CAE Virtual Durability Tests of Automotive Products in the Frequency Domain

ABSTRACT

Both NVH and durability performance of automotive products are mainly related to their structural frequency characteristics, such as their resonant frequencies, normal modes, stiffness and damping, and transfer function properties. During the automotive product development, product design validation test loads for NVH and durability are, therefore, often specified in the frequency domain, in terms of either swept sinusoidal vibration or random vibration in power spectral density function. This paper presents a procedure of CAE virtual design validation tests for durability evaluation due to the frequency domain vibration test loads. A set of frequency domain simulation techniques and durability evaluation methodologies, for material fatigue damage due to either random or sinusoidal vibration loads, are introduced as well. Finite element models of automotive products are developed along with their nonlinear frequency dynamic stiffness and damping elements and properties, such as those related to the mounts and rubber bushings. The dynamic stress simulation is realized by utilizing the frequency response analysis technique. Statistical properties are employed to account for the scatter nature of material fatigue S-N raw data, and a damage model for durability performance is then established by using the reliability and tolerance interval techniques. The durability life evaluation is based on the simulated dynamic stresses and the newly defined material fatigue damage model. Two examples of virtual durability evaluation tests of automotive products are also provided to illustrate applications of the proposed procedure and techniques, with respect to the random and swept sine vibration loads, respectively.

INTRODUCTION

It has been well established [1, 2] that the noise, vibration and harshness (NVH), and dynamic durability performance of a product are essentially related to its frequency characteristics, such as their resonant frequencies, normal modes, stiffness and damping, and transfer function properties between its inputs and outputs. Product dynamic loads and engineering response data, such as force, acceleration, stress and

strain commonly measured in terms of time histories, can be conveniently converted into the frequency domain for revealing their respective frequency spectrum, insight information and inherent properties [3], by using the fast Fourier transform (FFT) technique. For durability design validations of automotive products the vibration test loads are, therefore, often specified in the frequency domain, in terms of either swept sinusoidal vibration function or random vibration function in terms of power spectral density (PSD) [4,10,11].

In this paper, a procedure of virtual design validation tests for durability evaluation due to frequency domain vibration loads, by employing a set of computer aided engineering (CAE) simulation and durability techniques, is presented. The proposed virtual durability test procedure incorporates several frequency domain modeling and simulation technologies. The frequency domain test loads under the vehicle test environments, due to either the road load interface loads with the vehicle or the engine vibration loads, are simulated in the random or swept sine vibration format. The finite element models of automotive products are developed along with their nonlinear frequency dynamic stiffness and damping properties, as related to mounts and rubber bushings in vehicle structures. Dynamic stress response results are then simulated by using the frequency response analysis technique. The statistic properties are employed to account for the scatter nature of the material S-N fatigue raw data, and a damage model for durability evaluation is then established using the reliability and tolerance interval techniques. The durability evaluation is based on the dynamic stress results and the newly defined material damage model. The predicted durability life results of an automotive product under the frequency vibration test are then obtained for the given reliability parameters.

Two examples of durability evaluation of automotive products are provided to illustrate the applications of the proposed procedure and techniques. One example is on an axle structure system under a random vibration load based on the measured proving ground data. While the other is an engine suspension system under a swept sine vibration load, which was originated from dynamic

2008-01-0240

CAE Virtual Durability Tests of Automotive Products in the Frequency Domain

Hong Su Summitech Engineering, Inc.

Copyright 2008 SAE International

SAE Int. J. Passeng. Cars - Mech. Syst. | Volume 1 | Issue 1 165

Downloaded from SAE International by General Motors LLC, Tuesday, June 23, 2015

running conditions of an I4 gas engine. The virtual durability test results in the frequency domain are presented in terms of the estimated life for the given reliability target and a confidence level. It has shown that the virtual test approach can reveal the insight relationship into the design parameters, product design weak spots and durability life, provide a guidance to design improvement, and help to achieve our goal for only one successful physical design validation test.

VEHICLE LOADS IN THE FREQUENCY DOMAIN

There are generally two types of measured vehicle load raw data [10] employed in automotive industry for product development according to their original sources, namely, the proving ground load data for vehicle body mounted products, and the engine load data for engine mounted products. It has been well known that both the time and frequency domain data are mathematically related by using the fast Fourier transform technique [3, 4]. The commonly measured vehicle load data in terms of time histories are conveniently converted into the frequency domain for their respective frequency spectrum and then frequency load specifications.

In order to completely define the properties of a load in terms of the total damage [11], both the measured load data and the load schedule information are needed. The measured load data quantifies the load level and its frequency contents for each load event; while the load schedule information tells how the loads are applied to the product, such as duration and conditions. Therefore, the complete proving ground data and information include the measured vibration loads from vehicle due to road surface profile interfaces, and the corresponding schedule of road surface events, with defined vehicle speed, duration and road test schedule of a given type of vehicle. The engine load data, on the other hand, are measured vibrations mainly due to unbalanced inertial forces of the centrifugal and reciprocating components, corresponding to different engine speed RPM, in terms of harmonic contents of vibrations and a schedule with the engine duty cycle definition.

An engineering procedure along with its related methods for determination of a justified vibration test load specification in the frequency domain for automotive products, based on the measured vehicle load data, has been introduced in [11]. The resulted vibration test load specification for an automotive product can be presented in either swept sinusoidal or random vibration profile format, and both will have an equivalent durability damage level for the given test duration and reliability parameters.

BASIC EQUATIONS OF A STRUCTURE MODEL

In the automotive industry, vehicle structural systems commonly employ many plastic parts and rubber bushing components. A vehicle structural system, in general, demonstrates strong nonlinear response characteristics, with respect to the dynamic load level,

excitation frequency and temperature conditions, due to those nonlinear components. In order to model and simulate such a nonlinear structural system for virtual design validation (DV) tests of the automotive products, the modeling technique by using an array of locally linearized systems [13] is introduced. A brief development of the basic equations of the locally linearized model of a nonlinear automotive system is summarized as follows.

STRUCTURE MODEL IN TIME DOMAIN

For a general nonlinear dynamic structural system of an automotive product, such as a vehicle axle structure or engine suspension system with rubber mounts and bushings, which is under the dynamic proving ground or engine vibration loads and different temperature conditions, the mathematical model can be established as a system of nonlinear differential equations in the time domain [5, 6].

)(tP

txT,xx,KtxT,xx,CtxM

s 1

where [M] is the mass matrix of the dynamic system; {x(t)} is the generalized coordinate vector, as a function of time, t. [C] is the damping matrix, as a function of response {x}, time t and temperature, T. [K] is the stiffness matrix, as a function of response {x}, time t and temperature, T. {Ps(t)} is the dynamic force vector due to the vehicle loads.

The solution {x(t)} to the above system of equations is generally obtained by using the finite element integration simulation technique, corresponding to each time step. For the system due to dynamic vibration loads in the frequency domain, the steady state solution to the system of equations (1) is extremely difficult, if is not impossible, in terms of CPU time and resources.

FREQUENCY DOMAIN MODEL

An alternative approach to the above problem is to solve the system of equations in the frequency domain, in terms of transfer functions, using the Fourier transformation technique. A localized nonlinear model of a dynamic system, corresponding to the equation (1), in the frequency domain can be expressed as follows [14]:

)(PX

P,T,X,KP,T,X,CiM

s

jleqjleq

2

2

where =2f, f is the frequency (Hz), 1i , MjLl ...,,2,1,...,,2,1 , L is the total number of temperature cases, M is the total number of load level cases; [Ceq] is the equivalent local damping matrix, and [Keq] is the equivalent local stiffness matrix, as a function of response, frequency, temperature, and load level.

SAE Int. J. Passeng. Cars - Mech. Syst. | Volume 1 | Issue 1166


{X()} and {Ps()} are the structure response and load vectors in frequency domain, respectively.

RANDOM VIBRATION MODEL

When the input loads are in random vibration format and expressed as a matrix of the loading power spectral density (PSD) functions, [Sp()], the system of time domain differential equations of motion of the structure in (1) is then reduced to a system of algebra equations as follows.

3T nmmmpmnnnx HSHS

where m is the number of multiple input loads; 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.

412 eqeq KCiMH

The random response variables [Sx()], such as displacement, acceleration and stress response, in terms of power spectral density functions, are obtained by solving the system of the linear algebra equations in (3).

By employing equations (2) and (3), the solution to nonlinear differential equation (1) in the time domain has been translated to a solution to the system of localized linear algebra equations, in the frequency domain, for the given loads and conditions, in terms of sine sweep or random vibration format, respectively.

MATERIAL FATIGUE DAMAGE MODEL

MATERIAL FATIGUE DATA

The material fatigue properties are usually measured as S-N curve, which defines the relationship between the stress amplitude level, SA, versus the mean cycles to failure, N. For most high cycle fatigue durability problems (N 104), the S-N curve can be expressed as a simplified form:

5mASNB

where B and m are the material properties varying with material type, and its loading and environment conditions, such as mean stress, surface finishing, and temperature.

RELIABILITY PARAMETERS

It is well known [7] that material fatigue test data of sample pairs (Si, Ni) are randomly scattered on their stress (S) and number of cycles (N) plots. A mean S-N

curve can be estimated by using the least square analysis of all test data sample pairs of (Si, Ni) on log-log scales, based on a data sample size of W, such as illustrated in Figure 1. The material fatigue model in equation (5) is therefore only a medium S-N curve, estimated from the limited sample size (W). It is also known that the fatigue life log(N) of a set of S-N fatigue test data has a random distribution, corresponding to a given fatigue strength level log(S).

If we assume that log(N) is normally distributed for given log(S), and its variance is a constant, the uncertainties in S-N estimators can be then accounted for by using the tolerance interval technique. That is, the new design S-N curve can be shifted to the left (safe side of data) by amount of margin, which is determined from reliability requirements and statistical properties of the fatigue sample data. The new design S-N curve for the durability evaluation and the test load equivalency can then be defined as [16]:

60 s,p DWkBlogBlog

where B0 is the new fatigue strength coefficient, replacing B in equation (5), Ds is the standard deviation of the test sample data in log(N), and kp,(W) is the factor for a one-sided tolerance interval, corresponding to a proportion p of the population (reliability), confidence and sample size W. For a given set of reliability parameters, that is, the sample size W, confidence level and reliability goal p, the factor kp,(w) can be found in the table of factors for one-sided tolerance limits for normal distributions, such as in [18].

Fatigue S-N Data

1.E+01

1.E+02

1.E+03

1.E+04

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07log (N)

log

(S

)

(Ni,Si)

Distribution

New S-N

Figure 1: A reliability model for S-N fatigue data

MATERIAL FATIGUE IN FREQUENCY DOMAIN

RANDOM VIBRATION FATIGUE

Under random vibration loads, the fatigue damage of structures is estimated based on the statistical properties of the response stress PSD function. The statistical characteristics of the random vibration response stress can be obtained through the moments of the PSD function. The nth spectral moment, mn of the



stress PSD function S(f), frequency f in unit of Hz, is defined by the following equation [15]:

0

7dffSfm nn

Properties of a continuous stationary Gaussian process can be related to the above nth moments, mn of the PSD function. The root mean square (RMS) value, of PSD function is

82

1

21

00

dffSm

The average rate of the zero crossing with positive slope, E[0], which is also called as equivalent frequency in unit time, is expressed as:

902

1

0

2

mmE

The expected value of the accumulated damage, E[AD], due to fatigue random loading is evaluated based on the Palmgren-Miners rule [8], and expressed as:

1000 AA

AA

A

A dSSN

NSpdSSNSnADE

where n(SA) is the number of cycles applied at stress amplitude level SA; (SA) is the probability density function of the stress amplitude. Substituting equations (5) and (6) into (10), by integrating the equation, a general equation of fatigue damage from random stress response is obtained [2].

1112

20

0

mBETm,ADE

m !"

where T is time duration of random loading, (.) is the Gamma function, "(!, m) is the empirical rainflow correction factor, which distinguishes the effect of the bandwidth and shape of different PSD profiles.

12111 2 mbmamam, !!" mma 033.0926.0

323.2587.1 mmb

And ! is irregularity factor of PSD function defined as.

13402 mmm!

SWEPT SINE VIBRATION FATIGUE

Under swept sine vibration loads, the fatigue damage of structures is estimated based on the stress response level profile swept at all frequencies within the frequency range of the lower and upper ends. In swept sinusoidal vibration tests of automotive products, logarithmic sweep at a constant rate is commonly employed (Octave/min). That means that every swept octave will contain the same test loading duration, which is similar to the pink noise test.

An octave is the interval between one frequency and another with half or double its value. For example, for a sine frequency of 10 Hz, the frequency of an octave above it is at 20 Hz, and the frequency of an octave below is at 5 Hz. The ratio of frequencies of an octave apart is 2:1. The frequency range between two limit frequencies f1 and f2, in terms of octaves XO, is obtained as:

)(f

floglg

flgflgXO 142 12

212

The logarithmic swept frequency f with respect to the sweep time t, within the test frequency range between the lower limit fl and the upper limit fu, is expressed as

)(CtRflog 152 where R is the constant octave sweep rate per time of the swept sine test, and C is a constant for the given sweep speed and frequency range.

The accumulated damage, ADs, due to the swept sine fatigue loading is also evaluated based on the Palmgren-Miners rule, and expressed as follows:

)(dfflnBR

NSdSNSnAD u

l

f

f

mwA

A

As 162

2

00

where Nw is the total number of sinusoidal sweeps applied during the whole test time duration, (f) is stress amplitude as function of frequency f in Hz. In CAE virtual durability tests, the stress function (f) is usually obtained from the simulation of the automotive products, using the frequency response analysis technique.

EXAMPLES OF VIRTUAL DURABILITY TESTS

In the following sections, two examples of automotive products are provided as to illustrate the procedure of the virtual durability tests in the frequency domain. The applications of the related techniques outlined in the previous sections are also demonstrated as well. The first example is on an axle structure system under the random vibration load based on the proving ground data. The second example is for an engine suspension system under the swept sine vibration load originated from the engine dynamic work conditions.



AN AXLE STRUCTURE UNDER RANDOM VIBRATION LOAD

RANDOM VIBRATION TEST LOAD SPECIFICATION

The random vibration test loads to the axle structure are specified from the measured vehicle proving ground load data, by employing the engineering procedure for the vibration test load specification outlined in [11]. As a brief example, one channel data corresponding to the rear left upper control arm (LR UCA) load is illustrated here. The durability schedule of the proving ground for light trucks includes five major durability road surface events as indicated in Table 1. Three of the typical measured proving ground road load data for trucks, namely, the Hard Route, Silver Creek and Power Hop Hill, are shown in Figures 2 to 4, respectively.

Table 1: Schedule of a Light Truck Proving Ground Test

Duration

(%)1 Hard Route Clockwise - HRPH 3.88 65.96 50.32 Hard Route Clockwise - HRCB 2.06 35.02 26.73 Silver Creek Clockwise - SC20 1.12 19.04 14.54 Silver Creek Clockwise - SC40 0.52 8.84 6.75 Power Hop Hill 0.14 2.38 1.8

Total Time

(Hours)No. PG Road Surface

Time per

Pass

Figure 2: A time load of UCA bracket (Ch# 36, HR)

Figure 3: A time load of UCA bracket (Ch# 36, SCR)

Figure 4: A time load of UCA bracket (Ch# 36, PHH)

The resulted random vibration test load for the light truck axle structure is specified by employing the FFT and fatigue damage equivalency technologies. The key parameters in the time to frequency domain data conversion include those for the selection of data average methods, frequency resolution, frequency range, window function, and data buffer overlap. Figure 5 presents a random vibration test load specification, in terms of power spectral density (PSD) function, which is derived from the five (5) measured channel data and the proving ground durability test schedule. From the random vibration load PSD specification, it is easy to know that the vibration load energy of the proving ground roads to the axle structure is mainly distributed below the frequency of 1 Hz.

Load Force PSD (LR UCA (-Y))

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

0.01 0.10 1.00 10.00 100.00Frequency (Hz)

Lo

ad P

SD

(N

^2/H

z)

Figure 5: Random vibration load for UCA bracket

LIGHT TRUCK AXLE STRUCTURE MODEL

The finite element model of the light truck axle structure is shown in Figure 6. The major components of axle structure consist of the axle shaft, differential carrier, pinion and ring gear set, differential case, rear cover, tube assemblies, spring seats, various mounting brackets and rubber bushings. There are total 12 major loads, including the driveline torque and the suspension components reaction forces, which are applied upon the axle structure.

Figure 6: An axle structure FEA model

DYNAMIC STRESS SIMULATION



The frequency response analysis technique is employed to compute the dynamic stress response of the axle structure under the random vibration loads. The rubber bushing components are modeled as CBUSH elements with their respective stiffness and damping properties by employing the commercial FEA software package [19]. The additional structural damping is specified as modal damping () in the finite element model, typically of 2 to 5%. The random vibration loads are defined as a matrix of the loading power spectral density functions, [Sp()] as shown in equation (3). Each random load is specified by an auto-correlation PSD function table. The relationships among the loads are specified by the cross-correlation PSD function tables, respectively.

The high stress areas of the axle structure are identified using either directly sorting of the simulated element stress results or indirectly sorting of the modal strain energy density information approach. The later usually can save a lot of simulation efforts in terms of compute time and storage space. An example of the stress distribution profile of the axle brackets is shown in Figure 7. The high stress of the control arm bracket is located around pin support area of the bracket. The corresponding random vibration stress response in terms of PSD profile is illustrated in Figure 8.

Figure 7: A stress distribution of axle brackets

Element 53691 (Bracket, LR UCA (-Y))

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0.01 0.10 1.00 10.00 100.00Frequency (Hz)

Str

ess

PS

D (

MP

a^2/

Hz)

Figure 8: Dynamic stress PSD of UCA bracket

DURABILITY EVALUATION

The durability evaluation of the axle structure is based on the simulated random vibration response stress in PSD and the fatigue damage model. The fatigue damage model is derived from the original material fatigue S-N curve data, their statistic properties and reliability requirement. The original material S-N curve data of the steel bracket is illustrated in Figure 9. The parameters of the reliability requirement for the evaluation are selected as follows: Reliability target, R=95%, confidence level, CL=90%, and material fatigue S-N test data sample size, W=72.

Figure 9: Material fatigue S-N of an axle bracket

Based on the random vibration dynamic response stress PSD function of the rear axle left upper control arm bracket, for example, the root mean square (RMS) value of the random vibration stress PSD shown in Figure 8 is computed as 55.08 MPa. The equivalent frequency is 15.5 Hz, and the irregularity factor ! of the stress PSD function is 0.412. For the bracket made of the steel with fatigue properties shown in Figure 9, and based on the given reliability requirement, the estimated durability life of the UCA bracket is then computed as 2.3 times of the design life of the axle structure.

AN ENGINE SUSPENION UNDER SWEPT SINE VIBRATION LOAD

SWEPT SINE VIBRATION LOAD SPECIFICATION

The swept sine vibration test load to the engine suspension system is derived from the measured engine vibration data, by employing the engineering procedure for the vibration load specification in [11]. Typical measured engine loads are expressed in terms of harmonic contents of vibration accelerations with respect to the engine rotation speed (RPM). A measured I4 engine vibration response data, with the first several major harmonic orders, from the top of the engine is presented in Figure 10. The engine duty cycle schedule is used to define how an engine speed will be distributed during the life of the vehicle, based on the statistic database. An example of the engine duty cycle definition schedule is illustrated in Table 2.



Figure 10: A measured load contents of an I4 engine

Table 2: An I4 Engine Duty Cycle Definition

No.Duration Portion

1 Idle to 25% of redline 28.90%2 25% to 50% of redline 63.70%3 50% to 75% of redline 6.90%4 75% of redline to redline 0.40%

RPM Load Range

The resulted specification on the swept sine vibration test load for the engine suspension system is derived by employing the FFT and fatigue damage equivalency technologies. In this example, the test time duration is selected as 100 hours with frequency range from 20 to 1000 Hz. The total damage due to swept sine vibration will be equivalent to that of the engine with a design life of 5500 hours and correlate the engine duty cycle definition in Table 2.

The corresponding swept sinusoidal vibration test load specification is summarized in Table 3 and shown in Figure 11 as well.

Table 3: A Swept Sine Vibration Specification

Figure 11: A swept sine vibration engine load

From the sine vibration test load specification, it is easy to see that the vibration load energy of an I4 engine is mainly distributed below the frequency of 200 Hz.

ENGINE SUSPENSION SYSTEM MODEL

The finite element model of the engine suspension system is shown in Figure 12. The engine suspension system consists of a spring plate beam, four (4) mounting brackets and bushings. The mass inertial properties of both engine and transmission are also included in the FE model. Some mount bushings are made of rubber and other of fluid device. The major functions of the suspension system are two folds: (1) to support the engine and transmission weight and dynamic loads, and (2) to isolate engine vibration load from transmitting to the body structure.

Figure 12: An engine suspension FEA model

In order to enhance the vibration isolation performance, various mount bushings are employed in the engine suspension system. Typical rubber bushings with nonlinear stiffness and damping properties are specified for the suspension design in the frequency domain. Examples of the properties with the rear mount bushing used are illustrated in Figures 13 and 14 respectively. It can be seen that the damping value of a rubber bushing

Frequency Acceleration(Hz) (G)

20 1.165 9.56

200 9.56285 1.321000 1.32

Type of sweep = LogTime per sweep (min) = 20

Sweep speed (Oct/min) = 0.565Number of sweeps = 300

Test Duration (hours) = 100

Sine Vibration Load (Engine Mount, Z)

0

2

4

6

8

10

10 100 1000

Frequency (Hz)

Acc

e (g

)

Me asured Engine Vib Data (Z , M ount1)

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000 7000

Engine spe ed (RPM)

Acc

e (m

/s^2

)

OA

2nd

4th

6th

8th



is decreased with an increase of the vibration frequency. While the stiffness value of a bushing will however be increased with the increased frequency.

ES01 Rear-Bushing Dynamic Damping

0.00

0.05

0.10

0.15

0.20

0 200 400 600 800 1000Frequency (Hz)

Dam

pin

g (

N.s

/mm

)

R-Direction

P-Direction

Q-Direction

Figure 13: Damping properties of a bushing

ES01 Rear-Bushing Dynamic Stiffness

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000Frequency (Hz)

Sti

ffn

ess

(N/m

m)

R-Direction

P-Direction

Q-Direction

Figure 14: Stiffness properties of a bushing

DYNAMIC STRESS SIMULATION

The frequency response analysis technique is employed to simulate the dynamic stress response of the suspension structure under the swept sine vibration load, with the engine and transmission masses included in the FE model. All the mount bushings are modeled as CBUSH elements with their stiffness and damping properties as functions of the frequency, and defined in PBUSH tables [19], respectively. The structural damping is also specified as modal damping () in the finite element model.

The high stress areas of the suspension structure are identified using either directly sorting of the simulated element stress results or indirectly sorting of the modal strain energy density information approach. An example of the modal strain energy density distribution of the right mount bracket is shown in Figure 15. One of the high stress areas is located around the foot of the right mount bracket structure. The corresponding dynamic stress profile of the sine vibration response is illustrated in Figure 16.

Figure 15: A computed SED distribution of a bracket

Figure 16: Dynamic sine stress of a bracket

Figure 17: Material fatigue S-N of an engine bracket

DURABILITY EVALUATION

The durability evaluation of the bracket structure is based on the simulated swept sine vibration response stress and the fatigue damage model. The fatigue damage model is derived from the original material fatigue S-N curve data, their statistic properties and reliability requirement. The original material S-N curve data of the steel bracket is illustrated in Figure 17. The parameters of the reliability requirement for the evaluation are selected as follows: Reliability target,

Swept Sine Stress Response (E3017)

0

50

100

150

200

250

1 10 100 1000Frequency (Hz)

Str

ess

(MP

a)



R=95%, confidence level, CL=90%, and material fatigue S-N test data sample size, W=156.

For durability analysis of the right mount bracket, the frequency range of the fatigue damage evaluation is from 20 to 300 Hz. The total test duration is 100 hours. The maximum stress of the bracket within the swept frequency range is 159 MPa at 27 Hz, as shown in Figure 16. For the bracket made of the steel with fatigue properties shown in Figure 17, and based on the given reliability requirement, the estimated durability life of the bracket is then computed as 3.6 times of the design life of the mount bracket structure.

CONCLUSIONS AND DISCUSSIONS

An engineering procedure of CAE virtual design validation tests of automotive products for durability evaluation due to frequency domain vibration loads is presented in this paper. The basic theoretical backgrounds and a set of key technologies for frequency domain simulation techniques and durability evaluation methods, such as those for material fatigue damage due to either random or swept sinusoidal vibration loads, are introduced as well.

The finite element models of automotive products in the frequency domain are developed along with their nonlinear frequency dynamic stiffness and damping properties, such as the automotive components related to mounts and rubber bushings. The dynamic stress simulation, either in terms of random vibration or swept sine vibration, is realized by using the frequency response analysis technique. Statistical properties are employed to account for the scatter nature of material fatigue S-N raw data. And a fatigue damage model for durability evaluation is established using the reliability and tolerance interval techniques. The durability analysis is based on the simulated dynamic stress results and the newly defined material fatigue damage model. The durability life prediction of an automotive product under the frequency test load is determined with respect to the given reliability parameters.

Two examples of automotive products are provided as to illustrate the procedure of the virtual durability tests in frequency domain and the applications of related techniques. One example is on an axle structure system under the random vibration load, based on the proving ground data. While the other example is an engine suspension system under the swept sine vibration load, which is originated from the engine dynamic running conditions. Both examples demonstrate the whole process of the procedure applications, from the vibration load specification, to finite element modeling with nonlinear stiffness and damping characteristics, to frequency response analysis for dynamic stress results, to the material fatigue damage model with reliability requirement, and to the durability life evaluation of the product.

The results of the demonstration examples have shown that the CAE virtual test approach can, at an early stage of product development phase, identify weak spots and potential durability life issues of the product, reveal the insight relationship into the design parameters, and provide a guidance to design improvement. And that will help to achieve our goal for only one successful physical design validation test as well.

ACKNOWLEDGMENTS

The author would like to acknowledge the support and activities from various organizations, institutes and companies, related to the work on the vibration test specification and durability evaluation in the frequency domain for automotive products, especially the SAE Automotive Electronic System Reliability Standard Committee, the ISO/TC22/SC3/WG13 16750 Standard Committee for Environmental Conditions and Testing, the EWCAP working group of USCAR, Ford Motor Company, Visteon Corporation and Summitech Engineering, Inc. The author also wants to thank the support and assistance on many automotive design validation projects from the CAE and testing teams, especially Mr. Yuan Hua for his work on an engine suspension project, and the encouragement and help from our management and colleagues.

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16. Su, H., M. Ma, and D. Olson, "Accelerated Tests of Wiper Motor Retainers Using CAE Durability and Reliability Techniques," SAE paper 2004-01-1644, SP-1879, pp103-109, March 2004.

17. Su, H., J. Kempf, B. Montgomery and R. Grimes, "CAE Virtual Tests of Air Intake Manifolds Using Coupled Vibration and Pressure Pulsation Loads," SAE paper 2005-01-1071, SAE Transactions, Vol.114, Journal of Engines, pp.935-961, 2005.

18. Natrella, M.G., "Experimental Statistics," National Bureau of Standards Handbook 91, August, 1, 1963.

19. "MSC.Nastran Reference Manual, Sections for Coupled Acoustic Analysis," MSC Software publication, 2002.

CONTACT

Dr. Hong Su Summitech Engineering, Inc. Phone: (734)448-2312 E-mail: [email protected] NOMENCLATURE

B material fatigue property B0 new fatigue strength coefficient C constant, or damping coefficient [C] damping matrix [Ceq] equivalent local damping matrix CAE computer aided engineering CPU computer process unit

DV design validation Ds standard deviation of test sample data E[0] expected rate of zero crossing with positive slope equivalent frequency in unit time E[AD] expected value of accumulated damage f frequency (Hz) fl frequency range lower limit fu frequency range upper limit FEA finite element analysis FEM finite element model, or method [H()] transfer function i unit imaginary number, 1i [K] stiffness matrix [Keq] equivalent local stiffness matrix kp, factor for a one-sided tolerance interval L total number of temperature cases [L] geometry coupling matrix m material fatigue property M total number of load level cases [M] mass matrix mn nth spectral moment of a PSD function n total number of responses n(SA) number of cycles at level SA N number of fatigue life cycles Nw total number of sinusoidal sweeps NVH noise, vibration and harshness p propobility with respect to test population, reliability goal P() applied force in frequency domain {Pp()} coupling load vector in frequency domain {Ps(t)} force vector due to vehicle structure load {Ps()} structural load vector in frequency domain PSD power spectral density R constant octave sweep rate RMS root mean square value SAE Society of Automotive Engineers, Inc. SA stress amplitude level [Sp()] matrix of loading PSD functions [SX()] matrix of response PSD functions t time T temperature {x} response vector {x(t)} generalized coordinate vector {X()} structure response vector in frequency domain W sample size ! irregularity factor stress, or root mean square value " empirical rainflow correction factor

angular frequency (radian/second) confidence level (.) Gamma function modal damping



CAE Virtual Durability Tests of Automotive Products in the Frequency Domain

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