/ / Article

Research Article | Open Access

Volume 2018 |Article ID 1056397 | https://doi.org/10.1155/2018/1056397

Zhongmin Wang, Rongrong Li, "Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section", Shock and Vibration, vol. 2018, Article ID 1056397, 14 pages, 2018. https://doi.org/10.1155/2018/1056397

# Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

Accepted29 Jan 2018
Published28 Mar 2018

#### Abstract

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

#### 1. Introduction

The dynamical problem of rotating uniform and nonuniform solid beam is widely used in many practical engineering, such as helicopter rotor blades and wind turbine blades. Also, the dynamics of rotating tapered hollow beams is of practical significant, for example, rotating tank gun barrel (hollow circular cross-section). As pointed out in , in dynamical analysis, a rotating beam differs from a nonrotating beam because it also possesses centrifugal stiffness and Coriolis effects that influence its dynamical characteristics. Besides the above effects, there are some complicated factors, including the secondary coupling deformation term, coupling effect, and the variable coefficient differential equation. Therefore, the methodologies and solutions for rotating nonuniform beam turn out to be cumbersome.

The dynamic analysis of rotating uniform beams has been the subject of many articles and received much attention. Yoo and Shin  investigated the effect of centrifugal force for rotating uniform cantilever beams and used a modal formulation to obtain the natural frequencies and mode shape. Tsai et al.  proposed the corotational finite element method combined with floating frame method to derive differential equation of motion for the rotating inclined Euler uniform beams at constant angular speed and investigated the steady-state deformation and the natural frequencies of infinitesimal free vibration. Vinod Kumar and Ganguli  used the static part of the homogeneous differential equation of violin strings to obtain new shape functions for the finite element analysis of rotating Timoshenko beams. Aksencer and Aydogdu  studied flapwise vibration of rotating composite beams, which are used in different beam theories, including Euler-Bernoulli, Timoshenko, and Reddy beam theories, and obtained some results for different orthotropy ratios, rotation speed, hub ratio, length to thickness ratio of the rotating composite beam, and different boundary conditions. Li et al.  developed a new dynamic model of a planar rotating hub-beam system, where the beam is of an Euler-Bernoulli type and the deformation of the beam is described by the slope angle and stretch strain of the centroid line of the beam. They obtained four corresponding spatially discretized models, that is, ESA, FOSA, SOSA, and SSOSA model, and calculated natural frequencies and mode shapes of the system with the chordwise bending and stretching coupling effect. J. W. Lee and J. Y. Lee  investigated the effects of cracks on the natural frequencies of a rotating Bernoulli-Euler beam using a new numerical method in which these effects can be computed simply using the transfer matrix method.

In the above referenced articles, the model of rotating uniform beam and nonuniform beam have been considered, especially for rotating tapered beam, which has rectangular cross-section with linearly varying width and constant height, with linearly varying height and constant width, and with linearly varying width and height. However, to the best of the authors’ knowledge, no research work related to the dynamics of a rotating beam with varying hollow circular cross-section (or rotating tapered hollow beam) has been yet presented. The dynamical of the system is of practical significant because rotating tapered hollow beams are widely used as structural components in the engineering field.

In this paper, the investigation proceeds as follows. First the geometry parameters of a rotating tapered cantilever beam with hollow circular cross-section are described, and the governing differential equation of motion for transverse free vibration of a rotating tapered Rayleigh beam is derived using Hamilton variational principle. Next, for harmonic oscillation, the differential equation with variable coefficients is solved using the differential quadrature method, and an eigen-equation of the system for dimensionless parameters is formulated in explicit algebraic form. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

#### 2. Parameters of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

Figure 1 shows the schematic diagram of a rotating tapered cantilever beam with hollow circular cross-section, which has length L, elastic modulus , and density ρ and is fixed at point of a rigid hub with radius . The hub is rotating in the horizontal plane around point with a rotating angular speed . A fixed (inertial) planar coordinate system OXY through the fixed point and a floating coordinate system oxy that is tangent to the attachment point of the beam to the hub are prescribed, respectively. The latter relative to the former rotates with a rotation angle of large range motion.

The rotating beam with varying hollow circular cross-section is considered, whose outer diameter varies linearly and the inner diameter keeps unchanged along its longitudinal -axis, as shown in Figure 2. The beam has the root radius (at x = 0) and the end radius (at x = L), the wall thickness versus the coordinate , and the radius at the middle line of the wall thickness for any cross-section. A local coordinate system with a normal direction and tangential direction at the central line of hollow circular cross-section is adopted.

The average radius and the wall thickness can be expressed, respectively, as follows:where is called the taper ratio of cross-section. It is also stipulated that the section size of the beam decreases and increases linearly from the root to the end, that is, , in which is denoted by (called the ratio of inner radius to the root radius). There are two particular cases: one is a uniform beam when , that is, , and the other is a particular varying cross-section beam when , that is, .

The area of any cross-section and its moment of inertia with respect to axis can be expressed, respectively, as where and are the area and the moment of inertia with respect to axis of the root cross-section of the beam, respectively, , and are given by

Any hollow circular cross-section of the beam is shown as Figure 3. The vertical coordinate of any point M can be expressed, respectively, as

#### 3. Differential Equation of Motion

##### 3.1. The Description of the Deformation Field

Figure 4 shows that is a radius vector of the original point o of floating coordinate system oxy with respect to the point O of the inertial coordinate system OXY, is a radius vector of any point , which is on the axis of the beam before deformation, the point P is the positions of the point after deformation, and is a displacement vector of the point .

The vector of point relative to original point O of inertial coordinate system OXY can be expressed aswherein which and are the axial displacement and the transverse bending deflection, respectively; is axial displacement of any point caused by transverse bending, in which the prime denotes spatial derivatives with respect to ; is a second-order coupling term that represents longitudinal shrinking of the rotating tapered beam with hollow circular cross-section beam caused by the transverse displacement . It includes the coupling effect between the axial displacement and transverse displacement of rotating tapered hollow beam.

Taking the derivative with respect to time for (4), the velocity vector of the point at the inertial coordinate system can be obtainedwhere is a antisymmetric matrix relating to angular speed , in which the over dot denotes derivative with respect to time .

##### 3.2. Differential Equation of Motion
###### 3.2.1. Kinetic Energy of System

Kinetic energy of system consists of two parts: one is the kinetic energy of the hub and the other is the kinetic energy of the beam with hollow circular cross-section; namely,

The kinetic energy of the hub is given bywhere is rotary inertia of the hub with respect to central axis.

Neglecting the axis displacement of the beam with hollow circular cross-section, the kinetic energy of the per unit length of the beam can be expressed aswhere is the effective mass of the per unit length of the beam; it can be expressed as

The kinetic energy of the beam with hollow circular cross-section can be rewritten asThus, substituting (9) and (12) into (8), the total kinetic energy of the system can be expressed as

###### 3.2.2. Strain Energy of System

Neglecting the deformation energy caused by shear deformation, the strain energy of rotating beam with hollow circular cross-section is written aswhere is the elastic modulus of material; and represent the normal stress and normal strain in direction, respectively.

In (6), ignoring the axial displacement and nonlinear term, the normal strain can be got by the relationship between strain and displacement: namely,

According to Figure 3, a geometrical relationship is given by

Thus, the strain energy of the beam with hollow circular cross-section can be rewritten as

###### 3.2.3. Derivation of Differential Equation of Motion

In this paper, Hamilton variational principle for elastic system is used to derive the differential equation of motion. The basis form of Hamilton variational principle can be showed as

Substituting (13) and (17) into (18) and implementing a lot of variational operation and integration by parts, a variational expression is given by where .

Because second-order coupling deformation term   is a second-order small quantity, we can neglect some nonlinear terms and time-varying coupling terms in (19) to simplify the equation appropriately. Thus, the differential equation of motion of the rotating beam with hollow circular cross-section can be derived

Taking uniform rotation into consideration, that is, , , a differential equation of motion of the beam can be expressed asin which ,  /.

The boundary conditions of the cantilever beam are as follows:

##### 3.3. Dimensionless Method of the Equation

For simplicity, the following dimensionless quantities are introduced: , , , (called dimensionless angular speed of the hub), (called ratio of hub radius to beam length), and (called slenderness ratio).

Dimensionless expression of (21) can be expressed aswhere , +

Let the solution of (23) be ; a differential equation of mode shape can be written aswhere is dimensionless natural frequency.

The dimensionless forms of the boundary conditions (22) are rewritten as

In order to solve the differential equation with variable coefficients (24) and deal with the boundary conditions (25), the differential quadrature method (DQM) and the method are used, respectively. Selecting nonuniform nodes, the node coordinates are as follows [27, 28]:where is the numbers of nodes and is small parameter.

According to the DQM procedures, (24) can be discretized as follows:where ,  , , .

The boundary conditions (25) can also be discretized as follows:

Equations (27), (28a), (28b), (28c), and (28d) can be rewritten as a simpler matrix formwhere , , and are all square matrix; is column matrix.

Equation (29) denotes a generalized eigenvalue problem. Based on the linear algebra theory, the sufficient and necessary conditions of homogeneous linear algebraic equations which create the nonzero solution are that the determinant of coefficients equals zero; thus, one can arrive to the following generalized eigen-equation:where the square matrices , , and involve some parameters, such as ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section, each dimensionless natural frequencies, and dimensionless angular speed of the hub.

#### 5. Numerical Results and Analyses

##### 5.1. Rotating Tapered Cantilever Beam with Solid Circular Cross-Section

In this paragraph, setting in (24), we can obtain the differential equation of motion for rotating tapered beam with solid circular cross-section, where taper ratio is defined. It becomes evident that if the taper ratio and , they are varying solid circular cross-section beam with zero radius at free end and entirely uniform solid circular cross-section, respectively. Prior to the presentation of our numerical results, let us first consider three particular cases to confirm the effectiveness of the present approach: a simple uniform nonrotating cantilever beam, a rotating tapered Euler-Bernoulli cantilever beam, and a rotating tapered Rayleigh cantilever beam, which is given by setting parameters , , , and , parameters , , and parameters , , respectively. In three cases, we calculate the first three-order dimensionless natural frequencies by selecting for different taper ratio of cross-section and dimensionless angular speed, and some numerical results are tabulated in Tables 1 and 2  . From the two tables, we can see that the numerical results in the present coincide well with the existing ones . These verify that the method presented in this paper is efficient and accurate. The first three dimensionless mode shapes are shown as Figure 5 for , , , , and .

 Present Ref.  Present Ref.  Present Ref.  Present Ref.  Present 0 3.5164 3.5160 3.9570 3.9567 4.6257 4.6252 5.8293 5.8231 8.7445 22.0366 22.0345 20.8091 20.807 19.5514 19.5476 18.5183 18.480 20.1042 61.6960 61.6972 55.3500 55.3304 48.5953 48.5789 41.4611 41.321 36.5279 5 6.4499 6.4495 6.7732 6.7729 7.2905 7.2901 8.2653 8.2620 10.6634 25.4483 25.4461 24.0673 24.0660 22.6379 22.6360 21.4209 21.384 22.8601 65.2118 65.2050 58.6518 58.6364 51.7012 51.6918 44.4176 44.269 41.0619 10 11.2029 11.2023 11.4859 11.4856 11.9419 11.9415 12.7928 12.791 14.7477 33.6417 33.6404 31.8863 31.8895 30.0289 30.0299 28.3381 28.301 29.4213 74.6671 74.6493 67.5293 67.5316 60.0369 60.0399 52.2738 52.100 51.2478 15 16.1445 - 16.4165 - 16.8587 - 17.6745 - 19.4476 43.9323 - 41.7005 - 39.2756 - 36.9308 - 37.5106 87.9543 - 80.0153 - 71.7371 - 63.1548 - 59.9363
 Present Ref.  Present Ref.  Present Ref.  Present Ref.  Present 0 3.5073 3.5070 3.9486 3.9483 4.6174 4.6168 5.8198 5.8136 8.7267 21.6497 21.6477 20.5498 20.5475 19.3883 19.3846 18.4214 18.3834 20.0372 59.2069 59.2073 53.7093 53.6911 47.6173 47.6021 40.9588 40.825 36.5169 5 6.4254 6.4251 6.7524 6.7521 7.2720 7.2717 8.2469 8.2436 10.6360 24.9759 24.9737 23.7510 23.7497 22.4394 22.4375 21.3029 21.266 22.7786 62.5464 663.5392 56.8916 56.8774 50.6483 50.6397 43.8720 43.730 41.0472 10 11.1604 11.1598 11.4494 11.4490 11.9087 11.9083 12.7584 12.767 14.6997 32.9667 32.9654 31.4395 31.4425 29.7515 29.7521 28.1722 28.136 29.3002 71.5142 71.4977 65.4444 65.4472 58.7850 58.7878 51.6155 51.450 50.5772 15 16.0851 - 16.3650 - 16.8105 - 17.6223 - 19.3744 43.0010 - 41.0951 - 38.9051 - 36.7064 - 37.3335 84.0806 - 77.4631 - 70.2081 - 62.3423 - 59.2553

Furthermore, in the case of , for example, , that is, the radius of beam cross-section at free end is more than one at the cantilevered end, the first three-order dimensionless natural frequencies of a rotating tapered Rayleigh cantilever solid beam for three dimensionless angular speeds are tabulated in Table 3. It can be seen in Table 3 that as a whole, the first three-order dimensionless natural frequencies increase with the increase of dimensionless angular speeds.

 0 3.1877 2.9421 2.7461 2.5882 2.4704 2.4129 2.4613 2.6858 22.6579 23.5718 24.3952 25.1281 25.7580 26.2480 26.5270 26.4787 64.2417 68.9100 73.2615 77.3249 81.1317 84.7333 88.2071 91.6568 5 6.1987 6.0331 5.9156 5.8549 5.8827 6.0577 6.4683 7.2429 26.0836 27.0745 27.9483 28.6698 29.2062 29.4743 29.3543 28.6666 67.7177 72.5014 76.9672 81.1744 85.1868 89.0823 92.9551 96.9133 10 10.9632 10.8352 10.7906 10.8785 11.1850 11.8383 13.0323 15.1307 34.3001 35.4363 36.3539 36.9976 37.2721 37.0347 36.0682 33.9720 77.0509 82.1465 86.9066 91.4446 95.8797 100.3326 104.9216 109.7575 15 15.9065 15.8297 15.9030 16.2187 16.9153 18.1923 20.3830 24.3670 44.5894 45.8474 46.7372 47.1832 47.0645 46.1947 44.2444 40.3254 90.1006 95.6133 100.7463 105.6461 110.4616 115.3326 120.3820 125.7134

Figure 6 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different ratios of hub radius to beam length at , . It can be found from Figure 6 that, with the increase of the ratios of hub radius to beam length, the first three-order dimensionless natural frequencies increase. Figure 7 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different slenderness ratios at , . It can be seen from Figure 7 that, for two different , the increase of slenderness ratio has scarce influence on the first-order dimensionless natural frequencies; however, it has significant influence on the second- and the third-order dimensionless natural frequency. This shows that the increase of rotary inertia makes the system natural frequency decrease, and this conclusion is consistent with those given by Timoshenko beam. Figure 8 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different taper ratios of cross-section at , . It can be seen from Figure 8 that, with the increase of taper ratio, the first three-order dimensionless natural frequencies of the system decrease. It is noted that the increase of taper ratio has scarce influence on the first-order dimensionless natural frequency, it has slight influence on the second-order dimensionless natural frequency, and it has obviously an effect on the third-order dimensionless natural frequency. It should be pointed out that, as shown in Figures 68, in the case of the given ratios of hub radius to beam length, the slenderness ratio, and taper ratio of cross-section, the first three dimensionless natural frequencies of rotating tapered solid beam monotonically increase with dimensionless angular speed of the hub.

##### 5.2. Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

For a rotating tapered cantilever beam with hollow circular cross-section, its inner diameter , that is, , and the taper ratio is defined. When , the outer radius of beam at free end is the same as its inner radius. When , the beam is entirely uniform hollow circular cross-section. This section will mainly discuss the effect of ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three dimensionless natural frequencies of rotating tapered hollow beams.

As a particular case of hollow circular cross-section, Table 4 gives the variation of the first three dimensionless natural frequencies of rotating uniform thin-wall cross-section beams with dimensionless angular speed of the hub for three different slenderness ratios at , , and . It should be pointed out that the first three dimensionless natural frequencies of nonrotating beam in Table 4 equal the first three dimensionless natural frequencies 3.5156, 22.0336, and 61.7010  of cantilever beam multiplied by . We can see that the numerical results in the present coincide well with the existing ones . The first three dimensionless natural frequencies of the rotating uniform thin-wall cross-section beam decreased as the slenderness ratio was raised.

 Ω Present Ref.  0 4.7781 4.7771 4.7555 4.5844 29.9438 29.9393 28.9935 23.6666 83.8339 83.8393 77.8914 54.3817 5 7.2326 - 7.1843 6.8112 32.5368 - 31.4682 25.4442 86.4579 - 80.2848 55.9058 10 11.7433 - 11.6598 10.9967 39.2955 - 37.9215 30.0678 93.8425 - 87.0105 60.1779 15 16.6095 - 16.4939 15.5592 48.4647 - 46.6776 36.2668 104.8728 - 97.0268 66.4979
###### 5.2.1. Effect of the Hub Dimensionless Angular Speed and Taper Ratio of Cross-Section

Table 5 gives the variation of the first three dimensionless natural frequencies of rotating tapered hollow beams with dimensionless angular speed of the hub for two different slenderness ratios and three ratios of inner radius to the root radius at . With this table, it is obvious that the first three dimensionless natural frequencies increase with the ratio of inner radius to the root radius, except the second-order dimensionless natural frequencies at , .

 0 4.0743 4.4869 5.4689 4.0021 4.3928 5.3165 21.1137 22.8085 25.9958 19.1933 20.4539 22.8066 55.0759 59.0308 65.6330 45.1361 47.2503 50.6619 5 6.8283 7.0878 7.7613 6.6530 6.8756 7.4647 24.2176 25.6314 28.3193 21.8856 22.8463 24.6981 58.1504 61.7981 67.8694 47.5230 49.3294 52.2535 10 11.5023 11.6837 12.1584 11.1927 11.3104 11.6424 31.7436 32.6600 34.3608 28.4415 28.8297 29.6427 66.4593 69.3894 74.1474 53.9527 55.0169 56.7149 15 16.4109 16.5675 16.9725 15.9740 16.0415 16.2467 41.2695 41.7660 42.5327 36.7455 36.5930 36.3560 78.2143 80.3361 83.4935 62.9728 63.1570 63.3260

Figure 9 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for and at , , . With this figure, it is also understood that the first three-order dimensionless natural frequencies of rotating tapered hollow beam monotonically increase with dimensionless angular speed of the hub. Meanwhile, it is further noted in Figure 9(a) that, with the increase of the taper ratio of cross-section, the first-order dimensionless natural frequency of the system is reduced slightly. However, in Figures 9(b) and 9(c), with the increase of the taper ratio of cross-section, the values of the second- and third-order dimensionless natural frequencies of the system are increased.

In addition, it is also observed in Figure 9 that the increase of slenderness ratio makes three dimensionless natural frequencies of the system decrease, contrasting Figures 9(a), 9(b), and 9(c), obviously, and the effect of the slenderness ratio on the second- and third-order dimensionless natural frequencies is relatively greater than the first-order dimensionless natural frequency.

###### 5.2.2. Effect of Ratios of Hub Radius to Beam Length and Slenderness Ratio

As it was expected in Table 6, the change of ratios of hub radius to beam length has no effect on the natural frequencies of nonrotating tapered beam . It is also observed in Table 6 that, for rotating tapered beam, the dimensionless natural frequencies of the system increase with ratios of hub radius to beam length.

 0 4.0021 4.0021 4.0021 4.0743 4.0743 4.0743 19.1933 19.1933 19.1933 21.1137 21.1137 21.1137 45.1361 45.1361 45.1361 55.0759 55.0759 55.0759 5 6.9391 7.9790 9.1094 7.1163 8.1648 9.3065 22.2340 23.5717 25.1345 24.6043 26.0908 27.8311 47.8600 49.1793 50.7682 58.5679 60.2042 62.1795 10 11.8553 14.1905 16.6439 12.1676 14.5191 16.9959 29.4809 29.4809 37.4245 32.8996 37.1390 41.7938 55.0961 59.0961 64.2301 67.8847 73.2615 79.3845 15 17.0078 20.6152 24.3667 17.4498 21.0838 24.8719 38.5013 44.7447 51.3531 43.2329 50.2622 57.7703 65.0693 72.6549 80.8113 80.8610 90.5387 101.1210

Figure 10 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for at , . With this figure, the first three-order dimensionless natural frequencies of the system increase with dimensionless angular speed of the hub and ratios of hub radius to beam length.

Figure 11 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different taper ratios of cross-section , at , , . Figure 12 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different slenderness ratios at , , . It can be seen in Figures 11 and 12 that, for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of the system are almost linearly increased. Meanwhile, it is noted that the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.

#### 6. Conclusion

In this paper, a new type of transverse vibration of a rotating tapered cantilever beam with linearly varying solid and hollow circular cross-section, that is, rotating tapered beam, was presented. The rotating beam, which is considered a tapered cantilever beam, is modeled by the Rayleigh beam theory. Considering the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. A differential quadrature method to solve the above-mentioned differential equation with variable coefficients was employed to simulate the dynamical behaviors of tapered rotating beams. Also, for two types of rotating tapered beams with solid and hollow circular cross-section, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are depicted. The main results of this study are summarized as follows.

When the rotating angular speed is constant, in the case of the given ratios of the hub radius to beam length, the slenderness ratio, and the taper ratio of cross-section, the first three-order dimensionless natural frequencies of rotating tapered solid and hollow beams monotonically ascend as the hub dimensionless angular speed increases. For a rotating tapered hollow beam at a constant angular speed, the first-order dimensionless natural frequency of the system is reduced slightly with the increase of the taper ratio of cross-section, the values of the second- and the third-order dimensionless natural frequencies of the system are increased; for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of system almost linearly increase, and the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.

#### Conflicts of Interest

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

#### Acknowledgments

The authors are grateful for the support from the National Natural Science Foundation of China (Grant no. 11472211).

1. D. Adair and M. Jaeger, “Simulation of tapered rotating beams with centrifugal stiffening using the Adomian decomposition method,” Applied Mathematical Modelling, vol. 40, no. 4, pp. 3230–3241, 2016. View at: Publisher Site | Google Scholar
2. H. H. Yoo and S. H. Shin, “Vibration analysis of rotating cantilever beams,” Journal of Sound and Vibration, vol. 212, no. 5, pp. 807–828, 1998. View at: Publisher Site | Google Scholar
3. M. H. Tsai, W. Y. Lin, Y. C. Zhou, and K. M. Hsiao, “Investigation on steady state deformation and free vibration of a rotating inclined Euler beam,” International Journal of Mechanical Sciences, vol. 53, no. 12, pp. 1050–1068, 2011. View at: Publisher Site | Google Scholar
4. A. S. Vinod Kumar and R. Ganguli, “Violin string shape functions for finite element analysis of rotating Timoshenko beams,” Finite Elements in Analysis and Design, vol. 47, no. 9, pp. 1091–1103, 2011. View at: Publisher Site | Google Scholar | MathSciNet
5. T. Aksencer and M. Aydogdu, “Flapwise vibration of rotating composite beams,” Composite Structures, vol. 134, pp. 672–679, 2015. View at: Publisher Site | Google Scholar
6. L. Li, W. D. Zhu, D. G. Zhang, and C. F. Du, “A new dynamic model of a planar rotating hub-beam system based on a description using the slope angle and stretch strain of the beam,” Journal of Sound and Vibration, vol. 345, pp. 214–232, 2015. View at: Publisher Site | Google Scholar
7. J. W. Lee and J. Y. Lee, “In-plane bending vibration analysis of a rotating beam with multiple edge cracks by using the transfer matrix method,” Meccanica, vol. 52, no. 4-5, pp. 1143–1157, 2017. View at: Publisher Site | Google Scholar
8. J. B. Gunda and R. Ganguli, “New rational interpolation functions for finite element analysis of rotating beams,” International Journal of Mechanical Sciences, vol. 50, no. 3, pp. 578–588, 2008. View at: Publisher Site | Google Scholar
9. Y. Cheng, Z. Yu, X. Wu, and Y. Yuan, “Vibration analysis of a cracked rotating tapered beam using the p-version finite element method,” Finite Elements in Analysis and Design, vol. 47, no. 7, pp. 825–834, 2011. View at: Publisher Site | Google Scholar | MathSciNet
10. G. Bulut, “Effect of taper ratio on parametric stability of a rotating tapered beam,” European Journal of Mechanics - A/Solids, vol. 37, pp. 344–350, 2013. View at: Publisher Site | Google Scholar | MathSciNet
11. J. R. Banerjee and D. R. Jackson, “Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution,” Computers & Structures, vol. 124, pp. 11–20, 2013. View at: Publisher Site | Google Scholar
12. K. Sarkar and R. Ganguli, “Modal tailoring and closed-form solutions for rotating non-uniform Euler-Bernoulli beams,” International Journal of Mechanical Sciences, vol. 88, pp. 208–220, 2014. View at: Publisher Site | Google Scholar
13. K. Sarkar and R. Ganguli, “Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams,” Meccanica, vol. 49, no. 6, pp. 1469–1477, 2014. View at: Publisher Site | Google Scholar
14. A.-Y. Tang, X.-F. Li, J.-X. Wu, and K. Y. Lee, “Flapwise bending vibration of rotating tapered Rayleigh cantilever beams,” Journal of Constructional Steel Research, vol. 112, pp. 1–9, 2015. View at: Publisher Site | Google Scholar
15. L. Li and D. Zhang, “Dynamic analysis of rotating axially FG tapered beams based on a new rigid-flexible coupled dynamic model using the B-spline method,” Composite Structures, vol. 124, pp. 357–367, 2015. View at: Publisher Site | Google Scholar
16. Y. Huo and Z. Wang, “Dynamic analysis of a rotating double-tapered cantilever Timoshenko beam,” Archive of Applied Mechanics, vol. 86, no. 6, pp. 1147–1161, 2015. View at: Publisher Site | Google Scholar
17. V. Panchore, R. Ganguli, and S. N. Omkar, “Meshless local Petrov-Galerkin method for rotating Euler-Bernoulli beam,” CMES: Computer Modeling in Engineering & Sciences, vol. 104, no. 5, pp. 353–373, 2015. View at: Google Scholar
18. V. Panchore, R. Ganguli, and S. N. Omkar, “Meshless local Petrov-Galerkin method for rotating Timoshenko beam: a locking-free shape function formulation,” CMES: Computer Modeling in Engineering & Sciences, vol. 108, no. 4, pp. 215–237, 2015. View at: Google Scholar
19. M. Ghafarian and A. Ariaei, “Free vibration analysis of a system of elastically interconnected rotating tapered Timoshenko beams using differential transform method,” International Journal of Mechanical Sciences, vol. 107, pp. 93–109, 2016. View at: Publisher Site | Google Scholar
20. E. Ghafari and J. Rezaeepazhand, “Vibration analysis of rotating composite beams using polynomial based dimensional reduction method,” International Journal of Mechanical Sciences, vol. 115-116, pp. 93–104, 2016. View at: Publisher Site | Google Scholar
21. V. Panchore and R. Ganguli, “Quadratic B-spline finite element method for a rotating non-uniform Rayleigh beam,” Structural Engineering and Mechanics, vol. 61, no. 6, pp. 765–773, 2017. View at: Publisher Site | Google Scholar
22. L. Li, D. G. Zhang, and W. D. Zhu, “Free vibration analysis of a rotating hub-functionally graded material beam system with the dynamic stiffening effect,” Journal of Sound and Vibration, vol. 333, no. 5, pp. 1526–1541, 2014. View at: Publisher Site | Google Scholar
23. G. Zhao and Z. Wu, “Coupling vibration analysis of rotating three-dimensional cantilever beam,” Computers & Structures, vol. 179, pp. 64–74, 2017. View at: Publisher Site | Google Scholar
24. L. Li and D. G. Zhang, “Free vibration analysis of rotating functionally graded rectangular plates,” Composite Structures, vol. 136, no. 2, pp. 493–504, 2016. View at: Publisher Site | Google Scholar
25. H. Zafarmand and M. Kadkhodayan, “Nonlinear analysis of functionally graded nanocomposite rotating thick disks with variable thickness reinforced with carbon nanotubes,” Aerospace Science and Technology, vol. 41, pp. 47–54, 2015. View at: Publisher Site | Google Scholar
26. T. Dai and H.-L. Dai, “Thermo-elastic analysis of a functionally graded rotating hollow circular disk with variable thickness and angular speed,” Applied Mathematical Modelling, vol. 40, no. 17-18, pp. 7689–7707, 2016. View at: Publisher Site | Google Scholar
27. Y. F. Zhou and Z. M. Wang, “Application of the differential quadrature method to free vibration of viscoelastic thin plate with linear thickness variation,” Meccanica, vol. 49, no. 12, pp. 2817–2828, 2014. View at: Publisher Site | Google Scholar
28. Y. F. Zhou and Z. M. Wang, “Vibrations of axially moving viscoelastic plate with parabolically varying thickness,” Journal of Sound and Vibration, vol. 316, no. 1–5, pp. 198–210, 2008. View at: Publisher Site | Google Scholar
29. L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill, Singapore, 1986.