Shock and Vibration
Volume 2018 (2018), Article ID 1056397, 14 pages
https://doi.org/10.1155/2018/1056397

Research Article

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

Zhongmin Wang and Rongrong Li

School of Civil Engineering and Architecture, Xi’an University of Technology, Xi’an 710048, China

Correspondence should be addressed to Zhongmin Wang; wangzhongm@xaut.edu.cn

Received 26 October 2017; Accepted 29 January 2018; Published 28 March 2018

Academic Editor: Matteo Filippi

Copyright © 2018 Zhongmin Wang and Rongrong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1], 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 [2] 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. [3] 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 [4] 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 [5] 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. [6] 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 [7] 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 recent years, more studies related to transverse vibration of rotating nonuniform beams can be found in the following papers. Gunda and Ganguli [8] developed new interpolating functions which satisfy the static part of the homogenous governing differential equation for rotating uniform and tapered beams and imposed as a constraint equation in the derivation of the shape functions. Cheng et al. [9] investigated vibration characteristics of cracked rotating tapered beam by p-version finite element method and analyzed the effects of crack location, crack size, rotating speed, and hub radius on vibration characteristics of the beam. Bulut [10] considered out-of-rotation plane bending vibrations of rotating composite beam with periodically varying speed and further examined the effect of taper ratio on dynamic stability of this parametrically system. Banerjee and Jackson [11] addressed the free vibration problem of a rotating tapered Rayleigh beam by developing its dynamic stiffness matrix. In their analysis, the effects of centrifugal stiffening, an outboard force, an arbitrary hub radius, and importantly, the rotary inertia (Rayleigh beam) are included. Sarkar and Ganguli [12] proposed an inverse problem approach for dynamics of the rotating nonuniform Euler-Bernoulli beam and showed that there exists a certain class of rotating Euler-Bernoulli beam, having cantilever and pinned-free boundary conditions, which has a closed-form polynomial solution to its governing differential equation. At the same year, they also studied the free vibration of a nonhomogeneous rotating Timoshenko beam, having uniform cross-section, using an inverse problem approach, for both cantilever and pinned-free boundary conditions [13]. Tang et al. [14] studied free vibration of rotating tapered cantilever beams with rotary inertia using the integral equation method and analyzed the effects of the rotary inertia, angular speed, taper ratio, and hub radius. Li and Zhang [15] developed a new rigid-flexible coupled dynamic model to study dynamics of rotating axially functionally graded (FG) tapered beams by using the B-spline method (BSM) and observed some new interesting phenomena of frequency veering and mode shift in a rotating axially FG tapered beam when the B-S coupling effect is included. Huo and Wang [16] derived the nonlinear dynamic equations of a rotating, double-tapered, cantilever Timoshenko beam and analyzed the effect of angular speed, hub radius, slenderness ratio, and the height and width taper ratios on the natural frequencies of the rotating Timoshenko beam when the rotation beam is in a steady state, in which the extensional deformation of the beam is considered. Panchore et al. [17, 18] investigated free vibration problem of a rotating Euler-Bernoulli beam and a rotating Timoshenko beam using meshless local Petrov-Galerkin method and introduced a locking-free shape function formulation with an improved radial basis function interpolation. Ghafarian and Ariaei [19] presented a new procedure for determining natural frequencies and mode shapes of a system of elastically connected multiple rotating tapered beams through a differential transform method, which obey the Timoshenko beam theory, and discussed the effects of the rotational speed, hub radius, taper ratios, rotary inertia, shear deformation, slenderness ratio, and elastic layer stiffness coefficients on the natural frequencies. Ghafari and Rezaeepazhand [20] presented free vibration analysis of rotating composite beams with arbitrary cross-section using dimensional reduction method. Adair and Jaeger [1] used the computational approach of AMDM to analyze the free vibration of nonuniform Euler-Bernoulli beams under various boundary conditions, rotation speeds, and hub radii and simultaneously obtained the natural frequencies and corresponding closed-form series solution of the mode shape. Panchore and Ganguli [21] studied the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Other researchers also investigated the relevant second-order coupling term that represents longitudinal shrinking of the rotating beam caused by the transverse displacement. Li et al. [22] introduced a dynamic model of a rotating hub-functionally graded material beam system with the dynamic stiffening effect. In their work, the dynamic stiffening effect of the rotating hub-FGM beam system is captured by a second-order coupling term. Zhao and Wu [23] established the coupling equations of motion of a rotating three-dimensional cantilever beam to study the effects of Coriolis term and steady-state axial deformation on coupling vibration, which considered the longitudinal shrinkage caused by flapwise and chordwise bending displacement. At present, a large amount of articles relating to free vibration of rotating functionally graded plates or disk can be found (see, for instance, [24–26]).

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.

Figure 1: Schematic diagram of a rotating tapered cantilever beam with hollow circular cross-section.

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.

(a)

(b)

Figure 2: Geometry of rotating tapered cantilever beam with hollow circular cross-section and coordinate system.

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

(a)

(b)

Figure 3: Any hollow circular cross-section of the beam and infinitesimal arc length.

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 .

Figure 4: Schematic diagram of the deformation field.

The vector of point relative to original point O of inertial coordinate system OXY can be expressed as

where

in 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 obtained

where 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 by

where 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 as

where 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 as

Thus, 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 as

where 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 as

in 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 as

where , +

Let the solution of (23) be ; a differential equation of mode shape can be written as

where is dimensionless natural frequency.

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

4. Differential Quadrature Method

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 form

where , , 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 [14]. 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 .

Table 1: Comparison of the first three dimensionless natural frequencies of rotating solid tapered Euler-Bernoulli beams at , , , .

Present
Ref. [14]
Present
Ref. [14]
Present
Ref. [14]
Present
Ref. [14]
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

Table 2: Comparison of the first three dimensionless natural frequencies of rotating solid tapered Rayleigh beams for different at , , .

Present
Ref. [14]
Present
Ref. [14]
Present
Ref. [14]
Present
Ref. [14]
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

Figure 5: The first three dimensionless mode shapes for , , , , and .

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.

Table 3: The first three dimensionless natural frequencies of rotating solid tapered Rayleigh beams for different at , , .

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 6–8, 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.

Figure 6: Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for various at , .

Figure 7: Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for various , at , .

Figure 8: Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for various , , at , .

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 [29] of cantilever beam multiplied by . We can see that the numerical results in the present coincide well with the existing ones [29]. The first three dimensionless natural frequencies of the rotating uniform thin-wall cross-section beam decreased as the slenderness ratio was raised.

Table 4: The first three order dimensionless natural frequencies of rotating circular thin-wall cross-section beams at , , .

Ω

Present
Ref. [29]

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 , .

Table 5: The first three order dimensionless natural frequencies of rotating tapered hollow beams for various ratio of inner radius to the root radius .

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.

(a)

(b)

(c)

Figure 9: Variation of the first three dimensionless natural frequencies of rotating tapered hollow beam with dimensionless angular speed of the hub for different values of , and , at , : (a) 1st mode; (b) 2nd mode; and (c) 3rd mode.

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.

Table 6: The first three-order dimensionless natural frequencies of rotating tapered hollow beams for various 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 10: The first three-order dimensionless natural frequencies with dimensionless angular speed of the hub for different at .

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.

Figure 11: The first three-order dimensionless natural frequencies with ratios of hub radius to beam length for , at , , and .

Figure 12: The first three-order dimensionless natural frequencies with ratios of hub radius to beam length for , at , , and .

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).

References

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.

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.

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.

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.

T. Aksencer and M. Aydogdu, “Flapwise vibration of rotating composite beams,” Composite Structures, vol. 134, pp. 672–679, 2015.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

G. Zhao and Z. Wu, “Coupling vibration analysis of rotating three-dimensional cantilever beam,” Computers & Structures, vol. 179, pp. 64–74, 2017.

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.

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.

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.

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.

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.

L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill, Singapore, 1986.