International Journal of Mechanical and Materials Engineering (IJMME), Vol. (7), No. 1, 63-74. FINITE ELEMENT BASED VIBRATION FATIGUE ANALYSIS OF A NEW TWO- STROKE LINEAR GENERATOR ENGINE COMPONENT M. M. Rahan, A. K. Ariffin, and S. Abdullah Departent of Mechanical and Materials Engineering Faculty of Engineering, Universiti Kebangsaan Malaysia 436 UKM, Bangi, Selangor, Malaysia Phone: +(6)3-891-61, Fax: +(6)3-8959659 E-ail: ustafiz@eng.uk.y, kaal@eng.uk.y ABSTRACT This paper presents the finite eleent analysis technique to predict the fatigue life using the narrow band frequency response approach. Such life prediction results are useful for iproving the coponent design at the very early stage. This paper describes how this technique can be ipleented in the finite eleent environent to rapidly identify critical areas in the structure. Fatigue daage is traditionally deterined fro the tie signals of the loading, usually in the for of stress and strain. However, there are scenarios when a spectral for of loading is ore appropriate. In this case the loading is defined in ters of its agnitude at different frequencies in the for of a power spectral density (PSD) plot. A frequency doain fatigue calculation can be utilized where the rando loading and response are categorized using power spectral density functions and the dynaic structure is odeled as a linear transfer function. This paper investigates the effect of ean stress on the fatigue life prediction by using a rando varying load. The obtained results indicate that the Goodan ean stress correction ethod gives the ost conservative results copared with Gerber, and no ean stress correction ethod. The proposed analysis technique is capable of deterining preature products failure phenoena. Therefore, it can reduce cost, tie to arket, iprove product reliability and custoer confidence. Keywords: Fatigue, Fast Fourier Transfor; vibration; power spectral density function; frequency response; power density function. INTRODUCTION Structures and echanical coponents are frequently subjected to oscillating loads which are rando in nature. Rando vibration theory has been introduced for ore then three decades to deal with all kinds of rando vibration behaviour. Since fatigue is one of the priary causes of coponent failure, fatigue life prediction has becoe a ajor subject in alost any rando vibration [1-4]. Nearly all structures or coponents have been designed using tie based structural and fatigue analysis ethods. However, by developing a frequency based fatigue analysis approach, the true coposition of the rando stress or strain responses can be retained within a uch optiized fatigue design process. The tie doain fatigue approach consists of two ajor steps. Firstly, the nubers of stress cycles in the response tie history [5-7] are counted. This is conducted through a process called a rain flow cycle counting. Secondly, the daage fro each cycle is deterined, typically fro an S-N curve. The daage is then sued over all cycles using linear daage suation techniques to deterine the total life. The purpose of presenting these basic fatigue concepts is to ephasize that the fatigue analysis is generally thought of as a tie doain approach, That is, all of the operations are based on tie descriptions of the load function. This paper deonstrates that an alternative frequency doain [4,8-9] fatigue approach is ore appropriate. A vibration analysis is usually carried out to ensure that the structural natural frequencies or resonant odes are not excited by the frequencies of the applied load. It is often easier to obtain a PSD of stress rather than a tie history [1-11]. The dynaic analysis of coplicated finite eleent odels is considered in this study. It is beneficial to carry out the frequency response analysis instead of a coputationally intensive transient dynaic analysis in the tie doain. A finite eleent analysis based on the frequency doain can siplify the proble. The designer can carry out the frequency response analysis on the finite eleent odel (FEM) to deterine the transfer function between load and stress in the structure. This approach requires that the PSD of the load is ultiplied by the transfer function to the PSD of the stress. The ain purpose of the present paper is to derive forulas for the prediction of the fatigue daage when a coponent is subjected to statistically defined rando stresses. THEORETICAL BASIS The equation of otion of a linear structural syste is expressed in atrix forat in Equation 1. 63
[ M ]{ x& ( t) } + [ C]{ x& ( t) } + [ K ]{ x( t) } = { p( t) } & (1) where {x(t)} is a syste displaceent vector, [M], [C] and [K] are ass, daping and stiffness atrices, respectively, {p(t)}is an applied load vector. The syste of tie doain differential equations can be solved directly in the physical coordinate syste. When loads are rando in nature, a atrix of the loading power spectral density (PSD) functions [S p (ω)] can be generated by eploying the Fourier transfor of the load vector {p(t)}. This can be written as shown in Equation (). ( ω) Λ S ( ω) Λ S ( ω) S11 1i 1 M Λ M () [ S ( )] ( ) ( ) ( ) p ω = S ω ω ω i1 Sii Si ( ) ( ) ( ) M Λ M S1 ω Λ Si ω Λ S ω where is the nuber of input loads, M and Λ are the colun and row atrix respectively. The diagonal ter S ii (ω) is the auto-correlation function of load p i (t), and the off-diagonal ter S ij (ω) is the cross- correlation function between loads p i (t) and p j (t). Fro the properties of the cross PSDs, it can be shown that the ultiple input PSD atrix [S p (ω)] is a Heritian atrix. The syste of tie doain differential equation of otion of the structure in Equation (1), is then reduced to a syste of frequency doain algebra equations as shown in Equation (3), [ ( )] [ H( ω) ] S ( ω) [ ] H( ) [ ] T n S x = n n n p ω ω (3) where n is the nuber of output response variables. The T denotes the transpose of a atrix. [H(ω)] is the transfer function atrix between the input loadings and output response variables. It can be written as Equation (4) shown below, ( ) 1 [ H ( )] = [ M ] ω + i [ C] ω + [ K ] ω (4) The response variables [S p (ω)] such as displaceent, acceleration and stress response in ters of PSD functions are obtained by solving the syste of the linear algebra equations in Equation (3). The stress power spectra density [3-4,9-1] represents the frequency doain approach input into the fatigue. This is a scalar function describing how the power of the tie signal is distributed aong frequencies [13]. Matheatically, this function can be obtained by using a Fourier transfor of the stress tie history s auto-correlation function, and its area represents the signal s standard deviation. It is clear that the PSD is the ost coplete and concise representation of a rando process. There are any iportant correlations between the tie doain and frequency doain representations [14] of a rando process. In fact there is transforation, which can be used to ove fro the tie doain to the frequency doain as shown in Figure 1. The inforation extracted fro the frequency doain directly and used to copute fatigue daage, are the PSD oents used to copute all of the inforation required to estiate fatigue daage, in particular the probability density function (pdf) of stress ranges and the expected nubers of zero crossings and peaks per second. The nth oent of PSD area is coputed by Equation (5). M n n ) = f G( f df (5) where f is the frequency and G(f) is the single sided PSD at frequency f Hz. Tie doain Force Tie Inverse Fourier transfor Figure 1 The transforation between tie and frequency doains A ethod for coputing these oents is shown in Figure. Soe very iportant statistical paraeters can be coputed fro these oents. These paraeters are root ean square (rs), expected nuber of zero crossing with positive slope (E []), expected nuber of peaks per second (E [P]). The forulas in Equation (6) highlight these properties of the spectral oents. rs = 4 = ; E[] = ; E[ P] (6) f Fourier transfor Coplex FFT G(f) Figure Calculating oents fro a PSD Frequency Frequency doain 64
Another iportant property of the spectral oents is the fact that it is possible to express the irregularity factor as a function of the zero, second and fourth order spectral oents, as shown in Equation (7). E[ ] γ = = (7) E[ P] 4 The irregularity factor γ is an iportant paraeter that can be used to evaluate the concentration of the process near a central frequency. Therefore, γ can be used to deterine whether the process is narrow band or wide band. A narrow band process (γ 1) is characterized by only one predoinant central frequency indicating that the nuber of peaks per second is very siilar to the nuber of zero crossings of the signal. This assuption leads to the fact that the pdf of the fatigue cycles range is the sae as the pdf of the peaks in the signal (Bendat theory). In this case fatigue life is easy to estiate. In contrast, the sae property is not true for wide bend process (γ ). Bendat [13] has proposed first significant step towards a ethod of deterining fatigue life fro PSDs. Bendat showed that the probability density function (pdf) of peaks for a narrow band signal tended towards Rayleigh distributions as the bandwidth reduced. Furtherore, for a narrow banded tie history Bendat assued that all the positive peaks in the tie history would be followed by corresponding troughs of siilar agnitude regardless of whether they actually fored stress cycles. Using this assuption, the pdf of stress range would also tend to a Rayleigh distribution. To coplete his solution ethod, Bendat used a series of equations derived by Rice [15] to estiate the expected nuber of peaks using oents of area beneath the PSD. Bendat s narrow band solution for the range ean histogra is therefore expressed in Equation (8). E[ D] = i n S i t = N( S ) K E[ P] T = K i S b [ 4 S S b p( S) ds e S 8 ] ds S S 8 E [ P] T e (8) 4 = N ( S) = E[ P] T p( S) (9) where N(S) is the nuber of stress cycles of range S N/ expected in T sec. E[P] is the expected nuber of peaks. Paraeter p(s) in Equation (9) can be defined as shown in Equation (1) to Equation (1). Z Z Z D1 Q DZ R e + e + D 3Ze Q R p( S) = (1) D 1 ( x γ ) =, 1 + γ D 1 γ D1 D = 1 R 3 = 1 D1 D, γ (11) D = 1.5( γ D Q = D 3 1 D γ x D1 R = 1 γ D + D 1 1 1 x, Z = 4, R), S = (1) where x, D 1, D, D 3, Q and R are all functions of, 1, and 4 ; Z is a noralized variable. NUMERICAL EXAMPLE In the finite eleent odel of the cylinder block of the linear generator engine, there are several contact areas including cylinder head, gasket, and hole for bolt. Therefore constraints are eployed for the following purposes: (i) to specify the prescribed enforce displaceents, (ii) to siulate the continuous behavior of displaceent in the interface area, (iii) to enforce rest condition in the specified directions at grid points of reaction. Due to the coplexity of the geoetry and loading on the cylinder block, a three-diensional FEM was adopted as shown in Figure 3. The loading and constraints on the cylinder block are shown in Figure 4. 4 1, where N is the nuber of cycles of stress range (S) occurring in T seconds. The Dirlik solution [16] is expressed by the Equation (9) and details the specific literature reported in the Refs. [16-1]. 65
hole surfaces. In addition, preload was also applied on the gasket surface generating pressure of.3 MPa. The constraints were applied on the bolthole for all six degree of freedos. Multi-point constraints (MPCs) [] were used to connect the parts thru the interface nodes. These MPCs were acting as an artificial bolt and nut that connect each parts of the structure. Each MPC s will be connected using a Rigid Body Eleent (RBE) that indicating the independent and dependent nodes. The configuration of the engine is constrained by bolt at the cylinder head and cylinder block. In condition with no loading configuration the RBE eleent with six-degrees of freedo were assigned to the bolts and the head hole. The independent node was created on the cylinder block hole. Figure 3. 3D Finite eleent odel LOADING INFORMATION Several types of variable aplitude loading history were selected fro the SAE and ASTM profiles for the FE based fatigue analysis. The detailed inforation about these histories is given in the literature [3-4]. The variable aplitude loadtie histories are shown in Figure 5 and the corresponding the PSD plots are also shown in Figure 6. The ters of SAETRN, SAESUS, and SAEBRAKT represent the load-tie history for the transission, suspension, and bracket respectively. The considered load-tie histories are based on the SAE s profile. In addition, I-N, A-A, A-G, R-C and TRANSP are representing the ASTM instruentation and navigation typical fighter, ASTM air to air typical fighter, ASTM air to ground typical fighter, ASTM coposite ission typical fighter, and ASTM coposite ission typical transport loading history, respectively [3]. RESULTS AND DISCUSSION Figure 4. Loading and constraints. Three-diensional odel geoetry was developed in CATIA software. A parabolic tetrahedral eleent was used for the solid esh. Sensitivity analysis was perfored to obtain the optiu eleent size. These analyses were perfored iteratively at different eleent lengths until the solution obtained appropriate accuracy. Convergence of the stresses was observed, as the esh size was successively refined. The eleent size of. was finally considered. A total of 35415 eleents and 669 nodes were generated with. eleent length. Copressive loads were applied as pressure (7 MPa) acting on the surface of the cobustion chaber and preloads were applied as pressure (.3 MPa) acting on the bolt The odal analysis is usually used to deterine the natural frequencies and ode shapes of a coponent. It can also be used as the starting point for the frequency response, the transient and rando vibration analyses. Coercial finite eleent codes such as NASTRAN offer several ode extraction ethods. The Lanczos ode extraction ethod is used in this study because Lanczos is the recoended ethod for ediu to large odels. In addition to its reliability and efficiency, the Lanczos ethod supports sparse atrix ethods that substantially increase coputational speed and reduce disk space. This ethod also coputes accurately the eigenvalues and eigenvectors. The nuber of odes to be extracted and used to obtain the cylinder blocks stress histories, which is the ost iportant factor in this analysis type. This ethod is used to obtain the first 1 odes of the cylinder block. 66
6.993-3.465 5 1 15.415 SAE standard transission (SAETRN) Loading -6.993 5 1 15 5 5.166 SAE standard suspension (SAESUS) loading P ressu re (M P a ) -6.993 1 3 4 5 6 6.3 -.45 5 1 15 6.475-1.54 1 3 4 5 6 5.53-1.9 1 3 4 6.475 SAE standard bracket (SAEBKT) loading ASTM Instruentation & Navigation (I-N) typical fighter loading ASTM air to air (A-A) typical fighter loading ASTM air to ground (A-G) typical fighter loading -1.54 1 3 4 5 7 ASTM coposite ission (R-C) typical fighter loading -4.89 4 6 8 1 1 ASTM coposite ission (TRANSP) typical transport loading Tie (Seconds) Figure 5 Different tie loading histories 67
.338 3.88E-4-6 -4-4 6 SAE Standard Transission (SAETRN) loading.557 Probability Density.445.5446 5.473E-4.3967-6 -4-4 6 SAE Standard Suspension (SAESUS) Loading -6-4 - 4 6 SAE Standard Bracket (SAEBKT) Loading -6-4 - 4 6 ASTM Instruentation & Navigation (I-N) Typical Fighter Loading -6-4 - 4 6 ASTM Air to Air (A-A) Typical Fighter Loading.3587-6 -4-4 6 ASTM Air to Ground (A-G) Typical Fighter Loading.3573-6 -4-4 6 ASTM Coposite Mission (R-C) Typical Fighter Loading 5.59-6 -4-4 6 ASTM Coposite Mission (TRANSP) Typical Transport Loading Pressure (MPa) Figure 6 Power spectral densities response 68
Table 1 lists the results obtained and the ode shapes of the cylinder block are shown in Figure 7. Table 1. The results of the odal analysis Mode no. Frequency (Hz) 1 186.3 59.5 3 36.61 4 317.65 5 37.65 6 339.84 7 383. 8 46.17 9 65.87 1 71.8 the range of to 1%, with values of 1 to 5% as the typical range [5]. Zero daping ratio indicate that the ode is undaped. Daping ratio of one represents the critically daped ode. The result of frequency response finite eleent analysis i.e. the axiu principal stresses of the cylinder block is presented in Figure 8 for zero Hz. Fro the results, axiu principal stresses of 38. MPa and 55.8 MPa were obtained at node 4936. Maxiuprincipal stress 38. MPa at node 4936 Maxiuprincipal stress MPa Mode 1, 186.3 Hz Mode, 59.5 Hz Mode 3, 36.61 Hz Mode 4, 317.65 Hz Mode 5, 37.65 Hz Mode 6, 339.84 Hz Mode 7, 383. Hz Mode 8, 46.17 Hz Mode 9, 65.87 Hz Mode 1, 71.8 Hz Figure 7 The ode shapes of the cylinder block The frequency response analyses were perfored using MSC.NASTRAN finite eleent code. The frequency response analysis used daping ratio of 5% of critical. The daping ratio is the ratio of the actual daping in the syste to the critical daping. Most experiental odal analysis software packages report the odal daping in ters of nondiensional critical daping ratio expressed as a percentage [6-7]. In fact, ost structures have critical daping values in Figure 8 Maxiu principal stresses contour plotted at zero Hz It can be seen that the axiu principal stress varies when the plot are drawn the higher frequencies. This is due to dynaic influences of the first ode shape. The variation of axiu principal stresses with the frequency is shown in Figure 1. It can be seen that the axiu principal stress is obtained axiu at the frequency 3 Hz. The axiu principal stresses of the cylinder block at 3 Hz is presented in Figure 9. Fro the results, the axiu principal stresses of 38. MPa and 55.8 MPa were obtained at node 4936. Figures 11 to 14 show the applied tie-load histories, PSD s of narrow band signal (SAESUS loading condition), corresponding probability density function and cycle histogra, respectively.
MPa Maxiu principal stress 56.1 MPa at node 54 E4 Figure 11 Tie load histories Figure 9 Maxiu principal stresses contour plotted at 3 Hz 1.4E7 RMS Power (Newtons.H) Frequency (Hz) 5 Figure 1 Corresponding (Figure 11) power spectral density 1.E-4 Probability Density Force (Newton) -E4 Tie (Seconds) 5-1.533494E4 Force (Newtons) 1.59486 Figure 1 Maxiu principal stresses plotted against frequency Figure 13 Corresponding (Figure 11) probability density function 7
Maxiuheight : 458 Predicted iniu life is 1 9.44 seconds at node 4936 Log of Life (Seconds) 458 Cycles Z-Axis Range Newtons X-Axis 3.998E4-1.5366E4 Mean Newtons Y-Axis 1.536E4 Figure 14 Corresponding (Figure 11) cycle histogra The fatigue life contour result for the ost critical locations are shown at zero Hz and 3 Hz in Figures 15 and 16 using the SAETRN loading histories [3] respectively. The iniu life prediction in these cases is 1 7.67 seconds and 1 9.44 seconds for zero Hz and 3 Hz respectively. It can be seen that these two fatigue life contours are different and ost daage has been found at frequency of 3 Hz. The coparison results for this node are given in Table at different loading conditions. Dirlik ethod with ean stress correction is considered in this study. The area near the circular hole (as arked in Figure 15) shows the position of the shortest life. The Goodan ethod gives the best conservative prediction when copared with the Gerber ean stress correction. It can be seen that the AA4-T6 has the longest life than the AA661-T6 for all cases. Predicted iniu life is 1 7.67 seconds at node 4936 Log of Life (Seconds) Figure 16 Vibration fatigue life contour in log plotted for 3 Hz Frequency resolution of the transfer function is significant to capture the input PSD. The significances of the frequency resolutions of SAEBKT loading histories are also shown in Figures 17 and 18. Two types of Fast Fourier Transfor (FFT) buffer size width naely 819:.614 Hz and 16384:.35 Hz were used in this figures. The total area under each input PSD curve is deterined to be identical. However, the 16384:.35 Hz width has twice as any points copares to the 819:.614 Hz. The frequency resolution of the transfer function in the iportant areas of the input PSD is the doinant factor. This is shown in Figure 19 for two different cases. Figure 19 shows that utilizing any points in the input PSD can identify the daage ore accurately. However, the approach is not suitable for accurately identifying daage with a large spike occurred between two frequencies in the transfer function. For the worst case scenario the technique can entirely iss the daage. 1 RMS Power (MPa. Hz -1 ) Frequency (Hz) 1 Figure 15 Vibration fatigue life contour plot for zero Hz Figure 17. Power spectral density at FFT buffer size of 819:.614 Hz width 71
1 RMS Power (MPa. Hz -1 ) Frequency (Hz) 1 Figure 18. Power spectral density at FFT buffer size of 16384:.35 Hz width. Figure 19 Effect of frequency resolution 7
Table Fatigue life in seconds using the Dirlik ethod with ean stress correction Loading Predicted vibration fatigue life in seconds conditions 4-HV-T6 661-T6-8-HF No ean Goodan Gerber Non ean Goodan Gerber SAETRN.75E9.53E9.74E9 4.66E7 4.14E7 4.65E7 SAESUS 1.53E1 1.46E1 1.5E1.1E11 1.84E11.E11 SAEBKT 3.36E1 3.15E1 3.35E1 9.E8 8.19E8 8.99E8 I-N 1.47E1 1.39E1 1.46E1 4.4E8 4.4E8 4.39E8 A-A 3.6E9 3.37E9 3.59E9 7.89E7 7.18E7 7.88E7 A-G 9.14E8 8.47E8 9.13E8 1.6E7 1.43E7 1.59E7 R-C 1.96E9 1.8E9 1.95E9 3.8E7 3.46E7 3.81E7 TRANSP 1.51E11 1.44E11 1.5E11 7.3E9 6.74E9 7.9E9 CONCLUSIONS The concept of vibration fatigue analysis has been presented, where the rando loading and response are categorized using PSD functions. A state of art review of vibration fatigue techniques has been presented. The frequency doain fatigue analysis has been applied to a typical cylinder head of a free piston engine. Fro the results, it can be concluded that the Goodan ean stress correction ethod gives the ost conservative prediction for all loading conditions and aterials. The results clearly indicate that the AA4-HV-T6 is a superior aterial for all the ean stress ethods. The life predicted fro the vibration fatigue analysis is consistently higher except for the bracket loading condition. In addition, the vibration fatigue analysis can iprove the understanding of the syste behaviors in ters of frequency characteristics of the structures, loads and their couplings. ACKNOWLEDGMENTS The authors would like to thank the Departent of Mechanical and Materials Engineering, faculty of Engineering, Universiti Kebangsaan Malaysia. The authors are grateful to Malaysia Governent especially Ministry of Science, Technology and Innovation under IRPA project (IRPA project no: 3--- 56 PR5/4-3) for providing financial support. REFERENCES [1] Bolotin, V.V., 1984. Rando Vibrations of Elastic Systes, The Hague, The Netherlands: Martinus Nijhoff. [] Crandell, S.H. and Mark, W.D., 1973. Rando Vibration in Mechanical Systes. New York :Acadeic Press. [3] Soong, T.T. and Grigoriu, M., 1973. Rando Vibration of Mechanical and Structural Systes, UK: Prentice Hall International Ltd, London. [4] Newland, D.E., 1993. An Introduction to Rando Vibrations, Spectral and Wavelet analysis, UK: Longan Scientific and Technical, Essex. [5] Suresh, S.,. Fatigue of Materials, UK: Cabridge University Press. [6] Stephens, R.I., Fatei, A., Stephens, R.R. and Fuchs, H.O., 1. Metal Fatigue in Engineering, USA: John Wiley and Sons Inc. [7] Bishop, N.W.M. and Sherratt, F., 1999. A theoretical solution for the estiation of rainflow ranges fro power spectral density data, Fatigue Fracture, Engineering Materials Structure, Volue 13, pp. 311-36. [8] Wirsching, P.H., Paez, T.L. and Oritz, K., 1995. Rando Vibration, Theory and Practice, USA: John Wiley and Sons, Inc. [9] Bishop, N.W.M., 1988. The use of frequency doain paraeters to predict structural fatigue, Ph.D. Thesis, University of Warwick, UK. [1] Bishop, N.W.M. and Sherratt, F.,. Finite Eleent based Fatigue Calculations, UK: NAFEMS Ltd., Scotland. [11] Bishop, N.W.M. and Sherratt, F., 1989. Fatigue life prediction fro power spectral density data. Part-1, Traditional Approaches and part- Recent 73
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