63

International Journal of Mechanical and Materials Engineering (IJMME), Vol. 2 (2007), No. 1, 63-74.

FINITE ELEMENT BASED VIBRATION FATIGUE ANALYSIS OF A NEW TWO-STROKE LINEAR GENERATOR ENGINE COMPONENT

M. M. Rahman, A. K. Ariffin, and S. Abdullah

Department of Mechanical and Materials Engineering

Faculty of Engineering, Universiti Kebangsaan Malaysia 43600 UKM, Bangi, Selangor, Malaysia

Phone: +(6)03-8921-6012, Fax: +(6)03-89259659 E-mail: mustafiz@eng.ukm.my, kamal@eng.ukm.my

ABSTRACT This paper presents the finite element analysis technique to predict the fatigue life using the narrow band frequency response approach. Such life prediction results are useful for improving the component design at the very early stage. This paper describes how this technique can be implemented in the finite element environment to rapidly identify critical areas in the structure. Fatigue damage is traditionally determined from the time signals of the loading, usually in the form of stress and strain. However, there are scenarios when a spectral form of loading is more appropriate. In this case the loading is defined in terms of its magnitude at different frequencies in the form of a power spectral density (PSD) plot. A frequency domain fatigue calculation can be utilized where the random loading and response are categorized using power spectral density functions and the dynamic structure is modeled as a linear transfer function. This paper investigates the effect of mean stress on the fatigue life prediction by using a random varying load. The obtained results indicate that the Goodman mean stress correction method gives the most conservative results compared with Gerber, and no mean stress correction method. The proposed analysis technique is capable of determining premature products failure phenomena. Therefore, it can reduce cost, time to market, improve product reliability and customer confidence.

Keywords: Fatigue, Fast Fourier Transform; vibration; power spectral density function; frequency response; power density function.

INTRODUCTION Structures and mechanical components are frequently subjected to oscillating loads which are random in nature. Random vibration theory has been introduced for more then three decades to deal with all kinds of random vibration behaviour. Since fatigue is one of the primary causes of component failure, fatigue life prediction has become a major subject in almost any random vibration [1-4]. Nearly all structures or

components have been designed using time based structural and fatigue analysis methods. However, by developing a frequency based fatigue analysis approach, the true composition of the random stress or strain responses can be retained within a much optimized fatigue design process. The time domain fatigue approach consists of two major steps. Firstly, the numbers of stress cycles in the response time history [5-7] are counted. This is conducted through a process called a rain flow cycle counting. Secondly, the damage from each cycle is determined, typically from an S-N curve. The damage is then summed over all cycles using linear damage summation techniques to determine the total life. The purpose of presenting these basic fatigue concepts is to emphasize that the fatigue analysis is generally thought of as a time domain approach, That is, all of the operations are based on time descriptions of the load function. This paper demonstrates that an alternative frequency domain [4,8-9] fatigue approach is more appropriate. A vibration analysis is usually carried out to ensure that the structural natural frequencies or resonant modes are not excited by the frequencies of the applied load. It is often easier to obtain a PSD of stress rather than a time history [10-11]. The dynamic analysis of complicated finite element models is considered in this study. It is beneficial to carry out the frequency response analysis instead of a computationally intensive transient dynamic analysis in the time domain. A finite element analysis based on the frequency domain can simplify the problem. The designer can carry out the frequency response analysis on the finite element model (FEM) to determine the transfer function between load and stress in the structure. This approach requires that the PSD of the load is multiplied by the transfer function to the PSD of the stress. The main purpose of the present paper is to derive formulas for the prediction of the fatigue damage when a component is subjected to statistically defined random stresses.

THEORETICAL BASIS The equation of motion of a linear structural system is expressed in matrix format in Equation 1.

64

[ ]{ } [ ]{ } [ ]{ } { })()()()( tptxKtxCtxM =++ &&& (1)

where {x(t)} is a system displacement vector, [M], [C] and [K] are mass, damping and stiffness matrices, respectively, {p(t)}is an applied load vector. The system of time domain differential equations can be solved directly in the physical coordinate system. When loads are random in nature, a matrix of the loading power spectral density (PSD) functions [Sp(ω)] can be generated by employing the Fourier transform of the load vector {p(t)}. This can be written as shown in Equation (2).

( )[ ]

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

ΛΛ

Λ

Λ

ΛΛ

=×

ωωω

ωωω

ωωω

ω

mmmim

imiii

mi

mmp

SSS

MM

SSS

MM

SSS

S

1

1

1111

0

0 (2)

where m is the number of input loads, M and Λ are the column and row matrix respectively. The diagonal term Sii(ω) is the auto-correlation function of load pi(t), and the off-diagonal term Sij(ω) is the cross- correlation function between loads pi(t) and pj(t). From the properties of the cross PSDs, it can be shown that the multiple input PSD matrix [Sp(ω)] is a Hermitian matrix. The system of time domain differential equation of motion of the structure in Equation (1), is then reduced to a system of frequency domain algebra equations as shown in Equation (3),

( )[ ] ( )[ ] ( )[ ] ( )[ ] Tnmmmpmnnnx HSHS ××××

= ωωωω (3)

where n is the number of output response variables. The T denotes the transpose of a matrix. [H(ω)] is the transfer function matrix between the input loadings and output response variables. It can be written as Equation (4) shown below,

( )[ ] [ ] [ ] [ ]( ) 12 KCiMH−

++−= ωωω (4)

The response variables [Sp(ω)] such as displacement, acceleration and stress response in terms of PSD functions are obtained by solving the system of the linear algebra equations in Equation (3). The stress power spectra density [3-4,9-12] represents the frequency domain approach input into the fatigue. This is a scalar function describing how the power of the time signal is distributed among frequencies [13]. Mathematically, this function can be obtained by using a Fourier transform of the stress time history’s auto-correlation function, and its area represents the signal’s standard deviation. It is clear that the PSD is the most complete and concise representation of a random process. There are many important correlations between the time domain and frequency domain representations [14] of a random process. In fact there is transformation, which can be used to move from the time domain to the frequency

domain as shown in Figure 1. The information extracted from the frequency domain directly and used to compute fatigue damage, are the PSD moments used to compute all of the information required to estimate fatigue damage, in particular the probability density function (pdf) of stress ranges and the expected numbers of zero crossings and peaks per second. The nth moment of PSD area is computed by Equation (5).

dffGfM0

nn )(∫=

∞

(5)

where f is the frequency and G(f) is the single sided PSD at frequency f Hz.

Figure 1 The transformation between time and frequency domains

A method for computing these moments is shown in Figure 2. Some very important statistical parameters can be computed from these moments. These parameters are root mean square (rms), expected number of zero crossing with positive slope (E[0]), expected number of peaks per second (E [P]). The formulas in Equation (6) highlight these properties of the spectral moments.

2

4

0

20 ][;]0[;

m

mPE

m

mEmrms === (6)

Figure 2 Calculating moments from a PSD

Inverse Fourier transform

Fourier transform

For

ce

Tim

e do

mai

n

Frequency Time

Com

plex

F

FT

Fre

quen

cy d

omai

n

f

G(f)

65

Another important property of the spectral moments is the fact that it is possible to express the irregularity factor as a function of the zero, second and fourth order spectral moments, as shown in Equation (7).

40

2

mm

m

PE

0E==

][

][γ (7)

The irregularity factor γ is an important parameter that can be used to evaluate the concentration of the process near a central frequency. Therefore, γ can be used to determine whether the process is narrow band or wide band. A narrow band process (γ→1) is characterized by only one predominant central frequency indicating that the number of peaks per second is very similar to the number of zero crossings of the signal. This assumption leads to the fact that the pdf of the fatigue cycles range is the same as the pdf of the peaks in the signal (Bendat theory). In this case fatigue life is easy to estimate. In contrast, the same property is not true for wide bend process (γ→0). Bendat [13] has proposed first significant step towards a method of determining fatigue life from PSDs. Bendat showed that the probability density function (pdf) of peaks for a narrow band signal tended towards Rayleigh distributions as the bandwidth reduced. Furthermore, for a narrow banded time history Bendat assumed that all the positive peaks in the time history would be followed by corresponding troughs of similar magnitude regardless of whether they actually formed stress cycles. Using this assumption, the pdf of stress range would also tend to a Rayleigh distribution. To complete his solution method, Bendat used a series of equations derived by Rice [15] to estimate the expected number of peaks using moments of area beneath the PSD. Bendat’s narrow band solution for the range mean histogram is therefore expressed in Equation (8).

∫

∫∑

=

==

dSem

SS

K

TPE

dSSpSK

S

SN

nDE

m

S

b

bt

i i

i

]4

[][

)()(

][

08

2

0

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

= 08

2

04][

m

S

em

STPE (8)

where N is the number 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. [16-21].

)(][)( SpTPESN = (9)

where N(S) is the number of stress cycles of range S N/mm2

expected in T sec. E[P] is the expected number of peaks. Parameter p(S) in Equation (9) can be defined as shown in Equation (10) to Equation (12).

0

23

22

21

2)(

2

2

2

m

ZeDeR

ZDe

Q

D

Sp

Z

R

ZQ

Z −−−

++

= (10)

,1

1,

1

)(2 211

22

2

1 R

DDD

xD m

−

−−−=

+

−=

γ

γ

γ

40

2213 ,1

mm

mDDD =−−= γ (11)

,)(25.1

1

23

D

RDDQ

−−=

γ

,1 2

11

21

DD

DxR m

+−−

−−=

γ

γ

04

2

0

1

2,

m

SZ

m

m

m

mxm == (12)

where xm, D1, D2, D3, Q and R are all functions of m0, m1, m2

and m4; Z is a normalized variable.

NUMERICAL EXAMPLE In the finite element model 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 employed for the following purposes: (i) to specify the prescribed enforce displacements, (ii) to simulate the continuous behavior of displacement in the interface area, (iii) to enforce rest condition in the specified directions at grid points of reaction. Due to the complexity of the geometry and loading on the cylinder block, a three-dimensional FEM was adopted as shown in Figure 3. The loading and constraints on the cylinder block are shown in Figure 4.

66

Figure 3. 3D Finite element model

Figure 4. Loading and constraints.

Three-dimensional model geometry was developed in CATIA®

software. A parabolic tetrahedral element was used for the solid mesh. Sensitivity analysis was performed to obtain the optimum element size. These analyses were performed iteratively at different element lengths until the solution obtained appropriate accuracy. Convergence of the stresses was observed, as the mesh size was successively refined. The element size of 0.20 mm was finally considered. A total of 35415 elements and 66209 nodes were generated with 0.20 mm element length. Compressive loads were applied as pressure (7 MPa) acting on the surface of the combustion chamber and preloads were applied as pressure (0.3 MPa) acting on the bolt

hole surfaces. In addition, preload was also applied on the gasket surface generating pressure of 0.3 MPa. The constraints were applied on the bolthole for all six degree of freedoms. Multi-point constraints (MPCs) [22] 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 Element (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 element with six-degrees of freedom were assigned to the bolts and the head hole. The independent node was created on the cylinder block hole.

LOADING INFORMATION

Several types of variable amplitude loading history were selected from the SAE and ASTM profiles for the FE based fatigue analysis. The detailed information about these histories is given in the literature [23-24]. The variable amplitude load-time histories are shown in Figure 5 and the corresponding the PSD plots are also shown in Figure 6. The terms of SAETRN, SAESUS, and SAEBRAKT represent the load-time history for the transmission, suspension, and bracket respectively. The considered load-time histories are based on the SAE’s profile. In addition, I-N, A-A, A-G, R-C and TRANSP are representing the ASTM instrumentation and navigation typical fighter, ASTM air to air typical fighter, ASTM air to ground typical fighter, ASTM composite mission typical fighter, and ASTM composite mission typical transport loading history, respectively [23].

RESULTS AND DISCUSSION

The modal analysis is usually used to determine the natural frequencies and mode shapes of a component. It can also be used as the starting point for the frequency response, the transient and random vibration analyses. Commercial finite element codes such as NASTRAN offer several mode extraction methods. The Lanczos mode extraction method is used in this study because Lanczos is the recommended method for medium to large models. In addition to its reliability and efficiency, the Lanczos method supports sparse matrix methods that substantially increase computational speed and reduce disk space. This method also computes accurately the eigenvalues and eigenvectors. The number of modes to be extracted and used to obtain the cylinder blocks stress histories, which is the most important factor in this analysis type. This method is used to obtain the first 10 modes of the cylinder block.

67

Figure 5 Different time loading histories

0 500 1000 1500-3.465

6.993

0 500 1000 1500 2000 2500-6.993

2.415

0 1000 2000 3000 4000 5000 6000-6.993

5.166

0 50 100 150-0.45

6.3

0 100 200 300 400 500 600-1.54

6.475

0 100 200 300 400-1.029

5.523

0 100 200 300 400 500-1.54

6.475

0 200 400 600 800 1000 1200-4.829

7

Pr

es

su

re

(M

Pa

)

SAE standard transmission (SAETRN) Loading

SAE standard suspension (SAESUS) loading

SAE standard bracket (SAEBKT) loading

ASTM Instrumentation & Navigation (I-N) typical fighter loading

ASTM air to air (A-A) typical fighter loading

ASTM air to ground (A-G) typical fighter loading

ASTM composite mission (R-C) typical fighter loading

ASTM composite mission (TRANSP) typical transport loading

Time (Seconds)

68

Figure 6 Power spectral densities response

-6 -4 -2 0 2 4 63.88E-4

0.3038

SAE Standard Transmission (SAETRN) loading

-6 -4 -2 0 2 4 6

0.5057

SAE Standard Suspension (SAESUS) Loading

-6 -4 -2 0 2 4 60.2445

SAE Standard Bracket (SAEBKT) Loading

ASTM Instrumentation & Navigation (I-N) Typical Fighter Loading

-6 -4 -2 0 2 4 65.473E-4

0.5446

-6 -4 -2 0 2 4 6

0.3967

ASTM Air to Air (A-A) Typical Fighter Loading

-6 -4 -2 0 2 4 6

0.3587

ASTM Air to Ground (A-G) Typical Fighter Loading

-6 -4 -2 0 2 4 6

0.3573

ASTM Composite Mission (R-C) Typical Fighter Loading

-6 -4 -2 0 2 4 6

5.259

Pro

babi

lity

Den

sity

ASTM Composite Mission (TRANSP) Typical Transport Loading

Pressure (MPa)

Table 1 lists the results obtained and the mode shapes of the cylinder block are shown in Figure 7.

Table 1. The results of the modal analysis

Mode no. Frequency (Hz)

1 186.32 2 259.05 3 306.61

4 317.65 5 327.65

6 339.84 7 383.20

8 462.17 9 650.87

10 721.28

Figure 7 The mode shapes of the cylinder block

The frequency response analyses were performed using MSC.NASTRAN finite element code. The frequency response analysis used damping ratio of 5% of critical. The damping ratio is the ratio of the actual damping in the system to the critical damping. Most experimental modal analysis software packages report the modal damping in terms of non-dimensional critical damping ratio expressed as a percentage [26-27]. In fact, most structures have critical damping values in

the range of 0 to 10%, with values of 1 to 5% as the typical range [25]. Zero damping ratio indicate that the mode is undamped. Damping ratio of one represents the critically damped mode. The result of frequency response finite element analysis i.e. the maximum principal stresses of the cylinder block is presented in Figure 8 for zero Hz. From the results, maximum principal stresses of 38.0 MPa and 55.8 MPa were obtained at node 49360.

Figure 8 Maximum principal stresses contour plotted at zero Hz

It can be seen that the maximum principal stress varies when the plot are drawn the higher frequencies. This is due to dynamic influences of the first mode shape. The variation of maximum principal stresses with the frequency is shown in Figure 10. It can be seen that the maximum principal stress is obtained maximum at the frequency 32 Hz. The maximum principal stresses of the cylinder block at 32 Hz is presented in Figure 9. From the results, the maximum principal stresses of 38.0 MPa and 55.8 MPa were obtained at node 49360.

Figures 11 to 14 show the applied time-load histories, PSD’s of narrow band signal (SAESUS loading condition), corresponding probability density function and cycle histogram, respectively.

Maximum principal stress MPaMaximum principal stress

38.0 MPa at node 49360

Mode 1, 186.32 Hz Mode 2, 259.05 Hz Mode 3, 306.61 Hz Mode 4, 317.65 Hz

Mode 5, 327.65 Hz Mode 6, 339.84 Hz Mode 7, 383.20 Hz Mode 8, 462.17 Hz

Mode 9, 650.87 Hz Mode 10, 721.28 Hz

70

Figure 9 Maximum principal stresses contour plotted at 32 Hz

Figure 10 Maximum principal stresses plotted against frequency

Figure 11 Time–load histories

1.4E7

0250

RM

S Po

wer

(N

ewto

ns2 .H

)

Frequency (Hz)

Figure 12 Corresponding (Figure 11) power spectral density

Figure 13 Corresponding (Figure 11) probability density function

1.2E-4

01.529486-1.533494E4

Prob

abili

ty D

ensi

ty

Force (Newtons)

MPaMaximum principal stress 56.1 MPa at node 50420 2E4

-2E450000

Forc

e(N

ewto

n)

Time (Seconds)

71

Figure 14 Corresponding (Figure 11) cycle histogram

The fatigue life contour result for the most critical locations are shown at zero Hz and 32 Hz in Figures 15 and 16 using the SAETRN loading histories [23] respectively. The minimum life prediction in these cases is 107.67 seconds and 109.44 seconds for zero Hz and 32 Hz respectively. It can be seen that these two fatigue life contours are different and most damage has been found at frequency of 32 Hz. The comparison results for this node are given in Table 2 at different loading conditions. Dirlik method with mean stress correction is considered in this study. The area near the circular hole (as marked in Figure 15) shows the position of the shortest life. The Goodman method gives the best conservative prediction when compared with the Gerber mean stress correction. It can be seen that the AA2024-T6 has the longest life than the AA6061-T6 for all cases.

Figure 15 Vibration fatigue life contour plot for zero Hz

Figure 16 Vibration fatigue life contour in log plotted for 32 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 Transform (FFT) buffer size width namely 8192:0.06104 Hz and 16384:0.03052 Hz were used in this figures. The total area under each input PSD curve is determined to be identical. However, the 16384:0.03052 Hz width has twice as many points compares to the 8192:0.06104 Hz. The frequency resolution of the transfer function in the important areas of the input PSD is the dominant factor. This is shown in Figure 19 for two different cases. Figure 19 shows that utilizing many points in the input PSD can identify the damage more accurately. However, the approach is not suitable for accurately identifying damage with a large spike occurred between two frequencies in the transfer function. For the worst case scenario the technique can entirely miss the damage.

Figure 17. Power spectral density at FFT buffer size of 8192:0.06104 Hz width

0

3.0998E4 -1.5366E4

1.5326E4 0

458

Range NewtonsX-Axis

Mean NewtonsY-Axis

CyclesZ-Axis

Maximum height : 458

Log of Life(Seconds)

Predicted minimum life is 107.67 seconds at node 49360

Log of Life(Seconds)

Predicted minimum life is 109.44 seconds at node 49360

10

0100

RM

S Po

wer

(M

Pa2 . H

z-1)

Frequency (Hz)

72

Figure 18. Power spectral density at FFT buffer size of 16384:0.03052 Hz width.

Figure 19 Effect of frequency resolution

1 0

01 00

RM

S Po

wer

(M

Pa2 . H

z-1)

F r e q u e n c y ( H z )

73

Table 2 Fatigue life in seconds using the Dirlik method with mean stress correction

Predicted vibration fatigue life in seconds

2024-HV-T6 6061-T6-80-HF

Loading conditions

No mean Goodman Gerber Non mean Goodman Gerber SAETRN 2.75E9 2.53E9 2.74E9 4.66E7 4.14E7 4.65E7

SAESUS 1.53E12 1.46E12 1.52E12 2.01E11 1.84E11 2.00E11

SAEBKT 3.36E10 3.15E10 3.35E10 9.00E8 8.19E8 8.99E8

I-N 1.47E10 1.39E10 1.46E10 4.40E8 4.04E8 4.39E8

A-A 3.60E9 3.37E9 3.59E9 7.89E7 7.18E7 7.88E7

A-G 9.14E8 8.47E8 9.13E8 1.60E7 1.43E7 1.59E7

R-C 1.96E9 1.82E9 1.95E9 3.82E7 3.46E7 3.81E7

TRANSP 1.51E11 1.44E11 1.50E11 7.30E9 6.74E9 7.29E9

CONCLUSIONS The concept of vibration fatigue analysis has been presented, where the random loading and response are categorized using PSD functions. A state of art review of vibration fatigue techniques has been presented. The frequency domain fatigue analysis has been applied to a typical cylinder head of a free piston engine. From the results, it can be concluded that the Goodman mean stress correction method gives the most conservative prediction for all loading conditions and materials. The results clearly indicate that the AA2024-HV-T6 is a superior material for all the mean stress methods. The life predicted from the vibration fatigue analysis is consistently higher except for the bracket loading condition. In addition, the vibration fatigue analysis can improve the understanding of the system behaviors in terms of frequency characteristics of the structures, loads and their couplings.

ACKNOWLEDGMENTS The authors would like to thank the Department of Mechanical and Materials Engineering, faculty of Engineering, Universiti Kebangsaan Malaysia. The authors are grateful to Malaysia Government especially Ministry of Science, Technology and Innovation under IRPA project (IRPA project no: 03-02-02-0056 PR0025/04-03) for providing financial support.

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