# Free Vibration Analysis of Rings via Wave Approach

Authors: Wang Zhipeng; Liu Wei; Yuan Yunbo; Shuai Zhijun; Guo Yibin; Wang Donghua
Journal: Shock and Vibration
 (2018)

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## Abstract

Free vibration of rings is presented via wave approach theoretically. Firstly, based on the solutions of out-of-plane vibration, propagation, reflection, and coordination matrices are derived for the case of a fixed boundary at inner surface and a free boundary at outer surface. Then, assembling these matrices, characteristic equation of natural frequency is obtained. Wave approach is employed to study the free vibration of these ring structures. Natural frequencies calculated by wave approach are compared with those obtained by classical method and Finite Element Method (FEM). Afterwards natural frequencies of four type boundaries are calculated. Transverse vibration transmissibility of rings propagating from outer to inner and from inner to outer is investigated. Finally, the effects of structural and material parameters on free vibration are discussed in detail.


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## Body

## 1. Introduction

Noise and vibration problem of rings has been a hotspot that makes it attract lots of attention because of the wide applications, such as gear transmission system and aircraft structures. Since most of components within these structures can be regarded as a simple model of ring structure, dynamic properties of these structures in recent years have been the subject of present studies. Based on the Stodola–Vianello method, Gutierrez et al. [1] considered the free vibration of annular membranes with continuously variable density by using Rayleigh method, differential quadrature method, and finite elements simulation. By adopting the dynamic stiffness method, Jabareen and Eisenberger [2] investigated the natural characteristics of the nonhomogeneous annular structure. Wang [3] discussed the natural frequency with the boundaries of fixed, free, and simply supported. And the researches indicated clearly that complex modes would switch correspondingly with the radius increasing gradually. Based on the first-order shear deformation theory, Roshan and Rashmi [4] analyzed the free vibration of axisymmetric sandwich circular plate with relatively stiff thickness. Oveisi and Shakeri [5] constructed a sandwich composite circular plate containing piezoelectric material, finding that the feedback control gain had an effective control for suppressing the transverse vibration. Hosseini-Hashemi et al. [6] presented the free vibration of functionally graded circular plate with stepped thickness via Mindlin plate theory. By employing harmonic differential quadrature method for obtaining numerical solution of circular plate, Civalek and Uelker [7] made a further analysis for the behavior of the boundary conditions of fixed and simply supported. Moreover, the results obtained by finite difference method were compared with those calculated by harmonic differential quadrature method, whose feasibility is verified by Bakhshi Khaniki and Hosseini-Hashemi [8]. Liu et al. [9] construct the composite thin annular plate by wave approach. However, composite structures with multilayer are not studied. Different boundaries and parameter effects are also not analyzed.Additionally, in terms of the vibration of structures, there is an alternative method called “wave approach” which is very efficient and widely used for calculating the natural frequency of structures, such as beams, plates, rings, and periodic structure by describing waves in the matrix form. For example, as early as 1984, Mace [10] applied wave approach to analyze the wave behaviors in Euler beam. By dividing the waves into propagation and attenuation matrices, he derived the reflection matrices under three boundary conditions, which established a theoretical foundation for wave approach. Adopting wave approach, Mei [11] analyzed the flexural vibration with added mass for Timoshenko beam, and the influence of lumped mass on natural frequency is also discussed in detail. Kang et al. [12] divided the real and imaginary parts of wave solutions of curved beam into four cases, and they calculated the natural frequencies by combining propagation, transmission and reflection matrices. Lee et al. [13, 14] considered the power flow when wave propagated in curved beam. Furthermore, they applied the Flugge theory to analyze the free vibration of a single curved beam, and their result was compared with Kang et al., which verified the correctness of the numerical results. Huang et al. [15] investigated the free vibration of planar rotating rings. The effect of cross section on natural frequency was also discussed. Bahrami and Teimourian [16] studied the free vibration of composite plates consisting of two layers, and they also made a comparison between classical results and wave propagation results. Tan and Kang [17] concentrated on the free vibration of rotating Timoshenko shaft with axial force and discussed the effect of continuous condition and cross section on natural frequencies. From the wave point of view, Bahrami and Teimourian [18] analyzed the free vibration of nanobeams for the first time. Ilkhani et al. [19] studied the free vibration of thin rectangular plate. It should be noted that the above scholars have done lots of studies for free vibration of structures by using wave approach, while few reports for the analysis of natural frequency for transverse vibration of rings can be found. In fact, it is well known that the nature of vibration is the propagation of waves. Analyzing free vibration in terms of wave propagation and attenuating can have a better understanding for us. Moreover, one advantage of using wave approach to analyze the free vibration is its conciseness of matrices that makes the natural frequencies be calculated easily. Wave approach is a strong tool for studying the behavior of wave transmission and reflection in waveguides, providing a practical engineering application such as filters.The emphasis of this paper is focused on the free vibration of rings. This paper is organized into five parts. Section1 is introduction. In Section 2, propagation, coordination, and reflection matrices are deduced in forms of matrix. In addition, the characteristic equation of natural frequency is obtained using classical method and wave approach. In Section 3, natural frequencies of rings are calculated by combining these matrices. Meanwhile, vibration transmissibility of rings propagating from outer to inner and from inner to outer is obtained. In Section 4, the influence of structural and material parameters on natural frequencies is discussed. Section 5 is the conclusion.


## 2. Theoretical Analysis

### 2.1. Classical Method for Free Vibration

#### 2.1.1. Solution of Transverse Vibration

Consider sandwich rings consisting of two different materials depicted in Figure1. Adhesive can be employed for connecting the rings. Same material is selected for the first and third layers. The other material is selected for the second layer. Radius of the first and third layers is r0 and rc. Radius of the intermediate layers is ra and rb. Radial span of the first and third layers is a1. Radial span of the second layer is a2. w is bending deflection. h is thickness. At the boundaries of r=ra and r=rb, positive–going and negative–going wave vectors are b1+, b1-, a2+, a2-, b2+, b2-, a3+, a3-. Also, considering another two boundaries at r=r0 and r=rc, positive–going and negative–going wave vectors are a1+, a1-, b3+, b3-. In cylindrical coordinates, the radius is assumed to be large enough compared to thickness which means that it satisfies the small deformation theory. Transverse solution is given by Wang [3]:(1)W=A1J0kr+B2Y0kr+C3I0kr+D4K0kr,where A1, B2, C3, D4 are constants which are determined by boundaries. J0(kr) and Y0(kr) are Bessel functions of first and second kinds, respectively. I0(kr) and K0(kr) are modified Bessel functions of first and second kinds. k=(4π2f2ρh/D)0.25 is wave number, and D=Eh3/12(1-σ2) is stiffness.Figure 1
Composite rings.


#### 2.1.2. Solution of Classical Bessel Method

With regard to rings subjected to bending excitation, expression of transverse displacement, rotational angle, shear force, and bending moment within the first and third layers can be written as(2)W1r=A11J0k1r+B11Y0k1r+C11I0k1r+D11K0k1r,∂W∂r1r=-k1A11J1k1r-k1B11Y1k1r+k1C11I1k1r-k1D11K1k1r,M1r=DA11k1rJ1k1r-σ1k1rJ1k1r-k12J0k1r+B11k1rY1k1r-σ1k1rY1k1r-k12Y0k1r+C11k12I0k1r-k1rI1k1r+σ1k1rI1k1r+D11k12K0k1r+k1rK1k1r-σ1k1rK1k1r,Q1r=DA11k13J1k1r+B11k13Y1k1r+C11k13I1k1r-D11k13K1k1r.Applying fixed boundary condition atr=r0 obtains(3)J0k1r0Y0k1r0I0k1r0K0k1r0Ψ11=0(4)-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0Ψ11=0,where Ψ11=A11B11C11D11T.Free boundary condition is selected atr=rc; then(5)Dk1rcJ1k1rc-σ1k1rcJ1k1rc-k12J0k1rck1rcY1k1rc-σ1k1rcY1k1rc-k12Y0k1rck12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rck12K0k1rc+k1rcK1k1rc-σ1k1rcK1k1rcΨ13=0Dk13J1k1rck13Y1k1rck13I1k1rc-k13K1k1rcΨ13=0,where Ψ13=A13B13C13D13T.In order to obtain the natural frequencies, substituting (A.9) into (5) and combining (3)-(4), it reduces to(6)J0k1r0Y0k1r0I0k1r0K0k1r0-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0J1Y1I1K1k13J1k1rc×T13k13Y1k1rc×T13k13I1k1rc×T13-k13K1k1rc×T13Ψ11=0,where the specific theoretical derivation of T13 in (6) is presented in the Appendix. And each element is defined as(7a)J1=k1rcJ1k1rc-σ1k1rcJ1k1rc-k12J0k1rc×T13,(7b)Y1=k1rcY1k1rc-σ1k1rcY1k1rc-k12Y0k1rc×T13,(7c)I1=k12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rc×T13,(7d)K1=k12K0k1rc+k1rcK1k1rc-σ1k1rcK1k1rc×T13.Therefore, (6) can be written as a 4×4 determinant:(8)J0k1r0Y0k1r0I0k1r0K0k1r0-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0J1Y1I1K1k13J1k1rc×T13k13Y1k1rc×T13k13I1k1rc×T13-k13K1k1rc×T13=0,where (8) is the characteristic equation of natural frequency. By searching the root, natural frequency of rings can be calculated with a fixed boundary at inner surface and a free boundary at outer surface.


#### 2.1.3. Solution of Classical Hankel Method

The solution is obtained in (1). However, it also can be expressed in a Hankel form:(9)W=A1+H02k1r+A1-H01k1r+B1+K0k1r+B1-I0k1r,where H0(1)(k1r) and H0(2)(k1r) are the Hankel functions of second and first kinds, respectively. They can be defined as(10)H01k1r=J0k1r+iY0k1rH02k1r=J0k1r-iY0k1r.Similarly, expression of parameters within the first and third layers can be written as(11)Wr=A1+J0k1r-iY0k1r+A1-J0k1r+iY0k1r+B1+K0k1r+B1-I0k1r(12)∂W∂r1r=A1+-k1J1k1r+ik1Y1k1r+A1--k1J1k1r-ik1Y1k1r-B1+k1K1k1r+B1-k1I1k1r,(13)M1r=DA1+k1rJ1k1r-k12J0k1r+ik12Y0k1r-ik1rY1k1r+σr-k1J1k1r+ik1Y1k1r+A1-k1rJ1k1r-k12J0k1r-ik12Y0k1r+ik1rY1k1r+σr-k1J1k1r-ik1Y1k1r+B1+k12K0k1r+k1rK1k1r-σrk1K1k1r+B1-k12I0k1r-k1rI1k1r+σ1k1rI1k1r(14)Q1r=DA1+-ik13Y1k1r+k13J1k1r-B1+k13K1k1r+A1-ik13Y1k1r+k13J1k1r+B1-k13I1k1r.Natural frequencies of transverse vibration can be calculated using classical Hankel method. Characteristic equation of natural frequency can be deuced like the process of (3)–(8). In order to avoid repeating, herein, it is ignored.


### 2.2. Wave Approach for Free Vibration

In this section, the solution is presented in terms of cylindrical waves for this ring. Meanwhile, positive–going propagation, negative–going propagation, coordination, and reflection matrices are also deduced. By combining these matrices, natural frequencies are calculated using wave approach.


#### 2.2.1. Propagation Matrices

Wave propagates along the positive–going and negative–going directions when propagating within structures, as is shown in Figure1. Waves will not propagate at the boundaries but only can be reflected. Moreover, parameters are continuous for the connection. In recent years, many researchers describe the waves in the matrix forms [8–17].By considering (11), positive–going waves can be described as(15a)a1+=A1+J0k1r0-iY0k1r0B1+K0k1r0,(15b)b1+=A1+J0k1ra-iY0k1raB1+K0k1ra,(16a)a2+=A1+J0k2ra-iY0k2raB1+K0k2ra,(16b)b2+=A1+J0k2rb-iY0k2rbB1+K0k2rb,(17a)a3+=A1+J0k1rc-iY0k1rcB1+K0k1rc,(17b)b3+=A1+J0k1rc-iY0k1rcB1+K0k1rc.These wave vectors are related by(18a)b1+=f1+ra-r0a1+(18b)b2+=f2+rb-raa2+(18c)b3+=f3+rc-rba3+.Substituting matrices (15a), (15b), (16a), (16b), (17a), (17b) into (18a)–(18c), positive–going propagation matrices are obtained as(19a)f1+=J0k1ra-iY0k1raJ0k1r0-iY0k1r000K0k1raK0k1r0(19b)f2+=J0k1rb-iY0k1rbJ0k1ra-iY0k1ra00K0k1rbK0k1ra(19c)f3+=J0k1rc-iY0k1rcJ0k1rb-iY0k1rb00K0k1rcK0k1rb.Similarly, negative–going waves can be rewritten as(20a)a1-=A1-J0k1r0+iY0k1r0B1-I0k1r0(20b)b1-=A1-J0k1ra+iY0k1raB1-I0k1ra(21a)a2-=A1-J0k2ra+iY0k2raB1-I0k2ra(21b)b2-=A1-J0k2rb+iY0k2rbB1-I0k2rb(22a)a3-=A1-J0k1rc+iY0k1rcB1-I0k1rc(22b)b3-=A1-J0k1rc+iY0k1rcB1-I0k1rc.These wave vectors are related by(23a)a1-=f1-r0-rab1-(23b)a2-=f2-ra-rbb2-(23c)a3-=f3-rb-rcb3-.Substituting matrices (20a), (20b), (21a), (21b), (22a), (22b) into (23a)–(23c), negative–going propagation matrices are obtained:(24a)f1-=J0k1r0+iY0k1r0J0k1ra+iY0k1ra00I0k1r0I0k1ra(24b)f2-=J0k1ra+iY0k1raJ0k1rb+iY0k1rb00I0k1raI0k1rb(24c)f3-=J0k1rb+iY0k1rbJ0k1rc+iY0k1rc00I0k1rbI0k1rc.


#### 2.2.2. Reflection Matrices

Keeping the boundary condition ofr=r0 fixed, thus, displacements and rotational angle are taken as(25)A1+J0k1r0-iY0k1r0+A1-J0k1r0+iY0k1r0+B1+K0k1r0+B1-I0k1r0=0A1+-k1J1k1r0+ik1Y1k1r0+A1--k1J1k1r0-ik1Y1k1r0-B1+k1K1k1r0+B1-k1I1k1r0=0.The relationship of incident wavea1+ and reflected wave a1- is related by(26)a1+=R0a1-.Substituting (25) into (26), the reflection matrices can be obtained as follows:(27)R0=-J0k1r0-iY0k1r0J0k1r0-iY0k1r0K0k1r0K0k1r0-k1J1k1r0+ik1Y1k1r0J0k1r0-iY0k1r0-k1K1k1r0K0k1r0-1J0k1r0+iY0k1r0J0k1r0+iY0k1r0I0k1r0I0k1r0-k1J1k1r0-ik1Y1k1r0J0k1r0+iY0k1r0k1I1k1r0I0k1r0.Keeping the boundary condition ofr=rc free gives(28)DA1+k1rcJ1k1rc-k12J0k1rc+ik12Y0k1rc-ik1rcY1k1rc+σrc-k1J1k1rc+ik1Y1k1rc+A1-k1rcJ1k1rc-k12J0k1rc-ik12Y0k1rc+ik1rcY1k1rc+σrc-k1J1k1rc-ik1Y1k1rc+B1+k12K0k1rc+k1rcK1k1rc-σrk1K1k1rc+B1-k12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rc=0.DA1+-ik13Y1k1rc+k13J1k1rc-B1+k13K1k1rc+A1-ik13Y1k1rc+k13J1k1rc+B1-k13I1k1rc=0The relationship of incident waveb3+ and reflected wave b3- is(29)b3-=R3b3+.Substituting (28) into (29), reflection matrices are calculated as(30)R3=-k1/rcJ1k1rc-k12J0k1rc-ik12Y0k1rc+ik1/rcY1k1rc+σ/rc-k1J1k1rc-ik1Y1k1rcJ0k1rc+iY0k1rck12I0k1rc-k1/rcI1k1rc+σ1k1/rcI1k1rcI0k1rcik13Y1k1rc+k13J1k1rcJ0k1rc+iY0k1rck13I1k1rcI0k1rc-1×k1/rcJ1k1rc-k12J0k1rc+ik12Y0k1rc-ik1/rcY1k1rc+σ/rc-k1J1k1rc+ik1Y1k1rcJ0k1rc-iY0k1rck12K0k1rc+k1/rcK1k1rc-σ/rk1K1k1rcK0k1rc-ik13Y1k1rc+k13J1k1rcJ0k1rc-iY0k1rc-k13K1k1rcK0k1rc.


#### 2.2.3. Coordination Matrices

By imposing the geometric continuity atr=ra yields(31)J0k1ra-iY0k1raK0k1ra-k1J1k1ra+ik1Y1k1ra-k1K1k1rab1++J0k1ra+iY0k1raI0k1ra-k1J1k1ra-ik1Y1k1rak1I1k1rab1-=J0k2ra-iY0k2raK0k2ra-k2J1k2ra+ik2Y1k2ra-k2K1k2raa2++J0k2ra+iY0k2raI0k2ra-k2J1k2ra-ik2Y1k2rak2I1k2raa2-k1raJ1k1ra-k12J0k1ra+ik12Y0k1ra-ik1raY1k1ra+σ1ra-k1J1k1ra+ik1Y1k1rak12K0k1ra+k1raK1k1ra-σ1rak1K1k1ra-ik13Y1k1ra+k13J1k1ra-k13K1k1rab1++k1raJ1k1ra-k12J0k1ra-ik12Y0k1ra+ik1raY1k1ra+σ1ra-k1J1k1ra-ik1Y1k1rak12I0k1ra-k1raI1k1ra+σ1k1raI1k1raik13Y1k1ra+k13J1k1rak13I1k1rab1-=k2raJ1k2ra-k22J0k2ra+ik22Y0k2ra-ik2raY1k2ra+σ2ra-k2J1k2ra+ik2Y1k2rak22K0k2ra+k2raK1k2ra-σ2rak2K1k2ra-ik23Y1k2ra+k23J1k2ra-k23K1k2raa2++k2raJ1k2ra-k22J0k2ra-ik22Y0k2ra+ik2raY1k2ra+σ2ra-k2J1k2ra-ik2Y1k2rak22I0k2ra-k2raI1k2ra+σ2k2raI1k2raik23Y1k2ra+k23J1k2rak23I1k2raa2-.Equations (31) can be rewritten as(32)Ra1+b1++Ra1-b1-=Ta2+a2++Ta2-a2-Ra3+b1++Ra3-b1-=Ta4+a2++Ta4-a2-.According to the continuity atr=rb, shear force and bending moment are required that(33)J0k2rb-iY0k2rbK0k2rb-k2J1k2rb+ik2Y1k2rb-k2K1k2rbb2++J0k2rb+iY0k2rbI0k2rb-k2J1k2rb-ik2Y1k2rbk2I1k2rbb2-=J0k1r1-iY0k1rbK0k1rb-k1J1k1rb+ik1Y1k1rb-k1K1k1rba3++J0k1rb+iY0k1rbI0k1rb-k1J1k1rb-ik1Y1k1rbk1I1k1rba3-k2rbJ1k2rb-k22J0k2rb+ik22Y0k2rb-ik2r2Y1k2rb+σ2rb-k2J1k2rb+ik2Y1k2rbk22K0k2rb+k2rbK1k2rb-σ2rbk2K1k2rb-ik23Y1k2rb+k23J1k2rb-k23K1k2rbb2++k2rbJ1k2rb-k22J0k2rb-ik22Y0k2rb+ik2rbY1k2rb+σ2rb-k2J1k2rb-ik2Y1k2rbk22I0k2rb-k2rbI1k2rb+σ2k2rbI1k2rbik23Y1k2rb+k23J1k2rbk23I1k2rbb2-=k1rbJ1k2rb-k12J0k1rb+ik12Y0k1rb-ik1rbY1k1rb+σ1rb-k1J1k1rb+ik1Y1k1rbk12K0k1rb+k1rbK1k1rb-σ1rbk1K1k1rb-ik13Y1k1ra+k13J1k1rb-k13K1k1rba3++k1rbJ1k1rb-k12J0k1rb-ik12Y0k1rb+ik1rbY1k1rb+σ1rb-k1J1k1rb-ik1Y1k1rbk12I0k1rb-k1rbI1k1rb+σ1k1rbI1k1rbik13Y1k1rb+k13J1k1rbk13I1k1rba3-Equations (33) can be written as(34)Rb1+b1++Rb1-b1-=Tb2+a2++Tb2-a2-Rb3+b1++Rb3-b1-=Tb4+a2++Tb4-a2-.


#### 2.2.4. Characteristic Equation of Natural Frequency

Combining propagation matrices, reflection matrices, and coordination matrices derived in Section2.2, natural frequencies of composite rings can be calculated smoothly. Figure 1 presents a clear description of incident and reflected waves. Thus, the wave matrices described by (18a)–(18c), (23a)–(23c), (26), (29), (32), and (34) are assembled as(35)
b
1
+
=
f
1
+
r
a
-
r
0
a
1
+
a
1
-
=
f
1
-
r
0
-
r
a
b
1
-
b
3
+
=
f
3
+
r
c
-
r
b
a
3
+
a
1
+
=
R
0
a
1
-
b
2
+
=
f
2
+
r
b
-
r
a
a
2
+
a
2
-
=
f
2
-
r
a
-
r
b
b
2
-
a
3
-
=
f
3
-
r
b
-
r
c
b
3
-
b
3
-
=
R
3
b
3
+
R
a
1
+
b
1
+
+
R
a
1
-
b
1
-
=
T
a
2
+
a
2
+
+
T
a
2
-
a
2
-
R
b
1
+
b
2
+
+
R
b
1
-
b
2
-
=
T
b
2
+
a
3
+
+
T
b
2
-
a
3
-
R
a
3
+
b
1
+
+
R
a
3
-
b
1
-
=
T
a
4
+
a
2
+
+
T
a
4
-
a
2
-
R
b
3
+
b
2
+
+
R
b
3
-
b
2
-
=
T
b
4
+
a
3
+
+
T
b
4
-
a
3
-
.In order to obtain the natural frequency, (35) can be rewritten in a matrix form(36)-I2×2R000000000000-I2×20f1-0000000000000-I2×20f2-0000000000000-I2×20f3-f1+0-I2×20000000000000f2+0-I2×20000000000000f3+0-I2×2000Ra1+Ra1-Ta2+Ta2-00000000Ra3+Ra3-Ta4+Ta4-000000000000Rb1+Rb1-Tb2+Tb2-00000000Rb3+Rb3-Tb4+Tb4-000000000000R3-I2×2a1+a1-b1+b1-a2+a2-b2+b2-a3+a3-b3+b3-=Ffa1+a1-b1+b1-a2+a2-b2+b2-a3+a3-b3+b3-=0.F ( f ) is a matrix of 12×12. If (36) has solution, it requires that(37)Ff=0.By solving the roots of characteristic equation (37), one can calculate the real and imaginary parts. It is important here to note that the natural frequencies can be found by searching the intersections in x-axis.


## 2.1. Classical Method for Free Vibration

### 2.1.1. Solution of Transverse Vibration

Consider sandwich rings consisting of two different materials depicted in Figure1. Adhesive can be employed for connecting the rings. Same material is selected for the first and third layers. The other material is selected for the second layer. Radius of the first and third layers is r0 and rc. Radius of the intermediate layers is ra and rb. Radial span of the first and third layers is a1. Radial span of the second layer is a2. w is bending deflection. h is thickness. At the boundaries of r=ra and r=rb, positive–going and negative–going wave vectors are b1+, b1-, a2+, a2-, b2+, b2-, a3+, a3-. Also, considering another two boundaries at r=r0 and r=rc, positive–going and negative–going wave vectors are a1+, a1-, b3+, b3-. In cylindrical coordinates, the radius is assumed to be large enough compared to thickness which means that it satisfies the small deformation theory. Transverse solution is given by Wang [3]:(1)W=A1J0kr+B2Y0kr+C3I0kr+D4K0kr,where A1, B2, C3, D4 are constants which are determined by boundaries. J0(kr) and Y0(kr) are Bessel functions of first and second kinds, respectively. I0(kr) and K0(kr) are modified Bessel functions of first and second kinds. k=(4π2f2ρh/D)0.25 is wave number, and D=Eh3/12(1-σ2) is stiffness.Figure 1
Composite rings.


### 2.1.2. Solution of Classical Bessel Method

With regard to rings subjected to bending excitation, expression of transverse displacement, rotational angle, shear force, and bending moment within the first and third layers can be written as(2)W1r=A11J0k1r+B11Y0k1r+C11I0k1r+D11K0k1r,∂W∂r1r=-k1A11J1k1r-k1B11Y1k1r+k1C11I1k1r-k1D11K1k1r,M1r=DA11k1rJ1k1r-σ1k1rJ1k1r-k12J0k1r+B11k1rY1k1r-σ1k1rY1k1r-k12Y0k1r+C11k12I0k1r-k1rI1k1r+σ1k1rI1k1r+D11k12K0k1r+k1rK1k1r-σ1k1rK1k1r,Q1r=DA11k13J1k1r+B11k13Y1k1r+C11k13I1k1r-D11k13K1k1r.Applying fixed boundary condition atr=r0 obtains(3)J0k1r0Y0k1r0I0k1r0K0k1r0Ψ11=0(4)-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0Ψ11=0,where Ψ11=A11B11C11D11T.Free boundary condition is selected atr=rc; then(5)Dk1rcJ1k1rc-σ1k1rcJ1k1rc-k12J0k1rck1rcY1k1rc-σ1k1rcY1k1rc-k12Y0k1rck12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rck12K0k1rc+k1rcK1k1rc-σ1k1rcK1k1rcΨ13=0Dk13J1k1rck13Y1k1rck13I1k1rc-k13K1k1rcΨ13=0,where Ψ13=A13B13C13D13T.In order to obtain the natural frequencies, substituting (A.9) into (5) and combining (3)-(4), it reduces to(6)J0k1r0Y0k1r0I0k1r0K0k1r0-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0J1Y1I1K1k13J1k1rc×T13k13Y1k1rc×T13k13I1k1rc×T13-k13K1k1rc×T13Ψ11=0,where the specific theoretical derivation of T13 in (6) is presented in the Appendix. And each element is defined as(7a)J1=k1rcJ1k1rc-σ1k1rcJ1k1rc-k12J0k1rc×T13,(7b)Y1=k1rcY1k1rc-σ1k1rcY1k1rc-k12Y0k1rc×T13,(7c)I1=k12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rc×T13,(7d)K1=k12K0k1rc+k1rcK1k1rc-σ1k1rcK1k1rc×T13.Therefore, (6) can be written as a 4×4 determinant:(8)J0k1r0Y0k1r0I0k1r0K0k1r0-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0J1Y1I1K1k13J1k1rc×T13k13Y1k1rc×T13k13I1k1rc×T13-k13K1k1rc×T13=0,where (8) is the characteristic equation of natural frequency. By searching the root, natural frequency of rings can be calculated with a fixed boundary at inner surface and a free boundary at outer surface.


### 2.1.3. Solution of Classical Hankel Method

The solution is obtained in (1). However, it also can be expressed in a Hankel form:(9)W=A1+H02k1r+A1-H01k1r+B1+K0k1r+B1-I0k1r,where H0(1)(k1r) and H0(2)(k1r) are the Hankel functions of second and first kinds, respectively. They can be defined as(10)H01k1r=J0k1r+iY0k1rH02k1r=J0k1r-iY0k1r.Similarly, expression of parameters within the first and third layers can be written as(11)Wr=A1+J0k1r-iY0k1r+A1-J0k1r+iY0k1r+B1+K0k1r+B1-I0k1r(12)∂W∂r1r=A1+-k1J1k1r+ik1Y1k1r+A1--k1J1k1r-ik1Y1k1r-B1+k1K1k1r+B1-k1I1k1r,(13)M1r=DA1+k1rJ1k1r-k12J0k1r+ik12Y0k1r-ik1rY1k1r+σr-k1J1k1r+ik1Y1k1r+A1-k1rJ1k1r-k12J0k1r-ik12Y0k1r+ik1rY1k1r+σr-k1J1k1r-ik1Y1k1r+B1+k12K0k1r+k1rK1k1r-σrk1K1k1r+B1-k12I0k1r-k1rI1k1r+σ1k1rI1k1r(14)Q1r=DA1+-ik13Y1k1r+k13J1k1r-B1+k13K1k1r+A1-ik13Y1k1r+k13J1k1r+B1-k13I1k1r.Natural frequencies of transverse vibration can be calculated using classical Hankel method. Characteristic equation of natural frequency can be deuced like the process of (3)–(8). In order to avoid repeating, herein, it is ignored.


## 2.1.1. Solution of Transverse Vibration

Consider sandwich rings consisting of two different materials depicted in Figure1. Adhesive can be employed for connecting the rings. Same material is selected for the first and third layers. The other material is selected for the second layer. Radius of the first and third layers is r0 and rc. Radius of the intermediate layers is ra and rb. Radial span of the first and third layers is a1. Radial span of the second layer is a2. w is bending deflection. h is thickness. At the boundaries of r=ra and r=rb, positive–going and negative–going wave vectors are b1+, b1-, a2+, a2-, b2+, b2-, a3+, a3-. Also, considering another two boundaries at r=r0 and r=rc, positive–going and negative–going wave vectors are a1+, a1-, b3+, b3-. In cylindrical coordinates, the radius is assumed to be large enough compared to thickness which means that it satisfies the small deformation theory. Transverse solution is given by Wang [3]:(1)W=A1J0kr+B2Y0kr+C3I0kr+D4K0kr,where A1, B2, C3, D4 are constants which are determined by boundaries. J0(kr) and Y0(kr) are Bessel functions of first and second kinds, respectively. I0(kr) and K0(kr) are modified Bessel functions of first and second kinds. k=(4π2f2ρh/D)0.25 is wave number, and D=Eh3/12(1-σ2) is stiffness.Figure 1
Composite rings.


## 2.1.2. Solution of Classical Bessel Method

With regard to rings subjected to bending excitation, expression of transverse displacement, rotational angle, shear force, and bending moment within the first and third layers can be written as(2)W1r=A11J0k1r+B11Y0k1r+C11I0k1r+D11K0k1r,∂W∂r1r=-k1A11J1k1r-k1B11Y1k1r+k1C11I1k1r-k1D11K1k1r,M1r=DA11k1rJ1k1r-σ1k1rJ1k1r-k12J0k1r+B11k1rY1k1r-σ1k1rY1k1r-k12Y0k1r+C11k12I0k1r-k1rI1k1r+σ1k1rI1k1r+D11k12K0k1r+k1rK1k1r-σ1k1rK1k1r,Q1r=DA11k13J1k1r+B11k13Y1k1r+C11k13I1k1r-D11k13K1k1r.Applying fixed boundary condition atr=r0 obtains(3)J0k1r0Y0k1r0I0k1r0K0k1r0Ψ11=0(4)-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0Ψ11=0,where Ψ11=A11B11C11D11T.Free boundary condition is selected atr=rc; then(5)Dk1rcJ1k1rc-σ1k1rcJ1k1rc-k12J0k1rck1rcY1k1rc-σ1k1rcY1k1rc-k12Y0k1rck12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rck12K0k1rc+k1rcK1k1rc-σ1k1rcK1k1rcΨ13=0Dk13J1k1rck13Y1k1rck13I1k1rc-k13K1k1rcΨ13=0,where Ψ13=A13B13C13D13T.In order to obtain the natural frequencies, substituting (A.9) into (5) and combining (3)-(4), it reduces to(6)J0k1r0Y0k1r0I0k1r0K0k1r0-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0J1Y1I1K1k13J1k1rc×T13k13Y1k1rc×T13k13I1k1rc×T13-k13K1k1rc×T13Ψ11=0,where the specific theoretical derivation of T13 in (6) is presented in the Appendix. And each element is defined as(7a)J1=k1rcJ1k1rc-σ1k1rcJ1k1rc-k12J0k1rc×T13,(7b)Y1=k1rcY1k1rc-σ1k1rcY1k1rc-k12Y0k1rc×T13,(7c)I1=k12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rc×T13,(7d)K1=k12K0k1rc+k1rcK1k1rc-σ1k1rcK1k1rc×T13.Therefore, (6) can be written as a 4×4 determinant:(8)J0k1r0Y0k1r0I0k1r0K0k1r0-k1J1k1r0-k1Y1k1r0k1I1k1r0-k1K1k1r0J1Y1I1K1k13J1k1rc×T13k13Y1k1rc×T13k13I1k1rc×T13-k13K1k1rc×T13=0,where (8) is the characteristic equation of natural frequency. By searching the root, natural frequency of rings can be calculated with a fixed boundary at inner surface and a free boundary at outer surface.


## 2.1.3. Solution of Classical Hankel Method

The solution is obtained in (1). However, it also can be expressed in a Hankel form:(9)W=A1+H02k1r+A1-H01k1r+B1+K0k1r+B1-I0k1r,where H0(1)(k1r) and H0(2)(k1r) are the Hankel functions of second and first kinds, respectively. They can be defined as(10)H01k1r=J0k1r+iY0k1rH02k1r=J0k1r-iY0k1r.Similarly, expression of parameters within the first and third layers can be written as(11)Wr=A1+J0k1r-iY0k1r+A1-J0k1r+iY0k1r+B1+K0k1r+B1-I0k1r(12)∂W∂r1r=A1+-k1J1k1r+ik1Y1k1r+A1--k1J1k1r-ik1Y1k1r-B1+k1K1k1r+B1-k1I1k1r,(13)M1r=DA1+k1rJ1k1r-k12J0k1r+ik12Y0k1r-ik1rY1k1r+σr-k1J1k1r+ik1Y1k1r+A1-k1rJ1k1r-k12J0k1r-ik12Y0k1r+ik1rY1k1r+σr-k1J1k1r-ik1Y1k1r+B1+k12K0k1r+k1rK1k1r-σrk1K1k1r+B1-k12I0k1r-k1rI1k1r+σ1k1rI1k1r(14)Q1r=DA1+-ik13Y1k1r+k13J1k1r-B1+k13K1k1r+A1-ik13Y1k1r+k13J1k1r+B1-k13I1k1r.Natural frequencies of transverse vibration can be calculated using classical Hankel method. Characteristic equation of natural frequency can be deuced like the process of (3)–(8). In order to avoid repeating, herein, it is ignored.


## 2.2. Wave Approach for Free Vibration

In this section, the solution is presented in terms of cylindrical waves for this ring. Meanwhile, positive–going propagation, negative–going propagation, coordination, and reflection matrices are also deduced. By combining these matrices, natural frequencies are calculated using wave approach.


### 2.2.1. Propagation Matrices

Wave propagates along the positive–going and negative–going directions when propagating within structures, as is shown in Figure1. Waves will not propagate at the boundaries but only can be reflected. Moreover, parameters are continuous for the connection. In recent years, many researchers describe the waves in the matrix forms [8–17].By considering (11), positive–going waves can be described as(15a)a1+=A1+J0k1r0-iY0k1r0B1+K0k1r0,(15b)b1+=A1+J0k1ra-iY0k1raB1+K0k1ra,(16a)a2+=A1+J0k2ra-iY0k2raB1+K0k2ra,(16b)b2+=A1+J0k2rb-iY0k2rbB1+K0k2rb,(17a)a3+=A1+J0k1rc-iY0k1rcB1+K0k1rc,(17b)b3+=A1+J0k1rc-iY0k1rcB1+K0k1rc.These wave vectors are related by(18a)b1+=f1+ra-r0a1+(18b)b2+=f2+rb-raa2+(18c)b3+=f3+rc-rba3+.Substituting matrices (15a), (15b), (16a), (16b), (17a), (17b) into (18a)–(18c), positive–going propagation matrices are obtained as(19a)f1+=J0k1ra-iY0k1raJ0k1r0-iY0k1r000K0k1raK0k1r0(19b)f2+=J0k1rb-iY0k1rbJ0k1ra-iY0k1ra00K0k1rbK0k1ra(19c)f3+=J0k1rc-iY0k1rcJ0k1rb-iY0k1rb00K0k1rcK0k1rb.Similarly, negative–going waves can be rewritten as(20a)a1-=A1-J0k1r0+iY0k1r0B1-I0k1r0(20b)b1-=A1-J0k1ra+iY0k1raB1-I0k1ra(21a)a2-=A1-J0k2ra+iY0k2raB1-I0k2ra(21b)b2-=A1-J0k2rb+iY0k2rbB1-I0k2rb(22a)a3-=A1-J0k1rc+iY0k1rcB1-I0k1rc(22b)b3-=A1-J0k1rc+iY0k1rcB1-I0k1rc.These wave vectors are related by(23a)a1-=f1-r0-rab1-(23b)a2-=f2-ra-rbb2-(23c)a3-=f3-rb-rcb3-.Substituting matrices (20a), (20b), (21a), (21b), (22a), (22b) into (23a)–(23c), negative–going propagation matrices are obtained:(24a)f1-=J0k1r0+iY0k1r0J0k1ra+iY0k1ra00I0k1r0I0k1ra(24b)f2-=J0k1ra+iY0k1raJ0k1rb+iY0k1rb00I0k1raI0k1rb(24c)f3-=J0k1rb+iY0k1rbJ0k1rc+iY0k1rc00I0k1rbI0k1rc.


### 2.2.2. Reflection Matrices

Keeping the boundary condition ofr=r0 fixed, thus, displacements and rotational angle are taken as(25)A1+J0k1r0-iY0k1r0+A1-J0k1r0+iY0k1r0+B1+K0k1r0+B1-I0k1r0=0A1+-k1J1k1r0+ik1Y1k1r0+A1--k1J1k1r0-ik1Y1k1r0-B1+k1K1k1r0+B1-k1I1k1r0=0.The relationship of incident wavea1+ and reflected wave a1- is related by(26)a1+=R0a1-.Substituting (25) into (26), the reflection matrices can be obtained as follows:(27)R0=-J0k1r0-iY0k1r0J0k1r0-iY0k1r0K0k1r0K0k1r0-k1J1k1r0+ik1Y1k1r0J0k1r0-iY0k1r0-k1K1k1r0K0k1r0-1J0k1r0+iY0k1r0J0k1r0+iY0k1r0I0k1r0I0k1r0-k1J1k1r0-ik1Y1k1r0J0k1r0+iY0k1r0k1I1k1r0I0k1r0.Keeping the boundary condition ofr=rc free gives(28)DA1+k1rcJ1k1rc-k12J0k1rc+ik12Y0k1rc-ik1rcY1k1rc+σrc-k1J1k1rc+ik1Y1k1rc+A1-k1rcJ1k1rc-k12J0k1rc-ik12Y0k1rc+ik1rcY1k1rc+σrc-k1J1k1rc-ik1Y1k1rc+B1+k12K0k1rc+k1rcK1k1rc-σrk1K1k1rc+B1-k12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rc=0.DA1+-ik13Y1k1rc+k13J1k1rc-B1+k13K1k1rc+A1-ik13Y1k1rc+k13J1k1rc+B1-k13I1k1rc=0The relationship of incident waveb3+ and reflected wave b3- is(29)b3-=R3b3+.Substituting (28) into (29), reflection matrices are calculated as(30)R3=-k1/rcJ1k1rc-k12J0k1rc-ik12Y0k1rc+ik1/rcY1k1rc+σ/rc-k1J1k1rc-ik1Y1k1rcJ0k1rc+iY0k1rck12I0k1rc-k1/rcI1k1rc+σ1k1/rcI1k1rcI0k1rcik13Y1k1rc+k13J1k1rcJ0k1rc+iY0k1rck13I1k1rcI0k1rc-1×k1/rcJ1k1rc-k12J0k1rc+ik12Y0k1rc-ik1/rcY1k1rc+σ/rc-k1J1k1rc+ik1Y1k1rcJ0k1rc-iY0k1rck12K0k1rc+k1/rcK1k1rc-σ/rk1K1k1rcK0k1rc-ik13Y1k1rc+k13J1k1rcJ0k1rc-iY0k1rc-k13K1k1rcK0k1rc.


### 2.2.3. Coordination Matrices

By imposing the geometric continuity atr=ra yields(31)J0k1ra-iY0k1raK0k1ra-k1J1k1ra+ik1Y1k1ra-k1K1k1rab1++J0k1ra+iY0k1raI0k1ra-k1J1k1ra-ik1Y1k1rak1I1k1rab1-=J0k2ra-iY0k2raK0k2ra-k2J1k2ra+ik2Y1k2ra-k2K1k2raa2++J0k2ra+iY0k2raI0k2ra-k2J1k2ra-ik2Y1k2rak2I1k2raa2-k1raJ1k1ra-k12J0k1ra+ik12Y0k1ra-ik1raY1k1ra+σ1ra-k1J1k1ra+ik1Y1k1rak12K0k1ra+k1raK1k1ra-σ1rak1K1k1ra-ik13Y1k1ra+k13J1k1ra-k13K1k1rab1++k1raJ1k1ra-k12J0k1ra-ik12Y0k1ra+ik1raY1k1ra+σ1ra-k1J1k1ra-ik1Y1k1rak12I0k1ra-k1raI1k1ra+σ1k1raI1k1raik13Y1k1ra+k13J1k1rak13I1k1rab1-=k2raJ1k2ra-k22J0k2ra+ik22Y0k2ra-ik2raY1k2ra+σ2ra-k2J1k2ra+ik2Y1k2rak22K0k2ra+k2raK1k2ra-σ2rak2K1k2ra-ik23Y1k2ra+k23J1k2ra-k23K1k2raa2++k2raJ1k2ra-k22J0k2ra-ik22Y0k2ra+ik2raY1k2ra+σ2ra-k2J1k2ra-ik2Y1k2rak22I0k2ra-k2raI1k2ra+σ2k2raI1k2raik23Y1k2ra+k23J1k2rak23I1k2raa2-.Equations (31) can be rewritten as(32)Ra1+b1++Ra1-b1-=Ta2+a2++Ta2-a2-Ra3+b1++Ra3-b1-=Ta4+a2++Ta4-a2-.According to the continuity atr=rb, shear force and bending moment are required that(33)J0k2rb-iY0k2rbK0k2rb-k2J1k2rb+ik2Y1k2rb-k2K1k2rbb2++J0k2rb+iY0k2rbI0k2rb-k2J1k2rb-ik2Y1k2rbk2I1k2rbb2-=J0k1r1-iY0k1rbK0k1rb-k1J1k1rb+ik1Y1k1rb-k1K1k1rba3++J0k1rb+iY0k1rbI0k1rb-k1J1k1rb-ik1Y1k1rbk1I1k1rba3-k2rbJ1k2rb-k22J0k2rb+ik22Y0k2rb-ik2r2Y1k2rb+σ2rb-k2J1k2rb+ik2Y1k2rbk22K0k2rb+k2rbK1k2rb-σ2rbk2K1k2rb-ik23Y1k2rb+k23J1k2rb-k23K1k2rbb2++k2rbJ1k2rb-k22J0k2rb-ik22Y0k2rb+ik2rbY1k2rb+σ2rb-k2J1k2rb-ik2Y1k2rbk22I0k2rb-k2rbI1k2rb+σ2k2rbI1k2rbik23Y1k2rb+k23J1k2rbk23I1k2rbb2-=k1rbJ1k2rb-k12J0k1rb+ik12Y0k1rb-ik1rbY1k1rb+σ1rb-k1J1k1rb+ik1Y1k1rbk12K0k1rb+k1rbK1k1rb-σ1rbk1K1k1rb-ik13Y1k1ra+k13J1k1rb-k13K1k1rba3++k1rbJ1k1rb-k12J0k1rb-ik12Y0k1rb+ik1rbY1k1rb+σ1rb-k1J1k1rb-ik1Y1k1rbk12I0k1rb-k1rbI1k1rb+σ1k1rbI1k1rbik13Y1k1rb+k13J1k1rbk13I1k1rba3-Equations (33) can be written as(34)Rb1+b1++Rb1-b1-=Tb2+a2++Tb2-a2-Rb3+b1++Rb3-b1-=Tb4+a2++Tb4-a2-.


### 2.2.4. Characteristic Equation of Natural Frequency

Combining propagation matrices, reflection matrices, and coordination matrices derived in Section2.2, natural frequencies of composite rings can be calculated smoothly. Figure 1 presents a clear description of incident and reflected waves. Thus, the wave matrices described by (18a)–(18c), (23a)–(23c), (26), (29), (32), and (34) are assembled as(35)
b
1
+
=
f
1
+
r
a
-
r
0
a
1
+
a
1
-
=
f
1
-
r
0
-
r
a
b
1
-
b
3
+
=
f
3
+
r
c
-
r
b
a
3
+
a
1
+
=
R
0
a
1
-
b
2
+
=
f
2
+
r
b
-
r
a
a
2
+
a
2
-
=
f
2
-
r
a
-
r
b
b
2
-
a
3
-
=
f
3
-
r
b
-
r
c
b
3
-
b
3
-
=
R
3
b
3
+
R
a
1
+
b
1
+
+
R
a
1
-
b
1
-
=
T
a
2
+
a
2
+
+
T
a
2
-
a
2
-
R
b
1
+
b
2
+
+
R
b
1
-
b
2
-
=
T
b
2
+
a
3
+
+
T
b
2
-
a
3
-
R
a
3
+
b
1
+
+
R
a
3
-
b
1
-
=
T
a
4
+
a
2
+
+
T
a
4
-
a
2
-
R
b
3
+
b
2
+
+
R
b
3
-
b
2
-
=
T
b
4
+
a
3
+
+
T
b
4
-
a
3
-
.In order to obtain the natural frequency, (35) can be rewritten in a matrix form(36)-I2×2R000000000000-I2×20f1-0000000000000-I2×20f2-0000000000000-I2×20f3-f1+0-I2×20000000000000f2+0-I2×20000000000000f3+0-I2×2000Ra1+Ra1-Ta2+Ta2-00000000Ra3+Ra3-Ta4+Ta4-000000000000Rb1+Rb1-Tb2+Tb2-00000000Rb3+Rb3-Tb4+Tb4-000000000000R3-I2×2a1+a1-b1+b1-a2+a2-b2+b2-a3+a3-b3+b3-=Ffa1+a1-b1+b1-a2+a2-b2+b2-a3+a3-b3+b3-=0.F ( f ) is a matrix of 12×12. If (36) has solution, it requires that(37)Ff=0.By solving the roots of characteristic equation (37), one can calculate the real and imaginary parts. It is important here to note that the natural frequencies can be found by searching the intersections in x-axis.


## 2.2.1. Propagation Matrices

Wave propagates along the positive–going and negative–going directions when propagating within structures, as is shown in Figure1. Waves will not propagate at the boundaries but only can be reflected. Moreover, parameters are continuous for the connection. In recent years, many researchers describe the waves in the matrix forms [8–17].By considering (11), positive–going waves can be described as(15a)a1+=A1+J0k1r0-iY0k1r0B1+K0k1r0,(15b)b1+=A1+J0k1ra-iY0k1raB1+K0k1ra,(16a)a2+=A1+J0k2ra-iY0k2raB1+K0k2ra,(16b)b2+=A1+J0k2rb-iY0k2rbB1+K0k2rb,(17a)a3+=A1+J0k1rc-iY0k1rcB1+K0k1rc,(17b)b3+=A1+J0k1rc-iY0k1rcB1+K0k1rc.These wave vectors are related by(18a)b1+=f1+ra-r0a1+(18b)b2+=f2+rb-raa2+(18c)b3+=f3+rc-rba3+.Substituting matrices (15a), (15b), (16a), (16b), (17a), (17b) into (18a)–(18c), positive–going propagation matrices are obtained as(19a)f1+=J0k1ra-iY0k1raJ0k1r0-iY0k1r000K0k1raK0k1r0(19b)f2+=J0k1rb-iY0k1rbJ0k1ra-iY0k1ra00K0k1rbK0k1ra(19c)f3+=J0k1rc-iY0k1rcJ0k1rb-iY0k1rb00K0k1rcK0k1rb.Similarly, negative–going waves can be rewritten as(20a)a1-=A1-J0k1r0+iY0k1r0B1-I0k1r0(20b)b1-=A1-J0k1ra+iY0k1raB1-I0k1ra(21a)a2-=A1-J0k2ra+iY0k2raB1-I0k2ra(21b)b2-=A1-J0k2rb+iY0k2rbB1-I0k2rb(22a)a3-=A1-J0k1rc+iY0k1rcB1-I0k1rc(22b)b3-=A1-J0k1rc+iY0k1rcB1-I0k1rc.These wave vectors are related by(23a)a1-=f1-r0-rab1-(23b)a2-=f2-ra-rbb2-(23c)a3-=f3-rb-rcb3-.Substituting matrices (20a), (20b), (21a), (21b), (22a), (22b) into (23a)–(23c), negative–going propagation matrices are obtained:(24a)f1-=J0k1r0+iY0k1r0J0k1ra+iY0k1ra00I0k1r0I0k1ra(24b)f2-=J0k1ra+iY0k1raJ0k1rb+iY0k1rb00I0k1raI0k1rb(24c)f3-=J0k1rb+iY0k1rbJ0k1rc+iY0k1rc00I0k1rbI0k1rc.


## 2.2.2. Reflection Matrices

Keeping the boundary condition ofr=r0 fixed, thus, displacements and rotational angle are taken as(25)A1+J0k1r0-iY0k1r0+A1-J0k1r0+iY0k1r0+B1+K0k1r0+B1-I0k1r0=0A1+-k1J1k1r0+ik1Y1k1r0+A1--k1J1k1r0-ik1Y1k1r0-B1+k1K1k1r0+B1-k1I1k1r0=0.The relationship of incident wavea1+ and reflected wave a1- is related by(26)a1+=R0a1-.Substituting (25) into (26), the reflection matrices can be obtained as follows:(27)R0=-J0k1r0-iY0k1r0J0k1r0-iY0k1r0K0k1r0K0k1r0-k1J1k1r0+ik1Y1k1r0J0k1r0-iY0k1r0-k1K1k1r0K0k1r0-1J0k1r0+iY0k1r0J0k1r0+iY0k1r0I0k1r0I0k1r0-k1J1k1r0-ik1Y1k1r0J0k1r0+iY0k1r0k1I1k1r0I0k1r0.Keeping the boundary condition ofr=rc free gives(28)DA1+k1rcJ1k1rc-k12J0k1rc+ik12Y0k1rc-ik1rcY1k1rc+σrc-k1J1k1rc+ik1Y1k1rc+A1-k1rcJ1k1rc-k12J0k1rc-ik12Y0k1rc+ik1rcY1k1rc+σrc-k1J1k1rc-ik1Y1k1rc+B1+k12K0k1rc+k1rcK1k1rc-σrk1K1k1rc+B1-k12I0k1rc-k1rcI1k1rc+σ1k1rcI1k1rc=0.DA1+-ik13Y1k1rc+k13J1k1rc-B1+k13K1k1rc+A1-ik13Y1k1rc+k13J1k1rc+B1-k13I1k1rc=0The relationship of incident waveb3+ and reflected wave b3- is(29)b3-=R3b3+.Substituting (28) into (29), reflection matrices are calculated as(30)R3=-k1/rcJ1k1rc-k12J0k1rc-ik12Y0k1rc+ik1/rcY1k1rc+σ/rc-k1J1k1rc-ik1Y1k1rcJ0k1rc+iY0k1rck12I0k1rc-k1/rcI1k1rc+σ1k1/rcI1k1rcI0k1rcik13Y1k1rc+k13J1k1rcJ0k1rc+iY0k1rck13I1k1rcI0k1rc-1×k1/rcJ1k1rc-k12J0k1rc+ik12Y0k1rc-ik1/rcY1k1rc+σ/rc-k1J1k1rc+ik1Y1k1rcJ0k1rc-iY0k1rck12K0k1rc+k1/rcK1k1rc-σ/rk1K1k1rcK0k1rc-ik13Y1k1rc+k13J1k1rcJ0k1rc-iY0k1rc-k13K1k1rcK0k1rc.


## 2.2.3. Coordination Matrices

By imposing the geometric continuity atr=ra yields(31)J0k1ra-iY0k1raK0k1ra-k1J1k1ra+ik1Y1k1ra-k1K1k1rab1++J0k1ra+iY0k1raI0k1ra-k1J1k1ra-ik1Y1k1rak1I1k1rab1-=J0k2ra-iY0k2raK0k2ra-k2J1k2ra+ik2Y1k2ra-k2K1k2raa2++J0k2ra+iY0k2raI0k2ra-k2J1k2ra-ik2Y1k2rak2I1k2raa2-k1raJ1k1ra-k12J0k1ra+ik12Y0k1ra-ik1raY1k1ra+σ1ra-k1J1k1ra+ik1Y1k1rak12K0k1ra+k1raK1k1ra-σ1rak1K1k1ra-ik13Y1k1ra+k13J1k1ra-k13K1k1rab1++k1raJ1k1ra-k12J0k1ra-ik12Y0k1ra+ik1raY1k1ra+σ1ra-k1J1k1ra-ik1Y1k1rak12I0k1ra-k1raI1k1ra+σ1k1raI1k1raik13Y1k1ra+k13J1k1rak13I1k1rab1-=k2raJ1k2ra-k22J0k2ra+ik22Y0k2ra-ik2raY1k2ra+σ2ra-k2J1k2ra+ik2Y1k2rak22K0k2ra+k2raK1k2ra-σ2rak2K1k2ra-ik23Y1k2ra+k23J1k2ra-k23K1k2raa2++k2raJ1k2ra-k22J0k2ra-ik22Y0k2ra+ik2raY1k2ra+σ2ra-k2J1k2ra-ik2Y1k2rak22I0k2ra-k2raI1k2ra+σ2k2raI1k2raik23Y1k2ra+k23J1k2rak23I1k2raa2-.Equations (31) can be rewritten as(32)Ra1+b1++Ra1-b1-=Ta2+a2++Ta2-a2-Ra3+b1++Ra3-b1-=Ta4+a2++Ta4-a2-.According to the continuity atr=rb, shear force and bending moment are required that(33)J0k2rb-iY0k2rbK0k2rb-k2J1k2rb+ik2Y1k2rb-k2K1k2rbb2++J0k2rb+iY0k2rbI0k2rb-k2J1k2rb-ik2Y1k2rbk2I1k2rbb2-=J0k1r1-iY0k1rbK0k1rb-k1J1k1rb+ik1Y1k1rb-k1K1k1rba3++J0k1rb+iY0k1rbI0k1rb-k1J1k1rb-ik1Y1k1rbk1I1k1rba3-k2rbJ1k2rb-k22J0k2rb+ik22Y0k2rb-ik2r2Y1k2rb+σ2rb-k2J1k2rb+ik2Y1k2rbk22K0k2rb+k2rbK1k2rb-σ2rbk2K1k2rb-ik23Y1k2rb+k23J1k2rb-k23K1k2rbb2++k2rbJ1k2rb-k22J0k2rb-ik22Y0k2rb+ik2rbY1k2rb+σ2rb-k2J1k2rb-ik2Y1k2rbk22I0k2rb-k2rbI1k2rb+σ2k2rbI1k2rbik23Y1k2rb+k23J1k2rbk23I1k2rbb2-=k1rbJ1k2rb-k12J0k1rb+ik12Y0k1rb-ik1rbY1k1rb+σ1rb-k1J1k1rb+ik1Y1k1rbk12K0k1rb+k1rbK1k1rb-σ1rbk1K1k1rb-ik13Y1k1ra+k13J1k1rb-k13K1k1rba3++k1rbJ1k1rb-k12J0k1rb-ik12Y0k1rb+ik1rbY1k1rb+σ1rb-k1J1k1rb-ik1Y1k1rbk12I0k1rb-k1rbI1k1rb+σ1k1rbI1k1rbik13Y1k1rb+k13J1k1rbk13I1k1rba3-Equations (33) can be written as(34)Rb1+b1++Rb1-b1-=Tb2+a2++Tb2-a2-Rb3+b1++Rb3-b1-=Tb4+a2++Tb4-a2-.


## 2.2.4. Characteristic Equation of Natural Frequency

Combining propagation matrices, reflection matrices, and coordination matrices derived in Section2.2, natural frequencies of composite rings can be calculated smoothly. Figure 1 presents a clear description of incident and reflected waves. Thus, the wave matrices described by (18a)–(18c), (23a)–(23c), (26), (29), (32), and (34) are assembled as(35)
b
1
+
=
f
1
+
r
a
-
r
0
a
1
+
a
1
-
=
f
1
-
r
0
-
r
a
b
1
-
b
3
+
=
f
3
+
r
c
-
r
b
a
3
+
a
1
+
=
R
0
a
1
-
b
2
+
=
f
2
+
r
b
-
r
a
a
2
+
a
2
-
=
f
2
-
r
a
-
r
b
b
2
-
a
3
-
=
f
3
-
r
b
-
r
c
b
3
-
b
3
-
=
R
3
b
3
+
R
a
1
+
b
1
+
+
R
a
1
-
b
1
-
=
T
a
2
+
a
2
+
+
T
a
2
-
a
2
-
R
b
1
+
b
2
+
+
R
b
1
-
b
2
-
=
T
b
2
+
a
3
+
+
T
b
2
-
a
3
-
R
a
3
+
b
1
+
+
R
a
3
-
b
1
-
=
T
a
4
+
a
2
+
+
T
a
4
-
a
2
-
R
b
3
+
b
2
+
+
R
b
3
-
b
2
-
=
T
b
4
+
a
3
+
+
T
b
4
-
a
3
-
.In order to obtain the natural frequency, (35) can be rewritten in a matrix form(36)-I2×2R000000000000-I2×20f1-0000000000000-I2×20f2-0000000000000-I2×20f3-f1+0-I2×20000000000000f2+0-I2×20000000000000f3+0-I2×2000Ra1+Ra1-Ta2+Ta2-00000000Ra3+Ra3-Ta4+Ta4-000000000000Rb1+Rb1-Tb2+Tb2-00000000Rb3+Rb3-Tb4+Tb4-000000000000R3-I2×2a1+a1-b1+b1-a2+a2-b2+b2-a3+a3-b3+b3-=Ffa1+a1-b1+b1-a2+a2-b2+b2-a3+a3-b3+b3-=0.F ( f ) is a matrix of 12×12. If (36) has solution, it requires that(37)Ff=0.By solving the roots of characteristic equation (37), one can calculate the real and imaginary parts. It is important here to note that the natural frequencies can be found by searching the intersections in x-axis.


## 3. Numerical Results and Discussion

In this section, free vibration of rings is calculated by using wave approach, and the results are also compared with those obtained by classical method. Material RESIN is selected for the first and third layers. Material STEEL is selected for the middle layers. Material and structural parameters are given in Table1.Table 1
Material and structural parameters.
Material parameters Densityρ (kg/m3) Young modulusE (Pa) Poisson’s ratio I (RESIN) 1180 0.435 × 10 10 0.3679 II (STEEL) 7780 21.06 × 10 10 0.3  Structural parameters r a = 9 r 0 r c = r b + 40 h (mm) 45 125 1Based on Bessel and Hankel solutions calculated by classical method theoretically, natural frequency curves are presented by solving characteristic equation (8) depicted in Figure 2. Furthermore, (37) is calculated using wave approach. It can be seen that the real and imaginary parts intersect at multiple points simultaneously in x-axis. It is important, here, to note that the roots of the characteristic curves are natural frequencies when the values of longitudinal coordinates are zero.Figure 2
Natural frequency obtained by classical method and wave approach.In Figure2, two different natural frequencies can be clearly presented in the range of 450–1500 Hz, that is, 1244.22 Hz and 1443.31 Hz. However, the values are very small in the range of 0–450 Hz. In order to find whether the values in this range also intersect at one point, three zoomed figures are drawn for the purpose of better illustration about the natural frequencies of characteristic curves which are described in Figure 3.Figure 3
Characteristic curves in the range of 0–450 Hz.
(a)
0–40 Hz (b)
40–180 Hz (c)
180–450 HzNatural frequencies calculated by these two methods are compared. Modal analysis is carried out by FEM. The natural frequencies are presented in Table2 from which it can be observed that the first five-order modes calculated by these three methods are in good agreement. Obviously, it also can be found that natural frequencies obtained by ANSYS software are larger than the results calculated by classic method and wave approach, which is mainly caused by the mesh and simplified solid model in FEM. However, these errors are within an acceptable range, which verifies the correctness of theoretical calculations. To assess the deformation of rings, Figure 4 is employed to describe the mode shape. It can be found that the maximum deformations of the first three mode shapes occur in the outermost surface. The fourth and fifth mode shapes appear in the innermost surface.Table 2
Results calculated by classical method, wave approach, and FEM.
Method 1st mode 2nd mode 3rd mode 4th mode 5th mode Classical Bessel 37.65 Hz 167.54 Hz 414.27 Hz 1244.22 Hz 1443.31 Hz Classical Hankel 37.65 Hz 167.54 Hz 414.27 Hz 1244.22 Hz 1443.31 Hz Wave approach 37.65 Hz 167.54 Hz 414.27 Hz 1244.22 Hz 1443.31 Hz FEM 37.76 Hz 168.30 Hz 415.19 Hz 1247.90 Hz 1448.11 HzFigure 4
Mode shapes of natural frequencies. (a) First mode. (b) Second mode. (c) Third mode. (d) Fourth mode. (e) Fifth mode.
(a) (b) (c) (d) (e)Adopting FEM method, the first five natural frequencies are calculated for four type boundaries, as is shown in Table3. It shows that the first natural frequency is 37.76 Hz (Min) at the case of inner boundary fixed and outer boundary free. The first natural frequency is 143.40 Hz (Max) at the case of inner and outer boundaries both free.Table 3
Comparison of free vibration by FEM for four type boundaries.
Different boundaries 1st mode 2nd mode 3rd mode 4th mode 5th mode Inner free, outer free 143.40 Hz 334.10 Hz 569.55 Hz 1336.50 Hz 1845.16 Hz Inner fixed, outer free 37.76 Hz 168.30 Hz 415.19 Hz 1247.90 Hz 1448.10 Hz Inner free, outer fixed 70.44 Hz 321.96 Hz 572.94 Hz 1388.86 Hz 1848.67 Hz Inner fixed, outer fixed 101.07 Hz 413.28 Hz 1272.37 Hz 1479.74 Hz 2927.38 HzHarmonic Response Analysis of rings is carried out by using ANSYS 14.5 software. RESIN is chosen for the first and third layer. The second layer is selected as STEEL. Element can be selected as Solid 45, which is shown in red and blue in Figure5(a). Through loading transverse displacement onto the innermost layer and picking the transverse displacement onto the outermost layer, vibration transmissibility of rings propagating from inner to outer is obtained by using formula dB=20log(douter/dinner). Similarly, through loading transverse displacement onto the outermost layer and picking the transverse displacement onto the innermost layer, vibration transmissibility propagating from outer to inner is obtained by using formula dB=20log(dinner/douter).Figure 5 (a)
The meshing mode (b)
Vibration responseFigure5(b) indicates that there is no vibration attenuation in the range of 0–1500 Hz when transverse vibration propagates from outer to inner. Also, four resonance frequencies appear, namely, 70.44 Hz, 321.96 Hz, 572.94 Hz, 1388.86 Hz, which coincide with the first four-order natural frequencies in Table 3 at the case of innermost layer free and outermost layer fixed. Compared with the case of vibration propagation from outer to inner, there is vibration attenuation when vibration propagates from inner to outer. In addition, five resonance frequencies also appear, namely, 37.76 Hz, 168.30 Hz, 415.19 Hz, 1247.9 Hz, and 1448.1 Hz, which coincide with the results obtained by wave approach, classical Hankel, and classical Bessel methods shown in Table 2.


## 4. Effects of Structural and Material Parameters

### 4.1. Structural Parameters

The effects of structural parameters such as thickness, inner radius, and radial span are investigated in Figure6. Adopting single variable principle, herein, only change one parameter. Figure 6(a) shows clearly that, with thickness increasing, the first modes change from 37.76 Hz to 188.15 Hz, and the remaining three modes increase obviously, which indicates that thickness has great effect on the first four natural frequencies. In fact, characteristic equation of natural frequency is determined by thickness, density, and elastic modulus, which is shown by the expression of wave number k=(4π2f2ρh/D)0.25 and stiffness D=Eh3/12(1-σ2). Therefore, thickness is used to adjust the natural frequency directly through varying wave number k=(4π2f2ρh/D)0.25 in (36).Figure 6
Effect of structural parameters.
(a)
Thickness (b)
Inner radius (c)
Radial spanFrom the wave numberk=(4π2f2ρh/D)0.25, it can be found that inner radius is not related to the natural frequency. Thus, inner radius almost has no effect on the natural frequency shown in Figure 6(b).In Figure6(c), there are five different types analyzed for the radial span ratios of RESIN and STEEL, that is, a1 : a2=1×0.04/2.2 : 1.2×0.04/2.2, a1 : a2=1×0.04/2.1 : 1.1×0.04/2.1, a1 : a2=0.04 : 0.04, a1 : a2=1.1×0.04/2.1 : 1×0.04/2.1, a1 : a2=1.2×0.04/2.2 : 1×0.04/2.2, respectively. When radial span is equal to 1, this means that the size of RESIN and STEEL is 1 : 1, namely, a1:a2=0.04:0.04. For this case, the total size of composite ring is max, so the mode is min. Additionally, symmetrical five types cause the approximate symmetry of Figure 6(c). It also can be found that as radial span increases, natural frequencies appear as a similar trend, namely, decrease afterwards increase.


### 4.2. Material Parameters

Adopting single variable principle, density of middle material STEEL is replaced by the density of PMMA, Al, Pb, Ti. Similar with the study on effects of structural parameters, the effects of density and elastic modulus are studied for the case of keeping the material and structural parameters unchanged. Also, material parameters of PMMA, Al, Pb, and Ti are presented in Table4.Table 4
Material parameters.
Method Densityρ (kg/m3) Young modulusE (Pa) Poisson’s ratio PMMA 1062 0.32 × 1010 0.3333 Al 2799 7.21 × 1010 0.3451 Pb 11600 4.08 × 1010 0.3691 Ti 4540 11.7 × 1010 0.32Figure7(a) indicates that as density increases, the first mode decreases but not very obviously. However, the second, third, and fourth modes reduce significantly. Figure 7(b) shows that when elastic modulus increases gradually, the first mode increases but not significantly. The second, third, and fourth modes increase rapidly.Figure 7
Effect of material parameters.
(a)
Density (b)
Elastic modulus


## 4.1. Structural Parameters

The effects of structural parameters such as thickness, inner radius, and radial span are investigated in Figure6. Adopting single variable principle, herein, only change one parameter. Figure 6(a) shows clearly that, with thickness increasing, the first modes change from 37.76 Hz to 188.15 Hz, and the remaining three modes increase obviously, which indicates that thickness has great effect on the first four natural frequencies. In fact, characteristic equation of natural frequency is determined by thickness, density, and elastic modulus, which is shown by the expression of wave number k=(4π2f2ρh/D)0.25 and stiffness D=Eh3/12(1-σ2). Therefore, thickness is used to adjust the natural frequency directly through varying wave number k=(4π2f2ρh/D)0.25 in (36).Figure 6
Effect of structural parameters.
(a)
Thickness (b)
Inner radius (c)
Radial spanFrom the wave numberk=(4π2f2ρh/D)0.25, it can be found that inner radius is not related to the natural frequency. Thus, inner radius almost has no effect on the natural frequency shown in Figure 6(b).In Figure6(c), there are five different types analyzed for the radial span ratios of RESIN and STEEL, that is, a1 : a2=1×0.04/2.2 : 1.2×0.04/2.2, a1 : a2=1×0.04/2.1 : 1.1×0.04/2.1, a1 : a2=0.04 : 0.04, a1 : a2=1.1×0.04/2.1 : 1×0.04/2.1, a1 : a2=1.2×0.04/2.2 : 1×0.04/2.2, respectively. When radial span is equal to 1, this means that the size of RESIN and STEEL is 1 : 1, namely, a1:a2=0.04:0.04. For this case, the total size of composite ring is max, so the mode is min. Additionally, symmetrical five types cause the approximate symmetry of Figure 6(c). It also can be found that as radial span increases, natural frequencies appear as a similar trend, namely, decrease afterwards increase.


## 4.2. Material Parameters

Adopting single variable principle, density of middle material STEEL is replaced by the density of PMMA, Al, Pb, Ti. Similar with the study on effects of structural parameters, the effects of density and elastic modulus are studied for the case of keeping the material and structural parameters unchanged. Also, material parameters of PMMA, Al, Pb, and Ti are presented in Table4.Table 4
Material parameters.
Method Densityρ (kg/m3) Young modulusE (Pa) Poisson’s ratio PMMA 1062 0.32 × 1010 0.3333 Al 2799 7.21 × 1010 0.3451 Pb 11600 4.08 × 1010 0.3691 Ti 4540 11.7 × 1010 0.32Figure7(a) indicates that as density increases, the first mode decreases but not very obviously. However, the second, third, and fourth modes reduce significantly. Figure 7(b) shows that when elastic modulus increases gradually, the first mode increases but not significantly. The second, third, and fourth modes increase rapidly.Figure 7
Effect of material parameters.
(a)
Density (b)
Elastic modulus


## 5. Conclusion

This paper focuses on calculating natural frequency for rings via classical method and wave approach. Based on the solutions of transverse vibration, expression of rotational angle, shear force, and bending moment are obtained. Wave propagation matrices within structure, coordination matrices between the two materials, and reflection matrices at the boundary conditions are also deduced. Additionally, characteristic equation of natural frequencies is obtained by assembling these wave matrices. The real and imaginary parts calculated by wave approach intersect at the same point with the results obtained by classical method, which verifies the correctness of theoretical calculations.A further analysis for the influence of different boundaries on natural frequencies is discussed. It can be found that the first natural frequency is Min 37.76 Hz at the case of inner boundary fixed and outer boundary free. In addition, it also shows that there exists vibration attenuation when vibration propagates from inner to outer. However, there is no vibration attenuation when vibration propagates from outer to inner. Structural and material parameters have strong sensitivity for the free vibration.Finally, the behavior of wave propagation is studied in detail which is of great significance to the design of natural frequency for the vibration analysis of rotating rings and shaft systems.



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*Source: 6181204-2018-05-14.xml*