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Determination of Center of Gravity and Dynamic Stability Evaluation of a Cargo-type Tricycle

Ekuase Austin, Aduloju Sunday Christopher*, Ogenekaro Peter, Ebhota Williams Saturday, Dania David E.

National Engineering Design Development Institute, Nnewi, Nigeria

Abstract: Dynamic stability of vehicles is a major concern to vehicle manufacturers, as this determines how safe a vehicle will be on the road, to passengers and other road users. The location of centre of gravity (CG) on a vehicle determines its stability. The objective of this work is to evaluate the dynamic stability of a modeled cargo tricycle. The mass Properties capability of the SolidWorks software was used to determine the CG location on the tricycle. Result shows that the model will response to side load with a yaw motion and it’s an oversteer vehicle. Therefore it is unstable at high speed above its critical speed. The rollover threshold (Fc) for the tricycle model is 0.32g.

Keywords: dynamic stability, center of gravity, cargo tricycle, tipping treshold, yaw motion, oversteer

Received February 07, 2015; Revised March 02, 2015; Accepted March 13, 2015

Citation: Ekuase Austin, Aduloju Sunday Christopher, Ogenekaro Peter, Ebhota Williams Saturday, Dania David E., Determination of Center of Gravity and Dynamic Stability Evaluation of a Cargo-type Tricycle. American Journal of Mechanical Engineering, 3 (1). 26-31. January 2015. doi:10.12691/ajme-3-1-5

1. Introduction

Dynamic stability of vehicles is a major concern to vehicle manufacturers, as this determines how safe a vehicle will be on the road, to passengers and other road users. The location of centre of gravity (CG) on a vehicle determines its stability. In an attempt to safely exploit the available capacities of both the highway system and motor transport fleet, the Canadian government had proposed regulatory principles and recommended limits. The regulatory principles were formed based on a research conducted to understand the influence of heavy vehicle weights and dimensions on the stability and controllability of the vehicles which are used in the highway system. One of the seven performance indicators for stability recommended is the rollover threshold [1]. Systems Technology Inc [2] studied the rollover accident experience of small cars as a function of the rollover potential. Result shows that rollover accident rate (fatal accident per 100,000 new car years) decreases with increasing rollover threshold. Robertson and Kelly [3] conducted a methodical analysis of rollover accident experience for passenger cars and utility vehicles. In their work, they considered a wider range of vehicle and rollover was taken as the ‘first harmful event’ in accidents per 100,000 vehicle-years. Their findings reveal a direct relationship between rollover threshold and accident rates, and the inclusion of utility vehicle increased the accident rates. The high involvement of the utility vehicles made the Federal Motor Vehicle Safety Standard (FMVSS) to propose a minimum rollover threshold of 1.2 for new utility vehicle [4].

In Asian countries like India, Bangladesh, Thailand and Nepal, three wheeled vehicle TWV is commonly used as means of public transportation, utility vehicles and carrying freight. The use of TWV also known as tricycle in Africa is expected to increase over the next decades. Because of their low cost and maneuverability in small turn radii and crowded roads, three wheelers is preferred compared to 4-wheel vehicles, hence they are important means of transportation in developing countries [5]. Delise et al [6] however discussed the poor stability of the three-wheeled vehicle as an important factor for accidents. Relatively higher centre of gravity have being cited as contributing factor for rollover, therefore passenger safety is compromised. Nevertheless, Van Valkenburg et al [7], ascertain that it is possible to make a three wheel vehicle as stable as a 4-wheel car.

We have analyzed the body shape of tree wheelers for reduction in fuel consumption [8] but Patrick. J [9] finds it disturbing that most three wheeler designers do not know or follow the procedures for designing stability into these three wheelers even when it can be achieved. In order to guarantee the safety of drivers and road user, the dynamic stability of a modeled three wheel vehicle for manufacturing should be examined for better stability. Therefore the objective of this paper is to evaluate the dynamic stability of a modeled cargo tricycle.

2. Dynamic Stability Analysis of Tricycle

2.1. Lateral Stability

2.1.1. Vehicle Response to Side Loads at the CG: Static Margin

Figure 1 show three conditions of the bicycle model of a four wheeled vehicle. It is assumed that each end of the vehicle has single tire with double cornering stiffness of the actual tire in the bicycle model. The CG is assumed on the centerline of the vehicle. The horizontal forces which act upon the CG and the tire contact patches are of interest. Three centers of gravity positions are considered i.e CG, CG1 and CG2. A vehicle headed straight along path P, travelling at steady velocity V is represented in figure 1(a). The velocities of the axle line above the front and rear tire contact patches are VF and VR, and have same magnitude and direction as V.

Figure 1. Three condition of bicycle model (a) Straight Ahead (b) Apply side load (c) Response to side load (Source: [7])

Figure 1b shows the vehicle traveling along path P as the side load is applied. Lateral forces at the tires, FF and FR, would develop to counteract the side load. Corresponding slip angles αF & αR would be present as the velocities veer off their original paths. Figure 1a shows three vehicle responses after the application of the side load. The response is a sideways movement of the CG off the path P an angular rotation about the CG, which is the yaw response and indicated by angle θ. The point at which a side load is applied and there is no yaw response is called the Neutral Steer Point (NSP). The distance from the front axle line to the NSP is:

[picture] | (1)

Where = cornering stiffness values of a single front and rear tire

= total cornering stiffness values for the four wheeler bicycle

WB = wheel base

LG = distance from front axle line to CG.

Static Margin

The character of the yaw response of a vehicle is determined by the location of the NSP relative to the CG. The distance from the CG rearward to the NSP divided by the wheelbase is termed “static margin” (SM).

[picture] | (2)

If the same tires are used at each end, , equation (2) can be written as

[picture] | (3)

The value of SM can be positive, negative or zero and it’s an indicator of a vehicle yaw response.

For SM = 0, the center of mass is at one half of the wheelbase, LG = WB/2. In that case, a load at the CG will not cause any yaw response. The front and rear angle will make the vehicle sideslip. It is termed “Neutral steer”.

For SM = (+), the CG is ahead of the NSP, shown as CG1, then LG/WB is less than one half. Figure 4C shows the yaw response in which the front slip angle is larger than the rear and the vehicle is headed in the direction of applied force. This is considered “stable”.

For SM = (-), the CG is behind NSP, shown as CG2, then LG/WB is larger than one half. Figure 1C shows the rear slip angle is larger than the front and the vehicle is turned against the direction of the applied force. This is considered “unstable”.

2.1.2. Vehicle Response in Turn

Figure 2. Vehicle response in a turn Source: [7]

The Figure 2 shows the stability of a vehicle in a turn. If a vehicle is negotiating a corner of radius R, travelling at a velocity V and with a steer angle of δ. The vehicle has a Front axle velocity VF and Rear axle velocity VR with their corresponding slip angles ( and ) as indicated in the figure above.

The side load is a centrifugal force F which acts on the CG and it increases with the square of the velocity V. Angle is the angle between the vehicle centerline and the direction of the velocity of the CG. Hence the steering angle in degrees can be expressed as:

[picture] | (4)

[picture] | (5)

The term is lateral acceleration in terms of g’s.

Where v = velocity of CG along the path

R = radius of the CG path

G = acceleration of gravity

The term in the brackets has special significance and is called “understeer gradient” and denoted with symbol K

[picture] | (6)

Where = weight on front wheels

= weight on rear wheels

= total front cornering stiffness

= total rear cornering stiffness

Hence stability is analyzed in a turn as δ changes as the lateral acceleration ay increases. δ is the sum of a constant term, plus an expression involving one of the following terms which could be positive, zero, or negative

From Equation (4)

From Equations (6)

From Equation (2) .

The condition that change the signs of these terms are equivalent: if , then K > 0 and SM >0. The sign change of these terms determines stability and defines vehicle steer characteristics.

Neutral Steer: , K = 0 and SM = 0

Steering angle δ remains constant at the values 57.2 WB/R, degrees. As the vehicle negotiates a radius R and slowly increases the velocity causing lateral acceleration to increase, then lateral forces at each end of the vehicle increases to cause slip angles αF and αR to increase. For a neutral steer vehicle, the vehicle centerline will slightly rotate toward the direction of velocity V, hence decreasing the side slip angle β and thus increase both front and rear slip angles the same amount as δ remain constant.

Understeer: , K > 0 and SM > 0

As steering angle increases with speed, the front slip angle is larger than the rear. Though, it is self correcting. If the δ is not increase further, the positive yaw will turn the front of the vehicle toward the outside of the original path, thereby increasing the radius R and reducing the centrifugal force. Thus, a positive SM is termed “stable”.

Oversteer: , K < 0 and SM < 0

The rear slip angle is larger than the front, i.e the rear is side slipping more than the front, and hence the steering angle is reduced from its neutral steer value. If there is no correction, the vehicle moves outward from the original path, decreasing the radius R and increasing the centrifugal force requiring a greater correction. The correction is a turn in the direction of skid which can lead to the vehicle spinning. Thus a negative SM is termed “unstable”.

2.1.3. Application of Static Margin and Understeer Gradient in Three Wheelers

For three wheelers with two wheels at the rear and one wheel at the front, the expression for K and SM can be written in terms of single tire stiffness values as:

[picture] | (7)

[picture] | (8)

If the same tires are used, then each tire would have the same stiffness values, CF = CR

2.1.4. Critical Speed

Figure 3. Change of steer angle with speed (Source: [4])

Characteristic speed is experience by understeer vehicle which is the speed at which the steer angle required to negotiate a turn is twice the ackerman angle. While critical speed is associated with oversteer vehicle. It is the speed at the steer angle of a vehicle is zero and above which the vehicle become directionally unstable as shown in Figure 3. It can be represented by the equation below:

[picture] | (9)

Vcrit = critical speed, L = Wheelbase, g = acceleration due gravity, K = Understeer gradient

Rollover stability is another form of vehicle stability, which is the resistance to tipping over in a turn. The “quasi – static” model is used to examine the level of , that causes the inside tire(s) to have zero vertical load when negotiating a turn. This level of lateral acceleration is term the “tipping threshold” and can be represented as ().

2.2. Rollover Stability

The value of for a three wheeler involves the longitudinal placement of CG as well as its height and the vehicle track. Figure 4 shows a three wheeled vehicle negotiating a right turn with vertical reactions at the tire contact point and the lateral friction coefficient multiplied by their vertical loads represent the lateral forces. The Lateral acceleration, ay, multiplied by the vehicle weight, W represents the side load at the CG. The subscript “i” denotes the inside front wheel and “o” denoted the outside front wheel. HG represents the height of the CG, TR represents the front track, and LG is the distance from the front axle rearward to the CG. The CG is assumed to be on the centerline of the vehicle. The vehicle tips over at some value of the side load; however this may not actually occur since the vehicle slides before tipping over. Hence the largest value of lateral acceleration (ay) that the vehicle will experience is equal to the coefficient of lateral friction at the tires. The lateral forces are simply whatever is needed to keep the vehicle from sliding, for tipping to be examined. This will occur when the inside front wheel leaves the ground or when = 0. At this point, the vehicle is supported by the outside front and rear wheel. This can be expressed as:

[picture] | (6)

The expression shows that increasing the front track TR, decreasing LG and lowering the CG all contribute to a larger value.

Table 1. Rollover Threshold for vehicle types

Figure 4. Force diagram for tipping analysis, Right hard turn (Source: [7])

3. Simulation Procedures

A tricycle was drawn and modeled with Solidworks software and its components assembled to form the complete model. The engine, chassis, cabin, tires and bucket in the assembled model were configured using Solidworks material library to their appropriate material for manufacturing. This was to achieve a close estimate of the tricycle weight in reality to enhance efficient weight distribution on the tires.

Table 2. Modeled Tricycle mass Properties

The mass of the Tricycle was determined with the Mass Properties capability of the SolidWorks software. The mass properties and moments of inertia of the tricycle is shown in the table below. Also the position of the center of mass (gravity) was represented with the cursor appropriately located at the coordinates X9.75, Y-15.08 and Z -263.90 (Figure 5a&b).

4. Results

Figure 5. The position of the CG on the tricycle (a) Side View (b) Rear view

At the location of the center of gravity on the tricycle model, the following parameters value were determined in order to evaluate how stable the tricycle will be when manufactured. From Figure 5 above:

Table 3. Simulation results

5. Discussion

The tricycle remains stable at low speed below its critical speed. At its critical speed and beyond, because the CG of the vehicle falls behind the Neutral Steering Point (NSP) the rear wheel slip angle is larger than the front wheel which makes the Static margin of the tricycle negative. Hence the vehicle respond to side load with a yawing motion i.e. a sideways movement of the CG about its original path of travel is experienced. As a result the tricycle is unstable at speed above its critical speed.

From simulation results, analysis can be made when the tricycle negotiate a turn. At low speed cornering because the tricycle rear axle weight is greater than the front axle, the rear wheel slip angle is larger than the front wheel. Understeer gradient (K) is negative making the tricycle oversteer. This means as speed increases, the steering angle continually decreases until it become zero at the tricycle critical speed. Because tricycle is oversteer, it is unstable above its critical speed when negotiating a turn.

According to Patrick fenner [9], to achieved better rollover stability the CG height for a vehicle should be less than half the track length (HG < TR/2). The values from the tricycle model show that the CG height (HG) is larger than half of the track length (TR). Rollover threshold (Fc) for the tricycle model is 0.32g. Bigger value for Fc is better as it offer more stability.

For lateral and rollover stability to be improved on the tricycle model the value of LG has to be reduced and HG must be lowered which change the positioning of the center of gravity on the tricycle model. This can be achieved by reducing the total volume and weight of the model.

6. Conclusion

A design cargo tricycle model dynamic stability was evaluated. Solidworks software was used for solid modeling and stability analyses. The Mass Properties capability of the SolidWorks Software was used to determine the location of the center of mass on the model. The mass, CG height, track length, and wheelbase measurement were determined and used to calculate values for front & rear weight distribution, static margin, understeer gradient and lateral acceleration for the tricycle model. Result shows that the tricycle model will respond to side load with a yaw motion about its CG at high speed above the critical speed which makes it unstable. Also because the tricycle is an oversteer vehicle, it become unstable during cornering at speed above the critical speed. The tipping threshold for the tricycle model is 0.32g, but can be improve by reducing the value of LG and HG.

Acknowledgement

This project was supported by National Agency for Science and Engineering Infrastructure (NASENI) and National Engineering Design Development Institute (NEDDI).

References

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