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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
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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.
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{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
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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.
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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
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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.
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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
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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)
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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.
REFERENCES
1. J.S. Bendat and A.G. Piersol, Random Data: Analysis and
Measurement Procedures, John Wiley, New York, 1971.
2. P. Wirsching, T. Paez and K. Oritz, Random Vibration, Theory
and Practice, John Wiley and Sons, Inc., 1995.
3. D.E. Newland, An Introduction to Random Vibrations and
Spectral Analysis, 2nd Ed., Longman Inc., New York, 1984.
4. J.D. Robson and C.J. Dodds, "Stochastic Road Inputs and
Vehicle Response," Vehicle System Dynamics, No.5, pp1-13, 1975.
5. Meirovitch L., "Analytical Methods in Vibrations," The
Macmillan Co., NY, New York, 1967.
6. Thomson, W.T., "Theory of Vibration with Applications," 3rd
Edition, Prentice Hall, Englewood Cliffs, New Jersey, 1988.
7. Stephens, R.I., Fatemi, A., Stephens, R.R. and Fuchs, H.O.,
"Metal Fatigue in Engineering," John Wiley & Sons, New York,
NY, 2000.
8. Miner, M.A., "Estimating Fatigue Life with Particular
Emphasis on Cumulative Damage," from Metal Fatigue, ed. G. Sines
and J.L. Waisman, McGraw-Hill, 1959.
9. SAE J1099 Committee, Technical Report on Low Cycle Fatigue
Properties, SAE J1099, Society of Automotive Engineers, Warrendale,
PA, August 2002.
10. SAE J1211 Committee, Recommended Environmental Practices for
Electronic Equipment
SAE Int. J. Passeng. Cars - Mech. Syst. | Volume 1 | Issue 1
173
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-
Design, SAE J1211, Society of Automotive Engineers, Warrendale,
PA, November 1978.
11. Su, H. "Vibration Test Specification for Automotive Products
Based on Measured Vehicle Load Data," SAE 2006-01-0729, SAE
Transactions, Vol.115, Journal of Materials & Manufacturing,
pp.571-581, 2006.
12. Su, H., "A Comparison of Vibration Profiles and
Specifications for Automotive Components for USCAR/EWCAP," Visteon
Internal Tech Report, A935_C-99-006, 1999.
13. Su, H., Rakheja, S. and Sankar, T.S., Stochastic Analysis of
Nonlinear Vehicle Systems Using a Generalized Discrete Harmonic
Linearization Technique, Journal of Probabilistic Engineering
Mechanics, Vol.6, No.4, 1991, pp175-183.
14. Su, H., D. Steinert, K. Egle and B. Weipert, "Localized
Non-linear Model of Plastic Air Induction Systems for Virtual
Design Validation Tests," SAE paper 2005-01-1516, SP-1960,
pp.15-24, April 11, 2005.
15. Su, H. "Automotive CAE Durability Analysis Using Random
Vibration Approach," MSC 2nd Worldwide Automotive Conference,
Dearborn, MI, Oct. 2000.
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
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