Available online at www.sciencedirect.com ScienceDirect Materials Today: Proceedings 4 (2017) 997–1004 www.materialstoday.com/proceedings 5th International Conference of Materials Processing and Characterization (ICMPC 2016) Prediction of Natural Frequency of Gas Turbine Disc Kalapala Prasad1*, B.Anjaneya Prasad2, M.Ananda Rao3, 1.Asst. Prof, Department of mechanical Engg.,UCEK,JNTU KAKINADA, Email: prasad_kalapala@yahoo.co.in A.P, INDIA, 2..Prof., Department of mechanical Engg.,JNTU HYDERABAD, A.P, INDIA 3. Professor, MLRIT, HYDERABAD, A.P, INDIA ABSTRACT In the present paper, Natural frequencies of gas turbine disc determined by Holzer’s method as well as Fast Fourier Transform spectrum analyzer are compared. Hub is a circular disc made up of Aluminium alloy to whose outer periphery blades are welded. Disk’s inner end is coupled with a shaft. Thus the Natural frequencies of this component can be kept at a minimum range. Also this material exhibits good properties such as low embrittlement, low toxicity, easy formation of complex shapes and some other economical issues. Many gas turbine models have adopted mild steel hub. Reduction of torsional vibrations in the circular disc results in decreased Natural frequency of gas turbine assembly. This paper also discuss the information to find stiffness, damping ratio and amplitude. ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Key Words: Holzer’s method, torsional stiffness, damping ratio, FFT analyzer, Natural frequency, circular disc. 1. INTRODUCTION Gas turbine disc is a circular component with hollow structure. On the outer periphery of this disc, 6 blades were uniformly spaced around the dia. While a shaft is connected at the inner periphery. The main reasons behind the selection of Aluminium alloy.This hub is subjected to torsional vibrations while transmitting motion and these vibrations are reduced by low natural frequencies of component. Blades are casted around the hub on which flow of high pressure gasses takes place to produce work. Whose outer periphery blades are welded Hub’s inner end is coupled with a shaft. Thus the Natural frequencies of this component can be kept at a minimum range. Also this material exhibits advantages properties such as low embrittlement, corrosion. *Correspondence Author Telephone No: 9963993472 E-mail address: prasad_kalapala@yahoo.co.in 2214-7853 ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). 998 Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 Serious development of the gas turbine began only after the second world war with shaft power in mind, but attention was soon shifted to the turbojet engine for aircraft propulsion. Since then, the gas turbine made a progressively greater impact in an increasing variety of applications. However, only in the recent past much research effort has been focused on the design and development of efficient gas turbine units. Of the various means of producing mechanical power available today the gas turbine, in many respects, seems to be the most satisfactory power plant. It is mainly due to the absence of reciprocating and rubbing members which reduces the vibration and balancing problems, High reliability, Low lubricating oil consumption and High power to weight ratio. John V.T.Ramakrishna[11] and S.S.Rao [10] Told that the structural and thermal characteristics in gas turbine rotor assembly due to various operating conditions will be analyzed by varying the suitable materials. George Touchtontolds that the goal of bringing down the generation costs and mitigation risk of current gas turbine developments and future projects.Rao and Kolla[7] et al presented the optimal value of efficiency and mass of the axial flow gas turbine stage using GA. Rao and Kolla et al [8] discussed about he effects of constraints on the efficiency and mass of stage of axial flow gas turbine. Guha [5] analyzed the influence of various levels of mathematical modelling on gas turbine performance and determined the optimum pressure ratio as a formation of turbine entry temperature and component efficiencies. And developed a conceptual design optimization code to minimize the fluid dynamic losses in mixed flow pump impellers. Sakawa [6] formulated dduision making problems arising from optimal operation planning for hot parts scheduling of gas turbines of thermal power plant are taken as multi objective programming problem. Yokoyama [3] investigated the effect of introducing steam injected gas turbine into cogeneration plants. Angutiak and Andrtc [4] presented the an outline of a method which basing on mathematical modeling, can be used for sterlyuing the infhence of power and electricity exchange on the efficiency of the independent power proeluess IPE. Rao and Gupta [1] solved the problem of stage design of axial flow gas turbine with the objective of minimizing aerodynamic losses and mass of the stage. Rao and Gupta [2] considered the mechanical constraints and optimized the aerodynamics losses and mass of the stage in an axial flow turbine. Nomenclature ππ = Natural frequency π= Damping factor πΏ = Logarithmic decrement FFT = Fast fourier transform Xa = Minimum amplitude Xb = Maximum amplitude Kt = Torsional stiffens J = Mass moment of Inertia G = Shear modulus Ip = Polar moment of inertia L = Length M = Mass r = Radius d = Diameter Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 999 2. METHODOLOGY 2.1Importance of HUB Hub is a circular component with hollow structure. On the outer periphery of this disc, 6 blades were uniformly spaced around the dia. While a shaft is connected at the inner periphery. The main reasons behind the selection of Aluminium alloy are its low weight, availability, ductility, low embrittlement, high torsional stiffness, while on the other hand, elements like Titanium alloy, mild steel, nickel,chromium alloy have rust formation, low formability, availability. 2.2Holzer’s Method Holzer’s method is a theoretical approach using which the natural frequency of hub is determined by considering it as a three rotor system. Polar moment of inertia, Torsional stiffness, mass moment of inertia helps in determining the corresponding frequency value. 3. EXPERIMENTAL WORK The hub is fixed to the cantilever beam and an impact hammer force is applied to it. The natural frequency of the hub is obtained through the graph generated by Fast Fourier Transform Spectrum analyzer equipment, which is a well known experimental approach. Finally the values are compared for obtaining reduced vibration design model and for rapid optimization. 3.2 Natural frequency of a hub by holder’s method: From physical model given hub has following dimensions, length, l = 0.098 m Rigidity modulus, G = 27GPa = 27 x 109 N/m2 Polar moment of inertia, πΌπ = π 32 (π·04 − π·π4 ) (1) = 3 × 10−7 π4 Mass moment of inertia, 1 π½ = π(π02 − ππ2 ) (2) 2 = 5.34 × 10−4 πΎπ − π2 Torsional stiffness, πΎπ‘ = πΊπΌπ (3) πΏ = 82 × 103 π − π Considering Holzer’s method of finding Natural frequencies of rotor, take three rotor bars as in Torsional stiffness for each point will be half of the total value i.e., kt1, kt2 Mass moment of inertia is same for each rotor i.e., J1, J2, J3. Two definite frequencies in the three rotor system can be obtained from equation. 1 π12 , π22 = οΏ½οΏ½ 2 πΎπ‘1 π½1 + πΎπ‘1 +πΎπ‘2 π½2 Kt1 = Kt2 = 41 x103 N-m + πΎπ‘2 π½3 οΏ½ ± οΏ½( πΎπ‘1 π½1 + πΎπ‘1 +πΎπ‘2 π½2 + πΎπ‘2 2 ) π½3 − 4πΎπ‘1 πΎπ‘2 (π½1 +π½2 +π½3) π½1 π½2 π½3 οΏ½ (4) 1000 Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 J1 = J2 = J3 = 1.78 x 10-4 Kg-m2 π12 , π22 = 1 (41 × 103 )4 (41 × 103 )4 2 4(41 × 103 )2 × 3(1.78 × 10−4 ) οΏ½( οΏ½ ± ) − οΏ½ 1.78 × 10−4 (1.78 × 10−4 )3 2 (1.78 × 10−4 ) π12 , π22 = 1 οΏ½(921 × 106 ) ± οΏ½(8.48 ×× 1017 ) − (6.37 × 1017 )οΏ½ 2 1 π12 = (921 × 106 − 459 × 106 ) 2 π12 = 230 × 106 π1 = 15165 πππ/π ππ π1 = 2274 π»π Natural frequency Fig. 1 The above graph, Fig. 1, represents the frequency response curve of gas turbine Disc using FFT analyserThe accelerometer readings are considered on Y-axis as it is a dependent variable, while on the other hand as the natural frequency values are being independent variables they are taken on X-axis. The first or peak amplitude is registered at 2242.5Hz as 86.4m/s2 .The graph is subjected to a steep increment just before the peak amplitude, while it undergoes decrement after reaching the first amplitude. It remains constant over the range of 920Hz to 1280Hz.The second Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 1001 amplitude is registered with a range of about 86.399m/s2 at a frequency of around 2189Hz.The amplitudes are maintained almost constant over a frequency range of 200-680Hz .As the natural frequencies goes on increasing ,the amplitude values reduces after certain critical conditions leading to dead position, indicating zero natural frequency in the gas turbine Disc Boundary conditions for rotor are considered to be cantilever beam accelerometer 45 40 Amplitude (Mts) 35 30 25 accelerometer 20 15 10 5 0 -5 0 0.01 0.02 0.03 0.04 0.05 0.06 Fig. 2 Time Period (Secs) The graph, Fig. 2, obtained from the FFT analyzer where the hammer force is on the x-axis and the acceleration is on the y-axis, the highest peak gives a reading of 39.73 m/s2. From the above graphical notation we can calculate the damping factor of the component( ζ ) by using the logarithmic decrement formula. The δ is nothing but the log of the amplitudes of two successive peaks here the two peaks are taken from graphs. EXPERIMENTAL CALCULATION: πΏ= 2ππ οΏ½1−π 2 π€βππππΏ = πππ from the above graph Xa = 86.399 m/s2 Xb = 86.4 m/s2 π₯π π₯π (5) Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 1002 πΏ = πππ 86.4 86.399 5.02 *10-6 = 2ππ οΏ½1−π 2 π·ππππππππππ‘πππ = 0.00008 THEORITICAL CALCULATION: ωn = 4.987/(δ)1/2 (δ)½ = 4.987/(2274) (δ) 1/2 = 2.19*10^-3 (δ)= 4.809*10^-6 Also πΏ= 2ππ οΏ½1 − π 2 4.809*10^-6= π οΏ½1−π 2 π2 1−π 2 2ππ οΏ½1−π 2 = (4.809*10^-6)/(2Π) =5.859*10^-13 π 2 = 5.589 ∗ 10^ − 13 π = 7.65 ∗ 10^ − 7 π = 0.00007% RESULTS AND DISCUSSIONS An illustration of Theoretical as well as Experimental analysis of various parameters such as natural frequency, damping ratio and logarithmic decrements is represented in the table 1. Table:1Theoretical And Experimental Analysis Of Various Parameters ω Factors Affecting n (natural frequency) ζ (DAMPING FACTOR) δ( logarithmic decrement value) Theoretical Value Experimental Value Range % Error 2274 2242.5 32.5 1.38 0.00007 0.00008 0.00001 4.80 x 10-6 5.02 *10-6 0.22 x10-6 4.37 4.38 Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 1003 Theoreticalfrequency calculated using Holzer’s method is obtained as2274Hz ,while the frequency obtained through FFT analyser is noted as 2242.5Hz.So both of these theoretical and experimental values differ by a range of 31.5Hz ,which results in an error percent of 1.38. Similarly Damping factors for Theoretical and Experimental approaches are registered as 0.00007 and 0.00008 leading to an error percent of 4.37.The final values of logarithmic decrement are obtained to be 4.80×10-4 and 5.02×10-4for theoretical and experimental analysis respectively,resulting in an error percent of 4.38 with a range of 0.22×10-4. The software DEWES Soft 7.1 version we have used in FFT analyser proves more reliable compared to others such as OROS clear as well as reduced values of natural frequencies indicate the effectiveness of Aluminium alloy over other materials like mild steel,Titanium etc., which have less strength to weight ratio,lessformability,high economical cost than our material.Holzers method adopted by us is an accurate ,fast and efficient approach to obtain the natural frequencies over others as like Differential transform method so on.The hollow Disc we used in our analysis weights only 50 percent to solid disc so it plays a crucial role in obtaining reduced natural frequencies.As we can infer that the obtained value doesnot match the design value the effect of resonance can be neglected. CONCLUSIONS 1. 2. 3. 4. 5. The natural frequencies, tensional stiffnesses, damping ratios and logarithmic decrements are obtained through holzer’s method Fast fourier transform spectrum analyzer techniques are analyzed and compared. An identity in natural frequencies May be noted due to this comparison of employed methods. The aluminium alloy hub, is more efficient in reducing torsional vibrations under damped conditions. It should be reduced torsional vibrations of hub helps in controlling the vibrations of turbo machine efficiency. Based on the above results advanced optimization techniques may be employed. References: 1. 2. 3. 4. 5. 6. 7. 8. S.S.Rao and Gupta, 1980 optimum design of axial flow gas turbine stage, part 1: formation and analysis of optimization problem, journal of engineering for power no.1. 102, PP 782-789. S.S.Rao and R.S.Gupta 1950 optimum design of axial flow gas turbine stage part-II solution of the optimization problem and numerical results, journal of engineering power, Vol.102, PP-790-797. Yokoyama .RMatsumato Y.1995, optimal operation of cogeneration plants with steam injected gas turbines. Journal of engineering of gas turbines and power, Vol.117 No.1, P.P.60-66. Augusiak and And 1996, optimization of IPP projects with gas turbines, Conf. at Iraklia, Vol.3 PP 808-811. Performance and optimization of gas turbines with real gas effects, Proc. Instn. Mech. Engrs Vol.215 Part A, PP.507512. Sakawa, M.V.TakaJ.Inuigulri, M.ShronmaniI.Sugionohana N. and Inone, T 1993 Hot party operating schedule of gas turbines by genetic alogirthms and fuzzy satisfying methods, proceedings of the internal joint conference on deuraldetwod Pub. By IEE, PP 746-749. M.A.Rao, K.V.J.Rao, Ch. Penchalaiah, J.Srinivas and SrinivasKolla, 2002, optimum design of axial flow gas turbine stage using genetic algorithm approach. J.Engg for gas turbines and power M.A.Rao, KV.J.Rao, Chpenchalaiah, J.Srinivas and 2002 “Pptimum storage design in axial flor gas turbines” Journal of power an energy, Proc. Inst. Mech. Engg Port-A (communicated) 1004 9. Kalapala Prasad/ Materials Today: Proceedings 4 (2017) 997–1004 Current gas turbine developments and future projects. George touch ton “Gas turbine, leading technology for competitive markets, global gas turbine news Vol 36. No.1. 1996. 10. S.S.Rao “The finite element method in engineering, BH publications dew delhi 3rd edition 1999. 11. John V.T.Ramakrishna “The desing and analysis of gas turbine blade”, international journal of Advanced Research and studies Vol2 No.1, Dec 2012.
