Skip to main content
SpringerLink
Log in

A family base optimization of a developed nonlinear vehicle suspension model using gray family design algorithm

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The primary purpose of this paper is to present a novel algorithm for designing a family of variants. This platform base algorithm is utilized for designing suspension system for a family of Renault Logan cars. In this case, a gray penalty function is presented to minimize variations between suspension parameters and generate suspension variants which have most commonality between each other in different cars of the family. This study developed a nonlinear mathematical model in order to simulate the dynamic performance of suspension system. Therefore, a novel suspension model is presented based on combining vehicle vibration model and geometry suspension model which are subjected to a random vibration road profile excitation and gray family design algorithm. Geometry suspension model analyzes the effect of suspension parameters (i.e., hard points, length of arms, camber angle and caster angle) on the stability, ride and handling of the vehicle. Vehicle vibration model investigates the influence of seat location, damping and spring coefficients on the driver seat acceleration, roll angular and pitch angular acceleration, relative displacement and sprung mass acceleration. The results achieved by the simulation of the full vehicle in ADAMS/CAR and experimental test of Renault Logan car, which have been done in one of main Iranian automakers (SAIPA), demonstrate the accuracy of the novel model. Comparing the results of the family base suspension model and results of ADAMS/CAR simulation indicates the beneficial of this algorithm and its usage in designing suspension system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

\(m_\mathrm{s}\) :

Sprung mass (kg)

\(m_\mathrm{u1}\) :

Unsprung mass of front left suspension (kg)

\(m_\mathrm{u2}\) :

Unsprung mass of rear left suspension (kg)

\(m_\mathrm{u3}\) :

Unsprung mass of front right suspension (kg)

\(m_\mathrm{u4}\) :

Unsprung mass of rear right suspension (kg)

\(c_\mathrm{s1}\) :

Damping coefficient of front left suspension (N s/m)

\(c_\mathrm{s2}\) :

Damping coefficient of rear left suspension (N s/m)

\(c_\mathrm{s3}\) :

Damping coefficient of front right suspension (N s/m)

\(c_\mathrm{s4}\) :

Damping coefficient of rear right suspension (N s/m)

\(k_\mathrm{t1}\) :

Tire stiffness of front left tire (N/m)

\(k_\mathrm{t2}\) :

Tire stiffness of rear left tire (N/m)

\(k_\mathrm{t3}\) :

Tire stiffness of front right tire (N/m)

\(k_\mathrm{t4}\) :

Tire stiffness of rear right tire (N/m)

\(k_\mathrm{s1}\) :

Spring stiffness of front left suspension (N/m)

\(k_\mathrm{s2}\) :

Spring stiffness of rear left suspension (N/m)

\(k_\mathrm{s3}\) :

Spring stiffness of front right suspension (N/m)

\(k_\mathrm{s4}\) :

Spring stiffness of rear right suspension (N/m)

\(z_\mathrm{r1}\) :

Road profile excitation of front left tire (m)

\(z_\mathrm{r2}\) :

Road profile excitation of rear left tire (m)

\(z_\mathrm{r3}\) :

Road profile excitation of front right tire (m)

\(z_\mathrm{r4}\) :

Road profile excitation of rear right tire (m)

\(c_\mathrm{seat}\) :

Damping coefficient of driver seat (N s/m)

\(k_\mathrm{seat}\) :

Spring stiffness of driver seat (N/m)

\(\varphi \) :

Control arm rotation angle (deg)

\(\varphi _{0}\) :

Initial control arm rotation angle (deg)

\(z_\mathrm{s}\) :

Sprung mass displacement (m)

\(z_\mathrm{u}\) :

Unsprung mass displacement (m)

\(\xi _{a}\) :

Length of suspension arm (m) (Fig. 4)

\(\xi _{b}\) :

Length of suspension arm (m) (Fig. 4)

\(\xi _{c}\) :

Length of control arm (m) (Fig. 4)

\(\mathrm{acc.}_{zs}\) :

Sprung mass acceleration (\(\mathrm{m/s}^{2})\)

\(\mathrm{acc.}_{\varphi }\) :

Control arm angular acceleration \((\mathrm{deg/s}^{2})\)

\(\varphi \) :

Roll angle of control arm (deg)

d :

Relative displacement of sprung mass and unsprung mass (m)

\(\tilde{V}\) :

Relative velocity of sprung mass and unsprung mass (m/s)

\(\xi _{0}\) :

Dimension in reference frame \(\xi _{0}\tau _{0}\psi _{0}\)

\(\xi _{1}\) :

Dimension in reference frame \(\xi _{1}\tau _{1}\psi _{1}\)

\(\tau _{0}\) :

Dimension in reference frame \(\xi _{0}\tau _{0}\psi _{0}\)

\(\tau _{1}\) :

Dimension in reference frame \(\xi _{1}\tau _{1}\psi _{1}\)

\(\psi _{0}\) :

Dimension in reference frame \(\xi _{0}\tau _{0}\psi _{0}\)

\(\psi _{1}\) :

Dimension in reference frame \(\xi _{1}\tau _{1}\psi _{1}\)

u :

Value of point p in \(\xi \) direction

v :

Value of point p in \(\tau \) direction

w :

Value of point p in \(\psi \) direction

\(z_{w}\) :

Value of point w in z direction (m) (Fig. 4)

\(z_{p}\) :

Value of point p in z direction (m) (Fig. 4)

\(z_{q}\) :

Value of point q in z direction (m) (Fig. 4)

\(y_{w}\) :

Value of point w in y direction (m) (Fig. 4)

\(y_{p}\) :

Value of point p in y direction (m) (Fig. 4)

\(y_{q}\) :

Value of point q in y direction (m) (Fig. 4)

\(\sigma \) :

Angle between ow and y-axis (deg) (see Fig. 4)

\(F_{xk}\) :

The kth external force in x direction (N)

\(F_{yk}\) :

The kth external force in y direction (N)

\(F_{zk}\) :

The kth external force in z direction (N)

\(q_{j}\) :

The jth general coordinate in Lagrange equation

V :

Potential energy in Lagrange equation

T :

Kinetic energy in Lagrange equation

D :

Energy of damping in Lagrange equation

\(Q_{j}(n)\) :

The general force corresponding to the general coordinate \(q_{j}\)

t :

Time domain in Lagrange equation (s)

\(\lambda (t)\) :

White noise function

\(\psi \) :

Spectral density of white noise

\(\sigma ^{2}\) :

Variance of road roughness

\(\zeta _{0}\) :

Constant value in linear regression equation

\(\zeta _{1}\) :

Constant value in linear regression equation

\(\nu _{i}\) :

The regression coefficients in linear regression equation

References

  1. Nariman-Zadeh, N., Salehpour, M., Jamali, A., Haghgoo, E.: a Pareto optimization of a five-degree of freedom vehicle vibration model using a multi-objective uniform-diversity genetic algorithm (MUGA). Eng. Appl. Artif. Intell. 23, 543–551 (2010)

    Article  Google Scholar 

  2. Lopez, A., Olazagoitia, J.L., Moriano, C., Ortiz, A.: Nonlinear optimization of a new polynomial tyre model. Nonlinear Dyn. 78, 2941–2958 (2014)

    Article  Google Scholar 

  3. Guclu, R.: Fuzzy logic control of seat vibrations of a non-linear full vehicle model. Nonlinear Dyn. 40, 21–34 (2005)

    Article  MATH  Google Scholar 

  4. Fang, Z., Shu, W., Du, D., Xiang, B., He, Q., He, K.: Semi-active suspension of a full-vehicle model based on double-loop control. Procedia Eng. 16, 428–437 (2011)

    Article  Google Scholar 

  5. Joao, P.C., Goncalves, J.A., Ambrosio, C.: Optimization of vehicle suspension systems for improved comfort of road vehicles using flexible multibody dynamics. Nonlinear Dyn. 34, 113–131 (2003)

    Article  MATH  Google Scholar 

  6. Dangor, M., Dahunsi, O.A., Pedro, J.O., Ali, M.M.: Evolutionary algorithm-based PID controller tuning for nonlinear quarter-car electrohydraulic vehicle suspensions. Nonlinear Dyn. 78, 2795–2810 (2014)

    Article  Google Scholar 

  7. Jin, Y., Luo, X.: Stochastic optimal active control of a half-car nonlinear suspension under random road excitation. Nonlinear Dyn. 72, 185–195 (2013)

    Article  MathSciNet  Google Scholar 

  8. Bouazara, M., Richard, M.J.: An optimization method designed to improve 3-D vehicle comfort and road holding capability through the use of active and semi-active suspensions. Eur. J. Mech. A Solids 20(3), 509–520 (2001)

    Article  MATH  Google Scholar 

  9. Gundogdu, O.: Optimal seat and suspension design for a quarter car with driver model using genetic algorithms. Int. J. Ind. Ergon. 37(4), 327–332 (2007)

    Article  Google Scholar 

  10. Attia, H.A.: Dynamic modelling of the double wishbone motor-vehicle suspension system. Eur. J. Mech. A Solids 21, 167–174 (2002)

    Article  MATH  Google Scholar 

  11. Cao, D., Song, X., Ahmadian, M.: Vehicle system dynamics: international journal of vehicle mechanics and mobility. Veh. Syst. Dyn. 49(1–2), 3–28 (2011)

    Article  Google Scholar 

  12. Chen, K., Beale, D.G.: Base dynamic parameter estimation of a Macpherson suspension mechanism. Veh. Syst. Dyn. 39(3), 227–244 (2003)

    Article  Google Scholar 

  13. Hu, Y., Chen, M.Z.Q., Shu, Z.: Passive vehicle suspensions employing inerters with multiple performance requirements. J Sound Vib. 333, 2212–2225 (2014)

    Article  Google Scholar 

  14. Nemeth, B., Gaspar, P.: Set-based analysis of the variable-geometry suspension system. In: 19th World Congress The International Federation of Automatic Control, Cape Town, South Africa (2014)

  15. Yang, Q., Yu, S., Jiang, D.: A modular method of developing an eco-product family considering the reusability and recyclability of customer products. J. Clean. Prod. 64, 254–265 (2014)

    Article  Google Scholar 

  16. Ramadan, K., ElMaraghy, W.: Product families and platforms diversification: customer expectations, product variations, or self-competition? Procedia Cirp. 16, 104–109 (2014)

    Article  Google Scholar 

  17. Wei, W., Yixiong, F., Jianrong, T., Zhongkai, L.: Product platform two-stage quality optimization design based on multi-objective genetic algorithm. Comput. Math. Appl. 57, 1929–1937 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shijia, L., Yunkai, T., Shijian, L.A.: Study of product family design dna based on product style. In: 2009 IEEE 10th international conference on computer-aided industrial design and conceptual design, pp. 377–38 (2009). doi:10.1109/CAIDCD.2009.5375436

  19. Nayak, R.U., Chen, W., Simpson, T.W.: A variation-based method for product family design. Eng Optim. 34, 65–81 (2002)

    Article  Google Scholar 

  20. Fallah, M.S., Bhat, R., Xie, W.F.: New model and simulation of Macpherson suspension system for ride control applications. Veh. Syst. Dyn. 47(2), 195–220 (2009)

    Article  Google Scholar 

  21. Spong, M.W., Vidyasagar, M.: Robot Dynamics and Control. Wiely, London (2008)

    Google Scholar 

  22. Dai, Z., Scott, M.J.: Effective product family design using preference aggregation. J. Mech. Des. 128(4), 659–667 (2006). doi:10.1115/1.2197835

  23. Simpson, T.W., Jiao, J., Siddique, Z., Hölttä-Otto, K.: Advances in Product Family and Product Platform Design Methods and Applications. Springer, Heidelberg (2014)

    Book  Google Scholar 

  24. Raychaudhuri, S.: Introduction to Monte Carlo simulation. In: Proceedings of the 2008 Winter Simulation Conference Austin, Texas (2008)

  25. Uys, P.E., Els, P.S., Thoresson, M.: Suspension settings for optimal ride comfort of off-road vehicles travelling on roads with different roughness and speeds. J. Terramech. 44, 163–175 (2007)

    Article  Google Scholar 

  26. Jung, J.H., Kwon, W.T.: Optimization of EDM process for multiple performance characteristics using Taguchi method and Grey relational analysis. J. Mech. Sci. Technol. 24(5), 1083–1090 (2010)

    Article  Google Scholar 

  27. Ning, X., Zhao, C., Shen, J.: Dynamic analysis of car suspension using ADAMS/car for development of a software interface for optimization. Procedia Eng. 16, 333–341 (2011)

    Article  Google Scholar 

  28. Yang, Y., Ren, W., Chen, L., Jiang, M.: Study on ride comfort of tractor with tandem suspension based on multi-body system dynamics. Appl. Math. Model. 33, 11–33 (2009)

    Article  Google Scholar 

  29. Tian, Q., Liu, C., Machado, M., Flores, P.: A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody systems. Nonlinear Dyn. 64, 25–47 (2011)

    Article  MATH  Google Scholar 

  30. Shojaeefard, M.H., Khalkhali, A., Yarmohammadisatri, S.: An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson correlation coefficient. Veh. Syst. Dyn. 55(6), 827–852 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Automotive Simulation and Optimal Design Research Laboratory, School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran

    Sadegh Yarmohammadisatri, Mohammad Hasan Shojaeefard & Abolfazl Khalkhali

Authors
  1. Sadegh Yarmohammadisatri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Hasan Shojaeefard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abolfazl Khalkhali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abolfazl Khalkhali.

Appendix

Appendix

In Eq. (26):

$$\begin{aligned} T= & {} \frac{1}{2}\left( {m_\mathrm{s1} +m_\mathrm{u1}} \right) \dot{z}_\mathrm{s1}^2\\&+\frac{1}{2}m_\mathrm{u1} \xi _c^2 \dot{\varphi } ^{2}+m_\mathrm{u1} \xi _c \cos \left( {\varphi -\varphi _0} \right) \dot{\varphi } \dot{z}_\mathrm{s1}\\ V= & {} \frac{1}{2}k_\mathrm{s1} \left[ 2a_\xi -b_\xi \left( {\cos {\sigma } '+\cos \left( {{\sigma } '-\varphi } \right) } \right) \right. \\&\left. -2\left\{ {a_\xi ^2} \right. -a_\xi b_\xi \right. \left. \left. \left( \cos {\sigma } '+\cos \left( {{\sigma } '-\varphi } \right) \right. \right. \right. \\&\left. \left. \left. +b_\xi ^2 \cos {\sigma } '\cos \left( {{\sigma } '-\varphi } \right) \right) \right\} ^{1/2} \right] \\&+\frac{1}{2}k_\mathrm{t1} \left[ z_\mathrm{s1} +\xi _c \left( \sin \left( {\varphi -\varphi _0} \right) \right. \right. \\&\left. \left. -\,\sin \left( {-\varphi _0} \right) \right) -z_\mathrm{r1} \right] ^{2} \\ D= & {} \frac{1}{2}c_\mathrm{s1} \left[ {\frac{b_\xi \sin \left( {{\sigma } '-\varphi } \right) \dot{\varphi } }{2\left( {a_\xi -b_\xi \cos \left( {{\sigma } '-\varphi } \right) } \right) ^{1/2}}} \right] ^{2}\\ D= & {} \frac{c_\mathrm{s1} b_\xi ^2 \sin ^{2}\left( {{\sigma } '-\varphi } \right) \dot{\varphi } }{8\left( {a_\xi -b_\xi \cos \left( {{\sigma } '-\varphi } \right) } \right) }^{2} \end{aligned}$$

In Eq. (47)

$$\begin{aligned}&\frac{\partial f_1}{\partial x_1} =\frac{-k_\mathrm{t1} \xi _c \sin ^{2}\left( {-\varphi _0} \right) }{D_1}\\&\frac{\partial f_1}{\partial x_2} =0\\&\frac{\partial f_1}{\partial x_3} =-\frac{1}{D_1^2} \left\{ {\left[ {\frac{1}{2}k_\mathrm{s1} \left( {b_\xi {+}\frac{d_\xi }{\left( {c_\xi -d_\xi \cos \left( {{\sigma } '-\varphi } \right) } \right) ^{1/2}}} \right) } \right. } \right. \cos \left( {{\sigma }'+\varphi _0} \right) \\&\quad \left. -\, \frac{1}{2}k_\mathrm{s1} \sin {\sigma } '\cos \left( {-\varphi _0} \right) \left( {\frac{d_\xi ^2 \sin \left( {{\sigma } '} \right) }{2\left( {c_\xi -d_\xi \cos \left( {{\sigma } '} \right) } \right) ^{3/2}}} \right) \right. \\&\quad \left. -\, k_\mathrm{t1} \xi _c^2 \sin ^{2}\left( {-\varphi _0} \right) \cos \left( {-\varphi _0} \right) \right] \\&\quad \times \left. \left[ {m_\mathrm{s1} \xi _c +m_\mathrm{u1} \xi _c \sin ^{2}\left( {-\varphi _0} \right) } \right] \right. \\&\quad \left. +\, m_\mathrm{u1} k_\mathrm{s1} \xi _c \sin {\sigma } '\cos ^{2}\left( {-\varphi _0} \right) \left( {b_\xi +\frac{d_\xi }{\left( {c_\xi -d_\xi \cos \left( {{\sigma } '-\varphi } \right) } \right) ^{1/2}}} \right) \right\} \\&\frac{\partial f_1}{\partial x_4} =\frac{1}{D_1} \frac{c_\mathrm{s1} b_\xi ^2 \sin ^{2}\left( {{\sigma }^{'}} \right) }{4\left( {a_\xi -b_\xi \cos \left( {{\sigma }^{'}} \right) } \right) } \end{aligned}$$

In Eq. (47)

$$\begin{aligned} \frac{\partial f_2}{\partial x_2}= & {} 0\\ \frac{\partial f_2}{\partial x_1}= & {} \frac{-m_\mathrm{s1} k_\mathrm{t1} \xi _c \cos \left( {-\varphi _0} \right) }{D_2}\\ \frac{\partial f_2}{\partial x_3}= & {} -\frac{1}{D_2^2} \left\{ \left[ \frac{1}{2}\left( {m_\mathrm{s1} +m_\mathrm{u1}} \right) k_\mathrm{s1} \cos \left( {{\sigma } '} \right) \right. \right. \\&\left. \left. \times \left( {b_\xi +\frac{d_\xi }{\left( {c_\xi -d_\xi \cos \left( {{\sigma } '-\varphi } \right) } \right) ^{1/2}}} \right) \right. \right. \\&\left. -\frac{1}{2}\left( {m_\mathrm{s1} +m_\mathrm{u1}} \right) k_\mathrm{s1} \sin {\sigma } '\left( {\frac{d_\xi ^2 \sin \left( {{\sigma } '} \right) }{2\left( {c_\xi -d_\xi \cos \left( {{\sigma } '} \right) } \right) ^{3/2}}} \right) \right. \\&\left. -\,k_\mathrm{t1} \xi _c^2 \sin ^{2}\left( {-\varphi _0} \right) \cos \left( {-\varphi _0} \right) \right] \\&\times \left[ {m_\mathrm{s1} m_\mathrm{u1} \xi _c +m_\mathrm{u1}^2 \xi _c^2 \sin ^{2}\left( {-\varphi _0} \right) } \right] \\&-\,\frac{1}{2}\left( {m_\mathrm{s1} +m_\mathrm{u1}} \right) m_\mathrm{u1}^2 k_\mathrm{s1} \xi _c^2 \sin {\sigma } '\sin \left( {-\varphi _0} \right) \cos \left( {-\varphi _0} \right) \\&\times \left. {\left( {b_\xi +\frac{d_\xi }{\left( {c_\xi -d_\xi \cos \left( {{\sigma } '-\varphi } \right) } \right) ^{1/2}}} \right) } \right\} \\ \frac{\partial f_2}{\partial x_4}= & {} -\frac{1}{D_2} \frac{\left( {m_\mathrm{s} +m_\mathrm{u}} \right) c_\mathrm{s1} b_\xi ^2 \sin ^{2}\left( {{\sigma } '} \right) }{4\left( {a_\xi -b_\xi \cos \left( {{\sigma } '} \right) } \right) } \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yarmohammadisatri, S., Shojaeefard, M.H. & Khalkhali, A. A family base optimization of a developed nonlinear vehicle suspension model using gray family design algorithm. Nonlinear Dyn 90, 649–669 (2017). https://doi.org/10.1007/s11071-017-3686-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3686-8

Keywords

Search

Navigation