Vehicle centre of gravity. Forces acting on vehicle during ... · HEIßING, B. & ERSOY, M. 2010....

Vehicle center of gravity. Forces acting on vehicle during parking and

movement

Vehicle dynamics

1. Vehicle – driver – road interaction

2. Vehicle safety (not crashworthiness –

active safety)

3. Comfort of driving

4. Economics

Driver Vehicle

Responds

Effect

Vehicle dynamics

Mathematical Model(The vehicle)

Input(driver decision)

Output(Responds of the vehicle)

• Forces• Mass• Moments of inertia• Stiffens, damping

Newton Euler

Vehicle vs roadVehicle vs environment

Complexity of the model (DoF dependent)1. Simple 2. Complex

• Acceleration• Velocity

Vehicle dynamicsmodel complexity

GILLESPIE, T. D. 1992. Fundamentals of Vehicle Dynamics, Society of Automotive Engineers.

• Longitudinal

• Lateral

• Vertical

• Longitudinal vs vertical

• Lateral vs vertical

• Longitudinal vs lateral vs

vertical

Center of gravityCG (CoG)

The imaginary point at which the vehicle’s

entire mass can be concentrated

A low center of gravity results in:

• Good handling and driving safety;

• Reduced vehicle pitch and roll;

• Reduced wheel load fluctuations on

inclined surfaces;

A high center of gravity results in:

• Increased rear axle load on inclined surfaces.

MACEY, S., WARDLE, G. & GILLES, R. 2009. H-point: The Fundamentals of Car Design & Packaging, Design Studio Press.

Longitudinal dynamicsParked vehicle (simples approach)

JAZAR, R. N. 2008. Vehicle Dynamics: Theory and Application, Springer US.

• No movement – no acceleration • CG determination;• The normal force under each axis calculations;• A symmetric two-axel vehicle is equivalent to a

rigid beam having two supports.

𝐹𝑧1 =1

2𝑚𝑔

𝑎2

𝑎1+𝑎2=1

2𝑚𝑔

𝑎2

𝑙

𝐹𝑧2 =1

2𝑚𝑔

𝑎1

𝑎1+𝑎2=1

2𝑚𝑔

𝑎1

𝑙

Longitudinal dynamicsParked vehicle (simples approach)


Applying the planar static equilibrium equations

These equations provide the brake force and reaction

forces under the front and rear tires.

tilting angle – the weight vector mg goes through the contact point of the rear tire with

the ground

Maximum angle, the braking force is proportional

to the normal force 𝐹𝑧

CG determination• To determine the longitudinal position of C, we should measure the

total weight of the car as well as the force under the front or the rear wheels.

𝑎1 =2𝑙

𝑚𝑔𝐹𝑧2 =

2𝑙

𝑚𝑔𝑚𝑔 − 2𝐹𝑧1

𝐹𝑧2 =1

2(𝑚𝑔 − 2𝐹𝑧1)


• To determine the height of mass center C, we should measure the forceunder the front or rear wheels while the car is on an inclined surface.

• Assuming the force under the front wheels is 2𝐹𝑧1 , the height of the mass center can be calculated by static equilibrium conditions

CG determination

1. the tires are assumed to be rigid disks with radius 𝑅

2. fluid shift, such as fuel, coolant, and oil, are ignored

3. suspension deflections are assumed to be zero.

Suspension deflection generates the maximum effect on height determination error. To eliminate the

suspension deflection, we should lock the suspension, usually by replacing the shock absorbers with

rigid rods to keep the vehicle at its ride height


Acceleration performance

• 𝑾 is the weight of the vehicle acting at its CG with a magnitude equalto its mass times the acceleration of gravity

•𝑾

𝒈𝒂𝒙is a d' Alembert force. If the vehicle is accelerating along the road,

it is convenient to represent the effect by an equivalent inertial force• 𝑾𝒇,𝑾𝒓the dynamic weights carried on the front and rear wheels

• 𝑭𝒇, 𝑭𝒓 Tractive effort (for FWD 𝐹𝑟 = 0, RWD 𝐹𝑓 = 0)

WONG, J. Y. 2001. Theory of Ground Vehicles, Wiley.Moving vehicle must be strong enough to overcome all resisting forces

• 𝑹𝒓𝒇, 𝑹𝒓𝒓 rolling resistance of the front and rear tires

• 𝑹𝒂 aerodynamics resistance• 𝑹𝒅 drawbar load• 𝑹𝒈(𝑾 ∙ 𝒔𝒊𝒏𝜽) grade resistance

The axle loads determine the tractive effort obtainable at each axle, affecting the acceleration, gradeability, maximum speed, and

drawbar effort


WONG, J. Y. 2001. Theory of Ground Vehicles, Wiley.

The equation of motion along the longitudinal axis x of the vehicle is

By introducing the concept of inertia force, the above equation may be rewritten as

To evaluate the performance potential, the maximum tractive effort that the vehicle can develop has to be determined. There are two limiting factors to the maximum tractive effort of a road

vehicle: 1. the coefficient of road adhesion and the normal load on the

drive axle or axles;2. the characteristics of the power plant and the transmission.



• The loads carried on each axle will consist of a static component,plus load transferred from front to rear (or vice versa) due to the other forces acting on the vehicle;

• To predict the maximum tractive effort that the tire-groundcontact can support, the normal loads on the axles have to bedetermined;

• The load on the front axle 𝑊𝑓 can be found by summing

torques about the point "𝐴" under the rear tires

When the vehicle is climbing up a hill, the negative sign is used for the term 𝑊ℎ𝑠𝑖𝑛𝜃𝑠

When the vehicle is climbing up a hill, the positive sign is used for the term 𝑊ℎ𝑠𝑖𝑛𝜃𝑠



For small angles of slope, cos𝜃𝑠 ≈ 1,. For passenger cars, the height

of the point of application of the aerodynamic resistance 𝒉𝒂, and

that of the drawbar hitch 𝒉𝒅, may be assumed to be near the

height of the center of gravity 𝒉.

𝑹𝒈 = 𝑾 ∙ 𝒔𝒊𝒏𝜽



• There is a maximum tractive effort that the tire-ground contact can support;

• There is a maximal acceleration of a vehicle proportional to the friction under its tires;

• Max tractive effort depends of tire road interaction – coefficient of road adhesion;

• Rolling resistance is dependent upon the rolling resistance coefficient and the weight of the vehicle 𝑹𝒓 = 𝒇𝒓 ∙ 𝑾;

Braking performance


The braking force 𝐹𝑏, originating from the brake system and developed onthe tire-road interface is the primary retarding force.

When the braking force is below the limit of tire-road adhesion, the braking force 𝐹𝑏 is

𝑰 is the rotating inertia connected with the wheel being decelerated,

𝜶𝒂𝒏 corresponding angular deceleration

𝒓 the rolling radius of the tire

𝑻𝒃 the applied brake torque,

Braking performance


In addition to the braking force, the rolling resistance of tires,aerodynamic resistance, and grade resistance (when traveling on aslope) also affect vehicle motion during braking

• When the vehicle is moving uphill, the positive sign for the term 𝑾𝒔𝒊𝒏𝜽,

• During braking, there is a load transfer from the rear axle to the front axle;

• By considering the equilibrium of the moments about the front and rear tire ground

contact points, the normal loads on the front and rear axles are:

Braking performance


By considering the force equilibrium in the horizontal direction, the following relationship can be established

The maximum braking force that the tire-ground contact can support is determined by the normal load and thecoefficient of road adhesion. The maximal braking force. Any further increase in the braking force would cause the tiresto lock up

When the distribution of the braking forces between the front and rear axles is in exactly the same proportion as that of

normal loads on the front and rear axles will the maximum braking forces of the front and rear tires be developed at the

same time

Braking performance


• The braking forces between the front and rear axles is a function of the design of the brake system when no wheels are locked;

• By considering the force equilibrium in the horizontal direction, thefollowing relationship can be established.

• Not appropriate selection of front vs rear axe brakingperformance leads to decrease of maneuverability;

• Front tires lock first on roads with a coefficient of adhesionlower than a certain value.

Braking performance


The conditions under which the front or the rear tires will lock first canbe quantitatively determined.

The braking forces of the front and rear axles as determined by the brake system design are expressed by

Braking performance


The lock up depends of the tire road interaction, hence the front tires will lock up when

the vehicle deceleration rate (in g-units) associated with theimpending lock-up of the tires can be defined by

For a given vehicle with a particular braking force distribution on a given road surface, the front tires will lock first if

• The front and rear tires will lock at the samedeceleration rate only on a particular road surface -maximum braking forces of the front and rear axles;

• Under all other conditions, either the front or reartires will lock first, resulting in a loss of either steeringcontrol or directional stability;

• Ideally, the braking force distribution should beadjustable to ensure optimum braking performanceunder various operating conditions.

Gradability• Gradability is usually defined as the maximum grade a

vehicle can negotiate at a given steady speed.

• Slope at a constant speed, the tractive effort has toovercome grade resistance, rolling resistance, andaerodynamic resistance


• For small angles 𝒔𝒊𝒏𝜽𝒔 = 𝒕𝒂𝒏𝜽𝒔 so the grade resistance can be replaced by grad in %

• The limits of tractive effort set by the nature of tire-road adhesion usually determine the maximum gradability of the vehicle

Lateral force;Cornering

• When the vehicle is moving very slowly, there is a kinematic conditionbetween the inner and outer wheels that allows them to turn slip-free- Ackerman condition

Since triangle MAQ is similar to triangle MCF,

• When a vehicle is negotiating a turn at moderate or higher speeds, the effect of the centrifugal force (an inertia force arisingfrom the normal component of acceleration towards the center of the turn) acting at the center of gravity can no longer be neglected.

• To balance the centrifugal force, the tires must develop appropriate cornering forces – side slip angle


Lateral force;Side slip angle

HEIßING, B. & ERSOY, M. 2010. Chassis Handbook: Fundamentals, Driving Dynamics, Components, Mechatronics, Perspectives,

Vieweg+Teubner Verlag.

• Side slip angle The angle between the wheel’s velocity vector along the wheel center plane and thevehicle’s actual direction of displacement at the tire contact patch;

• A tire slip angle occurs mainly when lateral forces are acting on the center of tire contact;• The sideslip angle, together with the tire and road surface, determine the lateral force threshold of

each wheel;• The size of the slip angle depends on the force transmission properties of the tire and the disturbing

force;


• The relationship between the cornering force and the slip angle is of fundamental importance to

the directional control and stability of road vehicles;

• When the tire is moving at a uniform speed the side force 𝐹𝑠 applied at the wheel center and the

cornering force 𝐹𝑦𝛼 developed in the ground plane are usually not collinear;

• 𝐭𝒑 is called the pneumatic trail, and the product of the cornering force and the pneumatic trail

determines the self-aligning torque;

• The aligning or self-aligning torque help the steered tire return to the original position after

negotiating a turn;

• To provide a measure for comparing the cornering behavior of different tires, a parameter called cornering stiffness 𝑪𝜶 is established . It is defined as the derivative of the cornering force 𝑭𝒚𝜶with respect to slip angle 𝜶 evaluated at zero slip angle:

Lateral force;Side slip angle

WONG, J. Y. 2001. Theory of GroundVehicles, Wiley.



• The handling characteristics of the vehicle depend, to a great extent, on therelationship between the slip angles of the front and rear tires, 𝜶𝒇 and 𝜶𝒓

• The steer angle 𝜹𝒇 required to negotiate a given curve is a function of not only the

turning radius 𝑹, but also the front and rear slip angles 𝜶𝒇 and 𝜶𝒓

• The slip angles are dependent on the side forces acting on the tires and their cornering stiffness. The cornering forces on the front and rear tires can be determined from the dynamic equilibrium of the vehicle in the lateral direction.

The normal load on each of the front tires

Hence



Within a certain range, the slip angle and cornering force may be considered to be linearly related with a constant cornering stiffness,

For simplification of this example it was assumed that the vehicle has two tires with double stiffness

The cornering stiffness of a given tire varies with a number of operational parameters, including inflation pressure, normal load, tractive (or braking) effort, and lateral force. It may be regarded as a constant only

within a limited range of operating conditions

• The steer angle is

• Hence• 𝑲𝒖𝒔 the understeer coefficient and is expressed in radians. Function of weight

distribution and tire cornering stiffness;• Dependent on the values of the understeer coefficient 𝑲𝒖𝒔 or the

relationship between the slip angles of the front and rear tires, the steady-state handling characteristics may be classified into three categories: neutralsteer, understeer, and oversteer

• 𝑲𝒖𝒔 = 𝟎 equivalent to the slip angles of the front and rear tires

being equal𝛼𝑓 = 𝛼𝑟, 𝑎𝑛𝑑𝑊𝑓

𝐶𝛼𝑓=

𝑊𝑟

𝐶𝛼𝑟

• In such case the angle required to negotiate a given curve is independent of forward speed and is given by


Lateral force;Cornering; Neutral steer

• When a neutral steer vehicle is accelerated in a constant radius turn, the driver should maintain the same steering wheel position.

• When a neutral steer vehicle originally moving along a straight line is subjected to a side force acting at the center of gravity, equal slip angles will be developed at the front and rear tires. As a result, the vehicle follows a straight path at an angle to the original.

•𝑲𝒖𝒔 > 𝟎 which is equivalent to the slip angles of the front and rear tires

being equal 𝛼𝑓> 𝛼𝑟, 𝑎𝑛𝑑𝑊𝑓

𝐶𝛼𝑓>

𝑊𝑟

𝐶𝛼𝑟

• In such case the steer angle required to negotiate a given curve increases with the square of vehicle forward speed (or lateral acceleration).


Lateral force;Cornering; Understeer

• When a side force acts at the center of gravity of an understeer vehicle originally moving along a straight line, the front tires will develop a slip angle greater than that of the rear tires

• As a result, a yaw motion is initiated, and the vehicle turns away from the side force;• At the same steering wheel position and vehicle forward speed, the turning radius

of an understeer vehicle is larger than that of a neutral steer vehicle;• There is a characteristic speed for and understeer vehicles. It is the speed at which

the steer angle required to negotiate a turn is equal to 2𝐿/𝑅

•𝑲𝒖𝒔 < 𝟎 which is equivalent to the slip angles of the front and rear tires

being equal 𝛼𝑓< 𝛼𝑟, 𝑎𝑛𝑑𝑊𝑓

𝐶𝛼𝑓<

𝑊𝑟

𝐶𝛼𝑟


Lateral force;Cornering; Oversteer

• In such case the steer angle required to negotiate a given curve decreases with anincrease of vehicle forward speed (or lateral acceleration)

• when a vehicle is accelerated with the steering wheel fixed, the turning radiusdecreases,

• For the same steering wheel position and vehicle the turning radius of an oversteervehicle is smaller than that of a neutral steer vehicle.

• When a side force acts at the center of gravity of an oversteer vehicle originally movingalong a straight line, the front tires will develop a slip angle less than that of the reartires

• As a result, a yaw motion is initiated, and the vehicle turns into the side force;• For an oversteer vehicle, a critical speed 𝑽𝒄𝒓𝒊𝒕 can be identified. It is the speed at which

the steer angle required to negotiate any turn is zero, as shown. The critical speed also

represents the speed above which an oversteer vehicle exhibits directional instability.

Literature

1. GILLESPIE, T. D. 1992. Fundamentals of Vehicle Dynamics, Society of Automotive Engineers.

2. JAZAR, R. N. 2008. Vehicle Dynamics: Theory and Application, Springer US.

3. WONG, J. Y. 2001. Theory of Ground Vehicles, Wiley.

4. DIXON, J. 2009. Suspension Analysis and Computational Geometry, Wiley.

5. KARNOPP, D. 2004. Vehicle Stability, CRC Press.

6. RAJAMANI, R. 2011. Vehicle Dynamics and Control, Springer US.

7. BLUNDELL, M. & HARTY, D. 2004. The Multibody Systems Approach to Vehicle Dynamics, Elsevier Science.

8. REIMPELL, J., STOLL, H. & BETZLER, J. 2001. The Automotive Chassis: Engineering Principles, Elsevier Science.

9. LEON, P. 2008. Mechanika ruchu, Warszawa, WKŁ.

10. KARNOPP, D. 2004. Vehicle Stability, CRC Press.

Homework 1

A passenger car has a weight of 20.105 kN and a wheelbase of 2.8 m.The weight distribution on the front axle is 53.5%, and that on the rear axle is 46.5% under static conditions.

a) If the cornering stiffness of each of the front tires is 38.92 kN/rad and that of the rear tires is 38.25 kN/rad, determine the steady-state handling behavior of the vehicle.

b) If the front tires are replaced by a pair of radial-ply tires, each of which has a cornering stiffness of 47.82 kN/rad, and the rear tiresremain unchanged, determine the steady-state handling behavior of the vehicle under these circumstances

Homework 2

A passenger car weighs 21.24 𝑘𝑁 and has a wheelbase of 2.87 𝑚. Thecenter of gravity is 1.27 𝑚 behind the front axle and 0.508 𝑚 aboveground level. The braking effort distribution on the front axle is 60%.The coefficient of rolling resistance is 0.02. Determine which set of thetires will lock first on two road surfaces: one with a coefficient of roadadhesion 𝜇 = 0,8, and the other with 𝜇 = 0,2.

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