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22 QR of RTRI, Vol. 62, No. 1, Feb. 2021
Tadao TAKIGAMIYuki AKIYAMA
Parameter Determination Method of Three-dimensional Analytical Models for Elastic Vibration of Railway Vehicle Carbodies
Ken-ichiro AIDAVehicle Noise and Vibration Laboratory, Vehicle Structure Technology Division
Reducing elastic vibration of railway vehicle carbodies is required to improve ride com-fort. A high-accuracy numerical analysis model is required to study effective vibration reduc-tion methods. This paper describes a new numerical analysis model, and a method for deter-mining parameters for the proposed model by using particle swarm optimization. After the application of the proposed method to a Shinkansen-type test vehicle, the authors created an analytical model with a maximum natural frequency difference between measured and cal-culated results of the targeted six elastic vibration modes within 0.86%, which indicates the effectiveness of the proposed model and parameter determination method.
Keywords: carbody, elastic vibration, analytical model, parameter determination
1. Introduction
Carbody elastic vibrations caused by three-dimension-al (3D) elastic deformation have been shown to negatively affect passenger comfort, as the human body is sensitive to the frequencies resulting from vertical vibration. To reduce these vibrations and improve ride comfort, several analyti-cal models studying the elastic vibration of railway vehicle carbodies have been proposed. The beam model, which ex-presses the carbody as a uniform elastic beam [1], has been widely used to study vibration reduction methods. How-ever, this simplistic model cannot express the complex 3D elastic deformation of a carbody [2, 3]. Therefore, numeri-cal analysis models, such as a finite element (FE) model or a box-type model [4], have also been used.
While a FE model accurately represents the carbody structure, it requires detailed information, such as struc-tural drawings, and has high degrees of freedom (often ≥ 500,000 for an entire vehicle). When appropriate param-eters are set, accurate results can be obtained; however, the determination of the large number of input parameters is time consuming and costly. Therefore, to optimize com-ponent specifications requiring iterative calculations, it is unsuitable to use a FE model.
A box-type model [4] treats the carbody as a simple box-type structure in which each panel is modeled as an elastic plate or combination of elastic beams connected by springs, without considering the detailed structure, equip-ment, interior panels, etc. Compared with a FE model, a box-type model has a simpler structure and fewer degrees of freedom (1,000 or less), which makes it possible to build models and perform numerical calculations at lower cost. However, certain parameters such as plate and beam rigid-ity or artificial spring coefficients, must be determined by the model user. Furthermore, a box-type model does not consider in-plane (longitudinal and lateral) deformation of the floor and roof; therefore, lateral-vertical coupled vibra-tion modes are not accurately represented.
Therefore, this work proposes an extended box-type model as a new analytical model composed of the three-dimensional elastic bodies which can express vibration modes more appropriately than the original box type mod-
el. Moreover, an efficient method is proposed to update the model parameters automatically by using an optimization technique.
2. Extended box-type model
2.1 Carbody model
Figure 1 (a) shows the carbody model proposed in this study. In this model, the roof, floor, side and end panels are modeled using anisotropic three-dimensional elastic bodies of the continuous system. The movement of the six degrees of freedom (longitudinal, lateral, vertical, roll, pitch and yaw directions), in-plane deformation and out-of-plane deformation are considered for each component. In addi-tion, these components are connected with each other by artificial translational springs between them as shown in Fig. 1 (b). In order to adjust the coupling rigidity between each component about the translation and the rotation mo-tions, two parallel springs are introduced. The rigidities of these springs are decided by the parameter determination method described later. In this model, the elastic bodies are modeled as a continuous system in which the displacement function is expressed by a combination of the polynomials. Thereby, this model has a smaller number of degrees of freedom (3500 or less) than in a detailed FE model.
2.2 Bogie model
Figure 1 (c) shows the vehicle model including bogies. The bogie frames and wheelsets are modeled as rigid bod-ies. They have six degrees of freedom of the longitudinal, lateral, vertical, roll, pitch and yaw directions. The degree of freedom of the wheelset is constrained appropriately depending on the condition of the excitation. In this model, the wheelsets and the bogie frames are connected by the axle springs and the axle box supporting devices. The bogie frames and the carbody are combined by the air springs, the traction devices and the yaw dampers.
PAPER
23QR of RTRI, Vol. 62, No. 1, Feb. 2021
2.3 Equations of motion
This section describes the outline of the process to obtain the equations of motion; the detailed description is available in the literature [5]. The first step is to calculate the total of the kinetic energy TT, the strain energy UT, the potential energy stored in the springs VT and the dissipa-tion function DT. Then, these values are substituted into Lagrange’s equation:
d
d
T T T
T T
tT T U
V D
��
�
��
�
�� �
��
�
��
�
�� �
��
�
��
�
��
���
�
��
�
�� �
��
q q q
q q��
��
�
�� � 0
(1)
where q denotes the generalized co-ordinate vector and �
represents the time differential (d/dt). Thus, the following
Fig. 1 The extended box type model including bogies
equation of motion can be obtained:
Tq Dq U V q Q � � � �( ) (2)
where Q denotes a generalized force vector corresponding to q and ��
represents the second time differential (d2/dt2). Since the Symbolic Math Toolbox of MATLAB is used to perform these processes, the equation of motion is derived automatically by inputting each energy as the symbolic ex-pressions.
3. Vibration characteristics of the target vehicle
3.1 Shinkansen type test vehicle
A Shinkansen type test vehicle shown in Fig. 2 is used as the target vehicle in this study. This carbody vehicle shell is similar to typical Shinkansen vehicles. The vehicle is equipped with fundamental interior panels and seats. However, since it did not have underfloor equipment, large iron blocks were hung under the floor.
3.2 Stationary excitation tests
In order to understand the vibration characteristics of the target vehicle, a stationary excitation test was con-ducted. In this test, an electro dynamic exciter set on the ground was used to excite the carbody of the vehicle placed on the rail, as shown in Fig. 3. A band random wave with a uniform frequency component in the range of 5-50 Hz was used as an excitation signal. The excitation force and vertical acceleration at the measurement points shown in Fig. 4 were measured. The excitation force was measured by a load cell attached between the carbody and driving rod, as shown in Fig. 3. The measured data were recorded at a sampling frequency of 200 Hz; the cut-off frequency of the anti-alias filters (low-pass filter) was set to 80 Hz.
The major vibration modes of the target vehicle identi-fied by experimental modal analysis from the measured results are shown in Fig. 5, in which the numerical value with Hz represents the natural frequency of each vibra-tion mode. This paper creates the extended box-type model that can express these natural frequencies and deforma-tion shapes.
Fig. 2 Shinkansen type test vehicle
roll, pitch and yaw directions. The degree of freedom of the wheelset is constrained appropriately depending on the condition of the excitation. In this model, the wheelsets and the bogie frames are connected by the axle springs and the axle box supporting devices. The bogie frames and the carbody are combined by the air springs, the traction devices and the yaw dampers.
（a） Panels modeled as anisotropic three-
dimensional elastic bodies
（b） Cross section of connections between panel
（c） Vehicle model
Fig. 1 The extended box type model including bogies
2.3 Equations of motion
This section describes the outline of the process to obtain the equations of motion; the detailed description is available in the literature [5]. The first step is to calculate the total of the kinetic energy TT, the strain energy UT, the potential energy stored in the springs VT and the dissipation function DT. Then, these values are substituted into Lagrange's equation:
dd𝑡𝑡 �
𝜕𝜕𝑇𝑇�𝜕𝜕�̇�𝐪 � − �𝜕𝜕𝑇𝑇�𝜕𝜕𝐪𝐪 � + �𝜕𝜕𝑈𝑈�𝜕𝜕𝐪𝐪 �
+ �𝜕𝜕𝑉𝑉�𝜕𝜕𝐪𝐪 � + �𝜕𝜕𝐷𝐷�𝜕𝜕�̇�𝐪 � = 0 (1)
where q denotes the generalized co-ordinate vector and � ̇ � represents the time differential (d/d𝑡𝑡). Thus, the following equation of motion can be obtained:
𝐓𝐓�̈�𝐪 + 𝐃𝐃�̇�𝐪 + (𝐔𝐔 + 𝐕𝐕)𝐪𝐪 = 𝐐𝐐 (2)
where Q denotes a generalized force vector corresponding to q and � ̈ � represents the second time differential ( d�/d𝑡𝑡� ). Since the Symbolic Math Toolbox of MATLAB is used to perform these processes, the equation of motion is derived automatically by inputting each energy as the symbolic expressions. 3. Vibration characteristics of the target vehicle
3.1 Shinkansen type test vehicle
A Shinkansen type test vehicle shown in Fig. 2 is used as the target vehicle in this study. This carbody vehicle shell is similar to typical Shinkansen vehicles. The vehicle is equipped with fundamental interior panels and seats. However, since it did not have underfloor equipment, large iron blocks were hung under the floor.
3.2 Stationary excitation tests
In order to understand the vibration characteristics of the target vehicle, a stationary excitation test was conducted. In this test, an electro dynamic exciter set on the ground was used to excite the carbody of the vehicle placed on the rail, as shown in Fig. 3. A band random wave with a uniform frequency component in the range of 5–50 Hz was used as an excitation signal. The excitation force and vertical acceleration at the measurement points shown in Fig. 4 were measured. The excitation force was measured by a load cell attached between the carbody and driving rod, as shown in Fig. 3. The measured data were recorded at a sampling frequency of 200 Hz; the cut-off frequency of the anti-alias filters (low-pass filter) was set to 80 Hz.
The major vibration modes of the target vehicle identified by experimental modal analysis from the measured results are shown in Fig. 5, in which the numerical value with Hz represents the natural frequency of each vibration mode. This paper creates the extended box-type model that can express these natural frequencies and deformation shapes.
O
xy
zRoof
End panel
Rear floor
Front floor
Middle floor
Side panel
End panel
Side panel
y
Floor
Roof
Side panel
z
Floor
Roof
End panelx
z
Side panel
Side panel
End panelx
y
Rear floor
x
z
Front floor
Middle floor
Yaw damperTraction linkAir springAxle box supporting deviceAxle spring and damper
Wheelset
z
x
Bogie
Carbody
（a） Panels modeled as anisotropic three-dimensional elastic bodies
24 QR of RTRI, Vol. 62, No. 1, Feb. 2021
Fig. 3 The photo of the stationary excitation test
4. Parameter determination method
4.1 Outline
In the extended box-type model, measured values were used to determine the mass of the carbody, bogie frames and wheelsets. The spring coefficients and damping coef-ficients of the bogie were identified by the parameter esti-mation method proposed in a previous study [6]. However, there is no established method to determine the rigidity of the three-dimensional elastic bodies constructing the car-body and to determine the spring coefficient of the artificial springs which connect each panel of the carbody. To deter-mine these unknown parameters, the authors proposed a parameter determination method using particle swarm optimization (PSO) [7]. PSO is a numerical optimization technique modeled after the behavior of swarming animals, such as birds flocking or fish schooling, in which particles search for their best position by moving around in space. Here, the co-ordinates of the particle positions correspond to model parameter values and a combination of param-eters to minimize the evaluation function is sought.
4.2 Flow of the parameter determination
The flowchart in Fig. 6 presents the parameter deter-mination procedure. First, the base model is created and the first swarm, or group of particles, are generated in ran-dom positions. Next, the particle positions are converted into parameters of the extended box-type model and the natural frequencies and eigenvectors are calculated. Then, the evaluation function is obtained by:
EFfn fn
fnii
ii ii
ii
��
�max, ,
,
mea cal
mea
100 (3)
where fn denotes natural frequency and the subscripts mea, cal, and ii denote the measured value, calculated value, and the ii-th natural mode, respectively. This equation describes the evaluation function as the maximum value of the error about the natural frequencies of the targeted modes.
To obtain this evaluation function, the vibration modes that have the maximum correlation between the calculated and measured results of the extended box-type model must be selected. Thus, the modal assurance criterion (MAC) is used to select the vibration mode and is calculated by:
MAC ( , )( )
( ) ( )mea calmea
Hcal
meaH
mea calH
cal
�
2
'φφ
φ φφ
φφ φ
(4)
where φ is the eigenvector and the superscript H repre-sents the complex conjugate transpose. A MAC value close to 1 corresponds to a good correlation between the two modal vectors; a value of shows no correlation. The calcula-tion mode having the maximum correlation is automati-cally selected. Then, update of parameters and iteration process follow. Thus, model parameters allowing for good agreement in mode shapes and natural frequencies can be determined.
4.3 Validation of the proposed methodology
Following the flowchart shown in Fig. 6, the authors determined the parameters of the extended box-type
Fig. 4 Measurement points for vertical acceleration
Fig. 5 Results of the experimental modal analysis(Numerical values with Hz mean the naturalfrequency corresponding to the vibration mode)
(a) 9.88 Hz (b) 11.18 Hz
(c) 13.20 Hz (d) 15.79 Hz
(e) 16.23 Hz (f) 18.50 Hz
Fig. 5 Results of the experimental modal analysis
(Numerical values with Hz mean the natural frequency corresponding to the vibration mode)
4.2 Flow of the parameter determination
The flowchart in Fig. 6 presents the parameter determination procedure. First, the base model
is created and the first swarm, or group of particles, are generated in random positions. Next, the
particle positions are converted into parameters of the extended box-type model and the natural
frequencies and eigenvectors are calculated. Then, the evaluation function is obtained by:
𝐸𝐸𝐸𝐸 = max𝑖𝑖𝑖𝑖
|𝑓𝑓𝑓𝑓mea,𝑖𝑖𝑖𝑖 − 𝑓𝑓𝑓𝑓cal,𝑖𝑖𝑖𝑖𝑓𝑓𝑓𝑓mea,𝑖𝑖𝑖𝑖
| × 100 (3)
where 𝑓𝑓𝑓𝑓 denotes natural frequency and the subscripts mea, cal, and 𝑖𝑖𝑖𝑖 denote the measured
value, calculated value, and the 𝑖𝑖𝑖𝑖 -th natural mode, respectively. This equation describes the
evaluation function as the maximum value of the error about the natural frequencies of the targeted
modes.
Roof
Floor
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model by setting the target modes as the six measured modes shown in Fig. 5. The number of particles in the swarm was set to 1000 and PSO was performed until ei-ther the evaluation function reached 0.5% or less or 1000 iterations were performed. Figure 7 presents the rela-tionship between the number of iterations and the evalu-ation function value. Although the evaluation function value did decrease with increasing iterations, the evalu-ation function did not reach 0.5% before 1000 iterations had been performed.
Figure 8 shows the results of modal analysis of the proposed model using the determined parameters, in which the numerical values with Hz denote the natural frequency and the numerical values with % indicate the error between the measured and calculated results. These results reveal that the mode shapes are in good agreement and the natural frequencies errors are all within 0.86%. This indicates that, the proposed parameter determination method is effective.
The time needed to calculate the evaluation function with each parameter change was about 7 seconds per par-ticle, which is corresponding to the parameter set, with a commercial PC (Intel Core i9 7980XE CPU and 64 GB memory). In case the position of 1000 particles is updated for 1000 iterations, the calculation repeats 1 million times in total and takes one week by performing parallel com-puting of the Toolbox of MATLAB. In this calculation, a sufficiently large number of particles and iterations were specified. These settings can be changed according to the accuracy needed and the time allotted.
A frequency response analysis of the stationary excita-tion test was performed. Figure 9 shows measured results of the Shinkansen-type test vehicle and calculated results
Fig. 6 Flowchart for determining parameters
of the extended box-type model about the frequency re-sponse function (FRF) between the excitation force and the vertical acceleration of the carbody floor. From these results, good agreements were observed between the mea-sured and calculated FRF gain about the peak values and frequencies. Although only natural frequencies were used for parameter determination, the proposed model provides good agreement between measured and calculated results in terms of both the eigenvalues and the frequency re-sponse analysis obtained.
Fig. 7 Evaluation function value vs. iterations for PSO
Fig. 8 Results of modal analysis of extended box-type model (Numerical values with Hz denote natural frequency,
and values with % denote the error between the cal-culation result and measured results shown in Fig. 5)
Fig. 7 Evaluation function value vs. iterations for PSO
(a) 9.91 Hz
(0.30%) (b) 11.17 Hz
(0.09%)
(c) 13.21 Hz
(0.08%) (d) 15.71 Hz
(0.51%)
(e) 16.37 Hz
(0.86%) (f) 18.43 Hz
(0.38%) Fig. 8 Results of modal analysis of extended box-type model
(Numerical values with Hz denote natural frequency, and values with % denote the error between the calculation result and measured results shown in Fig. 5)
Mode #13(13.2073Hz)
26 QR of RTRI, Vol. 62, No. 1, Feb. 2021
5. Conclusions
This paper proposes an extended box-type model to express the 3D elastic vibration of a railway vehicle car-body, and a method for determining the parameters for the model.
The proposed model expresses the lateral-vertical cou-pled vibration modes that are not represented in the origi-nal box-type model by considering the carbody as a combi-nation of the 3D elastic bodies. Thus, the model provides a more accurate numerical analysis than the conventional box-type model at a lower computational cost than the FE model.
A parameter determination method using PSO was also proposed in which parameters are selected based on their ability to minimize natural frequency errors between measured and calculated results. Then, the parameters of the extended box-type model of a Shinkansen-type test vehicle were determined using this method. The natural
frequency errors of the target vibration modes were within 0.86% and the mode shapes agreed well between measured and calculated results. The measured and calculated FRFs were also in good agreement. Overall, the results indicate the effectiveness of the proposed model and parameter de-termination method.
In future work, the authors will use the proposed extended box-type model for studying multimodal vibra-tion reduction, such as use of dynamic vibration absorb-ers (DVAs) or for the control system design of active mass dampers (AMDs).
References
[1] Suzuki, Y. and Akutsu, K., “Theoretical Analysis of Flexural Vibration of Car Body,” Quarterly Report of Railway Technical Research Institute, Vol. 31, No. 1, pp. 42-48, 1990.
Fig. 9 Frequency response function gain from the input force to the acceleration on the floor of the measurement points in Fig. 4
27QR of RTRI, Vol. 62, No. 1, Feb. 2021
[2] Takigami, T. and Tomioka, T., “Modal Vibration Analy-sis of Recent Railway Vehicles,” Proceedings of the 12th Asia Pacific Vibration Conference (APVC2007), Sapporo, Japan, August 6-9, 2007, No.62.
[3] Tomioka, T., Takigami, T. and Aida, K., “Modal Vibra-tion Characteristics of Flexural Vibrations in Railway Vehicle Carbodies,” Proceedings of 1st intern. confer-ence on Railway Technology (Railways2012), Las Pal-mas de Gran Canaria, Spain, April 18-20, 2012, No.33.
[4] Tomioka, T., Takigami, T. and Suzuki, Y., “Numerical analysis of three-dimensional flexural vibration of rail-way vehicle car body,” Vehicle System Dynamics, Vol. 44, Supplement, pp. 272-285, 2006.
[5] Akiyama, Y., Tomioka, T., Takigami, T., Aida, K. and Kamada, T., “A three-dimensional analytical model and parameter determination method of the elastic vibration of a railway vehicle carbody,” Vehicle System Dynamics, Vol. 58, No.4, pp. 545-568, 2020.
[6] Ishida, H. and Miyamoto, M., “Parameter Estimation of Railway Vehicle by Experimental Results on Roll-ing Stock Test Stand,” Quarterly Report of Railway Technical Research Institute, Vol. 32, No. 2, pp. 81-89, 1991.
[7] Kennedy, J. and Eberhart, R., “Particle Swarm Opti-mization,” In Proceedings of IEEE International Con-ference on Neural Networks, pp. 1942-1948, 1995.
Authors
Yuki AKIYAMA, Dr. Eng.Assistant Senior Researcher, Vehicle Noiseand Vibration Laboratory, Vehicle StructureTechnology DivisionResearch Areas: Carbody Vibration Control,Vibration Analysis, Ride Comfort Evaluation
Tadao TAKIGAMI, Dr. Eng.Senior Chief Researcher, Head of VehicleNoise and Vibration Laboratory, VehicleStructure Technology DivisionResearch Areas: Flexural Vibration of Carbody,Vibration Analysis, Ride Comfort Evaluation
Ken-ichiro AIDAAssistant Senior Researcher, Vehicle Noiseand Vibration Laboratory, Vehicle StructureTechnology DivisionResearch Areas: Carbody Structure, RideComfort Evaluation
