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Modeling a Car Suspension System

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Car suspension system
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  PC Modeling and Simulation of Car Suspension System M. Prem Jeya Kumar 1* , K . Gopalakrishnan 2 , V. Srinivasan 3 , R. Anbazhagan 4  and J. Sundeep Aanand 5 1 Professor, Department of Automobile Engineering, Bharath University, Chennai-73; premjeyakumar.coe@bharathuniv.ac.in 2 Professor of Electronics, Department of Electronics, Bharath University, Chennai-73; gopikrishna2804@gmail.com 3 Professor, Department of Automobile Engineering, Bharath University, Chennai-73; srinivasan@v9669.yahoo.co.in 4 Professor, Department of Automobile Engineering, Bharath University, Chennai-73; anbu.mit@gmail.com 5 Professor, Department of Computer Science Engineering, Bharath University, Chennai-73; sundeepssn@hotmail.com Abstract The car suspension system of this model contains two parts. The irst part deals with the formulation of a mathematical model for a conventional full car passive suspension system. Typically, the mathematical modeling is done on the basis of mechanical network analysis. The second part deals with simulation of the mathematical model of the suspension system. Simulation is carried out using MATLAB. Program was carried out for MATLAB and the simulation results were obtained in the form of graphical plots. Keywords:   Passive Suspension Sprung Mass, Unsprung Mass, Dampers, Spring. 1. Introduction Te first step towards achieving a simulation or a passive suspension is to generate the basic equation o motions or the dynamic system. Te seven basic degrees o reedom considered or derive the equation o motion are, our lin-ear motion about each suspension ,one linear motion about the mass at the centre o gravity, one angular displacement about the longitudinal axis known as ‘Roll’ [1-3].Te mathematical model o car is to identiy the prob-lems which occur in the suspension o the cars and SUV’s. With the help o vehicle data’s, the exact physical char-acteristics o that car can be obtained by simulating the mathematical model. For that MALAB sofware is mainly used to simulate the car. 2. Mathematical Modeling Basics A mathematical model o a dynamic system is defined as a set o equations that represents dynamics o the system accurately or at least, airly well. A mathematical model is not unique to a given mechanical system. A system may be repre-sented in many different ways and thereore may have many mathematical models, depending on one’s perspective.Te dynamics o a mechanical system can be described in terms o differential equations. Such differential equa-tions may be obtained by using physical laws governing particular system, or example, Newton’s laws are used in case o a mechanical system. Mathematical models may assume many different orms. Depending on the particular system and the particular circumstances, one mathematical *Corresponding author: M. Prem Jeya Kumar (premjeyakumar.coe@bharathuniv.ac.in) Indian Journal of Science and Technology Supplementary Article 6_5_35.indd 46296/28/2013 2:08:08 PM  PC Modeling and Simulation of Car Suspension System 4630 Indian Journal o Science and echnology | Print ISSN: 0974-6846 | Online ISSN: 0974-5645www.indjst.org | Vol 6 (5S) | May 2013 model may be better suited than other models. Once a mathematical model o a system is obtained, various ana-lytical and computer tools can be used or analysis and synthesis purposes. 3. Full Car Model A ull car model constitutes o our unsprung masses con-nected to a single sprung mass about the centre o gravity o the vehicle and our dashpot and spring arrangements used to connect the sprung mass with the unsprung masses. Te sche-matic representations o the car model are given below [4, 5].Apart rom the our displacement measurements, ull car model also takes into account one vertical displace-ment about the centre o gravity, pitching motion about the transverse axis and rolling motion about the longitudinal axis or modeling o the system.Te highlight o the ull car mathematical model is that it can be used to include all type o motion like yaw, toe-in, toe-out, etc. 4. Dynamics of Full Car Model Based on the ull car model, the equations o motion or the ull car model have been derived, using network analysis. Te linearized equations o motions o the ull car model can be expressed as m Z U F   f u p r  1111   = +  (1) m Z U F   f u p r  2222 .    = +  (2)  m Z U F  r u p r  1333 .    = +  (3)  m Z U F  r u p r  2444 .    = +  (4)  m Z F U U U U  s s s p p p p   = − − − − 1234  (5)  Ι Τ q q q   .   Z U U U U   f p f p r p r p = + + − − 1111 1234  (6)  Ι Τ Φ Φ Φ .   Z t U t U t U t U   f p f p r p r p = − + − + 1234  (7)where, the passive suspension orces are given by   U C Z Z t Z Z k Z Z   p f u f f S f u s 11111 1 = + − − ( ) + − ( )     q    Φ   U C Z Z t Z Z k Z Z   p f u f f S f u s 22222 1 = + + − ( ) + − ( )     q    Φ  Figure 1. Full Car Model. 6_5_35.indd 46306/28/2013 2:08:22 PM  M. Prem Jeya Kumar, K . Gopalakrishnan, V. Srinivasan, R. Anbazhagan and J. Sundeep Aanand  4631 Indian Journal o Science and echnology | Print ISSN: 0974-6846 | Online ISSN: 0974-5645www.indjst.org | Vol 6 (5S) | May 2013   U C Z Z t Z Z k Z Z   p r u r r S r u s 31313 1 = − − − ( ) + − ( )     q    Φ   U C Z Z t Z Z k Z Z   p r u r r S r u s 42424 1 = − + − ( ) + − ( )     q    Φ and the tyre orces are given by: F k z z  r tf r u 1111 = − () F k z z  r tf r u 2222 = − () F k z z  r tf r u 3333 = − () F k z z  r tf r u 4444 = − () 5. State Space Representation Te linearized equations (1-7) are then converted into space orm by comparing the coefficients o state vectors and input vectors. Te state space model is given by    X˙  = Ax + Bu Y = Cx + Duwhere, A – state matrix B – Input Matrix C – Output Matrix D – Direct transmission matrix X – State vector Y – Output Vector Assuming state vector & input vector to be, x Z Z Z Z Z Z Z Z Z Z Z Z Z Z  u u u u u u u u s sT  =          11223344  q q    Φ Φ u z z z z  r r r r T  =   1234 Te values o other matrices in the state space model were obtained.aking state space model into account, a Mat lab pro-gram were written to convert the matrices into a model. Tese models were used or running simulation by provid-ing real constants value rom a typical SUV. Te parameters used are m s  = 1600 Kg I θ  = 1000 kg m 2  I φ  = 450 kg m 2 t    = t r  = 0.75m l    = 1.15m l r  = 1.35mm    = m r  = 50 kg k     = k  r  = 20 kN/m c    = c r  = 5 kN/mk  t   = k  tr  = 250kN/m 6. Simulation Results7. Conclusion Te dynamic equations o motion were derived, taking in to account the seven degrees o motion. Tese seven equations o motion orm the mathematical model o the car. Tis model is urther used or simulation by providing with real time parametric values o an SUV. Te simulation results obtained depict the exact response characteristics o the ull car model. For a positive road input at node 1 the suspension displacement turned out to be in positive direction, while a negative road input at node 2 depicted a similar output. Te main application o this simulation would be in the problem identification o the suspension system in any car. Te only modification that has to be made is to supply the parametric  values o the particular model which is being investigated. 8. Symbols and Abbreviation m  f  1, 2  = unsprung mass at the ront end m r  1, 2  = unsprung mass at the rear endm s  = sprung mass at the centre o gravity o the vehicleI Φ  = moment o inertia about the sprung mass due to rollI φ  = moment o inertia about the sprung mass due to pitch M = moment about the sprung massC 1,2  = damper rate at the ront endC r1,2  = damper rate at the rear endK 1,2  = spring stiffness at the ront endK r1,2  = spring stiffness at the rear endk  t1,2  = tyre stiffness at the ront endk  tr1,2  = tyre stiffness at the rear endU p1–4  = passive suspension orces Figure 2. Passive suspension displacements at node 1 and node 2 respectively. 6_5_35.indd 46316/28/2013 2:08:33 PM  PC Modeling and Simulation of Car Suspension System 4632 Indian Journal o Science and echnology | Print ISSN: 0974-6846 | Online ISSN: 0974-5645www.indjst.org | Vol 6 (5S) | May 2013 U c1,2  = damper orces at the ront endU cr1,2  = damper orces at the rear endU k1,2  = spring orces at the ront endU kr1,2  = spring orces at the rear endU m1,2  = opposing orces due to unsprung mass at ront endU mr1,2  = opposing orces due to unsprung mass at rear endU ms  = opposing torque due to sprung mass at centre o gravity. θi  = opposing torque due to pitch I Φ i  = opposing torque due to roll θ  = torque due to pitch φ   =   torque due to rollF s  = external load disturbances.F r  = orce exerted on unsprung massZ r1–4  = tyre displacementsZ u1,2  = unsprung mass deflection at the ront endZ u3,4  = unsprung mass deflection at the rear rontZ s  = sprung mass deflection at the centre o gravity Z θ  = angular displacement due to pitchZ φ  = angular displacement due to roll 9. References 1. Ogata K (2001). Modern control engineering, 3 rd  Edn., Prentice-Hall o India.2. Cebon D (1993). Interaction between heavy vehicles and roads, 39 th  Buckendale Lecture, 1 st  Edn., SAE Interenational.3. Smith M C, and Wang F (2002). Controller parameterization or disturbance response decoupling: application to vehicle active suspension control, IEEE ransactions on Control Systems echnology, vol 10(3), 393–407.4. Williams R A (1997). Automotive active suspensions, Part-1: basic principles, Proceedings o the Institution o Mechanical Engineers Conerence, vol 211, No. 6, 415–426. 5. Crouse W H, and Anglin D L (2002). Automotive Mechanics, 9 th  Edn., ata McGraw Hill Edition. 6_5_35.indd 46326/28/2013 2:08:33 PM
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