DETERMINATION OF VERTICAL CHARACTERISTICS OF RAIL VEHICLE
SUSPENSIONS BY DROP TEST ON THE TRACK
J.Chvojana, R.Mayerb, P.Polacha, J.Václavíka
a
ŠKODA VÝZKUM s.r.o., Tylova 1/57, 316 00 Pilsen, Czech Republic, [email protected]
b
Sciotech office, Department of Engineering, University of Reading, RG6 6AY,
[email protected]
ABSTRACT
This article follows up with the ideas of EUREKA Eurobogie and Footprint projects, the goal of which is to improve the
suspension quality of rail and road vehicles and to find techniques for the measurement of the impact of the vehicle to the
environment. Based on the tests realized on a shaker rig in the laboratory, the methodology for the track drop test was
proposed. This method is suitable for the determination of the natural frequency and critical damping ratio of suspension
system of railway wagons. Basic relations for the calculation of these values on the basis of time force and acceleration
histories are given in the article. An example of the drop test with HAA coal wagon provided with composite suspensions is
shown. Estimation of the dynamic loading coefficient based on the track drop test using MBS modeling is also introduced and
a simple relation between drop and track force time histories is derived. The MBS model improvement according to the data
obtained during the test is still in progress.
Introduction
The environmental impact of the vehicle involves the lifetime cost of the maintaining the infrastructure and the environment.
One of the principal objectives of EUREKA Footprint Project is to develop the techniques for the measurement of this impact
and to relate it to economic costs. The vehicle footprint is especially a function of the suspension system quality. To be
classified as a ‘rail-friendly‘ the suspension must isolate the wagon from rail unevenness so that the dynamic wheel loading is
minimized. The main factors influencing dynamic loading are unsprung and sprung masses of the vehicle, sprung mass
frequency and sprung mass damping ratio.
Drop test enables to determine the natural frequency and the critical damping of a suspension rather dynamically than quasistatically. The EU definition for ‘road friendly’ suspensions defines both the maximum value of the natural frequency of the
sprung mass (< 2 Hz) and the requisite dynamic damping ratio (20 %). Moreover there is a requirement that no more than
50 % of dynamic damping may come from Coulomb (friction) forces. The EC drop directive for a road vehicle was adopted.
Presented work follows the tests performed on a shaker rig with a simultaneous drop of actuators of one axle to excite the
natural frequency with the instrumented HAA coal wagon [1]. The drop test was simulated using hydraulic actuator and the
optimum drop height was determined. The approach was taken over from two previous projects, DIVINE and Eurospring, as
this technique had never been previously used for studying dynamic interactions of rail vehicles with the track.
Tests on the shaker rig are special very expensive procedures. That is why the drop test procedures were performed direct on
the railway. The purpose of this article is an easy test method proposal for the determination of the dynamic characteristics of
the suspension with full size wagons on the track by crossing an artificial bump.
By additional testing it was shown (but only reported for road in the past) that the mentioned tests could also be used to predict
which suspensions have good ride quality and low dynamic loading. A numerical model using alaska MBS software was
created and shaker rig tests were simulated and validated. Based on this model, simulations of wagon behavior on the real
track were performed to predict the track dynamic loading numerically based on the proposed track drop test.
Theory
The wagon is a dynamic system formed by unsprung and sprung masses separated by own suspensions. Another suspension
between the wheel and rail track subsoil influences the dynamic behavior of this system. For the simplification let us take one
quarter of the wagon. The influence of the subsoil is neglected in further considerations. After the wagon suspension is excited
due to the crossing of the vertical bump or dip without any other actuated force, the system performs so called natural viscous
dampened oscillation (Figure 1) according to the following equation of motion
q&& + 2br ⋅ ω o ⋅ q& + ω o2 ⋅ q = 0
,
K
m
ωo =
(1)
where m is sprung mass, β is damping coefficient of the suspension, K is dynamic stiffness of the suspension and ωo is natural
circular frequency of the no dampened system.
qme-brwo
q(t)
5
acceleration a [m.s-2]
displacement q [mm]
10
q(t+Td)
0
3
{q2,t2}
2
{q4,t4} {q6,t6}
1
0
-1
{q3,t3} {q5,t5} {q7,t7}
-2
-5
Td
-3
{q1,t1}
-10
-4
0
0.5
1
1.5 time t [s]
2
0.5
1
1.5
2
time t [s]
2.5
Figure 1. Theoretical displacement damping curve and the real one in acceleration domain (composite suspensions)
Solving this equation - amplitude of the oscillation - is described in the following relation
(
)
q (t ) = q m ⋅ e − brωot sin (ω d t +ψ o ) , ω d = ω o 1 − br2 ,
(2)
where br is relative damping coefficient, qm, ψo are initial amplitude and phase of the oscillation and ωd is circular frequency of
the oscillation of the dampened system. The decrease of the maximum amplitude of this oscillation during one oscillation
period Td is expressed by the logarithmic damping decrement δ
Td β
2πbr
q (t ) 
=
 = br ω oTd =
2m
1 − br2
 q (t + Td ) 

δ = ln 
.
(3)
The relative damping coefficient br and the natural frequency of the system fo can be expressed using δ from (2) and (3) as
follows, where ζ is critical damping ratio
 δ 
1+

1+ ξ 2
 2π 
fo =
=
Td
Td
2
br =
δ
4π 2 + δ 2
,
.
(4)
The displacement signals are necessary for damping characteristics extraction, as the equations (1), (2) are based on
displacement. The derivations, which are not presented here, showed that if the damping is not constant both evaluations
based on displacement and acceleration give a slightly different values.
Suspension Characteristics Extraction
The sense of drop test is the extraction of suspension characteristics on the base of measured suspension deformations and
sprung mass acceleration during damping oscillation process. All characteristics given in the last chapter can be extracted
from the recorded displacement or acceleration time histories. For further calculation, only extreme peaks and zero axes
crossing peaks can be used because only here the amplitude and the phase of the oscillation can be read out with the
sufficient accuracy. The best solution how to calculate the damping using all existing extreme peaks is to found the equation
for the damping envelope curve of the oscillation, which is expressed as the first, amplitude part of equation (2), giving an
exponential function. The best curve fit can be made by the approximation of the exponential function using the least square
method by means of the linear regression of following equation
ln (qi ) = ln (q m ) − br ω o t i
(5)
where {qi, ti} are the measured amplitude and time pairs of extreme coordinates, read out from the damping time history chart
(see Figure 1). Solved coefficients qm and brωo are expressed as follows
br ω o =
n∑ (t i ⋅ q i ) − ∑ t i ∑ qi
n∑ t i2 − (∑ t i )
2
,
q m = q − br ω o ⋅ t
(6)
where q , t are the mean values of data sets used for calculation. After calculating the envelope curve coefficients, the
logarithmic damping decrement δ can be calculated using (3) and the natural frequency can be calculated using (4) (notice
that it differs from the frequency of the damping oscillation). This approach is not quite correct, if the damping is not constant
with time. If it is like that the average values are obtained. Correct results of damping depending on time can be obtained using
non-linear regression.
The example of drop test evaluation is given for movable empty HAA coal wagon at real track with the height of the rectangular
bump of 20 mm, speed approximately 5 km⋅h-1, air temperature –5 °C. The wagon was equipped with 2-leaf composite
suspensions. The unsprung mass for one suspension (one quarter of the wagon) was 2700 kg and the static stiffness of the
suspension at empty state was 870 N⋅mm-1. This gives the natural frequency of the suspension 2.86 Hz. The sprung mass
acceleration time histories at left dropped wheel were used for this example (Figure 1 b).
acceleration a a [m.s -2]
100
aenv (t ) = 27 ⋅ e
−4.024 t
ζ a [1]
The time history is first filtered using Butterworth low-pass filter, cut off frequency 8 Hz, filter order 4. This type of filter gives
smooth waveform with unambiguous peaks and zero axes crossing points. The displacement time history has to be equalized.
During pulling the wagon through the bump the wagon may be tilted and the relative displacement can be changed. First, the
signal was approximated with tilted straight line, using linear regression; second the linear regression values were subtracted
from the raw data to obtain the equalized data. Then the extremes are to be extracted from the signal, using automatic peak
analysis. The number of points for further analysis needs to be corrected by hand (see Figure 1b). The absolute value of all
selected peaks is used for further calculation.
0.3
points
0.25
function
averrage
0.2
10
0.15
0.1
0.05
1
0
1
1.2
1.4
1.6
1.8
tim e t [s]
1
1.2
1.4
1.6
1.8
tim e t [s]
Figure 2. Envelope damping function (a) and evaluation of critical damping ratio (b)
1
n
δ i = ⋅ ln
δ
qi
, ξi = i
qi +1
2π
,.. δ
q
1
= ⋅ ln i ,
n
qi +n
ξ=
δ
2π
(7)
The peaks are approximated using exponential function according to (5) (Figure 2a). The evaluation of the critical damping
ratio ζ is performed in two ways. First using envelope curve, second from the discrete peaks. The average value using all
peaks is also drawn, both according to (7). It is shown in Figure 2b.
The relation of the critical damping ratio ζ on the displacement amplitude of the oscillation is shown in Figure 3. The evaluation
of natural frequency is shown also in Figure 3. Again this is made on the basis of the envelope curve and also using discrete
peak points.
0.25
ζ a [1]
f oa [Hz]
6.5
0.2
discrete values
envelope function
mean
6
5.5
5
0.15
4.5
4
0.1
discrete values
envelope function
averrage
3.5
3
2.5
0.05
0
2
4
6
8
am plitude qm [m m ]
0
2
4
6
8
am plitude q m [m m ]
Figure 3. Dependence of critical damping ratio (a) and natural frequency on the oscillation amplitude (b)
Realization of Track Drop Test
The following procedure is proposed for the track drop test. Three artificial wedge shaped bumps of the heights 10 mm, 15 mm
and 20 mm are, step by step, placed on each rail in front of the wagon and fixed to the rail (Figure 5). The reason for using
three bumps is to find the best conditions for excitation of the sprung mass oscillation. The wagon is then drawn over these
bumps at low speed of approximately 4 km·h-1 (Figure 5). Each bump over crossing is three times repeated due to the
statistical purposes. The example of this test is given in Figure 6 where the HAA coal wagon was used.
The GRP parabolic 2-leaf suspensions with rubber dampers were mounted to the wagon. Wagon was instrumented with lowfrequency inductive accelerometers, positioned just over both front wheels on the sprung mass. At the same time, inductive
displacement sensors were fitted between each front wheel and the chassis, to measure its relative displacement due to the
verification purposes. To acquire data the data recorder enabling the minimum sampling frequency of 400 Hz had to be used.
In this case the digital measuring amplifier SPIDER8 controlled via standard notebook using Catman 4.5 program (both HBM)
was used. The distance between the wagon and the bump highest point was 2000 mm before each test.
During the test, the acceleration or displacement time histories were recorded and evaluated according the relations given in
previous chapter.
For the classification of suspension behavior in horizontal transversal direction the same procedure as for the classification of
vertical suspension characteristics was prepared.
Figure 4. Composite suspension used for test
10 (15)
ACCELEROMETER
a
F
250
STRAIN GAUGES
Figure 5. Schematic set-up for the test (a) and used wedge-shaped bump (b)
Figure 6. Instrumented wheel and realization of the drop test
Dynamic Loading of the Rail Evaluation Based on MBS Support
It is supposed that the wear or damage of railroad is related to the dynamic loading coefficient DLC as it is known for
pavements. The coefficient DLC is calculated from the time history of the force acting between the wheel and the rail according
the relation
DLC =
sF
F
,
(8)
where s F is standard deviation of the dynamic vertical load F and Fm is mean load (static component of the dynamic signal,
which is equal the load of one wheel). The aim was to estimate the track DLC coefficient using the drop test results. For this
purpose the force history during the drop test had to be measured using strain gauges stuck to the rail (or with other device
measuring the bending of the rail), which is proportional to the vertical force (see Figure 5). After crossing the bump the
coefficient DLC was calculated for the drop test. Further it was supposed that the standard deviation of the force damped
oscillation is in some specific relation to the standard deviation of the force time history on real track for given profile quality. In
general, the track vertical force history is formed by several bumps at various heights, just forming the track spectra. The
dependence between both coefficients during drop and real track was studied using simulated processes with the help of MBS
model.
The drop test and the track test were simulated using MBS model. Drop test was simulated for three shapes of bumps (two
wedge-shaped and one rectangular-shaped) and for three speeds of bump over travel. The track test was simulated for four
track quality profiles BQ2, BQ4, BQ6 and BQ9 according to UK standard (Figure 7). Each profile was driven on at least at
three different speeds from the recommended interval for these band qualities, which was made in coincidence with previous
shaker rig tests. Both tests were simulated with HAA coal wagon at tare and three loading levels of the wagon using steel
parabolic 5-leaf suspensions and GRP 2-leaf suspensions. Together 48 track tests and 48 bump test were simulated for both
suspension types. During each simulation the acceleration of each wheel, acceleration of the sprung mass just over each
wheel and the vertical force between the rail and each wheel were monitored.
300
BQ 2
BQ 4
BQ 6
BQ 9
Displ. SPD G(n) [mm3]
Vertical alignm ent [m m ]
20
10
0
-10
350
400
450
BQ4
200
BQ6
150
BQ9
100
50
0
0.00
-20
300
BQ 2
250
500
0.05
distance travelled [m ]
0.10
0.15
0.20
0.25
0.30
wave number n [1/m]
F [kN]
Figure 7. Track profile four quality bands in displacement and frequency domain
0
-10
-20
-30
-40
-50
-60
-70
-80
0
20
40
60
80
100
time t [s]
Figure 8. Visualization of HAA coal wagon MBS model and an example of calculated track test force history
MBS Model
Multibody models of HAA coal wagon [2] were created in alaska (software for the investigation of kinematic quantities and
dynamic behavior of non-linear constrained mechanical systems consisting of the system of bodies). Multibody models were
formed by nine rigid bodies, coupled one to the other by nine kinematic joints (Figure 8). Rigid bodies corresponded to the
individual structural parts of the wagon or they were so called auxiliary bodies. Multibody models of the wagon had ten
degrees of freedom in total.
Contacts between wheel and rail were modeled by connecting the corresponding points with force spring-damper elements.
Non-linear deformation characteristics of both used types of leaf suspension were determined on the basis of static
characteristics measured in the laboratory. In simulating movement with multibody models in alaska software non-linear
equations of motion were generated using Lagrange’s method. The equations were solved by means of numerical time
integration. Resulted time histories were obtained using Shampine-Gordon iteration algorithm.
The value of the vertical stiffness and vertical damping coefficients of the leaf suspensions was increased in multibody models
to improve the coincidence of shaker rig tests and computer simulation results. Nevertheless, resulted computations still
confirm insufficient amount of input data for the correct interpretation of real suspension behavior.
Results Obtained from MBS Simulations
Standard statistical quantities as well as the DLC coefficient were calculated for each time history obtained from simulations
with wagon MBS models. Correlation analysis was performed to obtain some relations between the calculated values. After
first findings the obtained data were reduced in the following way. The track test statistics were averaged for each band quality
track with respect to the speed so that the averaged DLC coefficients were obtained for each BQ independent of the speed.
The reason for averaging was the fact that each band represents the specific characteristic mean speed. The dependence of
DLC truck coefficients averaged like that on the one wheel static vertical force is given in Figure 9.
After the comparison of the drop tests results it was decided to consider only the lowest bump over-travel speed, which was
v = 4 km·h-1. The reason was that the dependence on the wheel load seems to be nearly linear. This speed is also most
suitable for the real drop test performance. Presented results concerns the bump height h = 15 mm.
For each wheel load and track band quality the proportional coefficients were calculated as a ratio between obtained mean
track test DLC and mean drop test DLC
K (BQ, L ) =
DLCTrack (BQ, L )
DLC Bump (BQ )
(9)
Resulted coefficients K are plotted in Figure 9b. It is obvious that these coefficients are nearly insensitive to the wheel load.
Coefficients K are given numerically in Table 1 both for steel and GRP suspensions and its value averaged according to wheel
load K is also calculated. This averaged value can be used for estimating the DLC coefficient from the drop test for given
track quality BQ independent of the suspension type according following relation
DLCTrack (BQ ) = K (BQ ) ⋅ DLC Bump
(10)
3.0
K []
DLC []
0.4
.
0.35
2.5
0.3
2.0
0.25
0.2
1.5
0.15
1.0
0.1
0.05
BQ2
BQ4
BQ6
40
60
80
BQ9
0
0.5
BQ2
BQ4
BQ6
BQ9
0.0
20
100
20
40
60
w heel load F [kN]
Figure 9. Mean DLC coefficients (a) and proportional coefficients K (b)
80
100
w heel load F [kN]
Table 1. Derived proportional constants K
Track
quality
BQ2
BQ4
BQ6
BQ9
Steel suspensions
Wheel load [kN]
34.25
K [-]
1.69
2.14
2.32
1.37
56.03
K [-]
1.55
1.80
2.03
1.04
77.4
K [-]
1.47
1.75
2.38
1.12
97.7
K [-]
1.65
2.07
2.75
1.08
Composite suspensions
Wheel load [kN]
K [-]
1.59
1.94
2.37
1.15
34.25
K [-]
1.61
1.92
2.52
1.19
56.03
K [-]
1.45
1.81
2.35
1.15
77.4
K [-]
1.42
1.75
2.44
1.13
97.7
K [-]
1.54
1.64
2.34
0.97
K
[-]
1.51
1.78
2.41
1.11
K
[-]
1.55
1.86
2.39
1.13
Conclusion
In this paper the methodology for performing the track drop test was introduced. It is suitable for the determination of the
natural frequency and critical damping ratio of suspension system in the railway wagons. The proposed approach to the
dynamic loading coefficient estimation based on the track drop test using MBS modeling was introduced and the simple
relation between drop and track force time histories was derived. Further work will be focused on the improvement of the
numerical model, finding closer functional relation between the tests and modeling the wagon response to the transversal
excitation.
Acknowledgment
The authors thank the Ministry of Education, Youth and Sports (MSMT) of the Czech Republic for financial support of
Eurobogie and Footprint projects, which are filed under the numbers OE 196 and 197.
References
1.
2.
Chvojan, J., Cherruault, J.-Y., Kotas, M., Rayner, M., Václavík, J. “Dynamic investigations on freight wagon suspensions”,
Proceedings of 12th international conference ICEM 12, Bari, 2004.
Polach, P., Hajžman, M., Václavík, J., Chvojan, J. “Computer simulations of coal wagon excitations on a test stand and
comparison with experimental results”, Acta Mechanica Slovaca, EAN 2006, Vol. 10, 1/2006, pp. 405-412, 2006 (in
Czech).