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Article

Disturbance Propagation Model of Luggage Drifting Motion Based on Nonlinear Pressure in Typical Passenger Corridors of Transportation Hubs

by
Bingyu Wei
1,
Rongyong Zhao
1,*,
Cuiling Li
1,
Miyuan Li
1,
Yunlong Ma
1 and
Eric S. W. Wong
2
1
School of Electronic and Information Engineering, Tongji University, Shanghai 201804, China
2
Hongkong Institute of Water and Sanitation Safety, Hong Kong 999077, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(11), 4942; https://doi.org/10.3390/app14114942
Submission received: 1 April 2024 / Revised: 1 June 2024 / Accepted: 3 June 2024 / Published: 6 June 2024
Browse Figures
Figure 1
<p>Particle diagram of passengers and luggage in a transport hub.</p> "> Figure 2
<p>Diagram of a linear contact model. Note, the pink disk represents a luggage-laden particle, and the blue disk represents a non-luggage-carrying particle.</p> "> Figure 3
<p>Illustration of luggage drifting motion and passenger pressure when luggage drifts. (<b>a</b>) The higher the probability of drifting, the greater the sway amplitude of a luggage-laden passenger. (<b>b</b>) A big drift angle is caused by a slight turn at high speed. (<b>c</b>) Diagram of the pressure of surrounding passengers in the case of luggage drifting.</p> "> Figure 4
<p>Schematic diagram of the luggage case steering pressure characteristics.</p> "> Figure 5
<p>Passenger and luggage model in a simulation (Pathfinder 2022 software).</p> "> Figure 6
<p>Schematic illustration of particles in the simulation (Pathfinder 2022 software).</p> "> Figure 7
<p>A luggage-laden passenger made a sudden turn in an L-shaped corridor.</p> "> Figure 8
<p>Crowd density around the luggage-laden passenger particle on both sides in key frames.</p> "> Figure 9
<p>Disturbing force of the luggage-laden passenger particle on both sides in key frames.</p> "> Figure 10
<p>Simulation of a passenger carrying luggage going against the crowd.</p> "> Figure 11
<p>Crowd density around a luggage-laden passenger particle on both sides.</p> "> Figure 12
<p>Disturbing force of the luggage-laden passenger particle on both sides.</p> "> Figure 13
<p>Contact pressure distribution during luggage retrograding.</p> "> Figure 14
<p>Contact pressure distribution around points P and S in a passenger crowd at the moment of t = 8.10 s, step = 810.</p> "> Figure 15
<p>The change in the contact pressure at points P and S caused by luggage retrograding in a passenger crowd.</p> ">
Versions Notes

Abstract

:
In current transportation hubs, passengers travelling with wheeled luggage or suitcases is a common phenomenon. Due to the fact that most luggage occupies a certain space in dense passenger crowds with high mass inertia, its abnormal motion, such as drifting, can frequently trigger unavoidable local disturbances and turbulence in the surrounding pedestrian flows, further increasing congestion risk. Meanwhile, there still is a lack of quantitative disturbance propagation analysis, since most state-of-the-art achievements rely on either scenario-based experiments or the spatial characteristics of crowd distribution assessed qualitatively. Therefore, this study considers the luggage-laden passenger as a deformable particle. The resulting disturbance on surrounding non-luggage-carrying passengers is analyzed and quantified into a nonlinear pressure term. Subsequently, the disturbance propagation model of passenger-owned luggage is developed by adapting the classical Aw–Rascle traffic flow model with a pressure term. Simulation experiments of disturbances caused by luggage drifting and retrograding were conducted in Pathfinder 2022 Software. Experimental results showed that the disturbing force of a left-sided crowd can reach a peak of 238 N with a passenger density of 3.0 p / m 2 , and the maximum force difference between the left- and right-sided disturbing force can reach 153 N, as confirmed by a case study in an L-shaped corridor of a transportation hub. Furthermore, it is recommended that the proposed model can be applied in crowd flow analysis and intelligent decision-making for passenger management in transportation hubs.

1. Introduction

In recent decades, the advancement of modern public transportation systems, such as metros and railways, has led to increasingly crowded transportation hubs teeming with a vast number of passengers carrying luggage, trolley cases, or suitcases [1]. Since luggage serves as a moving obstacle to surrounding passengers with a relatively high mass inertia, the presence of luggage-laden passengers can heavily impact evacuation efficiency during security emergencies such as fires and stampedes in transportation hubs [2]. Studying the dynamics of luggage-laden passengers can help in the development of automatic decision-making systems in public management [3,4].
Most achievements focusing on luggage and passengers have uncovered the notable impact that luggage-laden passengers can have on emergency evacuation processes. Through field observations, studies have found that the walking speed of a luggage-laden passenger exhibits a distinct reduction compared to that of those not carrying luggage [5,6,7]. By investigating passenger flows in airport terminals, it was found that the size of the luggage actually affects the crowd distribution in building planes [8]. With the increase in the ratio of luggage carrying in bottleneck areas, evacuation efficiency can decrease, even resulting in evacuation failure [9]. These observation results preliminarily reveal how luggage affects normal crowd evacuation.
The influence of wheeled luggage or trolley cases on the movement of surrounding passengers is complex. Factors such as the proportion and size of the luggage, the passenger density, and the dynamics of luggage’s abnormal motions are crucial considerations [10]. Additionally, the presence of luggage leads much more frequently to collisions and friction for pedestrians in a crowd [11]. Traffic flow models have been applied to dense crowd modeling. In [12], a modified Aw–Rascle model was proposed to describe crowd convergence mechanisms, and it was found that internal pressure is positively correlated with velocity and density. However, since luggage is a moving obstacle, luggage-laden passengers and the surrounding passengers cannot be regarded as the same medium in a macroscopic view, and their interaction mechanisms cannot be described by conventional fluid models directly [13].
Moreover, [14] shows that the sway amplitude of luggage-laden passenger increases in highly dense pedestrian flows. As the wheeled luggage follows its owner with the help of a carry handle, the luggage will maintain a certain level of motion inertia during its movement. When its owner makes a sudden turn or sways, the suitcase will be driven by the passenger to turn, but the motion of the luggage experiences a certain lag following the passenger, which probably causes luggage drifting. Luggage drifting motion interferes with the expected movement of other passengers, and this crowd disturbance spreads out and dissipates with increasing time and distance [15]. Therefore, the mechanism of disturbance of luggage drifting on a crowd and the disturbance propagation dynamics are challenging issues for crowd safety, and are the primary focuses of this paper.
To address the issues above, the main contributions of this study are as follows: (1) We propose a dynamic disturbance propagation model for abnormal luggage states based on the conservation law of fluid dynamics and the Aw–Rascle model. (2) As an obvious novelty, the disturbance effect of luggage on the surrounding passengers is transferred quantitatively into a nonlinear pressure term based on a contact model and pressure characteristics analysis. (3) Simulations of luggage drifting and retrograde disturbance are conducted to validate the effectiveness of the proposed disturbance propagation model. The numerical results display a more realistic dynamic process of the impact of luggage-laden passengers, and the values of disturbing forces and the density of passengers on both sides can help in crowd dynamics analysis.
The outline of this paper is as follows: in Section 2, we conduct a literature survey on related works; in Section 3, we propose a disturbance dynamics model for luggage-laden passengers; in Section 4, the mechanism of the disturbance on surrounding passengers is studied, and we further propose the disturbance propagation model in a passenger–luggage flow; in Section 5, simulation experiments of the disturbance propagation induced by the retrograding motion and drifting motion of luggage are conducted based on scenarios that occur in a straight corridor and an L-shaped corridor in a transportation hub, respectively.

2. Related Works

2.1. Impact of Carried Luggage on a Crowd

The carrying of luggage, baggage, and suitcases is one of the prominent features of pedestrians in transportation hubs such as railway stations [16]. Since the luggage occupies a tangible space and its motion is not entirely synchronized with its owner, its presence can impede the smooth flow of other passengers in the vicinity [1].
In recent years, impact assessments of luggage-laden passengers on overall crowd flow have been conducted. For example, Fu et al. [17] analyzed the temporal features, passing performance, spatial distribution, and crowd danger of pedestrian trajectories from video recordings, and concluded that the average passing time of merging flow in a Y-shaped corridor increased with the increasing number of pedestrians carrying luggage. Shi et al. [13] used boundary distance, interpersonal distance, and distance headway to describe the distribution of pedestrians in the crowd with luggage. They found that luggage would enhance the distance headway for the whole crowd. Jian et al. [18] designed a series of experiments of luggage-laden pedestrians on the stairs of transportation terminals to study the influences of luggage on the longitudinal movement of a crowd. Deng et al. [19] experimentally investigated the flow of luggage-laden pedestrians through a bottleneck in both normal and emergency situations.
Based on pedestrian trajectories, a dual-peak spatial distribution of pedestrians was observed when no pedestrians carry luggage or when all pedestrians carry luggage. Typically, Huang et al. [20] experimented with one-dimensional luggage-laden pedestrian movement. It was found that the average distance headway of pedestrians was obviously enlarged and the acceleration was reduced when pedestrian switched their direction due to the presence of luggage. Li et al. [21] used the Aw–Rascle model to depict the disturbance propagation. In this study, luggage drifting can be taken as an individual anomaly. Therefore, we use an individual-crowd method to analyze the impact of luggage-laden passengers on other surrounding passengers.
Nevertheless, these aforementioned studies quantified the impact of carried luggage on crowd dynamics by conducting scenario-based experiments. These experiments analyzed the spatial and temporal distributions of crowds, as well as motion characteristics, comparing conditions with and without luggage-laden pedestrians. However, few prior studies have focused on the quantitative analysis of luggage-laden passengers. Only a limited number of research endeavors have explored the impact mechanisms of wheeled luggage on crowd dynamics. Therefore, this study defines the luggage and its carrier as a combined particle, modeling the influence of this combination on crowd dynamics through a comprehensive force analysis.

2.2. Disturbance Propagation in Crowd Flow

Due to physical environment disturbances, object disturbances, and individual behavior mutation, a crowd flow can possibly change from a stable and orderly state into a chaotic and unstable state [22]. The redistribution of crowd density and fluctuation in velocity propagate instability within the crowd, enhancing the risk of stampedes. This further emphasizes the importance of accurately quantifying crowd disturbances. For example, Wei et al. [23] improved a social force model by introducing the disturbance fluctuation force to describe the state change of pedestrian flow during evacuation. They provided the mathematical expression of disturbing forces. Researchers have recently made noteworthy progress in attracting attention to quantifying disturbances from a force perspective [24,25,26].
Achievements in the literature verified that an internal disturbance can result in varying degrees of dynamic propagation within a crowd flow. Lu [27] applied wave theory to analyze a crowd of pedestrians, regarded as a continuum, and observed that disturbances in the crowd could propagate in the form of waves. Qin et al. [28] designed a distributed parameter system to describe the disturbed crowd dynamics based on the mass conservation law. Hanseler et al. [29] described multiple pedestrian groups’ simultaneous and potentially conflicting propagation based on the continuum theory of pedestrian and cell transmission model. Wang et al. [30] proposed an extended car-following model to describe the dynamic features of mixed traffic, including pedestrians walking on adjacent lanes. They used the reductive perturbation method to analyze the nonlinear stability of the car flow model. Ma et al. [22] studied a stampede at a 2010 German music festival from the perspective of flow pattern, and their results showed that the force generated by the physical contact between pedestrians led to the unsteady crowd movement. The amplified and diffused energy of the disturbance ultimately led to severe turbulence and stampedes.
The research achievements in both crowd dynamics and traffic flow above indicate that a certain intensity of disturbances can induce congestion, even when density and flow are far from saturation. This comes about as both traffic flow and crowd flow transition from free flow to crowded flow in a complex and nonlinear process. It is significant to consider both the spatial and temporal features of disturbance and the impact of disturbance on crowd stability in the area of passenger safety management. In our previous studies, the panic propagation dynamics [31] and the disturbance propagation model for abnormal behavior of pedestrians were already proposed [32]. Thereby, we focus on the disturbance propagation model of luggage drifting motion in this study.

3. Disturbance Dynamics Model of Luggage-Laden Passengers

3.1. Dynamic Pressure between Luggage-Laden and Non-Luggage-Carrying Passengers

In a dense pedestrian flow, the incomplete pursuit movement of luggage relative to its owner can lead the neighboring passengers to avoid the luggage autonomously, which can be described from the perspective of contact force or pressure. In the field of rock mechanics, the interactions of granular material were modeled by varying contact models using the discrete element method (DEM) [33]. Analogously, the pressure between luggage-laden and non-luggage-carrying passengers can be regarded as the contact force between particles of different sizes. Meanwhile, for simplicity and emphasizing the dynamic pressure of two different media, non-luggage-carrying passengers with a certain range are assembled into one particle. In contrast, every luggage-laden passenger is regarded as a separate particle, as shown in Figure 1.
This study uses the contact stiffness model to analyze the contact forces between circular particles. The contact stiffness models can be divided into the following categories: the linear contact model [33] and the Hertz–Mindlin contact model [34]. While a rigorous non-linear model, such as the elastoplastic contact model, offers a more detailed representation of deformation during contact, it is computationally intensive and is often unnecessary for accurately simulating the collision deformation of pedestrian particles [35]. Therefore, we use the linear contact model to efficiently simulate the interactions between luggage-laden passengers and those around them.
In a contact stiffness model, although particles are regarded as rigid bodies, overlap between particles and a certain degree of collision deformation are permitted due to the interpersonal distance in a real crowd. Figure 2a illustrates the ball–ball-level contact of luggage-laden passengers and non-luggage-carrying passengers. In the linear contact model, the simplified relationship between the elastic force and displacement is linear [36]. The contact force in the contact position C is composed of linear force F l and dashpot force F d ; that is,
F c = F l + F d
The linear component provides linear elastic and frictional properties, while the damped component provides a viscous property. Linear and dashpot forces can be resolved into shear and normal forces. The linear force is generated by a linear spring with constant normal stiffness k n and shear stiffness k s , and the dashpot force is generated by a damper with a normal and shear critical damping ratio of β n and β s , respectively. That is,
F l = F n l + F s l , F d = F n d + F s d , F n l = k n g s , F s l = k s Δ δ , F n d = − β n Δ v n , F s d = − β s Δ v s
where the subindex s and n represent the shear and normal direction, respectively. g s is how deep a luggage-laden particle reaches into another particle’s personal space, resulting in normal pressure. Δ δ is the relative displacement increment at the boundary. Δ v n and Δ v s are the relative velocity increments in the normal and shear direction, respectively.
In a real scenario, the disturbance effect of luggage-laden passengers often differs significantly on one side from the other [37]. To simplify, when only one passenger carries luggage in a passenger flow, passengers on one side can be considered equivalent to a face, as shown in Figure 2b. Then, the ball–ball-level contact force is extended to the ball–facet level by elongating the contact surface. The left-side disturbing force in the normal direction can be modeled in Equation (3), where x n ( t ) is the overlap of normal direction in time t and Δ v ( ρ , t ) is the normal velocity of the luggage-laden passenger relative to the crowd on the left, which can be described by Equation (4), where n l e f t is the unit vector from the luggage in luggage-laden passenger to the crowd on the left at time t and ν l e f t ( ρ ) is the velocity of left side of the crowd relative to the luggage-laden passenger particle with crowd density ρ . The values of ν l ( ρ ) in different scenarios of transportation hubs are shown in Table 1.
F n _ l e f t ( ρ , t ) = F n l + F n d = − k n x n ( t ) − β n Δ v ( ρ , t )
Δ v ( ρ , t ) = ( ν i − ν l ( ρ ) ) n l e f t
The left-side disturbing force in the tangential direction is F s _ l e f t ( t ) :
F s _ l e f t ( ρ , t ) = − k s x s ( t ) − β s ν s _ l e f t ( t )
where x s ( t ) is the overlap in the tangentiall direction at time t. ν s _ l e f t ( t ) is the tangent velocity of the luggage-laden passenger relative to the left side of the crowd, which can be expressed as Equation (6):
ν s _ l e f t ( t ) = r ω × n l e f t
where r and ω i are the radius and the spin angular velocity of luggage, respectively. Thus, when the luggage-laden passenger crosses the crowd, its disturbing force on the crowd on the left side is
F d ι s t u r b _ l e f t ( ρ , t ) = F s _ l e f t ( ρ , t ) + F n _ l e f t ( ρ , t )

3.2. Disturbing Effect of Luggage Drifting Motion

Section 3.1 elucidates the dynamic disturbance caused by passengers carrying luggage on the surrounding passengers under relatively smooth flow conditions. The disturbing force primarily exhibits a linear relationship with both displacement and velocity. However, when trampling and stampedes occur in transportation hubs, wheeled luggage tends to drift with the accelerating turns of its owner or may even be abandoned during escape [39], further exerting turbulence on an over-congested crowd. Since this abnormal motion, i.e., drifting, of luggage can lead to a sudden impact on the neighboring pedestrians, the neighboring pedestrians can resist slight interference and readjust themselves back to a balanced state; therefore, the linear pressure in Section 3.1 is not sufficiently realistic. In [32], the main inter-impacting pressure of pedestrian fall behavior was analyzed along the longitudinal axis. Herein, the disturbing effects of both horizontal and vertical were quantified.
We consider luggage drifting left (or right) as an abnormal motion and describe its disturbing effect by analyzing its pressure characteristics. As shown in Figure 3c, when luggage abruptly drifts left, it affects the passengers i+1, i−1, j+1, and j−1. The right-side pressure of passengers on the left side of the luggage p j − 1 , r increases, while the left side pressure of passengers on the left side of the luggage p j − 1 , l remains unchanged. The luggage’s disturbing forces on the left side p i , l and p j − 1 , r constitute a pair of interacting forces. The p j + 1 , l (the left side pressure of passengers on the right side of the luggage), with an opposite trend to p j − 1 , r , decays first and returns to a balanced value at the completion of turning and with the movement of the crowd. The luggage’s disturbing force on the right side p i , r also constitutes a pair of interacting forces. As the luggage’s drifting increases with the time that the luggage-laden passenger stays in a disturbed position, the front pressure of the luggage p i , f and the rear pressure p i , b will first decay and climb to stable values, respectively. Therefore, the dynamic pressure of luggage drifting can be expressed non-linearly as follows (also shown in Figure 4):
(a)
For luggage i :
p i = p i , b + p i , f + p i , l + p i , r , p i , b = p 0 , t < t s p 0 e t − t s , t s ≤ t < t e k p 0 e t e − t s , t e ≤ t , k < 1 , p i , f = p 0 , t < t s p 0 e t s − t , t s ≤ t < t e α p 0 e t s − t e , t e ≤ t , α > 1 , p i , l = p 1 , t < t s p 1 e t − t s , t s ≤ t < t e p 1 e t e − t s , t e ≤ t , k < 1 , p i , r = p 1 , t < t s p 1 e t s − t , t s ≤ t < t e α p 1 e t s − t e , t e ≤ t , α > 1
where p 0 is the stable pressure value of p i , f and p i , f , p l is the stable pressure value of p i , l and p i , r , t s is the moment that the luggage starts drifting, corresponding to the position of “Start A” in Figure 7 and t e is the end moment that the luggage-laden passenger stop impacting on the passengers, corresponding to the position of “Endt B” in Figure 7.
(b)
For passengers in front:
p i + 1 = p i + 1 , b + p i + 1 , f , p i + 1 , b = p 0 , t < t s p 0 e t s − t , t s ≤ t < t e α p 0 e t s − t e , t e ≤ t , α > 1 p i + 1 , f = p 0 ,
(c)
For passengers behind:
p i − 1 = p i − 1 , b + p i − 1 , f , p i − 1 , f = p 0 , t < t s p 0 e t − t s , t s ≤ t < t e k p 0 e t e − t s , t e ≤ t , k < 1 , p i − 1 , b = p 0
(d)
For passengers on the left:
p j − 1 = p j − 1 , l + p j − 1 , r , p j − 1 , r = p 1 , t < t s p 1 e t − t s , t s ≤ t < t e p 1 e t e − t s , t e ≤ t , k < 1 , p j − 1 , l = p 1
(e)
For passengers on the right:
p j + 1 = p j + 1 , l + p j + 1 , r , p j + 1 , r = p 1 , p j + 1 , l = p 1 , t < t s p 1 e t s − t , t s ≤ t < t e α p 1 e t s − t e , t e ≤ t , α > 1

4. The Disturbance Propagation Model Based on the Aw–Rascle Model in Crowd Flow

4.1. Disturbance of a Luggage-Laden Passenger Particle on Surrounding Passengers

When a luggage-laden passenger encounters a sudden abnormal event, such as a rapid change in speed, abrupt turns, discarding suitcases, falls, seeking assistance for medical emergencies, brawls, or violent terrorist attacks, these unpredictable incidents can generate disturbing forces that disturb the balanced flow of the crowd. Regarding the form of pressure term of the Aw–Rascle traffic flow model [40], the crowd flow pressure term under an abnormal state is established in Equation (13).
P = f ( ρ , γ , ξ ) = ρ γ ⋅ ξ
where ρ is the crowd density, γ is the pressure coefficient, and ξ is the disturbance intensity. γ is used to reflect the resistance ability of the crowd under internal pressure. Hence, we define γ as the ratio of the disturbing force of luggage-laden passengers and their self-driving forces. The self-driving forces of pedestrians can refer to the social force model proposed by Helbing [41], or human centroid force [42]. Herein, we introduce desire force from the classic social force model, then γ is defined as
γ = p f d e s i r e
f d e s i r e = m i ν i 0 ( t ) e i 0 ( t ) − v i ( t ) τ i
where m i is the mass of the passenger i, ν i 0 is the desired velocity in a certain direction e i 0 , v i is the velocity of passenger i, and τ i is the relaxation term. Note that if luggage is drifting, then the pressure term is given by Equations (9)–(12). If not, we have
p = F d ι s t u r b _ l e f t ( or , F d ι s t u r b _ r i g h t )
If γ > 1 , the non-luggage-carrying passengers are squeezed at a level stronger than their own control, becoming prone to unsafe incidents. In [43], the pressure distribution around the disturbance source was assumed to follow a normal distribution. Thus, the disturbance intensity ξ of the x-axis can be expressed by
ξ x = ξ 0 ⋅ 1 2 π t e − ( x − a ) 2 2 t
Thus, the pressure term in the horizontal direction is
P h ( ρ , γ , ξ ) = ρ γ ⋅ ξ x = ρ γ ⋅ ξ 0 ⋅ 1 2 π t e − ( x − a ) 2 2 t
In an Aw–Rascle traffic flow model, the pressure describes how an “average” driver would respond to a variation in the density of cars in a limiting space. In this study, the pressure term is employed to describe the impact of disturbance on surrounding pedestrians caused by luggage drifting motion.

4.2. Disturbance Propagation Model Based on the Aw–Rascle Model

The propagation of disturbances in a passenger crowd is assumed to follow the mass conservation law of the macroscopic hydrodynamic model [12], that is
∂ ρ ∂ t + ∂ ( ρ v h ) ∂ x + ∂ ( ρ v l ) ∂ y = 0
where v h is the velocity in the horizontal direction and v l is the velocity in the lateral direction. In addition, the crowd to the rear can adjust their movement by directly observing the abnormal motion of the luggage-laden passenger, while the crowd in front cannot. This reflects the anisotropy of disturbance propagation in crowds. Thus, to describe the process of disturbance propagation, the Aw–Rascle model [40,44], a typical anisotropic traffic flow model, is employed. The classic Aw–Rascle model can be expressed by Equations (20) and (21):
∂ ( v h + P h ) ∂ t + v h ∂ ( v h + P h ) ∂ x = s 1 = 1 τ ( V e h − v h )
∂ ( v l + P l ) ∂ t + u ∂ ( v l + P l ) ∂ y = s 2 = 1 τ ( V e l − v l )
where τ is the relaxation time, V e h and V e l are the equilibrium velocities in the horizontal and lateral direction, respectively, s 1 and s 2 are the relaxation terms, and P h and P l refer to the horizontal and lateral pressure components, respectively. We replace the pressure term of the Aw–Rascle model with the pressure term of crowd interior disturbance in Equation (18), and then we obtain
∂ ( v h + P ξ h ) ∂ t + v h ∂ ( v h + P ξ h ) ∂ x = 1 τ ( V e h − v h )
∂ ( v l + P ξ l ) ∂ t + u ∂ ( v l + P ξ l ) ∂ y = 1 τ ( V e l − v l )
In the propagation process of a disturbance, motion damping exists. Due to a damping effect, energy is absorbed by passengers in the process of disturbance propagation until all the energy is dissipated. This phenomenon is called the damping wave elimination effect of crowds. The damping wave elimination of a crowd is carried out in the direction of the velocity of the disturbance.
To simulate the damping effect of crowds, it is reasonable to add the damping wave elimination terms to construct a disturbance propagation model in (22) and (23), as follows:
S x = − ρ μ 0 μ f ( x )
S y = − ρ μ 0 v f ( x )
where ρ is the crowd density and μ 0 μ and μ 0 v reflect the loss of disturbance during propagation. The damping distribution function is approximated by a linear damping function
f ( x ) = x − x s x e − x s
where x s , x e are the horizontal coordinates of the start and end of the wave elimination zone, respectively.
Multiplying both sides of Equation (22) by density and adding the damping wave elimination term, we obtain:
ρ ∂ ( v h + P ξ h ) ∂ t + ρ v h ∂ ( v h + P ξ h ) ∂ x − ρ μ 0 μ f ( x ) = ρ 1 τ ( V e h − v h )
For the vertical direction, similarly, we obtain:
ρ ∂ ( v l + P ξ l ) ∂ t + ρ u ∂ ( v l + P ξ l ) ∂ y − ρ μ 0 v f ( x ) = ρ 1 τ ( V e l − v l )
Equations (27) and (23) are the dynamic model for the disturbance propagation of a luggage-laden passenger. Note that the AR traffic flow model was originally designed for the analysis of car movement. When adopting the AR traffic flow into pedestrian flow modeling, several premises should be ensured: (1) mass conservation; (2) continuous flow; and (3) the velocity of pedestrian media is regarded as the equilibrium velocity of continuum pedestrian flow and is the reflection of the desired velocity of several pedestrians.

5. Case Study and Discussion

5.1. Experiment Parameters Setup

According to the combined characteristics of passengers and their luggage, the simulation models of passengers and luggage were established in Pathfinder 2022 software. The model of passengers and their luggage is shown in Figure 5. In [45], the mass of passengers and their luggage was assumed to conform to the normal distribution. The mass of adult passengers ranged from 38 to 85 kg, with a mean of 58 kg and a standard deviation of 7.8 kg. The mass of luggage ranged from 5 to 25 kg, with a mean of 16 kg and a standard deviation of 2.3 kg. Then, we initialized the passenger mass and luggage mass to 58 kg and 16 kg, respectively. As shown in Figure 6, we used a blue circle with a green triangle to represent non-luggage-carrying passengers and their motion direction in a crowd and used a brown circle with a green triangle to represent luggage-laden passengers.
The related parameters of the simulation and their values are set out in Table 2.

5.2. Case 1: Luggage Drifting Disturbance Propagation in an L-Shaped Corridor

A sudden turn by a luggage-laden passenger is usually accompanied by the drifting of wheeled luggage, which is commonly seen in transportation hubs, especially in L-shaped corridors, with much more potential risk. Therefore, a typical L-shaped corridor model in Shanghai Hongqiao railway station in China, with a width of a 10 m, was established in the Pathfinder 2022 software.
As shown in Figure 7, a luggage-laden passenger made a sudden turn in an L-shaped corridor and caused luggage drifting at the position of Start A. The crowd density and the disturbing force on both sides of the luggage-laden passenger were collected, respectively. In the fifth frame of the compared scenario video, the difference between the left side and right side disturbing force was 153 N. In the sixth frame of the video, when the luggage-laden passenger started turning left and drifting motion followed, the disturbing force of surrounding passengers on the left increased and reached a peak of approximately 238N, while the disturbing force on the right side decreased.
The crowd density of the luggage-laden passenger particle on both sides is plotted with key frames in Figure 8. Consequently, the disturbing force of the luggage-laden passenger particle on both sides with key frames is presented in Figure 9. With the completion of the passenger turning, the disturbing force on the passenger’s left side dropped and the disturbing force on the right side rose, meaning that the passenger’s left- and right-side disturbing forces were in a relatively balanced state. When the luggage suddenly drifted to the left, the passengers on the left side of the luggage would gather because of the obstruction of the luggage, thus increasing the density. After the passenger’s turn was completed, the crowd gradually readjusted its density on both sides back to a balanced value.
Applsci 14 04942 g007
Figure 7. A luggage-laden passenger made a sudden turn in an L-shaped corridor.
Figure 7. A luggage-laden passenger made a sudden turn in an L-shaped corridor.
Applsci 14 04942 g007
To emphasize and simplify the disturbance effect of luggage from a single luggage-laden passenger, we set the direction of all non-luggage-carrying passengers to be opposite to that of the luggage-laden passenger, as shown in Figure 6 and Figure 7. In this assumed situation, we obtain more salient results regarding density and disturbing force. In real scenarios, both non-luggage-carrying passengers and luggage-laden passengers always mix and move in a bidirectional flow. Future research or practical applications should consider this more complex situation.

5.3. Case 2: Luggage Retrograde Disturbance Propagation in a Straight Corridor

To further validate the proposed disturbance propagation model, the retrograde movement of a luggage-laden passenger was considered as the disturbance source in the forward-moving crowd. In Pathfinder 2022 software, a 20   m × 25   m hallway was created as the experiment scenario. For simplicity, only one luggage-laden passenger was set in this experiment. As shown in Figure 10, when the luggage-laden passenger crossed the channel in the opposite direction marked with a yellow curve, the normal movement of passing and surrounding passengers was consequently disturbed, which manifests as deviations from the previous forward direction and avoiding the luggage-laden passenger, and the disturbance continues to propagate further. The densities of the surrounding pedestrians on both sides in this process are shown in Figure 11, and the disturbing force on the surrounding pedestrians on both sides is shown in Figure 12. In Figure 12, F d i s t u r b − r   a n d   F d i s t u r b − l are the magnitude of the disturbing force of the right side and left side, respectively, with a slight fluctuation in time, but are roughly equal as the trajectory of the luggage-laden passenger is approximately straight.

5.4. Discussion of Disturbance Propagation

To connect the theoretical model and the experimental results further, the pressure of the crowd was imported and visualized in MATLAB R2019b. As depicted in Figure 13a, when the luggage-laden passenger was advancing and causing a disturbance within the crowd, the propagation of the disturbance consistently intensified, displacing surrounding passengers at an angle towards the rear. The peak of the disturbance wave formed on the rear side of the luggage’s motion direction, aligning with the direction of the luggage’s disturbing force on the surrounding passengers. Consequently, the dissemination of the disturbance caused by the luggage to the surrounding passengers progressed in a manner consistent with the direction of the disturbing force. In addition, a low-disturbance area formed behind the luggage.
Next, the change in the intensity of the disturbance directly behind the luggage as well as the side rear will be discussed. As shown in Figure 14, points P and S were chosen to represent the front and rear side, respectively. In Figure 15a, when the luggage-laden passenger particle passes through point P, the pressure of P is greater than that of point S, reaching a peak at the 400th frame. As the luggage-laden passenger particle moves away from point P, the pressure at point P oscillates at the frequency of the disturbance propagation and decays. The pressure at point S reaches the maximum at the 500th frame and remains at about 1, which is maintained in a low-equilibrium state with a long fluctuation time and a small fluctuation threshold. This is the reason that disturbances at different moments are superimposed at point S, making its fluctuation frequency longer. As it is too far from the trigger point of the disturbance, the initial pressure is less than the intensity of point P, but because of the propagation and superposition of the disturbance, the pressure attenuation speed is slower than that of point P.

6. Conclusions

In this study, we proposed a dynamics propagation model of luggage drifting motion based on nonlinear pressure and the Aw–Rascle model to analyze the disturbance of luggage-laden passengers quantitatively. To validate the proposed model, simulation experiments for luggage drifting and retrograding were conducted. It was proven that the proposed model can remain consistent with the real-world situation:
(1) During the process of the luggage drifting to the left, the disturbing force exerted on the passengers to its left remains weak. On the right side of the luggage, the disturbing force transitions from weak to strong, and then reverts to weak again.
(2) If a luggage-laden passenger moves against the flow, the disturbance propagated to the surrounding passengers advances in alignment with the direction of the disturbing force, and a low-disturbance zone is formed behind the luggage.
(3) The quantitative results of luggage drifting show that the disturbing force of passengers on the left rose to 238 N, and a maximum difference of 153 N was observed between the left side and the right side.
Considering future applications in transportation hubs, management personnel should control the density of passengers to prevent increasing safety risks due to excessive disturbance effects in corridor areas with a large proportion of luggage-laden passengers. Further research involving more realistic and complex disturbance quantification, as well as model calibration using measured data, is recommended. This effort can enhance pedestrian flow modeling and facilitate crowd management in transportation hubs.

Author Contributions

Conceptualization, methodology, software, validation, formal analysis and investigation, resources, data curation, writing—original draft preparation, B.W., M.L., R.Z. and C.L.; writing—review and editing, R.Z. and E.S.W.W.; visualization, B.W. and Y.M.; supervision and project administration, R.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (No. 72374154).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Particle diagram of passengers and luggage in a transport hub.
Figure 1. Particle diagram of passengers and luggage in a transport hub.
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Figure 2. Diagram of a linear contact model. Note, the pink disk represents a luggage-laden particle, and the blue disk represents a non-luggage-carrying particle.
Figure 2. Diagram of a linear contact model. Note, the pink disk represents a luggage-laden particle, and the blue disk represents a non-luggage-carrying particle.
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Figure 3. Illustration of luggage drifting motion and passenger pressure when luggage drifts. (a) The higher the probability of drifting, the greater the sway amplitude of a luggage-laden passenger. (b) A big drift angle is caused by a slight turn at high speed. (c) Diagram of the pressure of surrounding passengers in the case of luggage drifting.
Figure 3. Illustration of luggage drifting motion and passenger pressure when luggage drifts. (a) The higher the probability of drifting, the greater the sway amplitude of a luggage-laden passenger. (b) A big drift angle is caused by a slight turn at high speed. (c) Diagram of the pressure of surrounding passengers in the case of luggage drifting.
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Figure 4. Schematic diagram of the luggage case steering pressure characteristics.
Figure 4. Schematic diagram of the luggage case steering pressure characteristics.
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Figure 5. Passenger and luggage model in a simulation (Pathfinder 2022 software).
Figure 5. Passenger and luggage model in a simulation (Pathfinder 2022 software).
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Figure 6. Schematic illustration of particles in the simulation (Pathfinder 2022 software).
Figure 6. Schematic illustration of particles in the simulation (Pathfinder 2022 software).
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Figure 8. Crowd density around the luggage-laden passenger particle on both sides in key frames.
Figure 8. Crowd density around the luggage-laden passenger particle on both sides in key frames.
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Figure 9. Disturbing force of the luggage-laden passenger particle on both sides in key frames.
Figure 9. Disturbing force of the luggage-laden passenger particle on both sides in key frames.
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Figure 10. Simulation of a passenger carrying luggage going against the crowd.
Figure 10. Simulation of a passenger carrying luggage going against the crowd.
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Figure 11. Crowd density around a luggage-laden passenger particle on both sides.
Figure 11. Crowd density around a luggage-laden passenger particle on both sides.
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Figure 12. Disturbing force of the luggage-laden passenger particle on both sides.
Figure 12. Disturbing force of the luggage-laden passenger particle on both sides.
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Figure 13. Contact pressure distribution during luggage retrograding.
Figure 13. Contact pressure distribution during luggage retrograding.
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Figure 14. Contact pressure distribution around points P and S in a passenger crowd at the moment of t = 8.10 s, step = 810.
Figure 14. Contact pressure distribution around points P and S in a passenger crowd at the moment of t = 8.10 s, step = 810.
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Figure 15. The change in the contact pressure at points P and S caused by luggage retrograding in a passenger crowd.
Figure 15. The change in the contact pressure at points P and S caused by luggage retrograding in a passenger crowd.
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Table 1. Density-velocity of a crowd in key public areas [38].
Table 1. Density-velocity of a crowd in key public areas [38].
Key Area ν l ( ρ )
Entrance of corridor v ρ = − 0.3657 ρ + 1.6721
Middle of corridor v ρ = 0.806 e − 0.206 ρ
Exit of corridor v ρ = 0.0239 ρ 2 − 0.0121 ρ + 1.3195
Down stairs v ρ = 0.7909 e − 0.189 ρ
Up stairs v ρ = 0.8394 e − 0.212 ρ
Platform queuing area v ρ = 1.3016 e − 0.549 ρ
Table 2. Parameter setting in simulation.
Table 2. Parameter setting in simulation.
CategoryParametersValue
Non-luggage passenger particleInitial density ρ s p (ped/m2)Variable
Desire velocity v s p 0 ρ (m/s)Variable
Initial velocity v s p ρ (m/s)1.2 m/s
Particle radius r s p (m)1.1 m
Particle mass m s p (kg)64.08
ContactFraction coefficient μ0.3
Normal stiffness kn 1203.90
Initial shear stiffness ks 920.65
Normal damping coefficient β n β n ( ρ , k n )
Shear damping coefficient β s β s ( ρ , k s )
Luggage-laden passenger particleInitial density ρ (ped/m2)Variable
Desire velocity v 0 (m/s)Variable
Initial velocity v ρ (m/s)1.2 m/s
Particle radius r (m)0.75
Particle radius m (kg)74
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Wei, B.; Zhao, R.; Li, C.; Li, M.; Ma, Y.; Wong, E.S.W. Disturbance Propagation Model of Luggage Drifting Motion Based on Nonlinear Pressure in Typical Passenger Corridors of Transportation Hubs. Appl. Sci. 2024, 14, 4942. https://doi.org/10.3390/app14114942

AMA Style

Wei B, Zhao R, Li C, Li M, Ma Y, Wong ESW. Disturbance Propagation Model of Luggage Drifting Motion Based on Nonlinear Pressure in Typical Passenger Corridors of Transportation Hubs. Applied Sciences. 2024; 14(11):4942. https://doi.org/10.3390/app14114942

Chicago/Turabian Style

Wei, Bingyu, Rongyong Zhao, Cuiling Li, Miyuan Li, Yunlong Ma, and Eric S. W. Wong. 2024. "Disturbance Propagation Model of Luggage Drifting Motion Based on Nonlinear Pressure in Typical Passenger Corridors of Transportation Hubs" Applied Sciences 14, no. 11: 4942. https://doi.org/10.3390/app14114942

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