# Scale Models to Verify the Effectiveness of the Methods to Reducing the Vertical Bending Vibration of the Railway Vehicles Carbody: Applications and Design Elements

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Applications of Scale Models in Experimental Research on the Reducing of Vertical Vibrations of the Railway Vehicle Carbody

## 3. Demonstrative Experimental System for Testing the Functionality of a New Method of Reducing Vertical Bending Vibration of the Railway Vehicle Carbody

#### 3.1. Theoretical Concept of the Method

_{1...4}in the axles.

_{c}, and pitch, θ

_{c}, and the first carbody vertical bending, w

_{c}. In the case of bogies, the rigid vibration modes were considered, respectively, the bounce z

_{b}

_{1,2}and pitch θ

_{b}

_{1,2}.

_{1}and B

_{2}).

#### 3.2. Structure of the Demonstrative Experimental System for Laboratory Testing of the Method’s Functionality

#### 3.3. Theoretical Model of the Experimental Model of the Carbody with Anti-Bending Bars

_{c}, with rectangular cross-section of width b and height h (Figure 14).

_{c}= ρbhl

_{c}, where ρ is the density of the material, and the mass of the beam per unit length is m

_{c}= M

_{c}/l

_{c}. The bending stiffness of the beam is E

_{c}I

_{c}, where E

_{c}is the longitudinal modulus of elasticity, and I

_{c}is the moment of inertia of the cross-section of the beam. Additionally, the loss factor due to structural damping was introduced, denoted by η

_{c}. Anti-bending bars of length l

_{b}were fixed rigidly to the lower part of the experimental model of the carbody, at the distances l

_{b}

_{1,2}= (l

_{c}± l

_{b})/2 from the origin of the reference system (XOZ), respectively, at the distance h

_{b}from the neutral axis of the experimental model of the carbody.

_{b}is the longitudinal modulus of the bar material, and D

_{b}is the diameter of the cross-section of bar.

_{zc}and loss factor η

_{zc}. The longitudinal base of the elastic supports was 2a

_{c}, and the distances l

_{c}

_{1,2}= l

_{c}/2 ± a

_{c}fixed the support points of the experimental carbody model on the elastic supports.

_{c}(x, t), was the result of the superposition of the three modes of vibration,

_{c}is the displacement of the experimental model due to the bounce, θ

_{c}is the pitch angle, and T

_{c}is the time-coordinate of the first vertical bending mode; X

_{c}is the eigenfunction associated to the vertical bending mode

_{vb}is the angular frequency of the vertical bending of the experimental model of the carbody without anti-bending bars.

_{o}cosωt, of amplitude F

_{o}and angular frequency ω, acts upon the experimental model of the carbody at the distance x

_{o}from the origin of the reference system.

_{zci}are the forces in the elastic supports, and F

_{xbi}are the forces developed by the anti-bending bars, due to the rotation of the cross-sections where the bars were fixed to the carbody,

_{c}is the mass moment of inertia about the lateral axis of symmetry and, k

_{mc}and M

_{mc}are the modal stiffness and modal mass of the experimental model of the carbody,

#### 3.4. Dimensioning the Experimental Model of the Carbody

_{co}and T

_{co}are the amplitudes of the bounce and vertical bending vibrations.

- –
- bounce,$${f}_{b}=\frac{1}{2\mathsf{\pi}}\sqrt{\frac{4{k}_{zc}}{{M}_{c}}};$$
- –
- vertical bending,$${f}_{vb}=\frac{1}{2\mathsf{\pi}}\sqrt{\frac{{k}_{mc}}{{M}_{mc}}},$$$${f}_{xb}=\frac{1}{2\mathsf{\pi}}\sqrt{\frac{8{\mathsf{\psi}}^{2}{h}_{b}^{2}{k}_{xb}}{{M}_{mc}}},$$$$\mathsf{\kappa}=1+{\mathsf{\epsilon}}^{2}\frac{{M}_{c}}{{M}_{mc}}=1+{\mathsf{\epsilon}}^{2}\frac{{l}_{c}}{{\displaystyle \underset{0}{\overset{{l}_{c}}{\int}}{X}_{c}^{2}(x)\mathrm{d}x}},$$
_{c}_{1}and l_{c}_{2}corresponding to the wheelbase 2a_{c}obtained by scaling the passenger coach wheelbase.

_{xb}= 0 (f

_{xb}= 0), the natural frequencies of the experimental model of the carbody without anti-bending bars result

_{(b_vb)1}and f

_{b}, the vertical bending natural frequency of the experimental carbody model was obtained with the relation

_{c}= 4.7300407449 is the smallest nonzero solution to the characteristic Equation (5).

_{c}and did not depend on the width of the experimental model b.

_{c}, the natural frequency of the vertical bending f

_{vb}and the height of the model h. It followed from Equation (31)

_{c}= 2.6 m was obtained.

_{b}= 30 Hz and f

_{(b_vb)1}= 8 Hz, the frequency f

_{vb}of 10.92 Hz results from Equation (29). According to the relation (32), for f

_{vb}= 10.92 Hz, E

_{c}= 0.7·10

^{11}N/m

^{2}and ρ = 2.7·10

^{3}kg/m

^{3}(aluminium model), h = 0.013 m was obtained by rounding.

_{c}= 36.058 kg.

_{b}= 30 Hz, with the relation

#### 3.5. Dimensioning of Anti-Bending Bars

_{b}), the cross-section area (S

_{b}) of the anti-bending bar could be calculated using the relation (1)

_{b}is the longitudinal modulus of the anti-bending bar.

_{b}is the cross-section inertia moment of the anti-bending bar, ρ

_{b}its density, and α = 4,7300407449.

_{bmax}, the minimum section of the anti-bending bar was further determined,

_{b}

_{min}= 2(S

_{b}

_{min}/π)

^{1/2}.

_{b}= 2.1·10

^{11}N/m

^{2}and ρ

_{b}= 7.850·10

^{3}kg/m

^{3}).

_{b}= 40 mm. It was observed that both the maximum length of the anti-bending bars and their minimum diameter increased approximately linearly as a function of ${f}_{vb}^{b}$, but the increase was more pronounced in the case of the minimum diameter of the anti-bending bars. For example, the maximum length of the anti-bending bars increased according to the natural frequency of the experimental model of the carbody with anti-bending bars from 900 mm, at ${f}_{vb}^{b}=12$ Hz, to 1006 mm for ${f}_{vb}^{b}=18$ Hz, respectively, 11.78%. The increase in the minimum diameter of the anti-bending bars was 89%, respectively, from approx. 4.19 mm to 7.92 mm.

_{b}= 900 mm was imposed and adopted (value which is less than the limit of 966.3 mm, according to Figure 15), then D

_{b}= 6.22 mm results. Rounding to D

_{b}= 6 mm, a deviation of approx. 2.34% in relation to the imposed frequency was obtained.

_{b}= 900 mm, but with different diameters could be adopted. Table 4 shows the values of the natural frequencies of the vertical bending of the experimental model of the carbody with anti-bending bars that could be obtained for bars with a diameter between 5 and 8 mm, mounted at h

_{b}= 40 mm.

#### 3.6. Calculation of the Frequency Response Functions of the Experimental Carbody Model

_{o}cos(ωt), the following complex quantities were associated with the real ones:

_{c}and h

_{b}were the corresponding loss factors.

_{c}/2 iwa considered, and above the elastic supports, x = l

_{c}

_{1,2}.

#### 3.7. Results of the Numerical Simulations

_{zc}= 0.12—for elastic rubber elements and η

_{c}= η

_{b}= 0.01—for the experimental model of the carbody and anti-bending bars.

_{b}= 40 mm from the neutral axis of the experimental model of the carbody, and four diameter values, namely 5 mm, 6 mm, 7 mm and 8 mm.

^{2})/N in the middle of the experimental model, which was about 60% higher than above elastic supports where the value is 1.137 (m/s

^{2})/N.

_{b}= 40 mm and with the same four diameter values of the anti-bending bars as in the previous example. For the calculation of the frequency response function of the acceleration of the experimental model of the carbody without anti-bending bars, the relation (56) was used, adapted to this situation:

^{2})/N and decreased to 1.702 (m/s

^{2})/N in the case of the experimental model of the carbody with anti-bending bars with the diameter of 5 mm. For anti-bending bars with the diameter of 8 mm, the decrease was more important, reaching up to 0.735 (m/s

^{2})/N.

_{b}= 40 mm and at the distance h

_{b}= 50 mm. The presented results show that the increase in resonance frequencies of vertical bending was not only influenced by the diameter of the bar, but also by the distance h

_{b}. As can be seen, for the same diameter of the anti-bending bars, the resonance frequency of the vertical bending of the experimental model of the carbody increased as the distance h

_{b}was higher, and this can be explained by the increase in the bending moment of the longitudinal force in the anti-bending bars.

_{b}, two values were considered, 40 mm and 50 mm. On the one hand, it was also observed here that increasing h

_{b}had the effect of increasing the vertical bending frequency of the experimental model of the carbody by at least 2 Hz. On the other hand, a slight increase in the frequency of vertical bending of the experimental model of the carbody was observed when increasing the length of the anti-bending bar. For example, according to diagram (a), the vertical bending frequency increased from 13.30 Hz for l

_{b}= 600 mm to 14.64 Hz for l

_{b}= 900 mm. This result seems to be paradoxical, since when increasing the length of the anti-bending bars, their longitudinal stiffness decreased (Equation (1)), which led to the reduction in the vertical bending frequency of the experimental model of the carbody. Analysing the relation (24), with the help of which the vertical bending frequency of the experimental model of the carbody with anti-bending bars was calculated, it was observed that the bar length intervened through both stiffness of the anti-bending bars k

_{xb}and parameter ψ = dX

_{c}(l

_{b}

_{1})/dx.

_{xb}and ψ, on the vertical bending frequency is shown in Figure 21, for four values of the diameter of the anti-bending bars. Regardless of the diameter of the anti-bending bars, the vertical bending frequency curves of the experimental model of the carbody showed a maximum of 1565 mm. For smaller values of the length of the bars, the effect of the parameter ψ was dominant and the vertical bending frequency increased as the bars were longer. At larger lengths, the influence of the bar length was manifested mainly through the stiffness of the anti-bending bars and for this reason the vertical bending frequency decreased when the bars are longer.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Tomioka, T.; Takigami, T.; Suzuki, Y. Numerical analysis of three-dimensional flexural vibration of railway vehicle car body. Veh. Syst. Dyn.
**2006**, 44, 272–285. [Google Scholar] [CrossRef] - Huang, C.; Zeng, J.; Luo, G.; Shi, H. Numerical and experimental studies on the car body flexible vibration reduction due to the effect of car body-mounted equipment. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2018**, 232, 103–120. [Google Scholar] [CrossRef] - Yang, G.; Wang, C.; Xiang, F.; Xiao, S. Effect of train carbody’s parameters on vertical bending stiffness performance. Chin. J. Mech. Eng.
**2016**, 29, 1120–1126. [Google Scholar] [CrossRef] - Diana, G.; Cheli, F.; Collina, A.; Corradi, R.; Melzi, S. The development of a numerical model for railway vehicles comfort assessment through comparison with experimental measurements. Veh. Syst. Dyn.
**2002**, 38, 165–183. [Google Scholar] [CrossRef] - Fu, B.; Bruni, S. An examination of alternative schemes for active and semi-active control of vertical car-body vibration to improve ride comfort. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2022**, 236, 386–405. [Google Scholar] [CrossRef] - Jiang, Y.; Chen, B.K.; Thompson, C. A comparison study of ride comfort indices between Sperling’s method and EN 12299. Int. J.Rail Transp.
**2019**, 7, 279–296. [Google Scholar] [CrossRef] - Orvanäs, A. Methods for Reducing Vertical Carbody Vibrations of a Rail Vehicle. In Report in Railway Technology Stockholm; KTH Engineering Sciences Department of Aeronautical and Vehicle Engineering, Division of Rail Vehicles: Stockholm, Sweden, 2010. [Google Scholar]
- Dumitriu, M. On the critical points of vertical vibration in a railway vehicle. Arch. Mech. Eng.
**2014**, 61, 115–140. [Google Scholar] [CrossRef] - Dumitriu, M.; Dihoru, I.I. Influence of bending vibration on the vertical vibration behaviour of railway vehicles carbody. Appl. Sci.
**2021**, 11, 8502. [Google Scholar] [CrossRef] - Bokaeian, V.; Rezvani, M.A.; Arcos, R. The coupled effects of bending and torsional flexural modes of a high-speed train car body on its vertical ride quality. Proc. Inst. Mech. Part K J. Multi-Body Dyn.
**2019**, 233, 979–993. [Google Scholar] [CrossRef] - Dumitriu, M.; Cruceanu, C. Influences of carbody vertical flexibility on ride comfort of railway vehicles. Arch. Mech. Eng.
**2017**, 64, 119–238. [Google Scholar] [CrossRef] - Hui, C.; Weihua, Z.; Bingrong, M. Vertical vibration analysis of the flexible carbody of high speed train. Int. J. Veh. Struct. Syst.
**2015**, 7, 55–60. [Google Scholar] - Zhou, J.; Goodall, R.; Ren, L.; Zhang, H. Influences of car body vertical flexibility on ride quality of passenger railway vehicles. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2009**, 223, 461–471. [Google Scholar] [CrossRef] - Palomares, E.; Morales, A.L.; Nieto, A.J.; Chicharro, J.M.; Pintado, P. Comfort improvement in railway vehicles via optimal control of adaptive pneumatic suspensions. Veh. Syst. Dyn.
**2022**, 60, 1702–1721. [Google Scholar] [CrossRef] - Lewis, T.D.; Jiang, J.Z.; Neild, S.A.; Gong, C.; Iwnicki, S.D. Using an inerter-based suspension to improve both passenger comfort and track wear in railway vehicles. Veh. Syst. Dyn.
**2020**, 58, 472–493. [Google Scholar] [CrossRef] - Sugahara, Y.; Kojima, T. Suppression of vertical vibration in railway vehicle carbodies through control of damping force in primary suspension: Presentation of results from running tests with meter-gauge car on a secondary line. WIT Trans. Built Environ.
**2018**, 181, 329–337. [Google Scholar] - Zheng, X.; Zolotas, A.C.; Goodall, R.M. Combining active structural damping and active suspension control in flexible bodied railway vehicles. In Proceedings of the 35th Chinese Control Conference, Chengdu, China, 27–29 July 2016; pp. 8938–8944. [Google Scholar]
- Dumitriu, M. Influence of the suspension damping on ride comfort of passenger railway vehicles. UPB Sci. Bull. Ser. D Mech. Eng.
**2012**, 74, 75–90. [Google Scholar] - Sugahara, Y.; Watanabe, N.; Takigami, T.; Koganei, R. Development of primary suspension damping control system for suppressing vertical bending vibration of railway vehicle car body. In Proceedings of the 9th World Congress on Railway (WCRR 2011), Lille, France, 22–26 May 2011; p. 115148008. [Google Scholar]
- Sugahara, Y.; Watanabe, N.; Takigami, T.; Koganei, R. Vertical vibration suppression system for railway vehicles based on primary suspension damping control—system development and vehicle running test results. Q. Rep. RTRI
**2011**, 52, 13–19. [Google Scholar] [CrossRef] - Sugahara, Y.; Kazato, A.; Koganei, R.; Sampei, M.; Nakaura, S. Suppression of vertical bending and rigid-body-mode vibration in railway vehicle car body by primary and secondary suspension control: Results of simulations and running tests using Shinkansen vehicle. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2009**, 223, 517–531. [Google Scholar] [CrossRef] - Sugahara, Y.; Takigami, T.; Kazato, A.; Koganei, R.; Sampei, M. Suppression of vertical vibration in railway vehicles by damping force control of primary suspension using an LQG controller. J. Syst. Des. Dyn.
**2008**, 2, 251–262. [Google Scholar] [CrossRef] - Sugahara, Y.; Takigami, T.; Sampei, M. Suppressing vertical vibration in railway vehicles through primary suspension damping force control. J. Syst. Des. Dyn.
**2007**, 1, 224–235. [Google Scholar] [CrossRef] - Wu, P.; Zeng, J.; Dai, H. Dynamic response analysis of railway passenger car with flexible carbody model based on semi-active suspensions. Veh. Syst. Dyn.
**2004**, 41, 774–783. [Google Scholar] - Foo, E.; Goodall, R.M. Active suspension control of flexible bodied railway vehicles using electro-hydraulic and electromagnetic actuators. Control Eng. Pract.
**2000**, 8, 507–518. [Google Scholar] [CrossRef] - Gong, D.; Wang, K.; Duan, Y.; Zhou, J. Car body floor vibration of high-speed railway vehicles and its reduction. J. Low Freq. Noise Vibr. Act. Control
**2020**, 39, 925–938. [Google Scholar] [CrossRef] - Dumitriu, M. Study on improving the ride comfort in railway vehicles using anti-bending dampers. Appl. Mech. Mater.
**2018**, 880, 207–212. [Google Scholar] [CrossRef] - Graa, M. Modeling and control for vertical rail vehicle dynamic vibration with comfort evaluation. Int. J. Math. Model. Methods Appl. Sci.
**2017**, 11, 240–245. [Google Scholar] - Dumitriu, M. A new approach to reducing the carbody vertical bending vibration of railway vehicles. Veh. Syst. Dyn.
**2017**, 55, 1787–1806. [Google Scholar] [CrossRef] - Gong, D.; Zhou, J.; Sun, W. Passive control of railway vehicle car body flexural vibration by means of under frame dampers. J. Mech. Sci. Technol.
**2017**, 31, 555–564. [Google Scholar] [CrossRef] - Gong, D.; Zhou, J.; Sun, W.; Sun, Y.; Xia, Z. Method of multi-mode vibration control for the carbody of high-speed electric multiple unit trains. J. Sound Vib.
**2017**, 409, 94–111. [Google Scholar] [CrossRef] - Gong, D.; Zhou, J.S.; Sun, W.J. On the resonant vibration of a flexible railway car body and its suppression with a dynamic vibration absorber. J. Vib. Control
**2013**, 19, 649–657. [Google Scholar] [CrossRef] - Tomioka, T. Reduction of car body elastic vibration using high-damping elastic supports for under-floor equipment. Railw. Technol. Avalanche
**2012**, 41, 245–270. [Google Scholar] - Tomioka, T.; Takigami, T. Reduction of bending vibration in railway vehicle carbodies using carbody–bogie dynamic interaction. Veh. Syst. Dyn.
**2010**, 48, 467–486. [Google Scholar] [CrossRef] - Takigami, T.; Tomioka, T. Bending vibration suppression of railway vehicle carbody with piezoelectric elements (Experimental results of excitation tests with a commuter car). J. Mech. Syst. Transp. Logist.
**2008**, 1, 111–121. [Google Scholar] [CrossRef] - Schandl, G.; Lugner, P.; Benatzky, C.; Kozek, M.; Stribersk, M. Comfort enhancement by an active vibration reduction system for a flexible railway car body. Veh. Syst. Dyn.
**2007**, 45, 835–847. [Google Scholar] [CrossRef] - Dumitriu, M.; Cruceanu, C. Approaches for reducing structural vibration of the carbody railway vehicles. In Proceedings of the MATEC Web of Conferences—21st Innovative Manufacturing Engineering & Energy International Conference—IManE&E, 2017, Iași, Romania, 24–26 May 2017; Volume 112, p. 07006. [Google Scholar]
- Skachkova, A.; Trifonova, V.; Zaytseva, A.; Samoshkin, S. Methods of suppression of elastic oscillations of bodies of passenger railcars. Transp. Res. Procedia.
**2021**, 54, 522–529. [Google Scholar] [CrossRef] - Dumitriu, M. Numerical study on the influence of suspended equipments on the ride comfort in high speed railway vehicles. Sci. Iran.
**2020**, 27, 1897–1915. [Google Scholar] [CrossRef] - Dumitriu, M. Numerical analysis on the influence of suspended equipment on the ride comfort in railway vehicles. Arch. Mech. Eng.
**2018**, 65, 477–496. [Google Scholar] - Huang, C.; Zeng, J. Suppression of the flexible carbody resonance due to bogie instability by using a DVA suspended on the bogie frame. Veh. Syst. Dyn.
**2022**, 60, 3051–3070. [Google Scholar] [CrossRef] - Sun, Y.; Zhou, J.; Gong, D.; You, T.; Che, J. Research on multi-point connection of under-chassis equipment suspension system in high-speed trains. AIP Adv.
**2021**, 11, 125315. [Google Scholar] [CrossRef] - Wang, Q.; Zeng, J.; Wu, Y.; Zhu, B. Study on semi-active suspension applied on carbody underneath suspended system of high-speed railway vehicle. J. Vib. Control
**2020**, 26, 671–679. [Google Scholar] [CrossRef] - Guo, J.; Shi, H.; Li, F.; Wu, P. Field measurements of vibration on the car body-suspended equipment for high-speed rail vehicles. Shock Vib.
**2020**, 2020, 6041543. [Google Scholar] [CrossRef] - Guo, J.; Shi, H.; Luo, R.; Wu, P. Parametric analysis of the car body suspended equipment for railway vehicles vibration reduction. IEEE Access
**2019**, 7, 88116–88125. [Google Scholar] [CrossRef] - Zhu, T.; Lei, C.; Xiao, S.; Yu, J.P. Suspension stiffness selecting method of elastic suspension equipment under vehicle. J. Traffic Transp. Eng.
**2018**, 18, 111–118. [Google Scholar] - Sun, W.; Zhou, J.; Gong, D.; You, T. Analysis of modal frequency optimization of railway vehicle car body. Adv. Mech. Eng.
**2016**, 8, 1–12. [Google Scholar] [CrossRef] - Gong, D.; Zhou, J.; Sun, W. Influence of under-chassis-suspended equipment on high-speed EMU trains and the design of suspension parameters. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2016**, 230, 1790–1802. [Google Scholar] [CrossRef] - Shi, H.; Wu, P. Flexible vibration analysis for car body of high-speed EMU. J. Mech. Sci. Technol.
**2016**, 30, 55–66. [Google Scholar] [CrossRef] - Dumitriu, M. Influence of suspended equipment on the carbody vertical vibration behavior of high-speed railway vehicles. Arch. Mech. Eng.
**2016**, 63, 145–162. [Google Scholar] [CrossRef] - Aida, K.Y.; Tomioka, T.; Takigami, T. Reduction of carbody flexural vibration by the high-damping elastic support of under-floor equipment. Q. Rep. RTRI
**2015**, 56, 262–267. [Google Scholar] [CrossRef] [PubMed] - Luo, G.; Zeng, J.; Wang, Q. Identifying the relationship between suspension parameters of underframe equipment and carbody modal frequency. J. Mod. Transp.
**2014**, 22, 206–213. [Google Scholar] [CrossRef] [Green Version] - Shi, H.; Luo, R.; Wu, P. Influence of equipment excitation on flexible carbody vibration of EMU. J. Mod. Transp.
**2014**, 22, 95–205. [Google Scholar] [CrossRef] - Shi, H.L.; Luo, R.; Wu, P.B. Application of DVA theory in vibration reduction of carbody with suspended equipment for high-speed EMU. Sci. China Technol. Sci.
**2014**, 57, 1425–1438. [Google Scholar] [CrossRef] - Shi, H.L.; Luo, R.; Wu, P.B.; Zeng, J. Suspension parameters designing of equipment for electric multiple units based on dynamic vibration absorber theory. J. Mech. Eng.
**2014**, 50, 155–161. [Google Scholar] [CrossRef] - Shi, H.L.; Wu, P.B.; Luo, R. Coupled vibration characteristics of flexible car body and equipment of EMU. J. Southwest JiaoTong Univ.
**2014**, 49, 693–699. [Google Scholar] - Wu, H.C.; Wu, P.B.; Zeng, J. Influence of equipment under car on carbody vibration. J. Traffic Transp. Eng.
**2012**, 12, 50–56. [Google Scholar] - Sun, V.; Gong, D.; Zhou, J.; Zhao, Y. Influences of suspended equipment under car body on highspeed train ride quality. Procedia Eng.
**2011**, 16, 812–817. [Google Scholar] [CrossRef] - Sun, Y.; Gong, D.; Zhou, J. Study on vibration reduction design of suspended equipment of high speed railway vehicles. J. Phys.
**2016**, 744, 012212. [Google Scholar] [CrossRef] - Hansson, J.; Takano, M.; Takigami, T.; Tomioka, T.; Suzuki, Y. Vibration suppression of railway car body with piezoelectric elements (A study by using a scale model). JSME Int. J. Ser. C
**2004**, 47, 451–456. [Google Scholar] - Takigami, T.; Tomioka, T. Vibration suppression of scale model of railway carbody with piezoelectric elements (a study focused on designing shunt circuits). In Proceedings of the 12th SPIE International Symposium Smart Structures and Materials, San Diego, CA, USA, 17 May 2005; Volume 5760, pp. 329–340. [Google Scholar]
- Takigami, T. Investigation into suppressing the bending vibration of railway vehicle carbody with piezoelectric elements. Railw. Technol. Avalanche
**2005**, 9, 54–60. [Google Scholar] - Takigami, T.; Tomioka, T. Investigation to suppress bending vibration of railway vehicle carbodies using piezoelectric elements. Q. Rep. RTRI
**2005**, 46, 225–230. [Google Scholar] [CrossRef] - Takigami, T.; Tomioka, T.; Hansson, J. Vibration suppression of railway vehicle carbody with piezoelectric elements (A study by using a scale model of Shinkansen). J. Adv. Mech. Des. Syst. Manuf.
**2007**, 1, 649–660. [Google Scholar] [CrossRef] - Kamada, T.; Kiuchi, R.; Nagai, M. Suppression of railway vehicle vibration by shunt damping using stack type piezoelectric transducers. Veh. Syst. Dyn.
**2008**, 46, 561–570. [Google Scholar] [CrossRef] - Kamada, T.; Hiraizumi, T.; Nagai, M. Active vibration suppression of lightweight railway vehicle body by combined use of piezoelectric actuators and linear actuators. Veh. Syst. Dyn.
**2010**, 48, 73–87. [Google Scholar] [CrossRef] - Melero, M.; Nieto, A.J.; Morales, A.L.; Palomares, E.; Chicharro, J.M.; Ramiro, C.; Pintado, P. Experimental analysis of constrained layer damping structures for vibration isolation in lightweight railway vehicles. Appl. Sci.
**2022**, 12, 8220. [Google Scholar] [CrossRef] - Suzuki, E.; Watanabe, K.; Hoshino, H.; Yonezu, T.; Nagai, M. A study of Maglev vehicle dynamics using a reduced-scale vehicle model experiment apparatus. In Proceedings of the International Symposium on Speed-up, Safety and Service Technology for Railway and Maglev Systems (STECH’09), Niigata, Japan, 16–19 June 2009; p. 355984. [Google Scholar]
- Suzuki, E.; Watanabe, K.; Hoshino, H.; Yonezu, T.; Nagai, M. A study of Maglev vehicle dynamics using a reduced-scale vehicle model experiment apparatus. J. Mech. Syst. Transp. Logist.
**2010**, 3, 196–205. [Google Scholar] [CrossRef] - Kozek, M.; Bilik, C.; Benatzky, C. A PC-based flexible solution for virtual instrumentation of a multi-purpose test bed. Int. J. Online Biomed. Eng. Eng.
**2006**, 2, 1–5. [Google Scholar] - Popprath, S.; Benatzky, C.; Bilik, C.; Kozek, M.; Stribersky, A.; Wassermann, J. Experimental modal analysis of a scaled car body for metro vehicles. In Proceedings of the 13th International Congress on Sound and Vibration, Vienna, Austria, 2–6 July 2006. [Google Scholar]
- Kozek, M.; Benatzky, C.; Schirrer, A.; Stribersky, A. Vibration damping of a flexible car body structure using piezo-stack actuators. In Proceedings of the 17th World Congress the International Federation of Automatic Control, Seoul, Republic of Korea, 6–11 July 2008. [Google Scholar]
- Schirrer, A.; Kozek, M.; Schöftner, J. MIMO Vibration control for a flexible rail car body: Design and experimental validation. In Vibration Analysis and Control—New Trends and Development; Carbajal, F.B., Ed.; InTech: London, UK, 2011; pp. 309–336. [Google Scholar]
- Bokaeian, V.; Rezvani, M.A.; Arcos, R. A numerical and scaled experimental study on ride comfort enhancement of high-speed rail vehicle through optimizing traction rod stiffness. J. Vib. Control
**2021**, 27, 2548–2563. [Google Scholar] [CrossRef] - Mazilu, T.; Gheți, M.A. On the bending vibration of a train driving wheelset. In Proceedings of the IOP Conference Series: Materials Science and Engineering, 2019, ModTech International Conference—Modern Technologies in Industrial Engineering, Iași, Romania, 19–22 June 2019; Volume 591, p. 012059. [Google Scholar]
- Mazilu, T.; Răcănel, I.R.; Ghindea, C.L.; Cruciat, R.L.; Leu, M.C. Rail joint model based on the Euler-Bernoulli beam theory. Rom. J. Transp. Infrastruct.
**2019**, 8, 16–29. [Google Scholar] [CrossRef] - Mazilu, T. The dynamics of an infinite uniform Euler-Bernoulli beam on bilinear viscoelastic foundation under moving loads. Procedia Eng.
**2017**, 199, 2561–2566. [Google Scholar] [CrossRef] - UIC 513R; Guidelines for Evaluating Passenger Comfort in Relation to Vibration in Railway Vehicles. International Union of Railways: Paris, France, 1994.
- EN 12299; Railway Applications. Ride Comfort for Passengers. Measurement and Evaluation. British Standard: London, UK, 1999.
- Meirovitch, L. Elements of Vibration Analysis; McGraw-Hill International Edition: New York, NUY, USA, 1986; pp. 235–238. [Google Scholar]
- ROMARC SA. Available online: https://romarc.ro/2019/04/04/vagon-salon-ava-200/ (accessed on 11 February 2023).
- Sebeşan, I.; Dumitriu, M.; Tudorache, C. Contributions to the study of the elastic constants for suspensions used on high-speed steering bogies. Mech. Transp. Commun.
**2009**, 3, 30–34. [Google Scholar]

**Figure 2.**The commuter type test vehicle [35].

**Figure 4.**The scaled aluminium beam; mounting supports detail [67].

**Figure 5.**Types of scaled beams: (

**a**) without CLD; (

**b**) with CLD patch with uniform constraining layer; (

**c**) with CLD patch with honeycomb constraining layer [67].

**Figure 6.**CLD configurations analysed experimentally [67].

**Figure 8.**Overview of the test bench [70].

**Figure 9.**The rigid-flexible coupled model of the railway vehicle [29].

**Figure 10.**Effect of the anti-bending bars upon the carbody dynamic response at velocity of 270 km/h: (

**a**) at the carbody centre; (

**b**) above the front bogie, and (

**c**) above the rear bogie [29].

**Figure 11.**Influence of the anti-bending bars upon ride comfort: (

**a**) at the carbody centre; (

**b**) above the front bogie, and (

**c**) above the rear bogie [29].

**Figure 17.**Component of the acceleration response function of the experimental model of the carbody corresponding to the vertical bending: (

**a**) at middle; (

**b**) above elastic supports.

**Figure 18.**Maximum of the acceleration frequency response function along the experimental carbody model.

**Figure 19.**Influence of the diameter of the anti-bending bars in correlation with the distance h

_{b}on the frequency response function of the acceleration in the middle of the experimental mode of the carbody: (

**a**) h

_{b}= 40 mm; (

**b**) h

_{b}= 50 mm.

**Figure 20.**Influence of the length of the anti-bending bars in correlation with the distance h

_{b}on the frequency response function of the acceleration in the middle of the experimental model of the carbody: (

**a**) h

_{b}= 40 mm; (

**b**) h

_{b}= 50 mm.

**Figure 21.**Influence of the length of the anti-bending bars on the vertical bending frequency of the experimental model of the carbody.

**Table 1.**The main dimensions of the carbody of the AVA 200 passenger coach [81].

Length | 26 m |

Width | 2.825 m |

Height | 4.050 m |

Wheelbase | 19 m |

Length | l_{c} = 2.6 m |

Width | b = 0.4 m |

Height | h = 0.013 m |

Longitudinal distance between elastic supports (wheelbase) | 2a_{c} = 1.9 m |

Longitudinal modulus of elasticity | E_{c} = 0.7·10^{11} N/m^{2} |

Material density (aluminium) | ρ = 2.7·10^{3} kg/m^{3} |

Carbody mass | M_{c} = 36.504 kg |

Modal mass | M_{mc} = 37.816 kg |

Modal stiffness | k_{mc} = 151.244 kN/m |

Elastic support stiffness | k_{zc} = 324.252 kN/m |

Bounce | f_{b} = 30 Hz |

Vertical bending | f_{vb} = 10.06 Hz |

Natural frequencies of coupled vibration of bounce and vertical bending | f_{(b_vb)1} = 7.90 Hz |

f_{(b_vb)2} = 38.20 Hz |

**Table 4.**Frequency of the experimental model of the carbody with anti-bending bars as a function of the minimum diameter of the anti-bending bars.

The diameter of the anti-bending bars [mm] | 1 | 2 | 3 | 4 |

Vertical bending frequency of the experimental carbody model with anti-bending bars [Hz] | 13.07 | 14.64 | 16.20 | 17.71 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dumitriu, M.; Mazilu, T.; Apostol, I.I.
Scale Models to Verify the Effectiveness of the Methods to Reducing the Vertical Bending Vibration of the Railway Vehicles Carbody: Applications and Design Elements. *Appl. Sci.* **2023**, *13*, 2368.
https://doi.org/10.3390/app13042368

**AMA Style**

Dumitriu M, Mazilu T, Apostol II.
Scale Models to Verify the Effectiveness of the Methods to Reducing the Vertical Bending Vibration of the Railway Vehicles Carbody: Applications and Design Elements. *Applied Sciences*. 2023; 13(4):2368.
https://doi.org/10.3390/app13042368

**Chicago/Turabian Style**

Dumitriu, Mădălina, Traian Mazilu, and Ioana Izabela Apostol.
2023. "Scale Models to Verify the Effectiveness of the Methods to Reducing the Vertical Bending Vibration of the Railway Vehicles Carbody: Applications and Design Elements" *Applied Sciences* 13, no. 4: 2368.
https://doi.org/10.3390/app13042368