International Journal of Engineering Materials and Manufacture (2020) 5(1) 19-28 https://doi.org/10.26776/ijemm.05.01.2020.04 R. E. Jonjo and S. T. Nyalloma Mechanical and Maintenance Engineering Department Fourah Bay College, University of Sierra Leone Barham Road, Southern Central, 00232 Freetown, Sierra Leon E-mail: sahr.nyalloma@usl.edu.sl Reference: Jonjo, R. E. and Nyalloma, S. T. (2020). Modelling the Effect of Road Excitation on Vehicle Suspension System. International Journal of Engineering Materials and Manufacture, 5(1), 19-28. Modelling the Effect of Road Excitation on Vehicle Suspension System Robert Emmanuel Jonjo and Sahr Tamba Nyalloma Received: 19 March 2020 Accepted: 29 March 2020 Published: 30 March 2020 Publisher: Deer Hill Publications © 2020 The Author(s) Creative Commons: CC BY 4.0 ABSTRACT The vehicle suspension system serve a dual purpose – to provide passenger comfort and good road holding. In the design of a vehicle suspension system, these two contradictory criteria must be balanced out. Road irregularities are also a major source of anxiety amongst drivers and passengers alike. This research was undertaken to investigate the effect road irregularities will have on the vehicle structure especially the suspension system. In this study, the responses of different linear vehicle models are studied for step road input. The mathematical models considered are: a two degrees-of-freedom system (quarter car model) and a four degrees-of-freedom system (half car model). The equations of motion for both models were obtained using Newton’s method. These models are analysed using SIMULINK/Matlab. Different response parameters such as the acceleration of the vehicle body and the travel of the suspension are investigated for a passive suspension system. The responses of the vehicle suspension due to changes in parameters such as suspension stiffness and damping coefficients are investigated. The results show that road irregularities affect the vehicle structures and the response of the suspension system is dependent on the suspension parameters. Passive suspension systems do not satisfy road holding and passenger comfort at the same time. 1 INTRODUCTION The vehicle Suspension consists of springs, shock absorbers and other linkages that separates the car body from the car wheel. The functions of the automotive suspension on a vehicle includes; Isolating a car body from road irregularities, providing good handling by ensuring that roll and pitch motion are minimized and supporting the vehicle static weight. Vehicle suspension system should therefore isolate the vehicle body from road disturbances in order to give maximum passenger ride comfort whiles retaining continuous road-wheel contact so as to provide excellent road holding (Changizil and Rouhani 2011). It is therefore a challenge to design a vehicle to satisfy both ride comfort and road holding. Suspension systems are generally classified as passive, semi-active and active suspensions (Ghazaly and Moazz 2014, Tejas 2016) • Passive suspensions: Passive suspension systems usually consist of a non-controlled spring and a damper with fixed parameters (Agharkakli .A 2012). They lack enough energy absorption capability to sustain the load input into the vehicle systems (Gao et al 2016). They lack stability when compared with other suspension systems (Dahunsi 2011). • Semi-active suspensions: Semi active suspensions on the other hand provide a rapid change in rate of springs and damping coefficients. They usually have a fixed spring rate and a variable shock absorber. They do not provide any energy into suspension system. Compared to the active suspensions, these suspensions use less energy to operate and they are cheaper (Williams 1994). • Active suspensions: The active vehicle suspension system employs electronic control systems that monitor the operation of the suspension elements. In active suspension systems, the shock absorber is replaced by a force actuator. They reduce car body accelerations by allowing the suspension to absorb wheel accelerations using an actuator (Pedro and Ekoru 2013). Unlike passive and semi-active suspensions, active suspensions give dynamic compensation (Kashem et al. 2012). They are Superior in performance than passive and semi-active suspension systems. 2 MATHEMATICAL MODELING The three common vehicle models found in literature are the quarter car model (two degrees of freedom), the half car model (four degrees of freedom) and the full car model (seven degrees of freedom). Roll and pitch motions are Modelling the Effect of Road Excitation on Vehicle Suspension System 20 ignored in the quarter car model. A half car model on the other hand considers either roll motion or pitch motion. The full car model considers roll, yaw and pitch motions. 2.1 QUARTER CAR MODEL The quarter car model shown in Figure 1 is simple and widely used for dynamic performance analysis. It is a two degrees of freedom model. The quarter car model consists of one-fourth of the body mass, suspension components and one wheel. It represents the suspension system at any of the four wheels of the vehicle and the degrees-of- freedom are displacement of axle and displacement of the vehicle body at a particular wheel. The assumptions of a quarter car modelling are as follows: • The tire, spring and damper are modelled as a linear spring without damping, • Rotational motion in wheel and body are ignored, • The tire remains in contact with the road surface at all times • Effect of friction between road and wheel is neglected. Consider the free body diagram of the forces acting on the sprung mass as shown in Figure 2 with 𝑍𝑠 >𝑍𝑢>𝑍𝑟 .The equation of motion of the sprung mass 𝑀𝑆�̈�𝑆 + 𝐶𝑆(�̇�𝑆 − �̇�𝑈 ) + 𝐾𝑆(𝑍𝑆 − 𝑍𝑈 ) = 0 (i) Equation (i) can be rewritten as �̈�𝑠 = 1 𝑀𝑆 { 𝐶𝑆(�̇�𝑢 − �̇�𝑠) + 𝐾𝑆 (𝑍𝑢 − 𝑍𝑠)} (ii) Also, the free body diagram of the forces acting on the unsprung mass is shown in Figure 3. The equation of motion of the unsprung mass becomes: 𝑀𝑈�̈�𝑈 − 𝐶𝑆(�̇�𝑠 − �̇�𝑢) − 𝐾𝑆(𝑍𝑠 − 𝑍𝑢 ) + 𝐾𝑡 (𝑍𝑢 − 𝑍𝑅 ) = 0 (iii) Also, equation (iii) becomes; �̈�𝑈 = 1 𝑀𝑈 {−𝐶𝑆(�̇�𝑢 − �̇�𝑠 ) − 𝐾𝑆(𝑍𝑢 − 𝑍𝑠 ) − 𝐾𝑡 (𝑍𝑢 − 𝑍𝑅 ) (iv) Figure 1. Quarter car model Figure 2. Forces acting on sprung mass for Quarter car model Figure 3. Forces acting on the unsprung mass for the Quarter car model 2.2 HALF CAR MODEL The half car suspension model is a four degree of freedom system shown in Figure 4. This model considers both pitch and bounce motion of the vehicle. In this model, the following assumptions are made; • The vehicle undergoes both pitch and bounce motion separately. • The half car model can be a side, front or rear car model depending on the section of the vehicle considered. In this study, the side half car model will be considered. • The tire, springs and dampers are modelled as a linear spring. • The wheel is in contact with the road at all times. Jonjo and Nyalloma (2020): International Journal of Engineering Materials and Manufacture, 5(1), 19-28 21 The free body diagram for both the sprung and unsprung masses with bounce and pitch motion is shown in Figures 5 and 6. The equation of motion of the sprung mass exhibiting bounce motion is: 𝑀𝑠�̈�𝑠 + 𝐶𝑠𝑟 (�̇�𝑠 − �̇�𝑢𝑟 − 𝐶𝜃)̇ + 𝐶𝑠𝑓 (�̇�𝑠 − �̇�𝑢𝑓 + 𝑏𝜃)̇ + 𝐾𝑠𝑟 (𝑍𝑠 − 𝑍𝑢𝑟 − 𝐶𝜃) + 𝐾𝑠𝑓 (𝑍𝑠 − 𝑍𝑢𝑓 + 𝑏𝜃) = 0 (v) The equation of motion of the sprung mass exhibiting pitch motion is 𝐼�̈� − 𝐶𝑠𝑟 𝐶(�̇�𝑠 − �̇�𝑢𝑟 − 𝐶�̇�) + 𝐶𝑠𝑓𝑏(�̇�𝑠 − �̇�𝑢𝑓 + 𝑏�̇�) − 𝐾𝑠𝑟 𝐶(𝑍𝑠 − 𝑍𝑢𝑟 − 𝐶𝜃) + 𝐾𝑠𝑓 𝑏(𝑍𝑠 − 𝑍𝑢𝑓 + 𝑏𝜃) = 0 (vi) The equation of motion for both the front unsprung mass and rear unsprung mass are given respectively as, 𝑀𝑢𝑓 �̈�𝑢𝑓 − 𝐶𝑠𝑓(�̇�𝑠 − �̇�𝑢𝑓 + 𝑏�̇�) − 𝐾𝑠𝑓 (𝑍𝑠 − 𝑍𝑢𝑓 + 𝑏𝜃) + 𝐾𝑡𝑓 (𝑍𝑢𝑓 − 𝑍𝑟𝑓 ) = 0 (vii) 𝑀𝑢𝑟 �̈�𝑢𝑟 − 𝐶𝑠𝑟 (�̇�𝑠 − �̇�𝑢𝑟 − 𝐶�̇�) − 𝐾𝑠𝑟 (𝑍𝑠 − 𝑍𝑢𝑟 − 𝐶𝜃) + 𝐾𝑡𝑟 (𝑍𝑢𝑟 − 𝑍𝑟𝑟 ) = 0 (viii) Figure 4. Half car model Figure 5. Free body for sprung mass exhibiting bounce motion Figure 6. Free body diagram for sprung mass exhibiting (pitch motion) 3 SIMULATION 3.1 Simulation Parameters In the current research vehicle parameters to undertake simulation of the quarter car and half car models were obtained for a passenger sedan as presented by (Agostinacchio et al. 2014) and presented in Tables one and two respectively. Road excitation input is a step signal of amplitude 10 cm. Modelling the Effect of Road Excitation on Vehicle Suspension System 22 3.2 Simulation and Results The Simulink block diagram for the quarter car and half car models are shown in Figures 8 and 9 respectively. Both Figures represent the differential equations of motion for the two models investigated, that is, equations 2 and 4 for the quarter car model and equations 5, 6, 7 and 8 for the half car model. Using the parameters presented in Tables 1 and 2, simulation of the system models was undertaken and the results presented in graphical format. The displacement of the vehicle body, displacement of the unsprung mass(s), deflection of the wheels, suspension travel, acceleration of the car body in the vertical direction are the parameters that were investigated. The effect of the changes of various passive suspension parameters are investigated by keeping all other system parameters fixed and gradually varying the parameter being investigated. Figure 7. Free body diagrams for front and rear unsprung mass Table 1. Simulation parameters for Quarter car model Parameter Symbol Values Unit Sprung mass of the vehicle 𝑀𝑠 400 Kg Unsprung mass of vehicle 𝑀𝑢 40 Kg Stiffness of tire 𝐾𝑡 190000 N/m Spring constant of axle 𝐾𝑠 21000 N/m Damping coefficient of axle 𝐶𝑠 1000 Ns/m Table 2. Simulation parameters for half car model Parameter Symbol Values Unit Sprung mass of the vehicle 𝑀𝑠 400 𝑘𝑔 Moment of inertia of vehicle 𝐼𝑦 1100 𝑘𝑔𝑚 2 Unsprung mass of front axle 𝑀𝑢𝑓 40 𝑘𝑔 Unsprung mass of rear axle 𝑀𝑢𝑟 40 𝑘𝑔 Stiffness of front tire 𝐾𝑡𝑓 150000 𝑁 𝑚⁄ Stiffness of rear tire 𝐾𝑡𝑟 150000 𝑁 𝑚⁄ Spring constant of front axle 𝐾𝑢𝑓 21000 𝑁 𝑚⁄ Spring constant of rear axle 𝐾𝑢𝑟 21000 𝑁 𝑚⁄ Damping coefficient of front axle 𝐶𝑠𝑓 1500 𝑁𝑠 𝑚⁄ Damping coefficient of rear axle 𝐶𝑠𝑟 1500 𝑁𝑠 𝑚⁄ Front body length from centre of gravity 𝑐 1.47 𝑚 Rear body length from centre of gravity 𝑏 1.4 𝑚 Jonjo and Nyalloma (2020): International Journal of Engineering Materials and Manufacture, 5(1), 19-28 23 Figure 8. Simulink block diagram for quarter car model Figure 9. Simulink block diagram for half car model 3.2.1 Simulation for Quarter Car Model The displacement of the sprung and unsprung masses, acceleration of the sprung and unsprung masses, suspension travel and tire deflection are presented in Table 3. Figures 10, 11 and 12 show the displacement of the sprung mass, acceleration and displacement of the unsprung mass for the quarter car model. Modelling the Effect of Road Excitation on Vehicle Suspension System 24 Table 3. Simulation results for quarter car model No Parameters Values Maximum Sprung Mass Displacement Maximum Unsprung Mass Displacement Sprung Mass Acceleration Suspension Travel Wheel Deflection Rms Acceleration 1 Sprung Mass 400 0.1569 0.1268 16.776 -2.969 0.050 -0.115 0.0268 -0.100 0.5829 2 500 0.1601 0.1258 13.569 -2.439 0.053 -0.113 0.0258 -0.100 0.472 3 1000 0.1686 0.1238 6.920 -1.329 0.059 -0.1175 0.0238 -0.100 0.253 4 1500 0.1730 0.1231 4.643 -0.923 0.064 -0.1189 0.0231 -0.100 0.178 5 Unsprung Mass 50 0.1576 0.1309 15.796 -3.436 0.049 -0.1125 0.0309 -0.100 0.582 6 100 0.1584 0.1433 13.045 -5.056 0.051 -0.1161 0.0433 -0.100 0.586 7 150 0.1599 0.1461 11.452 -5.420 0.062 -0.1180 0.0426 -0.100 0.597 8 200 0.1631 0.1539 10.622 -6.335 0.067 -0.112 0.0534 -0.100 0.593 9 Suspension Stiffness 21000 0.1569 0.1268 16.776 -2.959 0.050 -0.1115 0.0268 -0.100 0.5829 10 30000 0.1646 0.1217 17.858 -4.346 0.054 -0.1059 0.0217 -0.100 0.639 11 40000 0.1709 0.1164 19.010 -5.892 0.055 -0.099 0.0164 -0.100 0.704 12 60000 0.1750 0.1197 20.139 -7.330 0.056 -0.095 0.0197 -0.100 0.791 13 Damping Coefficient 1500 0.1569 0.1268 16.776 -2.959 0.050 -0.115 0.0268 -0.100 0.5829 14 2000 0.1492 0.1188 18.998 -3.200 0.043 -0.099 0.0188 -0.100 0.659 15 2500 0.1421 0.1120 20.023 -3.195 0.037 -0.091 0.0120 -0.100 0.749 16 3000 0.1381 0.1100 22.478 -3.542 0.038 -0.0828 0.0108 -0.100 0.786 17 Tire Stiffness 150000 0.1569 0.1268 16.776 -2.959 0.050 -0.1115 0.0268 -0.100 0.5829 18 200000 0.1555 0.1310 19.880 -5.183 0.050 -0.1205 0.031 -0.100 0.629 19 250000 0.1542 0.1388 23.220 -7.252 0.050 -0.1278 0.0388 -0.100 0.680 20 300000 0.1535 0.1439 26.143 -8.821 0.049 -0.1324 0.0439 -0.100 0.747 Jonjo and Nyalloma (2020): International Journal of Engineering Materials and Manufacture, 5(1), 19-28 25 Figure 10. Displacement of the sprung mass Quarter car model Figure 11. Acceleration of sprung mass Quarter car model Figure 12. Displacement of the unsprung mass Modelling the Effect of Road Excitation on Vehicle Suspension System 26 3.2.2 Simulation for Half Car Model The simulation results for the Half- Car model is presented in Table 4. Figures 13, 14 and 15 show the displacement of the sprung mass, and the displacement of the unsprung masses for the half - car model. 4 RESULTS Simulations were carried out for the Quarter car and Half- car models. Different response properties were investigated such as vehicle sprung mass displacement, unsprung mass displacement, suspension travel, wheel deflection. From the simulation results presented in Tables 3 and 4, it is observed that the vehicle experiences a vertical acceleration as it passes over a road irregularity. The following general observations are made from the simulation results. From the simulation results it is evidently clear that an increase in the sprung mass leads to a lower vertical acceleration of the car body, an increase in the vertical displacement of the car body and an increase in the suspension travel. If all other parameters are kept constant whiles the unsprung mass(es) is gradually increased, similar trends are observed as in the case of the sprung mass. Vertical acceleration is reduced but suspension travel is increased. Increasing damping coefficient reduces car body displacement and suspension travel considerably but vertical acceleration increases whereas an increase in the stiffness of vehicle suspension leads to an increase in vertical acceleration. The suspension travel shows an increase and decrease pattern. Figure 13. Displacement of sprung mass of half car model Figure 14. Displacement of front unsprung mass Figure 15. Displacement of rear unsprung mass Modelling the Effect of Road Excitation on Vehicle Suspension System 27 Table 4. Simulation results for half car model No Parameters Values Sprung Mass Displace Unsprung Mass Displacement Sprung Mass Acceleration Suspension Travel Wheel Deflection RMS Accelerat ion Front Rear Front Rear Front Rear 1 Sprung Mass 400 0.148 0.1299 0.1301 32.034 -10.685 0.0415 -0.1045 0.0416 -0.1046 0.0299 0.030 3.80 2 500 0.151 0.1277 0.1278 26.057 -7.295 0.0443 -0.1074 0.0444 -0.1075 0.0278 0.0278 3.080 3 1000 0.1602 0.1231 0.1235 13.539 -2.434 0.0525 -0.1137 0.0527 -01138 0.0234 0.0236 1.595 4 1500 0.1652 0.1225 0.1226 9.129 -1.697 0.0570 -0.1159 0.0573 -0.1160 0.0225 0.02258 1.102 5 Front unsprung mass 50 0.148 0.1361 0.1299 30.318 -10.178 0.0419 -0.1023 0.0421 -0.1055 0.0361 0.0299 3.748 6 100 0.150 0.1485 0.1294 24.453 -7.746 0.0471 -0.0989 0.0434 -0.1080 0.0485 0.0294 3.430 7 150 0.148 0.1567 0.1297 23.178 -10.807 0.0613 -0.0930 0.0424 -0.1100 0.0568 0.0297 3.226 8 200 0.148 0.1607 0.1288 21.392 -12.042 0.0747 -0.0880 0.0438 -0.1110 0.0677 0.0288 3.090 9 Rear unsprung mass 50 0.1482 0.1298 0.1360 30.292 -10.172 0.0421 -0.1054 0.0419 -0.1020 0.0298 0.0362 3.748 10 100 0.1501 0.1292 0.1486 25.460 -7.748 0.0432 -0.1087 0.0474 -0.0989 0.0292 0.0486 3.436 11 150 0.1480 0.1295 0.1569 23.179 -10.823 0.0423 -0.1090 0.0613 -0.0939 0.0295 0.0569 3.227 12 200 0.1506 0.1286 0.1609 21.402 -12.052 0.0438 -0.1114 0.0747 -0.0889 0.0286 0.0609 3.090 13 Front suspension stiffness 21000 0.1483 0.1299 0.1301 32.034 -10.685 0.0415 -0.1045 0.0416 -0.1046 0.0299 0.0300 3.801 14 30000 0.1523 0.1257 0.1312 32.306 -9.342 0.0421 -0.1000 0.0452 -0.1036 0.0257 0.0312 3.917 15 40000 0.1547 0.1206 0.1315 33.043 -9.524 0.0412 -0.0958 0.0479 -0.1028 0.0206 0.0315 4.056 16 50000 0.1565 0.1200 0.1317 33.924 -10.805 0.0393 -0.0919 0.0506 -0.1023 0.0200 0.0317 4.179 17 Rear suspension stiffness 21000 0.1483 0.1299 0.1301 32.034 -10.085 0.0415 -0.1045 0.0416 -0.1046 0.0299 0.0300 3.801 18 30000 0.1521 0.1316 0.1267 32.203 -9.607 0.0449 -0.1014 0.0422 -0.0985 0.03163 0.0267 3.967 19 40000 0.1545 0.1316 0.1212 32.461 -9.477 0.0477 -0.1022 0.0408 -0.0956 0.03164 0.0212 4.071 20 50000 0.1559 0.1317 0.1202 33.609 -10.772 0.0501 -0.1019 0.0389 -0.0919 0.0317 0.0202 4.188 21 Front damping coefficient 1500 0.1483 0.1299 0.1301 32.034 -10.085 0.0.415 -0.1045 0.0416 -0.1046 0.0299 0.0300 3.801 22 2000 0.1440 0.1254 0.1318 33.159 -10.669 0.0387 -0.0922 0.0372 -0.1011 0.0254 0.0318 4.102 23 2500 0.1406 0.1230 0.1320 34.671 -11.702 0.0366 -0.0847 0.0336 -0.0997 0.0230 0.0322 4.272 24 3000 0.1375 0.1230 0.1314 35.598 -12.669 0.0349 -0.0800 0.0310 -0.0986 0.0231 0.0314 4.498 25 Rear damping coefficient 1500 0.1484 0.1299 0.1300 32.034 -10.685 0.0415 -0.1045 0.0416 -0.1046 0.0299 0.0300 3.807 26 2000 0.1440 0.1317 0.1255 33.148 -10.684 0.0370 -0.1011 0.0387 -0.0922 0.03170 0.0255 4.100 27 2500 0.1405 0.1319 0.1233 34.548 -11.761 0.0335 -0.0995 0.0367 -0.085 0.0317 0.0255 4.274 28 3000 0.0317 0.0319 0.1234 35.656 -12.700 0.0308 -0.0985 0.0350 -0.0802 0.0317 0.0255 4.495 Modelling the Effect of Road Excitation on Vehicle Suspension System 28 5 CONCLUSIONS In this work the effect of road excitations on a passive suspension for a passenger car was investigated. Mathematical modelling has been performed using a two and four degrees of freedom models. The differential equations of motion were simulated with Simulink/Matlab. The results from the simulation were used to analyse the performance of vehicle dynamics for step input road profile. The following conclusions are drawn. On the basis of the results, it can be concluded that road excitations affect the vehicle suspension which determines the ride comfort and stability of the vehicle. REFERENCES Agharkakli A., Ghobad Shafiei Sabet, Armin Barouz (2012). Simulation and Analysis of Passive and Active Suspension System using Quarter Car Model for Different Road Profile. International Journal of Engineering Trends and Technology, 3 (5). Agostinacchio, M., Ciampa D. and Olita, S. (2014). The Vibration Induced by Surface Irregulation in Road pavements– a MATLAB Approach. Springer, 6, 267-275. Changizi, N. and M. Rouhani, (2011). 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Mathematical Modeling and Simulation of a Simple Half – Car Vibration Model. International Journal for Scientific Research & Development, 4(2), 2321-0613. Williams R.A, (1994). Electronically controlled automotive suspensions. Computing & Control Engineering Journal, 5(3), 143-148. LIST OF ABBREVIATIONS 𝑀𝑠=sprung mass of vehicle 𝑀𝑢𝑓=unsprung mass of front axle 𝑀𝑢𝑟=unsprung mass of rear axle 𝐾𝑢𝑓 =suspension stiffness of front axle 𝐾𝑢𝑟 =suspension stiffness of rear axle 𝐶𝑠𝑓=damping coefficient of front axle 𝐶𝑠𝑟=damping coefficient of rear axle 𝐾𝑡𝑓=stiffness of front tire 𝐾𝑡𝑟 =stiffness of rear tire L=length of wheelbase b=distance to front axle from centre of gravity C=distance to rear axle from centre of gravity ɵ=pitch angle 𝑍𝑟𝑓=road excitation at the front wheel 𝑍𝑟𝑟=road excitation at the rear wheel