Abstract

In this study, mathematical models for conventional speed control humps, including circular, sinusoidal, and modified sinusoidal humps with lengths of 4m, 6.5m, and 9.5m, have been established. Additionally, mathematical models for circular, sinusoidal, normal distribution, trapezoidal, and flat-top sign humps applicable to any length and height are proposed. Simulations and comparisons have been conducted using five types of vehicles to evaluate the speed control humps and the maximum accelerations experienced by vehicles as they pass over them. The results demonstrate that the mathematical models for both conventional speed control humps and the proposed formulations—circular, sinusoidal, normal distribution, trapezoidal, and flat-top sign profiles—are effective for designing speed control humps of any length and height.

1. Introduction

Traffic control and safety is still a serious problem for almost all the countries in the world. As well known, one of the main accidents reason is excessive speed, thousands of people die or have severe injuries by traffic accidents each year. According to the report of the Department for Transport in the UK (2010), the risk increases slowly until crash speeds of around 30mph. Above this speed, risk increases rapidly, between 3.5 and 5.5 times from 30mph to 40mph ¹. Reference ^{2, 3} and ⁴ have shown that there is an important statistical relationship between speed variance and accident rate. Thus controlling the travelling speed of vehicles not to be excessive is very important.

Speed control humps are known as the most popular traffic calming devices, which are the raised area from the pavement, installed transversely to the traffic flow, are used for resident area for reducing the speed of vehicles which are traveling faster than the designated limit. According the hump profiles, speed control humps can be classified round-top and flat-top speed humps. In the round-top, there are circular, sinusoidal, parabolic, cycloidal and harmonic profile road humps. The original work on speed reducing road humps was carried out in the 1970’s at TRRL ⁵ on circular profile (round-top) humps of various dimensions. The oldest design shape is the Watts profile or called circular speed hump. It is circular section of a 3.7m long and a height of 0.075 to 0.1m extending over the whole width of the road ⁶.

Accordingly, another design developed in the United States is the Seminole profile or called flat-top hump. The design is characterized by the addition of a 3-meter flat section into the Watts profile hump with a total length of 6.7m ⁶. In the UK, the Road Hump Regulations ⁷, allow a maximum hump height of 100mm. Experience with round-top humps indicates that sinusoidal hump heights of greater than 100mm would cause grounding problems for cars with long wheelbases and low ground clearances when crossing humps of 3.7m or less in length ⁷. The Danish guidelines for speed reducing measures in urban traffic areas show a wide variety of different road hump profiles: round-top, dome-shaped and flat-top (Danish Road Directorate, 1991). For round-top and dome shaped humps, the recommended height is 100mm and the length is 3.0m, 4.0m, 6.5m and 9.5m for use on roads where the desired speeds are 20kph, 30kph, 40kph and 50kph respectively. Sinusoidal humps have also been tried in Denmark ⁸. Sinusoidal humps have also been used in New Zealand where it was reported ⁹ that ‘the crossing of the hump whilst causing a significant reduction in speed appeared to be more comfortable than the ‘Watts’ profile round-top hump’.

About of designing speed control humps, there are many researches focused on the optimization of speed humps profiles ^{10, 11, 12, 13, 14}. Maemori and Sakamoto ¹² applied multi-objective optimization methods with different car base lengths and various suspension systems. Maemori and Ando ¹⁰ determined the optimal circular hump dimensions in order to reduce the excessive shocks experienced by drivers of heavy vehicles passing current humps at normal speed and to compensate for the too few shocks experienced by light vehicles with semi-active suspensions passing over the same humps at excessive speeds. In an attempt to meet the intended objectives of these devices, many researchers have developed new optimal designs of speed control humps.

Other innovative solutions which can enhance the current speed humps could be found in the TER report (2004). Some of the solutions are smart speed bumps ¹⁵, intelligent speed humps ¹⁶, humps injected with smart materials (Barron and Gostout, 2013), and virtual humps ^{17, 18} proposed an optimization method of speed hump.

Experiments reported in Moinat, ¹⁹ and Jarvis ²⁰, have shown that as the hump length increases, peak accelerations tend to occur at higher speeds, and more linear dynamic effects are created. It has been concluded that longer humps exhibit better characteristics in terms of speed reduction.

Other studies have focused on the significance of the vehicle type and suggested that it should be taken into consideration in the course of hump profile design. As an instance, it has been mentioned that humps designed for transit buses at 25km/h would allow automobiles to cross them at 35km/h with no irritating shock ²¹.

The sinusoidal hump profile was based on a study by Delft Technical University using field experiments and a computer model that simulated the behaviour of a car and driver. The results indicated that traffic humps for 85th percentile crossing speeds of 30kph must be 4-5m long and that the sinusoidal hump profile was the most preferable of those tested ²². Sinusoidal humps have been used in Dutch 30kph zones and are the recommended road hump profile (height 120mm, length 4.5 to 5m) for use on cycle routes because they are less ‘annoying’ for cyclists than the round-top or flat-top profiles ²³.

In this study, mathematical models for conventional speed control humps, including circular, sinusoidal, and modified sinusoidal profiles with lengths of 4m, 6.5m, and 9.5m, have been developed.

Mathematical models for circular, sinusoidal, normal distribution, trapezoidal, and flat-top sign speed control humps for any length and height have been established.

Simulations and comparisons have been conducted using five types of vehicles to analyze speed control humps.

2. Conventional Speed Control Humps

In this section, nine types of humps, as shown in Table 1 and Figure 1, are used for formulation and simulation to compare the performance of speed control humps as vehicles (listed in Table 2) pass over them. Figure 1 illustrates the profiles of conventional round-top speed control humps listed in Table 1. All the speed humps have the same height of 0.1m, with lengths of 4m, 6.5m, and 9.5m.

We address speed control humps such as circular humps (dotted lines in Figure 1), sinusoidal humps (solid lines in Figure 1), and modified sinusoidal humps (dashed lines in Figure 1) to compare the performance of these three types through simulation.

The formulations of the conventional speed control humps are proposed approximate equations (1) to (6). The circular speed control humps are Quadratic Polynomial shown in equations (1), (4) and (7) for 4m, 6.5m and 9.5m long respectively. The sinusoidal and modified sinusoidal speed control humps are expressed as Quadratic Fourier Transform shown in equations (2) and (3) for 4m long, (5) and (6) for 6.5m long, (8) and (9) for 9.5m long humps. The height H is the function of the length for all kind of the humps. The formulations are quantitative approximate for the conventional speed control humps.

Circular speed control humps (L=4m, H=0.1m for 30kph)

(1)

Sinusoidal speed control humps (L=4m, H=0.1m for 30kph)

(2)

Modified Sinusoidal speed control humps (L=4m, H=0.1m for 30kph)

(3)

Circular speed control humps (L=6.5m, H=0.1m for 40kph)

(4)

Sinusoidal speed control humps (L=6.5m, H=0.1m for 40kph)

(5)

Modified Sinusoidal speed control humps (L=6.5m, H=0.1m for 40kph)

(6)

Circular speed control humps (L=9.5m, H=0.1m for 50kph)

(7)

Sinusoidal speed control humps (L=9.5m, H=0.1m for 50kph)

(8)

Modified Sinusoidal speed control humps (L=9.5m, H=0.1m for 50kph)

(9)

3. General Formulation of Speed Control Humps

The profile, length, and height of speed control humps are crucial factors in their design. The commonly used profiles for round-top speed control humps include sinusoidal, circular, and parabolic shapes. Each profile requires the optimization of two parameters: length 𝐿 and height 𝐻 as illustrated in Figure 1.

In order to do simulation and optimization of speed control hump easily, the formalization of all types hump need to be established. In this study, the qualitative formalizations are established as the equations (16) to (22). They are expressed by the functions of the hump length L and the height H. If input the data of L and H, the speed control hump with any length or height will be obtained. Example, if input L=4m, H=0.1m to equations (16), (17) and (18), the approximate circular hump profile with L=4m, H=0.1m will be expressed. The errors will be discussed in section 4.2.

Approximate formula of the circular hump shown in Figure 2 is expressed as Equation (10) and (11)

(10)

(11)

Whereandare the front and rear wheel vertical displacements, respectively,is the forward displacement of vehicles, L and H are the length and height of the hump (Figure 6), is the wheel base of vehicle (Figure 6). Those symbols are used in next section 3.2 to 3.5 as the same means.

Approximate formula of the sinusoidal hump is expressed as Equation (12) and (13)

(12)

(13)

Approximate formulations of the normal distribution hump is expressed as Equation (14) and (15)

(14)

(15)

Approximate formulations of the Trapezoidal hump is expressed as Equation (16) and (17)

(16)

(17)

Approximate formulations of the flat top sign hump is expressed as Equation (18) and (19)

(18)

(19)

4. Dynamic model, Equations of Motion and Parameters of a Half Vehicle

This section provides information on dynamics modeling of vehicle (Figure 6), equations of motion (equation20~24) and parameters (Table 2). In the dynamic model of vehicle (Figure 6), , , and represent the mass of the vehicle body (sprung mass), and the front and rear unsprung masses, respectively. Stiffnesses for the front and rear suspensions and the effective vertical stiffnesses of the front and rear tires are represented by , , and . Damping due to tires is ignored as usual. and are damping coefficients of the front and rear suspensions. Bounce dynamics of the sprung mass, wheel hop dynamics of the front unsprung mass, wheel hop dynamics of the rear unsprung mass, and pitch dynamics of the sprung mass of the half car model. and are the vertical displacements of front and rear tires. The equations of motion are given by equations of motion from equation (20) to (24).

Table 2 is the parameters of five kind of vehicles using in simulations. The parameters of the standard vehicle are using in optimization in the next study.

(20)

(21)

(22)

(23)

(24)

5. Simulations and Discussions

The results of the simulation for conventional speed control humps are shown in Figure 8, Figure 9, and Figure 10. The circular, sinusoidal, and modified sinusoidal humps are represented by solid, dashed, and dotted lines, respectively. The blue, green, red, and pink lines correspond to the simulation results for vehicles A, B, C, and D, respectively. The black lines represent the simulation results for a standard vehicle.

As is well known, the maximum vertical accelerations of vehicles increase with the speed at which they pass over a hump. There is a significant difference in maximum vertical accelerations between light (vehicle A) and heavy (vehicle D) vehicles. The maximum vertical accelerations for vehicle A (represented by blue lines) are more than twice as high as those for vehicle D (represented by pink lines) across all hump profiles. However, for all lengths of humps, vehicle D exhibits the smallest maximum vertical accelerations (see Figure 8, Figure 9, and Figure 10, pink lines). In contrast, vehicle A experiences the highest maximum vertical accelerations when passing over modified sinusoidal speed control humps (see Figure 8, Figure 9, and Figure 10, blue dotted lines).

To evaluate the qualitative formulas for the circular and sinusoidal speed control humps, as shown in equations 16 to 22, the curves of these speed control humps are compared with those of conventional speed control humps in Figure 8 and Figure 10. The red, blue, and black lines represent humps of lengths 4 meters, 6.5 meters, and 9.5 meters, respectively. The red, blue, and black lines correspond to hump lengths of 4 meters, 6.5 meters, and 9.5 meters, respectively. Solid lines represent the simulation results of the qualitative formulations, while dashed lines denote conventional speed control humps (see Table 1). For 4-meter length circular and sinusoidal humps, the profiles match perfectly between the qualitative formulations and conventional profiles (see Figure 11 and Figure 12, red solid and dashed lines). For 6.5-meter length circular and sinusoidal humps, the curves from the qualitative formulations (Figure 11 and Figure 12, blue solid lines) are smaller than the conventional profiles (Figure 11 and Figure 12, blue dashed lines) by 0.021% and 0.038%, respectively, in terms of section area error. For the 9.5m length circular and sinusoidal humps, the curves of the qualitative formulations profile (Figure 11 and Figure 12, black solid lines) are bigger than the conventional (Figure 11 and Figure 12, black broken lines) 0.014% and 0.032% (section area error) respectively. This means that the errors are enough smaller between the qualitative and conventional hump profiles.

Figure13 and Figure 14 show the maximum accelerations of five type vehicles pass though the speed control humps with length 4m, 6.5m and 9.5m, the passing speed are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80kph. Figure 13a,13b,13c are the results of the circular hump profiles with 4m, 6.5m and 9.5m length and 0.1m height, respectively. Figure 14a,14b,14c are the results of the Sinusoidal Hump profiles with 4m, 6.5m and 9.5m length and 0.1m height, respectively.

For the circular humps (Figure 13), the accelerations of all the vehicles passing over 4m and 6.5m length speed control humps are bigger than 9.5m length. For the vehicles, the accelerations of the vehicle D is smallest for all the passing over speeds.

For the sinusoidal humps (Figure 14), the accelerations of all the vehicles passing over 4m length speed control humps are greater than 6.5m and 9.5m length. For the vehicles, the accelerations of the vehicle D is smallest for all the passing speeds.

6. Conclusions

A mathematical model for five types of vehicles has been developed to study the dynamics of vehicles crossing speed control humps. Based on the simulation results, the following conclusions and recommendations can be made:

1. The mathematical models for the conventional speed control humps are effective. The section area errors of the curves are smaller than those of the conventional profiles by 0.021% and 0.038%, respectively, for the 6.5-meter length humps. For the 9.5-meter length circular and sinusoidal humps, the section area errors are 0.014% and 0.032%, respectively.

2. Based on the simulations, speed control humps are effective in controlling speeding vehicles. The profile of the humps—whether circular, sinusoidal, normal distribution, trapezoidal, or flat-top sign—have the same effect for controlling vehicle speed.

The mathematical models for circular, sinusoidal, normal distribution, trapezoidal, and flat top speed control humps are effective. These formulas can be used to develop and study speed control humps of various lengths and heights, as well as to analyze the impact on vehicle suspensions when passing over these humps.

References