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The elastic vibration of the wheelset is a potential factor inducing wheel-rail defects. It is important to understand the natural vibration characteristics of the flexible wheelset for slowing down the defect growth. To estimate the elastic free vibration of the railway wheelset with the multidiameter axle, the transfer matrix method (TMM) is applied. The transfer matrices of four types of elastic beam models are derived including the Euler–Bernoulli beam, Timoshenko beam, elastic beam without mass and shearing stiffness, and massless elastic beam with shearing stiffness. For each type, the simplified model and detailed models of the flexible wheelset are developed. Both bending and torsional modes are compared with that of the finite element (FE) model. For the wheelset bending modes, if the wheel axle is modelled as the Euler–Bernoulli beam and Timoshenko beam, the natural frequencies can be reflected accurately, especially for the latter one. Due to the lower solving accuracy, the massless beam models are not applicable for the analysis of natural characteristics of the wheelset. The increase of the dividing segment number of the flexible axle is helpful to improve the modal solving accuracy, while the computation effort is almost kept in the same level. For the torsional vibration mode, it mainly depends on the axle torsional stiffness and wheel inertia rather than axle torsional inertia.

Wheelsets are key moving components in the railway vehicle systems. Conventional rigid-body modeling methods only evaluate wheelset vibration characteristics in low frequency. However, many practical problems induced by wheel axle bending and torsional vibrations in the mid-high frequency domain have occurred frequently. These vibrations can be observed in all sorts of railway systems such as high-speed railways, subways, and heavy-haul railways. Some researchers have indicated that the rail corrugation, wheel-rail stick-slip vibration, and even wheel polygon wear are closely associated with wheelset elastic vibrations and track vibrations [

Potential influence of wheelset natural vibration on wheel-rail damage.

A number of methods can be used to solve the natural vibrating characteristics for elastic structures. Typical solving methods include finite element (FE) method and elastic vibration theory. Both of them have been widely used. To study the wheel-rail stick-slip problem in freight wagon systems, Sun and Simson [

Compared with other methods, the TMM has the advantages of simple programming, modularized establishment of vibration equation, and fast numerical calculation. However, the application of TMM in wheelset dynamics for different elastic-beam models has not been reported. How about the modal solving accuracy of wheelset models with different elastic beams? Is it necessary to consider the axle rotational inertia in wheelset torsional vibration? What are the influences of the vibration mass, shear stiffness, and number of wheel axle dividing segments on the natural vibration characteristics? Solving the above problems is of great significance for putting forward a scientific modeling scheme for the axle-disc system similar to the wheelset. In this paper, wheelsets with the multidiameter axle will be studied. They can be regarded as combination systems composed by rigid wheels and flexible axles. Axle bending vibration can be solved using various methods including the Timoshenko beam, Euler–Bernoulli beam, and massless elastic beam with or without shearing stiffness. The torsional vibration can also be solved by the elastic axle models with or without mass. Another point investigated in this paper is that what are the differences among various axle modeling methods in analyzing wheelset natural vibrations? A typical freight wagon wheelset will be used as an example. The transfer matrices for different elastic beam models are derived and then applied to calculate wheelset natural vibration characteristics. The corresponding FE models are also developed to verify the theoretical analysis results and examine the solving accuracies of different transfer matrix models.

For an actual wagon wheelset, as shown in Figure

Freight wagon wheelset with the multidiameter axle.

Wheelset structure and dimensions of the heavy-haul freight wagon.

For an axle with mass, the transfer matrix can be obtained by solving the vibration equations of an elastic beam. There are two typical beam theories for this case, i.e., Euler–Bernoulli beam and Timoshenko beam. The former does not consider the influence of rotational inertia and shear deformation whilst the latter does. Taking an infinitesimal axle segment as example (Figure _{rot}. In the Timoshenko beam theory, the factor _{rot} and both the deflection angles of _{g}, the inertial _{rot} can then be defined as _{rot} = _{g}^{2}. The force states of the bending moment

Force state of the flexible axle segment for the wheelset axle. (a) Flexible axle segment. (b) Force state.

For the axle segment, the vibration differential equation and force equilibrium relations can be deduced easily, which will not be explained in this paper. On this basis, the vibration equations of Euler–Bernoulli beam and Timoshenko beam can be expressed as equations (

Introduce the mathematic expression of deflection

Then, equations (

The general solution of equation (

Correspondingly, the general solution of equation (

In the modal coordinates, the state parameters of Θ,

The next key point is to determine the unknown coefficients

In free boundary conditions, the sate vector for

Hereby, the column matrix

So the state vector in position

For equations (

In some axle-disc type components, the axle can also be assumed as a massless elastic axle. The massless axle provides the main deformation energy but little inertial energy. If the axle vibrating mass is not taken into account, the force and displacement states at the input and output ends of the massless elastic beam have the relations, as shown in Figure

Force and deformation states of the massless beam element.

The elastic deformation and force conditions can be described by structural mechanics according to the deformation compatibility principle. The transfer relations can be expressed as

Changing the formation of equation (

Therefore, equations (

For the modeling of a flexible wheelset, the model can be divided into the elastically deformable part of axle segments and the nondeformable part of wheels. The rigid wheels are connected by the flexible axle with or without the vibrating mass. For the rigid wheel, its force and displacement state parameters at both sides are shown in Figure

Force analysis of the rigid wheel.

Two modeling schemes are proposed for the wheelset, as shown in Figure

Wheelset and its dynamic model of the piecewise flexible axle.

For the modeling schemes A and B, the corresponding transfer relations of state vectors from the left boundary to the right boundary can be calculated, respectively, by the following equations:_{T} is the global transfer matrix. _{2} and _{4} in equation (_{4} and _{8} in equation (

Using the assumption of free boundary condition, the shearing force and moments at left and right ends can be regarded as zero. Equation (

From equation (

For a vibration system, there should be nonzero solutions in equation (

The zero solutions of equation (_{0,1} = 1 in equation (

Then, the boundary condition can be expressed as

Finally, introducing _{0,1} into equations (

Following the modeling principle of TMM, the wheelset torsion vibration model can be simplified as a combined system of rigid wheels and torsional flexible axles, as shown in Figure _{i-1,i} and _{i-1,i} are the torsion angle and torque in the left side of

Wheelset torsional vibration models of the wheel and flexible axle. (a) Rigid wheel. (b) Flexible axle segment.

The rigid wheel has only the torsional DOF. The torsional angle and torque at the left side of the lumped mass are in accordance with those of wheel right side. The effect of wheel inertia should be taken into account in the torsional vibration equation. Transforming the torsional displacement into the form of modal coordinate as ^{iωt}, then the transfer matrix

With the same solving procedure described in Section _{p} is the polar moment of inertia determined by the axle cross-section dimensions.

From another perspective, in some applications, the axle can be regarded as a massless flexible axle with the transfer relations as

In the torsional vibration simulations, _{0,1} as given in equation (_{i} by _{i}:_{T} is the global transfer matrix. _{2} and _{4} in equation (_{4} and _{8} in equation (_{w} in equation (

To solve the natural vibration characteristics of the wheelset in Figure

Main material property parameters of wheelset.

Wheel type | Axle type | Material | Elasticity modulus | Shear modulus | Density | Poisson’s ratio |
---|---|---|---|---|---|---|

HFS | RF2 | Alloy steel | 2.06^{2} | 0.794^{2} | 7800 kg/m^{3} | 0.3 |

The natural frequencies of the flexible wheelset correspond to the nonzero solutions of equation (

Solution curves of 1st-order bending for different modeling schemes. (a) Simplified model. (b) Detailed mode.

Mode shapes of 1st-order bending for different modeling schemes. (a) Simplified model. (b) Detailed mode.

As the frequency increases, the modal frequency of 2nd-order bending can be found, as shown in Figure

Solution curves of 2nd-order bending for different modeling schemes. (a) Simplified model. (b) Detailed mode.

Mode shapes of 2nd-order bending for different modeling schemes. (a) Simplified model. (b) Detailed mode.

In the solution curve, the 3rd-order bending frequency is tried to be searched in Figure

Solution curves of 3rd-order bending for different modeling schemes. (a) Simplified model. (b) Detailed mode.

Mode shapes of 3rd-order bending for different modeling schemes. (a) Simplified model. (b) Detailed mode.

In addition, the torsional vibration characteristics of the wheelset modelled by flexible axles with and without the vibrating mass are compared. Referring to equations (

Natural torsional vibration characteristics. (a) Solution curve. (b) Mode shape.

To verify the accuracy of the wheelset transfer matrix models, a detailed FE model of the wheelset is established with the solid element in ANSYS, as shown in Figure

FE model of the wheelset.

Using free boundary conditions, the modal parameters are calculated. Within the frequency range of 500 Hz, the bending modes and torsional modes can be found. For instance, 1st-, 2nd-, and 3rd-order bending vibration occur at 116 Hz, 245 Hz, and 501 Hz, respectively. The mode shapes agree well with the results calculated by transfer matrices (Figures

According to the test and simulation results of the similar wheelset in [

Modal frequency comparison of the 1st-order bending mode between TMM and FE method.

For the 2nd-order bending mode of the wheelset, it has the same rule as reflected in Figure

Modal frequency comparison of the 2nd-order bending mode between TMM and FEM.

Modal frequency comparison of the 3rd-order bending mode between TMM and FE model.

As for the torsional mode shown in Figures

Bending and torsional modes of the wheelset solved by the FE model. (a) 1st-order bending: 116 Hz. (b) 2nd-order bending: 245 Hz. (c) 3rd-order bending: 501 Hz. (d) Torsional: 98 Hz.

Finally, the computational efforts are studied by comparing the computing times of different models, as shown in Table

Computing times of different models.

FE model | TMM simplified model | TMM detailed model | ||||||
---|---|---|---|---|---|---|---|---|

12.5 min | EB | Mless-B | Mless-BwS | TB | EB | Mless-B | Mless-BwS | TB |

2.6 s | 2.5 s | 2.5 s | 2.7 s | 2.8 s | 2.6 s | 2.6 s | 2.9 s |

In summary, for the TMM model of the wheelset, the computational time difference between different models is only below 0.3∼0.4 s as compared in Table

The elastic vibrating model and mode parameters of the wheelset are solved based on the transfer matrix method. The wheel axle with multidiameters contributes to the main elastic deformation, which is simulated by four types of elastic beam models including the Euler–Bernoulli beam, Timoshenko beam, elastic beam without mass and shearing stiffness, and the massless elastic beam with shearing stiffness. The modes of both bending and torsional vibrations are compared with the results of a FE model. Key conclusions are drawn as follows:

If the wheel axle is modelled by the Euler–Bernoulli or Timoshenko beam, the natural frequencies of the bending mode of the flexible wheelset can be reflected accurately. Generally, the frequency difference between the transfer matrix and FE results can be limited below 22% and 10% for the Euler–Bernoulli and Timoshenko beam models, respectively. If the vibrating mass of the axle is not considered, the frequency error can reach 20%∼40%, no matter the axle shearing stiffness is considered or not. Also, in this case, the bending mode with higher order cannot be obtained.

For the torsional vibration of the wheelset, there is a slight difference between the results of flexible axle models with and without the vibrating mass. The natural vibration frequency error between the transfer matrix model and FE model is about 3%. Both the torsional stiffness and wheel inertia influence its natural frequencies, while it has little relations with the rotational inertia of the axle.

For the detailed and simplified modeling schemes, the natural frequency solving accuracy of the former is 3% higher than that of the latter. More flexible axle segments divided means higher solving accuracy. The computational effort of the transfer matrix model is much lower than that of the FE model, which can be attributed to the low and invariant dimension of transfer matrices, whether in the simply or complex models.

The applicable conditions of different elastic beam models in wheelset modal analysis are revealed. The TMM may provide a convenient solution for modal analysis of the railway vehicle rigid-flexible coupling system.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant nos. 52072249, 11790282, and 12072208), and Self-Determined Project of State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures (Grant no. ZZ2021-10).