Force and Moment Diagram Acting on a Car When Standing on a Slope A. Car Turning Uphill B. Car Turning Downhill

The diagram in Figure 5.1a corresponds to a car standing on a slope turning up. When the slope angle  increases gradually until the front wheel lifts off the road surface, then the reaction force Z 1 = 0, the car will overturn around point O 2 . To determine the limiting angle at which the car will overturn, we set up the model equation

The moment of all forces with respect to point O 2 and then simplified with Z 1 = 0 will give:

Gbcos  l – Gh g. sin  l = 0 (5-1)

b

In there:

tg  l =

h

g

(5-2)

 1 - the limiting slope angle at which the vehicle will overturn when standing still and turning uphill. b, h g - the coordinate dimensions of the center of gravity (figure 5.1)

In the case of a car standing on a slope and turning down (Figure 5.1b), we do the same by taking the moment of the forces relative to point O 1 , then replacing Z 2 = 0 and simplifying, we get

a

h

tg  l =

g

(5-3)

In which:  l - the limit slope angle at which the vehicle will overturn when standing still and turning downhill.

Through the above expressions, we see that the static overturning limit slope angle only depends on the coordinates of the vehicle's center of gravity.

When the vehicle is on a slope, in addition to the instability caused by overturning, the vehicle can also slide down the slope due to insufficient braking force or poor grip between the wheels and the road surface... To prevent the vehicle from sliding down the slope, a handbrake system is often installed on the vehicle. In the case when the maximum braking force reaches the grip limit, the vehicle can slide down the slope. We have:

P Pmax =  .Z 2 = G.sin  l (5-4)

In which: P Pmax - Maximum braking force on the rear wheel;

- The coefficient of longitudinal adhesion of the wheel to the road

Z 2 - Perpendicular reaction force from the road acting on the rear wheel

Z 2 =

G . a .cos  G . h g .sin 

L

(5-5)

Substitute Z 2 into formula (5-4) and simplify to determine the limited slope angle when the vehicle is standing on a slippery slope (in the case of turning up).

tg  t = 

a

L   . h g

(5-6)

Limit slope angle when standing on a slope and turning back down will cause slipping:

tg  t = 

a

L   . h g

(5-7)

The condition to ensure safety for a vehicle standing on a slope is that the vehicle slides before it overturns. We have the expression:

tg  t < tg  l

a b

Simplify and we get;

L  h g h g

 < b (5-8)

h g

From the above formula, we can see that the limiting slope angle when a car is standing on a slope and skids or overturns depends only on the coordinates of the center of gravity and the coefficient of adhesion of the wheel to the road surface.

5.2.2. Dynamic longitudinal stability

When a car moves on a slope, it can become unstable (overturn or slide) under the influence of forces and moments acting on it. On the other hand, when a car moves at high speed on a flat road, it can also overturn. Below we will consider each case of vehicle instability in turn.

5.2.2.1. General case

Figure 5.2 shows the diagram of forces and moments acting on a car when moving uphill, unstable, with a trailer.

Figure 5.2. Diagram of forces and moments acting on a car when moving uphill

We use the formulas to determine the perpendicular reaction force from the line of action on the

The front wheels (Z 1 ) and the rear wheels (Z 2 ) studied in chapter 2 we have:

G .cos  b  f . r    G .sin  P  P  h  P . h 

Z 

bj gmm

1 

G .cos  a f . r

 

L

P

P  h

 P . h 

(5-9)

Z 2 

b G sin

L

j g

mm 





The method is similar to the static vertical stability part, we can immediately determine the slope angle.

that the vehicle overturns when moving uphill or downhill (in the case of the vehicle going uphill, Z 1 = 0 and downhill, Z 2 = 0)

For simplicity, let's consider the case of a car moving steadily uphill, without a trailer.

Therefore, the inertial force P j = 0, the trailer pulling force P m = 0. The limited slope angle when the vehicle is overturned

tg  d =

b f . r b

h g

P h g

(5-10)

5.2.2.2. In case the vehicle moves uphill at low speed, without trailer and moves steadily

In this case P j = 0; P m = 0, we will determine the angle of the slope at which the car overturns:

tg  d =

b f . r b

h g

(5-11)

In case of vehicle going downhill, the limiting slope angle at which the vehicle will overturn is determined as follows:

tg  d =

a f . r b

h g

(5-12)

The limiting slope angle at which the vehicle skids is determined as follows:

When the traction of the driving wheel reaches the limit of adhesion, the vehicle begins to slip. The value of the traction is determined as follows:

P kmax = P =  .Z 2 = G.sin  (5-13)

Substituting the above value of Z 2 into expression (5-4) and considering the small rolling resistance that can be ignored, we have:

P =  .Z 2 =  G

 a .cos   h g.sin  

L

(5-14)

Continue to substitute (5-14) into formula (5-13) and simplify to determine the limiting slope angle at which the vehicle skids when moving uphill:

In there:

time​​= 

a

L   . h g

(5-15)

P k max - maximum tangential traction force at the driving wheel; P - traction of the driving wheel

The conditions to ensure that the vehicle skids before it overturns are also determined as follows:

static stabilizer

5.2.2.3. In case the vehicle moves on a horizontal road at high speed,

trailer

Figure 5.3 shows the diagram of forces and moments acting on a car when moving at high speed.

In this case, when moving at high speed (such as passenger cars, ambulances...) on good roads, the effects of rolling resistance and inertia force P t = 0; P j = 0 and P m = 0 can be ignored. The value of air resistance is very large and will cause the car to overturn. When the car moves at a speed that reaches the limit value, the car will overturn around point O 2 (O 2 is the intersection of the vertical plane through the center of the rear wheel axle with the road) at that time the resultant force Z 1 = 0

To determine the limiting speed at which the car overturns, we use the calculation formula Z 1learned in chapter 2 as follows:

Figure 5.3. Diagram of forces acting on a car when moving at high speed

Z  G . b  P . h g

1 L

(5-16)

Replace the value of air resistance P = kFv 2 and then simplify, we can determine the dangerous speed at which the car overturns:

G . b

k . F . h g

v n =3.6 (5-17)

In which: v- is the vehicle speed in km/h;

v n - dangerous speed when the car overturns in km/h

From expression (5-17) we can see that the dangerous speed when the car overturns depends on the center of gravity of the car and the air resistance factor. Therefore, to increase the stability of the car, when designing, people often try to lower the center of gravity of the car.

On the other hand, for some special types of vehicles such as racing cars, people make the front of the car have a special shape so that a component of the air resistance P (reaction force P  ) has the effect of pressing the wheel down to the road surface, increasing the stability of the vehicle ( Figure 5.4 )

Figure 5.4. Shape of a car moving at high speed

5.3. Lateral stability of the car

5.3.1. Lateral stability of a car when moving on a horizontal slope

Figure (5.5) shows the diagram of forces and moments acting on a car when moving on a horizontal slope without pulling a trailer. In this case, it is assumed that the tracks of the front and rear wheels coincide, the center of gravity of the car is in the longitudinal plane of symmetry, the forces and moments acting on the car include:

-  is the horizontal angle of the line

- The weight of the car G is divided into two components according to the tilt angle 

- The moment of tangential inertial forces M jn acting in the horizontal plane when the vehicle moves unsteadily

- Reactions Z', Z'' and Y', Y''

Figure 5. 5. Diagram of forces acting on a car when moving on a horizontal inclined road

Under the effect of forces and moments, when the angle  gradually increases to the limit angle, the car overturns through point A (A is the intersection of the vertical plane through the center of the left wheel axle and the road surface), at that time the reaction force Z = 0. From the reaction force calculation formula learned in chapter 2, we have:

G c cos 

2d​

 Gh g sin d

M jn



Z” = 0

c

(5-18)

In formula (5-18) we consider M jn  0 because the small value can be ignored, from which we can determine the overturning limit angle when the vehicle moves on a horizontal inclined road.

In there:

tg  d =

c

2 hours​

(5-19)

 d - the limiting slope angle at which the vehicle overturns

When the vehicle's grip on the road is poor, the vehicle can also skid on an inclined road. To determine the limiting angle when the vehicle skids, we establish an equation projecting the forces onto a plane parallel to the road:

Gsin  = Y' + Y” =  y (Z” + Z') =  y Gcos  (5-20)

In there:

 - the limited slope angle at which the vehicle skids

 y - coefficient of adhesion between wheel and road surface

Simplifying the above formula, we get the limit slope angle at which the car skids.

tg  =  y (5-21)

Conditions for a vehicle to skid before tipping over when moving on a horizontal slope

tg  < tg  d or  y <

c

2 hours​

(5-22)

In case the car is stationary on a horizontal slope, in a similar way as above we can also determine the limiting angle of inclination at which it will overturn or slide.

Limit slope angle of overturning when moving on horizontal slope:

tg  t =

c

2 hours​

(5-23)

Limiting slope angle at which the vehicle skids:

tg  =  y (5-24)

Conditions for a vehicle to slide before it overturns

tg  < tg  d or  y <

c

2 hours​

(5-25)

5.3.2. Lateral stability of the car when turning on a horizontal inclined road

5.3.2.1. According to the overthrow conditions

When the vehicle turns, in addition to the forces acting as above, the vehicle is also subject to the centrifugal force P L located at the vehicle's center of gravity (Figure 5.6) with the rotation axis being YY and the pulling force on the trailer being P m . In this case, the direction of the force P m acting is considered to be in the horizontal direction. The forces P L and P m are both divided into two components due to the tilt angle  . When the angle 