FREQENCY DOMAIN FATIGUE ANALYSIS Peter Bigoš, Peter Bocko 1 Key words: frequency domain analysis, vibration fatigue, wind loading 1 INTRODUCTION Dynamical loading is a dominant factor causing the fatigue failure of constructions. Input values for evaluation of dynamical loading of constructions are material fatigue characteristics (S-N and E-N curves), stress or strain concentrator factor and loading history. This paper deals with methods for loading history processing of construction, especially in resonance domain. 2 COMPUTATION METHODOLOGY Structural analyses are in general divided on testing and analysis. Testing is usually experimental measuring of stress or strain. Alternative testing can include measuring of noise or vibration. Analysis usually means a computing of construction response on loading, e.g. using FEM. [1] In general we have at least two possibilities (domains), which can we use for testing or analysis. Analyses are usually performed in time domain, where all used input stresses and output responses are determined by time dependence. The alternative approach is analysis in frequency domain. In this case all loadings and responses are represented as an energetic dependency in different frequencies. In case of fatigue applications, time and frequency approach can be used. Historically older and more often applied is the time approach (Fig. 2.1). Input for this analysis is loading history characterized frequently in presented form. Dependence on these characteristics is usually linear. Spectrum of loading and response can be determined experimentally, analytically (e.g. using standards) or numerically (e.g. MSS mechanism system simulation). Complicated area for implementation of this method in time domain is creation of resonance vibration, which causes nonlinear dependency between loading intensity and construction response. Transformation function determination in this case is problematic and the most suitable methodology is experimental determination of construction response spectrum. Alternatively vibration fatigue analysis can be used. This approach is preferably used in situations where resonance effect is dominant. In this method the construction loading is evaluated by means of statistical methods. Process of this analysis is displayed on Figs. 2.2 and 2.3. 1 Prof. Ing. Peter Bigoš, CSc., Ing. Peter Bocko, Technical University of Košice, Faculty of Mechanical Engineering, Institute of Machine Design and Devices, Department of Constructions, Transport and Logistics, Letná 9, 040 01 Košice, Tel.: +421 55 602 2507, e-mail: peter.bigos@tuke.sk, pbocko@szm.sk 1
Fig. 2.1. Time domain fatigue analysis Fig. 2.2. Frequency domain fatigue analysis based on time history Fig. 2.3. Frequency domain fatigue analysis based on PSD 3 FATIGUE CALCULATION IN FREQUENCY DOMAIN The power spectral density (PSD) is the most common approach for representation of loading or response in frequency domain. PSD describes frequence dependency of time signal and it is an alternative to time signal. It is acquired by Fast Fourier Transformation. Transformation from frequency to time domain is also relatively simple task, which can be realized by inverse Fourier transformation. 2
Fig. 3.1. Fourier transform First step in fatigue computation in frequency domain is estimating of natural frequencies, which can be identified experimentally (e.g. strain-gage measurements, accelerometer,...), analytically (e.g. using standards), or by numerical methods (e.g. FEM). Next step is estimation of transform function, which creates dependency of stresses or strains on frequencies in particular range. This characteristic can be obtained upon the experiment, analytical or numerical computing. By using FEM is necessary to made dynamical frequency analysis, which is based on modal analysis. Important parameter of this analysis is logarithmic decrement of damping for construction. Based on PSD and transform function it is possible to define response spectrum of construction to loading. For this analysis it is necessary to use one of the following methods: Dirlik The best method in all cases, Narrow Band The original solution, Tunna Railway Engineering (UK), Steinberg Electronic Components (USA), Wirsching, Chaudhury & Dover,... [2] 3
4 APPLICATION OF FATIGUE DURABILITY CALCULATION FOR WIND LOADING 4.1 GENERAL Supporting structures of transport machines and a lot of supporting structures of buildings are exposed to wind effects. There are known occurrences of building structure crashes due to resonance loading and fatigue processes. For calculation of fatigue durability of wind loaded structures there is important a value of referential wind speed v ref,0 defined according to standards [3, 4] or submitter requirements. After then it can be determined a referential dynamic wind pressure q ref =(ρ/2).v ref 2 (ρ - air density). Next step is computing of resulting wind pressure p i =q ref.c e (z e ).c d.c f or wind force F wi =p i.a refi by means of wind exposition coefficient c e, roughness coefficient c r and topographic coefficient c t. In the case of dynamic oscillation caused by wind, i.e. if c d >1,2, it is necessary to perform a more detailed analysis. These above-mentioned considerations are concerning to blasts in the direction of wind. In the case of slender structures it is important to consider also other possible cases of instable behaviour. The most important are: cross resonance caused by vortex tear off, galloping, flutter, divergence and interferential galloping. In the case of wind loading it is necessary to take into consideration oscillations in the longitudinal and transversal direction related to wind speed. 4.2 HISTORY OF WIND LOADING History of wind loading can be determined by means of wind maps, standards and experimental measurement. For the final evaluation it is necessary to take into account wind speed and direction. Wind speed is given usually by means of average speed values, or by means of maximum speed values of wind blasts (Fig. 4.1). Wind direction can be obtained using a windrose chart (Fig. 4.2). This value is substantial for construction state of stress in the relationship with fatigue durability as well as with experimental methodology. Maximum gust (high values) 5 minute average wind speed Fig. 4.1 Wind history graph Fig. 4.2 Windrose charts 4
4.3 RESONANCE CAUSED BY VORTEX TEAR OFF On the Fig. 4.3 is given stress time behaviour in critical points of construction in the case of linear wind speed growth. Critical value of wind speed is 20 m/s(t=20 s), i.e. in this case a resonance effect and a strong stress accumulation is arising. Dependence between the stress in critical point and the wind speed is illustrated on Fig. 4.4. Chapter 4.4 presents a short description the above-mentioned stress values are determined. WIND SPEED [m/s] 30 25 20 15 10 5 0 0 5 10 15 20 25 30 TIME [s] STRESS [MPa] 220 180 140 100 60 20-20 -60-100 -140-180 -220 10 15 20 25 30 TIME [s] Fig. 4.3 Wind speed graph and corresponding stress - resonance 200 STRESS [MPa] 160 120 80 40 0 0 10 20 30 WIND SPEED [m/s] Fig. 4.4 Transform function Cross resonance caused by vortex tear off is a special case of vibration fatigue. In this case the actuated resonance is a consequence of turbulent gas flow around the construction. Critical speed value v crit is a main condition for resonance because in this case the vortex tear off frequency equals to natural frequency of construction. Critical wind speed, maximum oscillation amplitude and vortex tear off loading can be determined experimentally (real construction or model air tunnel), by means of analytical relations or from standards. For simple cross-sections and for typical structures can be used standards and analytical relations. For more complicated shapes it is better to apply some of experimental methods known from the fluid mechanics. In our case was applied an analysis of cross resonance in accordance with standards STN P ENV 1991-2-4 (1998-1-3) and DIN 4133 [3, 4] for dimensioning and strength control of various steel smokestack constructions. 5
Fig. 4.5. Frequency domain fatigue analysis Linear wind resonance Changing wind direction presents an obstruction for application of experimental methods because in such case it is not known critical point on construction and it is not possible to measure response of construction. For stable construction the critical point can be determined according to main wind direction, which causes resonance. 4.4 ANALYTICAL RELATIONS FOR RESPONSE CALCULATION - Critical wind speed v crit,i is calculated from the relation : b.n i,y vcrit,i = [m.s -1 ], St b - referential cross section width for cross oscillation [m], n i,y - natural frequency for i-shape of cross oscillation [Hz], St - Strouhal number [-], - maximum oscillation amplitude max y F can be obtained from the relation: 1 1 max y = F b K w K c lat 2 St Sc [m], K w - coefficient of effective correlative length [-], K - coefficient of natural shape [-], c lat - coefficient of actuating aerodynamic force f(re) [-], - Scruton number: 2m i,y δs Sc = [-], 2 ρ b m i,y - equivalent mass per unit length [kg.m -1 ], δ s - logarithmic decrement of construction damping [-], ρ - air density [kg.m 3 ], - vortex tear off loading F i,j is given from the relation: F = m 2 π n Φ maxy [N], ( ) 2 i,j j i,y i,y,j F 6
m j Φ i,y,j - oscillating mass in j-point [kg], - ratio of i-dynamic construction deviation at cross natural shape in j-point to the opposite cycle[-], 4.5 CALCULATION OF OPERATION DURABILITY FOR VORTEX TEAR-OFF - number of stress cycles N caused by cross resonance oscillation in the case of vortex tear off : v N= 6,3.10 T n ε e 7 crit i,y o vo 2 2 vcrit 9 v crit vo i,y vo N= 10 n e 2 2 vcrit vo [cycles] STN P ENV 1991-2-4, [cycles] DIN 4133 50 years, v o - referential wind speed value [m.s -1 ] (e.g. v o =5 m/s for q o =0,8-1,05 kn/m 2, v o =7 m/s for q o =1,3-1,7 kn/m 2 DIN 4133), T - durability in years [years], ε o - width of zone coefficient ε o =0,3 [-], - verification of operational strength: 1 N m A σ R = σa. N [MPa], σ < σ R, σ - stress range [MPa]. 5 CONCLUSION The paper describes possible ways for fatigue life estimation of civil and transport machine steel constructions which are loaded in frequency domain, especially under wind loading. Paper in this case is the contribution to the conception of loading history processing in time and frequency domain and their application for fatigue life estimation. The project is supported by Grant 9438/2005 of Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Science. 7
LITERATURE [1] Trebuňa, F., Bigoš, P.: Intenzifikácia technickej spôsobilosti ťažkých oceľových konštrukcií. Vienala, Košice, Slovakia 1998 [2] MSC.Fatigue 2004 On-line documentation. MSC.Software, 2004 [3] STN P ENV 1991-2-4 (1998-1-3). Zásady navrhovania a zaťaženia konštrukcií. Časť 2-4: Zaťaženia konštrukcií zaťaženie vetrom. Slovak Standard 1998 [4] DIN 4133 Schornsteine aus Stahl, Deutsche Norm. 1991 8