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47
ANALYSIS OF TWO STAGE SUSPENSION RAILWAY VEHICLES BASED ON OMA METHOD
Castañeda H. L. Martinod R. R. Betancur G. G.
Grupo de Estudios en Mantenimiento Industrial - GEMI.
Departamento de Ingeniería Mecánica.
Universidad EAFIT, Colombia.
1. Background
EAFIT University, through the Research Group GEMI, has decided to continue a series of Investigation Projects aimed at innovation, science and technology in the field of railway systems. The knowhow on trains constitutes the basis on which EAFIT University develops Investigation Project directed to companies in the transport sector and co-financed by the Administrative Department of Science, Technology and Innovation of Colombia.
It is in this direction that the work is focused on a methodological approach for the evaluation of the railway vehicles parameters, performed through a modal analysis under commercial conditions of the technical system. A methodology for the modal analysis of railway systems is formulated and applied. In order to comply with the objectives proposed for the project it is necessary to perform different activities which can be synthesized in the following fronts: (i) definition of the characteristics of the railway systems as the subject of study, (ii) determination of the security and comfort criteria for the evaluation of the railway system through the modal analysis, (iii) construction of the railway systems numerical model, and (iv) development of the case study.
2. Introduction
The modal properties of the railway systems are traditionally obtained using techniques known as EMA. EMA techniques have been widely documented [1-3], nonetheless it requires the system to be artificially excited and rarely does that excitation have the characteristics that exist in the railway system under commercial condition. As a matter of fact, the properties estimated by EMA may be poor relative to the response in service [4, 5].
The OMA technique is based on the measurement of the response signals only, using environmental excitations as unmeasured signals and obtaining the modal identification with high precision under typical operating conditions [6, 7]. The advantages
it possesses are the following: (i) evaluate highly non-linear systems (creepage phenomenon) for which the only parameter that can measured exactly are the response data [8, 9]; (ii) realize in situ tests without interruption and in parallel with other applications, it avoids the necessity of having an express test bank for the systems or its component and the dynamic system to be subject inactive time [10]; and (iii) it is a simple test procedure similar to the Operating Deflection Shapes (ODS) technique, which employs a reference transducers and various mobile response transducers. OMA does not use the reference transducers in the case that all the response are measured simultaneously, on the contrary, it uses one or various reference transducers depending of the number of repeated roots obtained. Any measurement can be employed as reference, meaning that OMA is a Multi-Input Multi-Output (MIMO) technique and allows estimating very proximal modes with high precision [8, 10].
OMA-LSCE technique has been adopted and in this approach the modal parameters of the system are identified using standard methods of identification in the domain of time known as LSCE [10, 11], which determines the relation between the Impulse Response Function (IRF) in a system of Multiple Degrees of Freedom (MDoF), its complex poles and residues through a complex exponential. IRF can be derived from the inverse of the Fourier Transform (FT) of a Frequency Response Function (FRF), through the process of Random Decrement (RD), or other methods. An Auto-Regressive (AR) model is built using the relationship between poles and residues; the solution of the AR model allows defining a polynomial expression in which the roots of the mathematical system belongs to the complex. Having established the roots (equivalent to the natural frequencies Q, and the damping rates §) the v may be derived by the AR model and obtain the modal shapes <p. [2].
With the calculation of the poles of the system it is possible to built de stabilization diagram that allows to represent graphical-
ly the poles when it is excited in one point (reference) and measurements are realized in another one (response). Once the poles have been selected it is possible to estimate the vibration shape [12, 13].
3. Description of the subject of study
The study is applied to the passenger vehicles belonging to a railway vehicle fleet of public transportation of Medellín city (Colombia), similar to a suburban train system (Fig. la) [13].
3.1. Background of the railway vehicle
The vehicle is a three-units cars, each of which have two (2) bogies. Each bogie is supported over two (2) axle-wheel set, and each car has two (2) suspension stages: primary and secondary.
The primary suspension stage is composed of external and internal helicoidal springs, parallel to each helicoidal spring there is an hydraulic damper in vertical
direction, and two (2) guide leafs are disposed on the end of the axle (Fig. 1b) providing the guide to the axlebox [14, 15].
The airsprings are part of the secondary suspension stage (Fig. 1b) and aim to fix the height of the carbox. Each airspring is mounted on an auxiliary spring that works in case of failure of the first one [15]. The vertical suspension is done through the hydraulic damper associated to each airspring.
This work is focused on the study of the dynamic vehicle response [11], which is influenced by the technical state of the four (4) identical vertical dampers that each car has <5. with j = 1,...,4. This type of component is characterized by a set of physical laboratory tests. The method of the test requires the application of an excitation of cyclical displacement (sinusoidal) in ranges of frequency and determined peak-to-peak amplitudes [4, 15]. The obtained relationship from the characterization of the
element is denominated nominal damping function, G10. From G10 a set of nine (9) hypothetical properties are established, e, with i = 1,...,9; which present a behavior similar to £., but affected by a coefficient that reduces the suspension properties of
the element d, e¡ - ^«io. In this way there
are different technical states of the component, generating a progressive degradation of the damping function (Fig. 1c).
3.2. Evaluation criteria
The instability of a vehicle must be avoided in the range of travel speed V, otherwise there will be an accentuated effect in the oscillation frequency rn that generates a dynamic negative behavior or Hunting that can even cause derailment. The problems of non-linear stability of railway vehicles have been subject of great amount of study and investigation [16-18].
a
(4) -*■
(6)<-(7)<-
(1) (2) (3)
(8) (9)
(1) Secondary vertical damper.
(2) Airspring.
(3) Central pivot.
(4) Helicoidal spring.
(5) Guide leaf
(6) Primary vertical damper.
(7) Axlebox
(8) Bogie frame.
(9) Wheel.
The damping must be adjusted to control the behavior of the vehicle operating at a speed inferior to the critical speed Vcr, Vcr > V [19, 20]. The stability must be determined through tests and simulations. Literature has established different limit values, relative to Z [4, 20, 21]: (i) danger limit, Zlim1 = 0 % a vehicle with a value Z ^ 0 % implies that the system excites itself, generating instability [20, 22]; and (ii) alarm limit Zlim2 = 5 %, a vehicle is considered stable if it has a value Z of 5 %, on the contrary it is denominated unstable (unsatisfactory) [23].
The unfavorable conditions that go in detriment of passengers comfort is defined according to the Z and ( criteria [24, 25]: (i) unsatisfactory limit, Zlim1 = 5 %, value Z > 5 % determines a positive perception of the passengers; (ii) deficient frequency range, (i>r = [8,10] Hz, the human body is sensible to vertical accelerations in this range; and (iii) limit of balance oscillation, (2 = 0,5 Hz, the vehicle must have a value ( superior to 0,5 Hz, on the contrary there is the risk to produce a movement that causes nausea to passengers.
3.3. Construction of a failure hypothesis
Three (3) hypothesis Zt are established in relation with the state of the suspension elements <5, defined parametrically through the progressive degradation of the damping fuction:
• hypothesis Z1: consists on the reduction of the damping function, changing the technical state £ of the all vertical hydraulic secondary dampers 5, with j = 1,...,4;
• hypothesis Z2: consists on the reduction of the damping function, changing the technical state £. for the leading dampers, this is the ones located to at a near to the driver (51, <52). The others dampers (<53, <54) conserve the nominal property £10; and
• hypothesis Z3: consists on the reduction of the damping function, changing the technical state £ for the training dampers, this is the ones located to at the far end opposite to the driver (<53, <54). The others dampers (< 1, < 2) conserve the nominal property e10.
4. Dynamic characterization of the vehicle applying the EMA method
The EMA technique has been used to determine the vehicle behavior in a given operating condition. The excitation of the vehicle is done trough a manual excitation on the carbox, on the location and direction necessary to induce a specific shape mode; to generate that such excitation, it is necessary to remove the damping elements from the vehicle suspension, condition denominated null damping condition, £0. The acquisition of the data is done with a minimum sampling frequency of 50Hz [13]. The set of signals is transformed to the domain of the frequency with the FFT algorithm to obtain the characteristic frequencies of the carbox and therefore the corresponding modal shape [4, 26]. Table 1 presents the results obtained with the EMA technique.
5. Development of numeric models
The railway vehicle model can be developed and running on a typical track and instrumented in a virtual environment, which allows investigating the effects of a wide range of possible variations of the vehicle parameters [4, 11]. The obtained results of a model can provide certain predictions of the dynamic behavior of the vehicle and the interaction with the track [27].
The focus of a virtual model is to numerically integrate the ordinary differential
equation that constitutes the model, using one or various integrating algorithms. This focus is usually known as simulation or numeric experimentation, which is equivalent to physics experimentation where the system is subject to given conditions and a response is recorded.
The numeric model is used to simulate the characteristics of the system and is considered a real virtual prototype. Virtual techniques allow the model to generate information about the dynamic behavior and the interaction between components, which are comparable only to physical prototypes. Simulations with numeric models are a valid source of necessary data to prove the formulated methodologies. The virtual techniques allow the model to generate a great quantity of information regarding not only the response and the dynamic behavior, but also about the interaction of the components, capacity details, and even esthetic qualities, which are only comparable to physical prototypes [3, 4, 12, 27].
The numeric model developed counts with 120 DoF [15] and comes from the union of the two (2) simplified models (Fig. 2a) and represents a complete motor car (Fig. 2b). It is based on theory of multi-body systems, using the analysis software VAMPIRE®. The values of the particular parameters of the model are exposed on Appendix 2.
6. Development of OMA-LSCE method in a virtual environment
It is necessary to point out the guidelines in order to be able to realize the test properly. The coherent design of the experiment allows a correct analysis of the data; this must be designed to evaluate the secondary suspension under controlled and normal operating conditions, without affecting the operation and security of the
Table 1. Modal parameters identified with the EMA technique, £0.condition.
Modal shape, <pr Lower roll, pt Bounce, p2 Yaw, p3 Pitch, p4
| Natural frequency, p [Hz] | 0,80 | 1,47 | 1,50 1 2,10 |
Fig. 2. Graphic representation of the numeric models: a. simple model; b. complete model 44 | «Транспорт Российской Федерации»
b
a
system. The test is defined following these conditions:
• the load condition of the vehicle (AW0), meaning the car is empty;
• the track section is straight line and track irregularities are considered;
• the vehicle speed, V= 80 km/h; and
• a variation of the technical state of secondary suspension dampers is assumed e. (Fig. 1c).
The measurement points are defined according to the guidelines of the railway standards [22]. Due to the fact that the interest of the work is focused on the study of the secondary vertical suspension, the signals to be registered must in the main bodies of the system, in this way three (3) signals in vertical direction are chosen. The signals are located:
• Z*, on the carbox floor, near to the
qP '
driver;..
• Z+ in the bogie frame near to the attack axle; and
• Zo, in the axlebox of the attack axle.
qi .. .. ..
The set of signals Z*, Z+, and Z° is processed by the OMA-LSCE method, based on the measurement of response of the system to the own excitations of the operation and from which the global modal parameters are estimated: 5, § and p [12]. The modal parameters are graphically represented by the stabilization diagrams. Appendix 3 exposes as an example a set of stabilization diagrams obtained from different operating conditions of the vehicle. In the diagrams it is possible to identify:
• two (2) local maximums in the range [1, 2.5] Hz, which correspond to the two (2) modal shapes (p2, and p4), and they are registered in the carbox by the sensor Zq*;
• three (3) local maximums in the range [4, 10] Hz, which correspond to the three (3) modal s.h. apes recorded in the bogie by the
sensor Z+; and
qi
• three (3) local maximums in the range [15, 25] Hz, which correspond to the vertical irregularities of the track, and they are obtained from the signal recorded by the sensor Zo.
ql
The work focuses on the analysis of the vehicle, more specifically in the carbox for which it has been possible to identify two (2) vertical modal shapes (p2, p4) under the load condition AW0. From the set of values of the modal parameters identified for the different hypothesis Zt it is possible to observe the existing relationship between the technical state of the damping set and the modal shapes p2 y p4, obtaining regressive lineal models with a correlation coefficient value of > 0,98 (Fig. 3).
The regressive models are considered valid given that the values yfH^ represent an association measurement of the statistical
model with the obtained data [26], which have an acceptable level for the reach of this work.
7. Validation of the numeric models
The regressive models belonging to Z1 and obtained by OMA-LSCE in a virtual environment are extrapolated to the value of the technical state of the damper with a null damping function, e0, for the modal shapes pi, meaning:
for p2, e (Q) = 9,577Q -
- 13,950 .•. if e = e0 ^ Q = 1,46 Hz,
for p4, e (Q) = 8,761Q -
- 18,326 .•• if e = e0 ^ Q = 2,09 Hz.
In this way the dynamic behavior of the vehicle without the secondary damper is obtained, emulating the total extraction of the damping elements of the suspension 5. with j = 1,...,4; which is the same testing condition realized to the vehicle with the EMA technique.
The values registered by the two (2) types of analysis, EMA and OMA, are compared (Table 2) which allows to observe that
Table 2. Parameters identified with the EMA and OMA techniques, condition e„
Modal analysis type Q, Hz
EMA 1,47 2,10
OMA 1,46 2,09
Error, £ [%] 0,91 0,04
Fig. 3. Development of regressive models Z■ obtained with the modal parameter Q
there is an estimation error of the modal parameter e < 1 %. Therefore, it is possible to consider valid the regressive models given that the e values represent an acceptable deviation level for the reach of this work.
8. Identification of the technical state of the suspension using the OMA-LSCE method
OMA-LSCE method is applied to the signals acquired during a test done to the three-car unit cars under typical operation
on a commercial track [4]. The measurement points are defined according to the guidelines of the railway standards. Given that the interest of this work focuses on the study of the secondary vertical suspension, the signals to be registered must be in the main bodies of the system, therefore three (3) signals in vertical directio.. n are chosen. The signals are located: (i) Z* Fig. 4a; (ii) Z+ Fig. 4b; and (iii) Z°p Fig. 4c.
There are three (3) straight commercial track sections denominated [15]: Rec-
ta58-63, Recta78-79 and Recta85-87. The signal Z°q] have a wide and soft band, meaning that the PS is constant and it has not poles or zeros in the frequency range of interest. Additionally, it is especially distributed in a uniform way, as [29]. The results of the section Recta58-63 (Fig. 5) are presented as an example. The respective discrete PSD function is calculated S (() (Fig. 5a), observing that the function S (() of the Z* signal only has one local maximum in the range of study [0, 25] Hz, which means that
7
6 ^
X
J %
1
О
F7 Show 1 П7 thow 3
Position Position
* I 1.9S13 I Э.1707
Y r~ 0.7457S I 0 66919
■ Current Line -j
[ |NON£Ch : NONE:0:
Ï
HI
a
J i I i öl У AVA?
Fig. 5 Result of the modal identification in the section track Recta58-63: a. PSD of Z*; and b. Diagram of stabilization of Z*
Table 3. Modal parameters identified by OMA
Track section Order Q, Hz Z, % P,
Recta 58-63 32 2,ll4 l2,38 PA
Recta 78-79 38 l,544 l9,38 P2
Recta 85-87 30 l,944 l5,97 Pa
the perturbation-response relation of the system is able to excite properly only one of the two vertical modal shapes (p2, or p4) depending on the particular irregularities of the track and the corrugation of the rail in the instant of the registration of the signals. Based on function S («) the stabilization diagram is obtained [10] (Fig. 5b). As S («) only registers one modal shape p,, the LSCE method identifies only the parameters associated to that shape (p2, o p4).
The results of the application of OMA-LSCE method are presented in Table 3, obtaining the following modal parameters: (i) identification of p2, Q = 1,53 Hz and Z = 19,3 %; and (ii) identification of p4, Q = 2,18 Hz; and Z = 14,1 %.
Based on the railway criteria for on motion security and passengers comfort, it is possible to perform an evaluation of the vehicle. The following relations are found: Q g «, Z >> ZlimJ, y Z > Zlim2; therefore, it's valid to affirm that the vehicle satisfies the requirements of the international standards for the two (2) criteria studied in the present work regarding the oscillation shapes P2, y P4.
Once the set of signal of one car have been obtained Zq*, Z+, y Z°p and the data processed through OMA-LSCE, the following identification of modal parameters is obtained: natural frequency: Q = 1,53 Hz, for p2 and Q = 2,18 Hz, for p4.
Starting from the identified modal parameters and based on the series of regressive models obtained for the vehicle, the probable state of the damper is located e,
for each one of the hypothesis Zi. The regressive models in the identification of the modal shape p2 are:
for Z1, e(Q) = 9,577Q -
- 13,950 .'. e(1.53) = 0,70,
for Z2, e (Q) = 13,161Q -
- 19,5 67 .'. e(1.53) = 0,57,
for Z3, e (Q) = 13,958Q -
- 20,£>73 .'. e(1.53) = 0,48;
In the modal shape p4 are:
for Z1, e(Q) = 8.761Q -
- 18.326 .'. e(2.18) = 0,77,
for Z2, e(Q) = 23.326Q -
- 50.240 .'. e(2.18) = 0,61,
for Z3, e(Q) = 25.431Q -
- 54.825. e(2.18) = 0,61.
The e values from the hypothesis Zt are tabulated (Table 4). The mean of the technical state is calculated, X. as well as their cor' 1
responding standard deviation <7, the later will be the criteria that defines the valid hypothesis Zt. The Z, that presents the least standard deviation is the one that adapts to the obtained dynamic characteristics of the system and therefore the value Xi of such hypothesis must be the technical state of the damper e, meaning Zt = Xi(e)Vmin(ffi).
The technical state of the damper is e = X"(e) + 2a with 95,45 % of confidence
factor. For the given case study, the value of the technical state is e = 0,716^0,1 in the four secondary vertical dampers (hypothesis Z1) with a certainty of 95,45 %. This means that the dampers are degraded 29 %.
9. Conclusions
The OMA-LSCE method is a suitable tool to establish the properties of the system components necessary for the evaluation of the vehicle according to the UIC, ISO and BS standards. Additionally, the formulated methodology is valid for establishing the technical state of the dampers e;, through dynamic signals obtained in the commercial service of the system.
A methodology has been proposed and applied for the evaluation of the technical state of the dampers e and the identification of hypothesis of suspension elements deterioration Zi through measurements of variables under operating conditions of the vehicle.
The vertical dampers of the secondary suspension stage present a direct influence on the dynamic behavior of the modal shapes p2 and p4, therefore the different technical states of the damper e. can be tested and estimated through the dynamic registration in the carbox. This means that from the sensors installed in the car-box that register appropriately the natural frequency Q it is possible to determine the variation of the damping function of dampers, in this way it is possible to infer their degradation or failure.
Table 4. Evaluation of the technical state e, in order to Z,.
Modal shape, tp . Modal parameter Damper technical state, e |
Description Value Zi Z2 Z5
P2 Q, Hz l.S3 0,70 0,57 0,48
2.90 0,70 0,58 0,42
PA Q, Hz 2.l8 0,77 0,6l 0,6l
2.20 0,67 0,75 0,72
Mean value, X. ' 1 0,7l 0,63 0,56
Standard deviation, a. ' 1 0,04 0,08 0,l4
The dynamic parameters, 5 and f, have a high dependence degree over the secondary suspension dampers <5..
Three (3) suitable sections track have been identified for the computation of the OMA-LSCE method through experimental registrations of the system in commercial operation, under an experimental model of three (3) measurement points, Z*, Z+, y Z°q] obtained from a set of sensors. The signals from the sensors can be used to identify the dynamic parameters, Q and f, in a vehicle with load condition AW0.
The experimental recording under load condition AW0 can be used to determine the technical state of the system components £. with the OMA-LSCE method. The 1
OMA-LSCE method it is not efficient for the identification of the technical state of the secondary dampers of a vehicle with load condition AW2 given that the perturbation-response relation is not enough to obtain a clear identification of the modal shapes ¡p..
The secondary suspension dampers have a high degree of dependence over the on motion security and passengers comfort criteria. International standards for railway vehicles define the range for deficient frequency ( = [8, 10] Hz. The human body is sensible to vertical accelerations; frequencies ( ~ 10 Hz cause excessive oscillations on p2, generating significant deficiency of comfort. Comparing with the values obtained for the analyzed vibration modes pr and under the different technical states of the damper £., the natural frequency is Q < 2 Hz. Therefore, the degradation of the damping function of the suspension elements does not incur per se in violation to the railway standards. □
References
1. Ewins D. J. Modal Testing: Theory, Prac-
tice and Application, Mechanical Engi-
neering Research Studies Engineering
Dynamics Series, Research Studies
Press, Second Ed, Hertfordshire, UK,
2000.
2. He J., Fu Z. F. Modal Analysis, Butter-
worth- Heinemann, UK, 2001.
3. Genta G. Vibration Dynamics and Con-
trol, Springer Science Business, Torino,
Italy, 2009.
4. Castañeda L. F., Zóttowski B. Estudio de
Explotación de vehículos Ferroviarios,
Fondo Editorial Universidad EAFIT, Co-
lombia, 2009.
5. Nagayama T., Abe M., Fujino Y., Ike-
da K. Structural Identification of a Non-
proportionally Damped System and its Application to a Full-Scale Suspension Bridge, Int J. Struct. Eng. 131. 2005. P. 1536-1545.
6. Jacobsen N. J., Thorhauge O. Data Acquisition Systems for Operational Modal Analysis, IOMAC - 3rd Int. Operational Modal Analysis Conf., Narum, Denmark, 2009.
7. Moller N. B., Herlufsen H., Gade S. Modal Parameters from a Wind Turbine Wing by Operational Modal Analysis', 32nd Int. Congress and Exposition on Noise Control Engineering, Seogwipo, Korea, 2003.
8. Yousaf E. Output Only Modal Analysis, MSc thesis, Mechanical Engineering Dep., Blekinge Inst. of Technology, Karlskrona, Sweden, 2007.
9. Jacobsen N. J., Andersen P., Brinck-er R. Using Enhanced Frequency Domain Decomposition as a Robust Technique to Harmonic Excitation in Operational Modal Analysis', ISMA - Int. Conf. on Noise and Vibration Engineering, Leu-ven, Belgium. 2006.
10.10. Uhl T., Kurowski P. Vioma User's Guide, University of Mining and Metallurgy in Kraków Press, Poland, 2002.
11. Goda K., Goodall R. Fault-Detection-and-Isolation System for a Railway Vehicle Bogie, in The Dynamics of Vehicles on Roads and on Tracks / ed. M. Abe. Taylor & Francis. Vol. 41. UK, 2003. P. 468-476.
12. Uhl T. The Inverse Identification Problem and its Technical Application, Springer-Verlag, 2006.
13. GEMI Research Group, Análisis Modal en Vehículos Ferroviarios Sistema de Transporte Metro de Medellín, Final Tech. Rep. Mechanical Project Research. EAFIT Univ., Colombia, 2010.
14. UTB, Mediciones Experimentales de Esfuerzos Dinámicos en Marcha. Metro de Medellín, Tech. Rep. Univ. Technshe Berlin, Colombia. 1997.
15. Martinod R. M., Castañeda L. F., Gallego J. A. Dynamic Analysis to Evaluate Stability, Safety and Confort in Railway Vehicles, ISPE - 23rd Int. Conf. on CAD/ CAM Robotics and Factories of the Future, Colombia, 2007. P. 403-408.
16. Zeng J., Wu P. Stability Analysis of High Speed Vehicles, JSME Int. J., Ser. C. Vol 47, N 2. Rec. 13th January, 2004.
17. Moelle D., Gasch R. Nonlinear Bogie Hunting, Proc. of 7th IAVSD Symp., The Dynamics of Vehicle on Road and Track / ed. Swets & Zeitlinger. UK, 1982.
18. Goodall R. M., Iwnicki S. Non-Linear Dynamic techniques v. Equivalent Conicity
Methods for Rail Vehicle Stability Assessment, in The Dynamics of Vehicles on Roads and on Tracks / ed. M. Abe. Taylor & Francis. Vol. 41. UK, 2003. P. 791-799.
19. Wickens A. H., Fundamentals of Rail Vehicle Dynamics. Guidance and Stability, Swets & Zeitlinger, USA, 2003.
20. CEN EN 14363, Railway Applications -Testing for Acceptance of Running Characteristics of Railway Vehicles - Testing of Running Behaviour and Stationary Test, Comité Européen de Normalisation, Brussels, 2005.
21. Dukkipati R. V. Lateral stability analysis of a railway truck on roller rig. Mechanism and Machine Theory 36, School of Engineering. Fairfield University. Fairfield, CT 06430, U.S.A. Rec. 3 June 1999, accepted 20 January 2000.
22. UIC 518, Testing and Approval of Railway Vehicle from the Point of View of Their Dynamic Behaviour - Safety - Track Fatigue - Ride Quality, Int. Union of Railways, Third Ed, France, 2005
23. Possel R., Beautefoy J., Matsudaira T. Papers Awarded Prizes in the Competition Sponsored, Office of Res. and Experiment (ORE) of Int. Union of Railways (UIC), ORE-Report RP2/SVA-C9, 1960.
24. BS 6841, Measurement and Evaluation of Human Exposure to Whole-Body Mechanical Vibration and Repeated Shock, British Standard, U.K, 1987.
25. ISO 2631-1, Mechanical Vibration and Shock - Evaluations of Human Exposure to whole Body Vibration - Part 4: Guidelines for the Evaluation of the Effects of Vibration and Rotational Motion on Passengers and Crew Comfort of Fixed Guideway Transport Systems, Int. Standards Org., Genève, Switzerland, 2001.
26. De Silva C. W. Vibration, Fundamentals and Practice, Taylor & Francis, Second Ed, USA, 2007.
27. Shabana A. A., Zaazaa K. E., Sugiya-ma H. Railroad Vehicle Dynamics a Computational Approach, CRC Press, Taylor & Francis Group, U.S.A, 3:978-1-42004581-9, 2008.
28. E.L. Grant, Statistical Quality Control, McGraw Hill, Third Ed, USA, 1964.
29. Chauhan S., Hansen M. H., Tcherniak D. Application of Operational Modal Analysis and Blind Source Separation/ Independent Component Analysis Techniques to Wind Turbines, Bruel & Kjœr Sound & Vibration Measurement, Denmark, 2009.
Appendix 1. Notation
Abbreviations, acronyms, coefficients and constants
AR Auto-regression model.
AW0 Anhängerladung Wägestück 0
(load weight 0; empty car).
DC Direct Current.
DoF Degrees of Freedom.
EMA Experimental Modal Analysis.
FFT Fast Fourier Transform.
FRF Frequency Response Function.
FT Fourier Transform.
IRF Impulse Response Function.
LSCE Least-Squares Complex Exponential.
MAN Maschinenfabrik Augsburg-Nüernberg.
MDoF Multiple Degrees of Freedom.
MIMO Multiple Input Multiple Output.
ODS Operating Deflection Shapes.
OMA Operational Modal Analysis.
FT Fourier Transform.
RD Random Decrement process.
A. rth modal constant.
r ij
hk kth IRF.
кД R2
VF
s
_r
X. i = 1, 2, 3
I ' '
V z.
Z*, Z+„ Z°,
ql ql ql
Q
a (w) ßkk = 0,1, ..., 2 <У = 1, .. 4
£
e. i=1,...,10
i ' '
fr
a. i = 1,2,3
i
Q
I
Zi = 1,2,3
Time interval.
Coefficient of determination.
Coefficient of correlation.
rth complex quantity root.
ith mean
Vehicle speed.
Conjugacy of the roots sr.
Vertical acceleration at carbody, leader bogie
and attack axlebox, respectively.
Natural frequency.
Receptance matrix
fcth real coefficient.
/th secondary vertical damper.
Error value.
ith damper technical state. rth mode shape. ith standard deviation Oscillation frequency. Damping rate. ith hyphotesis.
Appendix 2. Vehicle parameters
Element Quantity Value Unit
Mass I
Carbody 1 24486.36 kg
Bogie frame 2 6102.44 kg
Motor 4 5551 kg
Electromagnetic brake 4 704 kg
Traction link 2 - kg
Axle-wheel set 4 7083,52 kg
Stiffness
Lineal 1 kA = 1,00 kN/mm
Non-lineal 20 - kN/mm
k = 2,16 x ' kN/mm
Shear 8 k = 2,16 y ' kN/mm
k = 12,16 z ' kN/mm
Axi-directional 2 0,08 kNmm/s
Air stiffness 4 k = 4,52 kN/mm
k = 1,04 y kN/mm
Non-lineal damper 13 - kNmm/s
k = 67,32 kN/mm
k = 36E-5 y kN/mm
Busing 36 k = 72E-6 z kN/mm
k„ = 36E-5 MNm/rad
k, = 2E-6 MNm/rad
k = 1E-5 f MNm/rad
DOF 120 - -
Conicity non-lineal - -
Appendix 3. Stabilization diagrams
a. Hyphotesis Z1 with £1; b. Hyphotesis Z1 with £9; c Hyphotesis Z2 with £1; d. Hyphotesis Z2 with £9; e. Hyphotesis Z3 with £1; f. Hyphotesis Z3 with £9.
