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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()] can be generated by employing the Fourier transform of the load vector {p(t)}. This can be written as shown in Equation (2).

( )[ ]

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

=

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 are the column and row matrix respectively. The diagonal term Sii() is the auto-correlation function of load pi(t), and the off-diagonal term Sij() 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()] 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),

( )[ ] ( )[ ] ( )[ ] ( )[ ] T nmmmpmnnnx HSHS = (3)

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

( )[ ] [ ] [ ] [ ]( ) 12 KCiMH ++= (4)

The response variables [Sp()] 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 historys auto-correlation function, and its area represents the signals 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==

][

][ (7)

The irregularity factor is an important parameter that can be used to evaluate the concentration of the process near a central frequency. Therefore, can be used to determine whether the process is narrow band or wide band. A narrow band process (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 (0). Bendat [13] has proposed first significant