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Abstract

This paper investigates the modal parameter identification technique utilizing a drop impact load. A 12-DOF mathematical model of a Railway Vehicle System (RWVS) was constructed, followed by theoretical calculations, simulation analysis, and tests with four different drop impact loads. The responses to these loads were inspected through FFT, and the modal frequencies of the RWVS were determined by the Peak Picking (P-P) method. The simulation demonstrated that four types of drop impact loads can activate the bounce, pitch, and roll modes of the RWVS. The theoretical calculation and simulation analysis revealed that the maximum error of modal frequency identification is no greater than 6.5%. The experimental results verified that the drop impact excitation method can be used to identify the modal parameters of a complex railway vehicle system, which is highly beneficial for the design and verification of the vehicle structure.

Keywords

Railway vehicles drop impact modal parameter energy of modes modal contribution

Introduction

The identification of the modal parameters of the RWVS is essential to guide design, assess dynamic performance, and optimize structural parameters.^1–3 The implementation of vehicle lightweight technology leads to an increase in the elastic vibration of the carbody, resulting in a decrease in ride comfort, a decrease in the fatigue reliability of the mechanical structure, and an increase in the coupling vibration between internal vehicle system components.^4,5 Additionally, the application of vehicle power dispersion technology has caused the frequency of the vertical bending mode, which has a considerable impact on vehicle vibration, to be reduced to 7–12 Hz. That is, in the sensitive frequency range of the human body, the ride experience of passengers is further deteriorated.^6–8 The identification of the mode parameters of the RWVS is essential for improving the vehicle’s ride comfort, as well as the design and evaluation of vehicle structure.^9–12 The modal parameters of RWVS have been studied for many years, because they affect the dynamic performance of vehicles. Natural frequency, damping ratio, and mode vibration type can be estimated through modal parameter identification methods. Southwest Jiaotong University adopted a multi-channel excitation test system and used the impedance matrix method to conduct a modal test on an ordinary sleeping car. In this case, the input excitation was the axle box vibration signal measured in actual line operation. The State Key Laboratory of Traction Power established a rolling vibration test-bed for locomotives and vehicles, so as to realize the modal testing of vehicles under track excitation. Jin et al. simulated the ambient excitation with white noise excitation on the vibration test-bed, and used the multi-reference point LSCE method for modal parameters identification. This shows that the multi-reference point method based on environmental excitation has the same high identification accuracy as the traditional modal identification method.¹³ To solve the problem of low-frequency sloshing with the SW-160 bogie, Fu et al. identified the rigid-body modal parameters of the car body using the Random Subspace Method.¹⁴ Cheng et al. carried out multi-point excitation modal tests on the aluminum alloy car body of a high-speed train, and identified several main natural frequencies and vibration modes of the aluminum alloy car body.¹⁵ Zhang Tao identified the modal parameters of the gearbox of the railway vehicle through a bench test combined with a random load.¹⁶ Wang Wei obtained the working modes of the CRH2G high-speed train under the excitation environment of Lanxin Line.¹⁷ Li Wei employed the PolyMax method to determine the modal parameters of the scale-proportional car body of the high-speed train through hammering excitation.¹⁸ Liu Jia et al. used the combination of hammering method and the least-squares complex frequency domain method to obtain the frequency response function between the excitation point and the response point of the aluminum plate of the train waveform, and identified the modal parameters.^19,20 In recent years, the random decrement method has been used to identify the modal parameters of the RWVS.²¹ He Li used the information from limited measuring points to fit the displacement response and modal shape of a car body, then the equations of side wall, roof, and bottom modes of railway vehicles were obtained.²²

It should be noted that the majority of modal parameter identification methods require complex testing equipment, and are expensive to perform, or can only identify a few modal parameters, thereby limiting their practical application. To address this issue, this work attempts to explore a modal parameter identification method for vehicle structures based on a drop impact load, which is conducive to the design and verification of railway vehicle structures. The Wilson-θ method was then applied to calculate the free-decay response of the railway vehicle when subjected to a drop excitation. The responses to these loads were inspected through FFT, and the modal frequencies of the RWVS were determined by the Peak Picking (P-P) method. The simulation results are then compared with the experiments to identify the effectiveness of the additional mass technique in identifying the physical parameters of a real railway vehicle.

Three-dimensional model of the railway vehicle system

The 3D 12-DOF model of the RWVS is illustrated in Figure 1, with the capital letters $R_{X}$ , $Z$ and $R_{y}$ indicating the rolling, vertical and pitching directions, respectively. The stiffness and damping coefficients are represented by k and c, and the vertical stimulation of the ith wheel in the left and right are denoted by $η_{l i}$ and $η_{r i}$ . The carbody, bogie, and equipment are symbolized by the superscripts c, b, and e, respectively.

Figure 1.

Three-dimensional model of the RWVS. (a) Side view of the RWVS. (b) Front view of the RWVS.

Information regarding all parameters can be found in Table 1. Motion equations for the different components of the RWVS are derived from dynamic equilibrium and formulated as follows:

Table 1.

Parameters of the RWVS.

Description	Symbol	Unit	Value
Mass of carbody	M _c	ton	41.75
Carbody inertia moments	$J_{y c}$ , $J_{x c}$	ton·m²	2080, 23.2
Mass of bogie	$M_{b}$	ton	3.04
Bogie inertia moments	$J_{y b}$ , $J_{x b}$	ton·m²	3.934, 1.58
Mass of equipment	M _e	ton	6
Equipment inertia moments	$J_{y e}$ , $J_{x e}$	ton·m²	5.061, 1.62
Secondary suspension dimensions	$a_{c}$ , $h_{1}$ , $h_{2}$ , $b_{2}$ , $l_{1}$ , $l_{2}$	m	8.75, 0.75, 0.42, 1.23, 3.5, 21.5
Primary suspension dimensions	$h_{3}$ , $b_{1}$ , $a_{b}$	m	0.2, 1, 1.25
Secondary suspension stiffness	$k_{2 x}$ , $k_{2 z}$ , $k_{2 y}$	kN·m⁻¹	193.6, 265, 176
Secondary suspension damping	$c_{2 x}$ , $c_{2 z}$ , $c_{2 y}$	kNs·m⁻¹	43.12, 15.3, 39.2
Primary suspension stiffness	$k_{1 x}$ , $k_{1 z}$ , $k_{1 y}$	kN·m⁻¹	649, 2350, 590
Primary suspension damping	$c_{1 x}$ , $c_{1 z}$ , $c_{1 y}$	kNs·m⁻¹	64.75, 6.54, 58.86
Equipment suspension stiffness	$k_{e x}$ , $k_{e z}$ , $k_{e y}$	kN·m⁻¹	150, 125, 137.5
Equipment suspension damping	$c_{e x}$ , $c_{e z}$ , $c_{e y}$	kNs·m⁻¹	2.16, 1.8, 1.98
Equipment suspension dimensions	$b_{e}$ , $a_{e}$ , $l_{e}$	m	1,0.75, 9.5

The dynamic equilibrium of the carbody in the vertical direction is expressed as follows²³ $M_{c} {\ddot{z}}_{c} + 4 c_{2 z} {\dot{z}}_{c} + 4 k_{2 z} z_{c} - 2 c_{2 z} ({\dot{z}}_{2 b} + {\dot{z}}_{1 b}) - 2 k_{2 z} (z_{2 b} + z_{1 b}) - 4 c_{e z} {\dot{z}}_{e} - 4 k_{e z} z_{e} = 0$ (1)

The dynamic equilibrium of the pitching motion can be described as a balance of forces $J_{y c} {\ddot{R}}_{y c} + 4 (a_{c}^{2} c_{2 z} + h_{1}^{2} c_{2 x} + h_{c}^{2} c_{e x}) {\dot{R}}_{y c} + 4 (a_{c}^{2} k_{2 z} + h_{1}^{2} k_{2 x} + h_{c}^{2} k_{e x}) R_{y c} - 2 a_{c} c_{2 z} ({\dot{z}}_{2 b} - {\dot{z}}_{1 b}) - 2 a_{c} k_{2 z} (z_{2 b} - z_{1 b}) - 4 (a_{e}^{2} k_{e z} + h_{c} h_{e} k_{e x}) R_{y e} - 4 (a_{e}^{2} c_{e z} + h_{c} h_{e} c_{e x}) {\dot{R}}_{y e} - 2 (L - 2 l_{e 1} + 2 a_{e}) k_{e z} z_{e} - 2 (L - 2 l_{e 1} + 2 a_{e}) c_{e z} {\dot{z}}_{e} + 2 h_{1} h_{2} k_{2 x} (R_{y b 2} + R_{y b 1}) + 2 h_{1} h_{2} c_{2 x} ({\dot{R}}_{y b 2} + {\dot{R}}_{y b 1}) = 0$ (2)and the rolling motion, it is written as follows $J_{x c} {\ddot{R}}_{x c} + 2 (h_{1} h_{2} c_{2 y} - b_{2}^{2} c_{2 z}) ({\dot{R}}_{x b 1} + {\dot{R}}_{x b 2}) + 4 (b_{2}^{2} c_{2 z} + h_{1}^{2} c_{2 y}) {\dot{R}}_{x c} + 4 (b_{2}^{2} k_{2 z} + h_{1}^{2} k_{2 y}) R_{x c} + 2 (h_{1} h_{2} k_{2 y} - b_{2}^{2} k_{2 z}) ({\dot{R}}_{x b 1} + {\dot{R}}_{x b 2}) + 4 (h_{e} h_{c} c_{e y} - b_{e}^{2} c_{e z}) {\dot{R}}_{x e} + 4 (h_{e} h_{c} k_{e y} - b_{e}^{2} k_{e z}) R_{x e} = 0$ (3)

For the ith bogie, the vertical motion dynamic equilibrium is written as follows $M_{b} {\ddot{z}}_{1,2 b} + (2 c_{2 z} + 4 c_{1 z}) {\dot{z}}_{1,2 b} + (2 k_{2 z} + 4 k_{1 z}) z_{1,2 b} - 2 c_{2 z} {\dot{z}}_{c} - 2 k_{2 z} z_{c} + {(- 1)}^{t} 2 a_{c} c_{2 z} {\dot{R}}_{y c} + {(- 1)}^{t} 2 a_{c} k_{2 z} R_{y c} - k_{1 z} (η_{l j} + η_{l (j + 1)} + η_{r j} + η_{r (j + 1)}) = 0, t = 1,2, j = 1,3$ (4)the pitching motion, it is expressed as follows $J_{y b} {\ddot{R}}_{y 1,2 b} + (2 h_{2}^{2} k_{2 x} + 4 h_{3}^{2} k_{1 x} + 4 a_{b}^{2} k_{1 z}) R_{y 1,2 b} + (2 h_{2}^{2} c_{2 x} + 4 h_{3}^{2} c_{1 x} + 4 a_{b}^{2} c_{1 z}) {\dot{R}}_{y 1,2 b} + 2 h_{1} h_{2} k_{2 x} R_{y c} + 2 h_{1} h_{2} c_{2 x} {\dot{R}}_{y c} - a_{b} k_{1 z} (η_{l j} + η_{i j} - η_{l (j + 1)} - η_{r (j + 1)}) = 0, j = 1,3$ (5)and for the rolling motion, it can be expressed as follows $J_{x b} {\ddot{R}}_{x 1,2 b} + (2 b_{2}^{2} c_{2 z} + 2 h_{2}^{2} c_{2 y} + 4 h_{3}^{2} c_{1 y} + 4 b_{1}^{2} c_{1 z}) {\dot{R}}_{x 1,2 b} + (2 b_{2}^{2} k_{2 z} + 2 h_{2}^{2} k_{2 y} + 4 h_{3}^{2} k_{1 y} + 4 b_{1}^{2} k_{1 z}) R_{x 1,2 b} + 2 (h_{1} h_{2} c_{2 y} - 2 b_{2}^{2} c_{2 z}) {\dot{R}}_{x c} + 2 (h_{1} h_{2} k_{2 y} - 2 b_{2}^{2} k_{2 z}) R_{x c} - b_{1}^{2} k_{1 z} (η_{l j} + η_{l (j + 1)} - η_{r j} - η_{r (j + 1)}) = 0, j = 1,3$ (6)and the rolling motion, it is written as follows $J_{x c} {\ddot{R}}_{x c} + 2 (h_{1} h_{2} c_{2 y} - b_{2}^{2} c_{2 z}) ({\dot{R}}_{x b 1} + {\dot{R}}_{x b 2}) + 4 (b_{2}^{2} c_{2 z} + h_{1}^{2} c_{2 y}) {\dot{R}}_{x c} + 4 (b_{2}^{2} k_{2 z} + h_{1}^{2} k_{2 y}) R_{x c} + 2 (h_{1} h_{2} k_{2 y} - b_{2}^{2} k_{2 z}) ({\dot{R}}_{x b 1} + {\dot{R}}_{x b 2}) + 4 (h_{e} h_{c} c_{e y} - b_{e}^{2} c_{e z}) {\dot{R}}_{x e} + 4 (h_{e} h_{c} k_{e y} - b_{e}^{2} k_{e z}) R_{x e} = 0$ (7)for the ith bogie, the vertical motion dynamic equilibrium is written as follows $M_{b} {\ddot{z}}_{1,2 b} + (2 c_{2 z} + 4 c_{1 z}) {\dot{z}}_{1,2 b} + (2 k_{2 z} + 4 k_{1 z}) z_{1,2 b} - 2 c_{2 z} {\dot{z}}_{c} - 2 k_{2 z} z_{c} + {(- 1)}^{t} 2 a_{c} c_{2 z} {\dot{R}}_{y c} + {(- 1)}^{t} 2 a_{c} k_{2 z} R_{y c} - k_{1 z} (η_{l j} + η_{l (j + 1)} + η_{r j} + η_{r (j + 1)}) = 0, t = 1,2, j = 1,3$ (8)the pitching motion, it is expressed as follows $J_{y b} {\ddot{R}}_{y 1,2 b} + (2 h_{2}^{2} k_{2 x} + 4 h_{3}^{2} k_{1 x} + 4 a_{b}^{2} k_{1 z}) R_{y 1,2 b} + (2 h_{2}^{2} c_{2 x} + 4 h_{3}^{2} c_{1 x} + 4 a_{b}^{2} c_{1 z}) {\dot{R}}_{y 1,2 b} + 2 h_{1} h_{2} k_{2 x} R_{y c} + 2 h_{1} h_{2} c_{2 x} {\dot{R}}_{y c} - a_{b} k_{1 z} (η_{l j} + η_{i j} - η_{l (j + 1)} - η_{r (j + 1)}) = 0, j = 1,3$ (9)and for the rolling motion, it can be expressed as follows $J_{x b} {\ddot{R}}_{x 1,2 b} + (2 b_{2}^{2} c_{2 z} + 2 h_{2}^{2} c_{2 y} + 4 h_{3}^{2} c_{1 y} + 4 b_{1}^{2} c_{1 z}) {\dot{R}}_{x 1,2 b} + (2 b_{2}^{2} k_{2 z} + 2 h_{2}^{2} k_{2 y} + 4 h_{3}^{2} k_{1 y} + 4 b_{1}^{2} k_{1 z}) R_{x 1,2 b} + 2 (h_{1} h_{2} c_{2 y} - 2 b_{2}^{2} c_{2 z}) {\dot{R}}_{x c} + 2 (h_{1} h_{2} k_{2 y} - 2 b_{2}^{2} k_{2 z}) R_{x c} - b_{1}^{2} k_{1 z} (η_{l j} + η_{l (j + 1)} - η_{r j} - η_{r (j + 1)}) = 0, j = 1,3$ (10)

For the equipment, the vertical motion dynamic equilibrium is expressed as follows $M_{e} {\ddot{z}}_{e} + 4 c_{e z} {\dot{z}}_{e} + 4 k_{e z} z_{e} - 4 c_{e z} {\dot{z}}_{c} - 4 k_{e z} z_{c} - 2 (L - 2 l_{e 1} + 2 a_{e}) k_{e z} R_{y c} - 2 (L - 2 l_{e 1} + 2 a_{e}) c_{e z} {\dot{R}}_{y c} = 0$ (11)the pitching motion, the dynamic equilibrium is $J_{y e} {\ddot{R}}_{y e} + (4 a_{e}^{2} k_{e z} + 4 h_{e}^{2} k_{e x}) R_{y e} + (4 a_{e}^{2} c_{e z} + 4 h_{e}^{2} c_{e x}) {\dot{R}}_{y e} + (4 h_{c} h_{e} c_{e x} - 4 a_{e}^{2} c_{e z}) {\dot{R}}_{y c} + (4 h_{c} h_{e} k_{e x} - 4 a_{e}^{2} k_{e z}) R_{y c} = 0$ (12)and for the rolling motion, it is written as follows $J_{x e} {\ddot{R}}_{x e} + (4 b_{e}^{2} k_{e z} + 4 h_{e}^{2} k_{e y}) R_{x e} + (4 b_{e}^{2} c_{e z} + 4 h_{e}^{2} c_{e y}) {\dot{R}}_{x e} + (4 h_{c} h_{e} k_{e y} - 4 b_{e}^{2} k_{e z}) R_{x c} + (4 h_{c} h_{e} c_{e z} - 4 b_{e}^{2} c_{e z}) {\dot{R}}_{x c} = 0$ (13)In summary, the dynamic equation of the RWVS can be written in a general form as shown as follows $M \ddot{X} + C \dot{X} + K X = F$ (14)

The displacement, velocity, and acceleration of the RWVS are indicated by vectors $X$ , $\dot{X}$ , $\ddot{X}$ , respectively. The system is characterized by the total mass matrix M , damping matrix K , stiffness matrix C , and the external dynamic load vector F .

Implementation method for drop impact loads

As shown in Figure 2, the schematic diagram of the drop impact test for a rail vehicle is presented. The experiment involves obtaining a step impact excitation by dropping a railway vehicle off a wedge, followed by measuring the attenuation response of the vehicle’s free vibration. The movable wedge structure in the experiment is shown in Figure 2. The lower surface of the wedge section is curved, making it more stable in contact with the top of the track. There is a platform with a height of 200 mm at the top of the wedge, which is used for the vehicle to stop on before falling. The slope of the wedge is 1:20 and the hardness of the heat treatment is 260–280 HB. During the test, five test cross-sections were chosen on the chassis frame of the carbody, with accelerometers placed at the two vertices of each section, as illustrated in Figure 2. The exact procedures are as follows:

Step 1. The trailer is employed to haul the RWVS to the platform situated at the apex of the wedge, at a speed not exceeding 3 km/h;

Step 2. The RWVS is gently pushed away from the wedge, and then it will fall off from the wedge as a whole and hit the railway, leading to the RWVS experiencing a step-impact load;

Step 3. Upon the excitation of a drop impact load, the vehicle undergoes free vibration, and the time history signal of each measuring point is documented throughout the fall and subsequent free vibrations of the vehicle.

Figure 2.

Schematic illustration of a drop impact test on railway vehicles.

To completely excite the modes of the carbody, four drop impact test conditions are determined and the wedges are arranged as shown in Figure 3. As shown in Figure 3(a), eightwedges are used to support the eightwheels of the tested RWVS, where it can fall off and hit the railway to stimulate the bounce modes of the RWVS. As shown in Figure 3(b), twowedges are used to support the two wheels of the axle 1 to stimulate the pitch modes of the RWVS. In Figure 3(c), fourwedges are used to support the four wheels of axles 1 and 2 to stimulate the pitch modes of the bogie. Finally, in Figure 3(d), fourwedges are used to support the right four wheels of axles 1, 2, 3, and 4 to stimulate the roll modes of the RWVS.

Figure 3.

The four drop impact loads, with (a) the first excitation condition, (b) the second excitation, (c) the third excitation condition and (d) the fourth excitation condition.

Identification method of modal parameters based on impact loads

Motion-mode energy contribution ratio

According to the vehicle model in Figure 1, the step-by-step integration method is used to solve the impact vibration response of the vehicle. The Wilson-θ method is employed to calculate the free decay response of the railway vehicle when subjected to a drop excitation.²⁴ The MEM (Motion-Mode Energy Method) is employed to identify the predominant modes of the railway vehicle when subjected to four distinct excitation conditions.^25–27 By defining $Z = {[X \dot{X}]}^{T}$ , equation (14) can be arranged into a state-space form as $Z = A \dot{Z} + \bar{F}$ (15)where $A = [\begin{array}{c} 0 & I \\ - M^{- 1} K & - M^{- 1} C \end{array}], \bar{F} = [\begin{array}{c} 0 \\ - M^{- 1} F \end{array}]$ (16)

The matrix A is the characteristic matrix of the system, and its eigenvalues and eigenvectors are represented by U and Ω , respectively. These consist of 12 pairs of complex conjugate eigenvalues and 12 pairs of corresponding complex conjugate eigenvectors $Ω = diag (Λ Λ^{*}), U = [\begin{array}{l} Ψ & Ψ^{*} \\ Ψ Λ & Ψ^{*} Λ^{*} \end{array}]$ (17)

The asterisk (*) denotes the conjugation of a complex number, vector, or matrix; $Λ$ and $Ψ$ are the system eigenvalue and eigenvector matrices, respectively $Λ = diag (λ_{1} λ_{2} \dots λ_{12}), Ψ = [φ_{1} φ_{2} \dots φ_{12}]$ (18)

The modal analysis involves forming a 24 × 12 modal transition matrix $Γ$ from the eigenvalue matrix $Ψ$ and eigenvector matrix $Λ$ $Γ = {[Ψ Ψ Λ]}^{T}$ (19)

The 24 × 1 dimensional system state vector Z of the vehicle can be expressed as follows $Z = Γ q$ (20)where $q = {[q_{1} q_{2} \dots q_{i}]}^{T}$ is the modal amplitude vector (i = 1, 2, …, 12). Note that the modal amplitude $q_{i}$ is a complex number. Rewrite equation (18) as $q = pinv (Γ) Z$ (21)where $pinv (Γ)$ returns the Moore–Penrose pseudo inverse of $Γ$ .

The corresponding state vector component in the form of displacement and velocity is written as $[X_{i} {\dot{X}}_{i}]$ , (i = 1, 2, …, 12), derived from the following equations $X_{i} = r e a l (q_{i} φ_{i}), {\dot{X}}_{i} = r e a l (q_{i} φ_{i} λ_{i})$ (22)

The kinetic energy $e_{k i}$ and potential energy $e_{p i}$ stored in the ith mode are derived in equations (23) and (24). Their sum is defined as the motion-mode energy $e_{i}$ in equation (25) $e_{k i} = \frac{1}{2} {\dot{X}}_{i}^{T} M {\dot{X}}_{i} (i = 1,2, \dots, 12)$ (23) $e_{p i} = \frac{1}{2} {({L X}_{i})}^{T} K_{s} ({L X}_{i}) (i = 1,2, \dots, 12)$ (24) $e_{i} = e_{p i} + e_{k i} (i = 1,2, \dots, 12)$ (25)where L is the suspension node displacement conversion matrix of railway vehicle; and $K_{s}$ is the stiffness matrix for energy calculation, specified as $K_{s} = diag (k_{1 X}, k_{1 Y}, k_{1 Z}, k_{2 X}, k_{2 Y}, k_{2 Z}, k_{e X}, k_{e Y}, k_{e Z})$ (26)

The sum of the energy from all modes is then written as follows $E = \sum_{i = 1}^{12} e_{i}$ (27)Finally, the energy contribution ratio of the ith mode $η_{i}$ can be obtained $η_{i} = \frac{e_{i}}{E} \times 100 %$ (28)where $η_{i}$ is a function of time and is determined at each time instant.

Utilizing the theoretical formula, the energy contribution ratio of each motion-mode can be calculated for the four different drop impact excitations. The results are depicted in Figures 4–7, and the modal excitation state of the railway vehicle is outlined in Table 2.

Figure 4.

Mode contribution ratio of the RWVS in the first condition, with (a) carbody, (b) bogie, and (c) equipment.

Figure 5.

Mode contribution ratio of the RWVS in the second condition, with (a) carbody, (b) bogie, and (c) equipment.

Figure 6.

Mode contribution ratio of the RWVS in the third condition, with (a) carbody, (b) bogie and (c) equipment.

Figure 7.

Mode contribution ratio of the RWVS in the last condition, with (a) carbody, (b) bogie and (c) equipment.

Table 2.

Modal excitation state of the RWVS.

	First condition	Second condition	Third condition	Last condition	Evaluation
Carbody bounce	√	√	√	√	The effect of condition 1 is the same as condition 4; The effect of condition 2 is the same as condition 3
Carbody pitch	×	√	√	×	The effect of condition 2 is the same as condition 3
Carbody roll	×	×	×	√	/
Bogie bounce	√	√	√	√	The effects of the four conditions are basically the same
Bogie pitch	×	×	√	×	/
Bogie roll	×	×	×	√	/
Equipment bounce	√	√	√	√	The effect of condition 1 is the same as condition 4; the effect of condition 2 is the same as condition 3
Equipment pitch	×	√	√	×	The effect of condition 2 is the same as condition 3
Equipment roll	×	×	×	√	/

It can be inferred from Figure 4 that, under the first excitation condition, the energy contribution ratios of the bounce modes of the carbody, bogie, and equipment are not always zero. After 0.5 s, the energy contribution ratios of the bounce modes of the bogie and equipment diminish to zero. In this way, it can be deduced that the first excitation condition only stimulates the bounce mode of the carbody.

As shown in Figure 5, during the initial 0.5 s of the second excitation condition, the bogie-bounce mode is the most prominent, followed by the pitch modes of the carbody and the equipment, while other modes play a minor role. After 0.5 s, the carbody-bounce mode takes over as the dominant mode, while the bogie-bounce mode is relegated to a minor role.

In the third excitation condition, Figure 6 demonstrates that the pitch and bounce modes of the bogie are the primary modes in the initial 0.5 s; however, the carbody bounce mode soon becomes the leading mode after 0.5 s.

Under the last excitation condition, Figure 7 presents that the roll modes of the carbody, equipment, and bogie, as well as the bogie bounce modes, play an important role in the first 0.5 s, while the pitch modes of the carbody, equipment, and bogie have no effect. Then, after 0.5 s, the carbody bounce mode becomes the dominant one and the equipment bounce mode is the secondary one. The sum of the energy of the RWVS diminishes with time due to damping, and the motion-mode energy of the first 0.5 s can give us an accurate picture of the main energy source responsible for the physical motion of the RWVS in four excitation conditions. The results of energy contribution ratios demonstrate that these four excitation conditions can concentrate the excitation energy to stimulate the relative mode vibrations.

Damped natural frequency identification of the railway vehicle

Employing the theoretical formula mentioned above, the vibration acceleration of each motion-mode of the RWVS can be determined under the four different excitation conditions. The Fast Fourier Transformation (FFT) results of the vibration acceleration are illustrated in Figures 8–11, with the P-P method being used to estimate the damped natural frequency of each motion-mode of the RWVS, as shown in Table 3.

Figure 8.

The FFT results of the vertical acceleration in the first drop condition, with (a) carbody, (b) bogie and (c) equipment.

Figure 9.

The FFT results of the vertical acceleration in the second drop condition, with (a) carbody, (b) bogie and (c) equipment.

Figure 10.

The FFT results of the vertical acceleration in the third drop condition, with (a) carbody, (b) bogie and (c) equipment.

Figure 11.

The FFT results of the vertical acceleration in the last drop condition, with (a) carbody, (b) bogie and (c) equipment.

Table 3.

Results of modal parameter identification.

It can be concluded that the carbody-pitch and equipment-pitch damped natural frequencies can only be identified under the second excitation condition; the bogie-pitch can only be estimated under the third excitation condition, and the carbody-roll, equipment-roll, and bogie-roll damped natural frequencies can only be identified under the last excitation condition; as the response function of this motion-mode acceleration peaks in the corresponding excitation condition.

Table 3 demonstrates that the largest absolute value of percentage error is 6.5%, and two thirds of the absolute values of percentage error are less than 3%. This indicates that the P-P method is highly effective in identifying the damped natural frequency of the RWVS under the four excitation conditions.

Drop impact modal test of the railway vehicle system

To evaluate the efficacy of the drop impact load modal test for the modal parameter identification of the RWVS, a certain type of subway vehicle was tested in the line section of the factory, as demonstrated in Figure 12. During the testing process, a trailer is used to slowly tow the tested vehicle to the platform at the top of the wedge. Then, the vehicle is pulled down the steps of the wedge to generate impact vibrations. In the process, the vibration accelerations of the vehicle are monitored, and the wedge installation method and test conditions are conducted in accordance with the working conditions illustrated in Figure 3. Under the first excitation condition, Figure 13 displays the comparison between the simulation results and the measured results of vertical vibration acceleration response of the carbody. During the test, the acceleration sensor adopts built-in IC piezoelectric acceleration sensor (LC0142A) form LANCE MEASURE TECHNOLOGY CO., LTD

Figure 12.

The tested railway vehicle through a drop impact load.

Figure 13.

Comparison of the carbody vertical acceleration of testing and the simulation in a bounce drop test.

It is evident from Figure 13 that the measured results of vertical vibration under drop impact conditions are in line with the simulation results, indicating the effectiveness of the established mathematical model in simulating actual vehicle vibration. The modal frequencies of the carbody and bogie obtained by simulation and measurement are further compared, as shown in Table 4. It is clear that the simulation results are consistent with the measured results, indicating the validity of the mathematical model and the modal parameter identification method for drop impact load in identifying the modal parameters of the RWVS mechanism. This method is of great importance in the structural design and optimization of complex railway vehicles.

Table 4.

Comparison of modal frequency results.

Conclusion

This paper has explored the identification of modal parameters of railway vehicle systems, and the method discussed in this paper has been applied to identify the modal parameters of railway vehicles through a drop test. Therefore, the following conclusions can be drawn.

(1) The analysis of motion-mode energy revealed that the four drop excitation conditions can focus the excitation energy to induce the relevant mode vibration; however, the effects of the various working conditions on the modal excitation state of the system vary.

(2) Using the FFT results of the railway vehicle in four drop excitation conditions, the P-P method can precisely estimate the damped natural frequency of the RWVS.

(3) The experiment of modal parameter identification conducted on a railway vehicle shows that the largest absolute percentage error of identification is 6.5%, which confirms the effectiveness of the drop impact method in modal parameter identification of railway vehicles.

In comparison to the previous modal identification approach for railway vehicles, this technique requires less complex testing equipment and is more economical. Moreover, it can be used to ascertain the rigid modal parameters of railway vehicles when all parameters are unknown, thus possessing both theoretical research value and potential practical applications for railway vehicle modal parameter identification.

Footnotes

Declaration of conflicting interests

The author(s) declared that there were no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by The Open Foundation of State Key Laboratory of Traction Power (Grant no. TPL2106),The National Natural Science Foundation of China (Grant No.52005068),The Natural Science Foundation of Chongqing,China (Grant No. CSTB2022NSCQ-MSX0247),The Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No.KJQN202201312,No.KJQN202101325,No.KJ202201381395273) and The Opening Project of Engineering Research Center of Chongqing Education Commission of China in Running and Maintenance of Robots (Grant no. YWZX20220001).

ORCID iD

Zhengyong Duan

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