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Effects of Transverse Deformation on Free Vibration of 2D Curved Beams with General Restraints.

1. Introduction

Beams are one of the most extensively used structural components in a variety of branches of engineering applications, such as aircraft, civil construction, automobile, and naval vessel. The analytical evaluation of the vibration characteristics of beams has attracted much attention in the past decades because this information is very important for the low-vibration design and safety validation of engineering structures.

Strictly speaking, beams are three-dimensional (3D) blocks in physical sense for which the axial length is relatively larger than the other two dimensions. The 3D linear theory of elasticity may be applied in the theoretical modeling. However, such studies require high computing performance and lager storage capacities [1]. As a consequence, the beam problems are always simplified to a variety of one-dimensional (1D) representations by introducing several hypotheses in the kinetic relations and constitutive equations since the axial dimensions are relatively larger than the others. A variety of simplified 1D theories have been proposed so far, which are commonly divided into two aspects as follows: the classical beam theory (CBT) and the shear deformation beam theories (SDBTs). These specialties make them very attractive in the mechanics analysis of beams [2-10]. However, it is needed to be pointed out that the CBT is incapable of considering transverse deformation effect. The error of the calculating result is always great when dealing with moderately thick beams [11], since the shear effects on the cross section are more pronounced in moderately thick to thick beams and they are disregarded in the CBT. The FSDT overcomes this drawback and offers a more accuracy modeling theory since transverse deformation is further taken into account, even though the solutions based on the FSDTs are still not accurate due to the fact that the transverse normal components are still neglected. In addition, shear correction factors have to be incorporated in the FSDTs to adjust the transverse shear stiffness due to the fact that the transverse shear strains in the FSDTs are assumed to be constant in the thickness direction. The shear correction factors are difficult to determine because they depend not only on the geometric parameters, but also on the loading and boundary conditions. In order to obtain accurate solutions for thick beams, higher-order variation of axial displacement has been introduced into a wide variety of HSDTs. These theories are more accurate than the CBT and FSDTs without shear correction factors. But, unfortunately, the transverse normal effects are ignored in the conventional HSDTs. Thus, in order to analyze thick beams accurately, more advanced theories considering the through-thickness shear deformations are essentially required. Recently, Carrera [12, 13] developed the so-called Carrera Unified Formulation (CUF). According to the CUF, the obtained theories can have an order of expansion depending on the thickness functions that are used, which allows one to take into account the effects of the transverse normal effects.

The static and dynamic analysis of thick beam has been extensively investigated by many researchers. Chen et al. [14] proposed a mixed approach for the bending and free vibration of arbitrarily thick beams. In their method, the state space method and the differential quadrature method are combined to solve the problems. The method was further applied to the calculation of the elasticity solution of FGM beams by Ying et al. [15]. Hasheminejad and Rafsanjani [16] obtained semianalytical results for the transient dynamic response of thick simply supported beams through a powerful state space technique and the Laplace transformation. Thermoelastic behavior of arbitrarily simply supported beams subjected to thermomechanical loads is studied by Xu and Zhou [17, 18]. Zenkour et al. [19] studied the influence of transverse deformations on fiber reinforced viscoelastic beams. Malekzadeha and Karami [20] developed a mixed differential quadrature (DQ) and finite element (FE) approach for free vibration and buckling analysis of thick beams. This method applies a finite element discretization technique along axial direction while the thickness direction is discretized using DQM. The developments of studies of static and dynamic analysis of beams can be found in several monographs by Qatu [1], Rosen [21], Chidamparam and Leissa [22], Hodges [23], and Hajianmaleki and Qatu [24].

From the review of the literature, it is clear that although a lot of attention has been focused on static and dynamic analysis of thick beams, the extensive volume of literature on this subject was mainly limited to uniform straight beams with classical boundary conditions since their governing equation is much easier to be derived and tackled. The equations for a curved beam are more complicate and sophisticated because of curvilinear geometry. Inevitably, this introduces inherent complexity in finding their solutions [25]. In addition, the previous reviews showed that most beams are analyzed based on Euler-Bernoulli beam theory, Timoshenko beam theory, or the higher-order one-dimensional theory models which neglect the transverse normal deformation effect (thickness stretching). This appears quite inappropriate since the effect of transverse normal deformation on the static dynamic characteristics of thick beams is significant, especially at higher vibration modes of curved beams. Carrera et al. [26] and Koiter [27] recommended that a refinement of one-dimensional simplification theories is meaningless, unless the effects of transverse shear and normal deformations are all taken into account. Thus, seldom works are available that investigate the influences of transverse shear and normal deformations on the vibration characteristics of evident thick curved beams. The present work attempts to fill this gap.

In this paper, the modified Fourier series-based sampling surface method is further extended to the evaluation of elasticity solution of thick curved beams. The method was developed by Ye and Jin [28] based on a modified Fourier series technique proposed by Li [29] and SaS approach originally proposed by Kulikov et al. [30, 31]. The method combines the advantages of both approaches. A comprehensive numerical analysis and discussions are conducted to investigate the influence of transverse normal and shear deformations on the vibration characteristics of curved beams. The article is organized as follows: the theoretical formulation including model description, plan stress assumption, application of sampling surface method, and modified Fourier series approximation is presented in Section 2; convergence studies, results verification, and transverse deformation investigation are given in Section 3 and the concluding remarks are summarized in Section 4.

2. Theoretical Formulations

2.1. Model Description. A thick circular beam shown in Figure 1 is considered, in which b, h, and R represent the width, thickness, and inner radius of the beam. The beam is bounded along its edges by the boundaries [theta] = [[theta].sub.0] and [theta] = [[theta].sub.1]. The bottom surface of the beam is selected as the reference surface with the three orthogonal curvilinear coordinates [theta], y, and z; see Figure 1. In this paper, the beams are assumed to be isotropic and homogeneous and to vibrate freely in the [theta]-z plane. u, v, and w denote the three displacement components in the axial, lateral, and normal directions, respectively.

2.2. Plane Stress Assumption. As mentioned previously, the beam under consideration vibrates freely in the [theta]-z plane. Therefore, the plane stress hypothesis is adopted in the theoretical formulation for the purpose of improving the computational efficiency and maintaining the modeling precision synchronously.

For a curved beam, the 3D strain-displacement relations for any point in the domain of the beam can be found as [32]

[mathematical expression not reproducible], (1)

where [R.sub.z] = R + z, [[epsilon].sub.[theta]], [[epsilon].sub.y], and [[epsilon].sub.z] stand for the normal strains, and [[gamma].sub.yz], [[gamma].sub.[theta]z], and [[gamma].sub.[theta]y] are the shear components. In the case of homogeneous materials and linear small deformation assumptions, the 3D stresses can be derived according to Hooke's law:

[mathematical expression not reproducible], (2)

where [[sigma].sub.[theta]], [[sigma].sub.y], and [[sigma].sub.z] stand for the normal stresses; [[tau].sub.yz], [[tau].sub.[theta]z], and [[tau].sub.[theta]y] are the shear component. C' is the material stiffness matrix.

[C'.sub.11] = [C'.sub.2] = [C'.sub.23] = E(1 - v)/(1 + v)(1 - 2v),

[C'.sub.12] = [C'.sub.13] = [C'.sub.23] = v[C'.sub.11]/(1 - v),

[C'.sub.44] = [C'.sub.55] = [C'.sub.66] = E/2(1 + v), (3)

in which E means Young's module of the material and v represents Poisson's ratio.

Furthermore, (2) can be rearranged in the form of matrix as

[mathematical expression not reproducible], (4)

Therefore, the final stress-strain relations for the beam under plane stress hypothesis can be obtained as

[[sigma].sub.i] = C[[epsilon].sub.i], (5)

where C = [C'.sub.ii] - [C'.sub.io]([[C'.sub.00].sup.-1])[C'.sub.io].

2.3. Application of Sampling Surface Technique. The sampling surface technique was originally proposed by Kulikov et al. [33-35]. A brief resume and application of this technique are included in this section.

As shown in Figure 1, [S.sub.1], ..., [S.sub.j], ..., [S.sub.J] stand for the chosen sampling surfaces inside the transverse domain of the beam to introduce the displacement components of these surfaces as basic beam variables. J is the total number of the sampling surfaces. These surfaces are selected to be nonequally spaced and paralleled to the beam's middle surface. For application of the sampling surface technique, it was found that the distribution of the sampling surface has a great effect on the convergence and accuracy of the solutions. For the sake of the convergence, transverse coordinates of these surfaces are chosen as the roots of Chebyshev polynomial:

[z.sub.1] = 0,

[z.sub.J] = h,

[z.sub.j] = [z.sub.1] + [z.sub.J]/2 - h/2 cps ([pi]2j - 3/2J - 4);

2 [less than or equal to] j [less than or equal to] J - 1. (6)

Therefore, the basic variables in the axial and normal directions of an arbitrary sampling surface can be given by

u ([theta] [z.sub.j], t) = [u.sub.j] ([theta]) [e.sup.i[omega]t],

w ([theta], [z.sub.j], t) = [w.sub.j] ([theta]) [e.sup.i[omega]t];

1 [less than or equal to] j [less than or equal to] J, (7)

where [u.sub.j]([theta]) and [w.sub.j]([theta]) stand for the axial and transverse displacement components, respectively. t stand for the time variable, [omega] is the circular frequency. As a consequence, displacement field of the beam under vibration can be calculated by

[mathematical expression not reproducible], (8)

and [L.sub.j](z) is Lagrange's interpolation of degree J -1:

[mathematical expression not reproducible]. (9)

According to (1), strains on the jth sampling surface can be found as

[mathematical expression not reproducible], (10)

where [M.sup.r.sub.j] are determined by

[mathematical expression not reproducible], (11)

Similarly, strain distribution in the whole space should be represented as a linear combination of their corresponding strain components of the entire sampling surfaces as (8).

[mathematical expression not reproducible], (12)

The energy functional for the curved beam under the circumstance of free vibration is

[[PI].sub.S] = [T.sub.S] - [U.sub.s], (13)

where [U.sub.s] and [T.sub.s] denote the strain and kinetic energy function defined as follows:

[mathematical expression not reproducible], (14)

where [rho] stands for the material density. Substituting (5) and (12) into (14), the two energy functions can be further written as

[mathematical expression not reproducible]. (15)

2.4. Modified Fourier Series Approximation. The modified Fourier series approximation is introduced to represent the possible deformations of the curved beams. Particularly, each of the basic beam variables is mathematically described as a set of modified Fourier series including a standard cosine Fourier series as well as certain auxiliary functions [36-41]. The auxiliary terms are introduced for the purpose of removing the entire possible discontinuities with the basic beam variables and their derivatives at the edges to form a mathematically complete set and then ensure the convergence and speed up the calculation [39, 42-45]. In addition, the governing equations of the beams are derived and numerically solved by a modified variational principle for the sake of making arbitrary boundary conditions applicable.

As mentioned previously, the displacement variables at an arbitrary sampling surface in the modified form of Fourier series are

[mathematical expression not reproducible], (16)

where [u.sup.n.sub.j] and [w.sup.n.sub.j] (n = 0, 1, ..., N) are the expansion coefficients; [DELTA][theta] = [[theta].sub.1] - [[theta].sub.0] . N represents the truncation number.

The boundary conditions of the curved beams are supposed to be of essential type. The necessary boundary equations can be stated in functional form as follows by applying the penalty technique and Lagrange multipliers [6, 46-49]:

[mathematical expression not reproducible], (17)

where [[bar.u].sub.l] and [[bar.w].sup.l] denote the boundary values. [k.sup.l.sub.u] and [k.sup.l.sub.w] represent the penalty parameters. [[eta].sup.l.sub.u] and [[eta].sup.l.sub.w] are the parameters which define different restraint conditions. The boundary potential [[PI].sub.b1] is introduced by means of Lagrange multiplier technique while the boundary potential [[PI].sub.b2] is introduced by the aid of the penalty technique to ensure a uniform formulation to tackle general boundaries [6] and to ensure a computational stability in computational process. Taking the end of [theta] = [[theta].sub.0], for example, the values of the penalty parameters and boundary coefficients for different classical restraint conditions are shown in Table 1. For elastic boundary conditions, the boundary potentials [[PI].sub.b1] in (17) should be neglected and the penalty parameters will be determined at proper values [49].

Therefore, the final variational functional for the curved beam with general boundaries is defined as

[[PI].sub.total] ([u.sup.n.sub.j],[w.sup.n.sub.j]) = [[PI].sub.S] + [[PI].sub.b]; 0 [less than or equal to] n [less than or equal to] N, 1 [less than or equal to] j [less than or equal to] J. (18)

Finally, let the variation of the [[PI].sub.total] with respect to each coefficient ([u.sup.n.sub.j] and [w.sup.n.sub.j]) equal zero; the governing equations can be derived in a matrix form as

{K - [[omega].sup.2]M} G = 0, (19)

where K and M stand for the final stiffness and mass matrices of order 2(N + 1) * J. G denotes the vector of the unknown generalized displacements. Thus, solutions can be obtained directly by the eigenvalue decomposition of (19) and the roots of the decomposition are the square of eigenfrequency [omega]. The mode shape of the curved beam corresponding to each eigenfrequency can be constructed by substituting the corresponding eigenvector back into the displacement variables given in (16) and then substituting it in the displacement distribution formula given in (8).

3. Numerical Results and Discussion

Several examples for thick curved beams with different geometrical dimensions and boundary restraints are presented to verify the flexibility of the method. The transverse deformation effects are systematically investigated as well. To unify the discussion, character string X-Y (X/Y = F, S, C) is used to represent the boundary conditions of the beams. For example, C-F represents a circular beam with clamped and free restraints at the ends [theta] = [[theta].sub.0] and [theta] = [[theta].sub.1], respectively. To unify the discussion, the dimensionless variable of frequency is introduced in the calculation [OMEGA] = [omega][R.sup.2.sub.m] [square root of (12[rho]/E[h.sup.2])] (where [R.sub.m] = R + h/2). The beams are supposed to be made of steel (E = 210 GPa, v = 0.3, and [rho] = 7800 kg/[m.sup.3]).

3.1. Validation. Table 2 gives the first five nondimensional frequency parameters [OMEGA] of C-C supported circular beams. The numbers of the sampling surfaces and serious truncation are increased from 11 to 17 and 3 to 9, respectively. The geometric parameters of the beam are [R.sub.m] = 1 m, [DELTA][theta] = 2[pi]/3. For completeness, two thickness-to-radius ratios (i.e., h/[R.sub.m] = 0.1 and 0.2) corresponding to the moderately thick and thick beam configurations are considered in the study. As observed from Table 2, with the increase of truncated number, the natural frequencies tend to be constant values quickly. The maximum differences between the results based on the "11 x 3" and "17 x 9" computational schemes are less than 0.2%, which confirms the high convergence of the present method. The DQM results based on the FSDT [3] and the Ritz solutions with 2D elasticity theory [32] are also listed in the table. It is observed that the present solutions match well with those predicted by Malekzadeh et al. [3] and Jin et al. [32]. The slight differences between the three groups of results show the satisfied accuracy of the proposed approach. Table 3 compares the first six natural frequencies (Hz) of a circular beam with F-F, F-C, and C-C boundary conditions obtained by the current approach with those based on commercial FEM code. The results are calculated with the beam parameters [R.sub.m] = 1m, h/[R.sub.m] = 0.3 and with "17 x 9" truncation scheme. Calculations based on FEM commercial software ANSYS (PLANE82, 0.025 m) are used as the benchmark solutions. In Table 3, it is obvious that the present method produces good results comparing with FEM.

3.2. Transverse Deformation Effects. The effects of transverse deformation on the vibration characteristics of curved beams are investigated in this section. In Figures 2-9, relative deviations between frequency parameters [OMEGA] calculated by the CBT/FSDT theory models [5] and the present 2D approach for circular beams with various different geometries and boundary conditions are considered. The "deviations (%)" between the results are defined as

Deviations (%) = ([[OMEGA].sub.CBT/FSDT] -[[OMEGA].sub.2-D])/[[OMEGA].sub.2-D] x 100%. (20)

Figures 2-4 show the relative deviations of the 1st, 3rd, and 5th frequency parameters [OMEGA] for circular beams with different ratios of thickness-to-span length (h/[L.sub.[theta]]). The beam is supposed to be of unit span length; that is, [L.sub.[theta]] = .R[DELTA][theta] = 1. The ratio of h/[L.sub.[theta]] is varied between 0.01 and 0.2. F-F, C-C, and F-C boundaries are considered in the study. The results obtained by the current method of "N x J = 17 x 8" truncation scheme are selected as benchmark. From the figures, we can see that there is a clear increment of frequency parameter for the larger thickness-to-span length ratio and the increment becomes more prominent for higher modes. The maximum difference can be as much as 35%. Furthermore, results on the basis of CBT are generally higher than those based on FSDT model and the 2D elasticity theory because the effects of shear deformations are more significant in thick beams. It is due to the fact that hypotheses in the CBT will introduce additional stiffness in the modeling in fact. This investigation shows that the CBT can be grossly error for the modeling of moderately thick and thick curved beams. In addition, it is obvious that the results based on the FSDT are more accurate than those of CBT since the effects of traverse deformation are included.

Figure 5 shows a similar study for clamped circular beams with various thickness-to-radius ratios and span angles. Geometrical dimensions used in the study are [R.sub.m] = 1 m. "N x J = 17 x 9" displacement field is adopted for the 2D solutions in this study. As expected, the effects of transverse normal and shear deformations decrease as the span angle increases. The relative deviations between results based on the CBT model and the current 2D approach are also very big and the maximum difference can be as much as 50%. It can be observed that that the effects of the transverse normal and shear deformation varied with mode number and (span) length-to-radius ratio. Generally, lower (span) length-to-radius ratio values will lead to larger modeling deviation of vibration behavior since transverse effects are more significant for short beams.

Figures 6 and 7 consider the fundamental and fifth mode frequency parameters [OMEGA] of a circular beam based on the CBT and FSDT theory models, respectively. The thickness-to-span length ratio, h/[L.sub.[theta]], is varied from 0.01 to 0.2, corresponding to thin to thick beam configurations. Two boundary conditions, that is, F-F and C-C, are considered in the studies. The beam is supposed to be of unit span length and unit radius, that is, [L.sub.[theta]] = 1, [R.sub.m] = 1 m. From the figures, we can see that the effects of the shear deformation increase generally as the thickness-to-span length ratio increases. When the thickness-to-span length ratio is equal to 0.1, the difference between the CBT and FSDT results can be as many as 21.6% and 25.3% for the F-F and C-C boundary conditions, respectively. In addition, it is obvious that the transverse shear deformation has greater influence on the higher modes.

Elasticity solutions for circular beams with different geometrical dimensions and several sets of classical boundary conditions are presented in the following presentation. The lowest six frequencies considering three different thickness-to-radius ratios are presented in Table 4 ([DELTA][theta] = [pi]/2), where the classical boundary conditions are included. "N x J = 17x 9" displacement field is adopted in this study. From the results of Table 4, it becomes clear that the frequency parameters [OMEGA] decrease when the thickness of the beams is increased. However, it should be pointed out that the beam's natural frequencies (Hz) are increasing actually because the stiffness of a beam increases generally when its thickness is increased. For the sake of completeness, the first three modal shapes for the beam with C-C boundary conditions are presented in Figure 8. The figure indicates that the modal shapes of thick beams are characterized by complex coupling between the extension, bending, and shearing modes.

The lowest six frequency parameters [OMEGA] of the certain circular beams are presented in Table 5 with a variety of classical restraints and span angles. The thickness ratio (h/[R.sub.m]) of the beams is assumed to be constant as h/[R.sub.m] = 0.1, 20. Meanwhile, the span angle [DELTA][theta] is taken as [DELTA][theta] = [pi]/3, 2[pi]/3, and [pi], respectively. "N x J = 20 x 9" displacement field is adopted in this study. First of all, it is seen that the circular beams with C-C boundary conditions have highest frequency parameters among all boundary cases. The frequency parameters of the beam decrease when the span angle increases because when the span angle increases, the flexibility of the beam increases synchronously. Figure 9 gives the three lowest mode shapes for the circular beam of Table 6 with FC restraint. The figure reveals that the change in the span angle can directly affect the mode shapes of the beam. The modes of the beam are noted to be determined by bending, shear, and normal deformation, which could not be determined by the CBT and FSDT theory models.

Figure 10 shows the deviations between the first, third, and fifth frequency parameters [OMEGA] based on the CBT/FSDT theory models and those of the current 2D formulation for circular beams for various values of thickness-to-radius ratios and restraint rigidities. The beams ([L.sub.[theta]] = 1) are supposed to be clamped at one end ([theta] = [[theta].sub.0]) and elastically supported at the other end with stiffness rigidity [k.sup.l.sub.u] = [k.sup.l.sub.[]omega] = [eta] (where [eta]/D [member of] [[10.sup.-2], [10.sup.8]], D = E[h.sup.3]/12). "N x J = 17 x 9" displacement field is adopted in this study. From the figure, it is obvious that the effects of the transverse normal and shear deformation have a great influence on the frequencies of the beam when subjected to elastic boundary conditions. According to Figure 10, we can see that the errors of the CBT and FSDT are acceptable when the restraint rigidity [eta]/D is smaller than [10.sup.-1]. However, the error increases sharply when it is increased from [10.sup.-1] to [10.sup.5]. Then, the error decreases and remains the same when [eta]/D tends to be infinity. The error of the CBT/FSDT can be as much as 180%, 120%, and 85% for the worst case in each study.

Figures 11-13 show similar study for circular beams with different type of elastic boundary restraints. The following geometric parameters are used: [R.sub.m] = 1, h/[R.sub.m] = 0.2. The beam is clamped at the end of 0 = 06 and elastically restrained at the other end. The following three types of elastic supports are considered in the study: axially elastic restraint ([k.sup.l.sub.u] = [eta], [k.sup.l.sub.w] = 0), transversely elastic restraint ([k.sup.l.sub.u] = 0, [k.sup.l.sub.w] = and elastic restraint in both directions ([k.sup.l.sub.u] = [k.sup.l.sub.w] = [eta]). The changes of the relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models and those of the current 2D formulation with respect to elastic rigidity [eta]/D are the same as Figure 10.

Finally, Table 6 displays the lowest three nondimensional frequencies [OMEGA] for circular beams with a variety of geometric constants and restrained rigidities. The elastic boundary conditions studied in the table are the same as those of Figures 11-13. The geometrical dimensions used in the calculation are [R.sub.m] = 1m, [[theta].sub.0] = [pi]/2. "N x J = 17 x 9" displacement field is adopted in this study. The table reveals that the frequencies of the beam will increase when the rigidity of the restraint increases. This is because when the restraint rigidity increases, the stiffness of the beam increases synchronously while the mass remains unchanged. Table 7 shows similar studies for the beams with different span angles. The geometrical parameters and material properties used in the calculation are [R.sub.m] = 1 m, h/[R.sub.m] = 0.1. These results can be used to verify new 1D refined beam theories for further studies.

4. Conclusions

This paper proposes an accurate modified Fourier series-based sampling surface approach for the analytical evaluation of the vibration characteristics of thick curved beams. The approach is valid for arbitrary thickness configuration and maintains its simplicity and uniform in any type of boundary conditions (i.e., classical boundary condition, elastic support, or their combination). The theoretical models of the beams are based on the 2D theory of elasticity including the effects of both transverse shear and normal deformations. Under the current framework, the transverse beam domain is discretized by a set of nonequally spaced sampling surfaces and the displacement components coinciding with these surfaces are mathematically described as an set of modified Fourier series including the certain auxiliary terms which are used to form a mathematically complete set and guarantee the results convergent to the exact solutions. The governing equations of the beams are derived and numerically solved using a modified variational principle by the use of the penalty technique as well as Lagrange multipliers. Elasticity solutions including transverse shear and normal effects are compared with the corresponding one-dimensional results in terms of the classical and first-order shear deformation theories. The influences of transverse normal and shear deformation on the vibration characteristics are systematically evaluated. The results show that the proposed method is applicable for thick circular beams with arbitrary boundary conditions.

https://doi.org/10.1155/2017/2104015

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Xueren Wang, (1) Xuhong Miao, (1, 2) Di Jia, (1) and Shengyao Gao (1)

(1) Naval Academy of Armament, Beijing 100161, China

(2) College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Xueren Wang; wangxuerennaa@126.com

Received 1 July 2017; Revised 1 September 2017; Accepted 11 September 2017; Published 18 October 2017

Academic Editor: Nerio Tullini

Caption: FIGURE 1: Geometry and reference system for a curved beam and the diagrammatic sketch of sampling surface distribution.

Caption: FIGURE 2: Relative deviations between the first, third, and fifth frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for a complete free circular beam with various thickness ratios.

Caption: FIGURE 3: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (F-C boundary condition).

Caption: Figure 4: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for circular beams with various thickness ratios (C-C boundary condition).

Caption: Figure 5: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for circular beams with different thickness-to-radius ratios (C-C boundary condition).

Caption: FIGURE 6: Frequency parameters [OMEGA] of circular beams with various thickness ratios based on the CBT and FSDT theory models (F-F boundary condition).

Caption: FIGURE 7: Frequency parameters [OMEGA] of circular beams with various thickness ratios based on the CBT and FSDT theory models (C-C boundary condition).

Caption: FIGURE 8: Modal shapes relative to the C-C circular curved beams of Table 4.

Caption: FIGURE 9: Modal shapes relative to the F-C circular curved beams of Table 5.

Caption: FIGURE 10: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0]: clamped; [theta] = [[theta].sub.1]: ku = kw = [eta]).

Caption: Figure 11: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0] : clamped; [theta] = [[theta].sub.1]: [k.sup.l.sub.u] = [eta], [k.sup.l.sub.w] = 0).

Caption: FIGURE 12: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0]: clamped; [theta] = [[theta].sub.1]: [k.sup.l.sub.u] = 0, [k.sup.l.sub.w] = [eta]]).

Caption: Figure 13: Relative deviations between frequency parameters [OMEGA] based on the CBT/FSDT theory models to those of 2D theory for elastically restrained circular beams with different restraint rigidities ([theta] = [[theta].sub.0]: clamped; [theta] = [[theta].sub.1]: [k.sup.l.sub.u] = [k.sup.l.sub.w] = [eta]).
TABLE 1: Values of [[eta].sup.l.sub.u], [[eta].sup.l.sub.[omega]],
[k.sup.l.sub.u], and [k.sup.l.sub.w] for different classical
boundary conditions.

                                          Boundary
Boundary conditions                     coefficients

                                                   [[eta].sup.0.
                                   [[eta].             sub.
                                 sup.0.sub.u]        [omega]]

F (free): [[sigma].sub.              0                  0
  [theta]] = 0, [[tai].sub.
  [theta]z] = 0

S1 (simply supported):               0                  1
  [[sigma].sub.[theta]]
  =0, w = 0

S2 (simply supported):               1                  0
  u = 0, [[tai].sub.
  [theta]z] = 0

C (clamped): u =                     1                  1
  0, w = 0

                                           Penalty
Boundary conditions                       parameters

                            [k.sup.0.sub.u]    [k.sup.0.sub.w]

F (free): [[sigma].sub.            0                     0
  [theta]] = 0, [[tai].sub.
  [theta]z] = 0

S1 (simply supported):             0                 [10.sup.3]E
  [[sigma].sub.[theta]]
  =0, w = 0

S2 (simply supported):         [10.sup.3]E                0
  u = 0, [[tai].sub.
  [theta]z] = 0

C (clamped): u =               [10.sup.3]E            [10.sup.3]E
  0, w = 0

TABLE 2: Convergence of the lowest five frequency parameters [OMEGA]
for C-C supported curved beams ([R.sub.m] = 1 m, [DELTA][theta] =
2[pi]/3).

N       J                  h/[R.sub.m] = 0.1

                  1        2        3        4        5

11      3       12.067   21.555   35.782   40.788   63.430
        5       12.001   21.432   35.717   40.411   63.114
        7       11.998   21.428   35.716   40.401   63.104
        9       11.996   21.426   35.715   40.396   63.097

12      3       12.043   21.551   35.771   40.679   63.110
        5       11.984   21.421   35.706   40.300   62.772
        7       11.981   21.418   35.704   40.293   62.761
        9       11.980   21.416   35.703   40.288   62.754

13      3       12.042   21.531   35.759   40.668   63.069
        5       11.979   21.407   35.694   40.275   62.318
        7       11.976   21.405   35.693   40.268   62.308
        9       11.975   21.403   35.693   40.264   62.302

14      3       12.029   21.529   35.757   40.628   63.032
        5       11.969   21.401   35.692   40.250   62.260
        7       11.967   21.398   35.691   40.245   62.249
        9       11.966   21.397   35.690   40.241   62.243

15      3       12.027   21.518   35.752   40.625   62.966
        5       11.966   21.394   35.688   40.239   62.218
        7       11.964   21.392   35.687   40.233   62.210
        9       11.963   21.390   35.687   40.230   62.206

16      3       12.020   21.516   35.751   40.606   62.964
        5       11.960   21.390   35.686   40.226   62.208
        7       11.959   21.387   35.685   40.222   62.199
        9       11.958   21.386   35.685   40.219   62.195

17      3       12.019   21.510   35.749   40.603   62.943
        5       11.958   21.385   35.684   40.220   62.194
        7       11.957   21.384   35.683   40.214   62.187
        9       11.956   21.382   35.683   40.212   62.183

     FSDT [3]   11.391   20.392   34.001     --       --

     2D [32]    11.470   20.575   34.105   38.963   60.102

N       J                   h/[R.sub.m] = 0.2

                  1        2        3        4        5

11      3       10.959   14.434   24.614   28.932   38.069
        5       10.791   14.369   24.234   28.727   37.409
        7       10.788   14.368   24.231   28.724   37.403
        9       10.786   14.368   24.229   28.722   37.400

12      3       10.945   14.433   24.609   28.924   38.004
        5       10.781   14.367   24.224   28.714   37.338
        7       10.779   14.366   24.219   28.712   37.332
        9       10.778   14.366   24.217   28.711   37.330

13      3       10.943   14.432   24.596   28.921   37.993
        5       10.777   14.365   24.215   28.709   37.326
        7       10.774   14.365   24.211   28.705   37.317
        9       10.773   14.364   24.209   28.704   37.314

14      3       10.937   14.432   24.594   28.918   37.982
        5       10.772   14.365   24.212   28.705   37.316
        7       10.769   14.363   24.206   28.701   37.309
        9       10.769   14.363   24.205   28.701   37.307

15      3       10.936   14.431   24.588   28.918   37.981
        5       10.771   14.364   24.206   28.703   37.314
        7       10.767   14.363   24.201   28.698   37.304
        9       10.766   14.363   24.200   28.697   37.302

16      3       10.933   14.431   24.587   28.917   37.976
        5       10.767   14.363   24.205   28.700   37.308
        7       10.764   14.362   24.198   28.696   37.299
        9       10.763   14.362   24.197   28.695   37.298

17      3       10.933   14.431   24.585   28.916   37.976
        5       10.767   14.363   24.202   28.699   37.308
        7       10.763   14.362   24.196   28.694   37.297
        9       10.762   14.361   24.195   28.693   37.295

     FSDT [3]   10.271   13.622   23.107     --       --

     2D [32]    10.507   13.773   23.691   27.680   36.752

TABLE 3: The lowest six frequencies for circular beams with different
boundary conditions, geometrical properties, and modeling methods
(Hz, [R.sub.m] =1 m, h/[R.sub.m] = 0.3).

BC     Mode     [DELTA][theta] = [pi]/4

                Pre.     FEM     Diff. (%)

F-F     1      1861.2   1861.3     0.003
        2      3304.8   3304.8     0.001
        3      3653.6   3653.9     0.007
        4      5514.5   5515.4     0.016
        5      5873.7   5874.5     0.014
        6      6166.1   6166.4     0.005

F-C     1      373.90   373.99     0.024
        2      1420.4   1420.6     0.014
        3      1870.6   1870.8     0.009
        4      3475.3   3475.8     0.013
        5      4773.6   4774.1     0.010
        6      5133.8   5134.0     0.005

C-C     1      1683.9   1684.5     0.033
        2      2923.8   2924.6     0.028
        3      3461.9   3462.8     0.027
        4      4991.2   4992.3     0.022
        5      6282.3   6283.3     0.017
        6      6414.4   6414.6     0.003

BC    Mode   [DELTA][theta] = [pi]/2       [DELTA][theta] = 3[pi]/4

             Pre.     FEM     Diff. (%)    Pre.     FEM     Diff. (%)

F-F    1    545.85   545.85     0.001     239.19   239.19     0.000
       2    1333.9   1333.9     0.002     638.87   638.87     0.001
       3    1800.9   1800.9     0.001     1177.4   1177.4     0.001
       4    2285.0   2285.1     0.003     1355.5   1355.5     0.003
       5    3207.8   3207.9     0.004     1790.9   1790.9     0.002
       6    3344.4   3344.5     0.002     2289.5   2289.5     0.000

F-C    1    104.22   104.23     0.008     50.018   50.017     0.001
       2    431.29   431.32     0.008     182.16   182.15     0.003
       3    974.88   974.91     0.003     535.24   535.24     0.001
       4    1368.0   1368.1     0.005     898.42   898.43     0.001
       5    2131.7   2131.9     0.007     1173.9   1173.9     0.002
       6    2556.9   2556.9     0.001     1692.3   1692.3     0.002

C-C    1    850.21   850.24     0.004     512.94   512.93     0.002
       2    1079.5   1079.7     0.015     665.65   665.66     0.001
       3    1805.4   1805.5     0.007     1159.1   1159.1     0.002
       4    2056.6   2056.9     0.012     1227.9   1227.9     0.002
       5    2984.2   2984.5     0.009     1732.5   1732.5     0.002
       6    3280.8   3281.0     0.005     2148.3   2148.3     0.001

TABLE 4: The lowest six frequency parameters [OMEGA] of classically
restrained circular beams with various thickness ratios ([R.sub.m]
= 1 m, [DELTA][theta] = [pi]/2).

h/[R.sub.m]   Mode          Boundary condition

                      F-F      F-S1     F-S2     F-C

0.05           1     8.3653   5.2974   1.8360   1.4976
               2     23.694   18.869   11.065   7.1854
               3     46.884   40.247   28.619   22.477
               4     77.238   68.895   53.693   45.339
               5     114.23   104.30   85.785   74.297
               6     154.82   145.91   97.943   94.711

0.1            1     8.2897   5.2643   1.8326   1.4942
               2     23.048   18.453   10.919   7.0603
               3     44.492   38.450   27.682   21.456
               4     71.197   63.988   48.914   40.384
               5     77.261   77.261   50.605   49.486
               6     102.04   93.921   78.471   70.567

0.2            1     8.0147   5.1403   1.8195   1.4802
               2     20.994   17.077   10.404   6.6156
               3     37.663   33.367   24.305   17.818
               4     38.678   38.337   24.870   24.700
               5     57.082   52.145   42.532   36.793
               6     70.432   70.314   54.084   53.013

h/[R.sub.m]   Mode      Boundary condition

                     S1-S1     S1-S2     S1-C

0.05           1     2.6801   7.5652    4.3680
               2     14.464   23.307    17.702
               3     34.040   46.609    38.807
               4     60.960   77.028    66.582
               5     94.751   97.943    92.155
               6     134.88   114.06    106.77

0.1            1     2.6707   7.4945    4.3126
               2     14.213   22.672    17.053
               3     32.728   44.232    35.608
               4     57.039   48.917    47.535
               5     77.259   71.006    63.829
               6     86.002   101.899   92.113

0.2            1     2.6343   7.2362    4.1083
               2     13.354   20.658    14.846
               3     28.877   24.349    23.209
               4     38.320   37.789    32.799
               5     47.221   54.074    49.629
               6     67.144   57.039    54.498

h/[R.sub.m]   Mode      Boundary condition

                     S2-S2     S2-C     C-C

0.05           1     2.6800   9.5587   22.270
               2     14.464   26.891   39.349
               3     34.039   50.676   66.753
               4     60.959   66.953   74.701
               5     69.297   86.221   114.39
               6     94.749   123.04   143.50

0.1            1     2.6706   9.2748   21.246
               2     14.213   24.596   27.624
               3     32.727   34.057   46.328
               4     34.672   50.034   64.343
               5     57.038   73.445   79.316
               6     77.257   80.161   98.509

0.2            1     2.6343   8.2708   16.029
               2     13.354   16.094   18.192
               3     17.384   24.260   35.875
               4     28.876   36.821   36.990
               5     38.319   42.107   53.898
               6     47.220   59.788   68.390

TABLE 5: The lowest six frequency parameters [OMEGA] of classically
restrained circular beams with various span angles
([R.sub.m] = 1 m, h/[R.sub.m] = 0.1).

[DELTA][theta]    Mode           Boundary condition

                          F-F       F-S1       F-S2       F-C

[pi]/3             1     19.088    12.761     4.4804     3.2579
                   2     50.820    41.432     25.315     17.105
                   3     94.059    82.289     60.572     46.643
                   4     109.23    109.21     62.328     62.105
                   5     144.99    131.64     106.10     92.330
                   6     200.96    186.60     158.38     140.84

2[pi]/3            1     4.4805    2.6710     0.9739     0.8749
                   2     12.761    10.031     5.7637     3.6195
                   3     25.315    21.674     15.360     11.728
                   4     41.432    36.989     28.906     23.886
                   5     60.573    55.456     43.262     37.328
                   6     62.330    62.315     45.861     44.506

[pi]               1     1.8327    0.9507     0.4372     0.4349
                   2     5.2643    3.9432     2.1308     1.3692
                   3     10.919    9.1645     6.3020     4.6433
                   4     18.453    16.288     12.501     10.245
                   5     27.683    25.134     20.505     17.608
                   6     38.450    35.548     30.155     26.470

[DELTA][theta]    Mode        Boundary condition

                          S1-S1      S1-S2       S1-C

[pi]/3             1      7.4946     1.0378     11.416
                   2      32.728     18.234     38.524
                   3      71.007     50.481     60.019
                   4      109.21     62.312     81.544
                   5      118.59     93.844     128.08
                   6      172.44     144.88     158.76

2[pi]/3            1      1.0378     0.2623     1.9658
                   2      7.4946     3.6888     9.0668
                   3      18.234     12.365     20.381
                   4      32.728     25.042     34.259
                   5      50.482     41.228     42.524
                   6      62.313     43.263     55.402

[pi]               1      2.6707     1.0378     3.3476
                   2      7.4946     4.8346     8.4943
                   3      14.213     10.627     15.497
                   4      22.673     18.234     24.055
                   5      32.728     27.510     33.194
                   6      44.232     38.308     38.551

[DELTA][theta]    Mode      Boundary condition

                         S2-S2     S2-C     C-C

[pi]/3             1     7.4943   20.441   34.046
                   2     32.727   34.047   47.270
                   3     34.672   59.680   90.478
                   4     71.005   101.17   106.39
                   5     109.20   110.73   139.31
                   6     118.58   154.51   189.04

2[pi]/3            1     1.0378   4.7592   11.419
                   2     7.4945   14.121   20.445
                   3     18.234   26.580   34.049
                   4     32.727   34.065   38.533
                   5     34.672   45.344   59.694
                   6     50.481   59.944   60.023

[pi]               1     2.6706   1.5851   4.3142
                   2     7.4945   5.6604   9.2781
                   3     14.213   11.769   17.059
                   4     22.672   19.579   24.603
                   5     32.728   28.529   34.059
                   6     34.673   34.072   35.618

TABLE 6: Frequency parameters [OMEGA] for elastically supported
circular beams with different thickness ratios and restraint
rigidities ([R.sub.m] = 1 m, [[theta].sub.0] = [pi]/2).

                                 ku = [eta], kw = 0
h/[R.sub.m]     [eta]/D
                                  1          2         3

0.01           [10.sup.0]      1.5013     7.2263    22.817
               [10.sup.1]      1.5249     7.2266    22.817
               [10.sup.2]      1.7442     7.2296    22.818
               [10.sup.3]      3.1816     7.2654    22.825
               [10.sup.4]      6.8030     9.5884    22.907
               [10.sup.5]      7.0247     21.852    30.250

0.05           [10.sup.0]      1.5109     7.1856    22.477
               [10.sup.1]      1.6253     7.1873    22.477
               [10.sup.2]      2.4911     7.2050    22.483
               [10.sup.3]      6.0299     7.7159    22.538
               [10.sup.4]      7.0667     18.929    23.904
               [10.sup.5]      7.7839     22.923    44.333

0.10           [10.sup.0]      1.5207     7.0608    21.456
               [10.sup.1]      1.7408     7.0650    21.458
               [10.sup.2]      3.1733     7.1131    21.481
               [10.sup.3]      6.6891     9.3939    21.720
               [10.sup.4]      7.4853     19.902    25.792
               [10.sup.5]      8.7622     23.296    32.768

0.15           [10.sup.0]      1.5282     6.8657    19.865
               [10.sup.1]      1.8473     6.8741    19.872
               [10.sup.2]      3.7026     6.9755    19.942
               [10.sup.3]      6.7367     10.615    20.652
               [10.sup.4]      7.8630     17.816    24.479
               [10.sup.5]      8.6753     19.710    26.673

                                ku = 0, kw = [eta]
h/[R.sub.m]     [eta]/D
                                  1         2        3

0.01           [10.sup.0]      1.5040    7.2283   22.818
               [10.sup.1]      1.5515    7.2464   22.823
               [10.sup.2]      1.9406    7.4289   22.877
               [10.sup.3]      3.3673    9.2160   23.442
               [10.sup.4]      4.2514    15.578   29.613
               [10.sup.5]      4.3720    17.693   38.640

0.05           [10.sup.0]      1.5243    7.1955   22.480
               [10.sup.1]      1.7397    7.2863   22.506
               [10.sup.2]      2.8562    8.2070   22.780
               [10.sup.3]      4.1096    13.588   25.951
               [10.sup.4]      4.3410    17.262   36.620
               [10.sup.5]      4.3653    17.659   38.610

0.10           [10.sup.0]      1.5468    7.0804   21.461
               [10.sup.1]      1.9337    7.2624   21.514
               [10.sup.2]      3.3372    9.0383   22.066
               [10.sup.3]      4.1854    15.042   27.958
               [10.sup.4]      4.2996    16.861   34.871
               [10.sup.5]      4.3113    17.034   35.536

0.15           [10.sup.0]      1.5660    6.8947   19.872
               [10.sup.1]      2.0917    7.1678   19.946
               [10.sup.2]      3.5429    9.6553   20.751
               [10.sup.3]      4.1440    14.969   26.889
               [10.sup.4]      4.2159    15.961   28.964
               [10.sup.5]      4.2233    16.055   29.097

                                 ku = kw = [eta]
h/[R.sub.m]      [eta]/D
                                 1         2         3

0.01           [10.sup.0]      1.5066    7.2283    22.818
               [10.sup.1]      1.5770    7.2467    22.823
               [10.sup.2]      2.1392    7.4312    22.878
               [10.sup.3]      4.4370    9.2166    23.451
               [10.sup.4]      9.5523    15.677    29.817
               [10.sup.5]      17.584    25.942    40.813

0.05           [10.sup.0]      1.5374    7.1957    22.480
               [10.sup.1]      1.8517    7.2878    22.507
               [10.sup.2]      3.5131    8.2091    22.786
               [10.sup.3]      7.4008    13.649    26.053
               [10.sup.4]      16.954    18.929    37.704
               [10.sup.5]      18.203    33.998    52.083

0.10           [10.sup.0]      1.5725    7.0808    21.462
               [10.sup.1]      2.1335    7.2658    21.516
               [10.sup.2]      4.4148    9.0427    22.094
               [10.sup.3]      9.3746    15.112    28.313
               [10.sup.4]      17.240    20.798    37.990
               [10.sup.5]      19.364    26.704    43.372

0.15           [10.sup.0]      1.6040    6.8956    19.873
               [10.sup.1]      2.3677    7.1741    19.954
               [10.sup.2]      4.9852    9.6663    20.838
               [10.sup.3]      10.523    15.059    28.171
               [10.sup.4]      17.085    18.064    36.130
               [10.sup.5]      19.121    19.856    39.345

TABLE 7: Frequency parameters [OMEGA] for elastically supported
circular beams with different span angles and restraint rigidities
([R.sub.m] =1 m, h/[R.sub.m] = 0.1).

[[theta].sub.0]    [eta]/D       ku = [eta], kw = 0

                                 1        2        3

[pi]/3            [10.sup.0]   3.2671   17.105   46.643
                  [10.sup.1]   3.3484   17.110   46.647
                  [10.sup.2]   4.0665   17.154   46.688
                  [10.sup.3]   8.0451   17.639   47.081
                  [10.sup.4]   15.321   23.449   49.869
                  [10.sup.5]   18.965   31.838   55.957

2[pi]/3           [10.sup.0]   0.9258   3.6197   11.729
                  [10.sup.1]   1.2960   3.6210   11.731
                  [10.sup.2]   3.1142   3.6536   11.753
                  [10.sup.3]   3.6599   9.1322   12.165
                  [10.sup.4]   3.9841   12.050   23.150
                  [10.sup.5]   4.5641   13.485   25.680

[pi]              [10.sup.0]   0.5264   1.3719   4.6438
                  [10.sup.1]   1.0030   1.4177   4.6480
                  [10.sup.2]   1.3161   3.0900   4.7182
                  [10.sup.3]   1.3396   4.5807   9.8024
                  [10.sup.4]   1.4308   4.9059   10.622
                  [10.sup.5]   1.5524   5.4708   11.429

[[theta].sub.0]    [eta]/D       ku = 0, kw = [eta]

                                 1        2        3

[pi]/3            [10.sup.0]   3.3052   17.116   46.646
                  [10.sup.1]   3.6978   17.217   46.674
                  [10.sup.2]   6.0927   18.272   46.962
                  [10.sup.3]   10.265   27.077   50.196
                  [10.sup.4]   11.295   37.199   59.270
                  [10.sup.5]   11.403   38.393   59.956

2[pi]/3           [10.sup.0]   0.9197   3.6507   11.737
                  [10.sup.1]   1.2006   3.9285   11.815
                  [10.sup.2]   1.7696   5.9362   12.680
                  [10.sup.3]   1.9434   8.6156   18.070
                  [10.sup.4]   1.9635   9.0227   20.175
                  [10.sup.5]   1.9656   9.0622   20.360

[pi]              [10.sup.0]   0.4487   1.4211   4.6595
                  [10.sup.1]   0.5063   1.8014   4.8097
                  [10.sup.2]   0.5555   2.9075   6.2241
                  [10.sup.3]   0.5642   3.3007   8.2147
                  [10.sup.4]   0.5651   3.3429   8.4675
                  [10.sup.5]   0.5652   3.3471   8.4916

[[theta].sub.0]    [eta]/D        ku = kw = [eta]

                                 1        2        3

[pi]/3            [10.sup.0]   3.3143   17.117   46.646
                  [10.sup.1]   3.7784   17.222   46.679
                  [10.sup.2]   6.5964   18.303   47.008
                  [10.sup.3]   13.084   27.089   50.748
                  [10.sup.4]   23.265   37.875   68.723
                  [10.sup.5]   31.839   42.697   83.774

2[pi]/3           [10.sup.0]   0.9683   3.6508   11.737
                  [10.sup.1]   1.5359   3.9290   11.818
                  [10.sup.2]   3.3494   5.9983   12.716
                  [10.sup.3]   7.8507   9.3882   18.576
                  [10.sup.4]   9.2396   17.386   26.000
                  [10.sup.5]   10.492   19.461   32.929

[pi]              [10.sup.0]   0.5367   1.4241   4.6599
                  [10.sup.1]   1.0033   1.8520   4.8159
                  [10.sup.2]   2.3074   3.3600   6.3712
                  [10.sup.3]   3.1616   6.9331   9.9866
                  [10.sup.4]   3.4291   8.1148   15.259
                  [10.sup.5]   4.0046   8.7902   16.261
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Title Annotation:Research Article
Author:Wang, Xueren; Miao, Xuhong; Jia, Di; Gao, Shengyao
Publication:Shock and Vibration
Article Type:Report
Date:Jan 1, 2017
Words:10219
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