FINITE ELEMENT BASED VIBRATION FATIGUE

7/30/2019 FINITE ELEMENT BASED VIBRATION FATIGUE

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M. M. Rahman, A. K. Ariffin, N. Jamaludin, and C. H. C. Haron

April 2009 The Arabian Journal for Science and Engineering, Volume 34, Number 1B 231

FINITE ELEMENT BASED VIBRATION FATIGUE

ANALYSIS FOR A NEW FREE PISTON ENGINECOMPONENT

M. M. Rahman*

Faculty of Mechanical Engineering

Universiti Malaysia Pahang, Malaysia

A. K. Ariffin, N. Jamaludin, and C. H. C. Haron

Department of Mechanical and Materials Engineering

Faculty of Engineering

Universiti Kebangsaan Malaysia

43600 UKM, Bangi, Selangor, Malaysia

1. INTRODUCTION

Structures and mechanical components are frequently subjected to the 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 most important issue in almost any random vibration problem [14]. Nearly all structures or components

have been designed using the time based structural and fatigue analysis methods. However, by developing afrequency based fatigue analysis approach, the accurate composition of the random stress or strain responses can be

retained within a greatly optimized fatigue design process.

Fatigue analysis is generally thought of as a time domain approach, that is, all of the operations are based on timedescriptions of the load function. This paper demonstrates that an alternative frequency domain [4,8,9] fatigue

approach can be more appropriate.

A vibration analysis is generally carried out to ensure that structural natural frequencies or resonant modes are

not excited by the frequencies present in the applied load. Sometimes this is not possible and designers then have to

estimate the maximum response at the resonance caused by the loading. These are the best performed in thefrequency domain using the power spectral density functions of input loading and stress response. 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 valuable 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 perform the frequency response analysis on the finite element model

(FEM) to determine the transfer function between the load and stress in the structure. This approach requires that thePSD of load is multiplied by the transfer function to obtaining the PSD of stress. The main purpose of this paper is

to predict fatigue life when the component is subjected to statistically-defined random stresses.

Key words: fatigue life, vibration fatigue, power spectral density functions, frequency response, cylinder block, free

piston

____________________

* Corresponding Author

Faculty of Mechanical Engineering

Universiti Malaysia Pahang

Locked Bag 12, 25000 Kuantan, Pahang, Malaysia

Phone: +(6)09-5492239, Fax: +(6)09-5492244E-mail: [email protected]

Paper Received: 23 Apr il 2005; Revised 15 May 2007; Accepted 30 May 2007


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The Arabian Journal for Science and Engineering, Volume 34, Number 1B April 2009232

2. THEORETICAL BASIS

The stress power spectra density [34, 912] represents the frequency domain approach input into the fatigue.

This is a scalar function that describes 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 timedomain and frequency domain representations [14] of a random process. Bendat [13] showed that the probability

density function (PDF) of peaks for the narrow band signal tended towards Rayleigh distributions as the bandwidth

reduced. Furthermore, for the 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 the stress cycles. Using this technique, the PDF of stress range would also tend to the Rayleigh distribution.

Bendat used a series of equations derived by Rice [15] to estimate the expected number of peaks using the moments

of area beneath the PSD. Bendats narrow band solution for the range mean histogram is therefore expressed in

Equation (1). Assume p(S) is the Rayleigh distribution which is represented by a narrow band process and stress

amplitude (S) can be treated as a continuous random variable:

=

==

dSem

SS

k

TPE

dSSpSk

S

SN

nDE

m

S

b

bt

i i

i

0

2

8

04

][

)()(

][

(1)

where N(Si) is the number of cycles of stress range (S) occurring in Tseconds, ni is the actual counted number of

cycle, St is the total number of cycles equals to {E[P]. T},E[P] is the number of peak per seconds, andmo is the zero

order moment of area of the PSD. Parameters kandb are the materials constants that would be defined in the SNcurve.

The Dirlik solution [16] is expressed by Equation (2) [1621].

)(][)( SpTPESN = (2)

whereN(S) is the number of stress cycles of range (S) N/mm2expected in Tseconds.E[P] is the expected number of

peaks andp(S) is the probability density function.

0

23

22

21

2)(

2

2

2

m

ZeDeR

ZDe

Q

D

Sp

Z

R

Z

Q

Z

++

= (3)

40

2213

211

22

2

1 ,1,1

1,

1

)(2

mm

mDDD

R

DDD

xD m ==

+=

+

=

(4)

04

2

0

1

211

21

1

23

2,,

1,

)(25.1

m

SZ

m

m

m

mx

DD

DxR

D

RDDQ m

m ==+

=

=

(5)

wherexm, D1, D2, D3, Q, andR depend on the m0, m1, m2, andm4; Zis a normalized variable. m0, m1, m2, andm4

are the zeroth, 1st, 2

nd, and 4

thorder spectral moments area, respectively.

3. LOADING INFORMATION

Several types of variable amplitude loading history were selected from the SAE and ASTM profiles for the FE

based fatigue analysis. It is important to emphasize that these sequences are not intended to represent standard

loading spectra in the same way as in Carlos or Falstaf [12]. However, they do contain many features which are

typical of the automotive industries applications, and therefore, are useful in the evaluation of the life estimation

methods. [22, 23]. The variable amplitude loadtime histories are shown in Figure 1 and the corresponding power

spectral density response are shown in Figure 2. 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 profile. In addition, I-N, A-A, A-G, R-C, and TRANSP represent 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 [22]. The

abscissa is the time in seconds.


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Figure 1. Different timeloading histories


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ProbabilityDensity

Probability Density

-2 0 2 4 6

SAE Standard Transmission (SAETRN) Loading

-6 -4 -2 0

SAE Standard Suspension (SAESUS) Loading

-6 -4 -2 0 2 4

SAE Standard Bracket (SAEBKT) Loading

0 2 4 6

ASTM Instrumentation &Navigation (I-N) Loading

0 2 4 6

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

-1 0 1 2 3 4 5

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

0 2 4 6

ASTM Composite Mission (R-C) Typical Fighter Loading

-4 -2 0 2 4 6

ASTM Composite Mission (TRANSP) typical Transport Loading

MPa

MPa

MPa

MPa

MPa

MPa

MPa

MPa

0.575

2.10E-3

1.014

2.220E-5

0.2704

0.245E-4

0.5440

5.475E-4

0.5005

0.4002

0.585

04.07

Probability Density

Probability Density

Probability Density

Probability Density

Probability Density

Probability Density

Figure 2. Power spectral densities (PSD) responses

4. NUMERICAL EXAMPLE

4.2. Finite Element Modeling (FEM)

The cylinder block is the one of the important and safety-critical components of the free piston engine. A

geometric model is considered as an example parts in this study. There are several contact areas including the

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.


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A three-dimensional model of the free piston linear engine cylinder block was developed using CATIA

software. A 10 nodes 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 35 415 elements and 66 209 nodes were generated

with 0.20 mm element length. A pressure of 7.0 MPa was applied to the surface of the combustion chamber,generating a compressive load. A pressure of 0.3 MPa was applied to the bolt-hole surface, generating a preload.

This preload is obtained according to the RB&W recommendations [24]. In addition, 0.3 MPa pressure was applied

on the gasket surface.

Multi-Point Constraints (MPCs) were applied on the bolt-hole surface for all six degree of freedom, and [25]

were used to connect the parts thru the interface nodes. These MPCs acted as an artificial bolt and nut that connect

each part of the structure. Each MPC will be connected using a Rigid Body Element (RBE) that indicates the

independent and dependent nodes. The configuration of the engine is constrained by bolting between the cylinder

head and cylinder block. In the condition with no loading configuration, RBE elements with six degrees of freedom

were assigned to the bolts and the hole on the cylinder block. The independent node was created on the cylinder head

hole. 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 also shown in Figure 4.

Figure 3. Three dimensional finite element model


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Figure 4. Loading, constraints, and boundary conditions

5. RESULTS AND DISCUSSION

5.1. Modal Analysis

Modal analysis is usually used to determine the natural frequencies and mode shapes of a component. It can alsobe used as the starting point for the frequency response analysis. The finite element analysis codes usually used

several mode extraction methods. The Lanczos mode extraction method is used in this study Lanczos is the

recommended method for the medium to large models. In addition to its reliability and efficiency, the Lanczos

method supports sparse matrix methods that significantly increase computational speed and reduce the storage space.

This method also computes precisely the eigenvalues and eigenvectors. The number of modes was extracted and

used to obtain the cylinder block stress histories, which is the most important factor in this analysis. Using this

method to obtain the first 10 modes of the cylinder block, which are presented in Table 1 and the shape of the mode

are shown in Figure 5. It can be seen that the working frequency (50Hz) is far away from the natural frequency

(186.32 Hz) of the first mode.

Table 1. The Results of the Modal Analysis

Mode No. Natural Frequency (Hz)

1 186.32

2 259.05

3 306.61

4 317.65

5 327.65

6 339.84

7 382.20

8 462.17

9 650.8710 721.28

Constraints

(Bolt)

Pressure

(7 MPa)

Constraints

(Bolt)

Preload on bolt

(0.3 MPa)

Gasket pressure

(0.3 MPa)


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Figure 5. The mode shapes of the cylinder block

186.32 Hz, Mode 1 259.05 Hz, Mode 2

306.61 Hz, Mode 3 317.65 Hz, Mode 4

327.65 Hz, Mode 5339.84 Hz, Mode 6

383.20 Hz, Mode 7462.17 Hz, Mode 8

65.87 Hz, Mode 9 721.28 Hz, Mode 10


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Frequency Response Stress Analysis

The frequency response analyses were performed using the finite element analysis code. The frequency response

analysis used the damping ratio of 5% of critical. The damping ratio is the ratio of the actual damping in the system

to the critical damping. Most of the experimental modal reported that 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 [28]. Zero damping ratio indicates thatthe mode is undamped. Damping ratio of one represents the critically damped mode. The results of the pseudo-static

and the frequency response finite element analysis with zero Hz, i.e. the maximum principal stresses distribution of

the cylinder block are presented in Figures 6 and 7 respectively. From the results, the maximum and minimum

principal stresses of 380.0 MPa and 77.5 MPa for the pseudo-static analysis, and 380.0 MPa and 78.3 MPa for the

frequency response analysis for zero Hz were obtained respectively. These two maximum principal stresses contour

plots are almost identical.

Figure 6. Maximum principal stresses distribution for the pseudo-static linear analysis

3.80+02

3.50+02

3.19+02

2.89+02

2.58+02

2.28+02

1.97+02

1.67+02

1.36+02

1.06+02

7.50+01

4.45+01

1.40+01

-1.65+01

-4.70+01

-7.75+01

Maximum principal stress 380 MPa at node 49360Maximum principal stress

(MPa)

Default_Fringe

Max 3.80+02@Nd49360

Min -7.75+01@Nd13559


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Figure 7. Maximum principal stresses distribution for the frequency response analysis at zero Hz

The variation of the maximum principal stresses with the frequency is shown in Figure 8. It can be seen that the

maximum principal stress varies with higher frequencies. This variation is due to the dynamic influences of the first

mode shape. It is also observed that the maximum principal stress occurs at a frequency of 32 Hz. The maximum

principal stresses of the cylinder block for 32 Hz is presented in Figure 9. From the results, the maximum and

minimum principal stresses of 561.0 MPa and 207.0 MPa were obtained at node 49 360 and 47 782, respectively.

Figure 8. Maximum principal stresses plotted against frequency

3.80+02

3.49+02

3.19+02

2.88+02

2.58+02

2.27+02

1.97+02

1.66+02

1.35+02

1.05+02

7.44+01

4.38+01

1.33+01

-1.73+01

-4.78+01

-7.83+01

Maximum principal stress 380.0 MPa at node 49360Maximum principal stress

(MPa)

Default_Fringe

Max 3.80+02@Nd49360

Min -7.83+01@Nd13559

561 MPa

32 Hz

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz)

625

545

465

385

305

225

145

65

MaximumPrincipalstress(MPa)


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Figure 9. Maximum principal stresses contour for frequency response analysis at 32 Hz

Figures 1013 show the applied time histories, PSDs of narrow band signal (SAESUS), corresponding

probability density function (PDF) and cycle histogram, respectively

Figure 10. Timeload histories

1.4107

0250

RMSPower(N2.HZ)

Frequency (Hz)

5 10 15 20

Figure 11. Corresponding (Figure 10) power spectral density

5.61+02

5.10+02

4.59+02

4.07+02

3.56+02

3.05+02

2.54+02

2.03+02

1.51+02

1.00+02

4.91+01

-2.07+01

-5.33+02

-1.04+02

-1.56+02

-2.07+02

Maximum principal stress 561 MPa at node 49360Maximum principal stress

(MPa)

Default_Fringe

Max 5.61+02@Nd49360

Min -2.07+02@Nd47782

2104

-2104

50000

Force(Newton)

Time (Seconds)

40001000 2000 3000


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Figure 12. Corresponding (Figure 10) probability density function

Figure 13. Corresponding (Figure 10) cycle histogram

5.3. Vibration Fatigue Life Prediction Analysis

The results of the fatigue life contour for the SAETRN loading histories at the most critical locations for the

pseudo-static analysis and frequency response approach with 32 Hz are shown in Figures 14 and 15 respectively.

The minimum life prediction for the pseudo-static analysis and frequency response approach with 32 Hz are obtained

as 107.67

and 109.44

seconds respectively. It would be expected that the condition of lower stress would correspond tolonger life and vice versa. However, the results indicates the opposite, because is the frequency resolution of the

transfer function is selected for the low value then the results show the non-conservative prediction. From Figures 14

and 15, it can be seen that the fatigue life contours are different and most damage was found at a frequency of 32 Hz.

0

3.1104 -1.5310

4

1.531040

458

RangeNewtonsX-Axis

MeanNewtonsY-Axis

Cycles

Z-Axis

1.210-4

01.5310

4-1.5310

4

Probabilitydensity

Force (Newtons)


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Log of Life(Seconds)

Predicted minimum life is

107.67 seconds at node 49360

Figure 14. Predicted fatigue life contours plotted for pseudo-static linear analysis

Log of Life

(Seconds)Predicted minimum life is

109.44 seconds at node 49360

Figure 15. Predicted fatigue life contours plotted for frequency response analysis at 32 Hz


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The full set of the comparison results for the untreated polished cylinder block at critical location (node 49 360)

is given in Table 2 with various loading conditions. The narrow band solution is considered in this study. It isobserved that from Table 2, the SAESUS loading condition gives the longest life for all materials due to the loadhistories are predominantly compressive mean. The compressive mean loading beneficial for fatigue. In addition, theAG loading condition gives the lowest lives for all materials.

Table 2. Predicted Life in Seconds at Critical Location (Node 49 360) for Various Loading Conditions and

Materials

Predicted vibration fatigue life in seconds

2024-HV-T6 6061-T6-80-HFLoading

conditions No (zero)

meanGoodman Gerber

No (zero)

meanGoodman Gerber

SAETRN 2.75109 2.53109 2.74109 2.10107 2.02107 2.09107

SAESUS 1.531012 1.461012 1.521012 8.74109 7.97109 8.62109

SAEBKT 3.361010 3.151010 3.351010 1.06108 8.86107 1.03108

IN 1.471010 1.391010 1.461010 2.30108 2.21108 2.28108

AA 3.60109 3.37109 3.59109 3.93107 3.79107 3.89107

AG 9.14108 8.47108 9.13108 8.23106 8.17106 8.20106

RC 1.96109 1.82109 1.95109 1.83107 1.75107 1.81107

TRANSP 1.511011 1.441011 1.501011 2.27109 2.12109 2.24109

5.4. Effect of the Frequency Resolution

Frequency resolution of the transfer function is significant to capture the input PSD. The significances of thefrequency resolutions of SAETRN loading histories are also shown in Figures 16 and 17. Two types of Fast Fourier

Transform (FFT) buffer size width namely 8192:0.06104 Hz and 16384:0.03052 Hz were used in these figures. TheFFT buffer size defines the resolution of the power spectrum. The buffer must be a power of 2 and of course the

longer the buffer, the higher the resolution of the spectral lines. To calculate the resolution divide the Nyquistfrequency by half the FFT buffer size. If the Nyquist frequency is 250 Hz and the FFT buffer size selected is 8192,then the spectral lines are 250/(8192/2) = 0.06104 Hz apart. Another use of a smaller buffer size is for short data filesas these cannot be adequately analyzed with a big buffer, since there may not be enough data to give a good spectral

average. Using a smaller buffer size could give a better spectral average at the expense of spectral line width. Thetotal area under each input PSD curve is determined to be identical. However, the 16384:0.03052 Hz width has twiceas many points compares to the 8192:0.06104 Hz.

Figure 16. Power spectral density at FFT buffer size of 8192:0.06104 Hz width

10

0100

RMSPower(MPa2.H

z-1)

Frequency (Hz)


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Figure 17. Power spectral density at FFT buffer size of 16384:0.03052 Hz width

The frequency resolution of the transfer function in the important areas of the input PSD is the dominant factor.

This is shown in Figure 18 for two different cases. Figure 18 shows that utilizing many points in the input PSD canidentify the damage more accurately. However, the approach is not suitable for accurately identifying damage with alarge spike occurred between two frequencies in the transfer function. For the worst case scenario the technique canentirely miss the damage.

Figure 18. Effect of the frequency resolution

6. 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 of the vibration fatigue techniques has been presented. Frequencyresponse fatigue analysis has been applied to a typical cylinder block for a free-piston engine. From the results, it canbe concluded that the Goodman mean stress correction method gives the most conservative prediction for all loadingconditions and materials. The results clearly indicate that AA2024-HV-T6 is a superior material for all the meanstress correction methods. The life predicted from the vibration fatigue analysis is consistently higher except for thebracket loading condition. In addition, vibration fatigue analysis can improve the understanding of 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 the Malaysian Government, especially theMinistry of Science, Technology and Environment under IRPA project (IRPA project no: 03-02-02-0056PR0025/04-03) for providing financial support.

Potentially less damaging More damaging

Input PSD

Transfer

function

Response

PSD

10

0100

RMSPower(MPa2.Hz-1)

Frequency (Hz)


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REFERENCES

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