Constitución y reforma de la Constitución

None

Negative

Positive

Always Negative

Always Positive

Always Positive

Positive

Majority of the Displacements

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Always Negative

Negative

Majority of the Displacements

One of the most difficult parts of designing a suspension system is

compromising. There is no optimum suspension for all conditions; therefore, for every improvement there is a sacrifice. The key is to decide what is most

important for your particular application. In our case, we had to account for a race on a smooth track that contains many tight turns but can be subject to a variety of weather conditions.

To fulfill our goal of maintaining the largest accelerations possible, we examined the many components of a suspension design (the definitions of these terms are conveniently defined during the discussion). The first heavily debated design component examined was the roll axis, a line that connects the front and rear roll centers, around which the car body rotates when lateral forces are applied.

The roll center is defined as the effective center that the body will appear to rotate about. For roll centers with small radii, one could image a box suspended by two short strings at two corners suspended as something of a pendulum. The longer the string, the larger the pendulum, and the more minute the angular displacements the box will achieve. As shown below, the body roll Theta 1 is greater than Theta 2 with a longer roll center located below ground level.

TABLE 1

Geometries of Double Wishbone SLA Suspension

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A small amount of negative camber is desirable in a turn, as determined through a tire manufacturer‘s data.

However, the geometry must be such that the camber is zero for straight driving. If camber exists even when the car is not turning, the tire patch area is reduced and maximum possible traction is not attained. This also leads to uneven tire wear results. The desirable aspect of camber is that it can be used to increase tire patch area when the vehicle experiences body roll. To achieve this effect, camber must be positive when wheel displacement is negative (wheel droop), and negative when wheel displacement is positive (jounce).

Below is a graph for a formula car suspension system. It achieves slightly more than a degree of camber per inch of displacement. Comparing this slope to the tire data in, the results support that an increase in lateral force can be achieved.

Fig 24

Optimised Roll Centre Location

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Fig 25

Tire Data

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Many complications are added to the suspension geometry where the

steering control arms and rack are located. One of the biggest effects it can cause is ―toe-in/out‖, commonly known as ―bump steer‖. ―Toe-in‖ is when the front of the tires angle in towards each other, and ―toe-out‖ is when they angle away from each other. It is undesirable to have the tires independently steer the vehicle when the vehicle hits bump. This characteristic complicates tuning the vehicle by adding responses that are unpredictable to the driver. Both kinds of toe are a result of the position of the steering linkages. Since we are using an existing steering rack, its position has several constraints. To steer the vehicle, the control arms must be a distance from the axis about which the tire turns specifically the king pin

inclination (KPI) to induce a moment, thus turning the car. As the figure below demonstrates, the kingpin inclination affects scrub radius, which is pre-determined by re-using the vertical uprights.

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Bump steer is a very difficult characteristic to accommodate. In a majority of geometries tested, the amount of bump steer cannot be zero for all wheel

displacements. Therefore, we designed our suspension system with a minimal amount of Toe-In/Toe-Out by placing the rack where the pivots for inner and outer tie rod match the control arm pivot axis. In standard formula car designs, there has been as much as 3.5 degrees of toe-out over a 2-inch wheel displacement. Our results were a considerable improvement since our design has less than a 0.4-degree angle over 4-inch wheel displacement. This amount will not noticeably affect the handling of the vehicle.

Another aspect that must be considered is the caster angle. The combination of the caster angle and kingpin inclination greatly affects the handling of the car.

Both are very important since they influence the steering forces during lateral acceleration and the self-centering effect of the steering. As with Toe, it is desirable to minimize both the caster angle and the kingpin angle for all wheel displacements. The combination of the two has a large effect on the rate of camber change during wheel displacement.

The final design uses pneumatic trail to provide steering center effect at higher speeds. Pneumatic trail differs from the mechanical trail defined by the KPI and caster angle by specific tire characteristics. Pneumatic trail is a result of the tire patch area shape. The patch area roughly forms a triangle, thus providing a wedge effect with the ground and provides a horizontal centering force. Both types of trails act as weather vanes to the steering wheel but have varying effects at over a given range of speed. Mechanical trail is dominant at low speeds while pneumatic

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trail at high speeds. Skid warning is also maintained by minimizing mechanical trail, and since the effect of pneumatic trail is non-linear with vehicle speed, the driver will be able to sense when there is a significant decrease in pneumatic trail.

This provides an important source of driver feedback at higher speeds, and the vehicle will exhibit under steer and feel ―loose‖.

 Design Overview Static weight

The load transfer calculations use the following parameters to model the forces generated. The static forces are calculated using a driver weight of 80 Kg, vehicle weight of 240Kg estimated from existing vehicles, front-to-rear weight distribution

-3

Graph 4

Caster Angle v/s Wheel Displacement

Graph 5

Kingpin Angle v/s Wheel Displacement

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50/50, and left-to-right weight distribution 50/50. The height of the CG is estimated from the average of the moment of inertias of major components.

Height of CG =11” Wheel Base (WB) = 65”

Weight Driver (WD) = 80 kg Front Track – 50”

Weight Vehicle (WV) = 240 kg Rear Track – 51.6”

Total Vehicle Weight (TVW) = WD + WV = 320

Front Static Roll Center = 4

Front Bias = 0.5 Rear Static Roll Center = 6 Rear Bias = 0.5 Positive Acceleration = 1.5 Left Bias = 0.5 Negative Acceleration = 1.5 Right Bias - 0.5 Lateral Acceleration = 1.5

Lateral load transfer due to lateral acceleration

Table 2

Acceleration Data used for Calculations

Fig 27

Plot of Relevant Forces

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The lateral forces generated act through the center of gravity and are summed as a torque, or moment, to determine the vertical force on a tire. The moments are summed about the roll center axis when roll centers are determined at a static height. The two moments are summed to find Ftire and then split front to rear by multiplying by the bias ratio. The lateral acceleration has been set to a high value of 1.5 times the force of gravity, 1.5 g‘s. This number is used as a safety factor since the car will not encounter more than 1.2 g‘s of force.

Front Rear

L₁ = L_cg – L_rc,f = 6 L₃ = L_cg – L_rc,r = 4 L₅ = 11

L2 = front track/2 = 25 L4 = rear track/2 = 25.8 L6 = WB*front bias = 32.5 F₁ = a₁*(F_z,sfl + F_z,sfr) =521.25 F₂ = a₁*(F_z,srl + F_z,srr) = 521.25 L₇ = WB*rear bias = 32.5

Ft = (F1*L1 + F2*L3)/(L2 + L4) = 102.6082677

Fz, Lateral Acceleration Loads, steady state Front, 1 tire = 51.30413

Rear, 1 tire = 51.30413

Longitudinal weight transfer due to negative acceleration

Table 3

Vertical Tire Force Calculation

Table 4

Lateral Acceleration Loads

Fig 28

Tire Force Schematic

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The rear-to-front weight transfer due to braking is a sum of moments about the y-axis and is defined by the intersection of the x-coordinate of the CG projected on the ground.

Fz, Longitudinal weight transfer, negative acceleration, steady state Front = 264.6346 Front Left = 132.3173 Front Right = 132.3173

Rear = -264.635 Rear Left = -132.317 Rear Right = -132.317

Maximum loads achieved

The maximum vertical loads that could be reached correspond to the combined forces of negative and lateral accelerations with the static weight of the vehicle.

Fz, Maximum achievable loads; lateral + negative accelerations static + lateral + negative

Front Left = 357.3714 Front Right = 357.3714 Rear Left = 92.73683 Rear Right = 92.73683

Maximum Tractive Forces

The traction generated by a tire (Fx,y) is a function of vertical force exerted on a tire and the coefficient of friction () between the tire and ground. The coefficient of friction is a function of many variables including velocity, temperature, and tire wear. Wet, dry, and/or sandy surface conditions also serve as variables. The coefficient is estimated to be high at 1.5 so that the x and y components of the forces developed are high, since these forces will be used when determining component materials and dimensions as an added margin of safety.

Table 5

Longitudinal Weight Transfer

Table 6

Maximum Achievable Loads

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Horizontal tire force, Fx,y (lbs)

Fy: Cornering = u*(static + lateral acceleration loads)

Front Left = 337.5812 Front Right = 337.5812 Rear Left = 337.5812 Rear Right = 337.5812 Fx,y: Cornering & braking=u*(static + lateral + negative acceleration loads)

Front Left = 536.0572 Front Right = 536.05716 Rear Left = 139.1052 Rear Right = 139.10524

Fx: Braking = u*(static + negative acceleration loads)

Front Left = 459.101 Front Right = 459.10096 Rear Left = 62.14904 Rear Right = 62.149038

Factor of Safety Development

To determine the factor of safety we will assume worst-case scenarios for the loading of each component. Specifically the vehicle is under hard braking and hits a pothole or cone. This example exhibits a realistic case for a high performance

Fig 29

Schematic of tire with axes

Table 7

Horizontal Tire Force

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vehicle and parking lot track condition. The following is part of the analysis of the design of a front pushrod element.

The following forces are developed in the push-rod member during full spring compression and max damping setting on the Fox Racing shock:

F=n*(k*x + c*velocity)

Where, k is the spring constant, x is displacement, c is the damping coefficient, and n is the rocker ratio. The damping force is obtained through the manufacturer‘s supply of damper dyno charts and the spring rate is 39.55 N/m with maximum x displacement of two inches. To use a worst case scenario we will assume the vehicle bottoms out, that is max displacement of two inches is achieved, the damping is set at max 12 clicks closed, and the rocker ratio is three.

N F3(39.55*2409)1464.3

The pushrod is under buckling in a pin-pin configuration, thus using Bernoulli-Euler technique:

L2

P_crit EI

 (1)

Using this method with an aluminum ¾‖ diameter pushrod member 14‖ long will result in a factor of safety of four. For this worst-case scenario prediction, we are able to find a factor of safety by dividing the maximum load/static load to

essentially get a factor of safety for each member. Where the factor of safety was less than three, the member was re-designed until this criterion was met.

A-Arm Force Calculations

In order to compute the thickness necessary for each arm, the maximum forces had to be computed. After the forces were found, we used the largest one to calculate the diameter needed. A factor of safety of three was used. The relationship between the force in the arm and its area is as follows:

I

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The area can then be used to compute the needed diameter. The forces were found using standard static analysis equations. The four defining equations are as

follows:



^{Fx 0} ₍₃₎



^{Fy 0} ₍₄₎



^{Fz 0} ₍₅₎



^{Mo 0} ₍₆₎

They state that the sum of the forces in x, y, and z directions must be zero.

The last equation states that the sum of the moments about any point in the system must be equal to zero. The figure below shows the forces involved and their relationship to each other.

According to the force equations, we determined that the maximum force in any arm of the A-arms was 4009N. Our computations were completed using just a basic A-Arm with no truss braces. Therefore, when we discuss the maximum force in any of the arms, we mean the two major arms of the A-Arm. The truss design

Fig 30

Force Schematic & Truss Design

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was incorporated to cut down on the moments acting on the arm. Since that force was the maximum attained for any member, we used it for all calculations. This insured that all members would be able to handle the maximum possible force encountered. Using the maximum forces, the diameter needed for the A-arms was then calculated. The A-arms were treated as a pin-pin beam, since they are

connected with ball joints on each end. Ball joints do not act against moments, similar to the behavior of a pin.

The Bernoulli-Euler buckling criterion is given in Eq.1, where I is the moment of inertia. For a round member:

4

2 1 r I  

(7)

Using equation 2, we determined the thickness needed for each type of

material that we had considered. Also shown is the weight of an arm of the needed diameter. The results are shown in the figure below for the 14-inch arm. The results are based on a maximum force of 4009N and a factor of safety of 3.

Therefore, the force calculated for a pin-pin beam is less than 12.027 kN of force.

Material Modulus of Elasticity Diameter Weight

Aluminum 172.34 GPa 1.62 cm 2.02N

AZ91 146.14 GPa 1.82 cm 1.66N

Steel 308.167 GPa 1.242 cm 3.32N

Front Uprights

One of the critical factors we used to determine the best material was the strength to weight ratio of each material. We computed this by dividing the amount of force in the member by the weight required to hold that load. This determined that AZ91 alloy magnesium is the best of the three materials. In the end, other factors weighed into the decision of what material to use and AZ91 was not selected for the steering arms or the push rods. Further detail on this topic is

Table 8

Material Properities for a 14 inch pin-pin beam

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discussed in the steering arm and push rod sections. AZ91 was selected for the front and rear uprights as well as the A-Arms.

The front uprights are usually made of AZ91 magnesium to cut down on the weight of the vehicle. We modified the design heavily to incorporate a more adjustable steering arm. The uprights are the centerpieces of the suspension system; they transfer the forces from the tire to the A-arms. Their geometry is very important to the handling characteristics of the car.

Magnesium was used for the uprights since they are a part that is cast to the specific form that is needed. Casting was an excellent option since the uprights are not a simple shape that can be easily machined. They also can be cast very close to the exact shape needed. This leads to minimal machining which saves both cost and time.

Fig 31

Front Uprights

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Rear Uprights

Fig 33

Rear Uprights

Fig 32

Front Uprights CATIA V5 Model Static Load Test

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The rear upright must only travel within the range that the drive shafts of rear differential can handle. The suspension characteristics of the rear end were modified, but this was done in other ways. If the mounting points on the body are different a different geometrical configuration is obtained. We did this to customize the handling characteristics, as we wanted them. Magnesium was used for the same reasons discussed in the front upright section.

Rockers

The rocker design is a prototype at a design level rather than a final model.

The final model is based on the dynamic vehicle testing to optimize the wheel to shock travel ratio that can be changed from 1.3:3 to 3:1.8 in .5 increments. The design permits the use of a single two-inch travel spring and damper unit to perform in a spring rate range of 16.95N/m to 65.87N/m while retaining the required two inches of wheel travel.

This variable system is a design common to some teams and combines manufacturing time savings with material cost savings by using only one rocker to do the same work as six individually cast rockers.

Fig 34

Rear Uprights CATIA V5 Model Static Load Test

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Push rods

The pushrods are the member that transmits the vertical force of the tire to the spring/damper unit. As a result this member is subject to buckling loads in a pin-pin configuration.

If weight was the only factor we considered then we would have used AZ91 for the push rods. We ended up using aluminum push rods for a number of reasons.

A major reason is the availability of materials. Aluminum is readily available, but magnesium would have to be cast to a specific length. The availability factor also leads us to aluminum since we want components that can easily be repaired or replaced at the race. Repairs are very difficult with magnesium since it is not a weldable metal. Aluminum allowed us more versatility since we could make new arms quickly. We wanted to be able to change the rod lengths if needed to adjust the system. Aluminum is much less expensive than magnesium.

Fig 35

^Rockers

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The push rod design is based on keeping the system highly adjustable. The push rod is a rod with a left-handed thread on one side and a right-handed thread on the other. With this setup twisting one way enlarges the length of the rod and twisting the other decreases the length. A nut is tightened on each end to prevent it from adjusting while racing due to vibrations. The steering arms are designed with this same setup.

Fig 36

Rebound Damping

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Fig 37

Compression Damping

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Fig 38

^Pushrods

Fig 39

Bell Crank FEA Results

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Steering Arms

Fig 41

Steering Arms

Fig 40

Motion Ratio

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Like the pushrods, the steering arms are loaded to buckle in the pin-pin configuration as well. The force developed in this configuration is tabulated in the A-Arm force calculations section. The diameter on the rod is calculated using modulus of aluminum and required length of 12‖.

In document de la Constitución (página 34-39)