Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 8178703, 8 pages
https://doi.org/10.1155/2017/8178703

Research Article

Out-of-Plane Vibration of Curved Uniform and Tapered Beams with Additional Mass

Hamdi Alper Özyiğit,¹ Mehmet Yetmez,² and Utku Uzun²

¹Department of Mechatronics Engineering, Bulent Ecevit University, 67100 Zonguldak, Turkey
²Department of Mechanical Engineering, Bulent Ecevit University, 67100 Zonguldak, Turkey

Correspondence should be addressed to Mehmet Yetmez; yetmez@beun.edu.tr

Received 4 May 2017; Revised 14 June 2017; Accepted 18 June 2017; Published 18 July 2017

Academic Editor: Edoardo Artioli

Copyright © 2017 Hamdi Alper Özyiğit et al. 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.

As there is a gap in literature about out-of-plane vibrations of curved and variable cross-sectioned beams, the aim of this study is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. Out-of-plane free vibration of curved uniform and tapered beams with additional mass is also investigated. Finite element method is used for all analyses. Curvature type is assumed to be circular. For the different boundary conditions, natural frequencies of both symmetrical and unsymmetrical tapered beams are given together with that of uniform tapered beam. Bending, torsional, and rotary inertia effects are considered with respect to no-shear effect. Variations of natural frequencies with additional mass and the mass location are examined. Results are given in tabular form. It is concluded that (i) for the uniform tapered beam there is a good agreement between the results of this study and that of literature and (ii) for the symmetrical curved tapered beam there is also a good agreement between the results of this study and that of a finite element model by using MSC.Marc. Results of out-of-plane free vibration of symmetrically tapered beams for specified boundary conditions are addressed.

1. Introduction

Due to their importance and wide using areas in engineering, dynamics of curved beams have been prevalently investigated by many researchers. Particularly, vibration analysis of curved beams has been a remarkable research area in mechanics due to its various applications. For the complicated problems of many architectural and structural implementations, curved beams with variable cross-sections are generally main parts, such that beams can be used not only in the design of rib, curved continuous bridge, and ship, but also in gear, pump, turbine and so on. Kawakami et al. [1] present an approximate method to study the analysis for the in-plane and out-of-plane free vibration of horizontally curved beams with arbitrary shapes and variable cross-sections. It is stated that the characteristic equation for free vibration can be derived by applying the Green function, which is obtained as a discrete type solution of differential equations governing the flexural behavior of the curved beam under the action of a concentrated load. Krishnan and Suresh [2] investigate the effect of shear deformation and rotary inertia on natural and cross over frequencies of both curved uniform and nonuniform (varied cross-section) beams by using a simple cubic linear beam element. Results of free and forced in-plane vibrations of circular arches with variable cross-sections are given by Tong et al. [3] using the Kirchhoff assumptions for thin beams and taking the neutral axis as inextensible. Huang et al. [4] take into consideration out-of-plane dynamics of beams with arbitrarily varying curvature and cross-section by dynamic stiffness matrix method. Viola et al. [5, 6] investigate the in-plane linear free vibrations of nonuniform circular arches with damaged configurations by using analytical and generalized differential quadrature (GDQ) methods. Tornabene et al. [7] apply GDQ to solve linear dynamics of the arch with different geometrical and boundary conditions. Viola et al. [8, 9] present a solution of free harmonic vibration problem of multistepped and multidamaged arches by using analytical and GDQ with domain decomposition technique (GDQE) methods. In-plane free vibration of circular arches is worked by Liu and Wu [10] using the generalized differential quadrature rule (GDQR). Arches with uniform, continuously varying, and stepped cross-sections are presented to illustrate the validity and accuracy of the GDQR. Karami and Malekzadeh [11] analyze in-plane free vibration of circular arches with varying cross-sections by developing a differential quadrature method. A boundary element method is developed by Sapountzakis [12] for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary variable cross-section. Chen [13] uses the differential quadrature element method (DQEM) for in-plane vibration analysis models of arbitrarily curved beam structures. Tufekci and Dogruer [14] analyze free out-of-plane vibrations of a circular arch with uniform cross-section with respect to effects of transverse shear and rotary inertia due to the both flexural and torsional vibrations. Yang et al. [15] investigate free in-plane vibration of uniform and nonuniform curved beams with variable curvatures, including the effects of the axis extensibility, shear deformation, and rotary inertia by using extended-Hamilton principle. Shin et al. [16] solve the problem of vibration of a circular arch with variable cross-section using differential transformation and generalized differential quadrature. Similarly, on one hand, Malekzadeh et al. [17–19] present in-plane free vibration of laminated/functionally graded circular arches according to the differential quadrature method. On the other hand, Malekzadeh et al. [20] investigate out-of-plane free vibration of functionally graded circular curved beams in thermal environment. Another work about functionally graded circular curved beams is held by Piovan et al. [21]. In order to obtain natural frequencies for “bare” and “loaded” curved beams for different end conditions, Wu et al. [22] present an effective approach for free in-plane vibration analysis of a curved beam with various arbitrary concentrated elements. Özyiğit and Işık [23] investigate in-plane vibration of curved beams with variable cross-sections carrying additional mass to compute frequencies of symmetrical and nonsymmetrical tapered beams with different mass locations. Ni et al. [24] pay attention to in-plane and out-of-plane free vibration and stability of a curved rod in flow. Vibration suppression of curved beams is regarded by Rostam et al. [25]. The performance of two curved beam element models based on coupled polynomial displacement fields is taken into account by Ishaquddin et al. [26] for out-of-plane vibration of arches. The coupled polynomial interpolation fields are derived independently for Timoshenko and Euler-Bernoulli beam elements using the force-moment equilibrium equations. Lee and Jeong [27] consider flexural and torsional free vibrations of horizontally curved beams on Pasternak foundations.

Since out-of-plane vibrations of curved and variable cross-sectioned beams are not widely studied, the purpose of this work is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. In this analysis, the linear free out-of-plane vibrations of uniform and variable cross-section beams are considered by finite element method (FEM). The curvature of beams is circular and the cross-sections are taken circular and rectangular. The natural frequency is computed for different boundary conditions. An additional mass on beam is also considered and its effects on natural frequencies are investigated.

2. Modeling and Governing Equations

In applying finite element method to the approximate solution of curved beam problems, the following procedure can be considered [28–30]: Curved beam is modeled as a finite element as shown in Figure 1. , , and are global coordinates, and , , and are the tangential, radial, and out-of-plane displacements for the curved beam, respectively. Curved beam lies in plane, is tangential coordinate, and is the arch angle of one finite element. The out-of-plane elastic and kinetic energy equations of the curved beam can be expressed as follows:

where is the cross-sectional area, is modulus of elasticity, is modulus of rigidity, and and are mass moment and polar moment of inertia, respectively. Due to the fact that the beam is assumed to be of Bernoulli-Euler type, shear effect on the beam is neglected. In (1), denotes differentiation with respect to time . Out-of-plane curvature change, torsion, and slope terms are

where is the torsional displacement of the curved element. Out-of-plane displacement vector for one finite element is

Three degrees of freedom are taken for each node of elements. By following the finite element procedure, the stiffness and inertia matrices are obtained for out-of-plane vibrations.

Figure 1: Curved beam element.

3. Free Vibration Analysis: Results and Discussion

Matrix equation for the free vibrations of the curved beam starts with an equation of the form

where is a global displacement vector and and are global stiffness and inertia matrices, respectively. The solution of (4) is assumed to be

where , is natural frequency, and is displacement amplitude vector of all nodes. Then, one can obtain the eigenvalue equation giving the natural frequencies for both in-plane and out-of-plane vibrations

3.1. Convergence Study

A convergence study against the number of used elements is also put into consideration to see appropriate error reduction properties of the out-of-plane natural frequencies of the beam with or without mass under clamped end condition (C-C). Results of the convergence study are given in Tables 1 and 2. It can be noted that the experimental convergence starts with 10 elements and ends up with 100 elements. The number of elements for optimal convergence is in the range of 90 and 100.

Table 1: Convergence analysis for the out-of-plane natural frequencies of the beam with midpoint mass location (see midpoint position as Position 1 in Table 11) under C-C end condition.

Mode
# of elements

10
20
30
40
50
60
70
80
90
100

1
0.798
0.789
0.787
0.786
0.786
0.786
0.786
0.786
0.786
0.786

2
6.076
6.015
6.003
6.000
5.997
5.996
5.995
5.995
5.995
5.994

3
9.556
9.447
9.425
9.417
9.413
9.411
9.410
9.409
9.409
9.408

4
21.190
20.977
20.937
20.923
20.917
20.913
20.911
20.910
20.909
20.908

Table 2: Convergence analysis for the out-of-plane natural frequencies of the beam with bare position (see bare position as Position 6 in Table 11) under C-C end condition.

Mode
# of elements

10
20
30
40
50
60
70
80
90
100

1
2.220
2.197
2.193
2.191
2.190
2.190
2.190
2.190
2.189
2.189

2
6.076
6.015
6.003
5.999
5.997
5.996
5.995
5.995
5.995
5.994

3
12.436
12.307
12.282
12.273
12.269
12.267
12.266
12.265
12.264
12.264

4
21.198
20.979
20.938
20.924
20.917
20.913
20.911
20.910
20.909
20.908

3.2. Uniform Beams

According to the finite element model proposed, vibration of uniform circularly curved beams is compared with that of the literature. Results of the out-of-plane dimensionless frequencies of curved beams for clamped end condition (C-C) are given in Table 3. It can be noted that the results of the present study coincide with the results of the literature.

Table 3: The out-of-plane nondimensional natural frequency parameters of uniform and circularly curved beams with circular cross-sections under C-C end condition.

Mode
Present study
Viola et al. [8]
Malekzadeh and Setoodeh [17]
Piovan et al. [21]
Ni et al. [24]

60°
1
19.610
19.40190
19.398
19.442

2
55.070
54.02958
54.014
54.093

3
108.947
105.64828
105.61
105.707

4
180.868
172.77355

120°
1
4.490
4.451450
4.4515
4.471

2
12.970
12.82629
12.825
12.885

3
26.300
25.98937
25.984
26.064

4
44.205
43.57053

180°
1
1.8256
1.804340
1.8048
1.817
1.8108

2
5.2687
5.197995
5.1984
5.239
5.2359

3
11.0489
10.91819
10.918
10.984
11.0046

4
18.9317
18.72548

18.8837

3.3. Curved Tapered Beams

Actually, many studies about in-plane vibrations of curved tapered beams can be found in literature [2, 3, 5–11, 16–20]. However, out-of-plane vibrations of curved and variable cross-sectioned beams are not widely studied. The aim of this study is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered (see Figures 2 and 3).

Figure 2: Unsymmetrical tapered beam.

Figure 3: Symmetrical tapered beam.

It is assumed that (i) each beam possesses a constant width with as radius of the curvature, as arc angle, and as cross-section change parameter and (ii) the beam is circularly curved and its cross-section is rectangular with height of the cross-section as at the beginning, at the crown, and at the end for an unsymmetrical tapered beam and as at both ends and at the crown for a symmetrical tapered beam.

For the comparison of out-of-plane nondimensional fundamental natural frequency parameters, symmetrical tapered curved beam analysis under C-C end condition (see Figure 3) is assumed to be under plane strain condition and the general-purpose finite element code MSC.Marc (v2014, MSC Software, Santa Ana, CA, USA) is used. Four-node linear plane strain element (full integration 11) is considered for all five finite element beam models with constant 14400 elements, respectively. Results for the symmetrical curved tapered beam are given in Table 4. It is obvious that there is a good agreement between the results of this study and that of finite element model by MSC.Marc.

Table 4: Comparison of out-of-plane nondimensional fundamental natural frequency parameters of symmetrical tapered curved beams under C-C end condition.

Mode

FEM
Marc
FEM
Marc
FEM
Marc
FEM
Marc
FEM
Marc

1
1.82656
1.71268
1.99231
1.95735
2.16707
2.06221
2.37049
2.23698
2.55085
2.35337

2
5.2687
4.96329
5.62739
5.29291
5.97692
5.94196
6.35269
6.15867
6.70673
6.5816

3
11.0489
10.90525
11.63925
11.25478
12.23346
11.99578
12.85946
12.82765
13.44632
13.31699

4
18.9317
18.73466
19.8881
19.60848
20.90173
20.62211
21.87971
21.42602
22.83602
22.57947

3.3.1. Unsymmetrical Tapered Beams

In the first part of this section, analysis of an unsymmetrical tapered beam or beam with varying cross-section, change of arc angle at different end conditions is considered. Cross-section change parameter is taken as 0.2 and 0.4 while is same for both (). First natural frequency parameters for clamped (C-C), clamped-hinged (C-H), and hinged (H-H) end conditions are shown in Table 5. It is noted that (i) frequencies decrease with increasing of arc angle, (ii) as expected, the frequencies decrease from C-C to H-H end conditions, and (iii) increase of causes an increase for frequencies at C-H and mostly a decrease at the other two end conditions.

Table 5: Nondimensional fundamental natural frequency parameters of unsymmetrical tapered curved beams.

C-C
C-H
H-H

30°
80.43
78.28
58.19
59.81
34.51
33.12

60°
19.69
19.20
14.04
14.56
7.82
7.51

90°
8.47
8.29
5.90
6.21
2.896
2.782

120°
4.58
4.50
3.09
3.32
1.186
1.139

150°
2.811
2.779
1.826
2.017
0.409
0.393

180°
1.888
1.877
1.190
1.350
0.020
0.021

270°
0.873
0.876
0.593
0.668
0.446
0.402

360°
0.635
0.629
0.477
0.513
0.041
0.041

In the second part of this section, an additional mass is considered on some points of tapered beams. Corresponding to C-C boundary conditions and 90° arc angle, a curved beam with a concentrated mass (10 kg) at the midpoint is given in Figure 4(a). In Figure 4(b), the same beam is under free right end (C-F) and mass location at the free end. Dimensionless natural frequency parameters of unsymmetrically tapered curved beams at C-C boundary conditions are presented in Table 6. The results represent both bare and mass loaded cases with the range of α as 0.1–0.4 and constant .

Table 6: Nondimensional natural frequencies of unsymmetrical tapered curved beams with C-C end conditions.

Mode

Bare
Loaded
Bare
Loaded
Bare
Loaded
Bare
Loaded

1
8.51
4.76
8.47
4.75
8.40
4.74
8.29
4.72

2
24.16
24.08
24.00
23.73
23.73
23.13
23.33
22.32

3
48.04
38.59
47.73
38.78
47.17
39.05
46.34
39.34

4
79.94
79.44
79.42
77.59
78.48
74.85
77.09
71.48

Loaded with additional mass (10 kg) at midpoint of the beam (θ = 90°).

(a) C-C

(b) C-F

Figure 4: Unsymmetrically tapered beams with additional mass.

In the third part of this section, similar computation for C-F beam with the additional mass at the free end is completed with respect to bare and loaded beam cases. Results of first four frequency parameters are shown in Table 7.

Table 7: Nondimensional natural frequencies of unsymmetrical tapered curved beams with C-F end conditions.

Mode

Bare
Loaded
Bare
Loaded
Bare
Loaded
Bare
Loaded

1
1.84
0.87
2.04
0.92
2.25
0.97
2.48
1.00

2
8.43
5.97
8.80
6.16
9.14
6.32
9.46
6.44

3
24.28
19.77
24.56
19.85
24.74
19.82
24.81
19.69

4
48.13
41.66
48.25
41.57
48.15
41.29
47.81
40.77

Loaded with additional mass (10 kg) at free end of the beam (θ = 90°).

In the last part of this section, vibration for unsymmetrically tapered beams is considered for different arc angles. Again, according to bare and loaded beam cases, first natural frequencies are obtained for cross-section parameters 0.2 and 0.4, respectively, as given in Table 8.

Table 8: Nondimensional fundamental natural frequencies of unsymmetrical tapered curved beams with C-F end condition (10 kg additional mass at the free end).

θ

Bare
Loaded
Bare
Loaded

30°
16.41
7.167
20.21
7.529

60°
4.28
1.897
5.25
1.985

90°
2.04
0.925
2.48
0.961

120°
1.261
0.588
1.514
0.605

150°
0.906
0.435
1.071
0.442

180°
0.717
0.351
0.833
0.353

270°
0.481
0.226
0.533
0.226

360°
0.382
0.176
0.419
0.178

3.3.2. Symmetrical Tapered Beams

Although there are number of studies about in-plane vibration of curved and symmetrically tapered beams in literature [4, 5, 15], there is not any about out-of-plane vibration of that kind of structures. In this section, out-of-plane free vibration of symmetrically tapered beams at only C-C end conditions is put into consideration in such a way that (i) the beam is separated into 100 finite elements and (ii) a concentrated mass is located at the midpoint. The beam, namely, a half circled curved beam (), is shown in Figure 5.

Figure 5: Symmetrically curved beam with additional mass is at the top (crown).

In the first part of this section, the fundamental natural frequencies for different arc angles’ α values are obtained regarding bare and loaded cases. Results are presented in Table 9. One can clearly say that while increasing arc angle with additional mass decreases the frequencies, increasing of α with additional mass increases the frequencies.

Table 9: Fundamental frequencies of symmetrical tapered beams under C-C end condition.

θ

Bare
Loaded
Bare
Loaded
Bare
Loaded
Bare
Loaded

20°
196.753
94.883
210.796
101.640
224.883
108.433
239.017
115.266

40°
48.603
23.436
52.149
25.144
55.701
26.858
59.261
28.580

60°
21.205
10.222
22.801
10.992
35.411
11.764
25.997
12.538

80°
11.651
5.613
12.562
6.053
13.471
6.493
14.380
6.934

100°
7.261
3.495
7.851
3.780
8.439
4.065
9.025
4.350

120°
4.903
2.357
5.315
2.557
5.726
2.756
6.136
2.955

140°
3.501
1.680
3.805
1.826
4.108
1.974
4.409
2.121

160°
2.608
1.249
2.840
1.361
3.071
1.473
3.301
1.585

180°
2.008
0.959
2.189
1.047
2.370
1.135
2.551
1.222

Loaded with additional mass (50 kg) at midpoint of the beam.

In the second part of this section, again, the first five frequencies are computed for uniform and tapered () beams with the range of additional mass of 25–100 kg. As seen in Table 10, results indicate that the significant point is about the even modes (2–4). On one hand, these two modes keep their constancy independently of the amount of the additional mass. On the other hand, odd modes show regular decreasing behavior due to the increasing of load.

Table 10: Nondimensional out-of-plane natural frequencies of half circled uniform and symmetrical tapered beams (θ = 180°).

Type
Mode
Amount of the additional mass

Bare
25 kg
50 kg
75 kg
100 kg

Uniform
1
1.825
1.112
0.870
0.739
0.653

2
5.265
5.265
5.265
5.265
5.265

3
11.038
9.082
8.758
8.626
8.554

4
18.910
18.910
18.910
18.910
18.910

Tapered

1
2.370
1.449
1.135
0.963
0.851

2
6.353
6.353
6.353
6.353
6.353

3
12.860
10.457
10.066
9.908
9.822

4
21.880
21.880
21.880
21.880
21.880

Table 11: The out-of-plane natural frequencies of the beam with different mass locations under C-C end condition.

Mode
Position 1
Position 2
Position 3
Position 4
Position 5
Position 6

Midpoint (crown)

40th–60th node

30th–70th node

20th–80th node

10th–90th node

Bare

1
0.786
0.868
1.123
1.620
2.127
2.189

2
5.994
2.921
2.342
2.870
4.948
5.994

3
9.408
11.534
9.621
6.209
7.725
12.264

4
20.908
14.013
20.491
14.923
11.729
20.908

Additional mass: 100 kg, .

In the last part of this section, a change of location of concentrated mass is taken into account under bare beam case. As represented in Table 11, the following conditions are considered: (i) mass is considered at midpoint as Position 1 (100 kg) and (ii) the mass is separated into two equal pieces (50 kg each) and these pieces are located at two sides symmetrically as Positions 2–5, respectively. For the first natural frequency point of view, it is said that increasing position number from 1 to 6 increases fundamental frequency. For the rest of the modes, there is an up-and-down behavior due to the locations of additional masses. Also, when comparing Positions 5 with 6, first and second natural frequencies become close to each other.

4. Conclusions

Due to the fact that there is a gap in literature about out-of-plane vibrations of curved and variable cross-sectioned beams, this study is presented to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. Results conclude that the finite element solution proposed here is suitable for vibration analyses of curved and tapered beams with or without additional mass.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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