Automotive Suspension System Analysis and Simulationpeople.clarkson.edu/~estradjm/FinalProjectDNS.pdf · Automotive Suspension System Analysis and Simulation ... taking into account

Automotive Suspension System Analysis

and Simulation

Dynamical Systems, Final Project

Jenniffer Estrada _________________________

Danielle Maggio _________________________

Raymond Petosa _________________________

April 24, 2009

2

Table of Contents

Executive Summary 3

Statement of Problem and Assumptions 4

Results and Recommendations 5

Analysis 6

Discussion 7

Appendices 8-23

Appendix A – Passive Suspension 8-10

Appendix B – “Fault Mode” Suspension 11-13

Appendix C – Active Suspension 14-16

Appendix D – Block Diagram 17

Appendix E – Matlab Code 18-19

Appendix F – Free Body Diagram 20

Appendix G – Linear Circuit Model 21

Appendix H – Control Gain Values 22

Appendix I – Equations 23

3

Executive Summary

The main idea of this project was to model and simulate an automotive

suspension system containing linear electromagnetic actuators. Using this model to find

the best possible gains for the controls of the actuator to ensure comfort and handling

to the occupants in the vehicle. This problem was solved by using the equations of

motion that were able to be derived from the free body diagram (Appendix F), as well at

the circuit equations obtained by analyzing the circuit model given ( Appendix G). These

equations were input into Simulink as a block diagram (Appendix D) and simulated,

allowing us to perform the tasks asked in the project. From these simulations we

determined that an active suspension system with the control gains from Appendix H

gave us the most desirable outcome.

4

Statement of Problem and Assumptions

The main goal of this project was to simulate an automotive suspension system

containing linear electromagnetic actuators. These actuators would be able to react to

changing road conditions which would provide additional comfort and handling to the

occupants.

The project that was assigned involved analyzing and simulating a half car model

containing an active suspension system. We had to develop the equations of motion

necessary to put into a Simulink model, taking into account that the chassis of the

model had two separate degrees of freedom. We also had to find the output of the

circuit that represented the actuators, as they were part of the input to the equations of

motion for the chassis.

The following assumptions were made during our analysis of the active

suspension system: (1) any rotational degree of freedom would be of a small enough

angles and thus our equations were simplified, (2) the chassis is free to pivot about its

center of mass, (3) all motion in half car model is vertical in nature, (4) the car is moving

in a forward direction with a constant velocity of 10 m/s.

5

Results and Recommendations

From comparing the three different cases, we found that the active suspensions,

with the gains listed in Appendix H, had the most desirable results. This can be shown

by comparing the figures in Appendices A-C. By looking at Appendix C, figures 7 and 8,

which are the y direction displacements of wheels 1 and 2 respectively as the road

changes, you can see a substantial difference in the time it takes for each wheel to reach

a steady state. As shown in figure C-7, when wheel 1 reacts to step function input of the

road, the time it takes to settle out of oscillation is visibly less than the time it takes for

wheel 1 of the passive suspension system shown in A-7. Out of the 3 cases tested,

wheel 1 took the longest time to settle in the “fault mode” case, this can be seen in

figure B-7.

6

Analysis

In this project we were trying to determine how the half car model will react under

different circumstances. To begin the analysis of this model, we had to start with free body

diagrams of the elements of the system. We also needed to evaluate the circuit model

provided to solve for the output. This output from the circuit is the input of the force actuator.

From the free body diagram (Appendix F) we obtain four second order differential equations,

an additional two first order equations are derived from the circuit model (Appendix G). Using

these differential equations we developed a block diagram (Appendix D) to simulate our model.

Using the simulated model and given Matlab code (Appendix E) we were able to create the

three cases that were specified in the problem.

The first scenario that we tested was when the actuator forces were constrained to

zero, passive suspension. This resulted in a model that changed very abruptly, and lacked the

ability to settle in a desirable time period.

The second situation we tested was when just the actuator voltages were constrained to

zero, semi-active “fault mode” operation. When we ran this scenario through Simulink and

Matlab the results were of a similar nature to the passive system, but when looking at the

graphs of outputs (shown in Appendix B) the “fault mode” suspension appeared to be even

more erratic than the passive suspension system.

The last case that was tested was a well tuned actuator. To achieve this actuator we

had to toggle the K values of the controller until we reached an advantageous simulation, the

graphs of which are shown in Appendix C.

7

Discussion

After reviewing all of the data collected, we believe that the results are useful. It

becomes very visible that improvements in the suspension system make a substantial

difference in the motion of the model. For instance with the passive suspension system, the

model bounced around very abruptly, which would prove to be a very uncomfortable ride given

the road conditions that were input into the model Eqn(16) and Eqn(19). An active suspension

system has a substantial impact on a variety of things. It impacts the comfort level of the

passengers, as well as the handling ability of the car. An active suspension system also had an

impact on the car itself, for example a car takes bumps and changes in road conditions much

smoother than say a passive suspension which in the long run prevents long term wear and tear

on the car.

There were a number of assumptions that had to be made before beginning the project

regarding restraints on the motion of the model, as well as predictions on the way the model

would react to the inputs. These statements had to be made in order to simplify the model

down to something that could actually be analyzed in the time given. The assumptions that had

to be made definitely limited the accuracy of the results if compared to an actual car, but still

provided sufficient fluctuation to collect useful data.

8

Appendix A-Passive Suspension

Figure 1 - Yb1 Figure 2 – Yb2

Figure 3 – Yc Figure 4 – theta

Figure 5 - Yr1 Figure 6 – Yr2

(1) - Shows the deflection of the front end of the chassis with respect to Yb10 as the road conditions vary.

(2) - Shows the deflection of the back end of the chassis with respect to Yb20 as the road conditions vary.

(3) - Shows the deflection of the center of mass of the chassis with respect to Yc0 as the road conditions vary.

(4) - Shows the rotation of the chassis with respect to θc0 as the road conditions vary.

(5) - Shows the road at position one with respect to time.

(6) - Shows the road at position two with respect to time.

9

Appendix A-Passive Suspension continued

Figure 7 - Yw1 Figure 8 – Yw2

Figure 9 -Fe1 Figure 10 - Fe2

Figure 11 -I1 Figure 12 - I2

(7) - Shows the deflection of Yw1 with respect to Yw10 as the road conditions vary.


(9) - Shows the force of actuator 1.


(11) - Shows the current going to actuator 1.


10

Appendix A-Passive Suspension continued

Figure 13 - ei 1 Figure 14 - ei 2

Figure 15 - Ef1 Figure 16 - Ef2

Figure 17 - Ey1 Figure 18 - Ey2

(13) - Shows the voltage that is associated with actuator 1.


(15) - Shows control error for actuator 1.




11

Appendix B- “Fault Mode” Suspension










12

Appendix B- “Fault Mode” Suspension continued










13

Appendix B- “Fault Mode” Suspension continued










14

Appendix C-Active Suspension










15

Appendix C-Active Suspension continued










16

Appendix C-Active Suspension continued










17

Appendix D- Block Diagram

18

Appendix E- Sample Matlab Code

clc; clear;

alpha= 0; Kp=17000; Kd=50; Ki=.1; Ke=1; sim('DNS_Final_Project')

% Car Animation Script % This Matlab script is used in conjunction with your Simulink model to % produce an animation of the half-car body as it travels down the road % profile defined in the EE324, Spring 2009 project description. % % The script assumes your Simulink model has created the following vectors % related to the motion of the car half-body model: % yc = vertical position of the car body center of mass (CoM) % theta = angular position of the car body % yw1 = vertical position of the front wheel % yw2 = vertical position of the rear wheel % yr1 = vertical position of the road under the front wheel % yr2 = vertical position of the road under the rear wheel

% Paramter Setup for Car Animation xc=-1;yc0=1.25;dyc=0.25; D1=1;D2=2; % Define car body, e.g., center of mass,

height, length dxw=1;yw0=.25;dyw=0.1; % Define wheels rc1=sqrt(D1^2+dyc^2); ac1t=atan2(dyc,D1); ac1b=atan2(-dyc,D1); % Define front

of car in polar coordinates rc2=sqrt(D2^2+dyc^2); ac2t=atan2(dyc,-D2); ac2b=atan2(-dyc,-D2); % Define

rear of car in polar coordinates

% Clear the workshop screen clc;

% Get input to on status of active suspension control: off (0) or on (1) Active=input('Is active suspension control on (0=No, 1=Yes)? ');

% Enter Loop to Create Car Animation for i = 1:length(yc) figure(1);clf(1)

% Draw Car Body x=[xc+rc1*cos(ac1t+theta(i)), xc+rc1*cos(ac1b+theta(i)),

xc+rc2*cos(ac2b+theta(i)), xc+rc2*cos(ac2t+theta(i))]; y=[yc0+yc(i)+rc1*sin(ac1t+theta(i)), yc0+yc(i)+rc1*sin(ac1b+theta(i)),

yc0+yc(i)+rc2*sin(ac2b+theta(i)), yc0+yc(i)+rc2*sin(ac2t+theta(i))]; patch(x,y,'r'); hold on;

plot(xc,yc0+yc(i),'xk','MarkerSize',10,'linewidth',2);plot(xc,yc0+yc(i),'ok',

'MarkerSize',10,'linewidth',2)

% Draw Front Wheel

19

Appendix E- Sample Matlab Code continued

x=[xc+D1, xc+D1, xc+D1-dxw, xc+D1-dxw]; y=[yw0+yw1(i)+dyw, yw0+yw1(i)-dyw, yw0+yw1(i)-dyw, yw0+yw1(i)+dyw]; patch(x,y,'r')

% Draw Rear Wheel x=[xc-D2, xc-D2, xc-D2+dxw, xc-D2+dxw]; y=[yw0+yw2(i)+dyw, yw0+yw2(i)-dyw, yw0+yw2(i)-dyw, yw0+yw2(i)+dyw]; patch(x,y,'r')

% Draw Passive Suspension Connecting Body to Wheels plot([xc+D1-dxw/2, xc+D1-dxw/2],[yw0+dyw+yw1(i), yc0+yc(i)-(D1-

dxw/2)*tan(ac1t-theta(i))],'g-','MarkerSize',5,'linewidth',12); plot([xc-D2+dxw/2, xc-D2+dxw/2],[yw0+dyw+yw2(i), yc0+yc(i)+(D2-

dxw/2)*tan(ac2t-theta(i))],'g-','MarkerSize',5,'linewidth',12);

if Active % Draw Active Suspension Connecting Body to Wheels plot([xc+D1-dxw/2, xc+D1-dxw/2],[yw0+dyw+yw1(i), yc0+yc(i)-(D1-

dxw/2)*tan(ac1t-theta(i))],'b-','MarkerSize',5,'linewidth',5); plot([xc-D2+dxw/2, xc-D2+dxw/2],[yw0+dyw+yw2(i), yc0+yc(i)+(D2-

dxw/2)*tan(ac2t-theta(i))],'b-','MarkerSize',5,'linewidth',5); end

% Draw Passive Suspension Connecting Wheel to Road plot([xc+D1-dxw/2, xc+D1-dxw/2],[yr1(i),yw0-dyw+yw1(i)],'g-

','MarkerSize',5,'linewidth',12) plot([xc-D2+dxw/2, xc-D2+dxw/2],[yr2(i),yw0-dyw+yw2(i)],'g-

','MarkerSize',5,'linewidth',12)

% Draw Road x=[-3.5, -3.5, -1.5, -1.5, 0.5, 0.5]; y=[-1, yr2(i), yr2(i), yr1(i), yr1(i), -1]; patch(x,y,'k')

% Label Figure text(-3.4,3.35,'Car Body and Wheels (Red); Car Body Center of Mass (X); Road

(Black)') text(-3.4,3.1,'Suspension Components: Passive (Green); Active (Blue)') title('Response of Half-car Model with Active Suspension System') xlabel('Horizonal Position [m]') ylabel('Vertical Position [m]')

% Scale Plot Axes set(gca,'ylim',[-0.5,3.5]) set(gca,'xlim',[-3.5,0.5]) end

20

Appendix F- Mechanical Free Body Diagram

21

Appendix G- Linear Electromagnetic Actuator Schematic

ei emi

R

L

Ii

22

Appendix H- Control Gain Values

Kp = 17000

Kd = 50

Ki = 0.1

Ke = 1

23

Appendix I – Equations

Free Body Diagram Equations

(1) F1 = fe1 – Bb1(yb1’ – yw1’) – Kb1(yb1 – yw1)

(2) F2 = fe2 – Bb2(yb2’ – yw2’) – Kb2(yb2 – yw2)

(3) Θc’’ = (1/Jc)[D1F1 – D2F2]

(4) Yc’’ = (1/Mc)[F1 + F2]

(5) Yw1’’ = (1/Mw1)[-Kwi(yw1 – yr1) – F1]

(6) Yw2’’ = (1/Mw2)[-Kw2(yw2 – yr2) – F2]

(7) Yb1 = yc + D1 Θc

(8) Yb1’ = yc’ + D1 Θc’

(9) Yb2 = yc + D2 Θc

(10) Yb2’ = yc’ + D2 Θc’

Force Level Controller and Voltage Level Controller Equations

(11) 𝑓𝑐𝑖 = 𝐾𝑝𝐸𝑦𝑖 + 𝐾𝑖 𝐸𝑦𝑖 𝜆 𝑑𝜆𝑡

0+ 𝐾𝑑

𝑑𝐸𝑦𝑖

𝑑𝑡

(12) 𝐸𝑦𝑖 = 𝑦𝑏𝑖 − 𝑦𝑤𝑖

(13) 𝑒𝑖 = 𝐾𝑒𝐸𝑓𝑖

(14) 𝐸𝑓𝑖 = 𝑓𝑐𝑖 − 𝑓𝑒𝑖

Road Equations

(15) 𝑦𝑟1 𝑡 = .25 𝒰 𝑡 − 5 − 𝒰 𝑡 − 6 − .25 𝒰 𝑡 − 15 − 𝒰 𝑡 − 16

(16) 𝑦𝑟1 𝑡 = .25 𝒰 𝑡 − 𝑡1 − 𝒰 𝑡 − 𝑡2 − .25 𝒰 𝑡 − 𝑡3 − 𝒰 𝑡 − 𝑡4

(17) 𝑡𝑖 =𝑥𝑖

𝑣0

(18) 𝑦𝑟2 𝑡 = .25 𝒰 𝑡 − 5.4 − 𝒰 𝑡 − 6.4 − .25 𝒰 𝑡 − 15.4 − 𝒰 𝑡 − 16.4

(19) 𝑦𝑟2 𝑡 = .25 𝒰 𝑡 − 𝑡1 − 𝒰 𝑡 − 𝑡2 − .25 𝒰 𝑡 − 𝑡3 − 𝒰 𝑡 − 𝑡4

(20) 𝑡𝐷 =𝑥𝐷

𝑣0

Conversions

(21) θ c = 𝜔

(22) 𝜃 = 𝜔

Circuit Equation

(23) 𝑅𝐼𝑖 + 𝐿𝑑𝐼𝑖

𝑑𝑡+ 𝛼𝑣𝑖 = 𝑒𝑖

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Automotive Suspension System Analysis and Simulationpeople.clarkson.edu/~estradjm/FinalProjectDNS.pdf · Automotive Suspension System Analysis and Simulation ... taking into account