Nonlinear Vibration Analysis of Conventional Train A.Sridhar 1 J.Venkatesh 2 P.Pascal Jayaseelan 3 1 (Mechanical engineering, KSR College of Engineering, Namakkal, sridhar@ksrce.ac.in) 2 (Mechanical engineering, KSR College of Engineering, Namakkal, venkatksr1987@gmail.com) 3 (Mechanical engineering, KSR College of Engineering, Namakkal,jaiyu666@gmail.com) Abstract Trains are mass transport system and used to transport goods. People living in rural areas are shifting into cities for performing their daily task in the companies and offices, as the companies were switched in the inner part of the city. Because of this reason cities are overcrowding at a faster rate and high speed trains are required for people who are travelling long distance. Therefore, high speed trains need to manage in the traveling crowd and to minimize the travel time. The objective of this project is to improve the speed of the trains. The design of a high speed trains very complex due to the dynamics involved. Conventional linear theory is not adequate to deal with this kind of situation. The collapse of the Banana Bridge in the USA is the classical example of the failure of linear based design. The project deals with vibration analysis of the vehicle and track models by dynamics theory. The dynamic performance of the vehicle and track simulated using MATLAB and also to be compare the dynamic performance of the smooth surface wheel set and irregular surface wheel set using simple experimental setup. Finally, we give some recommendations for safe and effective design of conventional trains. Key words: High Speed Trains, Dynamics, Vibration Analysis, MATLAB, Wheels. 1. INTRODUCTION Trains are mass transport system and used to transport goods. People living in rural areas are shifting into cities for performing their daily task in the companies and offices, as the companies were switched in the inner part of the city. Because of this reason cities are overcrowding at a faster rate and high speed trains are required for people who are travelling long distance. So we need to improve the speed of the conventional trains. Due to the above reason, make safe and effective design of train by using nonlinear dynamics theory. Problem Identification The design of a high speed train is very complex due to the dynamics involved. The design of high speed trains based on the conventional linear theory is not adequate to deal with this kind of situation. The collapse of the Banana bridge in the USA is the classical example of the failure of linear theory based design. In this project work, a test for safe and effective high speed train design based on nonlinear theory will be performed. Another example, include the Jump phenomenon occurs in aircraft which can t capture by linear theory. 2. MATHEMATICAL MODEL By using the mathematical expression to define the system is known as mathematical model. It consists of the following elements, Car body Bogie frame Wheelset Primary suspension Secondary suspension Fig 1 Combined model 3. NUMERICAL SIMULATION In this phase discuss about the numerical simulation of the vehicle model and the track model. That model simulated using MATLAB (MATrix LABoratory). The mathematical model of the vehicle and the track converted into a Matlab function (script) file and program data. Equation of motion of the vehicle model in the form of second order differential equation and it s converted into two first order equations. a) Vehicle model Volume: 03 Issue: 01 2016 www.ijmtes.com 5
apply above condition then equation (3) becomes, m3*ẍ3+x3(k2+k3)+c2* ẋ3-k2*x2-c2* ẋ2=0 m3*ẏ6+y5*(k2+k3)+c2*y6-k2*y3-c2*y4=0 (12) equation (12) becomes, ẏ5=y6 (13) ẏ6=(1/m3)*[-y5(k2+k3)-c2*y6+k2*y3+c2*y4] (14) b) Track model Equation of motion of the track model in the form of second order differential equation and it s converted into two first order equations. Fig. 2 Vehicle model The above given free body diagram indicates the dynamic motion of the train. Let write the equation of motion, m1*ẍ1+c1*ẋ1+k1*x1-c1*ẋ2-k1*x2=0 (1) m2* ẍ2+ c2* ẋ2+k2*x2+k1*x2+ c1* ẋ2-k1*x1-c1* ẋ1- k2*x3-c2* ẋ3=0 m2*ẍ2+ẋ2(c2+c1)+x2(k2+k1)-k1*x1-c1* ẋ1-k2*x3-c2* ẋ3=0 (2) m3*ẍ3+k3*x3+k2*x3+c2* ẋ3-k2*x2-c2* ẋ2=0 m3*ẍ3+x3(k2+k3)+c2*ẋ3-k2*x2-c2*ẋ2=0 (3) Equations (1), (2) and (3) from mathematical model, m1*ẍ1+c1* ẋ1+k1*x1-c1* ẋ2-k1*x2=0 m2* ẍ2+ ẋ2(c2+c1)+x2(k2+k1)-k1*x1-c1* ẋ1-k2*x3- c2* ẋ3=0 m3*ẍ3+x3(k2+k3)+c2* ẋ3-k2*x2-c2* ẋ2=0 Let it converted into first order equation and Let, x1 = y1 ẋ1 = y2 = ẏ1 ẍ1 = ẏ2 x2 = y3 x3 = y5 ẋ2 = y4 = ẏ3 ẋ3 = y6 = ẏ5 ẍ2 = ẏ4 ẍ3 = ẏ6 apply above condition then equation (1) becomes, m1 ẏ2+c1* ẏ2+k1*y1-c1* ẏ4-k1*y3=0 m1ẏ2+c1*( ẏ2- ẏ4)+k1*(y1-y3)=0 (6) equation (6) becomes, ẏ1 = y2 (7) ẏ2 = (1/m1)*[c1(y4-y2)+k1*(y3-y1)] (8) apply above condition then equation (2) becomes, m2*ẍ2+ ẋ2*(c2+c1)+x2*(k2+k1)-k1*x1-c1* ẋ1-k2*x3-c2* ẋ3=0 m2*ẏ4+y4*(c2+c1)+y3*(k2+k1)-k1*y1-c1*y2-k2*y5- c2*y6=0 (9) equation (9) becomes, ẏ3 = y4 (10 ẏ4=(1/m2)*[-y4*(c2+c1)- y3(k2+k1)+k1*y1+c1*y2+k2*y5+c2*y6] (11) Fig.3 Track model Equations (4) and (5) from mathematical model, m1*ẍ1+c1*ẋ1+k1*x1-c1*ẋ2-k1*x2=0 (4) m2*ẍ2+ẋ2(c2+c1)+x2(k2+k1)-k1*x1-c1*ẋ1=0 (5) Let it converted into first order equation and Let, x1 = y1 ẋ1 = y2 = ẏ1 ẍ1 = ẏ2 and x2 = y3 ẋ2 = y4 = ẏ3 ẍ2 = ẏ4 apply above condition then equation (4) becomes, m1*ẍ1+c1* ẋ1+k1*x1-c1* ẋ2-k1*x2=0 m1ẏ2+c1*(ẏ2-ẏ4)+k1*(y1-y3)=0 (15) equation (15) becomes, ẏ1 = y2 (16) ẏ2 = (1/m1)*[c1(y4-y2)+k1*(y3-y1)] (17) apply above condition then equation (5) becomes, m2*ẍ2+ ẋ2*(c2+c1)+x2*(k2+k1)-k1*x1-c1* ẋ1=0 m2*ẏ4+y4*(c2+c1)+y3*(k2+k1)-k1*y1-c1*y2=0 (18) equation (18) becomes, ẏ3=y4 (19) ẏ4=(1/m2)*[-y4*(c2+c1)-y3(k2+k1)+k1*y1+c1*y2] (20) Finally, we get six first order differential equations from vehicle model and four first order differential equations from track model. After the above process, using the first order differential equations to compute the MATLAB function file and program data. Volume: 03 Issue: 01 2016 www.ijmtes.com 6
4.PARAMETERS a) Vehicle parameters To apply different values of parameters in function file andget consecutive responses. Then use that the different responses give the recommendations for the safe and effective design of conventional trains Parameters of vehicle model. Table 1 Parameters of vehicle model. Notations Parameters Values m1 Mass of the vehicle body (kg) 55790 Fig.4 (b) Vehicle Response m2 Mass of the bogie frame (kg) 2380 m3 Mass of the wheelset (kg) 2048 c1 Secondary damping (kns/m) 40 Fig.4 (c) Vehicle Response Different responses from track model are c2 Primary damping (kns/m) 20 k1 Secondary stiffness (MN/m) 2.45 k2 Primary stiffness (MN/m) 2.45 k3 Hertzian spring stiffness (N/m) 1.4e6 b)parameters of track model Taple 2 Parameters of track model Notations Parameters Values c) Responses m1 Mass of the Rail with pad (kg) 700 m2 Mass of the Sleeper (kg) 550 c1 Railpad damping (Ns/m) 2e5 c2 Sleeper damping (Ns/m) 1e5 k1 Railpad stiffness (N/m) 2.7e8 k2 Sleeper stiffness (N/m) 8e7 Responses from vehicle model with different conditions, Fig.5 (a) Track Response Fig.5 (b) Track Response Fig.5 (c) Track Response Fig.4 (a) Vehicle Response 5. EXPERIMENTAL SETUP This step is used to compare the dynamic performance of the smooth wheels and rail surface and the irregular surface wheels. Volume: 03 Issue: 01 2016 www.ijmtes.com 7
From the result and discussion, we give recommendations like to increase the speed of the train by reducing the mass of the vehicle or increase the value of the stiffness. From the above result to identify the following statements. To reduce mass by using light weight, high strength composite materials Fig.6 (a) Experimental Setup position Fig.6 (b) Experimental Setup position 6.RESULT From the above result, we give recommendations for the safe and effective design of conventional train. Using below table values to get multi responses. Table 4 Different parameters values of track S.No Mass (kg) Stiffness (N/m) m1 m2 k1 k2 1 700 550 2 800 700 3 500 400 2.7e8 8e7 4 1.7e7 7e7 5 700 550 3.7e7 9e7 6 4.7e7 10e7 Table 5 Different parameters values of vehicle S.No. Mass (kg) Stiffness (N/m) m1 m2 m3 k1 k2 k3 1 55790 2380 2048 2 60000 3000 2300 3 40000 2000 1800 2.45e6 2.45e6 1.42e8 4 2.45e6 2.45e6 1.42e8 5 55790 2380 2048 7e6 4e6 1.7e6 6 1.45e6 1.35e6 1.2e6 And maximize the stiffness of the vehicle. To use the experimental setup to evaluate surface and suspension performance of the wheel and track. 7. CONCLUSION This project presents the preparatory work carried out on the design and analysis of conventional trains. To start with, the available literature on conventional trains are carried out and studied. It is found that most of the studies are based on linear theory and it is actually an approximation of the real situation. The designs made based on appropriate analysis can lead to accidents. From the result and discussion, we give recommendations to increase the speed of the train by reducing the mass of the vehicle or increase the value of the stiffness. The experimental setup used to compare the dynamic performance of the normal surface wheelset and irregular surface wheelset. REFERENCES 1. Alexander Lovett and Greg Munden et al. (2013), High-Speed Rail Network Design and Station Location, Journal of the Transportation Research Board, No. 2374, pp. 1 8. 2. Ayman A. Aly and Farhan A. Salem Vehicle Suspension Systems Control: A Review, International journal of control, automation and systems vol.2 NO.2, ISSN 2165-8277. 3. C.H. Lee and M. Kawatani et al. (2006) Dynamic response of a monorail steel bridge under a moving train, Journal of Sound and Vibration 294,562 579. 4. Chunlei Yang and Fu Li et al. (2013) Comparative study on wheel rail dynamic interactions of side-frame cross-bracing bogie and sub-frame radial bogie, J. Mod. Transport. 21(1):1 8. 5. Davood Younesian and Amir Nankali (2009) Spectral Optimization of the Suspension System of High-speed Trains, International Journal of Vehicle Structures & Systems, 1(4), 98-103,ISSN: 0975-3060. 6. Dragan Sekulic and Vlastimir Dedovic (2011) The effect of stiffness and damping of the suspension system elements on the optimization of the vibrational behaviour of a bus, Faculty of Transport and Traffic Engineering. 7. D. Younesian and S. Mohammadzadeh et al. (2006) Dynamic Performance, System Identification and Sensitivity Analysis of the Ladder Tracks Journal of Sound and Vibration, Volume 199. 8. D. Mihai (2007) On the dependency between railway track discontinuities and wheel-rail rolling noise, SISOM and Homagial Session of the Commission of Acoustics. 9. D.P.Connolly a,n, G.Kouroussis b et al. (2014) Field testing and analysis of high speed rail vibrations, Soil Dynamics and Earthquake (2014)102 118. 10. Hossein Nia S and Stichel S et al. (2013) Investigation of Rolling Contact fatigue on the Wheels of a Three-Piece Bogie on the Swedish Iron ore Line via Multibody Simulation Considering Extreme Winter Condition, In: (pp. 357-363). 11. Ilaria Grossoni and Simon Iwnicki et al. (2013) Dynamic response of vehicle-track coupling system with an insulated rail joint, International Conference on Vibration Problems. Volume: 03 Issue: 01 2016 www.ijmtes.com 8
12. Javier Santamaria and Ernesto G. Vadillo (2004) Equivalent Conicity and Curve Radius Influence on Dynamical Performance of Unconventional Bogies.Comparison Analysis, Vehicle System Dynamics, Vol. 41 (suppl.), pp. 133-142. 13. Jin Shi and A.H. Chan et al. (2012) Influence of unsupported sleepers on dynamic responses of railroad embankment below a heavy haul railway line using simulation techniques, PRUKA annual conference. Volume: 03 Issue: 01 2016 www.ijmtes.com 9