virtual post rig

VIRTUAL POST RIG Mathematics and computing engineering

Development and implementation of a suspension

analysis system

Josep Mª Carbonell Oyonarte y Miguel Pareja Muñoz ETSEIAT-UPC

VIRTUAL POST RIG

JOSEP Mª CARBONELL OYONARTE Y MIGUEL PAREJA MUÑOZ 1

Contents 1- Introduction .......................................................................................................................... 2

1.1- Simplifications ............................................................................................................... 3

1.2- Modelization ................................................................................................................. 3

1.3- Goals .............................................................................................................................. 5

2- Program ................................................................................................................................. 6

2.1- Theoretical approach ......................................................................................................... 6

2.2- Performance Index ............................................................................................................. 7

3- Performance VS comfort ....................................................................................................... 8

3.1- Performance....................................................................................................................... 9

3.2- Comfort ............................................................................................................................ 10

4- Model validation ................................................................................................................. 11

5- Acknowledgements ............................................................................................................. 14

6- Literature ............................................................................................................................. 15

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1- Introduction

This project arises from the need to analyze and characterize the parameters and response of

a suspension system of a vehicle. Currently it is imperative to have a vision of the system and

in order to get it, there are plenty of facilities "Post Rig", a mechanism which transmits

vibrations to the vehicle and a sensor system monitors its response. These facilities have a set

of advantages and disadvantages:

Instead, what we propose is the creation of a virtual system in which, from the parameters of

the desired vehicle and the desired input, the temporary and frequency response of the

vehicle is obtained, where we can see the most important features. This improves both actual

Rig Post disadvantages discussed above:

It is very economical. Only the computational power required by the program is

needed.

It is extremely fast, being able to obtain system responses varying any parameter in a

simulation.

We obviously have the disadvantage that we are no longer working on the real car, but on a

simulation, where a number of simplifications and idealizations that we will discuss below have

been made.

Pros

• It works directly on the vehicle. Therefore, there are no modeling or simplification errors.

Cons

• It is extremely expensive, being a method available to very few companies.

• Few set up variations. During the time that last test, you can make a few changes in the system parameters because they are time consuming.

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1.1- Simplifications

As we only want to study the vertical response of the vehicle, we use the model called ¼. This

means that the system is treated as if it were only connected to the ground by one point, and

the entire mass rests on a suspension. This simplification prevents obtaining the

characterization of the longitudinal and lateral movements, outside our goal.

We do not consider the suspension geometry, it would be impossible to introduce to the

program the lots of existing different geometries.

1.2- Modelization

Model 1

For our program we have decided to rely on the most accurate and most used academically

model. It is as follows:

We see that in this case we model the tire as a spring + shock set, because from academic

studies to date is the model that has a response more similar to the actual tire.

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We have set out to assess the validity of two models more:

Model 2

In this model, also commonly used, the tire is modeled as a spring only. It is as follows:

Model 3

This is the most basic model. Hardly used in industry or in academic studies, only

educationally. It is as follows:

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1.3- Goals

Get both temporary and frequency response of a given suspension system. To study

their behavior.

To study the influence of all parameters in the response of the suspension system of a

vehicle.

To study the validity of the different possible models of suspension.

Performance Index calculation of assessing the quality of the suspension.

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2- Program

2.1- Theoretical approach We have obtained the transfer functions of the system, i.e., the relationship between the

vertical response of the suspended mass (car) or the unsprung mass (absorber) versus soil

variation laying move the car.

To do this, we extracted the dynamics equations for each of the models discussed above.

After, we have passed from the time domain to the frequency domain using Laplace

transforms and finally obtained the transfer function of the system.

Then the process followed for the model 1, displayed in Figure 1, is shown, by way of example:

𝑀𝑠�̈�𝑠 − 𝐶𝑠(𝑧�̇� − 𝑧�̇�) − 𝐾𝑠(𝑧𝑠 − 𝑧𝑢) = 0

𝑀𝑢�̈�𝑢 − 𝐶𝑠(𝑧�̇� − 𝑧�̇�) − 𝐾𝑠(𝑧𝑢 − 𝑧𝑠) + 𝐾𝑡(𝑧𝑢 − 𝑧𝑔) + 𝐶𝑡(𝑧�̇� − 𝑧�̇�) = 0

𝑍𝑠𝑀𝑠𝑠2 − 𝐶𝑠𝑠(𝑍𝑠 − 𝑍𝑢) − 𝐾𝑠(𝑍𝑠 − 𝑍𝑢) = 0

𝑍𝑢𝑀𝑢𝑠2 − 𝐶𝑠𝑠(𝑍𝑢 − 𝑍𝑠) − 𝐾𝑠(𝑍𝑢 − 𝑍𝑠) + 𝐾𝑡(𝑍𝑢 − 𝑍𝑔) + 𝐶𝑡𝑠(𝑍𝑢 − 𝑍𝑔) = 0

Unsprung mass versus ground

𝑍𝑢𝑍𝑔

(𝑠) =𝐶𝑠𝐶𝑡𝑠

2 + (𝐾𝑡𝐶𝑠 + 𝐾𝑠𝐶𝑡)𝑠 + 𝐾𝑡𝐾𝑠

𝑀𝑢𝑀𝑠𝑠4 + (𝑀𝑠𝐶𝑡 + 𝐶𝑠(𝑀𝑢 +𝑀𝑠))𝑠

3 + (𝐾𝑠(𝑀𝑢 +𝑀𝑠) + 𝐾𝑡𝑀𝑠 + 𝐶𝑠𝐶𝑡)𝑠2 + (𝐾𝑡𝐶𝑠 +𝐾𝑠𝐶𝑡)𝑠 + 𝐾𝑡𝐾𝑠

Sprung mass versus ground

𝑍𝑠

𝑍𝑔(𝑠) =

𝑀𝑠𝐶𝑡𝑠3+(𝐶𝑡𝐶𝑠+𝐾𝑡𝑀𝑠)𝑠

2+(𝐾𝑡𝐶𝑠+𝐾𝑠𝐶𝑡)𝑠+𝐾𝑡𝐾𝑠

𝑀𝑢𝑀𝑠𝑠4+(𝑀𝑠𝐶𝑡+𝐶𝑠(𝑀𝑢+𝑀𝑠))𝑠3+(𝐾𝑠(𝑀𝑢+𝑀𝑠)+𝐾𝑡𝑀𝑠+𝐶𝑠𝐶𝑡)𝑠2+(𝐾𝑡𝐶𝑠+𝐾𝑠𝐶𝑡)𝑠+𝐾𝑡𝐾𝑠

Dynamics equations

(time domain)

Dynamics equations

(frequency domain,

because s=jw)

Laplace transforms

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Overshoot

This value is the maximum value of the response to a step input less the end value of the

stabilized response, divided by the final value.

Bandwidth

This value is the frequency at which the response has a gain of about -3dB regarding the gain

at very low frequencies.

Steady state error

This value is the difference between the response of the system and the entrance to it, once

passed the transition period of stabilization.

.

2.2- Performance Index

We defined this value as the sum of distances from the response of the unsprung mass to an

input impulse regarding ground divided by the number of points assessed.

This parameter provides an assessment of the quality of the system, being higher as smaller

this value. In this way, we can compare two completely different suspensions just by looking at

this value.

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3- Performance VS comfort

One of the main goals of our project is to study the effect of the variables in the system

response. First, we will analyze how the value of the constants affects only in the response of

unsprung mass, and then do the same analysis for the suspended mass.

To do the analysis, we made a study with the program with the following ranges:

Kt

(N/m) Ks

(N/m) Ct

(Ns/m) Cs

(Ns/m) Ms (Kg)

Minimum 125000 27000 100 2000 250

Maximum 131000 28000 115 2115 265

The results of the simulation are as follows:

Case Overshoot Kt

(N/m) Ks

(N/m) Ct

(Ns/m) Cs

(Ns/m) Ms (Kg) Mu (Kg)

Unsprung maxim overshoot model 1 52.401 1.25e+05 27700 100 2000 260 40

Unsprung minim overshoot model 1 49.529 1.3e+05 27000 110 2100 250 40

Sprung maxim overshoot model 1 15.479 1.3e+05 27000 100 2000 250 40

Sprung minim overshoot model 1 13.112 1.25e+05 27700 110 2100 260 40

Unsprung maxim overshoot model 2 52.534 1.25e+05 27700 110 2000 260 40

Unsprung minim overshoot model 2 49.652 1.3e+05 27000 110 2100 250 40

Sprung maxim overshoot model 2 17.249 1.3e+05 27000 110 2000 250 40

Sprung minim overshoot model 2 14.945 1.25e+05 27700 110 2100 260 40

Sprung maxim overshoot model 3 38.467 0 27700 0 2000 260 0

Sprung minim overshoot model 3 36.038 0 27000 0 2100 260 0

Case Bandwid

th (rad/s)

Kt (N/m)

Ks (N/m)

Ct (Ns/m)

Cs (Ns/m)

Ms (Kg)

Mu (Kg)

Unsprung maxim bandwidth model 1 17.456 1.3e+05 27700 110 2100 260 40

Unsprung minim bandwidth model 1 17.02 1.25e+05 27000 110 2000 260 40

Sprung maxim bandwidth model 1 73.71 1.3e+05 27700 110 2100 260 40

Sprung minim bandwidth model 1 71.488 1.25e+05 27000 110 2100 250 40

Unsprung maxim bandwidth model 2 17.516 1.3e+05 27700 110 2100 260 40

Unsprung minim bandwidth model 2 17.076 1.25e+05 27000 110 2000 260 40

Sprung maxim bandwidth model 2 75.257 1.3e+05 27700 110 2100 260 40

Sprung minim bandwidth model 2 73.065 1.25e+05 27000 110 2100 250 40

Sprung maxim bandwidth model 3 17.72 0 27700 0 2100 260 0

Sprung minim bandwidth model 3 17.381 0 27000 0 2000 260 0

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The optimum values for the system to be analyzed are small Overshoot and large Bandwidth.

This will produce a low amplitude response and quickly eliminate oscillation.

If we want to maximize vehicle performance, we must optimize the response of the unsprung

mass, because we are interested in the tire contact as long as possible. But if we study the

vehicle's comfort, we optimize the response of the suspended mass, seeking to obtain the

smallest possible oscillation.

3.1- Performance For unsprung mass, we can evaluate the Overshoot decreases when:

Kt, that is, hardness of the tire increases.

Ks, that is, hardness of the spring decreases.

Ct

Cs

Ms

As for the Bandwidth, this increases when:

Kt

Ks

Ct

So in terms of optimize the performance, we use hard tires, shock absorbers with a great Cs,

and reduce to a minimum the suspended mass. Regarding the spring rate, high hardness

observed that provoke a larger Overshoot, but also increase the speed of dissipation. We

should study each case.

We present the graphs of the response of the unsprung mass of model 1 obtained with the

program:

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3.2- Comfort

When we look at the suspended mass, we note that to decrease the Overshoot must:

Kt

Ks

Ct

Cs

Ms

And to increase the Bandwidth we must:

Kt

Ks

Ct

Ms

We found that there is difference between the response of the sprung mass and unsprung. If

we want to optimize comfort, we assemble a stiffer springs, dampers with high Cs, increase

suspended mass, and particularly assess the choice between soft and hard tires.

Finally, to get a clearer idea of the system response, we present the graphs of the suspended

mass obtained from Model 1 with the program:

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4- Model validation

At the beginning of our program, we have taken as reference the most exhaustive ¼ vehicle

model. This is because most accurate studies use it, but we set as one of the objectives to

assess the accuracy of the assumptions and simplifications done in the other two models

studied. To do this, our program takes all graphics for each model, so you can make a visual

assessment of the differences in the response of each, and also calculate the error in the

values of maximum Overshoot and minimum Bandwidth (as they are the most damaging for

the system) committed by models 2 and 3. Presenting the values obtained in the previous

simulation:

Model Bandwidth Error Overshoot Error

Model 2 2.2054 11.44

Model 3 75.686 148.51

As we can see, the model 2 is acceptable in calculating the Bandwidth, because it has a

2.2054% error regarding the value obtained with the model 1, but differs much more on the

maximum Overshoot.

Model 3 presents a big mistake in calculating the Bandwidth minimum, but it is absolutely

useless to calculate the Overshoot.

Here are the graphs obtained for each model, to make a visual recognition of differences in the

response:

UNSPRUNG MASS

SPRUNG MASS

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As a conclusion, we can safely say that the model 3 is totally useless to make a minimally

serious study of the system, since it has some mistakes too high.

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As for the model 2, we see that the differences are not so great. We should make an

assessment in each individual case whether it is advantageous to win the precision you get

with Model 1, or if that is more advantageous to sacrifice precision in order to get a lower

complexity and computation time by eliminating the tire damper, using model 2.

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5- Acknowledgements

We want to finish the project by thanking all the help provided by Prof. Joseba Quevedo. It has

always been open to any consultation, encouraging us to get a job where you meet our goals

one step beyond our comfort zone.

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6- Literature

Estudio del comportamiento dinámico de un vehículo utilizando la herramienta

simmechanics de Matlab; Iván Mula, Universidad Carlos III de Madrid.

Comandos de Matlab útiles para la asignatura de Control; Analía Pérez Hidalgo;

Universidad Nacional de San Juan.

Los Sistemas de Suspensión Activa y Semiactiva: Una Revisión; Jorge Hurel Ezeta,

Anthony Mandow, Alfonso García Cerezo; Revista Iberoamericana de Automática e

Informática industrial.

Aerodinámica y Aero Post Rig aplicados al Diseño de Coches de Competicion; Timoteo

Briet

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