Surface vibrational modes in disk-shaped resonators

Ultrasonics xxx (2013) xxx–xxx

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Surface vibrational modes in disk-shaped resonators A.V. Dmitriev ⇑, D.S. Gritsenko, V.P. Mitrofanov Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia

a r t i c l e

i n f o

Article history: Received 24 August 2012 Received in revised form 2 December 2012 Accepted 7 November 2013 Available online xxxx Keywords: SAW resonators Whispering gallery modes Thin disks Electrostatic excitation

a b s t r a c t The natural frequencies and distributions of displacement components for the surface vibrational modes in thin isotropic elastic disks are calculated. In particular, the research is focused on even solutions for low-lying resonant vibrations with large angular wave numbers. Several families of modes are found which are interpreted as modified surface modes of an infinitely long cylinder and Lamb modes of a plate. The results of calculation are compared with the results of the experimental measurements of vibrational modes generated by means of resonant excitation in duraluminum disk with radius of 90 mm and thickness of 16 mm in the frequency range of 130–200 kHz. An excellent agreement between the calculated and measured frequencies is found. Measurements of the structure of the resonant peaks show splitting of some modes. About a half of the measured modes has splitting Dfsplit =fmode at the level of the order of 105. The Q-factors of all modes measured in vacuum lie in the interval (2. . .3)  105. This value is typical for duraluminum mechanical resonators in the ultrasonic frequency range. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction High quality factor (Q) microsphere, microtoroid and microdisk optical resonators with whispering gallery (WG) modes have received increasing interest in recent years in a wide range of applications [1,2]. Similar to the case of the optical resonators, one may expect numerous applications of mechanical resonators with surface vibrational modes. They can be used as signal filters for telecommunication and as biological and chemical sensors [3]. The sensitivity to surface layers makes them suitable for materials science related measurements and nondestructive testing. With the development of cavity optomechanics the growing interest to the interactions between high-Q optical and mechanical surface modes and to the possible usage of such interactions in various applications has appeared [4]. In nonpiezoelectrical materials broadband surface acoustic waves are usually generated thermoelastically by short laser pulses [5,6]. Broadband surface acoustic waves including cylindrical Rayleigh and WG modes propagating along cylindrical surfaces were investigated theoretically and experimentally [7]. In this paper the results of the experimental study of surface vibrational modes with large angular wave numbers generated by means of resonant electrostatic excitation in a thin duraluminum disk are presented. The results of measurements are compared with the results of theoretical calculation of the natural frequencies and distributions of the displacement components ⇑ Corresponding author. Tel.: +7 495 939 37 83; fax: +7 495 932 88 20. E-mail addresses: [email protected] (A.V. Dmitriev), mitr@hbar. phys.msu.ru (V.P. Mitrofanov).

for the surface vibrational modes in thin isotropic elastic disks. This allows us to identify the types of surface modes to which the measured modes belong. The Q-factors of the modes were measured in vacuum in order to exclude gas damping. Splitting of the resonant peaks was found for some modes. 2. Theoretical analysis Eigenfrequencies and mode shapes of vibrating cylinders can be calculated using different methods. Among the numerical methods one of the most wide-spread is the Ritz method [8], which has recently been used to study axisymmetric vibrations of short cylinders [9], flexural vibrations of cylinders under axial loads with angular wave number of n = 1 [10] and vibrations of cylinders with V-notches and sharp radial cracks [11]. Three-dimensional finite difference time domain method was used to study excitation and frequency response of WG modes in disk-shaped resonators [12]. In order to calculate eigenfrequencies and mode shapes of vibrating a vibrating circular cylinder with free boundaries the analytic method developed by various authors [13,14] has been used. Hutchinson [14] obtained the frequency spectra of vibrational modes with angular wave numbers from n = 0 to n = 4 with thickness-to-diameter ratio h varying in the range of 0 < h < 2. This method was also used in a recent work by Tamura [15], where the vibrations of a circular cylinder with high angular wave numbers n were studied. In Tamura’s work the vibrations of a thick cylinder with thickness-to-diameter ratio h  1 were mainly considered, but the case of a thin disk with h = 0.05 was also briefly described. In particular, the eigenfrequencies and mode shapes of vibrational modes with angular wave number n = 20 and the lowest resonant

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frequencies were calculated. Recently several modifications of this method were developed: to determine vibration response of a cylinder subjected to arbitrary distribution of axisymmetric excitation on its surfaces [16] and to study the vibrations of hollow cylinders with free and partially fixed boundaries [17]. In our calculations the method developed by Hutchinson is used. It involves the construction of the solutions as linear combinations of exact solutions of the differential equations of motion in three series which satisfy three of the six boundary conditions. The remaining three boundary conditions are then satisfied by orthogonalization of the corresponding stress tensor components on the boundaries. This leads to a homogeneous system of linear equations, from which the values of eigenfrequencies and the exact values of constant coefficients in solution series are calculated. The solution converges as more terms in the series are considered. The calculation result is thereby an approximate solution which identically satisfies the differential equations and approximately satisfies the boundary conditions. Nevertheless, some authors [14] prefer to classify this method as an ‘‘exact’’ one, so far as it allows to construct exact infinite series solutions. The calculation method is described in detail in Section 2.1 in order to provide the readership with a complete implementation-ready algorithm of calculation of resonant frequencies and displacement vector distributions for the vibrational modes of isotropic disks. 2.1. Method of computation The freely vibrating disk is considered as a circular cylinder with radius a and height (thickness) 2H and with stress–free boundaries. The equation of motion for the displacement vector U in isotropic elastic medium is

€ ¼ ðk þ 2lÞrðr  UÞ  lr  ðr  UÞ; qU

ð1Þ

where k and l are the Lame coefficients and q is the mass density. We introduce the dimensionless (divided by disk radius a) components of displacement vector u ¼ fur ; uh ; uz g and coordinates r, z; the dimensionless stress tensor components (divided by the shear modulus l) rik ; dimensionless wave numbers (multiplied by the disk radius a) a; b and d; dimensionless angular frequency x that is made dimensionless by multiplying angular frequency X by the disk radius a and dividing by the transverse wave velocity pffiffiffiffiffiffiffiffiffi v t ¼ l=q. Dimensionless thickness parameter is thickness-todiameter ratio h ¼ H=a. The six boundary conditions are

rzr ðr; h; hÞ ¼ 0; rzh ðr; h; hÞ ¼ 0; rzz ðr; h; hÞ ¼ 0; rrr ð1; h; zÞ ¼ 0; rrh ð1; h; zÞ ¼ 0; rrz ð1; h; zÞ ¼ 0:

ð2aÞ ð2bÞ ð2cÞ ð2dÞ ð2eÞ ð2fÞ

We choose the three boundary conditions (2a), (2b) and (2f) to be satisfied identically. This is an arbitrary choice, the only restriction of the following method is that we cannot satisfy three boundary conditions on one boundary identically. The solution is constructed as a linear combination of basic solutions [14]:

v ðr; h; zÞ ¼

Nz Nz Nr X X X Ai v Ai ðr; h; zÞ þ Bj v Bj ðr; h; zÞ þ C i v Ci ðr; h; zÞ; i¼1

j¼1

ð3Þ

i¼1

where v stands for displacement vector components ur ; uh ; uz and stress tensor components rlm with the same three series of con-

stants Ai ; Bj and C i . We exclude the time dependence by assumption that all components of the displacement vector and all components of the stress tensor vary in time sinusoidally with the same phase and frequency. Explicitly, the radial displacement is taken in the following form:

urAi ¼ ½2ðb2Ai  a2Ai Þvn ðbAi Þvn ðdAi rÞ þ 4a2Ai vn ðdAi Þvn ðbAi rÞ   cosðaAi zÞ= cosðaAi hÞ cos nh;  r n1 sinðaAi zÞ= sinðaAi hÞ " urBj ¼ 2ða

( 2 Bj

 b2Bj Þ

cosðdBj zÞsincðbBj hÞ

sincðdBj zÞcosðbBj hÞ   h n1 vn ðaBj rÞ  cosnh; r /n ðaBj Þ z

)

( þ4

ð4Þ

d2Bj cosðbBj zÞsincðdBj hÞ

)#

b2Bj sincðbBj zÞcosðdBj hÞ

ð5Þ

urCi ¼ ½4vn ðdCi Þ/n ðbCi rÞ þ 2/n ðbCi Þvn ðdCi rÞ   cosðaCi zÞ= cosðaCi hÞ cos nh;  nr n1 sinðaCi zÞ= sinðaCi hÞ

ð6Þ

for the axial displacement we take (we omit the indices i and j for simplicity; they relate to the terms u; a; b and d in the same way as in the previous three equations)

uzA ¼ ½2ðb2A  a2A Þvn ðbA Þ/n ðdA rÞ  4b2A vn ðdA Þ/n ðbA rÞ    sinðaA zÞ= cosðaA hÞ cos nh;  aA r n cosðaA zÞ= sinðaA hÞ

ð7Þ

     sincðdB zÞsincðbB hÞ sincðdB hÞsincðbB zÞ  4a2B uzB ¼ 2ða2B  b2B Þ cosðdB zÞcosðbB hÞ cosðdB hÞ cosðbB zÞ ( ) d2B hz n /n ðaB rÞ r  cos nh; /n ðaB Þ 1 ð8Þ uzC ¼ 2n/n ðbC Þ/n ðdC rÞaC r n 



 sinðaC zÞ= cosðaC hÞ



cosðaC zÞ= sinðaC hÞ

cos nh;

ð9Þ

and for the angular displacement we take

  uhA ¼  2ðb2A  a2A Þvn ðbA Þ/n ðdA rÞ þ 4a2A vn ðdA Þ/n ðbA rÞ   cosðaA zÞ= cosðaA hÞ sin nh;  nr n1 sinðaA zÞ= sinðaA hÞ " uhB ¼  2ða2B  b2B Þ 



( )#  cosðdB zÞsincðbB hÞ d2 cosðbB zÞsincðdB hÞ þ 4 B2 sincðdB zÞcosðbB hÞ bB sincðbB zÞcosðdB hÞ

  h / ðaB rÞ sin nh; nr n1 n /n ðaB Þ z

uhC ¼ ½4vn ðdC Þvn ðbC rÞ  2n2 /n ðbC Þ/n ðdC rÞr n1   cosðaC zÞ= cosðaC hÞ sin nh:  sinðaC zÞ= sinðaC hÞ

ð10Þ

ð11Þ

ð12Þ

The relations between dimensionless wave numbers a; b and d and frequency x are

a2 þ b2 ¼ x2 ; a2 þ d2 ¼ x2

ð13Þ

2

1  2r x ¼ g 2ð1  rÞ

;

ð14Þ

where r is the Poisson’s ratio and g ¼ v l =v t . The functions /n and vn introduced in Eqs. (4)–(12) are defined as

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A.V. Dmitriev et al. / Ultrasonics xxx (2013) xxx–xxx

/n ðxÞ ¼

"

J n ðxÞ ; xn

ð15Þ

(

J 0 ðxÞ vn ðxÞ ¼ nn1 : x

1

0

10

1

@ ðc þ 2Þ @r@ þ cr cr @h c @z@ rrr ur B C B1 @ CB C @ 1  0 uh A; @ rrh A ¼ @ r @h @ A @r r @ @ 0 rrz uz @z @r

0

rzr

1

0

@ @z

0

B C B @ rzh A ¼ @ 0 @

rzz

c

@r

þ

1 r



@ @z

c @ r @h

10

@ @r 1 @ r @h

ð17Þ



ði  1Þp=h ð2i  1Þp=ð2hÞ

 ;

ð19Þ

ð20Þ

so that the B terms in rrz are identically zero at r = 1. The constructed solution thereby identically satisfies the boundary conditions (2a), (2b) and (2f). The three remaining boundary conditions (2c), (2d) and (2e) are satisfied approximately by orthogonalization. Eq. (2d) can be satisfied only if rrr is orthogonal to a complete set of functions defined in the interval h 6 z 6 h. We choose such complete sets as cosðaAi zÞ and sinðaAi zÞ for even and odd solutions, respectively, with aAi defined as in (19):

  cosðaAi zÞ dz ¼ 0: rrr ð1; h; zÞ sinðaAi zÞ h h

ð21Þ

ð26Þ

wn ðixÞ ¼ xIn1 ðxÞ=In ðxÞ  n  1; and for x ¼ 0 we have wn ð0Þ ¼ n  1. Eq. (2c) can be satisfied if rzz is orthogonal to a complete set of functions defined in the interval 0 6 r 6 1. We choose this function set as J n ðaBj rÞ with aBj defined as in (20). Performing the integration in

Z

1

rzz ðr; h; hÞJn ðaBj rÞrdr ¼ 0

ð27Þ

Z

1

Z

" # dij n2 2 J n ðaBi rÞJ n ðaBj rÞrdr ¼ 1  2 J n ðaBj Þ; 2 aBj

1

J n ðrÞJ n ðaBj rÞrdr ¼

0

 a2Bj  2

ð28Þ

J 0n ðÞJ n ðaBj Þ;

ð29Þ

Nz Nz X X B2 MA2 M C2 ji Ai þ M jj Bj þ ji C i ¼ 0; i¼1

j ¼ 1; 2; . . . ; Nr ;

ð30Þ

i¼1

where

M A2 ji

"

¼ 2vn ðbA Þvn ðdA Þ  "

2 2 M B2 jj ¼ ðaB  bB Þ

2

  n2 h 1 2 ;



ð2d2A

#

a2 b2 x Þ 2 þ4 2A A2 ; 2 aB  dA aB  bA 2

b2A  a2A 

(

ð31Þ

sincðbB hÞ cosðdB hÞ d2 sincðdB hÞ cosðbB hÞ þ 4a2B B2 sincðdB hÞcosðbB hÞ bB sincðbB hÞ cosðdB hÞ

)#

ð32Þ

aB

M C2 ji ¼ 2n/n ðbA Þvn ðdA Þ

2d2A  x2

a2B  d2A

ð33Þ

:

Finally we require rrh to be orthogonal to the complete set of functions in the interval h 6 z 6 h. We choose the same function sets as for the boundary condition on rrr : h

rrh ð1; h; zÞ

h



cosðaAi zÞ sinðaAi zÞ

 dz ¼ 0:

ð34Þ

This leads to the equations

i ¼ 1; 2; . . . ; Nz ;

ð22Þ

j¼1

MA3 ii Ai

þ

Nr X C3 MB3 ij Bj þ M ii C i ¼ 0;

i ¼ 1; 2; . . . ; Nz ;

ð35Þ

j¼1

where M A1 ii

J 0n ðxÞ J ðxÞ  1 ¼ x n1  n  1: J n ðxÞ J n ðxÞ

Note that for pure imaginary argument values we have

Z

Performing the integration leads to the following set of equations: Nr X C1 M B1 ij Bj þ M ii C i ¼ 0;

wn ðxÞ ¼ x

leads to the equations

i ¼ 1; 2; . . . ; Nz ;

J 0n ðxÞ ¼ 0;

M A1 ii Ai þ

ð25Þ

where dik is the Kronecker delta. Here and below we omit indices i that relate to the symbols aA ; bA ; dA and indices j relating to aB ; bB and dB . We introduce the function

0

which makes the A and C terms in rzr and rzh identically zero at z ¼ h. In B series we choose aBj as the jth positive root of equation

Z

1 þ d1i

and taking into account relations

ð18Þ

where c ¼ k=l. One can easily make sure that the form of the solutions (4)–(12) makes the B terms in rzr and rzh identically zero at z ¼ h, and the A and C terms in rrz are identically zero at r = 1. In A and C series the values of a are chosen as:

aAi ¼ aCi ¼

  /n ðbA Þ 2vn ðdA Þwn ðbA Þ 1   þ/n ðdA Þ n2  1 þ a2A  x2 =2  wn ðdA Þ ;

MC1 ii ¼ 2hn



ð24Þ

0

1

ur CB C A@ uh A; @ ðc þ 2Þ @z uz

a2B  b2B n2  a2B þ 2b2B 2 2 2 dB  aA bB  a2A

) 2 d2B h sincðdB hÞsincðbB hÞ ; 4  cosðdB hÞ cosðbB hÞ

ð16Þ

Here J n ðxÞ is the Bessel function of first kind. As one can see from (13) and (14) the arguments of these functions can take either real or pure imaginary values as far as the values of a are real (see below Eqs. (19) and (20)). The convenience of these definitions is that their values are real in both of these cases: for pure imaginary argument values we take /n ðixÞ ¼ In ðxÞ=xn ; vn ðixÞ ¼ I0n ðxÞ=xn1 (and sincðixÞ ¼ sinhðxÞ=x; cosðixÞ ¼ coshðxÞ), where In is the modified Bessel (Infeld) function of first kind. For x ¼ 0 we take /n ð0Þ ¼ 1=ð2n n!Þ; vn ð0Þ ¼ 1=½2n ðn  1Þ! and sincð0Þ ¼ 1. The terms in curly brackets in Eqs. (4)–(12) represent two different solution sets. Following other authors [14,15], we refer to the top and bottom solutions as the ‘‘even’’ and ‘‘odd’’ solutions, respectively. This is an arbitrary choice since for the upper terms ur ; uh ; rrr ; rzz and rrh are the even functions of z, while uz ; rrz and rzh are the odd functions of z. The stress tensor components are derived from Eqs. (4)–(12):

0

2 2 2 M B1 ij ¼ ðn þ dB  x =2Þ

#

where



 1 þ d1i ¼ 2h ½ðb2A  a2A Þvn ðbA Þ/n ðdA Þ  ½n2  1 þ a2A  x2 =2  wn ðdA Þ 1 þ 2a2A vn ðdA Þ/n ðbA Þ  ½n2  1  b2A  wn ðbA Þ; ð23Þ

 2  2 2 M A3 ii ¼  ðbA  aA Þvn ðbA Þ/n ðdA Þwn ðdA Þ þ 2aA vn ðdA Þ/n ðbA Þwn ðbA Þ   1 þ d1i  2hn ; 1 ð36Þ

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A.V. Dmitriev et al. / Ultrasonics xxx (2013) xxx–xxx

"

M B3 ij

#

(

)

2 a2  b2B 2b2B d2B h sincðdB hÞsincðbB hÞ  ; ¼ 4n 2B þ dB  a2A b2B  a2A  cosðdB hÞ cosðbB hÞ

2,0

ð37Þ

Modes of an infinitely long cylinder Rayleigh WG1 WG2

1,8

ð38Þ

Ωna/ (n vt)= vn / vt

2 2 2 M C3 ii ¼ ½2vn ðdA Þ½n  1  bA =2  wn ðbA Þ  n /n ðdA Þwn ðdA Þ   1 þ d1i  2h /n ðbA Þ: 1

Eqs. (22), (30) and (35) can be grouped and written down in a matrix form as

0

MA1 B A2 MD ¼ @ M MA3

MB1 MB2 MB3

10 1 A MC1 CB C MC2 A@ B A ¼ 0; C MC3

1

1,4

1,2

ð39Þ 1,0

where M is a block square matrix of ð2Nz þ N r Þ  ð2N z þ Nr Þ size, D is a block vector of ð2Nz þ N r Þ size combined of vectors of coefficients A; B and C. In order to obtain the eigenfrequencies one should find the roots of equation det M ¼ 0. Instead of direct calculation of det M one can perform a condensation of matrix M into a matrix of N r  N r size. Because the matrices MA1 ; MC1 ; MA3 ; MC3 and MB2 are diagonal, such condensation requires no matrix inversions except the trivial inversions of diagonal matrices. Expressing C from Eq. (35) explicitly gives

C ¼ ðMC3 Þ ½MA3 A þ MB3 B:

Modes of a disk (h = 0.08864) Edge-like WG1-like Lamb-like WG2-like Other modes

1,6

ð40Þ

10

20

30

40

50

60

70

80

Angular wave number n Fig. 1. (colour online) Calculated normalized resonant frequencies versus angular wave number n in the duraluminum disk with thickness-to-diameter ratio h = 0.08864. Different mode families are denoted by solid symbols of different shapes. Only the low-lying even solutions are presented. Crosses represent the frequencies of the Rayleigh type surface modes in an infinitely long cylinder with the same radius a. The whispering gallery modes of such a cylinder are represented by open symbols.

Substituting C into (22) and (30) gives A1

m A þ mB1 B ¼ 0;

ð41Þ

2.2. Computation results

mA2 A þ mB2 B ¼ 0:

ð42Þ

The vibrational modes of a duraluminum circular disk with thickness-to-diameter ratio h = 0.08864 have been studied. This ratio is equal to the thickness-to-diameter ratio of a duraluminum disk resonator that is used in experiments (see below). The values of mass density, longitudinal and transverse sound velocities used for computation are typical for duraluminum (aluminum 2024): q ¼ 2:78 g=cm3 ; v l ¼ 6:375  105 cm=s and v t ¼ 3:150  105 cm=s, respectively. The corresponding value of Poisson’s ratio is r ¼ ðv 2l  2v 2t Þ=½2ðv 2l  v 2t Þ ¼ 0:3385. Some of the results provided by other authors [14,15] have also been reproduced in order to verify our implementation of the computational algorithm described in Section 2.1. A complete coincidence has been found. As it was shown [14], the optimal relation between the number of terms in series N r and N z is

Here

1 mA1 ¼ MA1  MC1 MC3 MA3 ;

ð43Þ

1 mA2 ¼ MA2  MC2 MC3 MA3 ;

ð44Þ

1 mB1 ¼ MB1  MC1 MC3 MB3 ;

ð45Þ

1 mB2 ¼ MB2  MC2 MC3 MB3 :

ð46Þ

Expressing A from (41) 1

A ¼ ðmA1 Þ mB1 B

ð47Þ

and substituting it into (42) gives

NB ¼ 0;

ð48Þ

where 1

N ¼ mB2  mA2 ðmA1 Þ mB1 :

ð49Þ

The size of the matrix N is N r  N r . The eigenfrequencies of the vibrating disk are the roots of the equation

det N ¼ 0:

ð50Þ

The algorithm of search for eigenfrequencies that is used includes setting values for r and h, computing coefficients of matrices M; mA1 ; mA2 ; mB1 ; mB2 and N and then evaluating the values of det N while taking steps in x until the value of det N changes sign. Then the bisection method (along with the tests on possible discontinuities) is used to evaluate the value of the eigenfrequency to any desired degree of accuracy. In order to obtain the mode shapes after finding the eigenfrequency the system of Eqs. (48) is solved for the values of B and then A and C are computed from Eqs. (47) and (40), respectively. The components of the displacement vector u are then calculated from Eqs. (4)–(12).

Nr  Nz =h:

ð51Þ

In calculations with h = 0.08864 we used Nz = 5 and Nr = 25. In this section results for even modes only are presented because in our experiments we study the waves excited by the force field that is symmetric relative to the plane z = 0 (see below), and therefore no odd modes are excited. The even modes with high angular wave numbers n J 10 are mainly considered. The frequencies xn of the lowest lying 7 families of modes divided by angular wave number n versus n are shown in Fig. 1 by solid symbols. We remind that frequency x in our calculations is dimensionless, so the values along the ordinate axis are the phase velocities of the waves vn relative to the velocity of transverse waves vt:

xn n

¼

Xn a nv t

¼

Xn kn v n ¼ ; 2pv t v t

ð52Þ

where kn and Xn are the wavelength and the dimensional angular frequency of the mode, respectively. The crosses and open symbols in Fig. 1 represent the spectra of the modes for which the components of displacement vector ur and uh are non-zero while uz component is identically zero for an infinitely long cylinder of radius a, which corresponds to a case of

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A.V. Dmitriev et al. / Ultrasonics xxx (2013) xxx–xxx

5

 ½J nþ2 ðx=gÞ  J n2 ðx=gÞ½J nþ2 ðxÞ  J n2 ðxÞ ¼ 0;

ð53Þ

where g ¼ v l =v t ¼ 2:024 for aluminum 2024. This equation has an infinite number of roots for each angular ðRÞ wave number n. The roots with the smallest frequency xn (crosses in Fig. 1) correspond to the Rayleigh cylindrical surface acoustic wave modes on an infinite cylindrical surface. All other ðWG1Þ ðWG2Þ roots xn ; xn , etc. correspond to the whispering gallery (WG) modes of different orders. As one can see, some parts of the spectra of vibrational modes of cylindrical disk behave in much like spectra of Rayleigh and WG waves in an infinitely long cylinder of the same radius a. To make this correspondence clear we analyze the shapes of vibrational modes (i.e. the distributions of the components of the displacement vector in the volume of the vibrating disk). The details are given in Appendix A. We conclude that the mode shapes in the lowest lying even family of modes (black solid squares in Fig. 1) are close to the shapes of Rayleigh-type cylindrical modes in an infinitely long cylinder for smaller values of n in the analyzed range (n K 20), and for the larger values of n the vibrations begin to localize more and more near the edges between the lateral and the end surfaces of the disk. The shape of the vibrations tends to the one for edge modes described by Tamura [15] for a cylindrical disk with h = 0.5. In the limit of large n and a these modes tend to the edge modes [19] localized at a right angle wedge. With increasing n the vibrations in all the modes considered in this section are localized in a thinner cylindrical layer close to the lateral surface of the disk. A possible interpretation of this fact is that the vibrations are ‘‘pushed’’ to the lateral surface similarly to the action of the centrifugal force. The green solid circles in Fig. 1 represent the family of the modes that can be interpreted in the region of 15 K n K 30 as modes formed by Lamb-like waves propagating along the circumference in the cylindrical layer located close to the lateral surface of the disk. With the further increasing of n (for n J 30) the vibrations in these modes begin to acquire properties that are typical for the Rayleigh-like surface vibrational modes. The distributions of the displacement vector components in the mode families that are denoted by solid red diamonds and blue pentagons in Fig. 1 authenticate that these modes should be labeled as the whispering gallery-like modes of the first and the second order, respectively. The shapes of the vibrations in the frequency families denoted by solid hexagons, triangles and stars are quite more complicated, although they still bear similarities to the Lamb modes of a plate.

Fig. 2. Schematic of the experimental setup. AC voltage of the desired frequency generated by the frequency synthesizer FS is applied through the high-voltage amplifier HVA to the comb-shaped electrode EE to excite the vibrations of the disk resonator DR. The vibration amplitude is monitored by measuring the variation of the capacitance between the disk resonator and the second electrode ME using the capacitive transducer CT.

Fig. 3. (colour online) Measured (solid symbols) and calculated (open symbols) normalized resonant frequencies of a duraluminum disk with thickness-to-diameter ratio h = 0.08864. The inverted triangles represent the calculated and measured frequencies of the lowest-lying odd solution (bending modes). The thick solid lines represent the borders of the area available for measurements in our setup.

z

½J nþ2 ðx=gÞ þ J n2 ðx=gÞ  2ðg2  1ÞJ n ðx=gÞ  ½J nþ2 ðxÞ þ J n2 ðxÞ

Ωna/ (n vt)= vn / vt

h ! 1. The eigenfrequencies in this case are the roots of the equation [18]

Fig. 4. The distribution of the vertical displacement component uz (in relative units) along the lateral disk surface (Dy ¼ aDh) measured for the lowest-lying even mode for n = 31 (squares) and its fit (solid line). The fit gives the angular wave number value n ¼ 31:1  0:4.

Fig. 5. The resonance curve of the lowest-lying even mode with n = 26.

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3. Experimental setup A schematic of the disk resonator together with excitation and measurement systems is shown in Fig. 2. The disk with diameter of 2a = 180.5 mm and thickness of 16 mm is made of polycrystalline duraluminum. Its material properties are similar to the properties of aluminum 2024. The disk is clamped to a stem which has a diameter of 20 mm. The comb-shaped electrodes used for electrostatic excitation of the disk vibration are installed on the movable platform so that the separation gap between lugs and disk is about 5 lm. The electrodes have three or two lugs. The electrode thickness is equal to the disk thickness (16 mm) in order to excite even vibrational modes of the disk resonator efficiently. To excite the disk vibration, an alternating voltage U a ¼ U 0 cos Xt is applied to the electrode. This voltage generates an electrostatic force F between the electrode and the grounded disk:

F¼

1 dC 2 U 20 dC ð1 þ cos 2XtÞ; U ¼ 2 dr a 4 dr

ð54Þ

comprising component at resonant frequency 2X, where dC=dr is the change in electrode-to-resonator overlap capacitance per radial displacement. For U0 = 300 V each lug of the electrode generates the attractive force with an amplitude of 103 N. Such an electrode is suitable for excitation of the disk vibrational modes that have a sizeable displacement in the radial direction. The condition for the optimal excitation of the mode with an angular wave number n is that the spatial period of the comb electrode is 2pa=n. Other modes with close wave numbers are also excited by such an electrode. Disk modes with n ranged from 17 to 37 are excited using two pairs of comb-shaped electrodes with spatial periods of 16 mm and 22 mm. The radial component of the disk vibration produces time-varying capacitance between the disk and the second (output) electrode, which has the same size and shape as the excitation electrode. This capacitance serves as the component of an LC circuit with resonant frequency of about 10 MHz. The circuit is pumped with voltage alternating at a frequency of the maximum slope angle of its resonance curve, and its output signal is demodulated in order to obtain the variation of the resonant frequency of the circuit. Thepdisplacement sensitivity of the capacitive sensor is about ffiffiffiffiffiffi 0:1 nm= Hz. 4. Experimental results and discussion In a first set of measurements resonant frequencies of the disk vibrations in the frequency band of 130–200 kHz are found. Among the large number of resonances those which formed the families of modes in accordance with the theoretical computations are chosen. They are shown in Fig. 3. The difference between the computed frequencies of the disk modes and measured frequencies does not exceed 1%. In order to check whether these modes do really belong to the corresponding mode families the distribution of the vertical component uz of the disk displacement along the circumference on the upper flat surface of the disk is measured by means of an additional capacitive sensor which is placed near that surface at a distance of about 30 lm from it. The typical displacement distribution measured for one of the edge modes is shown in Fig. 4. So the angular wave number n for these modes can be determined. The vertical component of the disk displacement along the disk radius have also been measured by means of the same technique in order to recognize the mode types. To measure the modes with relatively low amplitude of the ur component of the displacement vector near the circumferential surface (e.g. the Lamb-like mode family) an additional excitation electrode that is placed above the top sur-

face of the disk near the lateral surface is used. This in addition allows us to excite some of the odd (antisymmetric) modes. The lowest-lying family of the odd modes is also shown in Fig. 3 (red inverted triangles). These modes behave like the bending (flexural) modes described by Tamura [15]. Measurements of the shape of the resonance curves show splitting of some modes. The splitting of the mode can be determined if it is bigger or of the order of the width of its resonance peak. Such splitting is observed for about a half of the measured modes. The typical shape of the split mode resonance curve is shown in Fig. 5. The mode splitting is likely attributed to the inhomogeneity of the disk that breaks the symmetry and hence leads to the formation of symmetric and antisymmetric standing waves along the azimuthal direction. The Q-factors of the modes are derived from the resonance curves that are measured in vacuum under residual pressure of about 104 Torr. The resonance curves of modes without the evident splitting are fitted to a Lorenz function which allows calculation of the mode Q-factor. It is found that the values of the Qfactors of all measured modes are lying within (2. . .3)  105. This value is typical for duraluminum mechanical resonators in the ultrasonic frequency range [20]. It is likely determined by dislocation/grain boundary losses in polycrystalline duraluminum. 5. Conclusions Natural frequencies and distributions of displacement components for the surface vibrational modes in thin isotropic elastic disk have been calculated. In particular, the research has been focused on even solutions for low-lying resonant vibrations with large angular wave numbers. Several families of modes have been found which have been interpreted as modified Rayleigh and whispering gallery modes of an infinitely long cylinder and Lamb modes of a plate. The results of calculation have been compared with the results of the experimental measurements of vibrational modes generated by means of resonant electrostatic excitation in duraluminum disk with diameter of 180.5 mm and thickness of 16 mm in the frequency range of 130–200 kHz. In order to check whether these modes do really belong to the corresponding mode families the distribution of the vertical component uz of the disk displacement along the circumference on the upper flat surface of the disk have been measured by means of an additional capacitive sensor. An excellent agreement between the calculated and measured frequencies has been found. Measurements of the structure of the resonant peaks have shown that about a half of the measured modes had the splitting at the level of the order of Dfsplit =fmode  105 . We suggest that the mode splitting is caused by the inhomogeneity of the disk that breaks the symmetry and hence leads to the formation of symmetric and antisymmetric standing waves along the azimuthal direction. The Q-factors of all modes measured in vacuum were within ð2 . . . 3Þ  105 . This value is typical for duraluminum mechanical resonators in the ultrasonic frequency range. The theoretical calculations have been compared with the results of measurements which were carried out using duraluminum disk of relatively large size. Doing this simplified the identification of the observed modes and the comparison of theoretical and experimental results. In future these results can be used for development of high frequency microdisk mechanical resonators. Acknowledgements The authors gratefully acknowledge the support of the Russian Fund of Fundamental Researches under Grant 11-02-00383 and the United States National Science Foundation and Caltech under Grant PHY-0967049.

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 n xn uh ¼ AV J n r  f ðxn Þxn J 0n ðxn rÞ sin nh; r g

Appendix A. Displacement vector distributions in disk resonators The distributions of the components of the displacement vector are calculated using Eqs. (4)–(12). In this section the graphical distributions of the displacement vector components of some modes are presented. More specifically, the presented figures are the 2D graphics of the functions

^ r ðr; zÞ ¼ ur ðr; h; zÞ= cos nh; u

ðA:1Þ

^ h ðr; zÞ ¼ uh ðr; h; zÞ= sin nh; u

ðA:2Þ

^ z ðr; zÞ ¼ uz ðr; h; zÞ= cos nh: u

ðA:3Þ

^r ; u ^h ; u ^ z is set For each mode the maximal of all the values of u equal to unity. The mode shapes in the lowest-lying family of even modes for n = 10 are shown in Fig. A.6. One can see that the distribution of ur and uh for these modes for small values of n (n K 15) are close to the corresponding distributions of Rayleigh-type modes in an infinitely long cylinder, while the uz component is relatively small. Indeed, for the Rayleigh and WG modes of an infinitely long cylinder with frequencies given by (53) the non-zero components of the displacement vector are [18]

ur ¼ AV









xn 0 xn J r  nf ðxn ÞJ n ðxn rÞ cos nh; g n g

ðA:4Þ

7

ðA:5Þ

where AV is an amplitude constant and

f ð xÞ ¼

1 J n2 ðx=gÞ  J nþ2 ðx=gÞ : J n2 ðxÞ þ J nþ2 ðxÞ

g2

ðA:6Þ

^ r ðr; hÞ and u ^ h ðr; hÞ for the disk The calculated distributions of u ^ ðRÞ ^ ðRÞ and u r ðrÞ ¼ ur = cos nh and u h ¼ uh = sin nh for the infinitely long cylinder are compared in Fig. A.7. For the large values of n the disk vibrations in the lowest-lying even mode family are concentrated in small regions near the edges between the lateral and the end surfaces of the disk (Fig. A.8), and the component uh becomes small as compared with ur and uz . This type of modes was described by Tamura [15] as the edge modes. These modes are formed by the waves that tend to the waves propagating along a right angle wedge in the limit of large n and a. One can see that in all the range of angular wave numbers n shown in Fig. 1 for each n there are modes (denoted by red solid diamonds) which frequencies are very close to the eigenfrequenðWG1Þ cies of the WG modes of the first order xn of an infinitely long cylinder (red open circles). We denote this sequence of frequencies ðWG1-likeÞ as xn below. The distribution of the components of the displacement vector in these modes (by the example of n = 15) is given in Fig. A.9. The components of the displacement vector in this mode along with the ones in a WG mode of first order (given by Eqs. (A.4) and (A.5)) in an infinitely long cylinder are shown in Fig. A.10, from which one can conclude that this family of modes is actually the family of WG-like modes of the first order. Similar comparison of the vibration components in the WG modes of the second order in an infinitely long cylinder and the vibration components in the modes that are denoted by solid blue pentagons in Fig. 1 in the disk reveal that the latter is the family of

Fig. A.6. The distribution of the components of the displacement vector for the lowest-lying even mode with n = 10 (Rayleigh-like mode).

Fig. A.8. The distribution of the components of the displacement vector for the lowest-lying even mode with n = 80 (edge-like mode).

Fig. A.7. The distribution of the components of the displacement vector ur (black squares), uh (red circles) and uz (green triangles) along the radial direction on the top end surface of the disk (z = h) in the Rayleigh-like mode with n = 10. Solid black and dashed red lines represent ur and uh in a Rayleigh type surface mode with n = 10 in an infinitely long cylinder, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. A.9. The distribution of the components of the displacement vector for the ðWG1-likeÞ mode x15 .

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Fig. A.12. The distribution of the components of the displacement vector for the second lowest even mode with n = 25 (Lamb-like).

Fig. A.10. The distribution of the components of the displacement vector ur (black squares), uh (red circles) and uz (green triangles) along the radial direction on the top end surface of the disk (z ¼ h) in the WG-like mode of the first order with n = 15. Solid black and dashed red lines represent ur and uh in a WG mode with n = 15 in an infinitely long cylinder, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. A.13. The distribution of the components of the displacement vector for the second lowest even mode with n = 80 (Rayleigh-like).

Fig. A.11. The distribution of the components of the displacement vector for the ðWG2likeÞ mode x28 .

the WG-like modes of the second order (see Fig. A.11). We denote ðWG2likeÞ them xn below. The distributions of the components ur and uh of the displaceðWG1-likeÞ ment vector in the WG-like modes of the disk xn and ðWG2likeÞ xn remain qualitatively the same for all considered values of n and are close to the analogous distributions in the WG modes of an infinitely long cylinder. The vibrations in z-direction are relatively small. Besides the families of Rayleigh-like and WG-like modes one can see in Fig. 1 a bunch of families of modes which have greater slope as compared with Rayleigh-like and WG-like modes. The vibrations in the lowest-lying of them, denoted by green solid circles in Fig. 1, are close to the vibrations in the lowest-lying Lamb mode of an infinite plate [18] in the range of 15 K n K 30, so that these modes can be interpreted as modes formed by Lamb-like waves propagating along the circumference in the cylindrical layer located close to the lateral surface of the disk. The resonant frequencies of these modes in this range are also close to the resonant frequencies of the Lamb waves in an infinite plate of the same thickness 2H. In this type of modes the uz and uh components of the displacement vector dominate the ur component. The distribution of these components is shown in Fig. A.12. With the further increasing of n (for n J 30) the vibrations in this family of modes begin to acquire properties that are typical for the Rayleigh-like surface vibrational modes. The distributions of the components of the displacement vector for this type of

modes for n = 80 are shown in Fig. A.13. The shape of the vibrations in these modes differs from previously discussed Rayleigh-like modes in the lowest-lying even family: the distribution of the components of the displacement vector in this case is close to the case of Rayleigh-like modes in an infinitely long cylinder only in regions close to the z = 0 plane, and near the edges (z  h and r  1) the uz component dominates uh and ur ; moreover, the ur component tends to zero on the edges, so this type of modes possesses the properties of a Rayleigh-like wave near the central plane z = 0 and the properties of a Lamb-like wave near the edges. This is consistent with Tamura’s result [15] who showed that in the limit of h ! 1 the frequencies of this (second lowest even) family of modes converge to the frequencies of a Rayleigh-type surface waves in an infinitely long cylinder. The shape of the vibrations in the remaining frequency families that are denoted by solid triangles, hexagons and stars in Fig. 1 is quite more complicated, although they still bear similarities to ðL2likeÞ ðL3likeÞ the Lamb modes of a plate. We denote them as xn ; xn

Fig. A.14. The distribution of the components of the displacement vector for the ðL2likeÞ mode x35 .

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A.V. Dmitriev et al. / Ultrasonics xxx (2013) xxx–xxx ðWG2likeÞ

ðL3likeÞ

near n ¼ 23 (namely x23 and x23 ) are given in Figs. A.15, A.16. The relative difference between the resonant frequencies of these modes  8  104 . References

Fig. A.15. The distribution of the components of the displacement vector for the ðWG2likeÞ mode x23 (near the ‘‘intersection’’ of mode families).

Fig. A.16. The distribution of the components of the displacement vector for the ðL3likeÞ mode x23 (near the ‘‘intersection’’ of mode families).

ðL4likeÞ

and xn , respectively. The distribution of the components of ðL2likeÞ the displacement vector in the mode x35 is shown in Fig. A.14. One can see that the uz component dominates in these modes too, although the ur component also takes high values near the edges of the disk. The shape of the vibrations in modes that are situated close to the ‘‘intersections’’ of mode families in Fig. 1 can be quite messy, especially if the resonant frequencies of the modes are close to each other. The distributions of the components of the displacement vector in the two modes of the mode families that ‘‘intersect’’

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