Development of Ride Dynamics Mathematical Model for the Military Vehicle with Active Suspension System

Jakati, Ambarish; Jebaraj, C.; Banerjee, Saayan

doi:10.1007/978-981-15-5693-7_37

Ambarish Jakati¹¹,
C. Jebaraj¹¹ &
Saayan Banerjee¹²

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

592 Accesses

Abstract

The current paper aims to develop a ride dynamics math model of the entire military vehicle, with incorporation of the desired controller technique. The vehicle comprises 18 degree of freedom, viz., sprung mass bounce, pitch and roll about its CG, bounce for each of the 14 unsprung masses as well as for the driver’s seat. Coupled governing differential equations of motion for passive as well as active systems have been derived using state-space approach, and solved in Matlab. Various options are explored for optimising location of the controller such as PID and LQR, in order to obtain the desired vibration reduction at the driver’s location. Comparative dynamic analyses are carried out between the passive and active controlled system over standard terrain profiles, from which the optimum controller location is selected. This mathematical study would play a key role in fine-tuning the suspension properties as well as for deciding upon the optimum implementation of the controller with enhanced crew comfort at reduced cost.

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

Access this chapter

Log in via an institution

Chapter: USD 29.95; Price excludes VAT (USA)

eBook: USD 259.00; Price excludes VAT (USA)

Softcover Book: USD 329.99; Price excludes VAT (USA)

Hardcover Book: USD 329.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

Banerjee S, Balamurugan V, Krishnakumar R (2014) Ride dynamics mathematical model for a single station representation of tracked vehicle. J Terramech 53:47–58
Google Scholar
Ekoru JE, Pedro JO (2013) Proportional-integral-derivative control of nonlinear half-car electro-hydraulic suspension systems. J Zhejiang Univ Sci A 14(6):401–416
Article Google Scholar
Esmailzadeh E, Fahimi F (1997) Optimal adaptive active suspensions for a full car model. Veh Syst Dyn 27(2):89–107
Article Google Scholar
Gillespie TD (1992) Fundamentals of vehicle dynamics. Society of Automotive Engineers, USA
Book Google Scholar
Guclu Rahmi (2004) Active suspension control of eight degrees of freedom vehicle model. Math Comput Appl 9(1):1–10
MATH Google Scholar
Hu Jia-Wei, Lin Jung-Shan (2008) Nonlinear control of full-vehicle active suspensions with backstepping design scheme. IFAC Proc Vol 41(2):3404–3409
Article MathSciNet Google Scholar
Jakati A, Banerjee S, Jebaraj C (2017) Development of mathematical models, simulating vibration control of tracked vehicle weapon dynamics. Def Sci J 67(4):465–475
Google Scholar
Kaleemullah M, Faris WF, Hasbullah F (2011) Design of robust H∞, fuzzy and LQR controller for active suspension of a quarter car model. In: 2011 4th international conference on mechatronics (ICOM). IEEE, pp 1–6
Google Scholar
Mathworks; Help Literature; Matlab R2013a
Google Scholar
Mouleeswaran S (2012) Design and development of PID controller-based active suspension system for automobiles. PID controller design approaches-theory, tuning and application to frontier areas. InTech
Google Scholar
Taghirad HD, Esmailzadeh E (1998) Automobile passenger comfort assured through LQG/LQR active suspension. J Vib Control 4(5):603–618
Article Google Scholar
Venkatasubramanian N, Banerjee S, Balamurugan V (2016) Non-linear seventeen degrees of freedom ride dynamics model of a full tracked vehicle in simMechanics. Procedia Eng 144:1086–1093
Article Google Scholar

Download references

Acknowledgements

Authors would like to sincerely thank Dr. V Balamurugan, Sc. ‘G’, Additional Director, AP division (CVRDE) and Dr. P Sivakumar, Director (CVRDE), Chennai, for entrusting with the challenging though interesting work. We would like to thank each and every one at CEAD division, CVRDE for limitless support and encouraging us continuously. Finally, we would also like to thank all the Guides for their unending wishes, blessings and believing us at all times because of which we were able to complete the task.

Author information

Authors and Affiliations

School of Mechanical and Building Sciences, VIT University, Chennai, 600127, India
Ambarish Jakati & C. Jebaraj
CVRDE, DRDO, Avadi, Chennai, 600054, India
Saayan Banerjee

Authors

Ambarish Jakati
View author publications
You can also search for this author in PubMed Google Scholar
C. Jebaraj
View author publications
You can also search for this author in PubMed Google Scholar
Saayan Banerjee
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ambarish Jakati .

Editor information

Editors and Affiliations

Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, India
Subashisa Dutta
Department of Civil Engineering, Işık University, Istanbul, Turkey
Esin Inan
Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India
Santosha Kumar Dwivedy

Appendix

See Table 2.

Table 2 Parameters pertaining to 18 DOF of tracked vehicle

Full size table

1.
PID controller Gains:

$K_{Pil} = K_{Pir} = - 30{,}000$,
$K_{Dil} = K_{Dir} = 0$,
$K_{Iil} = K_{Iir} = 0$,
where $i = 1 \,\,{\text{to}}\,\, 7$

2.
LQR control parameters:

The ‘$Q$’ and ‘$R$’ are chosen by trial and error estimation:

$$x = 1e12 ; x_{1} = 1e15;$$

Q1 = diag([1e10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]);

R1 = diag([1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]);

$\begin{aligned}Q2 = diag ( [ & 0\;0\;x\;0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0]); \end{aligned}$

$R2 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned}Q3 = diag( [& x\;0\;0\;0\;x\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0 ]); \end{aligned}$

$R3 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned}Q4 = diag( [ & 0\;0\;0\;0\;0\;0\;x_{1} \;x_{1} \;x_{1} \;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0 ]);\end{aligned}$

$R4 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned}Q5 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0 ]); \end{aligned}$

$R5 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q6 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0\;0]); \end{aligned}$

$R6 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q7 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0]); \end{aligned}$

$R7 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q8 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0]; \end{aligned}$

$R8 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned}Q9 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0]);\end{aligned}$

$R9 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q10 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;\\ &\quad0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0 ]); \end{aligned}$

$R10 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q11 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0]); \end{aligned}$

$R11 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q12 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;x\;0\;0\;\\ &\quad 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0]);\end{aligned}$

$R12 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q13 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad 0\;x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0]); \end{aligned}$

$R13 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q14 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad x\;0\;0\;0\;0\;0\;0\;0\;0\;0]); \end{aligned}$

$R14 = diag\left( {\left[ {1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1\;1} \right]} \right);$

$\begin{aligned} Q15 = diag( [ & x\;0\;0\;0\;0\;0\;0\;0\;0\,0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;\\ &\quad x\;0\;0\;0\;0\;0\;0\;0 ]); \end{aligned}$

$R15 = diag\left( {\left[ {1\,1\,1\,1\,1\,1\,1\,1\,1\,1\,1\,1\,1\,1\,1} \right]} \right);$

where

$$\begin{aligned} A & = K_{1l} + K_{2l} + K_{3l} + K_{4l} + K_{5l} + K_{6l} + K_{7l} \\ & \quad + K_{1r} + K_{2r} + K_{3r} + K_{4r} + K_{5r} + K_{6r} + K_{7r} + K_{s} . \\ \end{aligned}$$

$$\begin{aligned} B & = - K_{1l} l_{1l} - K_{2l} l_{2l} - K_{3l} l_{3l} - K_{4l} l_{4l} + K_{5l} l_{5l} + K_{6l} l_{6l} + K_{7l} l_{7l} - K_{1r} l_{1r} \\ & \quad - K_{2r} l_{2r} - K_{3r} l_{3r} - K_{4r} l_{4r} + K_{5r} l_{5r} + K_{6r} l_{6r} + K_{7r} l_{7r} - K_{s} e_{p} . \\ \end{aligned}$$

$$\begin{aligned} C & = \left( { - K_{1l} - K_{2l} - K_{3l} - K_{4l} - K_{5l} - K_{6l} - K_{7l} } \right)a \\ & \quad + \left( {K_{1r} + K_{2r} + K_{3r} + K_{4r} + K_{5r} + K_{6r} + K_{7r} } \right)b + K_{s} e_{r} . \\ \end{aligned}$$

$$\begin{aligned} D & = - K_{1l} l_{1l} - K_{2l} l_{2l} - K_{3l} l_{3l} - K_{4l} l_{4l} + K_{5l} l_{5l} + K_{6l} l_{6l} + K_{7l} l_{7l} - K_{1r} l_{1r} - K_{2r} l_{2r} \\ & \quad - K_{3r} l_{3r} - K_{4r} l_{4r} + K_{5r} l_{5r} + K_{6r} l_{6r} + K_{7r} l_{7r} - K_{s} e_{p} . \\ \end{aligned}$$

$$\begin{aligned} E & = K_{1l} l_{1l}^{2} + K_{2l} l_{2l}^{2} + K_{3l} l_{3l}^{2} + K_{4l} l_{4l}^{2} + K_{5l} l_{5l}^{2} + K_{6l} l_{6l}^{2} + K_{7l} l_{7l}^{2} \\ & \quad + K_{1r} l_{1r}^{2} + K_{2r} l_{2r}^{2} + K_{3r} l_{3r}^{2} + K_{4r} l_{4r}^{2} + K_{5r} l_{5r}^{2} + K_{6r} l_{6r}^{2} + K_{7r} l_{7r}^{2} + K_{s} e_{p}^{2} . \\ \end{aligned}$$

$$\begin{aligned} F & = K_{1l} l_{1l} a + K_{2l} l_{2l} a + K_{3l} l_{3l} a + K_{4l} l_{4l} a - K_{5l} l_{5l} a - K_{6l} l_{6l} a \\ & \quad - K_{7l} l_{7l} a - K_{1r} l_{1r} b - K_{2r} l_{2r} b - K_{3r} l_{3r} b - K_{4r} l_{4r} b + K_{5r} l_{5r} b \\ & \quad + K_{6r} l_{6r} b + K_{7r} l_{7r} b - K_{s} e_{p} e_{r} . \\ \end{aligned}$$

$$\begin{aligned} G & = \left( { - K_{1l} - K_{2l} - K_{3l} - K_{4l} - K_{5l} - K_{6l} - K_{7l} } \right)a \\ & \quad + \left( {K_{1r} + K_{2r} + K_{3r} + K_{4r} + K_{5r} + K_{6r} + K_{7r} } \right)b + K_{s} e_{r} . \\ \end{aligned}$$

$$\begin{aligned} H & = K_{1l} al_{1l} + K_{2l} al_{2l} + K_{3l} al_{3l} + K_{4l} al_{4l} - K_{5l} al_{5l} - K_{6l} al_{6l} - K_{7l} al_{7l} \\ & \quad + \left( { - K_{1r} l_{1r} - K_{2r} l_{2r} - K_{3r} l_{3r} - K_{4r} l_{4r} + K_{5r} l_{5r} + K_{6r} l_{6r} + K_{7r} l_{7r} } \right)b - K_{s} e_{r} e_{p} . \\ \end{aligned}$$

$$\begin{aligned} I & = K_{1l} a^{2} + K_{2l} a^{2} + K_{3l} a^{2} + K_{4l} a^{2} + K_{5l} a^{2} + K_{6l} a^{2} + K_{7l} a^{2} + K_{1r} b^{2} + K_{2r} b^{2} \\ & \quad + K_{3r} b^{2} + K_{4r} b^{2} + K_{5r} b^{2} + K_{6r} b^{2} + K_{7r} b^{2} + K_{s} e_{r}^{2} . \\ \end{aligned}$$

Mass matrix is given by,

$$\begin{aligned} \left[ {M_{matrix} } \right] = diag( [& m_{s} \,M \,I_{p} \, I_{r} \, m_{1l} \, m_{2l} \, m_{3l} \,m_{4l} \,m_{5l} \,m_{6l} \,m_{7l} \, m_{1r} \, m_{2r} \, \\ & \quad m_{3r} \, m_{4r} \, m_{5r} \, m_{6r} \, m_{7r} ] ). \end{aligned}$$

Rights and permissions

Reprints and permissions

Copyright information

About this paper

Cite this paper

Jakati, A., Jebaraj, C., Banerjee, S. (2020). Development of Ride Dynamics Mathematical Model for the Military Vehicle with Active Suspension System. In: Dutta, S., Inan, E., Dwivedy, S. (eds) Advances in Rotor Dynamics, Control, and Structural Health Monitoring . Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-5693-7_37

Download citation

DOI: https://doi.org/10.1007/978-981-15-5693-7_37
Published: 30 August 2020
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-5692-0
Online ISBN: 978-981-15-5693-7
eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Development of Ride Dynamics Mathematical Model for the Military Vehicle with Active Suspension System

Abstract

Access this chapter

References

Acknowledgements

Author information

Authors and Affiliations

Corresponding author

Editor information

Editors and Affiliations

Appendix

Appendix

Rights and permissions

Copyright information

About this paper

Cite this paper

Download citation

Share this paper

Publish with us

Search

Navigation