shape pertaining to the highest eigenvalue
To illustrate the surface motion, for particular values of D and ro the radial and tangential motions have been calculated as a function of r.These displacements have been normalised to give U max r ¼ 1. Figures 4 and 5 show these displacements for k 1 ¼ 1:428 and r o ¼ 10 and correspond to the highest eigenvalue of Eq. ( 46), that is D ¼ 0:02 The curves also show how the calculation converges as the number of terms used in each summation, (K, P) is increased from 1 to 20.For values greater than 5 there is very little difference between the calculated values.The case K ¼ P ¼ 1 is surprising since a single term in each summation appears to give a good approximation solution.The points on the curve indicated by a dot represent displacement values determined from Eq. ( 46), once the integration over g had been performed, by carrying out a direct numerical evaluation of the infinite integral over n This numerical integration gives a result which matches the values obtained from the closed form expressions (53)-(57).
Figure 5 also shows that the tangential displacement U h has a nodal circle within ro < 10.
At values of r > 10; the curves confirm that both displacements converge rapidly to zero.
Figures 6 and 7 show the displacements for the same values of k 1 and ro but in this case they correspond to next (lower) eigenvalue, i.e., D ¼ 0:03187.This layer is thicker than that given above and the displacement U r now shows the presence of a single nodal circle within the range ro < 10.Displacement U h is similarly affected and two nodal circles are now shown.The response is typically that of a "higher" mode of vibration.As before the displacements have the features of a trapped response.In this case, the displacements predicted by setting K ¼ P ¼ 1 do not give a good approximation of the motion and it is necessary to include more terms in the summations.

VIII. SUMMARY OF RESULTS
The paper has shown that a trapped, degenerate mode resonator can be designed from a simple elastic plate which has identical circular layers of a different material symmetrically deposited on its upper and lower surfaces.If the layer thickness is assumed to be small compared with the thickness of the plate, the theory developed shows that a circumferentially harmonic mode of general order n exists and that its motion is always perpendicular to the unit normal of plate.This "pure" shear motion is derived from the general equations of motion of the plate and for displacements, which are skew symmetric about the mid plane is shown to be given by the solution to a dual integral equation.The kernels of these equations are determined by the dispersion relationships of the layered and un-layered plate.A solution, based upon the discontinuous integral properties of Bessel functions, is proposed to satisfy the un-layered portion of the plate and this is extended to the layered region to yield a Fredholm equation of the second kind.This equation is  solved by the method of Galerkin and Bobov by first specifying the frequency of the response and then by assuming that the solution can be represented as a finite Fourier-Bessel series over the radius defining the layer.This procedure reduces Freholm's equation to a standard eigenvalue problem, where the eigenvalue is the thickness of the layer.Results, based upon gold on fused quartz, have been calculated for the frequency region determined by the lowest dispersion curves and these show how the frequency of the resonator is determined by the layer thickness and the layer radius.For a given frequency and layer radius the normalised radial and tangential displacements-mode shapes-have been plotted for layer thicknesses corresponding to the two smallest layer thicknesses determined by the calculated eigenvalues.The mode shapes for both thicknesses have been proved to exhibit trapping behaviour and the motion decays beyond the edge of the layer.It is observed that the motion corresponding to the thicker layer appears to be that corresponding to a "higher" mode.This is suggested by noting that the motion contains an additional nodal circle.

APPENDIX A
The stresses r zz ; r zr ; and r zh at the surface z ¼ 1 follows from Eq. ( 13) and for the case U ¼ V ¼ A and B ¼ W ¼ 0 are given by r zz ¼ 0; r rz ¼ n r Since these stresses are zero for r > r o ; the above proves that the form chosen for vðrÞ guarantees the continuity of the surface stresses at r ¼ r o .

APPENDIX B
To evaluate integral Eq. ( 48) it is noted, from Ref. 15, (5.42), that J 2 nþ1 ðnr o Þ can be written as This, together with the integral representation allows Eq. ( 48) to be given by The inner integral can be evaluate using Cauchy's formula by following the contour shown in the coefficients C j;k can be written as Because of Eq. ( 45), the evaluation of the integral, when j ¼ k; needs careful consideration.The situation is best approached by letting J nþ1 ðb j r o Þ !0 only after taking the limit b j !b k in the general case of b j 6 ¼ b k .
V. THE RESPONSE OF THE PLATEEquations (25), (26), and (28) define the behaviour of the plate in the layered region and apply for all values of r r o .In the un-plated region, r > r o , the thickness D ¼ 0 and Eqs.(25)-(27) reduce to ð 1 0 dA dz z¼1 J n nr ð Þdn ¼ 0; (29)

FIG. 3 .
FIG. 3. (Color online) Layer thickness versus frequency parameters for gold on fused quartz for the case n ¼ 2.