OAD LOADS
R
oad loads are the external forces acting on a vehicle during operation. They arise from gravity, the interaction between tires and the road surface, and from air flowing past the moving vehicle. Understanding them is essential for the vehicle designer, as they affect almost all aspects of the design, from vehicle dy-namics and handling to power-speed models and structural analyses.
Figure 7-1 illustrates road loads acting on a bicycle riding in a straight line on level ground. Aerodynamic drag, rolling resistance of the tires, traction forces and braking forces all act parallel to the longitudinal axis of the bicycle. For a bicy- cle traveling at a constant speed on a level road, the sum of the drag and rolling resistance will exactly equal the tractive force produced by the driving wheel. If the road is not level, a component of the weight will also act in the longitudinal direction, pushing the bike downhill or restraining it during a climb.
Review of Equilibrium Equations Review of Equilibrium Equations
Frequently throughout this book equilibrium equations will be solved to deter- mine unknown reaction forces and moments. These equations stem from Newton’s second law, which states that the sum of forces acting on a body is equal to the mass of the body times its acceleration, or
F =ma
∑
In a Cartesian coordinate system, this equation can be applied in each of the coordinate directions x, y, and z. For static equilibrium, the acceleration, and hence the right-hand side, is equal to zero. Quasi-static analyses treat the neg- ative of the right-hand side (– ma) as if it were an externally applied force. This quantity is often called the inertial force. Quasi-static analyses are quite useful for many problems related to human-powered vehicles.
R
R
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Newton’s second law is valid for moments as well as forces provided the right- hand side is replaced with the product of the mass moment of inertia and the angular acceleration. Hence:
M = αI
∑
As with forces, this equation is valid for moments about each coordinate axis, giving three more equations. There are a total of six equilibrium equations for three-dimensional systems, and hence up to six unknown forces or moments can be found. When all forces lie within a common plane, three of the six equations are eliminated, leaving two force equations and one moment equation. The six equilibrium equations are:
x x x xx x y y y yy y z z z zz z F ma M I F ma M I F ma M I = = α = = α = = α
∑
∑
∑
∑
∑
∑
SAE Vehicle Coordinate System for Vehicle Dynamics SAE Vehicle Coordinate System for Vehicle Dynamics
Figure 7-2 depicts the vehicle coordinate system defined per SAE J670e, which is used throughout this book for all topics related to vehicle dynamics (with the exception of Patterson’s method for predicting bicycle handling). The vehicle Figure 7-1
Figure 7-1 Forces acting on a bicycle during level riding at constant speed (arrows are not proportional to magnitude)
Road Road LoadsLoads
coordinate system is located at the center of gravity with the x-axis directed for- ward, y-axis to the driver’s right, and the z-axis downward. These correspond to the longitudinal, lateral, and vertical directions respectively. The correspond- ing rotations about these axes are termed roll, pitch, and yaw. Rotation about the x-axis, such as a lean to either right or left, is known as roll. Yaw is a rotation about the vertical z-axis, for example from a heading of east to one of north. Pitch is rotation about the lateral y-axis. Vehicles with suspension systems may pitch forward during heavy braking or when riding over a bump.
Static Loads on Level Ground Static Loads on Level Ground
For the simple case of a stationary vehicle on level ground, the total weight of the vehicle, including the rider, and the vertical forces at each wheel are the parameters of interest. The static weight fraction is ratio of weight on one wheel to that of the entire loaded vehicle. Weight fraction affects handling and stability, and frequently crops up in many analyses, so it is good to understand the concept and how to calculate it.
Gravity acts on the entire vehicle, but for analytical purposes may be modeled as a point force acting through the center of gravity of the vehicle. The center of gravity is located at a location b forward of the rear axle and height h above the ground. Referring to Figure 7-3, it is seen that the total vehicle weight W is reacted by vertical forces at each wheel. The loads on each wheel can be found using Newton’s second law. First, the load on the front wheel is found by summing moments about the contact patch of the rear wheel. Since the vehicle is station- ary, there are no accelerations and the right-hand side is zero.
Figure 7-2
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Wb – W = 0 f L (7-1)
Given the total vehicle weight, the wheelbase, and the center of gravity loca- tion, the unknown force on the front wheel can be found
f
Wb mgb
W
L L
= = (7-2)
With the vertical load on the front wheel known, the load on the rear wheel can be found either by summing moments about the front tire patch or by summing forces in the vertical (z) direction.
( ) ( ) ( ) r f W L b mg L b W W W L L − − = = = − (7-3)
As noted, the weight fractions are obtained by dividing each wheel load by the total vehicle weight. Usually, the weight fraction is specified as the front axle weight fraction or rear axle weight fraction, and may be expressed as a percent- age. The front and rear weight fractions are given by:
100% 100% R R R R F F F F W W f f W W W W f f W W = × = × (7-4) Figure 7-3
Road Road LoadsLoads
On level ground, the wheel loads W F and W R are always directed upward, op-
posing gravity. In this case, they are perpendicular to the road, and are also known as the normal forces on the tires. By definition, W F and W R are taken to be normal
to the road surface, or acting in the z-direction relative to the vehicle coordinate system. In this case, the sum of the wheel loads is less than the total weight of the vehicle, as will be shown.
Example 7-1 Example 7-1
A bicycle with a 104 cm wheelbase is weighed with a rider by placing scales under each wheel. The rear wheel scale reads 491 N, while the scale under the front wheel reads 415 N. Find the total weight of the bicycle and rider and determine the horizontal location of the center of gravity.
Solution:
The total weight is found by W = W R + W F = 491 + 415 = 906 N. The hori-
zontal location of the center of gravity can be found solving Equation 7-2 for b:
415 1.04 .476 m 906 F W L b W × = = =
The center of gravity is 0.476 meters forward of the rear wheel axle. The front and rear static weight fractions are useful indicators of the static weight distribution between the front and rear axles. The weight fractions for this example are given by:
491 .542 906 415 .458 906 R R R R F F F F W W f f W W W W f f W W = = = = = =
This shows that 54% of the weight of the vehicle is on the rear axle, and 46% is on the front axle.
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Static Loads on a Grade Static Loads on a Grade
The weight fraction shifts when a vehicle is sitting on a slope. The analysis is similar except that longitudinal as well as vertical loads occur at the wheels. The vertical wheel forces are reduced and the horizontal force is produced by brakes or by applying pressure to the pedals so as to prevent rolling backwards. Slopes on roadways are usually specified as percent grade. Thus a road that rises 50 m over 1 km has a moderately steep grade of 0.05 or 5%. The inclination angle of the road is the arctangent of the grade.
q
q = tan–1(G) (7-5)
Typically, road grades are fairly small. Grades exceeding 12% are rare, even in mountainous regions. For small grades, the inclination angle and its tangent are approximately equal, and with little loss of accuracy one may assume qq = G (whereqq is measured in radians). This simplification incurs less than 0.5% error for a 12% grade and only 1.3% error for an extremely steep 20% grade.
The forces are shown in Figure 7-4. The longitudinal force shown on the rear wheel could occur at either wheel, depending on which brakes are applied. This analysis is not affected by which wheel is used, although it may be important in other contexts.
The wheel loads W f and W r are found by summing moments about the rear tire
patch.
(W cos(qq))b – (W sin(qq))h = W f L
Figure 7-4
Road Road LoadsLoads
This equation can be solved for the front wheel load W f as before. Likewise, the
reaction at the rear wheel patch can be found by summing forces about the front tire patch. The resulting equations are:
( c os( ) sin( )) (( )cos( ) sin( )) f r W b h W L W L b h W L θ − θ = − θ + θ = (7.6)
Equations 7-6 are the general equation for calculating vertical loads on the front and rear axles. If the grade is zero, they reduce to Equations 7-2 and 7-3. However, they can be simplified by using small angle approximations sin(qq)@qq @
G, (with the angle expressed in radians), and cos(qq)@ 1.
( ) (( ) ) f r W b hG W L W L b hG W L − = − + = (7.7)
Once the front and rear wheel loads are known, the weight fraction can be calculated using Equation 7-4.
Going up a hill increases the load on the rear wheel and reduces the load on the front wheel. Conversely, going down a hill increases the load on the front wheel. Vehicles with high centers of gravity will experience more of a weight shift. An upright road bike climbing a 10% grade may experience a 20% to 25% decrease in the front wheel load. (This assumes the rider stays in the saddle. Standing on the pedals can shift the CG forward significantly during hill climbing.) A low-racer on the same grade will probably experience less than 10% decrease in front wheel load compared with that on level ground.
Figure 7-5 shows the fractional reduction in front axle load as a function of grade and the dimensionless ratio of the height to position from rear tire patch of the vehicle center of gravity (h/b). Typical values of h/b for upright bicycles, recumbent, and very low, laid-back recumbents are indicated.
When riding up a steep hill with poor traction conditions, there is a possibil- ity of losing steering control due to the loss of load on the front wheel. This is particularly true with vehicles that have lightly loaded front axles, such as many long-wheelbase recumbents. Long-wheelbase delta tricycles that drive a single rear wheel are likely candidates for this problem, due to the low front weight
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fraction coupled with the turning moment produced by the single drive wheel under heavy torque conditions.
More severe problems arise when riding downhill. Braking also shifts the ver- tical load to the front wheel. Braking on a downhill slope can reduce the load on the rear wheel to zero, potentially resulting in a dangerous accident with the rider pitching over the handlebars. A high center of gravity exacerbates this problem, which is more common with upright bicycles than recumbents.
Steady Motion Road Loads Steady Motion Road Loads
A vehicle moving in a straight line at constant velocity is said to have steady motion. That is, all vehicle acceleration terms are zero. There are four groups of forces acting on a vehicle in steady motion—propulsive forces, retarding forces, weight and normal forces. These forces are shown in Figure 7-1. Pro- pulsive forces provide the forward thrust on the vehicle. For vehicles driven by wheels, this is a tractive force, which acts at the tire patch of each driving wheel. Retarding forces include all forces resisting forward motion of the vehicle, such as aerodynamic drag, rolling resistance, and braking forces. Weight can be con- sidered to act downward through the center of gravity of the vehicle. The normal reaction forces act through the tire contact patches in a direction normal to the road surface. As any cyclist has experienced, if the vehicle is moving up or down a grade, a component of the weight becomes a significant retarding or propulsive force. - 0 . 4 5 - 0 . 4 - 0 . 3 5 - 0 .3 5 - 0 . 3 - 0 .3 - 0 . 2 5 - 0 . 2 5 - 0 . 2 - 0 . 2 - 0 .2 - 0 . 1 5 - 0 .1 5 - 0 .1 5 - 0 . 1 - 0 . 1 - 0 .1 - 0 .1 - 0 . 0 5 - 0 . 0 5 - 0 .0 5 GRADE, %
FRACTIONAL CHANGE IN FRONT TIRE LOAD WITH GRADE
UPRIGHT RECUMBENT LOW RACER 0 2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 O G h e i g h t t o l o c a t i o n r a t i o h / b Figure 7-5
Road Road LoadsLoads
For a vehicle traveling at constant speed, the propulsive forces must equal the retarding forces. The tractive force is a function of the driving wheel torque T and drive wheel rolling diameter d, and is given by:
2 x T F d = (7-8)
At typical cruising speeds for most vehicles, aerodynamic drag is the most im- portant retarding force. For example, a 70-kg man riding an unfaired long-wheel base recumbent bicycle over asphalt would expend a total of 158 Watts in order to maintain 8.4 m/s on level ground without wind. He would expend 117 Watts to overcome aerodynamic drag, but only 35 Watts to overcome rolling resistance. If our rider switched to an upright touring bike, the power required to overcome drag would increase to 165 Watts due to the greater drag force.
The drag force depends strongly on the speed of the vehicle, increasing with the square of the speed. It also depends on the shape and size of the vehicle.
2
2
AERO D
F =ρC AV (7-9)
Whereρ is air density, A is the frontal or surface area of the vehicle, andC D is
known as the drag coefficient. The drag coefficient takes into account the shape and surface texture of the vehicle. V is the air velocity, which is equal to the vehi- cle speed if there is no wind. The drag force acts through the aerodynamic center of pressure which is not easily computed. Aerodynamic forces can also produce vertical forces, which can be significant at high speeds. If the vertical force is directed up, it is termed lift; otherwise it is called a downforce. Drag and drag coefficients are discussed in detail in Chapter 9.
There is some resistance to motion as tires roll over pavement. On very smooth surfaces, tires deform to support the load, creating a flat patch of contact between the tire and the flat road. Tires also deform as they roll over irregularities in the road surface. In addition, if the surface is soft, such as sand or soft dirt, the sur- face itself is deformed. The energy dissipated in these deformations gives rise to a retarding force known as rolling resistance. Rolling resistance is directly propor- tional to the vertical load on the wheel, but also depends on wheel diameter, tire construction and tread, inflation pressure, and the roughness and properties of the road surface. Larger diameter wheels can roll over surface irregularities eas- ier, and thus have lower rolling resistance. Thin-walled tires with slick tread have significantly less rolling resistance than do knobby tires designed for off-road
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traction. The tread, particularly knobby tread patterns, deforms significantly, in- creasing the hysteretic energy losses and rolling resistance. Rolling resistance is given by:
F RR = WC RR (7-10)
WhereW is the total weight of the vehicle and C RR is the coefficient of roll-
ing resistance. It can range from less than .002 for high-pressure racing tires on smooth track surfaces to almost .02 for wide, low-pressure tires on rough roads. Typical road bike tires on typical asphalt may have a C RR value around .005 or
so. Equation 7-10 is applicable to either a single wheel or the entire vehicle. If all wheels are the same size, use the same tires, and are inflated to the same pressure, C RR will be the same for all wheels. In that case, if W is the total vehicle
weight, the calculated force applies to the whole vehicle; if it is the vertical load on one wheel, the force applies to that wheel only. For vehicles with different size wheels or different tires, the rolling resistance coefficient C RR will vary between
wheels. In that case, the rolling resistance force is the sum of the product of the vertical force on a wheel with the coefficient of rolling resistance for that wheel.
, ,
RR z i RR i i
F =
∑
F C (7-11)In practice Equation 7-10 can be used for a vehicle with differing tire sizes/ types as long asC RR is determined for the entire vehicle. Generally, Equation 7-11
is only used if it is known that the coefficients of rolling resistance vary signifi- cantly between wheels.
Braking forces F B are retarding forces that act at the tire contact patch in a
direction parallel with the road surface. As expected, they can be considerably greater in magnitude than any of the other retarding forces. Braking forces are treated in more detail later in this chapter.
Problems involving steady-motion road loads frequently involve determination of the tractive force required to maintain steady speed on level roads or climbing hills. For steep descents, the braking force required to maintain constant speed can also be found. Usually, a cyclist is not braking and pedaling simultaneously, so either F x or F B is taken to be zero. For vehicles with steady motion on level
ground, the equilibrium equation for forces in the longitudinal direction is F x – F AERO – F = 0 RR – F B (7-12)
Road Road LoadsLoads
On a grade, an additional term is required in Equation 7-12 to account for the grav- itational force pushing the vehicle downhill. The complete term is mg sin(tan–1 G),
but the small angle approximation mgG again incurs less than a 2% error at a grade of 20%. Thus, for steady motion up or down a hill Equation 7-12 becomes:
F x – F AERO – F RR – F B – mgG = 0 (7-13)
Equation 7-13 describes the longitudinal forces acting on a vehicle traveling at constant speed. It is an important equation for modeling vehicle loads acting on a vehicle, and is the basis for the power-speed models developed in Chapter 8. In conjunction with Equations 7-6 both longitudinal and lateral loads can be analyzed. Lateral forces are developed during cornering, and are the subject of the next section.
Basic Loads in a Steady Turn Basic Loads in a Steady Turn
Lateral forces act on a vehicle to produce and maintain a turn. In accordance with Newton’s first law, a body moving in a curvilinear path must be acted on by an external force. For a turning vehicle, the tires sustain cornering forces that cause the vehicle to turn. These forces depend on tire properties and angles, and are discussed in detail in Chapter 11. A steady turn is one in which neither the vehicle speed nor the turn radius changes. During a steady turn the tire forces produce the lateral acceleration required for curvilinear motion. This force is di- rected toward the inside of the turn, and is opposed by centrifugal force, which is just the inertial force –may. The lateral acceleration ay is related to the vehicle
speed and turn radius by:
2 y V a R = (7-14)
Where V is the speed of the vehicle and R is the turn radius, measured from the turn center to the center of gravity of the vehicle. The total side load on the