11 th International Conferene on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 9-1 September 013 IN-PLANE VIBRATIONS OF CURVED BEAMS WITH VARIABLE CROSS-SECTIONS CARRYING ADDITIONAL MASS H.Alper Özyiğit* 1, Mustafa Işık 1 Bülent Eevit University, Engineering Faulty, Zonguldak, Turkey hamdialper@beun.edu.tr General Diretorate of State Airports Authority, Ankara, Turkey isikmus@yahoo.om Keywords: Variable Cross-Setion, Natural Frequeny, Free Vibrations. Abstrat. Linear in-plane vibrations of urved beams are investigated using finite elements method. Curvature of beams is taken irular and the ross-setions are taken irular and retangular. Uniform, unsymmetrial, and symmetrially tapered beams are onsidered and the natural frequenies are obtained for different boundary onditions. Elongation and bending effets are taken for in-plane vibrations. An additional mass on beam is also onsidered and its effet to natural frequenies is investigated for varying ross-setions. Espeially, symmetrially tapered beams an be used as a arrier leg in industry. The load on beam an be a mahine, a mehanial devie or a motor. If the load produes a harmoni vibration, the frequeny of it an be onsidered as a foring frequeny and if that frequeny oinides the natural frequeny of the beam resonane ours. So, the natural frequenies should be known in order to avoid the high amplitudes and damages of the resonane. In this work, the amount and the loation of the additional load is hanged and the effets on natural frequenies are investigated for symmetrially tapered beams whih an be used as arrier legs. The hanges of natural frequenies are determined and interpreted.
1 INTRODUCTION Vibration analysis of urved beams is an important subjet in mehanis due to its various appliations. Espeially urved beams with variable ross-setions have been widely used to satisfy modern arhitetural and strutural requirements. They an be used as gears, pumps and turbines, ships, in horizontally urved ontinuous bridges or in the design of ribs, et. There have been many studies about the urved beam vibrations. However, some of them are about the urved beams with variable ross-setions. Kawakami et al. [1] presented an approximate method to study the analysis for the in-plane and out-of-plane free vibrations of horizontally urved beams with arbitrary shapes and variable ross-setions. They indiate that the harateristi equation for free vibration an be derived by applying the Green funtion, whih is obtained as a disrete type solution of differential equations governing the flexural behaviour of the urved beam under the ation of a onentrated load. Krishnan and Suresh [] investigated the effet of shear deformation and rotary inertia on natural end ross over frequenies of urved beams by using a simple ubi linear beam element. They analyzed both uniform and non-uniform (varied ross-setion) beams. Free and fored in plane vibrations of irular arhes with variable ross-setions are investigated by Tong et al. [3] using the Kirhoff assumptions for thin beams. In-plane free vibrations of irular arhes are investigated by Liu and Wu [4] using the generalized differential quadrature rule (GDQR). Arhes with uniform, ontinuously varying and stepped ross-setions are presented to illustrate the validity and auray of the GDQR. Karami and Malekzadeh [5] analyzed in plane free vibration of irular arhes with varying ross-setions. By developing a differential quadrature method they aimed to obtain the higher order natural frequenies more aurately. Shin et al. [6] analyzed the vibration of a irular arh with variable ross-setion using differential transformation and generalized differential quadrature. They applied the generalized differential quadrature method (GDQM) and differential transformation method (DTM) for vibration analysis. Yang et al. [7] investigated free in-plane vibration of uniform and nonuniform urved beams with variable urvatures, inluding the effets of the axis extensibility, shear deformation and rotary inertia by using extended-hamilton priniple. Firouz-Abadi et al. [8] presented an asymptoti solution to transverse free vibrations of variable-setion beams by using Wentzel, Kramers, Brillouin (WKB) approximation. Ee et al. [9] studied vibration of an isotropi beam whih has a variable ross-setion. The results show that the non-uniformity in the ross-setion influenes the natural frequenies and mode shapes. Chen [10] developed the differential quadrature element method (DQEM) in-plane vibration analysis model of arbitrarily urved beam strutures. Rafezy and Howson [11] investigated the vibration of doubly asymmetri, three-dimensional strutures omprising wall and frame assemblies with variable ross-setion. A boundary element method is developed by Sapountzakis [1] for the non-uniform torsional vibration problem of doubly symmetri omposite bars of arbitrary variable ross-setion. Tüfekçi and Doğruer [13] analyzed free out-of-plane vibrations of a irular arh with uniform ross-setion by taking into effets of transverse shear and rotary inertia due to the both flexural and torsional vibrations by using the initial value method. Huang et al. [14] investigated out-of-plane dynamis of beams with arbitrarily varying urvature and rosssetion by dynami stiffness matrix method. In this study, the linear free in-plane vibrations of uniform and variable ross-setion beams are analyzed by Finite Element Method. The urvature of beams is irular and the ross-setions are taken irular and retangular. The natural frequenies and mode shapes
are obtained for different boundary onditions. An additional mass on beam is also onsidered and its effet to natural frequenies is investigated. MODELLING AND GOVERNING EQUATIONS Curved beam is modeled as a finite element as shown in Figure 1. X, Y, and Z are global oordinates, and u,v, w are the tangential, radial and out-of-plane displaements for the urved beam respetively. Curved beam lies in X-Y plane, s is tangential oordinate and is the arh angle of one finite element. v u β s w R Y γ X Z Figure 1: Curved beam element. The in-plane elasti and kineti energy equations of the urved beam an be expressed as follows 1 1 U EA I ds T A u v I ds ( ) (1) s where A is the ross-setional area, E is modulus of elastiity, I is mass moment of inertia. The beam is assumed as Bernoulli-Euler type, so the shear and rotary inertia effets are not taken into aount. denotes differentiation with respet to time t. In-plane strain, net ross-setional rotation and urvature hange terms are; u v, s R v u, v 1 u s R s s R s () Displaement vetor for one finite element is taken as follows T V u v u v e i (3) i i i i1 i1 i1 i1 s where i u i s i v i s w i i (4) s Four degrees of freedom are taken for eah node of elements. By following the finite element proedure, the stiffness and inertia matries are obtained for in-plane vibrations. 3
3 FREE VIBRATIONS The total energy in the system is onstant as follows T T V M V V KV 0 (5) where {V} denotes global displaement vetor, [K] and [M] are global stiffness and inertia matries. Then, one obtains the eigenvalue equation giving the natural frequenies K M 0 (6) n At this part, the in-plane free vibrations of uniform, unsymmetrial (Figure ) and symmetrial (Figure 3) tapered urved beams are analyzed. The beams are retengular ross-setioned and irularly urved. The height of the ross-setion is taken h 0 at the beginning, h at the rown, and h 1 at the end of the unsymmetri beam, while it is h 0 at both the beginning and the end, h at the rown of the symmetrially tapered beam. The width is taken onstant. R is the radius of the urvature, and θ 0 is the ar angle. By using a MATLAB omputer program whih was performed by the authors, the dimensionless inplane natural frequenies of uniform and tapered beams are obtained. 3.1 The frequenies of uniform and tapered beams First five frequenies of uniform urved beams at different end onditions are shown in Table 1. The ar angle is 90 0. v hh h 0 h C u h 1 h C = (h 0 +h 1 ) / R h 0 >h 1 => α=(h 0 -h C )/h C h 0 < h 1 => α=(h 1 -h C )/h C θ 0 θ Θ Figure. Unsymmetrially tapered beam Figure : Unsymmetrial tapered beam. 4
V h C u α=(h 0 -h C )/h C R θ 0 h 0 h 0 Figure 3: Symmetrial tapered beam. H-H C-C C-H Mode Present Ref () Ref (5) Present Ref () Present Ref () 1 33.9600 33.93 33.958 55.841 55.8 44.091 44.05 79.9433 79.4 79.851 106.6959 104.8 9.9838 91.8 3 15.163 15.14 193.009 171.8175 4 37.8999 37.06 84.6115 60.8579 5 349.497 349.39 409.7734 378.865 Table 1: Lowest frequeny oeffiients (Θ o λ 1 ) of a uniform irular arh. θ 0 =90 0 In Table, non-dimensional fundamental frequenies of a uniform irular urved beam are seen at different arh angles. The results are represented with the results of literature. The vibration of unsymmetri tapered beam is also analyzed. As seen in Figure, α is the taper parameter. In Table 3, the fundamental frequenies of unsymmetri tapered beam are presented for lamped-lamped and hinged-hinged boundary onditions. α is taken 0.1 and 0.. The dimensionless frequenies are obtained with the general formula of is the retangular ross-setion's area of the urved beam at the rown and is the moment of inertia of the beam at the rown as well. In Table 4 and Table 5, first two frequenies of unsymmetrial tapered beam are shown for 0. and 0.4 taper parameters. The end onditions are lamped-free, hinged-free, and free-free, respetively. Symmetrial tapered beam (Figure 3) is also analyzed for lamped-lamped and hinged-hinged end onditions. The results are presented at Table 6, at different ar angles. is taken 0.1, 0. and 0.3. The results are good enough in omparison with the literature. 3. The additional mass Symmetrially tapered beam an be used as a arrier leg in industry. The load on beam an be a mahine, a mehanial devie or a motor. If the load produes a harmoni vibration, the frequeny of it an be onsidered as a foring frequeny and if that frequeny oinides the natural frequeny of the beam resonane ours. So, the natural frequenies should be known in order to avoid the high amplitudes and damages of the resonane. Of ourse the amount and the loation of the load effets the natural frequenies. These fators are analyzed at this part of this work. (7) 5
C-C θ 0 Present Ref (4) H-H Present Ref (4) C-F Present Ref (4,5) H-F Present Ref (4,5) F-F Present Ref (4,5) 10 o 01.9736 01.9893 0 o 503.5457 503.5498 30 o.3699.3718 40 o 13.9753 13.9764 60 o 53.7396 53.740 80 o 9.17 9.175 90 o.648 100 o 17.959 17.96 10 o 11.8474 11.8476 140 o 8.314 8.315 160 o 5.973 5.974 180 o 4.3844 4.3844 193.4951 193.5037 31.516 31.5148 141.530 141.5331 78.5574 78.5580 33.658 33.661 17.9639 17.9641 13.7635 13.7637 10.7760 10.7761 6.967 6.968 4.6533 4.6534 3.179 3.179.667.667 115.495 115.4953 8.974 8.974 1.8964 1.8964 7.857 7.857 3.784 3.784 1.8763 1.8763 1.498 1.498 1.78 1.79 0.876 0.876 0.6647 0.6647 0.58 0.58 0.435 0.435 503.985 503.9888 14.49 14.438 54.199 54.1996 9.690 9.6905 1.3433 1.3434 6.4318 6.4319 4.8867 4.8868 3.8080 3.8080.4564.4564 1.6885 1.6885 1.0 1.0 0.9188 0.9188 733.6396 733.6490 18.7995 18.8019 80.8016 80.806 45.1131 45.1137 19.6498 19.6501 10.778 10.7730 8.3911 8.391 6.6960 6.6961 4.5088 4.5088 3.11 3.1.388.388 1.837 1.837 Table : Non-dimensional fundamental frequenies of a uniform irular urved beam under different boundary onditions. α = 0.1 α = 0.3 C-C θ 0 Present Ref (4,5) Ref(6) H-H Present Ref (4,5) Ref (6) C-C Present Ref (4,5) H-H Present Ref (4,5) 10 o 015.9664 016.983 017.0 0 o 50.0516 50.3033 50.30 30 o 1.710 1.816 1.8 40 o 13.6069 13.6698 13.67 50 o 78.186 78.575 78.58 60 o 53.5804 53.6075 53.607 80 o 9.1308 9.1456 90 o.5581 10 o 11.819 180 o 4.370 189.058 190.485 190.5 30.6068 30.7631 30.76 141.166 141.01 141.0 78.3355 78.3731 78.373 49.871 49.313 49.31 33.588 33.5461 33.546 17.9114 17.906 13.73 6.9059.595 197.9043 1975.996 491.3307 49.0978 16.9793 17.316 10.9698 11.159 76.5497 76.6699 5.4386 5.509 8.5118 8.5564.0797 11.5643 4.84 163.8007 165.767 314.1095 314.6084 138.680 138.4839 76.739 76.8588 48.787 48.3545 3.8387 3.8904 17.5356 17.5649 13.433 6.7560.058 Table 3: Non-dimensional fundamental frequenies of unsymmetri and irular urved beam under different boundary onditions. 6
C-F θ 0 Mode 1 Mode 10 o 88.405 88.4809 660.410 660.9595 0 o.1600.1631 163.0379 163.150 40 o 5.5759 5.5839 38.9308 38.9737 60 o.509.5140 16.198 16.108 80 o 1.4356 1.4399 8.4366 8.4461 10 o 0.6741 3.139 180 o 0.3360 1.1906 H-F Mode 1 Mode 47.477 47.9768 1598.8379 1600.4856 116.3778 116.5046 397.8714 398.80 7.5538 7.5837 97.7553 97.8563 11.3417 11.3540 4.318 4.3656 5.8519 5.8585 3.077 3.0516.001 9.4094 0.8111 3.5455 F-F Mode 1 Mode 731.9557 73.6887 009.7 011.7617 18.3894 18.5740 501.6036 50.1133 45.07 45.068 14.5819 13.7981 19.6177 19.659 54.774 54.4349 10.7601 10.7708 30.3543 30.1703 4.5081 1.9454 1.8395 5.788 Table 4: Frequenies of first two modes of an unsymmetrial and irular urved beam under different boundary onditions. (α = 0. ) C-F H-F F-F θ 0 Mode 1 Mode 10 o 63.4309 63.6881 587.8368 589.368 0 o 15.8938 15.9544 144.801 145.151 40 o 4.0057 4.011 34.849 34.3693 60 o 1.8077 1.8116 14.1085 14.1441 80 o 1.0344 1.0385 7.743 7.93 10 o 0.4846.770 180 o 0.447 0.9979 Mode 1 Mode 433.10 434.458 1535.31 1538.5761 106.438 106.679 381.7646 38.5978 4.9676 5.094 93.5785 93.7838 10.1639 10.189 40.4061 40.4953 5.1893 5.08 1.9311 1.9798 1.904 8.915 0.6984 3.3379 Table 5: Frequenies of first two modes of an unsymmetrial and irular urved beam under different boundary onditions. (α = 0.4) Mode 1 F-F Mode Mode 1 Mode 731.9557 73.6887 009.7 011.7617 18.3894 18.5740 501.6036 50.1133 45.07 79.0191 45.068 730.4943 14.5819 1971.6458 13.7981 1975.8165 19.6177 181.6943 19.659 18.0619 54.774 49.118754.4349 493.1599 10.7601 44.8839 10.7708 44.9747 30.3543 1.43930.1703 1.507 4.5081 19.5796 19.6187 1.9454 53.7595 53.873 1.8395 10.754 10.7756 5.788 9.801 9.865 α = 0.1 α = 0. α = 0.3 θ 0 C -C H -H 10 o 149.707 149.7593 1357.08 1357.106 0 o 535.4369 535.4500 337.3869 337.3889 30 o 36.513 36.5183 148.5481 148.5490 40 o 131.9054 131.9088 8.4730 8.4735 50 o 83.5051 83.5073 51.9091 51.9095 90 o 4.1437 14.4791 180 o 4.710.3948 C - C H - H 75.3034 75.3957 1419.041 1419.0535 566.7961 566.8193 35.7956 35.7985 50.400 50.4304 155.3584 155.3598 139.7048 139.7107 86.737 86.745 88.4771 88.4809 54.3167 54.317 5.6396 15.1739 5.0545.5195 Table 6: Non-dimensional fundamental frequenies of a symmetrial tapered urved beam for different α values. C -C H - H 399.099 399.1619 1479.441 1479.608 597.6898 597.79 367.7964 367.8006 64.13 64.1371 161.9885 161.9904 147.3900 147.3984 89.9738 89.9750 93.3769 93.384 56.6606 56.6613 7.1153 15.8505 5.3851.6411 7
As the first appliation, the load is onsidered as a point mass at the rown of the half irle symmetrially tapered beam (Figure 4). The first five in-plane natural frequenies of different amount of additional mass are shown in Table 7 for different boundary onditions. The beam is divided into 150 finite elements. The mid-point is the right side of the 75 th element. So the numbers lie from 0 to 150 from left to right as well. 75 0 150 Figure 4: Additional mass is at the rown ( mid-point ) Boundary Amount of the Additional Mass Mode Condition No mass 50 kg 100 kg 150 kg 00 kg 1 5.0546 4.8388 4.6485 4.4789 4.366 10.8698 9.6371 8.733 8.0186 7.4565 C-C 3 19.9084 19.9019 19.8965 19.8918 19.8878 4 30.3660 8.8794 7.9844 7.4014 6.9957 5 43.8876 43.6700 43.4888 43.3355 43.04 1.5195.416.318.341.1585 7.6897 6.947 6.3731 5.9156 5.5414 H-H 3 15.4044 15.3790 15.3577 15.3395 15.337 4 5.0495 3.6614.7863.197 1.7777 5 37.87 37.149 36.9936 36.887 36.7878 Table 7: The dimensionless in-plane natural frequenies of half irled tapered beam ( α = 0. ) As it s seen in the tables, the amount of additional mass slightly redues the natural frequenies. But if we look at Figure 5 whih shows the hanges of Mode 1 and Mode of the lamped-lamped beam, the distane between the frequenies dereases due to the amount of additional mass. 11 10 Mode 1 Mode Dimensionless natural frequenies 9 8 7 6 5 4 0 0 40 60 80 100 10 140 160 180 00 Amount of the Additional Mass (kg) Figure 5: The hanges of natural frequenies of first two modes ( C - C ) In Table 8, the natural frequenies of in-plane vibrations are investigated for the mass is at rown (Position 1) and as another appliation, the mass is divided into two symmetri points. 8
Mode Position 1 Position Position 3 Position 4 Position 5 Position 6 Mid-point (rown) 70th-80th node 60th-90th node 50th-100th node 40th-110th node 30th-10th node 1 4.7865 4.7558 4.5848 4.4904 4.605 4.8948 8.456 8.6856 9.9943 11.0010 10.4653 10.1418 3 0.8586 19.6894 17.01 18.3488 0.1304 18.6387 4 8.6019 9.6699 30.7484 4.815 7.8153 30.7533 5 45.3085 41.37 4.0384 44.7369 38.4451 44.601 Additional mass: 150 kg - α=0.3 Table 8: The in-plane natural frequenies of the beam with different mass loations (C-C) If we assume that the beam is a arrier leg and the mass is a motor or any devie whih auses vibration, the natural frequenies of system should be known to avoid resonane. First and seond modes are more important for the system beause it is easier for the foring frequeny to oinide the first and seond natural frequenies more than the further modes. So, as seen at Table 8, loating the half masses at different nodes hanges the natural frequenies. At this point, we determine that the gap between 1 st and nd modes is wider than the others at Position 4 (Figure 7). So this loation is more onvenient than the others in order to avoid resonane. 1 11 Mode 1 Mode Dimensionles Natural frequenies 10 9 8 7 6 5 4 1 3 4 5 6 Mass Position Figure 7. Natural Frequenies due to the mass loation (C - C), α=0.3 As making alulations, similar situation is observed for hinged-hinged boundary ondition. Position 4 is still more onvenient to avoid the resonane. But the results are quite different for Clamped-Sliding beam system. The gap between first two frequenies makes minimum at Position 1 (mass at the rown) whih seems the best loation at this ase. (Table 9) Mode Position 1 Position Position 3 Position 4 Position 5 Position 6 1 1.3398 1.343 1.3591 1.38 1.3976 1.393 4.8963 4.866 4.6643 4.5480 4.6677 4.9783 3 9.769 9.9836 11.1339 11.6540 10.7891 10.6117 4 1.0054 19.8949 17.5057 18.898 0.495 18.8599 5 8.6733 9.8406 31.6007 6.4699 9.6316 31.107 Additional mass: 150 kg - α=0.3 Table 9: The in-plane natural frequenies of the beam with different mass loations (C - S) 9
4 CONCLUSIONS The uniform and tapered urved beams are analyzed. Free in-plane natural frequenies are obtained. Symmetrial and unsymmetrial tapered beam vibrations are examined separately and the results of different boundary onditions are ompared with the literature. As a new approah, symmetrial tapered beam is assumed as a arrier leg and an additional load is loated on the beam. The load an be supposed as a devie or a motor whih an produe vibration. The foring frequeny an oinide with the natural frequeny of the system. So, this situation is resonane whih an ause damages on the system beause of high amplitudes. So, for that reason, it is important to make vibration analysis on the system. The analysis is made by hanging the amount and the loation of the mass. The hanges of natural frequenies are investigated and interpreted. The most onvenient mass positions are determined to avoid resonane. REFERENCES [1] M. Kawakami, T. Sakiyama, H. Matsuda, C. Morita, In-plane and out-of-plane free vibrations of urved beams with variable setions. Journal of Sound and Vibration, 187(3), 381-401, 1995. [] A. Krishnan, Y.J. Suresh, A simple ubi linear element for stati and free vibration analyses of urved beams, Computers and Strutures, 68, 473-489, 1998. [3] X. Tong, N. Mrad, B. Tabarrok, In-plane vibration of irular arhes with variable ross-setions, Journal of Sound and Vibration, 1(1), 11-140, 1998. [4] G.R.Liu, T.Y. Wu, In-plane vibration analyses of irular arhes by the generalized differential quadrature rule, Int. Journal of Mehanial Sienes, 43, 597-611, 001. [5] G. Karami, P. Malekzadeh, In-plane free vibration analysis of irular arhes with varying ross-setions using differential quadrature method, Journal of Sound and Vibration, 74, 777-799, 004. [6] Y.J. Shin, K.M. Kwon, J.H. Yun, Vibration analysis of a irular arh with variable ross-setion using differential transformation and generalized differential quadrature, Journal of Sound and Vibration, 309, 9-19, 008. [7] F. Yang, R. Sedaghati, E. Esmailzadeh, Free in-plane vibration of general urved beams using finite element method, Journal of Sound and Vibration, 318, 850-867,008. [8] R.D.F.Abadi, H.Haddadpour, A.B.Novinzadeh, An asymptoti solution to transverse free vibrations of variable-setion beams, Journal of Sound and Vibration, 304, 530-540, 007. [9] M.C. Ee, M. Aydogdu, V. Taskin, Vibration of a variable ross-setion beam, Mehanis Researh Communiations, 34, 78-84, 007. [10] C.N. Chen, DQEM analysis of in-plane vibration of urved beam strutures, Advanes in Engineering Software, 36, 41-44, 005. [11] B. Rafezy, W.P. Howson, Vibration analysis of doubly asymmetri, three-dimensional strutures omprising wall and frame assemblies with variable ross-setion, Journal of Sound and Vibration, 318(1-), 47-66, 008. [1] E.J. Sapountzakis, Torsional vibrations of omposite bars of variable-ross-setion by BEM, Computer Methods in Applied Mehanis and Engineering, 194, 17-145, 005. [13] Out-of-plane free vibration of a irular arh with uniform ross-setion: Exat solution, Journal of Sound and Vibration, 91, 55-538, 006. [14] C.S. Huang, Y.P. Tseng, S.H. Chang, C.L. Hung, Out-of-plane dynami analysis of beams with arbitrarily varying urvature and ross-setion by dynami stiffness matrix method, International Journal of Solids and Strutures, 37, 495-513, 000. 10