Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle
This study aims to establish a practical method for simulating the spindle coupled multi-axis loading fatigue test of a rear axle. A dynamic finite-element model of the rear axle was constructed and validated using a static calibration test. Based on the theory and methodology of the Schenck ITFC sy...
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| Опубліковано в: : | Проблемы прочности |
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| Дата: | 2017 |
| Автори: | , , , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2017
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/173749 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle / Z. Dong, X. Wang, W. Lou, Y. Huang, M. Zhong, H. Fan, L. Peng // Проблемы прочности. — 2017. — № 6. — С. 165-192. — Бібліогр.: 23 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859795427476373504 |
|---|---|
| author | Dong, Z. Wang, X. Lou, W. Huang, Y. Zhong, M. Fan, H. Peng, L. |
| author_facet | Dong, Z. Wang, X. Lou, W. Huang, Y. Zhong, M. Fan, H. Peng, L. |
| citation_txt | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle / Z. Dong, X. Wang, W. Lou, Y. Huang, M. Zhong, H. Fan, L. Peng // Проблемы прочности. — 2017. — № 6. — С. 165-192. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | This study aims to establish a practical method for simulating the spindle coupled multi-axis loading fatigue test of a rear axle. A dynamic finite-element model of the rear axle was constructed and validated using a static calibration test. Based on the theory and methodology of the Schenck ITFC system, a simulation process was devised which includes system identification, calculation of the input loading signals for the finite-element model, calculation of the response stress signals based on this model, calculation of the response strain signals from the corresponding stress signals, and finally, a comparison of the desired and achieved signals. The corresponding data processing programs were made using Matlab, ensuring their easy reproducibility. The desired signals were measured on the Hainan proving ground for a duration of 2441.216 s, using strain gauges and rosettes placed in important stress-prone locations of the rear axle. The results indicate that the desired signals can be reproduced comparatively accurately, ensuring that the strain distribution of the rear axle in the field can be reasonably predicted.
Предложен практический метод моделирования многоосного циклического нагружения задней оси автомобиля с учетом взаимодействия различных факторов. Разработана динамичная конечноэлементная модель задней оси и выполнена ее верификация с помощью калибровочных тестов при статическом нагружении. С использованием испытательной системы Schenck ITFC моделируется процесс, включающий идентификацию системы, вычисление входных сигналов нагрузки для конечноэлементной модели, выходных сигналов напряжения и деформации, а также сравнение расчетных сигналов с реальными. С помощью программного обеспечения Matlab реализованы легко воспроизводимые программы обработки данных. Проведено измерение динамических сигналов напряжения в течение 2,441.216 с нагружения на автомобильном стенде Hainan посредством тензодатчиков, размещенных в ключевых точках задней оси автомобиля. Получено удовлетворительное согласование между расчетным и экспериментальным распределением деформации в задней оси автомобиля.
Запропоновано практичний метод моделювання багатовісного циклічного навантаження задньої вісі автомобіля з урахуванням взаємодії різних чинників. Розроблено динамічну скінченноелементну модель задньої вісі та виконано її верифікацію за допомогою калібрувальних тестів при статичному навантаженні. Із використанням випробувальної системи Schenck ITFC моделюється процес, який включає ідентифікацію системи, обчислення вхідних сигналів навантаження для скінченноелементної моделі, вихідних сигналів напружень та деформації, а також порівняння розрахункових сигналів із реальними. За допомогою програмного забезпечення Matlab реалізовано програми обробки даних, що легко відтворюються. Проведено вимірювання динамічних сигналів напружень протягом 2441,216 с навантаження на автомобільному стенді Hainan за допомогою тензодатчиків, розміщених у ключових точках задньої вісі автомобіля. Отримано задовільний збіг розрахункового розподілу деформації у задній вісі автомобіля з експериментальним.
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| first_indexed | 2025-12-02T13:00:06Z |
| format | Article |
| fulltext |
UDC 539.4
Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a
Rear Axle
Z. Dong,
a,1
X. Wang,
a
W. Lou,
a
Y. Huang,
b
M. Zhong,
b
H. Fan,
b
and L. Peng
b
a State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering,
Tsinghua University, Beijing, China
b SAIC GM Wuling Automobile Co., Ltd., Liuzhou, China
1 dongzc13@mails.tsinghua.edu.cn
ÓÄÊ 539.4
×èñëåííîå ìîäåëèðîâàíèå ìíîãîîñíîãî öèêëè÷åñêîãî íàãðóæåíèÿ çàäíåé
îñè àâòîìîáèëÿ ñ ó÷åòîì âçàèìîäåéñòâèÿ ðàçëè÷íûõ ôàêòîðîâ
Ç. Äîíã
à
, Ê. Âàíã
à
, Â. Ëîó
à
, ß. Õóàíã
á
, Ì. Æîíã
á
, Õ. Ôàí
á
, Ë. Ïåíã
á
à Ëàáîðàòîðèÿ äèàãíîñòèêè, áåçîïàñíîñòè è ýíåðãèè àâòîìîáèëåé, Óíèâåðñèòåò Öèíõóà, Ïåêèí,
Êèòàé
á ÎÎÎ SAIC Motor, General Motors, Liuzhou Wuling Motors Co Ltd., Ëþ÷æîó, Êèòàé
Ïðåäëîæåí ïðàêòè÷åñêèé ìåòîä ìîäåëèðîâàíèÿ ìíîãîîñíîãî öèêëè÷åñêîãî íàãðóæåíèÿ çàä-
íåé îñè àâòîìîáèëÿ ñ ó÷åòîì âçàèìîäåéñòâèÿ ðàçëè÷íûõ ôàêòîðîâ. Ðàçðàáîòàíà äèíàìè÷íàÿ
êîíå÷íîýëåìåíòíàÿ ìîäåëü çàäíåé îñè è âûïîëíåíà åå âåðèôèêàöèÿ ñ ïîìîùüþ êàëèáðîâî÷íûõ
òåñòîâ ïðè ñòàòè÷åñêîì íàãðóæåíèè. Ñ èñïîëüçîâàíèåì èñïûòàòåëüíîé ñèñòåìû Schenck
ITFC ìîäåëèðóåòñÿ ïðîöåññ, âêëþ÷àþùèé èäåíòèôèêàöèþ ñèñòåìû, âû÷èñëåíèå âõîäíûõ
ñèãíàëîâ íàãðóçêè äëÿ êîíå÷íîýëåìåíòíîé ìîäåëè, âûõîäíûõ ñèãíàëîâ íàïðÿæåíèÿ è äåôîð-
ìàöèè, à òàêæå ñðàâíåíèå ðàñ÷åòíûõ ñèãíàëîâ ñ ðåàëüíûìè. Ñ ïîìîùüþ ïðîãðàììíîãî
îáåñïå÷åíèÿ Matlab ðåàëèçîâàíû ëåãêî âîñïðîèçâîäèìûå ïðîãðàììû îáðàáîòêè äàííûõ. Ïðî-
âåäåíî èçìåðåíèå äèíàìè÷åñêèõ ñèãíàëîâ íàïðÿæåíèÿ â òå÷åíèå 2,441.216 ñ íàãðóæåíèÿ íà
àâòîìîáèëüíîì ñòåíäå Hainan ïîñðåäñòâîì òåíçîäàò÷èêîâ, ðàçìåùåííûõ â êëþ÷åâûõ òî÷êàõ
çàäíåé îñè àâòîìîáèëÿ. Ïîëó÷åíî óäîâëåòâîðèòåëüíîå ñîãëàñîâàíèå ìåæäó ðàñ÷åòíûì è
ýêñïåðèìåíòàëüíûì ðàñïðåäåëåíèåì äåôîðìàöèè â çàäíåé îñè àâòîìîáèëÿ.
Êëþ÷åâûå ñëîâà: ìíîãîîñíîå öèêëè÷åñêîå íàãðóæåíèå, çàäíÿÿ îñü, ìåòîä êîíå÷íûõ
ýëåìåíòîâ, ÷àñòîòíàÿ ôóíêöèÿ, âîñïðîèçâåäåíèå èñòîðèè äåôîðìèðîâàíèÿ.
Introduction. In addition to meeting the relevant performance requirements, such as
ride comfort and handling stability, the suspension mounted between the wheel and body
(or frame) of the vehicle is also supposed to meet stringent strength and service-life
requirements. The load affecting the tire on the ground can be decomposed into a vertical,
lateral, longitudinal, and a brake force, as well as moments around corresponding axes
when the vehicle is moving. The axles transmit these forces and moments to the
suspension, leading to a complicated multi-axial random loading condition. The durability
of the axle and suspension is mainly influenced by the vertical, lateral, longitudinal and
brake force (as shown in Fig. 1), according to engineering experience. Therefore, meeting
the requirements for durability and strength is one of the most significant problems to be
solved during the design and development of a suspension.
In order to ensure that a suspension system can achieve the desired strength and
longevity, but also to shorten the test times needed, durability tests performed in laboratories
© Z. DONG, X. WANG, W. LOU, Y. HUANG, M. ZHONG, H. FAN, L. PENG, 2017
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 165
have been widely applied in the worldwide automotive industry. The laboratory road test of
a vehicle, which aims to replicate the conditions experienced by a vehicle in service [1–3],
is a comprehensive process. Nowadays, the most realistic road durability simulators for
suspensions are computer-controlled spindle coupled multi-axis loading test facilities, in
which computationally intensive iterative control techniques are applied to diminish the
simulation errors through an iterative process, so that the strain distribution experienced in
the field can be accurately reproduced. Additionally, commercial software suites have been
devised by the major servo-hydraulic test system suppliers, such as the initial release of
Remote Parameter Control (RPC) by MTS in 1977 [4] and of Transfer Function
Compensation (ITFC) by SCHENCK in 1979 [5]. In the late 1990’s, INSTRON produced
SPiDAR and LMS International developed TWR [6–8]. A spindle coupled four-axis
loading simulator in which the inputs are the longitudinal force Fx , vertical force Fz ,
lateral force Fy , and moment about the wheel axis M y [9, 10] is commonly used.
Here, the theory and methodology of the ITFC algorithm by SCHENCK is introduced,
which includes the following three main steps.
Step 1. System Identification. Figure 2 shows the principle outline of system
identification. In order to identify the dynamic characteristics of the whole test system,
including the test specimen, hydraulic system, filters, D/A, A/D and response-measuring
system), the computer sends an identification drive signal, which is close to a white noise,
to each input channel in turns, and the response signals of all channels are measured. The
identification signals yi (i m�1 2, , ... , , where m is the total number of all input channels)
in the time domain are generated by the computer. When an identification signal is sent to
an input channel, the input signals to the other channels are kept at zero, with the measured
response signals being xij ( , , , .. , )i j m�1 2 in the time domain.
Fig. 1. Loads and forces acting on a car wheel.
Fig. 2. Principle outline of system identification in ITFC.
166 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
The frequency response functions are calculated using the following equations:
H f
X f
Y f
ij
i j
i
( )
( )
( )
,
� ( , , , ... , ),i j m�1 2 (1)
where X fi j, ( ) is the Fourier transformation of xi j, ,
X f FFT xi j i j, ,( ) ( ),� (2)
Y fi ( ) is the Fourier transformation of yi ,
Y f FFT yi i( ) ( ),� (3)
whereby FFT represents the fast Fourier transformation, f is the frequency, H fij ( ) is
the frequency response function for the input signal yi and response signal xi j, caused
by yi , and m is the number of the input and output channels.
The system frequency response matrix H f( ) is devised as follows:
H f H fij( ) [ ( )].� (4)
As shown in Fig. 2, one group of input signals yi (i m�1 2, , ... , ) will generate one
corresponding group of response signals xi (i m�1 2, , ... , ), with the response signals
being calculated using the formula
X f Y f H fT T( ) ( ) ( ),� (5)
where T represents matrix transposition and X fT ( ) is the row matrix of response
signals:
X f X f X f X fT
m( ) [ ( ) ( ) ( )],� 1 2 � (6)
whereby X fi ( ) is the Fourier transformation of the response signal xi , and Y fT ( ) is the
row matrix of input signals, and
Y f Y f Y f Y fT
m( ) [ ( ) ( ) ( )],� 1 2 � (7)
where Y fi ( ) is the Fourier transformation of the input signal yi .
The following formula can be obtained from formula (5):
Y f X f H fT T( ) ( ) ( ),� �1
(8)
where H f�1 ( ) is the inverse matrix of the frequency response function,
H f H fij m m
�
�
��1 1( ) [ ( )] . (9)
Step 2. Determining the Input Signals through an Iterative Process. The response
signals in the time domain, which are measured on the road, are defined as the desired
response signals x t x x xs
T
s s sm( ) [ , , ... , ],� 1 2 which must be accurately reproduced in order
to make the rig tests realistic.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 167
Simulation of the Spindle Coupled Multi-Axis Loading Fatigue Test ...
In step 2, the aim is to determine the input signal y t y y yT
m( ) [ , , ... , ],� 1 2 which
ensures that the desired response signal x t x x xs
T
s s sm( ) [ , , ... , ]� 1 2 can be reproduced with
acceptable accuracy. In order to improve the accuracy of this reproduction, an iterative
process is applied to compensate for the non-linearity of the test system.
Figure 3 shows a diagrammatic representation of the iterative process for determining
the input signal from the desired response signal. In Fig. 3, x t x x xT
m( ) [ , , ... , ]� 1 2 is the
achieved response signal, whereas � � � �T
mt( ) [ , , ... , ]� 1 2 is the error signal between the
response signal x t x x xs
T
s s sm( ) [ , , ... , ]� 1 2 and obtained signal x t x x xT
m( ) [ , , ... , ]:� 1 2
�T
s
T T
s s sm mt x t x t x x x x x x( ) ( ) ( ) [ , , ... , ] [ , , ... ,� � � �1 2 1 2 ]. (10)
The term � � � �T
mt( ) [ , , ... , ]� 1 2 is the correcting signal, y t y y yT
m( ) [ , , ... , ]� 1 2
is the input signal, and E E E ET
m� [ , , ... , ]1 2 is the weighted error signal, which is
calculated using the formula
E
x x dt
x dt x
i
si i
t
si
t
i
si
�
�
�
�
�
08 02
2
0
2
0
.
( )
.
| |
.
max
max
�
(11)
The First Iteration Step. The aim of the first iteration is to determine the initial input
signal y1 from the desired response signal xs . The last-step response signal x t0 ( ) and
input signal y t0 ( ) are all zero row matrices. Thus,
�0 0( ) ( ) ( ) ( ),t x t x t x ts s� � � (12)
X f FFT x ts s( ) [ ( )],� (13)
� 1
1 1T
s
Tt FFT X f H f( ) [ ( ) ( )],�
� �
(14)
where FFT�1 represents the inverse fast Fourier transformation, and is the attenuation
matrix:
Fig. 3. Diagram of the iterative process used in the ITFC test system.
168 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
�
�
�
�
�
�
�
�
�
1
2
�
m
( , , , ... , ). i i m� �1 1 2 (15)
Considering the non-linearity of the system, [ ], which is a diagonal matrix, should be
applied in order to avoid iteration divergence and improve the accuracy.
The input signal obtained in the first iteration step is as follows:
y t y t t FFT X f H fT T T
s
T
1 0 1
1 1( ) ( ) ( ) [ ( ) ( )].� � �
� �� (16)
Subsequently, y1 is sent to the test system and the first achieved response x1 is
measured. The error signal between xs and x1, �1 1� �x xs , and the weighted error
E E E ET
m1 11 12 1� [ , , ... , ] are calculated via formulas (10) and (11), respectively. Presuming
that Es is the desired weighted error, if E Ei s1 � , the iteration will be stopped and y1 is
the driving signal needed. If, on the other hand, E Ei s1 � , the next iteration step will be
conducted.
The nth Iteration. In the nth iteration, the nth input signal yn is determined from the
desired response signal xs , the achieved response signal xn�1 and the input signal yn�1,
which are obtained in the last step:
�n s nt x t x t� �� �1 1( ) ( ) ( ), (17)
� �n nf FFT t� ��1 1( ) [ ( )], (18)
� �n
T
n
Tt FFT f H f( ) [ ( ) ( )],�
�
�
�1
1
1
(19)
y t y t tn
T
n
T
n
T( ) ( ) ( ).� ��1 � (20)
After this, yn is sent to the test system and the achieved response xn is measured.
The error signal between xs and xn , �n s nx x� � , and the weighted error signal En
T �
� [ , , ... , ]E E En n nm1 2 are again calculated using the formulas (10) and (11), respectively.
If E Eni s� , the iteration process will be stopped, with yn being the input signal needed.
If E Ei s1 � , the next iteration step will be conducted, until the weighted errors converge to
an acceptable tolerance level.
Step 3. Performing the Durability Test. Figure 4 shows a diagrammatic representation
of the durability test. The test system operates in an open-loop control mode, whereby the
final input signal y tn ( ), obtained at the end of the above iteration process, is repeatedly
sent to the test system until the test specimen is broken or the requirement for durability is
fulfilled. In such a test, the strain distribution in the specimen experienced on the road can
be accurately reproduced, yielding realistic test results.
In recent years, many vehicle manufacturers are moving towards virtual test simulations
instead of the laboratory or proving ground tests, because they need to shorten the
development period of new product and reduce costs in the competitive market environment.
At the time of writing, there are two main virtual test simulation methods: the virtual
proving ground (VPG) and virtual test rig (VTR) [11–14]. Their main advantage lies in the
fact that the quality of the design can be evaluated even before a prototype is available [15].
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 169
Simulation of the Spindle Coupled Multi-Axis Loading Fatigue Test ...
The methodology of VPG is as follows: (i) building a geometry model of the test tracks of
the proving ground [13, 14]; (ii) building a finite element model of the whole vehicle
including a tire model [13]; (iii) simulation of the vehicle driving on the proving ground
and calculation of the strain, stress and acceleration response histories; (iv) calculation of
longevity and fatigue damage. However, the VPG technology imposes truly vast demands
on computer resources. The time period and corresponding road length which can be
simulated within the limits of available computer resources and calculating time is thus
quite short, and consequently cannot represent the test on the proving ground with
sufficiently high statistical significance.
VTR technology is mainly used to evaluate the durability of the vehicle’s body or
frame. Figure 5 shows a typical virtual four-poster test rig model with four vertical
displacements as inputs [9]. The VTR methodology is as follows: (i) building a model of
the actuators [9]; (ii) building a finite element model of the whole vehicle including a tire
model [13]; (iii) simulating the vehicle driving on the proving ground with the control
signals being the acceleration signals measured on the proving ground, or with the
geometry model of the test tracks of the proving ground as the input signals and calculating
the strain, stress and acceleration response histories; (iv) calculating the vehicle’s life and
fatigue damage. The limitations of such VTR technology are similar, in principle, to those
encountered with VPG [16–18].
This paper aims to establish a simplified virtual test rig method which can accurately
reproduce the strain distribution in a selected specimen, thus avoiding the problems
associated with constructing a complicated finite-element model of the whole vehicle,
including the tire, and consequently prolonging the time interval, which can be simulated.
Fig. 4. Diagrammatic representation of the durability test.
Fig. 5. A virtual four-poster test rig model [19].
170 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
The mini-bus rear axle illustrated in Fig. 6 was selected as the specimen. A virtual rig test
or spindle coupled four-axial loading rig test based on the theory and methodology of ITFC
were simulated using the strain signals of the strain gauges Li and Ri (i�1, 2, 3, 4)
measured on the Hainan proving ground according to the specifications of the SAIC GM
Wuling Automobile Co., Ltd. (SGMW, for short) durability test. We used the desired
response signals, together with the vertical force Fz , lateral force Fy , longitudinal force
Fx , and moment M y of both sides of the axle as inputs.
1. Simulation of the Spindle Coupled Multi-Axial Loading Rig Test of the Rear
Axle. The simulation of the spindle coupled multi-axial loading rig test of the rear axle is
based on ITFC methodology. The first step comprised system identification, and Fig. 7
shows a flowchart of this process. A dynamic finite-element model of the rear axle was
used in place of the real, physical axle, servo-hydraulic loading system and signal
measurement system. A program for calculating the identification input forces and moments
was compiled, with which the input signals and the response stress signals were calculated
based on this finite-element model. The strain signals, on the other hand, were calculated
Fig. 6. A model of the mini-bus rear axle used as specimen in this study.
Fig. 7. Flowchart of the system identification process.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 171
Simulation of the Spindle Coupled Multi-Axis Loading Fatigue Test ...
from the response stress signals, using a strain calculation program. The strain, input force
and moment signals were then used to calculate the system frequency response function
matrix. Finally, an inverse system frequency function matrix was obtained. All calculation
programs were compiled in Matlab.
In another step, the input force and moment signals, used to reproduce the proving
ground strain signals, were determined (see Fig. 8 for a flowchart). The dynamic finite-
element model of the rear axle was again applied in place of the real axle, servo-hydraulic
loading system and signal measurement system.
In a final step, the stress signals of critical points or elements of the rear axle
finite-element model were obtained and processed to predict their fatigue damage and
longevity using software based on the local stress-strain theory.
2. Finite-Element Model of the Rear Axle. The dynamic finite-element model is the
key for a successful simulation of the spindle coupled multi-axis loading test of the rear
axle, and thus must be properly constructed. The model, illustrated in Fig. 9, was
constructed and analyzed using Altair HyperWorks 12.0 (Altair Engineering, Inc.). The rear
axle was modeled using solid elements (CTETRA) and shell elements (CTRIA3), with
dimensions ranging from 2 to 5 mm. Revolving joints were simulated using RBE2. Since
the axle housing is the focus of this study, the dimensions of the elements for modeling this
component, encompassing the upper and bottom housing carriers, were 2 mm, while the
dimensions of the elements for all other components were 5 mm. Table 1 lists the types,
dimensions and thicknesses of the elements used to model each component.
The axle shafts were modeled using the solid CTETRA elements, with element
dimensions of approximately 5 mm. The outboard bearings of the axle shaft, bearing the
largest forces, were also modeled using the solid CTETRA elements, which dimensions of
5 mm, as shown in Fig. 9b. The inboard bearings of the axle shafts, which bear much
smaller forces than the outboard ones, were modeled using the CTRIA3 plate and shell
elements, and their dimensions and thicknesses are given in Table 1 and shown in Fig. 9c.
The leaf springs were simplified as linear, by building leaf-end links and leaf-mid
links between the three leaf springs, as shown in Fig. 9a, d, and e. This models the leaf
spring in its fully loaded state. In order to obtain the correct stiffness of the leaf spring, a
group of virtual material properties was selected, with the virtual elastic modulus E�
�100 GPa, Poisson’s ratio �� 0.3, and density �� 7900 kg/m3. This treatment was
Fig. 8. Flowchart for determining the input force and moment signals used to reproduce the proving
ground strain signals.
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T a b l e 1
Types, Dimensions, and Thicknesses of the Elements Used to Model the Components
of the Rear Axle
Component Element type Element dimension
(mm)
Element thickness
(mm)
Loading bar CTRIA3 5 50
Leaf-end link CTRIA3 5 1
Leaf-mid link CTRIA3 5 5
Inboard bearings CTRIA3 5 2
Revolving joints RBE2 – –
Housing CTETRA 2 –
Axle carrier CTETRA 2 –
Upper housing carrier CTETRA 2 –
Bottom housing carrier CTETRA 2 –
Axle shaft CTETRA 5 –
Outboard bearing carrier CTETRA 5 –
Outboard bearing CTETRA 5 –
Rubber bushing CTETRA 5 –
Lifting lug CTETRA 5 –
U bolt CTETRA 5 –
a b
c d
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chosen because it is beneficial for calculation speed. The types, dimensions and thicknesses
of the elements used for the leaf-end and leaf-mid links are also listed in Table 1.
Rubber bushings (as shown in Fig. 9d and e) were used to connect the steel components
in the model, and it was assumed that the deformation of each rubber bushing is small and
that their material properties could be approached by a virtual elastic material with an
elastic modulus of E�1 MPa, Poisson’s ratio �� 0.48, and density ��1650 kg/m3,
which again was beneficial for calculation speed.
The axle carrier is made of cast iron, and the material properties of the elements used
to model it were thus as follows: elastic modulus E�170 GPa, Poisson’s ratio �� 0.25,
and density �� 7900 kg/m3. The material properties of the elements for modeling other
axle components were as follows: elastic modulus E� 203 GPa, Poisson’s ratio �� 0.3,
and density �� 7900 kg/m3.
The rotating centers of the front and rear lifting-lugs were modeled using nodes A, B,
C, D, E, and F, as shown in Fig. 9a, and their translational degrees of freedom along the X,
Y, and Z axes and the rotational degrees of freedom around these axes were restrained.
Therefore, only rotation around the Y axis was allowed. The revolving joints were also
simulated in this way.
The loading bars simulating the tires were modeled using CTRIA3 shell elements with
dimensions of 5 mm and a thickness of 50 mm. The distance between the left and right
loading bars corresponded to the track width of 1480 mm. The vertical length of the
loading bar was equal to the tire radius of 305 mm, as shown in Fig. 9a. FY was applied
on the lower center point of the loading bar, whereas FZ , FX , and MY were all applied
on the axle shaft center point, as shown in Fig. 9f.
Thus, the dynamic finite-element model used in this study is a fully linear one. In
order to confirm its accuracy, a static calibration test was performed using the test system
shown in Fig. 10. Seven strain gauges (L1, L2, L4, R1, R2, R4, and R5) and two 45�
rosettes (L3 and R3) were attached to the surface of the axle housing, as shown in Fig. 11.
The corresponding element ID numbers are L1–L4 and R1–R4, as shown in Fig. 6. The
strain gauges L1, L2, L4, R1, R2, R4, and R5 were used to measure the strains in the Y
direction, whereas L3 and R3 were used to measure the strains at an angle of 45� to Y.
L1 and R1 measured the longitudinal force FX , whereas L2, L4, R2, and R4 were
sensitive to the vertical force FZ and lateral force FY . L3 and R3 reacted to the brake
force FB , as shown in Fig. 10.
The static calibration test was performed under four loading conditions: (i) only
vertical force FZ is applied; (ii) lateral force FY and vertical force FZ are applied;
(iii) longitudinal force FX and vertical force FZ are applied; (iv) brake force FB and
e f
Fig. 9. Details of the dynamic-finite element model of the rear axle.
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vertical force FZ are applied. The strain signals were measured using the strain gauges
and rosettes placed in the positions shown in Fig. 11.
Fig. 10. Static calibration test system.
a
b
c
Fig. 11. Positions of strain gauges attached to the surface of the axle housing: (a) front view; (b) top
view; (c) bottom view.
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In order to check the accuracy of the dynamic finite element model shown in Fig. 9, a
corresponding static model was constructed, which was basically the same as the dynamic
one except for the input forces and moments. Figure 12 shows the input forces of the static
element model including the vertical force FZ , lateral force FY , longitudinal force FX ,
and brake force FB .
The stress distribution of the rear axle under each load condition was calculated using
the static finite-element model. The RADIOSS module of Altair HyperWorks 12.0 was
used to perform the model analysis. The normal Y strain � yc , which corresponds to that
measured using the strain gauges L1, L2, L4, R1, R2, R4, or R5 was calculated using
formula (21):
� � � � �yc y x z
E
� � �
1
[ ( )], (21)
where � x , � y , and � z represent the calculated normal X, Y, and Z stresses of the element
on which the corresponding strain gauge is placed. Presuming the strain �T is the strain
measured with the rosette L3 or R3, then
� � �T � �
1
2
45 135( ), (22)
whereby �45 and �135 are the strains measured by the two strain gauges in the direction
with an angle of 45� to Y in the strain rosette, respectively. The strains corresponding to
�45 and �135 were calculated using formulas (23) and (24), respectively:
�
� �
� �
� � �
�45
1
2
2
2
, ,c
x y
xy
x y z
xy
E
�
�
� �
� �
�
�
�
��
�
�
��
�
�
�
�
� (23)
�
� �
� �
� � �
�135
1
2
2
2
, ,c
x y
xy
x y z
xy
E
�
�
� �
� �
�
�
�
��
�
�
��
�
�
�
�
� (24)
where �45, c and �135, c are the calculated strains corresponding to �45 and �135,
respectively, � x , � y , and � z are the calculated normal X, Y, and Z stresses of the element
Fig. 12. Input forces of the static finite-element model of the rear axle.
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on which the corresponding strain rosette is placed, respectively, and � xy is the calculated
shear stress of the element. The strain �Tc , which corresponds to �T is calculated using
formula (25):
� � �
�
�Tc c c
xy
E
� � �� �
1
2
145 135( ) ( )., , (25)
Tables 2–5 show a comparison between the strains measured in the static calibration
test and the corresponding strains calculated using the formulas (21), (23), (24), and (25),
with the error calculated using formula (26):
Error
S S
S
c m
m
�
�
| |
%,100 (26)
where Error is the relative error between the measured strain S m and the corresponding
calculated strain S c . As can be seen in these tables, one error was 24.11% which is the
error corresponding to the strain gauge L1 under the loading condition in which a
longitudinal and a vertical force are applied, with the remaining errors all being smaller
than 16%. Thus, the accuracy of the finite-element model of the rear axle is acceptable.
T a b l e 2
Comparison of Strains Measured and Calculated for the Loading Condition
in which only a Vertical Force FZ � 5000 N is Applied
L1 R1 L2 R2 L3 R3 L4 R4
Sm ( )�� – – 171.094 135.395 – – �219.090 �180.835
Sc ( )�� – – 153.655 152.201 – – �185.069 �186.535
Error – – 10.19% 12.41% – – 15.53% 3.15%
T a b l e 3
Comparison of Strains Measured and Calculated for the Loading Condition
in which a Lateral Force FY � 5000 N and Vertical Force FZ � 5000 N are Applied
L1 R1 L2 R2 L3 R3 L4 R4
Sm ( )�� – – �215.863 �205.371 – – 195.042 175.021
Sc ( )�� – – �230.586 �228.246 – – 202.132 202.605
Error – – 6.82% 11.14% – – 3.63% 15.76%
T a b l e 4
Comparison of Strains Measured and Calculated for the Loading Condition
in which a Longitudinal Force FX � 5000 N and Vertical Force FZ � 5000 N are Applied
L1 R1 L2 R2 L3 R3 L4 R4
Sm ( )�� �113.841 �125.714 – – – – – –
Sc ( )�� �141.283 �144.517 – – – – – –
Error 24.11% 14.96% – – – – – –
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3. Measurement of Rear Axle Strain Signals on the Hainan Proving Ground. To
obtain the desired response signals for the virtual laboratory simulation, the strain signals of
the strain gauges L1, L2, L4, R1, R2, R4, and R5, as well as the strain rosettes L3 and
R3 (Figs. 6 and 11) of the rear axle were measured on the Hainan proving ground,
according to the specifications of the SGMW durability test. The signal of R5 was used to
verify the strain distribution of the rear axle, which will be discussed later, with other
measured strain signals used as the desired response signals. Figure 13 shows the measured
strain time history of the strain gauges L1 and R1 in a full test-drive cycle, at a sampling
frequency of 500 Hz, a low-pass filter cut-off frequency of 50 Hz and a duration of
2,441.216 s (1220,608 data points).
T a b l e 5
Comparison of Strains Measured and Calculated for the Loading Condition
in which the Brake Force FB � 3000 N and Vertical Force FZ � 5000 N are Applied
L1 R1 L2 R2 L3 R3 L4 R4
Sm ( )�� – – – – 111.740 111.740
Sc ( )�� – – – – 116.431 110.184
Error – – – – 4.20% 1.39%
a
b
Fig. 13. Normal Y strain histories of the strain gauges L1 (a) and R1 (b) measured in a full test cycle
on the Hainan proving ground according to the specifications of the SGMW durability test.
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4. System Identification. In order to obtain the input forces and moments for the
dynamic finite-element model of the rear axle, starting from the desired response signals, it
was necessary to identify the system frequency response function matrix. The input forces
and moments used in the dynamic finite element model of the rear axle include (1) the
left vertical force FZ 1, lateral force FY 1, longitudinal force FX 1, and moment MY 1; and
(2) the right vertical force FZ 2, lateral force FY 2, longitudinal force FX 2, and moment
MY 2. The output signals of the dynamic finite-element model are the stresses � x , � y , � z ,
and � xy of each strain gauge, with the formula (21) used to calculate the corresponding
normal Y strain � yc and formulas (23), (24), and (25) used to calculate the strain �Tc
corresponding to the strain signals measured by the strain rosettes L3 and R4. In
accordance with formula (5), this yields the following formula:
X t Y t H tT
in
T
� ( ) ( )[ ( )] ,� �8 8 (27)
X t x x x x x x x xT
� � � � � � � � �( ) [ ]� �1 2 3 4 5 6 7 8
� [ ],� � � � � � � �L R L R L R TL TR2 2 4 4 1 1 (28)
� � � � �L y L x L z L
E
2 2 2 2
1
� � �[ ( )],, , , (29)
� � � � �R y R x R z R
E
2 2 2 2
1
� � �[ ( )],, , , (30)
� � � � �L y L x L z L
E
4 4 4 4
1
� � �[ ( )],, , , (31)
� � � � �R y R x R z R
E
4 4 4 4
1
� � �[ ( )],, , , (32)
� � � � �L y L x L z L
E
1 1 1 1
1
� � �[ ( )],, , , (33)
� � � � �R y R x R z R
E
1 1 1 1
1
� � �[ ( )],, , , (34)
�
�
�TL
xy L
E
�� �
,
( ),
3
1 (35)
�
�
�TR
xy R
E
�� �
,
( ),
3
1 (36)
Y t y y y y y y y yin
T
in in in in in in in in( ) [ ], , , , , , , ,� 1 2 3 4 5 6 7 8 �
� [ ],F F F F F F M MZ Z Y Y X X Y Y1 2 1 2 1 2 1 2 (37)
wherein [ ( )]H t 8 8� is the system strain-force relationship matrix, T represents matrix
transposition, X tT
� ( ) is the response strain row matrix, x j� ( j�1, 2, …, 8) are the
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elements of X tT
� ( ), and �L1, �L 2, �L 4, �R1, �R2, and �R4 are the calculated normal Y
strains of the elements whose positions correspond to the strain gauges L1, L2, L4, R1, R2 ,
and R4 (as shown in Figs. 6 and 11), respectively. The terms � y L, 1, � x L, 1, and � z L, 1
stand for the calculated normal stresses of the element corresponding to strain L1, with L2
and L4, as well as R1, R2, and R4 substituted in a similar manner for the others. �TL and
�TR are the calculated normal strains of the elements whose positions correspond to the
strain rosettes L3 and R3 (as shown in Figs. 6 and 11), respectively, with � xy L, 3 and
� xy R, 3 being the calculated shear stresses corresponding to the strain rosettes L3 and R3.
Y tin
T ( ) represents the input forces and moments given as a row matrix, and yin i, (i�1, 2,
…, 8) are the elements of Y tin
T ( ). As shown in Fig. 14, FZ 1, FX 1, FY 1, and MY 1 are the
forces and the moment applied on the left end of the rear axle, whereas FZ 2, FX 2, FY 2,
and MY 2 are the forces and the moment applied on the right end of the rear axle. The
stresses mentioned above are all calculated using the dynamic finite element model of the
rear axle, with the above-mentioned input forces and moments used as the inputs to the
model. The dynamic finite element model was analyzed in Altair HyperWorks 12.0, using
the modal transient analysis module from RADIOSS.
In order to calculate the system frequency response-function matrix it was first
necessary to calculate the identification signal. The identification signal used here is a kind
of impulse signal called pseudo impulse, the expression of which can be described with the
following formula:
S t
A
N
f tpT
pp
n
n n
n
( ) cos( ),�
�
� 2
1
2
(38)
f
n
T
n � , (39)
T
N
f
FFT
s
� , (40)
N n n� � �2 1 1, n n2 1 0! ! , (41)
Fig. 14. Input forces and moment on the left end of the finite element model of the rear axle.
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where S tpT ( ) is the sampling value of the pseudoimpulse signal, A pp is the amplitude
( ),A pp � 400 N FFT is the N of a N-point FFT (N FFT � 8192), fs is the sampling
frequency ( fs � 500 Hz), T is the duration of the identification signal, T N fFFT s� �
� �8192 500/ 16.384 s, fn is the frequency at which a sampling value is calculated, n1
is the first value of n, n1 1� , n2 is the last value of n, n2 200� , and N is the total
number of fn . Figure 15a shows the time history of the calculated identification signal.
The reason why the pseudo impulse signal is used as the system identification signal
is that it has good characteristics, with its corresponding real part of the frequency spectrum
being a horizontal line, and its phase also being a horizontal line corresponding to a value
of zero, as shown in Fig. 15b and c. In this way, good system identification results can be
obtained.
Subsequently, the stress response signals were calculated using the dynamic finite
element model of the rear axle. Taking the calculated pseudo impulse signal as the signal of
the ith element yin i, of the row matrix Y tin
T ( ) [formula (37)], with other elements in
Y tin
T ( ) kept at zero, the stress response signals of the elements in the dynamic finite-
element model which correspond to the strain gauges and rosettes L1–L4 and R1–R4
(Fig. 6), were calculated using the model. Figure 16 shows the time histories of � x , � y ,
� z , and � xy for each of these elements. The ID numbers of the elements were 2382,643,
a b
c
Fig. 15. Pseudo impulse signal and spectrum of the pseudo impulse signal.
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2549,992, 2695,043, 1908,821, 2587,018, 2690,602, 2539,658, and 2043,102. The elements
x j i� , ( j i, ,�1 2, …, 8) of the row matrix X tT
� ( ) [formula (28)] were calculated using
formulas (28)–(36), with x j i� , ( j�1, 2, …, 8) being exclusively caused by yin i, .
The system frequency response function matrix was also calculated. From formulas
(27), (28), and (37), the formulas (42), (43), and (44) were obtained as follows:
X f Y f H fT
in
T
� ( ) ( )[ ( )] ,� �8 8 (42)
X f X X X X X X X XT T
� � � � � � � � �( ) [ ] ,� 1 2 3 4 5 6 7 8 (43)
Y f Y Y Y Y Y Y Y Yin
T
in in in in in in in in( ) [ ], , , , , , , ,� 1 2 3 4 5 6 7 8
T , (44)
where X fT
� ( ) and Y fin
T ( ) are the Fourier transformations of X tT
� ( ) and Y tin
T ( ),
respectively, X j� ( j�1, 2, …, 8) and Yin i, (i�1, 2, …, 8) are the transformations of x j�
( ,j�1 2, …, 8) in formula (28) and yin i, (i�1, 2, …, 8) in formula (37), respectively,
[ ( )]H f 8 8� is the system frequency response function matrix:
[ ( )]H f
H H H H H H H H
H H H H H H
8 8
11 12 13 14 15 16 17 18
21 22 23 24 25
� �
26 27 28
31 32 33 34 35 36 37 38
41 42 43 44 45 46
H H
H H H H H H H H
H H H H H H H H
H H H H H H H H
H H H H H H H
47 48
51 52 53 54 55 56 57 58
61 62 63 64 65 66 67 68
71 72 73 74 75 76 77 78
81 82 83 84 85 86 87
H
H H H H H H H H
H H H H H H H H88
�
�
�
�
�
�
�
�
�
�
�
�
�
�
, (45)
Fig. 16. Time histories of �x , � y , � z , and �xy of the eight elements, caused by each input force or
moment.
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H
X
Y
ij
w j i
win i
�
� ,
,
( , , , ... , ),i j�1 2 8 (46)
where X w j i� , and Ywin i, are the Fourier transformations of xw j i� , and ywin i, in the time
domain:
x n x n S nw j i j i st� �, ,( ) ( ) ( ),�
(47)
y n y n S nwin i in i st, ,( ) ( ) ( ),�
(48)
S n
A
A
n
f T
st
w
w
s
( ) cos ,�
�
�
�
�
��
�
�
��
�
�
�
�
�
1
1
1
2
(49)
X f A e FFT xw j i n w
j n
w j i�
�, ,( ) ( ) ( ),� �
1 (50)
Y f A e FFT ywin i n w
j n
win i, ,( ) ( ) ( ),� �
1
(51)
f
n
N
fn s�
�1
, n N�1 2, , ... , . (52)
Herein, FFT represents the fast Fourier transformation, x nj i� , ( ) and y nj i� , ( ) are the nth
sampling values of x j i� , and yin i, , n N�1 2, , ... , (N is the total number of the sampling
values of the identification signal), S nst ( ) is the lift window function or Hanning window
function for the case that Aw �1 is accepted, T is the duration of the identification signal,
fs is the sampling frequency, and fn is the frequency with which the sampling value of
the Fourier transformation is calculated. As an example, the real and image parts of H11
are shown in Fig. 17.
Fig. 17. Real and imaginary part of H11.
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5. Calculating the Input Forces and Moments to Reproduce the Desired Strain
Response Signals. According to formulas (47)–(52), the desired signals were transferred
from the time domain to the frequency domain via formulas (53)–(57):
x n x x x x x x xs
T
s s s s s s s, , , , , , , ,( ) [ , , , , , ,� � � � � � � �� 1 2 3 4 5 6 7 8, ],xs
T
� �
� [ , , , , , , ], , , , , , , , ,� � � � � � � �s L s R s L s R s L s R s TL s TR2 2 4 4 1 1
T , (53)
x n x n S nws
T
s
T
st, ,( ) ( ) ( ),� ��
(54)
X f A e FFT xs
T
n w
j n
ws
T
, ,( ) ( ) ( ),�
�� �
1 (55)
X f X X X X X X Xs
T
n s s s s s s s, , , , , , , ,( ) [ , , , , , ,� � � � � � �� 1 2 3 4 5 6 � �7 8, ] ,,X s
T
(56)
f
n
N
fn s�
�1
, n N�1 2, , ... , , (57)
where x ns
T
, ( )� is the desired signal row matrix in the time domain, xs i, � (i�1, 2, …, 8) are
the elements of x ns
T
, ( ),� �s L, 1, �s L, 2, �s L, 4, �s R, 1, �s R, 2, and �s R, 4 are the normal Y
strains of the elements whose positions correspond to the strain gauges L1, L2, L4, R1, R2 ,
and R4, �s TL, and �s TR, are the normal strains of the elements whose positions
correspond to the strain rosettes L3 and R3 (as shown in Figs. 6 and 11), X fs
T
n, ( )� is the
desired signal row matrix in the frequency domain, and X s i, � (i�1, 2, …, 8) are the
elements of X fs
T
n, ( ).�
According to formula (42), the following formulas can be obtained:
F f H f X fT
n s
T
n( ) [ ( )] ( ),,�
�
�
8 8
1
� (58)
F f F F F F F F M MT
n Z Z Y Y X X Y Y
T( ) [ , , , , , , , ] ,� 1 2 1 2 1 2 1 2 (59)
where F fT
n( ) is the row matrix of forces and moments used to reproduce the desired
strain row matrix X fs
T
n, ( )� in the frequency domain, FZ 1 , FZ 2 , FY 1 , FY 2 , FX 1 , FX 2,
MY 1 , and MY 2 are the elements of F fT
n( ), and [ ( )]H f 8 8
1
�
� is the inverse of the system
frequency response function matrix [ ( )]H f 8 8� .
According to formula (51), the force and moment row matrix [ ]F T , used to produce
the desired strain row matrix x ns
T
, ( )� in the time domain, can be obtained as follows:
[ ]
( )
( )
( )
,F
S n
IFFT
F f
A e
T
st
n
w
j n
�
�
�
�
�
�
�
1
1 (60)
[ ] [ , , , , , , , ] ,F F F F F F F M MT
Z Z Y Y X X Y Y
T� 1 2 1 2 1 2 1 2 (61)
where IFFT represents the inverse fast Fourier transformation, FZ 1, FZ 2, FY 1, FY 2,
FX 1 , FX 2 , MY 1 , and MY 2 are the elements of [ ]F T .
184 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
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As indicated in formula (58), the input loading signals in the frequency domain were
obtained via the inverse of the system frequency response function matrix and the desired
signals in the frequency domain. The number of accepted FFT points was 8192. Because
the number of sampling points in each channel of the desired strain signal measured during
an entire test cycle on the Hainan proving ground was 1220,608, the desired signal of each
channel was divided into 298 sections, with each section having 8192 data points. In order
to ensure the accuracy of calculation in the adjacent parts of the neighboring sections, they
overlapped, as shown in Fig. 18, with the input loading signal points used to reproduce the
desired signals calculated in one section after another, and the middle half of the obtained
data points in each section connected and accepted as the input loading signal data points.
The number of obtained data points for each channel of the input loading signal was also
1220,608.
Subsequently, the input loading signal data points were applied to the dynamic
finite-element model of the rear axle to reproduce the desired strain signals and the correct
stress and strain distributions in the structure. Since the model is a fully linear one, an
iteration process was not necessary.
Because the number of input loading data points which can be handled with the modal
transient analysis module of RADIOSS is about 80,000, each channel input loading signal,
comprising 1220,608 data points, was divided into 16 sections. In order to ensure the
accuracy of calculation in the adjacent parts of the neighboring sections, they were
overlapped by 1000 data points, so that the initial response values of the Modal transient
module were always zeros. The numbers of data points in each of the 16 sections are shown
in Table 6. It should be pointed out that the response data points corresponding to the
preceding 1000 data points in each section, which are the last 1000 points in the last
section, were discarded. The remaining response data points corresponding to the effective
data points in each section were consequently connected and accepted as the achieved
signal.
6. Comparisons between the Achieved and Desired Signals. Figure19 shows the
desired and achieved signals of the strain gauge or rosette L1, L2, L3, L4, R1, R2, R3,
and R4 shown in Figs. 6 and 11. Figure 19i and j shows the zoom-in of the signals in
Fig. 19g and h. The simulation error Ers is defined as
E
y n y n
y n
y n y n
rs
d
d
d�
�
�
��
�
08 02
2
2
.
[ ( ) ( )]
( )
.
max(| ( ) ( )| )
max(| ( )| )
,
y nd
(62)
where y nd ( ) is the desired signal and y n( ) is the achieved signal.
Fig. 18. Connections of the loads.
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Simulation of the Spindle Coupled Multi-Axis Loading Fatigue Test ...
T a b l e 6
The Number of Data Points in Each Loading Signal Section
Section Number of data points
No. 1 0 + 78,000
No. 2 1000 + 78,000
No. 3 1000 + 78,000
No. 4 1000 + 78,000
No. 5 1000 + 78,000
No. 6 1000 + 78,000
No. 7 1000 + 78,000
No. 8 1000 + 78,000
No. 9 1000 + 78,000
No. 10 1000 + 78,000
No. 11 1000 + 78,000
No. 12 1000 + 78,000
No. 13 1000 + 78,000
No. 14 1000 + 78,000
No. 15 1000 + 78,000
No. 16 1000 + 50,608
Number of effective data points 1220,608
a (reproduction of L1) b (reproduction of R1)
c (reproduction of L2) d (reproduction of R2)
e (reproduction of L3) f (reproduction of R3)
186 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
The simulation errors for the strain gauges or rosettes are given in Table 7, with the
maximum error being 26.02% (R3) and the minimum error 7.24% (R1), which are small
enough according to the experience gained in doing the multi-axial loading testing and
acceptable. As mentioned above, the strain gauge R5 is not a control sensor for obtaining
the input loading signals but used to check if the strain distribution of the axle is correctly
reproduced, whose desired and achieved signals are shown in Fig. 20, with the error being
22.58%. Figure 20b shows the zoom-in of Fig. 20a. Thus, the error is small enough,
indicating that the desired signal of strain gauge R5 and the strain distribution in the axle
are correctly reproduced.
g (reproduction of L4) h (reproduction of R4)
i (zoom in of the reproduction of L4) j (zoom in of the reproduction of R4)
Fig. 19. Signals of strain gauge or rosette L1, L2, L3, L4, R1, R2, R3, and R4.
T a b l e 7
Simulation Errors for Strain Gauge or Rosette L1, L2, L3, L4 , R1, R2, R3, and R4
L1 R1 L2 R2 L3 R3 L4 R4
Ers 10.10% 7.24% 10.98% 10.06% 11.66% 26.02% 13.56% 15.45%
Fig. 20. Reproduction of R5.
a b
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 187
Simulation of the Spindle Coupled Multi-Axis Loading Fatigue Test ...
7. Fatigue Life Prediction of the Rear Axle Housing. The local strain approach
[20–22] is applied to predict the fatigue life of the rear axle housing, whose basic equations
are the cyclic stress–strain curve,
" " "� � �
2 2 2
1
� �
#
�
�
�
�
�
�
#
E K
n/
, (63)
and the strain–life equation,
"� � �
�
2
2 2
0
�
# �
� #f
f
b
f f
c
E
N N( ) ( ) , (64)
where E is the cyclic elastic modulus, #K is the cyclic strength coefficient, #n is the
cyclic strain hardening exponent, #� f is the fatigue strength coefficient, b is the fatigue
strength exponent, #� f is the fatigue ductility coefficient, c is the fatigue ductility
exponent, "� is the strain range of the stress-strain hysteresis loop caused by the loading
history in the material, "� is the stress range of the loop, and � 0 is the mean stress of the
loop. "�, "� , and � 0 for each stress-strain hysteresis loop in the material can be obtained
by local stress–strain response analysis [22]. The housing of the axle is made with a
material equivalent to 1020, a USA steel, whose fatigue properties are listed in Table 8
[23].
Neuber’s rule is applied to predict the local stress-strain response in the critical
elements from the strain histories of the elements close to the critical elements. Neuber’s
rule is expressed with formula (65),
K f C C A A
2" " " "� � � �� , (65)
where "�C and "�C are the local stress and strain range of the hysteresis loop formed in
element C whose strain signal is known, respectively, "� A and "� A are the local stress
and strain range of the corresponding hysteresis loop in element A (critical element),
respectively, and K f is the fatigue notch factor.
Neuber’s rule can also be expressed with formula (66),
K S e K E e K
S
E
f C c f C f
C
A A
2 2 2 2
2
" " "
"
" "� � � � � , (66)
T a b l e 8
Fatigue Properties of 1020 Steel [23]
Order No. Parameter Data
1 Cyclic elastic modulus E, GPa 203
2 Cyclic strength coefficient #K , MPa 1221
3 Cyclic strain hardening exponent #n 0.24
4 Fatigue strength coefficient #� f , MPa 895
5 Fatigue strength exponent b �0.11
6 Fatigue ductility coefficient #� f 0.29
7 Fatigue ductility exponent c �0.47
188 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
where "SC and "eC are the nominal stress and strain range of element C, respectively,
which are obtained by processing the calculated nominal stress signal with the rainflow
counting arithmetic. Because the nominal stress signals of each element can be obtained
with the dynamic finite element analysis model above, formula (66) is more useful in
predicting the fatigue life of each element of the finite element model, with A C� and
K f �1, which implies that the nominal stress signal of each element is used to predict its
fatigue life and the stress concentration effect has been taken into account in the finite
element analysis.
A software based on the local stress-strain approach [22] has been made to process the
strain histories and to predict the fatigue lives and damages caused by them. Tables 9 and
10 show the fatigue life prediction results of the calculated and measured strain histories of
the strain gauges L1, L2, L3, L4, R1, R2, R3, and R4 , as shown in Fig. 11, with formula
(65) and K f �1 applied in the life predictions. Let Dc and Dm be the predicted fatigue
damages caused in a whole Hainan driving cycle with the calculated and measured strain
signal, respectively. The corresponding predicted fatigue lives are
L
D
c
c
�
1
, (67)
L
D
m
m
�
1
. (68)
The error Err is defined as
Err
L L
L
c m
m
�
�
�100%, (69)
where Lc and Lm are the predicted fatigue lives or the numbers of the whole Hainan
driving cycles the component can experience before the fatigue failure occurs.
T a b l e 9
Fatigue Life Prediction Results of the Calculated and Measured Strain Histories
of the Strain Gauges L1, L2, L3, and L4
Strain signal L1 L2 L3 L4
Measured Max. strain/�� 298.08 491.78 134.77 262.07
Min. strain/�� �238.32 �672.78 �100.37 �417.22
Dm 5.746e–11 4.002e–8 2.527e–14 3.040e–10
Lm/Hainan
drive cycle
1.740e+10 2.499e+7 3.957e+13 3.290e+9
Calculated Max. strain/�� 297.05 498.65 133.90 263.02
Min. strain/�� –238.30 –672.44 –100.46 –417.77
Dc 5.718e–11 4.124e–8 2.590e–14 3.234e–10
Lc /Hainan
drive cycle
1.749e+10 2.425e+7 3.860e+13 3.092e+9
Err 0.52% 2.96% 2.45% 6.02%
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 189
Simulation of the Spindle Coupled Multi-Axis Loading Fatigue Test ...
As shown in Tables 9 and 10, the predicted life errors between the calculated and
measured strain histories of strain gauge L1, L2, L3, L4, R1, R2, R3, and R4 are
between 0.17 and 9.34%, which are quite small, indicating that the accuracy of the
simulation is quite high. Thus, the forces and moments obtained above for reproducing
the desired strain response signals can be used for fatigue simulation.
Conclusions. The spindle coupled multi-axial loading rig test of the rear axle is
comparatively accurately simulated with the method presented, with the simulation errors
between 7.24 and 26.02% which are small enough according to the experience gained in
doing the multi-axial loading testing and acceptable. The method focuses on accurately
reproducing the strain distribution in the selected specimen experienced in the fields, with
which the problems with constructing the complicated whole vehicle finite element model
including the tire model and too short time duration which can be simulated are avoided.
Each channel desired signal adopted has 1,220,608 data points and a duration of 2441.216 s,
which is measured in a whole test cycle on Hainan proving ground according to the
specifications of durability test of SAIC GM Wuling Automobile Co., Ltd. In fact, the
number of the data points or the time duration of the desired signal which can be simulated
with the method is not limited. Thus, the method presented is practical and efficient.
Ð å ç þ ì å
Çàïðîïîíîâàíî ïðàêòè÷íèé ìåòîä ìîäåëþâàííÿ áàãàòîâ³ñíîãî öèêë³÷íîãî íàâàíòà-
æåííÿ çàäíüî¿ â³ñ³ àâòîìîá³ëÿ ç óðàõóâàííÿì âçàºìî䳿 ð³çíèõ ÷èííèê³â. Ðîçðîáëåíî
äèíàì³÷íó ñê³í÷åííîåëåìåíòíó ìîäåëü çàäíüî¿ â³ñ³ òà âèêîíàíî ¿¿ âåðèô³êàö³þ çà
äîïîìîãîþ êàë³áðóâàëüíèõ òåñò³â ïðè ñòàòè÷íîìó íàâàíòàæåíí³. ²ç âèêîðèñòàííÿì
âèïðîáóâàëüíî¿ ñèñòåìè Schenck ITFC ìîäåëþºòüñÿ ïðîöåñ, ÿêèé âêëþ÷ຠ³äåíòèô³-
êàö³þ ñèñòåìè, îá÷èñëåííÿ âõ³äíèõ ñèãíàë³â íàâàíòàæåííÿ äëÿ ñê³í÷åííîåëåìåíòíî¿
ìîäåë³, âèõ³äíèõ ñèãíàë³â íàïðóæåíü òà äåôîðìàö³¿, à òàêîæ ïîð³âíÿííÿ ðîçðàõóí-
êîâèõ ñèãíàë³â ³ç ðåàëüíèìè. Çà äîïîìîãîþ ïðîãðàìíîãî çàáåçïå÷åííÿ Matlab ðåàë³-
çîâàíî ïðîãðàìè îáðîáêè äàíèõ, ùî ëåãêî â³äòâîðþþòüñÿ. Ïðîâåäåíî âèì³ðþâàííÿ
äèíàì³÷íèõ ñèãíàë³â íàïðóæåíü ïðîòÿãîì 2441,216 ñ íàâàíòàæåííÿ íà àâòîìîá³ëü-
T a b l e 1 0
Fatigue Life Prediction Results of the Simulation and Measured Strain Histories
of the Strain Gauges R1, R2, R3, and R4
Stain signal R1 R2 R3 R4
Measured Max. strain/�� 306.69 430.65 77.00 246.50
Min. strain/�� �267.45 �557.47 �110.6 �364.30
Dm 1.182e–10 1.776e–8 4.791e–15 1.774e–10
Lm/Hainan
drive cycle
8.459e+9 5.630e+7 2.087e+14 5.640e+9
Calculated Max.strain/�� 307.98 431.03 88.05 247.24
Min.strain/�� �267.49 �557.44 �107.42 �364.53
Dc 1.184e–10 1.791e–8 5.285e–15 1.883e–10
Lc /Hainan
drive cycle
8.445e+9 5.583e+7 1.892e+14 5.310e+9
Err 0.17% 0.34% 9.34% 5.85%
190 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
íîìó ñòåíä³ Hainan çà äîïîìîãîþ òåíçîäàò÷èê³â, ðîçì³ùåíèõ ó êëþ÷îâèõ òî÷êàõ
çàäíüî¿ â³ñ³ àâòîìîá³ëÿ. Îòðèìàíî çàäîâ³ëüíèé çá³ã ðîçðàõóíêîâîãî ðîçïîä³ëó äåôîð-
ìàö³¿ ó çàäí³é â³ñ³ àâòîìîá³ëÿ ç åêñïåðèìåíòàëüíèì.
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Received 12. 01. 2017
192 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Dong, X. Wang, W. Lou, et al.
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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| id | nasplib_isofts_kiev_ua-123456789-173749 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-02T13:00:06Z |
| publishDate | 2017 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Dong, Z. Wang, X. Lou, W. Huang, Y. Zhong, M. Fan, H. Peng, L. 2020-12-19T15:44:32Z 2020-12-19T15:44:32Z 2017 Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle / Z. Dong, X. Wang, W. Lou, Y. Huang, M. Zhong, H. Fan, L. Peng // Проблемы прочности. — 2017. — № 6. — С. 165-192. — Бібліогр.: 23 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173749 539.4 This study aims to establish a practical method for simulating the spindle coupled multi-axis loading fatigue test of a rear axle. A dynamic finite-element model of the rear axle was constructed and validated using a static calibration test. Based on the theory and methodology of the Schenck ITFC system, a simulation process was devised which includes system identification, calculation of the input loading signals for the finite-element model, calculation of the response stress signals based on this model, calculation of the response strain signals from the corresponding stress signals, and finally, a comparison of the desired and achieved signals. The corresponding data processing programs were made using Matlab, ensuring their easy reproducibility. The desired signals were measured on the Hainan proving ground for a duration of 2441.216 s, using strain gauges and rosettes placed in important stress-prone locations of the rear axle. The results indicate that the desired signals can be reproduced comparatively accurately, ensuring that the strain distribution of the rear axle in the field can be reasonably predicted. Предложен практический метод моделирования многоосного циклического нагружения задней оси автомобиля с учетом взаимодействия различных факторов. Разработана динамичная конечноэлементная модель задней оси и выполнена ее верификация с помощью калибровочных тестов при статическом нагружении. С использованием испытательной системы Schenck ITFC моделируется процесс, включающий идентификацию системы, вычисление входных сигналов нагрузки для конечноэлементной модели, выходных сигналов напряжения и деформации, а также сравнение расчетных сигналов с реальными. С помощью программного обеспечения Matlab реализованы легко воспроизводимые программы обработки данных. Проведено измерение динамических сигналов напряжения в течение 2,441.216 с нагружения на автомобильном стенде Hainan посредством тензодатчиков, размещенных в ключевых точках задней оси автомобиля. Получено удовлетворительное согласование между расчетным и экспериментальным распределением деформации в задней оси автомобиля. Запропоновано практичний метод моделювання багатовісного циклічного навантаження задньої вісі автомобіля з урахуванням взаємодії різних чинників. Розроблено динамічну скінченноелементну модель задньої вісі та виконано її верифікацію за допомогою калібрувальних тестів при статичному навантаженні. Із використанням випробувальної системи Schenck ITFC моделюється процес, який включає ідентифікацію системи, обчислення вхідних сигналів навантаження для скінченноелементної моделі, вихідних сигналів напружень та деформації, а також порівняння розрахункових сигналів із реальними. За допомогою програмного забезпечення Matlab реалізовано програми обробки даних, що легко відтворюються. Проведено вимірювання динамічних сигналів напружень протягом 2441,216 с навантаження на автомобільному стенді Hainan за допомогою тензодатчиків, розміщених у ключових точках задньої вісі автомобіля. Отримано задовільний збіг розрахункового розподілу деформації у задній вісі автомобіля з експериментальним. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle Численное моделирование многоосного циклического нагружения задней оси автомобиля с учетом взаимодействия различных факторов Article published earlier |
| spellingShingle | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle Dong, Z. Wang, X. Lou, W. Huang, Y. Zhong, M. Fan, H. Peng, L. Научно-технический раздел |
| title | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle |
| title_alt | Численное моделирование многоосного циклического нагружения задней оси автомобиля с учетом взаимодействия различных факторов |
| title_full | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle |
| title_fullStr | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle |
| title_full_unstemmed | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle |
| title_short | Simulation of the Spindle Coupled Multi-Axial Loading Fatigue Test of a Rear Axle |
| title_sort | simulation of the spindle coupled multi-axial loading fatigue test of a rear axle |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173749 |
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