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Evaluating the reliability of two mathematical models of the pneumatic suspension via two various theoretical approaches

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Abstract

Two various mathematical approaches of the pneumatic suspension (PS) are proposed and researched to evaluate the reliability and performance of the mathematical models, advantages-disadvantages, and scope of application of each method on the PS of the vehicles. The experimental study of the PS model is also presented under the different harmonic excitations to further reinforce the reliability of the research results. The investigation results indicate that the accuracy of the mathematical approach I (MA-I) determined based on the lumped parameter of the PS model and the mathematical approach II (MA-II) determined via the mass flow rate of the PS is the same results. Besides, the MA-I can easily determine the initial stiffness and damping parameters of the PS and optimize or control the PS, while the MA-II can easily apply for the multi-airbag suspensions using the parallel and long pipes. Additionally, the MA-I should be used for the multi-axles vehicles of the heavy trucks or semi-trailer trucks using the PS that each PS consists of only one air bag connected with a reservoir to improve the ride comfort and road-friendly, conversely, the MA-II should be applied for the two-axle vehicles equipped with the hydraulically interconnected suspensions to enhance the ride comfort and the lateral/roll stability. The research results not only reinforce the reliability and give the advantages-disadvantages of each MA-I and MA-II but also clarify the effectiveness and scope of application of each method on different vehicles.

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Abbreviations

AME:

Advanced modeling environment

MA-I:

Mathematical approach I

MA-II:

Mathematical approach I

MTS:

Mechanical testing systems

RMS:

Root mean square

PSD:

Power spectral density

PS:

Pneumatic suspension

A s :

Pipe’s cross-section area/m2

A e :

Air bag’s effective area/m2

b ir,l :

Horizontal distance of suspensions/m

b air,l :

Horizontal distance of wheels/m

C β z :

Nonlinear damping parameter/Ns2 m−2

C s :

Damping coefficient of the pipe/Ns m−1

d s :

Pipe’s diameter/m

f :

Frequency of vibration excitation/Hz

F :

Air bag’s acting force/N

K x :

PS’s nonlinear elastic stiffness/N m−1

K v z :

Air bag’s viscous stiffness/N m−1

K e z :

Air bag’s static stiffness/N m−1

K tir,l :

Stiffness of right/left wheels/N m−1

l s :

Pipe length/m

l j :

Longitudinal distance of vehicles/m

M d,t,ai :

Mass of tractor driver, trailer, axles/kg

M b,air,l :

Mass of car and right/left wheels/kg

m s :

Air mass in pipe/kg

n :

Air’s polytropic constant

p a :

Standard atmospheric pressure/Pa

p b :

Air bag pressure/Pa

p r :

Reservoir pressure/Pa

p 0 :

PS’s initial pressure/Pa

Q :

Mass flow rate of air

q tir,l :

Vibration excitation at right/left wheels/m

V 0b :

Air bag’s initial volume/m3

V 0r :

Reservoir’s initial volume/m3

V b :

Air bag’s volume/m3

V r :

Reservoir's volume/m3

v :

Vehicle moving velocity/m s−1

z :

Air bag’s vertical displacement/m

Z d,t,ai :

Vertical vibration of tractor driver, trailer, axles/m

Z air,l :

Vertical vibrations at right/left car wheels/m

z(t):

Vibration excitation of PS model/m

Z b :

Vertical vibration of car body/m

φ d,t :

Angular vibration of tractor driver, trailer/rad

θ d,t,ai :

Angular vibration of tractor driver, trailer, axles/rad

φ b :

Pitching angle of car body/rad

θ b :

Rolling angle of car body/rad

δ s :

Air displacement in pipe/m

ρ :

Air density/kg m−3

With the heavy truck model:

i = 1 − 3 and j = 1 − 5

With the car model:

i = j = 1 − 2

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Acknowledgements

This research is supported by the Open Fund Project of Hubei Key Laboratory of Intelligent Transportation Technology and Device, Hubei Polytechnic University (Nos. 2021XZ107, 2020XZ107), the Key Scientific Research Project of Hubei Polytechnic University (No. 21xjz02A), and the Key Project of Scientific Research Plan of Education Department of Hubei Province (No. D20204501).

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Authors and Affiliations

  1. School of Mechanical and Electrical Engineering, Hubei Polytechnic University, Huangshi, 435003, China

    Renqiang Jiao & Vanliem Nguyen

  2. Faculty of Automotive and Power Machinery Engineering, Thai Nguyen University of Technology, Thai Nguyen City, 23000, Vietnam

    Vanliem Nguyen & Vanquynh Le

  3. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, China

    Xiaoyan Guo

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  1. Renqiang Jiao
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  2. Vanliem Nguyen
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  4. Vanquynh Le
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Correspondence to Vanliem Nguyen.

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Technical Editor: Samuel da Silva.

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Jiao, R., Nguyen, V., Guo, X. et al. Evaluating the reliability of two mathematical models of the pneumatic suspension via two various theoretical approaches. J Braz. Soc. Mech. Sci. Eng. 43, 482 (2021). https://doi.org/10.1007/s40430-021-03193-0

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