IJR International Journal of Railway Vol. 6, No. 4 / December 13, pp. 155-159 The Korean Society for Railway Vibration Analysis of Railway Tracks Force by Distribute Moving Loas Sinyeob Lee*, Dongkyu Kim*, Sangkeun Ahn* an Junhong Park Abstract The purpose of this stuy was to evelop a theoretical moel to analyze the vibration of finite railways force by istribute moving loas. The vibration characteristics of compliantly supporte beam utilizing compressional amping moel were investigate through the Rayleigh-Ritz metho. The istribute moving loa was analyze as the cross correlation function on railways. This allowe the use of statistical characteristics for simulation of the moving train wheels on the rail. The results showe there is a critical velocity inucing resonant vibration of the rail. The mass spring resonance from the rail fastening systems exhibite significant influence on the resulting vibration response. In particular, the effect of the viscoelastic core amping was investigate as an efficient metho for minimizing rail vibration. The ecrease of the average vibration an rolling noise generation by the amping core was maximize at the mass-stiffness-mass resonance frequency. Keywors : Rayleigh-ritz metho, Rolling noise, Sanwich panel 1. Introuction A complex structures compose of multi-layer panels is use to issipate vibration energy in many engineering applications. Damping material inserte as core in complex structures reuces resonant responses an noise raiation. Damping treatments have avantages on costs an maintenance compare to other vibration reuction methoologies. The moal characteristics of entire systems are etermine from interaction between each layer for the complex structures. Moving loa cause by train inuces track vibration an is an important source of environmental noise on the nearby resiential areas. Theoretical moel for vibration of the complex structure with core amping material is require for optimal esign for ecrease structural vibration in the frequency range of noise pollution. Sanwich moels to analyze vibration of complex structures have been stuie for ecaes. Rao an Nakra [1] propose the governing equation require for analyzing * Corresponing author: Hanyang University E-mail : parkj@hanyang.ac.kr Hanyang University cthe Korean Society for Railway 13 http://x.oi.org/1.778/ijr.13.6.4.155 vibration of sanwich panel with viscoelastic layer between two rigi panels. Douglas an Yang [] analyze frequency which influences translational eformation of viscoelastic layer in sanwich beam an compare ifferent vibration moels. The translational vibration was ominant as the thickness of the upper stiff panels increases. [3] Although the multi-layer construction is common in railway tracks, the numerical metho for investigation the ranom vibration in auio frequency ranges has not been propose, especially on analysis of the effects of the core ynamic properties. In this paper, rail-slab complex structure is analyze by the compressional amping moel. The influence of the core amping material on the vibration characteristics of the rail structures was analyze through the Rayleigh-Ritz metho. The istribute moving force using statistical characteristics was utilize for simulation of loas from operating trains to the rail structures. The influence of the core amping layer on resulting vibration was investigate.. Free Vibration Analysis of a Rail on a Slab.1 Rayleigh-Ritz methos As shown in Fig. 1 the numerical approaches for Vol. 6, No. 4 / December 13 155
Sinyeob Lee / IJR, 6(4), 155-159, 13 The potential energy by the bening stiffness of the beam an plate were given as follows: Fig. 1 The schematic of the rail-slab complex structures use for analyzing the vibration of railway tracks unerstaning the railway track vibration ware performe. The moving loas from the operating train are applie to the each beam. The Rayleigh-Ritz metho was applie for the vibration analysis of the track moel. Shear eformation of the core amping materials between beam an plate was ignore. This assumption is typically applie when the bening stiffnesses of the beam an the plate are large an the moulus of amping material is small. In aition, changes in the ynamic properties of the amping material which typically shows variation of 5-1% with frequency was ignore. Displacements of each structural element were assume as follows using the N trial functions for the beam an N trial functions for the plate. Lagrangian equations were applie to these equations as --- L ----------- L (1a) t = = 1 N mn --- ----------- L L = = 1 N t mn (1b) --- L ----------- L = = 1 N (1c) t mn where L=T V p V K is the system Lagrangian. The isplacements of the plates (w 1,w 4 ) an the beams (w,w 3 ) represente as the prouct of trial function an normal coorinate (, ): w 1 w w 3 = m 1 n 1 m m 3 m m. () The kinetic an potential energies of the system were calculate for the application of Rayleigh-Ritz metho. The kinetic energy was calculate as ab 1 T -- h w 1 -------- 1 y x -- h w x r1 -------------------------- = x t + t a + a 1 --. (3) s h w 3 x r s -------------------------- t x.(4) The potential energy by the supporting stiffness between the slab an rail, k s, an stiffness between the plate an floor, k f, were calculate as follow: a b 1 a 1 V K = --kˆfw 1y x + --kˆs w 1 w x a 1 + --kˆs w. (5a) 1 w 3 x. Selection an application of trial function for analysis Selection of the appropriate trial functions is important for minimizing numerical errors an computation times. In this stuy, beam functions are applie for trial functions after assuming that the structures are fixe at eges in y irections an free in x irections. When the beam length is large, the simulation result is similar to infinite beam, an is preferre to represent the actual railway structures. The length an with of the moel were assume as.5 m an 3m, respectively. 3. Response of Railway Track Moel Force by Distribute Moving Loa 3.1 Transfer function Transfer function to harmonic excitation was use to calculate the vibration response of railway track moel. The transfer function is etermine by the moal shape functions an natural frequencies obtaine from free vibration analysis of the Rayleigh-Ritz metho. The isplacements of the railway track using moal shape functions are given as: N + N w 13 t Re j 13 qˆ e j t = j. (6) j Cross correlation ensity function is efine as the Fourier transform of the prouct of the force an time (spaceelaye force) as follows: A S pp' kx 1 x ----- e j V k x x 1 =. (7) The equations of motion in the moal coorinates are given as follows + j q ˆ j = ˆ j f. j = 1 N + N (8) 156
Vibration Analysis of Railway Tracks Force by Distribute Moving Loas 3. Calculation of average vibration response from istribute moving loas The vibration response by the istribute moving loa was analyze through the cross correlation function. The cross correlation function for the translational vibration responses of the railway track was efine as follows using the transfer response function of forces, Spp, acting to each plates as Table 1 Applie properties of rail an slab 1 1 Rail Slab Length(a) 3 m 3 m With(b).5 m.45 m Thickness(h).518 m Density( ) 41 kg/m 7 kg/m3 Young's Moulus (E) 1 GPa 5 GPa Loss factor ( ).1.1 Moment of inertia (I) 1-4 m4 Syy r1 r = H r1 s1 H r s Spp s1 s s1 s, Properties (9) kf 187 MPa/m ks 8 MPa/m where r1 = xp1 yp1 xb1 1 xb1 an r = xp yp xb 1 x xb. Similarly, the variables s1 an s are positions of force acting on the rail. Spatial average vibration is represente as follows Syy r1 r A = A N + N ab Hj Hj Vj 1 3Spp. (1) j=1 The wavenumber components of the force is represente as A j V k x x1. Spp k x1 x = -----e (11) From equation (1), the vibration response from arbitrary moving loas is obtaine by superposing responses of impulse forces, which has wavenumber k. As a result, when the loa moving with velocity V is applie to the railway, loas applie to each plate an beam are analyze. 3.3 Response of rail-slob moel by moving loa When the loa moves on the railway, the responses of the rail an the slab were calculate. With the rail excite by the wheel of the train, it was assume that the external excitation was inuce by the pressing force at the contacting area. The mechanical properties of the rails assuming 41GPU rail an concrete slabs applie for the numerical analysis is summarize in Table 1. Analysis was conucte with N = 3 to warrant the convergence up to Hz. The spee of the loa was increase from m/s by.5 m/s to investigate the effects of moving velocity. The average vibration response increases significantly with the increasing velocity of the loa as shown in Fig.. The frequency of maximum vibration response also increase with a constant slope epening on the velocity. When the velocity of the maximum vibration response an Fig. Average velocity response of (a) the rail an (b) the slab with moving loa spee. The vibration response increase with the increasing moving velocity the natural frequency of the railways are ientical, it resulte in very resonant response. For the rail, the resonances at 8 Hz an 1 Hz were observe. The vibration magnitue of the rail was much larger than those of the slab, as expecte from irect application of the force on the rail. 157
Sinyeob Lee / IJR, 6(4), 155-159, 13 Fig. 3 Comparison of average velocity of (a) the rail an (b) the slab with change in the stiffness of the core amping material Fig. 4 Effects of amping in the core material on the average velocity of (a) the rail an (b) the slab. The amping showe to have significant influence on the resonant responses 3.4 Change of structural vibration characteristics by core amping materials The rail pas in railway tracks are use to reuce vibration an noise by moving train wheels on the rail an to prevent rails from excessive corrugations. The system ynamic characteristics are influence by inserte rail pas. The effect of the core amping material on the complex structure is require to be unerstoo. Fig. 3 shows the velocity response of the rail an slab with the change in the stiffness of the rail pas. Two harmonic forces of 5 kn were assume to move with the velocity of 1 m/s an.3 m apart. When the stiffness increase from 8 MPa to 14 MPa, the natural frequency change, an consequently, the frequency range of maximum vibration response showe cyclic variation. The slab response showe a ifferent tren compare to that of the rail with the increasing frequency. Whereas the rail vibration magnitues were reuce with increasing frequency from to 3 Hz, the slab vibration responses increase in the same frequency range. The resonant frequency increase for both the rail an slab with the increasing stiffness. This suggeste that their interaction shoul be incorporate to stuy the rolling noise generation. The effects of the core amping material on the vibration response from the moving loas were analyze as shown in Fig. 4. With the increasing amping ratio, the vibration magnitue was reuce especially at resonant frequencies. This frequency range correspons to the mass-stiffness resonance region assuming the rail as the mass an the rail pa as the spring. In esign process, the mass-stiffness resonant frequency is an important factor on estimating the effect of the rail pa amping. This amping is also influence by the fastening system as well as the viscoelastic properties of the pas. 158
Vibration Analysis of Railway Tracks Force by Distribute Moving Loas 4. Conclusions This stuy investigate the vibration of the sanwich structures excite by istribute moving loas. The effects of the core ynamic properties of the sanwich complex structure representing the railway tracks on the force response were analyze using the Rayleigh-Ritz metho. Cross correlation functions for the isplacement responses from the moving ynamic loas were applie to calculate response from the istribute loas. Through parametric stuies of the moving velocity an the core amping properties, the important parameters having influence on the rolling noise generation were ientifie. The natural frequency increase an vibration of rail ecrease with the increasing stiffness of the core amping material, but those of the slab increase. With the increasing amping in the core, the vibration responses of the rail an slab were reuce in the range near the mass-spring resonance frequency. The propose analysis proceures are require when stuying the vibration an noise reuction systems on complex railway structures equippe with various fastening an floating systems or functional concrete slabs. References 1. Saasiva, Y. V. K. an Nakra, B. C. (1974). Vibration of Unsymmetrical Sanwich Beams an Plates with Viscoelastic Cores, Journal of Soun an Vibration, Vol. 34, pp. 39-36.. Douglas, B. E. an Yang, J. C. S. (1978). Transverse Compressional Damping in the Vibratory Response of Elastic- Viscoelastic-Elastic Beams, The American Institute of Aeronautics an Astronautics, Vol. 16, No. 9, pp. 95-93. 3. Sisemore, C. L. an Darvennes, C. M. (). Transverse vibration of Elastic-Viscoelastic-Elastic Sanwich Beams, Journal of Soun an Vibration, Vol. 5, pp.155-167 4. Nguyen, V. H. an Duhamel, D. (8). Finite Element Proceures for Nonlinear Structures in Moving Coorinates. Part II: Infinite Beam Uner Moving Harmonic Loas, Computers & Structures, Vol. 86, pp. 56-63 5. Wu, J. S. an Shin, P. Y. (). Dynamic Responses of Railway an Carriage Uner the High-spee Moving Loas, Journal of Soun an Vibration, Vol. 36, pp. 61-87 6. Park, J.,.Mongeau, L. an Siegmun, T. (3). Analysis of the Flow-inuce Vibrations of Viscoelastically Supporte Rectangular Plates, Journal of Soun an Vibration, Vol. 64, pp. 5-45. 7. Park, J. an Mongeau, L. (1999). Effects of Seal Mechanical Properties on Soun Raiation from Roa Vehicle Sieglass Winows, Proceeings of Inter-noise 99, Fort Lauerale FL, pp. 795-8. 8. Lee Y. Y., Kim J. C. an Lee H. S. (11). A Stuy to Estimate the Emitte Soun Power from the Rolling Stock, Journal of the Korean Society for Railway, Vol. 14, No. 3, pp 11-15. 9. Newlan, D. E. (1993) An Introuction to Ranom Vibrations, Spectral & Wavelet Analysis, Aison-Wesley Longman, Essex UK. 159