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Coriolis effect on responses of rotating thin piezoelectric hollow cylinder
Theoretical & Applied Mechanics Letters 6 (2016) 274–276
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters journal homepage: www.elsevier.com/locate/taml
Letter
Coriolis effect on responses of rotating thin piezoelectric hollow cylinder Wenyao Liu a,∗ , Weiqiu Chen b a
Haitian School, Ningbo Polytechnic, Ningbo 315800, China
b
Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
highlights • Coriolis effect is considered for the first time in the analysis of a rotating piezoelectric hollow cylinder. • A different strategy is employed to derive the equation governing the radial displacement, which is then solved approximately but analytically when the shell is thin enough.
• Numerical examples show that the Coriolis effect can be significant under certain conditions in active control of the shell.
article
info
Article history: Received 22 September 2016 Accepted 7 October 2016 Available online 4 November 2016 *This article belongs to the Solid Mechanics Keywords: Coriolis effect Rotating cylinder Piezoelectric material Analytical solution
abstract Coriolis effect is considered in the analysis of a rotating piezoelectric hollow cylinder. An inhomogeneous Bessel equation governing the radial mechanical displacement is derived, which can be approximated as an Euler type differential equation when the cylinder is very thin. Numerical examples show that the Coriolis effect can be significant under certain conditions. © 2016 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Rotating structures have long been of significant research interest because of their practical importance in various engineering applications [1–4]. With the emergence of smart structures [5,6], efforts have also been made on the study of rotating piezoelectric plates and shells [7,8]. In a relatively recent paper, Galic and Horgan [9] (denoted as GH hereafter for brevity) derived a closed form exact solution for a rotating solid/hollow cylinder and some interesting issues were addressed and highlighted. In this letter, we further take account of the Coriolis effect in the investigation on the response of a rotating piezoelectric hollow cylinder. A relatively simple derivation of the equation governing the radial displacement is presented, which leads to a Bessel equation with inhomogeneous term. For the sake of simplicity, only very thin cylindrical shells are considered in this letter. Thus, the Bessel equation can be treated approximately as an Euler
∗
Corresponding author. E-mail address: [email protected] (W. Liu).
type differential equation, of which the solution is well known. Numerical calculation is finally performed and some interesting observations are discussed. The basic equations of an infinite radially polarized piezoelectric cylinder for the axisymmetric problem can be easily found. When the cylinder is rotating about its axis at constant angular velocity ω, the equation of motion with Coriolis effect is [10,11] dσrr
1
+ (σrr − σθ θ ) + ρd ω2 r + ρd ur ω2 = 0,
(1) dr r where σrr and σθ θ are respectively the radial and circumferential normal stress components, ur is the radial displacement, ρd is the material density, and r is the radial coordinate in the cylindrical coordinate system (r , θ , z ) with its origin located on the axis. In this letter, we present a quite different derivation. First, we can obtain the electric displacement Dr directly from the electric r equilibrium equation dD + Drr = 0 as follows dr Dr = Ar −1 ,
(2)
http://dx.doi.org/10.1016/j.taml.2016.10.001 2095-0349/© 2016 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
W. Liu, W. Chen / Theoretical & Applied Mechanics Letters 6 (2016) 274–276
275
where A is an arbitrary constant. Second, from the constitutive relations
σrr = c33
dur
σθ θ = c13
dr dur
dr dur
Dr = e33
+ c13
dr
ur
+ c11
+ e31
r ur
dϕ
+ e31
r
ur r
+ e33
− ε33
,
dr dϕ
dr dϕ dr
,
(3)
,
where ϕ is the electric potential, cij , eij , and εij are respectively the elastic, piezoelectric, and dielectric constants, it is obtained that dϕ dr
=
e33 dur
ε33 dr
σrr = k1
dur
σθ θ = k2
dr dur dr
+
+ k2
e31 ur
ε33 r ur
+ k3
r ur r
1
− e33
ε33
Ar −1 ,
Ar −1 ,
−
ε33
−
ε33
e31
(4)
Ar −1 ,
where k1 = c33 + e233 /ε33 , k2 = c13 + e31 e33 /ε33 , k3 = c11 + e231 /ε33 . (5) Substituting Eq. (4) into Eq. (1), yields
ρ2
d 2 ur dρ 2
+ρ
dur dρ
+ (ρ 2 Ω 2 − κ 2 )ur + Ω 2 aρ 3 − B = 0,
(6)
where ρ = r /a and Ω = ρd ω2 a2 /c33 are the dimensionless radial coordinate and angular velocity, respectively, with a being the inner radius of the hollow cylinder, and
κ 2 = k3 /k1 , B = (e33 − e31 )A/(k1 ε33 ).
Equation (6) is an inhomogeneous Bessel equation whose solution consists of two parts. The general solution part is a linear combination of the well known Bessel functions of the first and second kinds, while the particular part is represented by the so-called Lommel functions [12,13]. The solution is a little complicated, and from Eq. (4), the expression for the electric potential ϕ will further contain integrals involving Bessel functions and Lommel functions. In this letter, however, we will just pay our attention to the effect of Coriolis acceleration on very thin hollow cylinders, for which Eq. (6) can be approximated as
ρ2
d 2 ur dρ 2
+ρ
dur dρ
+ (ρ02 Ω 2 − κ 2 )ur + Ω 2 aρ 3 − B = 0,
Fig. 1. Responses of the rotating cylinder in Case A (Ω = 1).
(7)
(8)
where ρ0 = R/a, and R = (a + b)/2 is the mean-radius of the cylinder, with b being the outer radius of the hollow cylinder. Note that such an approximation technique has been widely used and validated in many branches of applied mechanics [10,14–16]. Obviously, when the thickness of the hollow cylinder decreases, the solution based on Eq. (8) will become more and more accurate. Equation (8) is an Euler equation, whose solution is very straightforward and omitted here for brevity. Actually, the solution contains three arbitrary constants: Two are the integral constants directly related to the general solution of Eq. (8), while the another one is A, appearing in the particular solution through the constant B. The three unknown constants can be determined from the boundary conditions at the inner and outer surfaces of the cylinder (one mechanical at each surface and one electric). It is noted that, when the electric potential is specified at the surfaces, one can only determine the electric difference between the two surfaces, since the constant electric potential plays a role like the rigid body displacement. On the other hand, when electric displacement is specified, only the condition at one surface should be considered,
and the one at another surface becomes a natural result because of the requirement of self-equilibrium of electric displacement. It is obvious that when the Coriolis effect is neglected, Eq. (8) is exact without any approximation. This equation is simpler than that of the fourth-order one as mentioned in GH. Although our solution contains only three arbitrary constants, and that of GH contains four arbitrary constants, the two solutions are exactly the same because the additional constant in GH is related to the constant electric potential only, which contributes nothing to the electroelastic field in the cylinder. For numerical illustration, we assume that there are no external mechanical forces acting on the inner and outer surfaces of the cylinder. In addition, we consider two types of electric boundary conditions: Case A is just Case 1 in GH (i.e. closed-circuit on both the inner and outer surfaces), while in Case B, the electric condition in Case A is replaced with zero electric displacement at either surface. From the previous studies [14,16], we can conclude that when the thickness-to-mean radius ratio of the cylinder h/R = 1/200, the error due to the approximation introduced in this note can be neglected, provided that the dimensionless angular velocity Ω is not very large. In fact, we have also performed the calculation according to the exact solution of Eq. (5), and found the two are almost identical. Results are presented in Figs. 1–3 for a lead zirconate titanate (PZT-4) hollow cylinder with h/R = 1/200, where two values of Ω are considered, i.e., Ω = 1 and Ω = 3. A new dimensionless radial coordinate ζ = (r − a)/(b − a) is used such that the inner and outer surfaces correspond to ζ = 0 and ζ = 1, respectively. The material constants of PZT-4 can be found in Ref. [17] as c33 = 1.15 × 1011 Pa, c11 = 1.39 × 1011 Pa, c13 = 7.43 × 1010 Pa, e31 = −5.2 C/m2 , e33 = 15.1 C/m2 , and ε33 = 5.62 × 10−9 F/m. For both cases, we assume that the electric potential at the inner surface is zero. The distribution of σrr is not given in Figs. 1 and 2
276
W. Liu, W. Chen / Theoretical & Applied Mechanics Letters 6 (2016) 274–276
because the Coriolis effect on it is completely negligible. Although the Coriolis effect on σθ θ is also little, it does lead to a more uniform distribution along the thickness direction, as shown in Figs. 1(a) and 2(a), which is favorable for the practical structure. The significance of Coriolis acceleration effect is clearly shown in Figs. 1(b) and 2(b) when considering the electric potential. In Fig. 3, we give the distributions of electric potential for two values of Ω . The phenomena shown in Fig. 3(a) and (b) are both of practical importance because the electric potential difference is generally used as a primary parameter in the feedback control of smart structures. Figure 3(a) indicates that the practical control may be overfull when the Coriolis effect is not considered in the theoretical prediction. Furthermore, Fig. 3(b) implies the possibility of a completely opposite action that may be induced in practice. In this letter, we considered the response of a rotating piezoelectric hollow cylinder by taking the Coriolis effect into consideration. A different strategy was adopted to derive the governing equation, which seems to be simpler than that reported in the literature when the Coriolis effect is absent. For cylinders with very small thickness, the equation was simplified and an analytical solution was derived. It was shown numerically that the Coriolis effect may become very important in the active control of rotating structures due to the coupling between the elastic and electric fields. Acknowledgments The work was supported by the National Natural Science Foundation of China (11321202) and the Fundamental Research Funds for the Central Universities (2016XZZX001-05). Fig. 2. Responses of the rotating cylinder in Case A (Ω = 3).
References
Fig. 3. Electric potential distributions in the rotating cylinder in Case B.
[1] G.H. Bryan, On the beats in the vibration of a revolving cylinder or bell, Proc. Cambridge Philos. Soc. 7 (1890) 101–111. [2] Y. Chen, H.B. Zhao, Z.P. Shen, Vibrations of high speed rotating shells with calculations for cylindrical shells, J. Sound Vib. 160 (1993) 137–160. [3] X.M. Zhang, Parametric analysis of frequency of rotating laminated composite cylindrical shells with the wave propagation method, Comput. Methods. Appl. Mech. Engrg. 191 (2002) 2029–2043. [4] Y.X. Chen, P.Q. Liu, Z.H. Tang, et al., Wind tunnel tests of stratospheric airship counter rotating propellers, Theor. Appl. Mech. Lett. 5 (2015) 58–61. [5] I. Chopra, Review of state of art of smart structures and integrated systems, AIAA J. 40 (2002) 2145–2187. [6] A. Abdelkefi, A. Hasanyan, J. Montgomery, et al., Incident flow effects on the performance of piezoelectric energy harvesters from galloping vibrations, Theor. Appl. Mech. Lett. 4 (2014) 022002. [7] W.Q. Chen, H.J. Ding, Exact static analysis of a rotating piezoelectric spherical shell, Acta Mech. Sin. 14 (1998) 257–265. [8] W.Q. Chen, H.J. Ding, J. Liang, The exact elasto-electric field of a rotating piezoceramic spherical shell with a functionally graded property, Int. J. Solids Struct. 38 (2001) 7015–7027. [9] D. Galic, C.O. Horgan, The stress response of radially polarized rotating piezoelectric cylinders, J. Appl. Mech. 70 (2003) 426–435. [10] R.A. DiTaranto, M. Lessen, Coriolis acceleration effect on the vibration of a rotating thin-walled circular cylinder, J. Appl. Mech. 31 (1964) 700–701. [11] C.T. Loy, K.Y. Lam, Vibrations of rotating thin cylindrical panels, Appl. Acoust. 46 (1995) 327–343. [12] I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, Series, and Products, seventh ed., Academic Press, New York, 2007. [13] C.F. Lü, J.S. Yang, J. Wang, et al., Power transmission through a hollow cylinder by acoustic waves and piezoelectric transducers with radial polarization, J. Sound Vib. 325 (2009) 989–999. [14] J.Q. Ye, K.P. Soldatos, Three-dimensional stress analysis of orthotropic and cross-ply laminated hollow cylinders and cylindrical panels, Comput. Methods Appl. Mech. Engrg. 117 (1994) 331–351. [15] Y. Tanigawa, Some basic thermoelastic problems for nonhomogeneous structural materials, Appl. Mech. Rev. 48 (1995) 287–300. [16] J. Zhu, W.Q. Chen, G.R. Ye, et al., Waves in fluid-filled functionally graded piezoelectric cylinders: A restudy based on the reverberation-ray matrix formulation, Wave Motion 50 (2013) 415–427. [17] M.L. Dunn, M. Taya, Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids, J. Appl. Mech. 61 (1994) 474–475.
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters journal homepage: www.elsevier.com/locate/taml
Letter
Coriolis effect on responses of rotating thin piezoelectric hollow cylinder Wenyao Liu a,∗ , Weiqiu Chen b a
Haitian School, Ningbo Polytechnic, Ningbo 315800, China
b
Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
highlights • Coriolis effect is considered for the first time in the analysis of a rotating piezoelectric hollow cylinder. • A different strategy is employed to derive the equation governing the radial displacement, which is then solved approximately but analytically when the shell is thin enough.
• Numerical examples show that the Coriolis effect can be significant under certain conditions in active control of the shell.
article
info
Article history: Received 22 September 2016 Accepted 7 October 2016 Available online 4 November 2016 *This article belongs to the Solid Mechanics Keywords: Coriolis effect Rotating cylinder Piezoelectric material Analytical solution
abstract Coriolis effect is considered in the analysis of a rotating piezoelectric hollow cylinder. An inhomogeneous Bessel equation governing the radial mechanical displacement is derived, which can be approximated as an Euler type differential equation when the cylinder is very thin. Numerical examples show that the Coriolis effect can be significant under certain conditions. © 2016 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Rotating structures have long been of significant research interest because of their practical importance in various engineering applications [1–4]. With the emergence of smart structures [5,6], efforts have also been made on the study of rotating piezoelectric plates and shells [7,8]. In a relatively recent paper, Galic and Horgan [9] (denoted as GH hereafter for brevity) derived a closed form exact solution for a rotating solid/hollow cylinder and some interesting issues were addressed and highlighted. In this letter, we further take account of the Coriolis effect in the investigation on the response of a rotating piezoelectric hollow cylinder. A relatively simple derivation of the equation governing the radial displacement is presented, which leads to a Bessel equation with inhomogeneous term. For the sake of simplicity, only very thin cylindrical shells are considered in this letter. Thus, the Bessel equation can be treated approximately as an Euler
∗
Corresponding author. E-mail address: [email protected] (W. Liu).
type differential equation, of which the solution is well known. Numerical calculation is finally performed and some interesting observations are discussed. The basic equations of an infinite radially polarized piezoelectric cylinder for the axisymmetric problem can be easily found. When the cylinder is rotating about its axis at constant angular velocity ω, the equation of motion with Coriolis effect is [10,11] dσrr
1
+ (σrr − σθ θ ) + ρd ω2 r + ρd ur ω2 = 0,
(1) dr r where σrr and σθ θ are respectively the radial and circumferential normal stress components, ur is the radial displacement, ρd is the material density, and r is the radial coordinate in the cylindrical coordinate system (r , θ , z ) with its origin located on the axis. In this letter, we present a quite different derivation. First, we can obtain the electric displacement Dr directly from the electric r equilibrium equation dD + Drr = 0 as follows dr Dr = Ar −1 ,
(2)
http://dx.doi.org/10.1016/j.taml.2016.10.001 2095-0349/© 2016 The Author(s). Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
W. Liu, W. Chen / Theoretical & Applied Mechanics Letters 6 (2016) 274–276
275
where A is an arbitrary constant. Second, from the constitutive relations
σrr = c33
dur
σθ θ = c13
dr dur
dr dur
Dr = e33
+ c13
dr
ur
+ c11
+ e31
r ur
dϕ
+ e31
r
ur r
+ e33
− ε33
,
dr dϕ
dr dϕ dr
,
(3)
,
where ϕ is the electric potential, cij , eij , and εij are respectively the elastic, piezoelectric, and dielectric constants, it is obtained that dϕ dr
=
e33 dur
ε33 dr
σrr = k1
dur
σθ θ = k2
dr dur dr
+
+ k2
e31 ur
ε33 r ur
+ k3
r ur r
1
− e33
ε33
Ar −1 ,
Ar −1 ,
−
ε33
−
ε33
e31
(4)
Ar −1 ,
where k1 = c33 + e233 /ε33 , k2 = c13 + e31 e33 /ε33 , k3 = c11 + e231 /ε33 . (5) Substituting Eq. (4) into Eq. (1), yields
ρ2
d 2 ur dρ 2
+ρ
dur dρ
+ (ρ 2 Ω 2 − κ 2 )ur + Ω 2 aρ 3 − B = 0,
(6)
where ρ = r /a and Ω = ρd ω2 a2 /c33 are the dimensionless radial coordinate and angular velocity, respectively, with a being the inner radius of the hollow cylinder, and
κ 2 = k3 /k1 , B = (e33 − e31 )A/(k1 ε33 ).
Equation (6) is an inhomogeneous Bessel equation whose solution consists of two parts. The general solution part is a linear combination of the well known Bessel functions of the first and second kinds, while the particular part is represented by the so-called Lommel functions [12,13]. The solution is a little complicated, and from Eq. (4), the expression for the electric potential ϕ will further contain integrals involving Bessel functions and Lommel functions. In this letter, however, we will just pay our attention to the effect of Coriolis acceleration on very thin hollow cylinders, for which Eq. (6) can be approximated as
ρ2
d 2 ur dρ 2
+ρ
dur dρ
+ (ρ02 Ω 2 − κ 2 )ur + Ω 2 aρ 3 − B = 0,
Fig. 1. Responses of the rotating cylinder in Case A (Ω = 1).
(7)
(8)
where ρ0 = R/a, and R = (a + b)/2 is the mean-radius of the cylinder, with b being the outer radius of the hollow cylinder. Note that such an approximation technique has been widely used and validated in many branches of applied mechanics [10,14–16]. Obviously, when the thickness of the hollow cylinder decreases, the solution based on Eq. (8) will become more and more accurate. Equation (8) is an Euler equation, whose solution is very straightforward and omitted here for brevity. Actually, the solution contains three arbitrary constants: Two are the integral constants directly related to the general solution of Eq. (8), while the another one is A, appearing in the particular solution through the constant B. The three unknown constants can be determined from the boundary conditions at the inner and outer surfaces of the cylinder (one mechanical at each surface and one electric). It is noted that, when the electric potential is specified at the surfaces, one can only determine the electric difference between the two surfaces, since the constant electric potential plays a role like the rigid body displacement. On the other hand, when electric displacement is specified, only the condition at one surface should be considered,
and the one at another surface becomes a natural result because of the requirement of self-equilibrium of electric displacement. It is obvious that when the Coriolis effect is neglected, Eq. (8) is exact without any approximation. This equation is simpler than that of the fourth-order one as mentioned in GH. Although our solution contains only three arbitrary constants, and that of GH contains four arbitrary constants, the two solutions are exactly the same because the additional constant in GH is related to the constant electric potential only, which contributes nothing to the electroelastic field in the cylinder. For numerical illustration, we assume that there are no external mechanical forces acting on the inner and outer surfaces of the cylinder. In addition, we consider two types of electric boundary conditions: Case A is just Case 1 in GH (i.e. closed-circuit on both the inner and outer surfaces), while in Case B, the electric condition in Case A is replaced with zero electric displacement at either surface. From the previous studies [14,16], we can conclude that when the thickness-to-mean radius ratio of the cylinder h/R = 1/200, the error due to the approximation introduced in this note can be neglected, provided that the dimensionless angular velocity Ω is not very large. In fact, we have also performed the calculation according to the exact solution of Eq. (5), and found the two are almost identical. Results are presented in Figs. 1–3 for a lead zirconate titanate (PZT-4) hollow cylinder with h/R = 1/200, where two values of Ω are considered, i.e., Ω = 1 and Ω = 3. A new dimensionless radial coordinate ζ = (r − a)/(b − a) is used such that the inner and outer surfaces correspond to ζ = 0 and ζ = 1, respectively. The material constants of PZT-4 can be found in Ref. [17] as c33 = 1.15 × 1011 Pa, c11 = 1.39 × 1011 Pa, c13 = 7.43 × 1010 Pa, e31 = −5.2 C/m2 , e33 = 15.1 C/m2 , and ε33 = 5.62 × 10−9 F/m. For both cases, we assume that the electric potential at the inner surface is zero. The distribution of σrr is not given in Figs. 1 and 2
276
W. Liu, W. Chen / Theoretical & Applied Mechanics Letters 6 (2016) 274–276
because the Coriolis effect on it is completely negligible. Although the Coriolis effect on σθ θ is also little, it does lead to a more uniform distribution along the thickness direction, as shown in Figs. 1(a) and 2(a), which is favorable for the practical structure. The significance of Coriolis acceleration effect is clearly shown in Figs. 1(b) and 2(b) when considering the electric potential. In Fig. 3, we give the distributions of electric potential for two values of Ω . The phenomena shown in Fig. 3(a) and (b) are both of practical importance because the electric potential difference is generally used as a primary parameter in the feedback control of smart structures. Figure 3(a) indicates that the practical control may be overfull when the Coriolis effect is not considered in the theoretical prediction. Furthermore, Fig. 3(b) implies the possibility of a completely opposite action that may be induced in practice. In this letter, we considered the response of a rotating piezoelectric hollow cylinder by taking the Coriolis effect into consideration. A different strategy was adopted to derive the governing equation, which seems to be simpler than that reported in the literature when the Coriolis effect is absent. For cylinders with very small thickness, the equation was simplified and an analytical solution was derived. It was shown numerically that the Coriolis effect may become very important in the active control of rotating structures due to the coupling between the elastic and electric fields. Acknowledgments The work was supported by the National Natural Science Foundation of China (11321202) and the Fundamental Research Funds for the Central Universities (2016XZZX001-05). Fig. 2. Responses of the rotating cylinder in Case A (Ω = 3).
References
Fig. 3. Electric potential distributions in the rotating cylinder in Case B.
[1] G.H. Bryan, On the beats in the vibration of a revolving cylinder or bell, Proc. Cambridge Philos. Soc. 7 (1890) 101–111. [2] Y. Chen, H.B. Zhao, Z.P. Shen, Vibrations of high speed rotating shells with calculations for cylindrical shells, J. Sound Vib. 160 (1993) 137–160. [3] X.M. Zhang, Parametric analysis of frequency of rotating laminated composite cylindrical shells with the wave propagation method, Comput. Methods. Appl. Mech. Engrg. 191 (2002) 2029–2043. [4] Y.X. Chen, P.Q. Liu, Z.H. Tang, et al., Wind tunnel tests of stratospheric airship counter rotating propellers, Theor. Appl. Mech. Lett. 5 (2015) 58–61. [5] I. Chopra, Review of state of art of smart structures and integrated systems, AIAA J. 40 (2002) 2145–2187. [6] A. Abdelkefi, A. Hasanyan, J. Montgomery, et al., Incident flow effects on the performance of piezoelectric energy harvesters from galloping vibrations, Theor. Appl. Mech. Lett. 4 (2014) 022002. [7] W.Q. Chen, H.J. Ding, Exact static analysis of a rotating piezoelectric spherical shell, Acta Mech. Sin. 14 (1998) 257–265. [8] W.Q. Chen, H.J. Ding, J. Liang, The exact elasto-electric field of a rotating piezoceramic spherical shell with a functionally graded property, Int. J. Solids Struct. 38 (2001) 7015–7027. [9] D. Galic, C.O. Horgan, The stress response of radially polarized rotating piezoelectric cylinders, J. Appl. Mech. 70 (2003) 426–435. [10] R.A. DiTaranto, M. Lessen, Coriolis acceleration effect on the vibration of a rotating thin-walled circular cylinder, J. Appl. Mech. 31 (1964) 700–701. [11] C.T. Loy, K.Y. Lam, Vibrations of rotating thin cylindrical panels, Appl. Acoust. 46 (1995) 327–343. [12] I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, Series, and Products, seventh ed., Academic Press, New York, 2007. [13] C.F. Lü, J.S. Yang, J. Wang, et al., Power transmission through a hollow cylinder by acoustic waves and piezoelectric transducers with radial polarization, J. Sound Vib. 325 (2009) 989–999. [14] J.Q. Ye, K.P. Soldatos, Three-dimensional stress analysis of orthotropic and cross-ply laminated hollow cylinders and cylindrical panels, Comput. Methods Appl. Mech. Engrg. 117 (1994) 331–351. [15] Y. Tanigawa, Some basic thermoelastic problems for nonhomogeneous structural materials, Appl. Mech. Rev. 48 (1995) 287–300. [16] J. Zhu, W.Q. Chen, G.R. Ye, et al., Waves in fluid-filled functionally graded piezoelectric cylinders: A restudy based on the reverberation-ray matrix formulation, Wave Motion 50 (2013) 415–427. [17] M.L. Dunn, M. Taya, Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids, J. Appl. Mech. 61 (1994) 474–475.