Design and Modeling of Viscoelastic Layers for Locomotive Wheel Damping

Abstract

Rail–wheel interaction is one of the most significant and studied aspects of rail vehicle dynamics. The vibrations caused by rail–wheel interaction can become critical when the radial, lateral and longitudinal loads of the vehicle, cargo and passengers are experienced while the vehicle is in motion along winding railroad paths. This mainly causes an excessive production of vibrations that may lead to discomfort for the passengers and shortening of the life span of the vehicle’s body parts. The use of harmonic response analysis (HRA) shows that the wheel experiences high vibrational amplitudes from both radial and lateral excitation. The present study describes a numerical and experimental design procedure that allows mitigation of the locomotive wheel resonance during radial and lateral excitations through viscoelastic layers. It is proven that these high frequencies can be reduced through the proper design of damping layer mechanisms. In particular, three parametric viscoelastic damping layer arrangements were analyzed (on the web of both wheel sides, under the rim of both wheel sides and on the web and under the rim of both wheel sides). The results demonstrate that the correct design and dimensions of these viscoelastic damping layers reduce the high-amplitude resonance peaks of the wheel successfully during both radial and lateral excitation.

1. Introduction

The wheel–rail contact dynamics of trains in motion are a complex phenomenon that requires in-depth analysis. Currently, this analysis is expressed by numerical modeling in some available commercial software with the computational ability to analyze complex problems. For instance, Thomson utilized a 2D model to examine the transient vibration characteristics of a train wheel [1]. Sato et al. [2] investigated the vibration and sound radiation characteristics of rail wheels by conducting experiments and validating their results through modeling the wheel and performing Finite Element Analysis (FEA). Zhong et al. [3] studied the vibration and sound radiation of a rotating train wheel subjected to vertical harmonic wheel–rail force (with and without rotation) using 2.5DFinit Element Method (FEM) to calculate the wheel vibration characteristics and validating their findings with the experimental results. They concluded that the wheel rotation speed splits most peaks of the vertical receptance at the wheel–rail contact point [4]. Mazilu [5] examined the radial and axial wheel vibration by using both theoretical and experimental approaches and concluded that larger resonance peaks of frequencies at radial excitation would start at frequencies larger than 1500 Hz. It was also observed that radial excitation would experience higher amplitude resonance peaks than lateral excitation.

It is known that vibrations in rails are caused by vehicle force on the railroad tracks from the wheels and from irregularities at the wheel–rail interference (corrugation). As a prevalent consequence, rail–wheel vibration can result in rolling noise, shorten the life cycle of the rolling surfaces from wear or cause discomfort to passengers, tonal noise and damage to the vehicle’s structural components. Additionally, excessive vibration on a rail wheel has long been acknowledged to be caused by rail wheel corrugation [3,4,5,6,7]. Balekwa and Kallon [8] investigated resonance frequencies via experimental and Finite Element Modal Analysis (FEMA) of the class D39 200 locomotive wheelset, correlating their results with corrugation frequencies around the track curves [9]. The locomotive class D39 200 is normally operated on the Belfast to Steelpoort railway line. This study highlighted how the locomotive experiences rail corrugation when passing through track curves, resulting in excessive rail wheel vibrations [10,11,12].

The main focus of the present manuscript was to develop a design procedure that allowed for mitigating locomotive wheel resonance during radial and lateral excitations through parametric modeling of viscoelastic layers. The originality of the study lies in the optimization method adopted. Through the application of computational methods and the experimental characterization of the damping, this allowed for finding the optimal geometry of the layers that minimized wheel vibrations in a wide range of frequencies.

In short, three viscoelastic damping layers were parametrically designed to be mounted on the locomotive wheel structure, being in contact at a point where the structure experiences high deformation in such a way that the viscoelastic damping layer cannot but dissipate energy as much as possible.

A case study of the locomotive class D39 200 wheel offered tangible results to support the methodology by validating it with experimental data. The resonance amplitude peaks observed in the wheels of this locomotive [8,9,10,11,12,13] were damped using this new viscoelastic passive damping design technique.

To be precise, the viscoelastic damping layer, which is applied on structures to mitigate vibrations, can be either an unconstrained (free) damping layer or a constrained damping layer. The difference is that a constrained damping layer consists of a steel or aluminum layer with a viscoelastic high-damping polymer sheet mounted between the structure and steel, while an unconstrained (free) damping layer is used instead when the viscoelastic high-damping polymer is mounted on the structure [14,15,16]. It is clear that the advantages of viscoelastic damping can be identified in the material properties, which consist of elastic and viscous characteristics at the same time.

Lopez [17] studied a wheel fitted with a steel rod ring inside the groove cut under the rim. The motion between the wheel and ring reduced vibration gradually through friction and interference by converting the vibration into heat. The conclusion was that by also increasing the ring damper mass mounted on the wheel, the damping efficiency on the wheel increased. The problem is that the theoretical design mechanism of this type of damping procedure, called a ring damper, is still not well understood [18]. As an advantage, ring dampers are simple and cost-effective compared with other wheel dampers. Figure 1 shows the rail wheel mounted with a damping ring under the rim of both sides of the wheel [19].

It should be noted that many researchers have investigated tuned mass dampers (TMDs) for different applications like rail wheels, beams and railway bridges [20,21,22]. A TMD, first developed by Frahm in 1909 [23,24], is a device having a mass–spring–damper system integral to the structure to reduce the dynamic response. Later in his book on mechanical vibrations, Den Hartog used a tuned mass damper to mitigate vibrations in systems [25]. The frequency of the damper is tuned to a structural frequency so that during excitation, the TMD resonates out of phase with the structural motion, reducing unwanted energy depending on the modal displacement in the system. The mass damper consists of a viscous elastic layer which can turn the vibration into heat energy [26] (Figure 2).

Suarez et al. [27] compared the vibration and noise of resilient and solid monobloc railway wheels on underground lines. Based on their results, the resilient wheels demonstrated better performance. It is the resilient wheel which is commonly designed for light transit systems in order to mitigate the dynamic vibration of the wheel, rolling noise produced by wheel–rail contact and wheel squeal noise on sharp curves [28] (Figure 3).

In light of these considerations, in the present study, the viscoelastic damping layer material 3M ISD112 [29] was chosen because of its widespread use and commercial availability [30,31,32,33].

Introduction excluded, the manuscript is organized into four sections. In the first section, the experimental and numerical analysis methodology are illustrated. In the second section, the principal results obtained for different viscoelastic layer positions and arrangements are discussed. The final considerations and conclusions are discussed in sections three and four, respectively.

2. Materials and Methods

The dynamic characterization of the D39 200 locomotive wheel and the design of the viscoelastic layers to damp wheel vibrations were performed by means of numerical analysis validated through experimental modal analysis (EMA). EMA was carried out by a roving hammer impact test with a Brüel & Kjær piezoelectric charge hammer (Brüel & Kjær, Nærum, Denmark) and a small-sized charge accelerometer (Figure 4). Frontend data and the specific software Test Xpress Version 10.0 and TestLab Version 16 from Siemens/LMS (Plano, TX, USA) were used for the acquisition of signals, post-processing, FRF calculations and complete identification of the main modes [34].

Here, the results of the EMA allowed us to fine-tune and validate a nonlinear finite element (FE) model of the wheel with energy absorption capabilities. Harmonic response analyses with the FE model of the wheel were performed through radial and lateral excitations, with the wheel assumed to be rigidly mounted on the axle. Viscoelastic damping layers were introduced into the FE model in three different arrangements of the wheel, and an optimization of vibration amplitude reduction was performed.

2.1. Finite Element Modal Analysis

The FE model of the locomotive wheel was meshed using 54,496 nodes and 32,606 tetrahedral 10-node elements (SOLID187) to determine the frequency at which the wheel would experience high-amplitude oscillations that could lead to resonance. The characteristics of the wheel, as well as the finite element modal and harmonic analysis, were performed with ANSYS 2020 R1 (ANSYS Inc., Canonsburg, PA, USA) software (Figure 5).

The wheel’s geometric profile, containing curves on the web and under the rim, makes tetrahedron mesh elements particularly suitable to fit the wheel’s geometry [35]. Thus, using Autodesk Inventor code a parametrization of the wheel and damping layer profile was carried out. This process provided the parametric geometric profiles of the cross sections of the three types of layers (on the web of both wheel sides, under the rim of both wheel sides and on the web and under the rim of both wheel sides) used to dampen the vibrations of the wheel (Figure 6a).

The initial values of the geometric parameters of the layers’ section profiles were obtained by approximating the profiles both under the rim and for the web of both wheel sides with polycentric curves. In particular, four circular arcs were used to approximate the section profile of the under the rim layer, and five circular arcs were used to approximate the section profile on the web of both wheel sides with an accuracy of over 99%. The values of the radii of these arcs constituted the design variables for optimization (Figure 6a).

After the mesh refinement procedure, which allowed us to validate the FE model based on the EMA, the best geometry of the layers and their best arrangement, which gave the optimal results in terms of vibration damping in the frequency range studied, were calculated. The design values, the optimal values assigned to the radii (design variables) in the parametric optimization study carried out, are shown in

Table 1. Layer profile geometric parameters.

Parameter	R1rim	R2rim	R3rim	R4rim	R1w.i.	R2w.i.	R3w.i.	R4w.i.	R5w.i.	R1w.e.	R2w.e.	R3w.e.	R4w.e.	R5w.e.
On design values	72.1	54.5	62.3	67.4	167.2	223.6	173.4	216.8	289.4	423.4	216.8	354.7	189.2	102.7
Max. variation	12	12	12	12	16	16	16	16	16	14	14	14	14	14
Optimal values	64.1	46.5	54.3	59.4	157.2	213.6	163.4	206.8	279.4	413.4	206.8	344.7	179.2	92.7

. The optimal values adopted for the radii of the profiles are the best compromise between the maximum allowed values (Table 1) and the space requirements of the wheel. They provided optimal thickness values of 8 mm and 10 mm for the viscoelastic damping layer under the rim and the viscoelastic damping layer, respectively.

Figure 6b shows the mesh of the three optimal viscoelastic layer arrangements. The internal web viscoelastic damping layer was meshed using 14,710 nodes and 4459 tetrahedron elements (Figure 6b₁). The viscoelastic damping layer under the rim was meshed using 7025 nodes and 2778 tetrahedron elements (Figure 6b₂). The external web viscoelastic damping layer was meshed using 12,025 nodes and 3781 tetrahedron elements (Figure 6b₃). The mesh size for all three configurations of viscoelastic damping layers was 2 mm.

The mesh refinement procedure was performed by setting 5% as the maximum admissible limit of frequency variation with respect to the values calculated through EMA. Mesh convergence was measured on the sixth deformation mode only for all mesh sizes. Mesh convergence is a good tool to verify if the mesh results are adequate, since the accuracy of a simulation depends on the geometry accuracy, material properties and loads and boundary conditions.

Evident from

Table 2. Mesh convergence.

Frequency (Hz)	Frequency Variation	Nodes	Elements
543.31	Experimental value	-	-
540.75	−0.47	4214	1531
541.56	−0.32	8672	4531
541.83	−0.27	54,496	32,606
541.05	−0.41	65,395	39,127

is that the results did not change with continued mesh refinement. Hence, it can be deduced that convergence was reached using 54,496 nodes and 32,606 tetrahedral elements.

Table 3. Material properties of the FEA model.

Material	Young’s Modulus (GPa)	Density (kg/m3)	Poison’s Ratio
Steel	210	7850	0.3

shows the material properties adopted in the FE analysis.

2.2. Harmonic Response Analysis

Harmonic Response Analysis (HRA) was conducted from 20 Hz to 5 kHz at 20-Hz intervals. The wheel was set free radially (y-axis) and laterally (x-axis). Two directions of the wheel, radial (vertically), and lateral (axially), were excited by a unitary force to investigate the frequency which would experience maximum amplitudes, leading to resonance of the locomotive wheel [36]. Figure 6a shows the wheel model with the load being applied radially (Figure 7a). Hence, the impact would represent the vertical harmonic wheel–rail force at the wheel–rail contact point. Figure 7b shows the load being applied laterally on the wheel model instead. The lateral impact represents the axial harmonic wheel–rail instability that results when the wheel travels through sharp curves while the wheel is experiencing lateral creepage impacts.

2.3. Viscoelastic Damping Layers

Viscoelastic materials have very effective characteristics for controlling vibrations in structural systems. The use of a viscoelastic damping layer depends on the material properties [37,38]. The viscoelastic material considered in this research was 3M ISD112 from the 3M™ Viscoelastic Damping Polymers manufacturing company (Tokyo, Japan) because of its widespread use and its commercial availability [31,32,33]. However, the procedure utilized in the scientific literature can obviously be applied to any viscoelastic material. The properties of the viscoelastic material studied in this paper were approximated by the elastic properties listed in

Table 4. The viscoelastic layer’s material properties.

Material Type	Young’s Modulus (E)	Density (ρ)	Poisson’s Ratio	Shear Modulus (G)
Viscoelastic material 3M ISD 112	833.78 × 104 N/m2	1350 kg/m3	0.49	277.93 × 104 N/m2

, and viscous damping would be added additionally by Rayleigh beta damping. ANSYS was adopted to investigate the harmonic response of the wheel, with a viscoelastic damping layer mounted in the three different arrangements shown in Figure 6a–c.

It was assumed that the viscoelastic damping material only underwent small deformations and hence had linear behavior. The element type used to characterize the viscoelastic damping layer material was the same used in the wheel model in order to ensure good boundary definition. It was proven by Pascon et al. [39] that the use of tetrahedral elements to solve problems with high polymetric layer materials using Finite Element Analysis does not present locking nor results making the layer too stiff, even if the material layer is applied with other different materials inside the analyzed domain.

For the viscoelastic layers in three optimal configurations, FEMA and HRA of the wheel were performed, and the natural frequencies for the three different positions (Figure 8a–c) were calculated.

The viscoelastic damping layer was mounted with a strong bonding agent to be in contact with the vibrating structural surface (Figure 9c). During the vibration of a structure, the viscoelastic damping layer material dissipates vibrations by means of alternate extension and compression of the viscoelastic damping layer as demonstrated in Figure 9a,b.

The mesh convergence was conducted by decreasing the element size for the viscoelastic damping layers of the web and under the rim. The viscoelastic damping layers on the web and under the rim were chosen to track the convergence since they consisted of both web and under the rim damping components.

Table 5. Mesh convergence of damping layers on the web and under the rim.

Deformation (mm)	Variation	Nodes	Elements
54.182	-	12,025	3781
54.191	0.13611	64,432	21,037

shows the details of the convergence results of the viscoelastic damping layer.

Figure 9c and

Table 6. Wheel design dimensions.

Description	Metric Value (mm)	Description	Metric Value (mm)
Wheel diameter	1026	Front of hub diameter	317.5
Rim thickness New wheel diameter	104.8	Back of hub diameter	336.6
Hub offset	18	Bore diameter	228.6
Back of rim to front of hub	130.3	Plate thickness at rim	25.4
Hub length	148.3	Plate thickness at hub	30.2
Rim width	141

show the final optimized model of the wheel for a class D39 200 locomotive and its design dimensions, respectively.

2.4. Harmonic Excitation of a Wheel with Viscoelastic Damping Layers

Finally, the boundary conditions of the wheel were set free radially (y-axis) and laterally (x-axis). Two directions of the wheel, radial and lateral, were excited by unitary force.

3. Results

The results are outlined in the following order: FEA modal analysis of the wheelset and the Harmonic Response Analysis of the wheel with and without viscoelastic damping layers.

3.1. Finite Element Modal Analysis

Table 7. First natural frequencies of the wheel.

Mode Number	Frequency (Hz)	Mode Shapes	Classification and Descriptions
1	222.15		Lateral Mode: High deformation at the wheel flange and rim.
2	223.61		Lateral Mode: Bending of the web.
3	302.9		Lateral Mode: High deformation at the wheel flange and rim.
4	459.92		Lateral Mode: Larger deformed area on the wheel flange.
5	460.91		Radial Mode: Cross-section of the wheel starts to alter.
6	543.31		Lateral Mode: Wheel flange and rim are laterally twisted.
7	1137		Radial Mode: Larger deformations on the rim, flange and under the rim.

shows the results obtained with FE modal analysis. The mode of vibrations that were relevant to the lateral mode were identified due to out-of-plane displacement of the web [40]. The legends illustrate the displacements in millimeters. It was noted from the fourth mode, for example, that the nodes of the wheel situated around the light blue color code of the deformed shape displaced the least (about 0.36743 mm). The nodes situated around the green and yellow color codes experienced an average displacement of 2.1252 mm, while the modes around the orange and red color codes experienced a larger displacement of 3.3069 mm. The second and third modes had no complex mode shape, but from the fourth mode, the shape of the wheel started to deform to different eigenmodes. Finite element modal and harmonic analysis of the wheel mounted with viscoelastic damping layer were performed, and the first 20 natural frequencies for 3 different positions are outlined in Appendix A, Appendix B, Appendix C, Appendix D. Table 7 shows the first seven natural frequencies and the relative mode shapes.

The radial modes were easily identified by observing the axial bending vibrations of the wheel. The nodes situated around the green and yellow colors experienced, on average, the smallest displacement, while the modes around the orange and red colors experienced larger displacements.

3.2. Harmonic Response Analysis of a Wheel without Viscoelastic Damping Layers

3.2.1. Lateral Harmonic Response of the Wheel

Figure 10 shows the behavior of the wheel’s lateral harmonic response.

The x-axis of the graph represents the frequencies, while the y-axis represents the amplitudes in terms of the normalized amplitude (mm/N). Figure 10 illustrates the lateral harmonic response results. The wheel structure had a maximum amplitude of 741.669 × 10⁻⁹ mm at a frequency of 518 Hz.

3.2.2. Radial Harmonic Response of the Wheel

Figure 11 shows the wheel’s radial harmonic response. The wheel structure had a maximum amplitude of 640.024 × 10⁻⁹ mm at a frequency of 1265 Hz. It was evident from the harmonic response results in the radial and lateral frequency responses that the wheel vibrated at different amplitudes and frequencies, so it can be deduced that the wheel experienced high amplitudes in the lateral direction. Moreover, the results from Figure 12 reveal that the wheel would reach a maximum radial amplitude of 640.24 × 10⁻⁹ mm at a frequency of 1265 Hz. However, during lateral excitation, it would reach a maximum amplitude peak of 741.669⁻⁹ mm at a frequency of 518 Hz.

3.2.3. Viscoelastic Material Property Estimation

The final step was to predict the characteristics of the viscoelastic material behavior, whereby the loss factor and modulus of the viscoelastic materials were determined using the modal frequency data of the wheel from

Table 8. Wheel complex natural frequencies, shear storage modulus and loss factor.

Mode Number	Complex Frequencies (Hz)	Complex Shear Storage Modulus (MPa)	Complex Loss Factor (ɳ)
1	222.15	10.20	0.078
2	223.61	10.03	0.082
3	302.91	11.02	0.091
4	459.92	10.50	0.120
5	460.91	10.31	0.170
6	543.31	10.11	0.198
7	1137.01	1.90	0.355
8	1137.20	1.89	0.356
9	1223.40	1.91	0.361
10	1224.72	1.91	0.362
11	1769.63	1.95	0.372
12	1847.82	1.92	0.377
13	1848.54	1.92	0.378
14	2032.72	1.88	0.389
15	2038.20	1.88	0.390
16	2163.21	1.80	0.401
17	2169.22	1.80	0.403
18	2575.42	1.85	0.434
19	2578.02	1.84	0.436
20	2939.02	1.82	0.501

. The procedure was carried out using curve-fitting of the 3M ISD112 viscoelastic material master curve provided by the manufacturer data at 30 °C in the frequency range from 20 Hz to 5 kHz.

As it was reported in [29,41], these procedures of determining the viscoelastic material using the master curves provided by manufacturer minimized the difference between the experimental measured and estimated data (Figure 13).

In our application, the primary task was to measure the storage modulus and loss factor of the 3M ISD112 viscoelastic material at the dominating vibrational frequency resonances, as it was indicated that viscoelastic material properties strongly depended on the frequency and temperature [42]. Moreover, the temperature and natural frequency ranges were employed with the nomograph provided by manufacturer of the 3M ISD112 to determine the viscoelastic material behavior of the performance (complex high loss factor and storage modulus) over the temperature and frequency range of the application [29]. The natural frequency, mode shape and modal masses were assumed to be affected little by the addition of the damping layers, and consequently, the finite element model for the undamped structure was used to derive these parameters [13]. The complex modal analysis was determined using the natural frequencies and the mode shapes of the wheel from Table 7 and were used to predict the modal damping of the complete structure. The procedure’s combinations of temperatures and frequencies are depicted in Figure 13. As an example, the natural frequency was found to be 518 Hz (at the modal analyses section), and the temperature was 30 °C (which was assumed to be constant). The point for 518 Hz was found on the right-hand axis. Following the point horizontally to the line for a temperature of 30 °C. At this intersection, the line was drawn to be a vertical line. Then, the modulus and loss factor values were read off the appropriate graph at the point of intersection with the vertical line. The complex storage modulus and loss factor were found to be 10.1 MPa and 0.198, respectively.

Hence, Table 8 demonstrates the numerical loss factor, storage modulus and shear modulus, which were calculated using the nomograph in Figure 13 with the natural frequencies of the locomotive rail wheel that were examined from Appendix A, Appendix B, Appendix C, Appendix D. In Appendix E, instead, there is the workflow the procedure followed to design viscoelastic layers.

The rail wheel’s equation of motion was

$$\left[\mathrm{M}\right]\left\{\mathrm{ü}\right\}+\left[\mathrm{C}\right]\left\{\mathrm{ú}\right\}+\left[\mathrm{K}\right]\left\{\mathrm{u}\right\}=\left\{\mathrm{F}\right\}$$

(1)

where [M] and [K] are the mass and stiffness matrices of the given element, respectively, and [F] is the load vector acting on the element, while [C] is the damping matrix having the following general form:

$$\left[\mathrm{C}\right]=\mathsf{\alpha }\left[\mathrm{M}\right]+\mathsf{\beta }\left[\mathrm{K}\right]$$

(2)

The equation for the damping ratio is given by two Rayleigh coefficients:

$$\mathsf{\zeta }=\frac{\mathsf{ɳ}}{2}$$

(3)

where ɳ is the complex loss factor, ζ is the damping ratio, α is the Rayleigh mass damping coefficient and β is the Rayleigh stiffness damping coefficient. The Rayleigh stiffness damping coefficient was applied and introduced directly in the material properties differently for each material.

The Rayleigh coefficients were determined using the complex natural frequencies [43]. In this study, the dominating vibrating frequencies with the high amplitude resonance peaks, which were 1265 Hz in the radial excitation from Figure 11 and 518 Hz for the lateral excitation from Figure 10, were used to determine the damping ratio and Rayleigh damping coefficients. However, the Rayleigh mass damping coefficient was neglected to avoid an error drag effect on the wheel.

3.3. Harmonic Response Analysis of Wheel with Viscoelastic Damping Layers

3.3.1. Radial Harmonic Analysis of a Wheel Mounted with Viscoelastic Layer on the Web of Both Sides of the Wheel

Figure 14 shows the differences of the wheel without viscoelastic layer (undamped wheel) and with the wheel mounted with viscoelastic layer on the web (damped wheel).

Figure 14b clearly demonstrates that the undamped maximum amplitude peak was 640.24 × 10⁻⁹ mm at a frequency of 1265 Hz, and with viscoelastic damping layer on the web, the amplitude was reduced to 239.54 × 10⁻⁹ mm at a frequency of 1265 Hz.

3.3.2. Radial Harmonic Analysis of a Wheel Mounted with Viscoelastic Layer on Both Sides under the Rim

Figure 15 demonstrates the difference between the wheel without viscoelastic layer (undamped wheel) and with the wheel mounted with viscoelastic layer (damped wheel) under the rim on both sides of the wheel. Figure 15b clearly demonstrates that the undamped maximum amplitude peak was 640.24 × 10⁻⁹ mm at a frequency of 1265 Hz, and after the viscoelastic damping layer was mounted under the rim on both sides of the wheel, the amplitude was reduced to 264.4 × 10⁻⁹ mm at a frequency of 1265 Hz.

3.3.3. Radial Harmonic Analysis of a Wheel Mounted with Viscoelastic Layer on Both Sides of the Web and under the Rim

Figure 16 shows the difference between the wheel without viscoelastic layers (undamped wheel) and the wheel mounted with viscoelastic layers (damped wheel) on the web and under the rim on both sides of the wheel during radial excitation. Evident from Figure 16b is that the undamped wheel during excitation would experience a maximum amplitude of 640.24 × 10⁻⁹ mm at a frequency of 1265 Hz. After the viscoelastic damping was mounted, the results undoubtedly reveal that the maximum peak was completely reduced.

In Figure 17 was highlighted that the wheel, when mounted with a viscoelastic damping layer on both sides of the web and under the rim, would completely reduce the maximum amplitude resonance compared with when it was mounted with a viscoelastic damping layer on the web or under the rim during radial excitation.

3.3.4. Lateral Harmonic Analysis of a Wheel Mounted with Viscoelastic Layer on the Web of Both Sides of the Wheel

Figure 18 demonstrates the difference between the wheel without viscoelastic layer (undamped wheel) and the wheel mounted with viscoelastic layer on the web (damped wheel). The results from Figure 18b demonstrate that the undamped wheel during lateral excitation would experience a maximum amplitude peak of 741.669 × 10⁻⁹ mm at a frequency of 518 Hz. After the wheel was mounted with viscoelastic damping layer on the web and excited laterally, the wheel experienced a maximum amplitude of 741.64 × 10⁻⁹ mm at a frequency of 537 Hz.

3.3.5. Lateral Harmonic Analysis of a Wheel Mounted with Viscoelastic Layer under the Rim on Both Sides of the Wheel

Figure 19b demonstrates the detailed analogy of the wheel without viscoelastic layer (undamped wheel) and with the wheel mounted with viscoelastic layer under the rim (damped wheel). The evidence from Figure 19b demonstrates that the undamped wheel would experience a maximum amplitude of 741.669 × 10⁻⁹ mm at a frequency of 518 Hz, while when the viscoelastic layer was mounted under the rim and excited laterally, the maximum amplitude the wheel experienced was 817.35 × 10⁻⁹ mm at a frequency of 518 Hz.

3.3.6. Lateral Harmonic Analysis of a Wheel Mounted with Viscoelastic Layer on the Web and under the Rim on Both Sides of the Wheel

Figure 20 demonstrates the detailed analogy of a wheel without viscoelastic layer (undamped wheel) and with a wheel mounted with viscoelastic layer on the web and under the rim (damped wheel). The evidence from Figure 20b demonstrates that the undamped wheel would experience a maximum amplitude of 741.669 × 10⁻⁸ mm at a frequency of 518 Hz, while when the viscoelastic layer was mounted under rim and excited laterally, the maximum amplitude the wheel experienced was 465.166 × 10⁻⁸ mm at a frequency of 518 Hz.

Finally, Figure 21 shows the resonance peaks for lateral excitation for the three cases investigated. The evidence from Figure 21 proves that during lateral excitation, mounting the viscoelastic damping layer on both the web and under the rim on both sides of the wheel was more effective than when the viscoelastic damping layer was mounted under the rim on both sides of the wheel.

4. Discussion

In this section, some critical observations are presented to comprehensively understand the outputted results that were explored under the circumstances described above.

The results from Figure 12 and Figure 17 show that the wheel experienced high amplitudes both from radial and lateral excitations. Hence, the frequency value, which was found to have a high amplitude peak on the wheel during excitation, was investigated as indicated in Table 7, which tabulates the natural frequencies and modal vibration numbers. It was found that the wheel resonance corresponded with the wheel mode vibration number 6 for lateral harmonic response and mode vibration number 10 for radial harmonic response. Based on these results, it can be concluded that the wheel would experience resonances at a frequency of 1265 Hz during radial excitation and 518 Hz during lateral excitation, because the harmonic responses of the wheel frequencies at resonance were equal to the natural frequencies of the wheel at mode vibration numbers 6 and 10, respectively.

After the viscoelastic damping layer was mounted in three different configurations on the wheel during radial excitation, the evidence from Figure 17 demonstrates that the viscoelastic damping layer, when mounted on the web and under the rim on both sides of the wheel, exhibited a better damping efficiency than the other cases. After the viscoelastic damping layer was mounted on the web and under the rim on both sides of the wheel, the maximum amplitude was found to be reduced by 442.02 × 10⁻⁹ mm. In the second configuration investigated, when the viscoelastic layer was mounted on the web on both sides of the wheel, the maximum amplitude peak at the resonance frequency was reduced by 400.7 × 10⁻⁹ mm. The final case investigated when the viscoelastic layer was mounted under the rim on both sides of the wheel, and it showed that the maximum amplitude peak at the resonance frequency was reduced by 375.84 × 10⁻⁹ mm. Furthermore, when the damping viscoelastic layer was mounted in three different cases on the wheel during lateral excitation, the evidence from Figure 21 demonstrates that the viscoelastic damping layer, when mounted on the web and under the rim on both sides of the wheel, exhibited a better damping efficiency than the other cases, because the undamped wheel’s maximum amplitude peak was found to be reduced from 741.669 × 10⁻⁹ mm to 465.14 × 10⁻⁹ mm. That means that the layer was able to reduce the amplitudes by 737.02 × 10⁻⁵ mm from the vibrating peak value. In the second case (under lateral excitation), when the viscoelastic damping layer was mounted on the wheel web on both sides of the wheel, the maximum amplitude peak at the resonance frequency was reduced from 741.669 × 10⁻⁹ mm to 741.64 × 10⁻⁹ mm, indicating that the layer was able to reduce the amplitudes by 290 × 10⁻¹³ mm from the vibration peak value. In the last case (under lateral excitation), when the viscoelastic damping layer was mounted under the rim on both sides of the wheel, the maximum amplitude peak was not reduced, since it was found to increase from 741.669 × 10⁻⁹ mm to 817.33 × 10⁻⁹ mm. These findings have good agreement with the findings from Rizwan et al. [7], as they concluded that at certain points during excitation, the damping layer could not be effective on the structure. However, the technique shows good efficiency for suppressing both the radial (vertically) and lateral (horizontally) dynamics of the wheel, except during lateral excitation, when the viscoelastic layer was mounted on the web and under the rim.

Cervello et al. [16] used the same design procedure proposed in this paper to analyze and design a low-noise railway wheel but adopted a constrained damping layer treatment. Jones and Thompson [39] used a combination of finite element complex modal analysis and TWINS software Version 1.1 to calculate the mode shapes and modal masses of the wheel. The constrained damping layer was designed to mount the wheel with different configurations. It is clear that much work was performed to suppress rail wheel vibration using a constrained damping layer. In the present paper, a viscoelastic damping layer (unconstrained damping layer) was selected to reduce excessive vibrations on the locomotive wheel set.

It is clear from Table 7 that the mode shapes of the wheel’s natural frequency during both radial and lateral excitations had an extensive impact on the wheel deformation. The displacement of the wheel depended upon the change of direction and modal shape of the wheel.

It is clear from Figure 9b and Appendix A, Appendix B, Appendix C that the nodes of the wheel mounted with a viscoelastic layer on the web, under the rim and both on the web and under the rim were situated around the light blue color in the code legend. That correlated with the least displacement of the deformed shape and was dominating with respect to the average (yellow and green) and large (red) displacement. The only deformation taking place was on the viscoelastic damping layer, and it can be assumed that this occurred during flexure of the wheel, causing the viscoelastic damping layer to dissipate vibrations by means of compression and tension.

5. Conclusions

The mitigation of vibrations on locomotive wheels by using a numerical and experimental modeling method to design a viscoelastic mechanism was investigated in this manuscript. Three different arrangements were designed and analyzed to examine the effectiveness of the viscoelastic damping layers mounted at different positions in the locomotive wheel. The damping mechanism employed in the study showed good efficiency in suppressing both the radial (vertical) and lateral (horizontal) vibration dynamics of the wheel. The interesting findings deduced from this manuscript can be helpful to train wheel design engineers to mitigate rail wheel vibrations. The configurations from Figure 16 showed a better efficiency in reducing the maximum amplitude peaks than the configurations from Figure 14 and Figure 15 during radial excitation. During lateral excitation, the configurations from Figure 18 demonstrated a better efficiency in mitigating the vibration modes than the configurations from Figure 20 and Figure 21.

The results from Figure 12 clearly validate that the locomotive wheel experienced higher amplitude resonance peaks during lateral excitation than during radial excitation, in agreement with the findings from Balekwa and Kallon [9], when the wheelset of a D39 200 locomotive was excited using a hammer to investigate the vibrational resonance in the lateral, longitudinal and radial directions. Figure 17 and Figure 21 give evidence for the viscoelastic damping layer to be a good damper to mitigate the resonance peaks of the wheel when properly mounted. Furthermore, the viscoelastic damping layer was revealed to be more effective under radial excitation than lateral excitation.