An Optimal Braking Force Distribution in the Rigid Drawbar Trailers with Tandem Suspension

Kamiński, Zbigniew

doi:10.2478/ama-2025-0012

Figures & Tables

Forces acting on a farm tractor (ISO coordinate system [35])

Forces acting on a single-axle semi-trailer

Forces acting on a tandem-axle semi-trailer

Forces acting on a walking beam (a) and bogie suspension (b)

Forces acting on a two leaf spring suspension

Forces acting on a two leaf-two rod suspension

Forces acting on two leaf spring suspension with equalization

Limits of adhesion utilization in accordance with Commission Delegated Regulation (EU) 2015/68 [8]: a – first solution, b – second solution

A block diagram of an algorithm for the optimization of brake forces of a semi-trailer with tandem suspension using the Monte Carlo method (OFs ‒ initial value of the objective function, Nd ‒ number of draws, Ngood – number of good solutions, meeting inequality constraints, Nbetter ‒ number of better solutions, reducing the value of the objective function)

Adhesion utilization curves fi(z) for an optimal distribution of brake forces in a single axle semi-trailer: a – an unladen trailer (I solution), b – a laden trailer (II solution)

Adhesion utilization curves fi(z) for an optimal distribution of brake forces in a tandem semi-trailer (considering the weight of the tandem suspension): a – an unladen trailer with air suspension (I solution), c – a laden trailer with air suspension (II solution), b – an unladen trailer with bogie suspension (I solution), d – a laden trailer with bogie suspension (II solution)

Adhesion utilization curves fi(z) for an optimal distribution of brake forces in a tandem axle semi-trailer (considering the weight of the tandem suspension): a – an unladen trailer with two leaf-two rod suspension (I solution), c – a laden trailer with two leaf-two rod suspension (II solution), b – an unladen trailer with two leaf suspension (I solution), d – a laden trailer with two leaf suspension (I solution)

Rigid drawbar trailer and tandem suspension technical data [37–38]

Semi-trailer with tandem axle		Tandem suspension
unladen	laden	bogie (3.1)	2 leaf spring (3.2)	2 leaf 2 rod (3.3)	2 leaf equal. (3.4)	air susp. (3.5)
m=3900 kg	m=19800 kg	d₁=0.705 m	c₁=0.454 m	c1=0.497 m	c1=0.454 m	c1=0.5 m
L₁=3.94 m	L₁=3.94 m	d₂=0.645 m	c=0.93 m	c=0.97 m	c=0.93 m	c=0.88 m
L₂=1.35 m	L₂=1.35 m	h_s=0.567 m	h_s=0.717 m	h_r1=h_r1=0.467 m	h_s=0.717 m	hs=0.717 m
a=4.26 m	a=4.055 m	h₂=0.547 m	h₂=0.567 m	h₂=0.567 m	h₂=0.567 m	h₂=0.567 m
h=1.19 m	h=1.62 m	b₂=0.03 m	d₁=d₂=0.21m	d₁=d₂=0.19 m	d₁=d₂=0.675 m
h_h=0.59 m	h_h=0.59 m			α₁= α₂=15º

The technical data and results of the optimization of brake force distribution in single axle semi-trailer: L1=4_45 m, hh=0_7 m (L ‒ laden, U ‒ unladen, I, II – first and second solution)

	m kg]	a [m]	h [m]	OF	β₁	i_P

U_I-U_II	2250	3.895	0.98	0.0947–0.0947	0.2079–0.2079	3.8097–3.8097
L_I - L_II	7250	3.840	1.25	0.1272–0.1272	0.2445–0.2445	3.0901–3.0901

The results of the optimization of brake force distribution in a tandem axle semi-trailer (L ‒laden, U ‒ unladen, Lw, Uw – laden and unladen with weight of suspension)

Suspension		OF	β₁	β₂₁	β₂₂	i_P	i_S

Bogie (3.1)	U-U_w	0.3040–0.3115	0.1629–0.1588	0.5740–0.5793	0.2631–0.2619	5.1402–5.2971	0.4583–0.4521
I and II solution	L-L_w	0.3006–0.3015	0.2301–0.2301	0.5286–0.5286	0.2412–0.2412	3.3456–3.3456	0.4564–0.4564

2 leaf 2 rod (3.3)	U-U_w	0.6716–0.8831	0.2313–0.2307	0.1803–0.1527	0.5884–0.6166	3.3239–3.3354	3.2635–4.0377
I solution	L-L_w	0.5841–0.6991	0.3117–0.3267	0.1674–0.1644	0.5209–0.5189	2.2086–2.1574	3.1116–3.1573
II solution	U-U_w	0.9446–1.2169	0.2075–0.2065	0.1705–0.1444	0.6220–0.6490	3.8198–3.8417	3.6483–4.4935
	L-L_w	0.7531–0.7893	0.2806–0.2818	0.1577–0.1525	0.5617–0.5657	2.5635–2.5490	3.5631–3.7096

2 leaf equal. (3.4)	U-U_w	0.2512–0.2512	0.1951–0.1951	0.4031–0.4031	0.4018–0.4018	4.1265–4.1265	0.9969–0.9969
I solution	L-L_w	0.2099–0.2099	0.2605–0.2605	0.3710–0.3710	0.3685–0.3685	2.8388–2.8388	0.9935–0.9935
II solution	U-U_w	0.3002–0.3002	0.1803–0.1803	0.4040–0.4040	0.4156–0.4156	4.5449–4.5449	1.0288–1.0288
	L-L_w	0.2117–0.2117	0.2561–0.2561	0.3721–0.3721	0.3717–0.3717	2.9046–2.9046	0.9989–0.9989

air susp. (3.5)	U-U_w	0.2511–0.2511	0.1951–0.1951	0.4031–0.4031	0.4018–0.4018	4.1265–4.1265	0.9969–0.9969
I solution	L-L_w	0.2098–0.2098	0.2605–0.2605	0.3710–0.3710	0.3685–0.3685	2.8388–2.8388	0.9935–0.9935
II solution	U-U_w	0.3003–0.3003	0.1803–0.1803	0.4040–0.4040	0.4156–0.4156	4.5449–4.5449	1.0288–1.0288
	L-L_w	0.2117–0.2117	0.2561–0.2561	0.3721–0.3721	0.3717–0.3717	2.9046–2.9046	0.9989–0.9989

2 leaf (3.2) only	U-U_w	5.3013–3.5221	0.2271–0.2337	0.0017–0.0218	0.7712–0.7446	3.4025–3.2798	458.23–34.199
I solution	L-L_w	4.6761–4.5855	0.3230–0.3202	0.0018–0.0030	0.6752–0.6768	2.0957–2.1233	376.30–222.04

An Optimal Braking Force Distribution in the Rigid Drawbar Trailers with Tandem Suspension

Figures & Tables

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Rigid drawbar trailer and tandem suspension technical data [37–38]

The technical data and results of the optimization of brake force distribution in single axle semi-trailer: L1=4_45 m, hh=0_7 m (L ‒ laden, U ‒ unladen, I, II – first and second solution)

The results of the optimization of brake force distribution in a tandem axle semi-trailer (L ‒laden, U ‒ unladen, Lw, Uw – laden and unladen with weight of suspension)

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