INTRODUCTION
A free piston linear generator integrates a combustion engine and a linear
electrical machine into a single unit without a crankshaft. This provides an
unconventional solution for series hybrid vehicles and distributes/emergency
power units (Blarigan, 2000; Goertz and Peng, 2000; Arshad et al., 2003;
Cosic et al., 2003). Blarigan (2000) developed the free piston alternator
in the 30 kW range. The author implemented a nonconventional combustion technique
known as homogeneous charge compression ignition (HCCI). The schematic diagram
of free piston linear engine is shown in Fig. 1. The absence
of the crankshaft has benefits in the reliability, efficiency, fuel consumption
and environmental emissions (Arshad et al., 2004). The used of free piston
generators to produce electricity with the Stirling engines has been around
for quite some times. However, applications with the internal combustion engines
are relatively new. Although many patents have been reported by Gray (2003),
Berlinger et al. (2004), Schaeffer (2005) and Wood (2005) and the reported
research works were rather sparse (Blarigan, 2000; Goertz and Peng, 2000).

Fig. 1: 
Free piston linear engine 
Arshad et al. (2004) investigated the use of transverse flux machines in a free piston generator. The authors concentrated only on the electric generator. The past research at West Virginia University of United States of America demonstrated the stable operation of a free piston engine generator by Famouri et al. (1999). Sparkignited linear engine and an aircore linear alternator system manage to produce 500 W of output power. The engine system was based on a pair of opposed cylinders, operating under a twostroke cycle. For the linear alternator, moving permanent magnets are located within a series of stationary coils for voltage induction.
In the twostroke free piston engine two horizontally opposed pistons are mounted
on a connecting rod, which is allowed to oscillate between two endmounted cylinders.
Most of the previous studies (Aichlmayr, 2002; Arshad et al., 2003, 2004),
which are related to the engine, combustion occurred in both cylinders, in order
to get the linear movement of the piston. For this linear engine, crank and
camshaft are eliminated and there is no rotary movement involved. Additionally,
the linear system of the engine should prove to be more efficient as the frictional
losses associated with the crack and rod bearings are eliminated. However, in
previous study by Aichlmayr (2002) determined that a single piston engine with
rebound device and the unbalanced situation. These need to be encounter for
this new free piston generator engine.
Mounting of the twostroke free piston engine structures is commonly fatigue loading. Fatigue durability has important issues in the design of free piston linear generator engine structures. In engineering applications, the purpose of fatigue research consists of predicting the fatigue life of structures, increasing fatigue life and simplifying fatigue tests especially fatigue tests of fullscale structures under a random load spectrum. The fatigue life of an engineering structure principally depends upon that of its critical structure members and also depends on the nature of loading, type of the materials, vibration effects and invariants i.e., in automotive the road profiles etc. There is an increasing interest within the internal combustion engine industry in the ability to produce designs that are strong, reliable and safe, whilst also light in weight, economic and easy to produce. These opposing requirements can be satisfied by analytically optimizing components of linear generator engine. In the engine system design, the mounting structure is among the most critical parts. Numerical techniques are necessary to simulate the physical behavior and to evaluate the structural integrity of the different designs. The objectives of the current study are to calculate the fatigue life for a mounting of linear engine using total life and crack initiation methods, to investigate the effect of mean stress on fatigue life and the probabilistic nature of fatigue on the SN curve via the design criteria.
Finite element based fatigue analysis: The fatigue analysis is used
to compute the fatigue life at one location in a structure. For multiple locations
the process is repeated using geometry information applicable for each location.
Necessary inputs for the fatigue analysis are shown in Fig. 2.
The three input information boxes are descriptions of the material properties,
loading history and local geometry. All of these inputs are discussed following
sections.

Fig. 2: 
Fatigue analysis prediction strategy 
• 
Material informationcyclic or repeated material data based on constant
amplitude testing. 
• 
Load histories informationmeasured or simulated load histories applied
to a component. The term loads is used to represent forces, displacements,
accelerations, etc. 
• 
Geometry informationrelates the applied load histories to the local stresses
and strains at the location of interest. The geometry information is usually
derived from finite element (FE) results. 
An integrated FE based durability analysis is considered a complete analysis
of an entire component. Fatigue life can be estimated for every element in the
finite element model and contour plots of life. Geometry information is provided
by FE results for each load case applied independently, i.e., the FE results
define how an applied load is transformed into a stress or strain at a particular
location in the component. Appropriate material data are also provided for the
desired fatigue analysis method. The schematic diagram of the integrated finite
element based fatigue life prediction analysis is shown in Fig.
3.
Fatigue analysis methods: Fatigue analyses can be carrying out one of three basic methodologies, i.e., the totallife approach, the crack initiation approach and crack propagation approach. The totallife (stresslife) approach was first applied over a hundred years ago (Wöhler, 1867) and considers nominal elastic stresses and how they are related to life. The crack initiation (strainlife approach) considers elasticplastic local stresses and strains. It represents more fundamental approach and is used to determine the number of cycles required to initiate a small engineering cracks. Crack propagation or linear elastic fracture mechanics (LEFM) approach is used to predict how quickly preexisting cracks grow and to estimate how many loading cycles are required to grow these to a critical size when catastrophic failure would occur. First two methods are used in this study and briefly discussed these two methods in the following sections.
The fatigue totallife (SN) approach is usually used for the life prediction
of components subjected to high cycle fatigue, where stresses are mainly elastic.

Fig. 3: 
Integrated finite element based fatigue life prediction analysis 
This approach emphasizes nominal stresses rather than local stresses. It uses
the material stresslife curve and employs fatigue notch factors to account
for stress concentrations, empirical modification factors for surface finish
effects and analytical equations such as modified Goodman and Gerber equation
to account for mean stress effects. The modified Goodman and Gerber equations
are given by Eq. 1 and 2, respectively.
where, σ_{a}, S_{e}, σ_{m} and S_{u }are
the alternating stress in the presence of mean stress, alternating stress for
equivalent completely reversed loading, the mean stress and the ultimate tensile
strength, respectively. The typical representation of these mean stress correction
methods is shown in Fig. 4.
The Basquin (1910) showed that alternating stress versus number of cycles to
failure (SN) in the finite life region could be represented as a loglog linear
relationship. Basquin equation was then used to obtain the fatigue life using
the material properties listed in Table 1. SN approach uses
to estimate the fatigue life for combined loading by determining an equivalent
axial stress (Zoroufi and Fatemi, 2004) using one of the common failure criteria
such as Tresca, von Mises, or maximum principal stress. The SN equation is
mathematically given by:
Table 1: 
Mechanical and cyclic properties of the materials (Boardman,
1982) 


Fig. 4: 
Comparison of the mean stress correction method 
where, S_{e}, σ΄_{f}, 2N_{f} and b are the stress amplitude, the fatigue strength coefficient, the reversals to failure and the fatigue strength exponent, respectively.
An important aspect of the fatigue process is plastic deformation. Fatigue cracks initiate from the plastic straining in localized regions. Therefore, cyclic straincontrolled fatigue method could better characterize the fatigue behaviour of the materials than cyclic stresscontrolled fatigue. Particularly in notched members where the significant localized plastic deformation is often present. In the crack initiation approach the plastic strain is directly measured and quantified. The totallife approach does not account for plastic strain. One of the main advantages of this method is that it accounts for changes in local mean and residual stresses.
When the load history contains large overloads, significant plastic deformation can exist, particularly at stress concentrations and the load sequence effects can be significant. In these cases, the crack initiation approach is generally superior to the totallife approach for fatigue life prediction analysis. However, when the load levels are relatively low such that the resulting strains are mainly elastic, the crack initiation and totallife approaches usually result in similar predictions.
The crack initiation approach to fatigue problems is widely used at present especially when the linear generator engine are started or stopped then it is subjected to a very high stress range. The fatigue crack initiation approach involves the techniques for converting load history, geometry and material properties (monotonic and cyclic) input into the fatigue life prediction. The operations involved in the prediction must be performed sequentially. First, the stress and strain at the critical site are estimated and rainflow cycle counting method (Amzallang et al., 1994) is then used to reduce the loadtime history based on the peakvalley sequential. The next step is to use the finite element method to convert a reduced loadtime history into a straintime history and calculate the stress and strain in the highly stressed area. Then the crack initiation methods are employed for predicting fatigue life. Following this, the simple linear damage hypothesis proposed by Palmgren (1924) and Miner (1945) is used to accumulate the fatigue damage. Finally, the damage values for all cycles are summed until a critical damage sum (failure criteria) is reached.
In order to perform the fatigue analysis and to implement the stressstrain
approach in complex structures, Conle and Chu (1997) used strainlife results
which is simulated using the three dimensional models to assess fatigue damage.
After the complex load history was reduced to an elastic stress history for
each critical element, a Neuber plasticity correction method was used to correct
for plastic behaviour. Elastic unit load analysis, using strength of material
and an elastic finite element analysis model combined with a superposition procedure
of each load point’s service history was proposed. Savaidis (2001) verified
the local strain approach for fatigue evaluation.

Fig. 5: 
Typical total strainlife curve 
In this study, it was observed that the local strain approach using the SmithWatsonTopper
(SWT) strainlife model is able to represent and to estimate many factors explicitly.
These include mean stress effects, load sequence effects above and below the
endurance limit and manufacturing process effects such as surface roughness
and residual stresses and also stated in the book by Juvinall and Marshek (1991).
The fatigue resistance of metals can be characterized by a strainlife curve.
These curves are derived from the polished laboratory specimens tested under
completely reversed strain control. The relationship between the total strain
amplitude (Δε/2) and reversals to failure (2N_{f}) can be
expressed in the following form (Coffin, 1954; Manson, 1953). Figure
5 represents the typical total strainlife curves.
where, N_{f} is the fatigue life; σ΄_{f} is the fatigue strength coefficient; E is the modulus of elasticity; b is the fatigue strength exponent; ε΄_{f} is the fatigue ductility coefficient and c is the fatigue ductility exponent.
Morrow (1968) suggested that mean stress effects are considered by modifying
the elastic term in the strainlife equation by mean stress (σ_{m}).
Smith et al. (1970) was introduced another mean stress model which is
called SWT mean stress correction model. It is mathematically defined as:
where, σ_{max} is the maximum stress and ε_{a} is the strain amplitude.
Material information: The material data is one of the major input, which
is the definition of how a material behaves under the cyclic loading conditions
it typically experiences during service operation. Cyclic material properties
are used to calculate elasticplastic stress strain response and the rate at
which fatigue damage accumulates due to each fatigue cycle. The materials parameters
required depend on the analysis methodology being used. Normally, these parameters
are measured experimentally and available in various handbooks. Two different
materials were used for this component, SAE 1045QT and SAE 1045595QT. Figure
6 shows a comparison between the two materials with respect to SN behaviour.
It can be seen that these curves exhibit different life behaviour depending
on the stress range experienced. From the figure, it is observed that in the
long life area (high cycle fatigue), the difference is lower while in the short
life area (low cycle fatigue), the difference is higher. Figure
7 shows the cyclic stressstrain curve of these two materials. It is seen
than how these two materials behave under cyclic loading conditions. It can
also be seen that how they behave relative to each other. It is also observed
that SAE 1045595QT is much higher strength steel with its yield point well
above that of SAE 1045450QT.
Figure 8 represents the strainlife curve indicating that
different fatigue life behaviour for both materials. It is plotted based on
the CoffinManson relationship. From the figure, it can be seen that in the
long life area (high cycle fatigue) the difference is lower while in the short
life area (low cycle fatigue) the difference is higher. Figure
9 and 10 also show that another strainlife curves those
are based on SWT and Morrow models, respectively.
Loading information: Loading is another major input for the finite element
based fatigue analysis. The component was loaded with three random time histories
corresponding to typical histories for transmission, suspension and bracket
components at different load levels. The detailed information about these loading
histories was contained in the literature (Tucker and Bussa, 1977). These loading
histories were scaled to two peak strain levels and used as fulllength histories.
Raw loadtime histories of the component are shown in Fig. 11.
The terms of SAETRN, SAESUS and SAEBRAKT represent the loadtime history for
the transmission, suspension and bracket, respectively. The considered loadtime
histories are based on the SAE’s profile. The abscissa uses the time in
seconds.

Fig. 6: 
Stresslife (SN) plot 

Fig. 7: 
Cyclic stressstrain plot 
Finite element analysis: Threedimensional model of linear generator
engine mounting was developed using the CATIA^{®} software. A parabolic
tetrahedron element was used for the solid mesh. Sensitivity analysis was performed
in order to obtain the optimum element size. These analyses were performed iteratively
at different element lengths until the solution obtained appropriate accuracy
Convergence of stresses was observed, as the mesh size was successively refined.

Fig. 9: 
SWT strainlife plot 

Fig. 10: 
Morrow’s strainlife plot 

Fig. 11: 
Loadtime histories 
A total of 35415 elements and 66209 nodes were generated at 0.20 mm element
length. The constraints were applied on the bolthole for all six degree of
freedom. The objective of the FE analysis was to investigate of the mounting.
Figure 12 shows the geometry, loading and boundary conditions
used for the FE analysis of mounting. From the resulting stress contours, the
state of stress can be obtained and consequently used for uniaxial life predictions.
RESULTS AND DISCUSSION
The linear static finite element analysis was performed using MSC.NASTRAN^{®}
finite element code. The equivalent von Mises stress contours and critical locations
are shown in Fig. 13. The bolt holes and fillet areas were
found to be areas of high stresses. The von Mises equivalent stresses are used
for subsequent fatigue life analysis and comparisons. From the FE Analysis results,
the maximum von Mises stresses of 178 MPa at node 11810 was obtained. The fatigue
life of the free piston engine mounting is obtained using variable amplitude
loading conditions by means of SAETRN, SAESUS and SAEBRAKT data set. The fatigue
life prediction results of mounting corresponding to 99.5% reliability value
are shown in Fig. 14. It can be seen that the predicted fatigue
life at most critical location near the bolt hole edge (node 11810) is 10^{6.88}
seconds when using SAE 1045450QT material and Goodman mean stress correction
method. The fatigue life is in terms of seconds using the variable amplitude
SAESUS loading. The fatigue equivalent unit is 3000 cpm (cycle per min) of time
history. The critical locations are also shown in Fig. 15
when using the SAESUS loading histories. It is found that bolt edge is the most
critical positions among the mounting.
Most realistic service situations involve nonzero mean stresses. It is, therefore,
very important to know the influence that mean stress has on the fatigue process
so that the fullyreversed (zero mean stress) laboratory data are usefully employed
in the assessment of real situations. Four types of mean stress correction method
are considered in this study i.e., Goodman and Gerber correction methods for
totallife approach and SWT and Morrow methods for crack initiation approach.
The predicted fatigue life at most critical location (node 11810) using different
loading histories are tabulated in Table 2 and 3,
respectively, using different materials and approaches.
It is difficult to categorically select one procedure in the preference to
the other. However, in Table 2, it can be seen that when using
the loading sequences are predominantly tensile in nature, the Goodman approach
is more conservative.

Fig. 13: 
Von mises stresses distribution contours 

Fig. 14: 
Life contour plotted in log of life 
In the case where the loading is predominantly compressive, particularly for
wholly compressive cycles, the no correction can be used to provide more realistic
life estimates and Gerber mean stress correction has been found to give conservative
when the time histories predominantly zero mean.
Table 2: 
Predicted fatigue life using totallife approach 

Table 3: 
Predicted fatigue life using crack initiation approach 


Fig. 15: 
Critical locations of mounting 
From Table 3, it is also seen that the two mean stress methods,
SWT and Morrow give lives less than that achieved using no mean stress correction
with the SWT method being the most conservative for loading sequences which
are predominantly tensile in nature. In the case where the loading is predominantly
compressive (SAESUS), particularly for wholly compressive cycles SWT and Morrow’s
methods have been found higher lives than no mean stress correction. When using
the time history has a roughly zero mean (SAEBRAKT) then all three methods have
been given approximately the same results. It can also be seen that SAE 1045595QT
is consistently higher life than SAE 1045450QT for all loading conditions.
The threedimensional cycle histogram and corresponding damage histogram for
materials SAE 1045450QT using SAESUS loading histories is shown in
Fig. 16 and 17, respectively. Figure 16
shows the results of the rainflow cycle count for the critical location on the
component.

Fig. 16: 
Rainflow cycle counting histogram at critical location (node
11810) using SAESUS data set 

Fig. 17: 
Damage histogram at the critical location (node 11810) using
the SAESUS data set 
It can be seen that a lot of cycles with a low stress range and fewer with
a high range. The height of each tower represents the number of cycles at that
particular stress range and mean. Each tower is used to obtain damage on the
SN curve and damage is summed over all towers. Figure 17
shows the lower stress ranges produced zero damage. It is also showed that the
high stress ranges were found to give the most of the damage and a fairly wide
damage distribution at the higher ranges which mean that it cannot point to
a single event causing damage.

Fig. 18: 
Effect of probabilistic nature of fatigue 
In this research investigate the effect of probabilistic nature on the fatigue
life. It can be seen that 1045595QT is the higher strength steel, gives a
much higher life prediction (Table 2, 3)
for all mean stress corrections. This means 1045595QT is a better material
to use. It is also observed at the SN curve as shown in Fig.
6. Figure 18 shows the effect of probabilistic nature
of fatigue using the SAESUS data set for totallife approach. From the Fig.
15, it can be seen that when increases the design criterion up to 99.9 (certainty
of survival 99.9%) then the life decreases as compared to lower certainty of
survival. This is due to the probabilistic nature of fatigue and the scatter
associated with the SN curves themselves. The material parameters associated
with SN curves take this into consideration with the standard error of Log(N)
(SE) determined by the regression analysis of the raw data. It is recommended
that 99.9 (99.9% certainty of survival) as the design criterion. The larger
the scatter in the original SN data that makes up the curve, the less certain
will be of survival. For both cases, it can be seen that 99.9% certainty of
survival is obtained more conservative results and also determined that SAE
1045595QT was a better material at all lives based on the SN curves. This
is due to the probabilistic nature of SN curves where the scatter in the SN
data for SAE 1045595QT is much more variable than for SAE 1045450QT.
CONCLUSION
A computational numerical model for the fatigue life assessment for mounting
of the linear generator engine is presented in this study. Through the study,
several conclusions can be drawn with regard to the fatigue life of a component
when subjected to complex variable amplitude loading conditions. The fatigue
damage estimated based on the PalmgrenMiner rule is nonconservative and SWT
correction and Morrow’s damage rule can be applied to improve the estimation.
It can be concluded that the influence of mean stress correction is different
for compressive and tensile mean stress. Failure appears to be more sensitive
to tensile mean stress, than compressive mean stress for total life approach.
It is also concluded for crack initiation approach that when the loading is
predominantly tensile in nature, SWT approach is more sensitive and is therefore
recommended. However, when the loading is compressive, the Morrow correction
can be used more realistic life estimates. Therefore, it can be used an efficient
and reliable means for the signoff of durability of a prototype engine with
actual service environments in the earlydeveloping stage. It can be also seen
that SAE 1045595QT is consistently higher life than SAE 1045450QT for all
cases.
ACKNOWLEDGMENTS
The authors would like to thank the Department of Mechanical and Materials Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia to provide the laboratory facilities. The authors would like to express their especial thanks to Universiti Malaysia Pahang for providing financial support under the project (No. RDU070346).