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.
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. [
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.
In applying finite element method to the approximate solution of curved beam problems, the following procedure can be considered [
Curved beam element.
Matrix equation for the free vibrations of the curved beam starts with an equation of the form
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
Convergence analysis for the out-of-plane natural frequencies of the beam with midpoint mass location (see midpoint position as Position 1 in Table
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 |
Convergence analysis for the out-of-plane natural frequencies of the beam with bare position (see bare position as Position 6 in Table
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 |
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
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. [ |
Malekzadeh and Setoodeh [ |
Piovan et al. [ |
Ni et al. [ |
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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 | ||||
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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 | ||||
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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 |
Actually, many studies about in-plane vibrations of curved tapered beams can be found in literature [
Unsymmetrical tapered beam.
Symmetrical tapered beam.
It is assumed that (i) each beam possesses a constant width with
For the comparison of out-of-plane nondimensional fundamental natural frequency parameters, symmetrical tapered curved beam analysis under C-C end condition (see Figure
Comparison of out-of-plane nondimensional fundamental natural frequency parameters of symmetrical tapered curved beams under C-C end condition.
Mode |
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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 |
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
Nondimensional fundamental natural frequency parameters of unsymmetrical tapered curved beams.
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C-C | C-H | H-H | |||
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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
Nondimensional natural frequencies of unsymmetrical tapered curved beams with C-C end conditions.
Mode |
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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 |
Unsymmetrically tapered beams with additional mass.
C-C
C-F
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
Nondimensional natural frequencies of unsymmetrical tapered curved beams with C-F end conditions.
Mode |
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Bare | Loaded |
Bare | Loaded |
Bare | Loaded |
Bare | Loaded |
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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 |
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
Nondimensional fundamental natural frequencies of unsymmetrical tapered curved beams with C-F end condition (10 kg additional mass at the free end).
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|
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||
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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 |
Although there are number of studies about in-plane vibration of curved and symmetrically tapered beams in literature [
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’
Fundamental frequencies of symmetrical tapered beams under C-C end condition.
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Bare | Loaded |
Bare | Loaded |
Bare | Loaded |
Bare | Loaded |
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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 |
In the second part of this section, again, the first five frequencies are computed for uniform and tapered (
Nondimensional out-of-plane natural frequencies of half circled uniform and symmetrical tapered beams (
Type | Mode | Amount of the additional mass | ||||
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Bare | 25 kg | 50 kg | 75 kg | 100 kg | ||
Uniform | 1 | 1.825 | 1.112 | 0.870 | 0.739 | 0.653 |
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3 | 11.038 | 9.082 | 8.758 | 8.626 | 8.554 | |
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Tapered |
1 | 2.370 | 1.449 | 1.135 | 0.963 | 0.851 |
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3 | 12.860 | 10.457 | 10.066 | 9.908 | 9.822 | |
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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 |
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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
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.
The authors declare that they have no conflicts of interest.