Strojniški vestnik - Journal of Mechanical Engineering 60(2014)4, 241-249 © 2014 Journal of Mechanical Engineering. All rights reserved.
D0I:10.5545/sv-jme.2013.1348	Original Scientific Paper
Received for review: 2013-07-31 Received revised form: 2013-11-15 Accepted for publication: 2013-12-11
Application of Constant Amplitude Dynamic Tests for Life Prediction of Air Springs at Various Control Parameters
Tomaž Bešter* - Matija Fajdiga - Marko Nagode University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
Air spring manufactures use constant amplitude tests for the quality validation of air springs. The tests are very simple and the only information we get from them is that a spring is adequate if it passes the test and inadequate if it does not. One of the objectives of this article is to use these tests to make life predictions based on the standardised load spectrum. This prediction is made with force as the damage parameter. The second objective is to determine if it is possible to use experimental results obtained at one control parameter, e.g. force, to make life predictions for another control parameter, e.g. stress. With equations it is proved that such transformation is possible. Keywords: vehicle suspension, air spring, load spectrum, dynamic tests, fatigue life
0 INTRODUCTION
1 STANDARD LOAD SPECTRUM
Air spring assembly consists of a piston, bellows, a bed plate and a bumper (Fig. 1). Pressure inside the bellows and the piston shape determines air spring characteristics. An air spring can have progressive spring characteristic, which is most suitable for transport vehicles that are loaded with various loads during exploitation [1].
Fig. 1. Air spring
Air spring manufacturers have been testing springs with various static and dynamic experiments in order to verify spring quality. Dynamic tests usually have a constant amplitude and require a certain number of load cycles without critical damage on the air spring. If the spring successfully endures the test, critical damage on the spring does not occur and hence test results do not give exact information about fatigue life. In air spring fatigue tests, critical damage usually occurs on air spring bellows. In this article, the possibilities of obtaining SN-curves with modified constant amplitude tests will be examined. To make fatigue life predictions, loads on air springs and appropriate SN-curves must also be determined.
For transport vehicles, standard load spectra were determined [2] and [3] which define dynamic wheel force Fz,dyn on transport vehicles. Standard load spectrum has 1.5*108 load cycles, which corresponds to 500000 km driving distance with 300 load cycles per kilometre. Standardised load spectra have three driving modes: straight driving, cornering and braking. Based on wheel force measurements during exploitation, standard load spectrum was determined with the level crossing method. Dynamic force Fz,dyn in standardized load spectra is expressed with dynamic load factor nz which represents ratio between dynamic and static load nz=Fz,dyn/ Fz,sta. Dynamic load factor nz depends on number of load cycles N and driving mode (Table 1, Fig. 2). For arbitrary static load Fz,sta dynamic load Fz,dyn can be determined with following equation:
(1)
F = n • F
z ,dyn z z ,sta'
Load ratio:
Rf =
FL,.
(2)
where Fmin is minimal force, and Fmax is maximal force. During exploitation this ratio is changing. Load ratio range for all driving modes has been presented in Table 2.
When deformations are small and materials have linear characteristic, load ratio is equal to stress ratio:
R = R = R = ^
<J„
(3)
where omin and omax are minimum and maximum stress. For high cycle fatigue, life calculation stress is
*Corr. Author's Address: Univesity of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, SI-1000 Ljubljana, Slovenia, tomaz.bester@fs.uni-lj.si
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usually used as the control parameter. When load ratio equals stress ratio, force can be used as the control parameter as well. On air springs, large deformations occur during exploitation and air spring bellows are made of polymer materials with nonlinear stress strain diagram [4] to [7]. Due to the geometric and material nonlinearities, Eq. (3) is not valid for air springs in general. This means fatigue life calculations with different control parameters will have different results. To compare fatigue life calculations with different parameters, SN-curves and Goodman diagrams would have to be determined for all control parameters in question. At least four experiments must be performed to design one Goodman diagram (Fig. 3). In this article, it will be shown how the Goodman diagram for one control parameter could be converted into the Goodman diagram for another control parameter that is not linearly dependent from the former parameter.
Table 1. Standard load spectra [2]
Driving mode	Dynamic load factor	Load spectrum parameters Hg = 1.5x108		
	nz	H	He	SHAPE
Straight driving	2.0	0.96-H,	H-10-6	Linear distribution
Cornering Outer wheel Inner wheel	1.5 0.4	0.04-H,	50	Normal distribution
Breaking	2.0	5-105	104	Normal distribution
2 DYNAMIC EXPERIMENTS AND FATIGUE LIFE
In the standardized load spectrum, load forces on wheels are defined. Wheel forces are transferred to air springs by suspension mechanism. As air spring forces are easily calculated, the easiest way to calculate fatigue life is to use force as a control parameter for the determination of SN-curves and the Goodman diagram. As load ratios of the standard load spectrum are in the range between 0 and 1, experiments for the determination of SN-curves and Goodman diagram must be in the same range.
For constant amplitude loads FaiRi, Fa2Ri, Fam2 and Fa2R2 the number of load cycles when critical damage occurs: N1R1, N2R1, N1R2 and N2R2 must be experimentally determined (Fig. 3). Those results allow to calculate SN-curves slope:
kR1 =
kR 2
lOg NlRl - lOg Na 2 R1 log Fa1R1 - log Fa 2R1 '
lOg Na1R 2 - lOg N 2 R 2 log Fa1R2 - log Fa2R 2
(4)
(5)
If fatigue limit is assumed to be at ND = 2*106, amplitude fatigue strength can be calculated for both dynamic factors:
FaDR1 Fa1R1
( N ^ k«<
D
V N1R1 J
(6)
101 102 103 104 105 106 107 108
N
Fig. 2. Standard load spectra Table 2. Load ratio ranges for various driving modes
Driving modes	Rf
Straight driving	0 to 1
Cornering	0.27 to 1
Breaking	0.5 to 1
FaDR 2 FalR 2
D
N
ViviR2 y
(7)
From amplitude forces and load ratio, medium forces are calculated (for development of Eqs. (8) and (9) see the appendix):
,, _ FaDR\ (1 + R )

(1 - 4 )
FaDR2 I1 + R2
(1 - R2 ) '
and the gradient of the Goodman diagram:
(8) (9)
F — F
M = aDR2 aDRl FmDR2 — FmDRl
(10)
When the gradient of the Goodman diagram is known, equivalent amplitude load Fa1R1 with load ratio R1 can be calculated (Fig. 4, for development of the Eq. (11) see the appendix):
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)4, 241-249
Fa
a\R\
Fa2R\
Fa\R2 FalRX Fqdrx
FaDRI
. R=-1 Fa. R= 0 FmLL R i												
								v				
\ /				R2 \x --~~"					X			
				___—-7		\				\		
												
		----—_									\	
		_____—-71	——									
FmDR\		FmDR2 Fm\R\ Fm2R\ Fm\R2 Fm N\R\ N2R\								N2R2 Nd N		
Fig. 3. Design of Goodman diagram
1 + M
F — F —
a1R1 a1Ri f
(R +1) (R -1)
1+M
(R. +1) (R1 -1)
(11)
		R=l			
R=-ao \			/		R=0
	FalRi				\ Rl
					
	FaiR\				
				//	\ ^ V
Fig. 4. Equivalent loads based on Goodman diagram [8]
Air spring manufacturers use various constant amplitude dynamic tests to validate quality and endurance of their air springs [9] and [10] . Two durability tests have been used in our research. In the first durability test, an air spring is loaded with the displacement amplitude ±50 mm, at the frequency 3.3 Hz. The air spring must endure 2*106 load cycles without critical damage. In the second durability test, the displacement amplitude is ±75 mm at the frequency 2 Hz, with the air spring having to endure 106 load cycles without critical damage. Both tests are stopped when the required number of load cycles is reached unless critical damage occurs first. If springs are tested until the critical damage occurs, two points on the SN-curve can be obtained. Air spring
characteristic was measured (Fig. 5) to determine the appropriate force amplitude for any displacement amplitude.
Fig. 5. Air spring characteristic
In the first durability test, the minimum required number of load cycles is the same as the fatigue limit. In the second durability test the minimum required number of load cycles is relatively near the fatigue limit. It is reasonable to use tests with higher amplitudes for determination of the remaining SN-curve points in order to reduce test time.
With two modified existing and two additional constant amplitude tests, two SN-curves and a Goodman diagram can be determined. This enables the transformation of any load to an equivalent load with the load ratio RF of a known SN-curve.
An SN-curve can be approximately determined with the tensile strength force FM and one dynamic test in the high cycle fatigue range because SN-curves with various load factors intersect near the tensile strength
Application of Constant Amplitude Dynamic Tests for Life Prediction of Air Springs at Various Control Parameters
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N2 N\ Nd Fig. 6. Approximate SN-curves [11]
force (Fig. 6). An air spring's maximum displacement is limited, therefore it is not possible to load a spring with a high enough load to cause the bellows to burst. The air spring bellows tensile strength cannot be directly determined, but it is possible to evaluate the tensile strength if pressure is increased in the spring's bellows until it bursts and the maximum force is used as the tensile strength [11].
Once SN-curves are determined, fatigue life can be calculated with one of the damage accumulation rules [8] and [11]. The most frequently used linear accumulation rules are the original and elementary Palmgren-Miner rule and the Haibach rule (Fig. 7). In all linear accumulation rules, damage d is equal to the inverse number of load cycles when critical damage occurs. Total damage is:
1
d = Y—. tr Nm
(12)
spectrum (Table 3). The results show significant differences between accumulation rules.
When material has stress in the linear region of the stress strain diagram and deformations are small, the stress ratio is equal to the load ratio. Air springs are made of material with nonlinear stress strain diagram and are submitted to large deformations in most load cycles of the standard load spectrum. To evaluate the significance of those nonlinearities, life predictions with stress as the damage parameter should be made.
Table 3. Fatigue life
	Basic Palmgren-Miner rule	Elementary Palmgren-Miner rule	Haibach rule
Numumber of load cycles	9020073	3670064	5155576
Relative number of load cycles	1	0.40688	0.57157
In the described manner, damage was calculated for an air spring loaded with standardized load
3 LIFE PREDICTION WHEN DYNAMIC EXPERIMETS HAVE DIFFERENT LOAD RATIOS
3.1 Determination of the Goodman Diagram from Four Dynamic Experiments with Different Load Ratios
A pair of experiments with equal load ratios when force is used as the control parameter (Fig. 3) will not have equal stress ratios (Fig. 8) due to the nonlinear relation between force and stress. It is not possible to determine SN-curves and the Goodman diagram directly with four experiments that have four different stress ratios. It will be shown that it is possible to solve the problem with a system of equations. The solution of this system can give all the data needed to determine the SN-curves and the Goodman diagram
Nj	Nd	N(log)
Fig. 7. SN-curves for original Palmgren-Miner, elementary Palmgren-Miner and Haibach rule [8]
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Strojniski vestnik - Journal of Mechanical Engineering 60(2014)4, 241-249
c mlfill
Fig. 8. Determination of the Goodman diagram from four dynamic tests
even if only four experiments with different stress ratios are available. Goodman diagram gradient M depends on two fatigue limit medium and amplitude stresses oaDmh aaDmh omDRn and amDR21 (Table 4). To determine those stresses, two SN-curve gradients kR11 and kR21 are needed. To determine those gradients, it is necessary to obtain equivalent stresses with stress
ratios Rn and R21: Om2R11, Om2R21, °a2R11 and aa2R21-
To calculate equivalent stresses the gradient of the Goodman diagram M is needed. There are eleven unknowns. Eleven equations can be set, hence the unknowns can be determined by the equation system solution:



'a 2¿12 - Mam 2 ¿12	+ 1)
a„
a„
((¿11 -1) + M (¿11 +1)) '
(a 2 R 22 - MOm 2 R 22 ) ' ( + 1) (( -1) + M(R21 +1)) 5
,(1 - Rll)

k
R11
11 ( + 1) '
= am1R 21 (1 - R21 ) 21 _ ( + 1) '
l0g N1R11 - l0g N2ri: l0g^fl1r11 - log°a2r1
(13)
(14)
(15)
(16)
(17)
TaD,R11 ®a1R11
' aDR 21 ^ alR 21
N ^
N
1r11

N kR 21 D



'ir21 y
°aDR11 (1 + R11 )
o.
M = -
(1-R11) '
dr21 (l + R21 )
(1 -R21 ) '
<aDR 21 -<aDR11 JmDR 21 ~<mDR11
(19)
(20) (21) (22) (23)
It is difficult to determine stresses on air spring bellows because large deformations and material nonlinearity must be considered. Despite these difficulties, precise enough stress-strain analyses were made [12] to [14]. As stresses in air spring bellows can be determined, known quantities in the system of equations are medium and amplitude stresses (Table 4). With part of equations being nonlinear, the system of equations does not have a simple analytical solution, but it can be solved numerically.
k
R 21
l0g N1R 21 - l0g N2 log ^a1R 21 - log ^a2
(18)
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Table 4. Determination of the Goodman diagram from four dynamic tests- used quantities
	Known quantities		Unknown quantities
°a1R11 [MPa]	Amplitude stress at Fa1R1 [N] and Fm1R1 [N]	&aDR11 [MPa]	Amplitude fatigue strength at R11
°a2R12 [MPa]	Amplitude stress at Fa2R1 [N] and Fm2R1 [N]	aaDR21 [MPa]	Amplitude fatigue strength at RR21
°a1R21 [MPa]	Amplitude stress at Fa2R1 [N] and Fm2R1 [N]	ffmDRH [MPa]	Medium stress at aaoR11 [MPa]
ffa2R22 [MPa]	Amplitude stress at Fa2R2 [N] and Fm2R2 [N]	0mDR21 [MPa]	Medium stress at oaDR221 [MPa]
0^1 [MPa]	Medium stress at Fm1R1 [N] and Fa1R1 [N]	kR11	SN-curve gradient at R11
Vm2R12 [MPa]	Medium stress at Fm2R1 [N] and Fa2R1 [N]	kR21	SN-curve gradient at R21
0m1R21 [MPa]	Medium stress at Fm1R2 [N] and Fa1R2 [N]	0a2R11 [MPa]	Equivalent amplitude stress for ffa2R12 [MPa] at R11
°m2R22 [MPa]	Medium stress at Fm2R2 [N] and Fa2R2 [N]	0a2R21 [MPa]	Equivalent amplitude stress for oa2R22 [MPa] at R21
R11	Stress ratio at Fa1R1 [N] and Fm1R1 [N]	0m2R11 [MPa]	Equivalent medium stress for am2R12 [MPa] at R11
R12	Stress ratio at Fa2R1 [N] and Fm2R1 [N]	0m2R21 [MPa]	Equivalent medium stress for om2R22 [MPa] at R21
R21	Stress ratio at Fa2R1 [N] and Fm2R1 [N]	M	Goodman diagram gradient
R22	Stress ratio at Fa2R22 [N] and Fm2R2 [N]		
	Number of load cycles when critical damage occurs at Fa1R1	[N] and Fmm [N]	
N2R12	Number of load cycles when critical damage occurs at Fa1R1	[N] and Fm1R1 [N]	
N1R21	Number of load cycles when critical damage occurs at Fa1R1 [N] and Fm1R1 [N]		
N2R22	Number of load cycles when critical damage occurs at Fa1R1 [N] and Fm1R1 [N]		
Nd	Fatigue limit		
3.2 Determination of the Goodman Diagram from Three Dynamic Experiments with Different Load Ratios
Although it is possible to determine the Goodman diagram from four dynamic tests with different dynamic factors, it is easier to use just three dynamic tests, but we have to omit the calculation in the first step to determine caD1, kR1 and M. Equivalent stress ofli2. omU and caU, amU (Fig. 9, Table 5) can be calculated using following equations:

f N ^ k
D
o„
o„
O „
O „
N
Nd
v N y
o„
i
(24)
(25)
(26)
3 J
As oa12 and aa13 are not known in Eqs. (25) and (26), they must be expressed with known quantities. Based on the Goodman diagram equations can be written.
= M (»12 ) = M („13 -^»3 ).
Medium stress can be written:
(27)
(28)
^12 (1 + R) (1 - R)
^13 (1 + R)
(1 - R)
(29)
(30)
If Eqs. (29) and (30) are inserted in Eqs. (27) and (28), equations for aa12 and aa13 can be written using aa2, oa3, om2, am3, R1 and M, where M is the only unknown:
(2 -Mam2)(1 -R) (1 -r)-m(1 + R) '
( -Mam3)(1 -R) (1 -R)-M(1 + R) •
(31)
(32)
If logarithms from Eqs. (24), to (26) are calculated and Eqs. (31) and (32) are used for ca12 and oa13, a system of three equations and three unknowns is obtained:
log^, - log o^ = -1 (logNd - logN), (33)
log^«Di - log
(2 - Maml )(1 - Ri) (1 - Ri)-m (1 + R)
= - (log Nd - log N2),
(34)
D
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Bester, T. - Fajdiga, M. - Nagode, M.
Strojniski vestnik - Journal of Mechanical Engineering 60(2014)4, 241-249
o a , "a 13 <*a\	, r=-1 r=0				r2		
	--^						R
Gal OflDl	--7						
							
	omD\ & m 12 Om\				Om2 Om 3		
"ml3
Fig. 9. Determination of the Goodman diagram from three dynamic tests
Table 5. Determination of the Goodman diagram from three dynamic tests - used quantities
	Known quantities		Unknown quantities
oil [MPa]	Amplitude stress with stress ratio R	ki	SN-curve gradient at R
o32 [MPa]	Amplitude stress with stress ratio R2	M	Goodman diagram gradient
Oa3 [MPa]	Amplitude stress with stress ratio R3	OiDi [MPa]	Amplitude fatigue strength at R
om1 [MPa]	Medium stress with stress ratio R^	OmDi [MPa]	Amplitude fatigue strength at R
om2 [MPa]	Medium stress with stress ratio R2	Oai2 [MPa]	Equivalent amplitude stress for oa2 at R
Om3 [MPa]	Medium stress with stress ratio R3	Oai3 [MPa]	Equivalent amplitude stress for oa3 at R
Ri	Stress ratio at oa1 and cm1	Omi2 [MPa]	Equivalent medium stress for om2 at R
R2	Stress ratio at oa2 and &m2	Omi3 [MPa]	Equivalent medium stress for &m3 at R
R3	Stress ratio at oa3 and &m3		
Ni	Number of load cycles when critical damage occurs at oa1 and om1		
n2	Number of load cycles when critical damage occurs at oa2 and om2		
«3	Number of load cycles when critical damage occurs at oa3 and &m3		
Nd	Fatigue limit
Xog°D1 - l0g
((3 -Mam3 )(1 -Ri) (1 - r )-m (1 + Ri )
= - (log Nd - log N3 ).
K,
(35) —
iog N1+iog l0g | + l0g
(c 2 - Mu 2 )(l - R)(
(-R)-M(U) ) = ^ (37)
(3 -Mam3)(1 -R)) i (1-R)-M(1 + R) } = l0gaai' (38)
From Eq. (33) logoaD1 can be calculated:
tog^m = log^fli + y(logNd -logN), (36)
and inserted into Eqs. (34) and (35). Thus two equations with two unknowns are obtained:
There is no simple analytical solution for Eqs. (37) and (38) but it is possible to calculate k1 and M numerically. When k1 is calculated, fatigue limit oaD1 can be calculated with Eq. (36). With the Goodman diagram gradient M, any stress ca1Ri with arbitrary stress ratio Ri can be transformed to equivalent stress oa1R1 with stress ratio R1 and known SN-curve gradient k1:
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1
k
Strojniski vestnik - Journal of Mechanical Engineering 60(2014)4, 241-249
(
1 + M
®a\R\ ®a1Ri /
(R+1) (R -1)
1+M
(R +1) (R -1)
(39)
This enables to calculate equivalent stress for any load, e.g. cfll2, Oau, etc. (Fig. 9).
6 APPENDIX
Development of the medium force equation (Eqs.
(8) and (9). Analogue equations are used for medium
stress (Eqs. (29) and (30)).
Fd max - maximum force
Fd min - minimum force
FmDRi - medium force with load ratio Ri
FaDRi - amplitude force with load ratio Ri
4 CONCLUSION
Unchanged durability tests cannot be used to make fatigue life predictions, because unchanged tests are stopped when the required number of load cycles is reached. If those tests are performed until critical damage is reached and additional constant amplitude tests are made, two SN-curves, Goodman diagram and fatigue life prediction can be made. Fatigue life prediction was made with force as damage parameter using three damage accumulation rules. Various damage accumulation methods gave significantly different results. The elementary Palmgren-Miner rule and the Haibach modification take into account even loads that are smaller than the fatigue limit therefore those rules give shorter fatigue life predictions than the basic Palmgren-Miner rule. No tests were made with loads smaller than the fatigue limit therefore no estimation about the accuracy of damage accumulation rules in this range can be made.
Due to geometric and material nonlinearities, fatigue life calculations with force as the control parameter may be significantly different than fatigue life calculations with stress as the control parameter. The comparison of life predictions using different control parameters would be very time consuming and expensive as we would have to experimentally determine Goodman diagrams for all control parameters in question. With the equations we proved it is possible to transform the Goodman diagram for one control parameter, e.g. force, into the Goodman diagram for another control parameter, e.g. stress. The transformation of the Goodman diagram to arbitrary control parameters makes further research of the influence of control parameters on fatigue life much quicker and less expensive.
5 ACKNOWLEDGMENT
The authors appreciate the support provided by the Veyance Technologies Europe, d.o.o.
F F - F
R _ / D min _ mDRi 1 aDRi
F	F + F
D max r mDRi aDRi
R ■ F + R ■ F = F - F
1 mDRi^ 1 aDRi 1 mDRi 1 aDRi>
Ri ' FmDRi FmDRi Ri ' FaDRi FaDRi,
FmDRi (Ri - 1 ) = -FaDRi{Ri + l),
F = — F
mDRi raDRi
(R +1)= F (1+ R)
(r -1)	aDRi(i - R y
Development of the equivalent amplitude force equation (Eq (11), Fig. 4). Analogue equation is used for equivalent amplitude stress (Eq. (39)).
Fa1Ri - amplitude load with load ratio R,
Fa1R1 - equivalent amplitude load with load ratio R1
Fm1Ri - medium load with load ratio R,
Fm1R1 - equivalent medium load with load ratio R1
F — F
M — alRl a\Ri
F — F
mlRi mlRi
F - f = M (F - F )
1 a1R1 1 a1Ri ly± V m1R1 1 mlRi } '
F - F = M
1 a1R1 1 a1Ri ly±
-F
(R + 1)
(Ri -1)
-F
F + M ■ F (R' + ^ = F - M ■ F
1 alRl	1 a'R' /n	1 a'Ri 1V1 1 m'Ri ■
(R -1)
(
1 + M
(Ri +1) (R -1)
= FlRi + M ■ Fa\Ri
(Ri+1) (R- -1)
r
T>i F
1 + M
\
(Ri+1) (R- -1)
a1R1 ± a1Ri f
1+M
(R1 + 1) (Ri -1)
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