Dynamic model of a logistic train with different steering systems and tire models

Paszkowiak, Wojciech; Bartkowiak, Tomasz

doi:10.1590/1679-78256147

Wojciech Paszkowiak ^** Corresponding author

Institute of Mechanical Technology, Poznan University of Technology, Pl. M. Sklodowskiej-Curie 5, 60-965 Poznan, Poland. E-mail: wojciech.z.paszkowiak@doctorate.put.poznan.pl, tomasz.bartkowiak@put.poznan.pl

http://orcid.org/0000-0003-2571-6737

Tomasz Bartkowiak

Institute of Mechanical Technology, Poznan University of Technology, Pl. M. Sklodowskiej-Curie 5, 60-965 Poznan, Poland. E-mail: wojciech.z.paszkowiak@doctorate.put.poznan.pl, tomasz.bartkowiak@put.poznan.pl

http://orcid.org/0000-0002-4715-6358

* Corresponding author

Author’s Contribuitions:

Conceptualization, W Paszkowiak and T Bartkowiak; Methodology, W Paszkowiak and T Bartkowiak; Investigation, W Paszkowiak; Writing - original draft, W Paszkowiak and T Bartkowiak; Writing - review & editing, W Paszkowiak and T Bartkowiak; Funding acquisition, W Paszkowiak and T Bartkowiak; Resources, W Paszkowiak and T Bartkowiak; Supervision, T Bartkowiak.

Editor:

Jan Awrejcewicz.

Point	Coordinate	Expression
$B_{0}$	$x_{B 0}$	$x$
$B_{0}$	$y_{B 0}$	$y$
$P_{0}$	$x_{P 0}$	$x + c_{0} \cos (β_{0})$
$P_{0}$	$y_{P 0}$	$y - c_{0} \sin (β_{0})$
$A_{0}$	$x_{A 0}$	$x + h_{0} \cos (β_{0})$
$A_{0}$	$y_{A 0}$	$y - h_{0} \sin (β_{0})$
$D_{0}$	$x_{D 0}$	$x - d_{0} \cos (β_{0})$
$D_{0}$	$y_{D 0}$	$y + d_{0} \sin (β_{0})$
$E_{0}$	$x_{E 0}$	$x + \frac{1}{2} v_{0} \sin (β_{0})$
$E_{0}$	$y_{E 0}$	$y + \frac{1}{2} v_{0} \cos (β_{0})$
$F_{0}$	$x_{F 0}$	$x - \frac{1}{2} v_{0} \sin (β_{0})$
$F_{0}$	$y_{F 0}$	$y - \frac{1}{2} v_{0} \cos (β_{0})$

Point	Coordinate	Expression
$B_{i}$	$x_{B i}$	$x - \sum_{j = 0}^{i - 1} d_{j} \cos (β_{j}) - \sum_{j = 1}^{i} (h_{j} + d_{f j}) \cos (β_{j})$
$B_{i}$	$y_{B i}$	$y + \sum_{j = 0}^{i - 1} d_{j} \sin (β_{j}) + \sum_{j = 1}^{i} (h_{j} + d_{f j}) \sin (β_{j})$
$P_{i}$	$x_{P i}$	$x_{B i} + c_{i} \cos (β_{i})$
$P_{i}$	$y_{P i}$	$y_{B i} - c_{i} \sin (β_{i})$
$A_{i}$	$x_{A i}$	$x_{B i} + h_{i} \cos (β_{i})$
$A_{i}$	$y_{A i}$	$y_{B i} - h_{i} \sin (β_{i})$
$D_{i}$	$x_{D i}$	$x_{B i} - d_{i} \cos (β_{i})$
$D_{i}$	$y_{D i}$	$y_{B i} + d_{i} \sin (β_{i})$
$E_{i}$	$x_{E i}$	$x_{B i} + \frac{1}{2} v_{i} \sin (β_{i})$
$E_{i}$	$y_{E i}$	$x_{B i} + \frac{1}{2} v_{i} \cos (β_{i})$
$F_{i}$	$x_{F i}$	$x_{B i} - \frac{1}{2} v_{i} \sin (β_{i})$
$F_{i}$	$y_{F i}$	$x_{B i} - \frac{1}{2} v_{i} \cos (β_{i})$

Point	Coordinate	Position equation
$B_{i}$	$x_{B i}$	$x - d_{0} \cos (β_{0}) - \sum_{j = 1}^{i - 1} d_{j} \cos (β_{j 2}) - \sum_{j = 1}^{i} d_{p j} \cos (β_{j 1}) - \sum_{j = 1}^{i} h_{j} \cos (β_{j 2})$
$B_{i}$	$y_{B i}$	$y + d_{0} \sin (β_{0}) + \sum_{j = 1}^{i - 1} d_{j} \sin (β_{j 2}) + \sum_{j = 1}^{i} d_{p j} \sin (β_{j 1}) + \sum_{j = 1}^{i} h_{j} \sin (β_{j 2})$
$P_{i 1}$	$x_{P i 1}$	$x_{A i} + c_{i 1} \cos (β_{i 1})$
$P_{i 1}$	$y_{P i 2}$	$y_{A i} - c_{i 1} \sin (β_{i 1})$
$P_{i 2}$	$x_{P i 1}$ $x_{B i} + c_{i 2} \cos (β_{i 2})$	$y_{P i 2}$
$P_{i 2}$	$y_{B i} - c_{i 2} \sin (β_{i 2})$	$A_{i}$
$x_{A i}$	$x_{B i} + h_{i} \cos (β_{i 2})$	$y_{A i}$
$x_{A i}$	$y_{B i} - h_{i} \sin (β_{i 2})$	$D_{i}$
$x_{D i}$	$x_{B i} - d_{i} \cos (β_{i 2})$	$y_{D i}$
$x_{D i}$	$y_{B i} + d_{i} \sin (β_{i 2})$	$E_{i}$
$x_{E i}$	$x_{B i} + \frac{1}{2} v_{i} \sin (β_{i 2})$	$y_{E i}$
$x_{E i}$	$y_{B i} + \frac{1}{2} v_{i} \cos (β_{i 2})$	$F_{i}$
$x_{F i}$	$x_{B i} - \frac{1}{2} v_{i} \sin (β_{i 2})$	$y_{F i}$
$x_{F i}$	$y_{B i} - \frac{1}{2} v_{i} \cos (β_{i 2})$	$G_{i}$
$x_{G i}$	$x_{A i} - \frac{1}{2} v_{i} \sin (β_{i 2})$	$y_{G i}$
$x_{G i}$	$y_{A i} - \frac{1}{2} v_{i} \cos (β_{i 2})$	$H_{i}$
$x_{H i}$	$x_{A i} + \frac{1}{2} v_{i} \sin (β_{i 2})$	$y_{H i}$
$x_{H i}$	$y_{A i} + \frac{1}{2} v_{i} \cos (β_{i 2})$	$β_{i 2}$

PARAMETERS		Conventional Clutch	Virtual Clutch	Double Ackermann
Mass [kg]	$m_{b}$	1100	1100	1100
	$m_{w 0}$	2	2	2
	$m_{d p i}$	-	-	5
	$m_{t i}$	50	50	45
	$m_{w i}$	2	2	2
Moment of inertia $[k g m^{2}]$	$I_{b}$	30	30	30
	$I_{w 0}, I_{w i}$	0.02	0.02	0.02
	$I_{d p i}$	-	-	0.1
	$I_{t i}$	1	1	1
Dimensions [m]	$h_{0}$	1	1	1
	$v_{0}$	0.5	0.5	0.5
	$c_{0}$	0.3	0.3	0.3
	$c_{i}, c_{i 1}$	0.4	0.3	0.35
	$c_{i 2}$	-	-	0.15
	$r_{0 f}$	0.2	0.2	0.2
	$r_{0 r}$	0.1	0.1	0.1
	$d_{0}, d_{i}$	0.2	0.6	0.2
	$h_{i}$	0.7	0.7	0.7
	$d_{f i}, d_{p i}$	0.6	0.2	0.6
	$v_{i}$	0.3	0.3	0.3
	$r_{i r}$	0.08	0.08	0.08

Coefficient	Name	Parameters	Formula
$p_{d f n}$	Normalized change in vertical load	-	$\frac{F_{n} - F_{n N}}{F_{n N}}$
$p_{C}$	Shape factor	$ρ_{0}$	$p_{C} = ρ_{C Y 1}$
$p_{D}$	Peak factor	$ρ_{1}, ρ_{15}$	$p_{D} = \frac{p_{μ} F_{n}}{1000}$
$p_{μ}$	Peak friction coefficient		$(ρ_{D Y 1} + ρ_{D Y 2} p_{d f n}) (1 - P_{D Y 3} γ^{2})$
$p_{B C D}$	Stiffness	$ρ_{3}, ρ_{4}, ρ_{5}$	$\frac{F_{n N} ρ_{K Y 1}}{1000} \sin (2 a r c t g (\frac{F_{n}}{ρ_{K Y 2} F_{n N}})) (1 - ρ_{K Y 3} \| γ \frac{180}{π} \|)$
$p_{B}$	Stiffness factor	$p_{B C D}, p_{C}, p_{D}$	$\frac{p_{B C D}}{p_{C} p_{D}}$
$p_{E}$	Curvature factor	$ρ_{6}, ρ_{7}, ρ_{16}, ρ_{17}$	$(ρ_{E Y 1} + ρ_{E Y 2} p_{d f n}) (1 - (ρ_{E Y 3} + ρ_{E Y 4} γ \frac{180}{π}) s i g n (α \frac{180}{π} + p_{H}))$
$p_{H}$	Horizontal shift	$ρ_{8}, ρ_{9}, ρ_{10}$	$ρ_{H Y 1} + ρ_{H Y 2} p_{d f n} + ρ_{H Y 3} γ \frac{180}{π}$
$p_{V}$	Vertical shift	$ρ_{11}, ρ_{12}, ρ_{13}, ρ_{14}$	$\frac{F_{n}}{1000} (ρ_{V Y 1} + ρ_{V Y 2} p_{d f n})$

Parameter	Role	Value	Parameter	Role	Value
$ρ_{C Y 1}$	Shape factor	1.3	$ρ_{K V 1}$	Change of stiffness with slip	1100
$ρ_{D Y 1}$	Lateral friction	$μ$	$ρ_{K Y 2}$	Change of progressivity of stiffness / load	10
$ρ_{D Y 2}$	Load influence on lateral friction coefficient	0	$ρ_{K Y 3}$	Camber influence on stiffness	0
$ρ_{D Y 3}$	Camber influence on lateral friction coefficient	0	$ρ_{H Y 1}$	Horizontal shift at load equal to 0 and camber equal to 0	0
$ρ_{E Y 1}$	Curvature factor	-2	$ρ_{H Y 2}$	Load influence on horizontal shift	0
$ρ_{E Y 2}$	Curvature change with load	0	$ρ_{H Y 3}$	Camber influence on horizontal shift	0
$ρ_{E Y 3}$	Curvature shift	0	$ρ_{V Y 1}$	Vertical shift	0
$ρ_{E Y 4}$	Curvature change with camber	0	$ρ_{V Y 2}$	Vertical shift at load equal to 0	0
$F_{n N}$	Nominal load	4000	$γ$	Camber angle	0

Parameter	Description
$M (q)$	System inertia matrix
$C (q, \dot{q})$	Centripetal and Coriolis matrix
$Q (q)$	Generalized forces vector
$L$	Lagrangian
$q$	Generalized coordinate vector
$T$	Kinetic Energy
$ψ$	Steering angle
$α$	Side-slip angle
$β$	Yaw angle
$F$	Force
$m$	Mass
$I$	Moment of inertia around the vertical axis
$μ$	Lateral friction coefficient
$τ$	Torque applied to the front wheel of the tractor
$t$	Time
$S u b s c r i p t$
$i$	Unit number
$f$	Front
$r$	Rear
$R$	Right
$L$	Left
$w$	Wheel
$n$	Normal (force)
$b$	Tractor
$t$	Trailer
$τ$	Driven

Brasil

Brasil

Dynamic model of a logistic train with different steering systems and tire models

Abstract

Graphical Abstract