Numerical and Experimental Investigations of Lateral Cantilever Shaft Vibration of Passive-Pitch Vertical-Axis Ocean Current

Transcription

1 R. Hantoro, et al. / International Enery Journal 1 (011) Numerical and Experimental Investiations of Lateral Cantilever Shaft Vibration of Passive-Pitch Vertical-Axis Ocean Current R. Hantoro *1, I.K.A.P Utama +, A. Susetyono + and Erwandi # Abstract Simulation and experiment of lateral shaft vibration of passive variable-pitch vertical-axis ocean current turbine have been performed. A cantilever type of shaft has been used and modeled usin finite element method, and simulated usin consistent mass matrix to obtain the vibration characteristics and responses. Variations of incomin fluid velocity and the correspondin rpm were used to identify the pattern of lateral displacement responses. Analysis of displacement responses at all nodes in x and y-direction at the same time was carried out, and confirmed with the presents mode shapes. The repeated pattern of periodic displacement responses due to the functions of force actin on the foils was identified. Correlation of critical azimuth position of the foils and displacement responses was presented. Experiment was conducted as vadation of the simulation results at a node of the finite element model. Periodic pattern responses resulted from simulation and experiment at the vadated node shows the suitabity with averae error value for all variations were 11.6% in x-direction and 1.% in y-direction. Keywords Cantilever, finite-element, lateral vibration, passive variable-pitch, vertical-axis. 1. INTRODUCTION Research on the effect of fluctuations of the force on the output power enerated by enery conversion system from renewable enery sources (wind, ocean currents) has become a serious concern amon researchers [1],[]. In the case of ocean current turbine system, the force of the fluid causes structure to rotate and chane its orientation to the incomin fluid flow. Force fluctuations which follow the turbine durin rotation become potency of vibrations of the turbine shaft. For dynamic response analysis, a nearized finite element model was employed to estabsh a control scheme for rotor systems []. The finite element method was also apped to a complex rotor system to evaluate its vibration response due to fluid forces [4], and yroscopic moments [5]. General purpose finite element codes [6] were adapted to simulate the dynamics of some complex rotor systems. This paper discussed the characteristics of lateral vibration on the main shaft of vertical-axis ocean current turbine with the use of passive variable-pitch. A cantilever type of shaft was used and modeled usin finite element method. Simulation was carried out usin consistent mass matrix in the Matlab packae to obtain the natural frequencies, mode shapes and deflection of vibration response. Fourier force function used in the *Post-Graduate Proram, Faculty of Marine Technoloy, Kampus ITS Sukolo Surabaya, Indonesia, Department of Naval Architecture and Shipbuildin Enineerin Kampus ITS Sukolo Surabaya, Indonesia, # BPPT-LHI (Indonesian Hydrodynamic Laboratory) Kampus ITS Sukolo Surabaya, Indonesia, Correspondin author; Tel: , Fax: hantoro@ep.its.ac.id. simulation was taken from force fluctuations data obtained at previous CFD simulation [7], [8]. Test was conducted as vadation of the simulation results at a node of the finite element model.. VIBRATIONS MODELING USING FINITE ELEMENT AND CONSISTENT MASS MATRIX Althouh early dynamic models of rotor systems were formulated either analytically [9] or usin transfer matrix approach [10], the potential of the powerful finite element technique was reconized at a very early stae [11]. In eneral, a structure is analyzed as a system of continuous or discrete systems (lumped system). A uniform structure ke rod can be more appropriate if treated as a continuous system. Finite element method in fact can be called a combination of two methods, namely the continuous and discrete elements in the level of eneral coordinates. In this study, the vertical axis turbine is modeled as a system consistin of the shaft which is divided into 10 elements. Three foils as producer of excitation force rests at two points on the shaft (node- and node-10), as shown in Fiure 1. With the total lenth of 1400 mm, each element has lenth of 140 mm. Parameters of Cantilever shaft turbine used in simulation and experiments are: Table 1. Material specifications. - shaft material : SS04 - modulus elasticity (E) : 00 GPa - shear modulus (G) : 86 GPa - shaft dimension : Lenth 1400mm, diameter 44.5 mm - density : 8000 k/m

2 19 The element stiffness matrix can be developed usin basic strenth of materials techniques to analyze the forces required to displace each deree of freedom a unit value in the positive direction. Usin the derees of freedom of element stiffness matrix results in the followin element stiffness matrix: K e;, i 1 6 = Ei I i l i l i 6 l i l i 6 4 Cantilever beam with two element model with one end not allowed to move is necessary to eminate the deree of freedom. To eminate the constrained derees of freedom, the rows and columns which correspond to the constrained lobal derees of freedom were eminated, reducin the lobal stiffness matrix to a 4x4 matrix, resultin as in Equation. 1E 1I1 1EI 6E1I1 6EI 1EI + + l 6E 1I 6EI 4E1I 1 4EI 6EI + + K = l1 l l1 l l 1EI 6EI 1EI l l l 6EI EI 6EI l l l 6E I E I l 6EI l 4EI l The same method can be performed on the number of element more than two with considerin of the computin capabity to perform hih-order matrix operations. For a beam which is modeled with finite element method there are several method to form the mass matrix, namely: (a) lumped mass for the translation, (b) lumped mass for the translation and rotation, and (c) consistent mass - distributed mass effects. The parameters of mass and inertial mass matrix element connectin rods in the inertial load point to point and iven the acceleration in the diaonal matrix. Equation shows the lumped mass matrix (LM), includin translation and rotation, (1) () R. Hantoro, et al. / International Enery Journal 1 (011) l 54 1l ml l 4l 1l l (4) = e ;, l 156 l 1l l l 4l m i With the same method to obtain the matrix k in Equations 1 and, the matrix CM for two elements m become, 1ml 0 54ml 1ml 1 0 8ml 1ml ml (5) m = 40 54ml 1ml 156ml ml 1ml ml ml 4ml With the acquisition of m and k then the eienvalue problem for homoeneous equations of motion can be written as, [ 0] & (6) m z + k z = & It is not obvious to weather the lumped mass matrices or consistent mass matrices yield more accurate result for eneral dynamic response problem [1]. The lumped mass matrices are approximate in the sense that they do not consider the dynamic coupn present between the various displacement derees of freedom of the element. However, since the lumped mass matrices are diaonal, they required less storae space durin computation. On the other hand, the consistent mass matrices are not diaonal and hence required more storae space. They too are approximate in the sense that the shape functions, which are derived usin static displacement patterns, are used even for the solution of dynamics problems. ml ml mli y () A ml 0 0 ml mli y A Lumped mass formulations were state of the art in structural dynamics until Archer s classic paper [1] introduced the consistent mass matrix in 196. Consistent mass (CM) Matrix for a beam element is, Fi. 1. Finite element model of vertical-axis turbine.. FOURIER FORCE FUNCTION MODELING Equations obtained by Fourier-force function are used to obtain the vibration response respect to time and the external forces actin on the shaft. It is obtained by makin Equation 6 become,

3 R. Hantoro, et al. / International Enery Journal 1 (011) [ m k ][ z] [F ] = (7) and all nodes in the deviation of z is, 1 [ z] [ m k ] [ F ] = (8) CFD simulation performed on previous research by Hantoro et al. [7] have resulted the pattern of force fluctuations in variations of flow velocity and rotation speed of turbine. Variations that are performed in the CFD simulations are as follows: Force fluctuation patterns appear in a full rotation at all variations classified in two directions as defined in Fiure. Simulation of lateral vibration on the main shaft for every variation was performed in two directions, namely in x and y-direction. Force in x- direction is the force actin on a rotatin turbine in the 19 same direction with the incomin fluid flow, while the y-direction is for the force that is perpendicular to the incomin fluid flow. The resulted force fluctuations of turbine for all variations which are actin on the shaft provides periodic pattern as shown in Fiure. Fourier force function moden performed by takin a period (T) of fluctuation patterns. Results of moden achieve areement to fit after the 6 th -order iterations. Plots of Fourier force function fluctuations performed with Matlab provides ood areement compared with the results of CFD data simulation. Fiure 4 shows an example of the suitabity at Var- variation. Table. Towin tank variations test Variation U (m/s) RPM Var Var Var Fi.. Definition of the lateral direction of the force and the vibration on the main shaft of turbine. (a)

4 194 R. Hantoro, et al. / International Enery Journal 1 (011) (b) Fi.. Fluctuation pattern of the force at all variations, (a) x-direction, (b) y-direction. (a) (b) Fi. 4. The suitabity of the force fluctuation pattern at Var- variations, (a) x-direction, (b) y- direction.

5 R. Hantoro, et al. / International Enery Journal 1 (011) TESTING OF VIBRATION DISPLACEMENT Experiments included the manufacture and testin of the work piece was carried out at the towin tank facity at Hydrodynamics Laboratory, Faculty of Marine Technoloy ITS, with specifications: 50 meters lenth, meters wide, and meters deep. The foil chord was set at 100 milmeters, with span of 1000 mm ivin aspect ratio of 10, and 500 mm arm to the shaft. The turbine was desined with three foils. The NACA 0018 profile was chosen as the foil section with data from [14]. This section is commonly used for Darrieus turbines. Its relatively hih thickness to chord ratio ives it ood strenth in bendin. The radial arms of the turbine were made from hih strenth aluminum. 195 Turbine shaft used cantilever type with one end fixed by the bearin and the other end free (overhanin) as shown in Fiure 6. Data collection was conducted with the same variation as mention in Table 1. The use of passive variable-pitch allowed foil to chane its relative position to the arm, where the position of the foil are iven the freedom to move within the interval to 10 0, as shown in Fiure 7. Manetic probe sensor (eddy currents) is placed on node-1 in the x and y-direction with a distance of 140 mm from the bearin in order to obtain displacement data (Fiure 8). Data collection was performed after towin tank carriae speed has stable with time sampn of second. Fi. 5. Towin tank facity at FTK ITS. Fi. 6. Vertical-axis turbine with three straiht foils. Fi. 7. Passive variable-pitch position relative to arm. Fi. 8. Installation of Eddy current sensors to the turbine shaft. 4. RESULTS AND DISCUSSION The natural frequency (ωn) was obtained by solvin the roots of the determinant of the equations of motion usin, [ m k ][ z] = [0] (9) n = det[ m k ] ω (10) Coupn between elements ives the derees of freedom two times the number of elements, resultin 0 varieties of natural frequency as shown in Fiure 9 and Table. Resulted displacement from simulation at all nodes at the same time indicatin the possibity of mode shape

6 196 occurrence. Fiure 10 shows the displacement at all nodes in x-direction at Var- for t i=, t i=8, and t i=6 (Fiures 10a-10c). Similar mode of displacement at all node was presented for all variations in x and y- direction, and these modes reconized to be areed to the 0th mode shape resulted from simulation (Fiure 10d). Chanes in inter-elements in the node-9 to ive sinificant difference when compared to the other nodes. This indicates that the use of the cantilever shaft for R. Hantoro, et al. / International Enery Journal 1 (011) vertical-axis ocean current turbine ive potential problems on the tip of shaft. It is obvious due to half of the force received on turbine concentrated at cantilever tip. Deflection at node-1 in a sinle rotation at all variations in x and y-direction shown in Fiures 11 to 1. Vibration response enerated in the simulation and testin in all variations ive a periodic pattern follows the force pattern on the turbine shaft. Table. Natural frequency of turbine shaft. Mode ω n (Hz) Mode ω n (Hz) Fi. 9. Natural frequency of turbine shaft. Responses of displacement appearin in x and y- direction are always positive. Accordin to the definition of the direction of displacement which has been described previously, the position displacement is always in the reion of x + and y +. Displacement at this area enerated by the excitation force resulted from interaction between the incomin fluid and foils accordin to the position at the azimuth as the turbine rotates. The hihest deflection occurs at each variation accordin to the azimuth position ( ) of the turbine shown in Table 4. The increase in turbine rotation speed is enerally dever enhanced value of the lateral deviation in x and y-direction. However, the maximum deviation is not always occur at the same azimuth position despite the azimuth anle intervals are repeated consistently in value close or equal to The averae error between the simulation and measurement at each of variation are iven in Table 5. Greater averae error resulted from experiment compared to simulation results. The averae value of the resultin error for all variations in x-direction is 11.6%, whereas for y-direction is 1.%.

7 R. Hantoro, et al. / International Enery Journal 1 (011) Fi. 10. Displacement all nodes at, (a) t i=, (b) t i=8, (c) t i=6, (d) 0 th mode shape. (a) (b) Fi. 11. Responses at node-1 at Var-1, (a) x-direction, (b) y-direction.

8 198 R. Hantoro, et al. / International Enery Journal 1 (011) (a) (b) Fi. 1. Responses at node-1 at Var-, (a) x-direction, (b) y-direction. (a) (b) Fi. 1. Responses at node-1 at Var-, (a) x-direction, (b) y-direction.

9 R. Hantoro, et al. / International Enery Journal 1 (011) Table 4. Position of azimuth of the hihest displacement. U (m/s) Direction Azimuth (deree) 0.6 x 80 0, 00 0, 0 0 y 60 0, 180 0, x 110 0, 0 0, 50 0 y 50 0, 190 0, x 110 0, 0 0, 0 0 y 50 0, 160 0, 80 0 Table 5. Averae error for variations U (m/s) Averae Error (%) x-direction Averae Error (%) y-direction CONCLUSION Simulation and testin of lateral vibration of passive variable-pitch vertical-axis ocean current turbine has been performed. Conclusions obtained from the research presented as follows: Resulted displacement from simulation at all nodes at the same time indicatin the present of the 0th of mode shape with natural frequency of 10 Hz. Potential problems on the tip of shaft obviously due to half of the force of turbine received concentrated at cantilever tip. The pattern of periodic displacement response repeat every 1/ of turbine rotation and follows the functions of force actin on the foils. Critical position of foils for the occurrence of lare displacement in x- direction is iven when foil-1 positioned at the azimuth , foil- at , and foil- at While on the y-direction is iven when the foil-1 at azimuth positions , foil- at , and foil- at Comparison of simulation and test results at node-1 ave an averae error value for all variations of 11.6% in x-direction and 1.% in y-direction. ACKNOWLEDGEMENT The authors expresses their ratitude to DPM Dikti and LPPM ITS for fundin the research under research scheme known as Postraduate Research Grand (HPTP) of the year 009 and 010. The authors also thank the technicians at the ITS towin tank for their help in the experimental work. REFERENCES [1] Kiho, S., Shiono, M. and K. Suzuki, The power eneration from tidal currents by Darrieus turbine. WRFC. [] Reuter, R.C. and M.H. Worstell, Torque ripple in a vertical axis wind turbine, Sandia Report SAND l Unmited Release UC-60. [] Firoozian, R. and R. Stanway, Moden and control of turbomachinery vibrations. Journal of Vibration and Acoustics, Stress, and Reabity in Desin 110: [4] Diewald, W. and R. Nordmann, Dynamic analysis of centrifual pump rotors with fluidmechanical interactions. ASME Journal of Vibration and Acoustics, Stress, and Reabity in Desin 111: [5] Sakata, M., Kimura, K., Park S.K. and Ohnabe, H., Vibration of bladed flexible rotor due to yroscopic moments. Journal of Sound and Vibration 11: [6] Rouch, K.E., Mcmaines, T.H., Stephenson, R.W. and Ermerick, M.F., Moden of complex rotor systems by combinin rotor and substructure models. In 4 th International ANSYS Conference and Exhibition, Part, [7] Hantoro. R., Utama. I.K.A.P., Erwandi, 009. Unsteady load analysis on a vertical axis ocean current turbine. In the 11 th International Conference on QIR (Quaty In Research), -6 Auust, Indonesia University. [8] Hantoro, R., Utama, I.K.A.P., Erwandi, Susetyono, A., 009. Unsteady load and fluidstructure interaction of verical-axis ocean current turbine. Journal of Mechanical Enineerin, Petra University Surabaya 11(1): 5-. [9] Dimentber, F., Flexural Vibrations of Rotatin Shafts. London: Butterworth. [10] Black, H., A nearized model usin transfer matrix to study the forced whirn of centrifual pumps rotor systems Journal of Enineerin for Industry. [11] Ruhl, R.L. and J.F. Booker, 197. A finite element model for distributed parameter turborotor systems. ASME Journal of Enineerin for Industry 94, [1] Archer, J.S., 196. Consistent mass matrix for distributed mass systems. Journal of the Structural Division, pp.161.

10 00 [1] Rao, S.S., Mechanical Vibration, rd Edition, Addison-Wesley Pubshin Co., Inc. [14] Sheldahl, R. and P. Kmas, Aerodynamic characteristics of seven symmetrical airfoil sections R. Hantoro, et al. / International Enery Journal 1 (011) throuh 180-deree anle of attack for use in aerodynamic analysis of vertical axis wind turbines. Technical Report SAND80-114, Sandia National Laboratories.