final project suspension+designode.engin.umich.edu/me555reports/2002/555-02-09.pdf · Term Project Final Report OF OPTIMIZATION OF SUSPENSION DESIGN ... In the process of developing

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Term Project Final Report OF

OPTIMIZATION OF SUSPENSION DESIGN

Youngwon Hahn & Seungjoo Lee

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Index

Abstract 4

1. Introduction 5

2. Definition of the Optimization Problem 7

2.1 Performance Index Optimization 8

2.1.1 Toe 8

2.1.2 Definition of Design Variables and Parameters 9

2.2 Force Optimization 12


2.3 Suspension Component Size Optimization 15


3. Mathematical Model for Optimization 17


3.1.1 Objective Function 17

3.1.2 Constraints 17




3.3 Suspension Component Size Optimization 21


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4. Sensitivity Analysis for Performance Index

And Force Optimization 22



5. Procedures for Performance Index and Force Optimization 25



6. Results and Trade-off for Performance Index

And Force Optimization 26



6.3 Trade-off 29

7. Procedures and Results

for Suspension Component Size Optimization 34

7.1 Procedure 34

7.2 Results 36

8. Conclusion 37

References 38

Appendix 39

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ABSTRACT

In the process of developing a new car, the design of the platform is one of the most important parts.

Especially, the suspension design is one of the complicated ones in the design of the platform. Because the

suspension design is affected severely by other components’ space, such as engine and frame, it is difficult

to set each end point of the components in the suspension freely. (Those points are called “hard points”) For

example, These points are lower arm ball joint point, damper upper point and etc in the suspensions system.

In this project, the decision of hard points in the suspension and the simple suspension size design

were performed by using the ADAMS and NASTRAN, engineering commercial software. First, the optimal

hard points of the suspension were checked considering major suspension parameter, toe using ADAMS

software. This parameter is the most important parameter among the suspension’s indices which are

functions of the design variables and parameters and have to be measured for the handling performance.

The target criterion for this suspension index was set by other vehicles’ experimental data. Second, through

the load analysis using ADAMS, the optimal suspension hard points will be considered to minimize the

reaction forces of the damper and lower arm end points. Third, based on these two results, the trade-off

points were set. In this project, only the toe and minimum reaction force of the damper were considered for

this trade-off and optimization to simplify analysis. Fourth, after setting hard points, the dimension of the

suspension components, such as diameter of the lower arm, was optimized using NASTRAN for the

structural analysis. The shape of the suspension components was assumed as an annular rod.

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1. Introduction

The suspension design is one of the complicated ones of the total vehicle design because it is related

with other part, such as engine and frame. Even though there is not enough space inside vehicle to design

the suspension system, the engineers have to meet the need of the best handling performance and design the

suspension component to have low stress in order to have the longest fatigue life. Due to severe layout

constraints, the optimization is needed desperately. Though many researches have been done on the

decision of “hard points” [ref. 2,3], there is little research including component design that is needed for

whole design process.

In this project, the optimal hard points were set by computer simulation using ADAMS because it is

difficult to express the relationship explicitly between these points and design factors.

Hard points are the set of joint positions for the suspension components, such as wheel center, damper

upper point and etc. As seen at chart 1., two different simulations were conducted. One is for optimization

Performance Index Optimization Force Optimization

Chart 1.

Initial Hard Point Setting

Hard Point Setting

Satisfy the target region of suspension parameters?

Kinematic Analysis

Optimization

Initial Hard Point Setting

Hard Point Setting

Minimize the reaction force of Damper & Spring?

Load Analysis

Optimization

Assuming the annular rod as the component of the suspension, optimize the size of the rod to get minimum weight constrained with stress.

Change Hard Point

Change Hard Point

Subsystem 1 Subsystem 2

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of the suspension index (in this project, only toe) – let’s called as “Performance Index Optimization” and

the other is for optimization of the reaction forces of the suspension components – let’s called as “Force

Optimization”. Considering handling performance, we have to know how the suspension index (toe)

change. We also have to minimize the reaction forces of the suspension components. The optimal hard

points were set considering these two results. The reason that we divided the problem into two separate sub-

problem was that if we attack this problem as one, the problem would be very complex to handle and it

would take too much processing time due to the large number of design variables. But intrinsically these

two problems were coupled each other, because the position of the hard points affects both Performance

Index (toe) and Force. We needed the trade-off between the two design decisions. In this project, as before-

mentioned, we only focused on the toe characteristic and the minimum reaction force of the specific hard

points for simplicity of simulations. The weighting factors were applied to these two objective functions in

order to get optimal hard points.

After setting all suspension hard points, the optimal size of the suspension component was calculated.

The shape of all suspension components was assumed as an annular rod. Using the hard points and the

reaction forces from the optimization results, we was able to optimize the size of the suspension component,

such as lower arm, using NASTRAN.

Our model was the Wishbone type front suspension and the vehicle is frame type passenger car.

Normally, there are serious layout constraints for this type of cars.

Design variables and parameters in each sub-problem are presented in Chapter 2 in detail, and the brief

explanation about toe that is the most important index for handling performance in suspension system is

also given in Chapter 2. In Chapter 3, mathematical model for optimization is given including the

explanation of their meanings. Sensitivity Analysis was done for the Performance Index Optimization and

Force Optimization in Chapter 4. In Chapter 5, detail optimization procedures for both Performance Index

and Force are explained and in Chapter 6, numerical results for each optimization and the result of the trade-

off between two sub-problems are given.

In Chapter 7, the procedure and result of the size optimization of a suspension component (in this project,

we selected only a lower arm component for simplifying the problem) are presented.

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2. Definition of the Optimization Problem

Using ADAMS, we made the quarter model of Wishbone type front suspension of frame type vehicle

as the figure 1.

Figure 1

This quarter model consists of several suspension components, such as lower arm, upper arm, damper,

tie rod and etc. In the both cases, the Performance Index Optimization and Force Optimization, this quarter

model was used.

As mentioned before in the introduction, we mainly focused on the critical suspension performance

index (toe) in the Performance Index Optimization because the effect of the toe is the most important in the

vehicle handling, and also try to minimize the reaction force on the damper and lower arm bush point in the

Force Optimization. Those were our sub-problems in which the trade-off has to be conducted. After we get

the optimal positions of the hard points, we optimized the size of the suspension component. In this project,

we simplified the component as an annular rod and only lower arm among the suspension components was

considered to simplify the problem. Using the optimization routine of NASTRAN, we decided the outer and

inner diameters of the lower arm component.

Wheel

Upper Arm

Lower Arm

Tie Rod

Drag Link

Damper

Spring

Bush

Ground Patch

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2.1 Performance Index Optimization

Because as before mentioned, toe is the most important suspension parameter, only this parameter was

optimized. For those who have not accustomed to this terminology, here brief explanation was given.

2.1.1 Toe

TOE IN (+)

TOP VIEW

BICYCLE MODEL MODEL

PATH

REAL VEHICLE DIRECTION

TOE OUT (-)

Figure 2

PATH REAR VEHICLE DIRECTION

A A

TOP VIEW REAR VIEW(A-A)

VEHICLE BODY

LATERAL ACCELERATION

VEHICLE BODY

REAL VEHICLE MOTION

Figure 3. EQUIVALENT VEHICLE MOTION

VEHICLE BODY

LATERAL ACCELERATION

BUMP

REBOUND

Toe Curve

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-80 -64 -48 -32 -16 0 16 32 48 64 80

Bump Distance

Toe(

+:in

, -:o

ut)

Org_toe

BUMP REBOUND

O/S

U/S

Figure 4

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In figure 2, you can see the definition of toe and the bicycle model of the vehicle. Because of lateral

weight transfer, the bicycle model is usually used when the vehicle turns (figure 3). In real, the body is

rolling at turning moment, but in the simulation we used the concept that the suspension is rolling and the

body is fixed (figure 3,4). If you think of the bumping wheel, the toe of that wheel changes continuously as

the wheel goes up, because of the suspension linking system. In figure 4, you can see the toe characteristic

of that wheel. Usually, if the toe is negative as the wheel goes up, this vehicle is under-steer. If not, it is

over-steer. When the vehicle turned, the real radius of turning path of the vehicle is bigger than the path that

the vehicle should follow, it is called that the vehicle is under-steer. If not, over-steer. Under-steer/over-steer

is one of the most important characteristic s of the vehicle. The vehicle design of the suspension in industries

should have the cars the under-steer characteristic for safety except for special purpose car such as racing

vehicle.

The desired toe curve can be set from the other vehicle experimental data or engineer’s experience. In

this project, the target toe curve was set from the engineer’s experience (one of this project member who

had experienced in suspension design for 5 years). Compared with the simulated bump result, the hard

points were decided in order to get the target toe curve.

2.1.2 Definition of Design Variables and Parameters

Design variables and parameters for Performance Index Optimization were defined in the Figure 5

and Table 1 and 2.

A

B

C

D

E

F

G

H I

Figure 5

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Symbol Comment Unit Picture Notation

lba_x Lower arm ball joint (x coordinate value) mm

lba_y Lower arm ball joint (y coordinate value) mm

lba_z Lower arm ball joint (z coordinate value) mm

A

uba_x Upper arm ball joint (x coordinate value) mm

uba_y Upper arm ball joint (y coordinate value) mm

uba_z Upper arm ball joint (z coordinate value) mm

B

udt_x Damper upper joint (x coordinate value) mm

udt_y Damper upper joint (y coordinate value) mm

udt_z Damper upper joint (z coordinate value) mm

C

fkbt_x Damper lower joint (x coordinate value) mm

fkbt_y Damper lower joint (y coordinate value) mm

fkbt_z Damper lower joint (z coordinate value) mm

D

lbf_x Lower arm bush front joint (x coordinate value) mm

lbf_y Lower arm bush front joint (y coordinate value) mm

lbf_z Lower arm bush front joint (z coordinate value) mm

E

lbr_x Lower arm bush rear joint (x coordinate value) mm

lbr_y Lower arm bush rear joint (y coordinate value) mm

lbr_z Lower arm bush rear joint (z coordinate value) mm

F

ubf_x Upper arm bush front joint (x coordinate value) mm

ubf_y Upper arm bush front joint (y coordinate value) mm

ubf_z Upper arm bush front joint (z coordinate value) mm

G

ubr_x Upper arm bush rear joint (x coordinate value) mm

ubr_y Upper arm bush rear joint (y coordinate value) mm

ubr_z Upper arm bush rear joint (z coordinate value) mm

H

Tt Target data for desired toe deg

Ts Original data for current toe deg

Table 1

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Design Parameters Design Variables Notes

lba_x

lba_y

lba_z

uba_x

uba_y

uba_z

udt_x

udt_y

udt_z

fkbt_x

fkbt_y

fkbt_z

lbf_x

lbf_y

lbf_z

lbr_x

lbr_y

lbr_z

ubf_x

ubf_y

ubf_z

ubr_x

ubr_y

ubr_z

- All given as the Design Variables and

Parameters are only position vectors for

Performance Index Optimization.

- Unimportant parameters were omitted in

this table such as bush stiffness.

- Selection of the design variables and

parameters was based on the sensitivity

analysis (Ref. Chapter 4)

- Tt : target Data for desired Toe angle

- Ts : current toe angle which is achieved

only from simulation result

- Tt and Ts are used only in objective

function for Performance Index

Optimization(Ref. Chapter 3)

- Ts is the function of Design Parameters

and Variables.

There are several layout constraints because the space of the vehicle is limited. To make the

suspension components not being in the space of the engine and frame, each hard point can be changed only

in the limited ranges. We applied these ranges as layout constraints on each hard points position vector.(Ref.

Chapter 3)

Table 2

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2.2 Force Optimization


Obviously, the maximum reaction force occurs at the damper top point for the upward external force

and at the lower arm bush point for the external lateral external force considering the conventional

suspension structure. Minimizing this reaction force is important in the point of view of the vibration and

noise because this unwanted reaction force can excite the floor vibration.

The objective function was to minimize this reaction force at the damper top and lower arm bush point (Ref.

Chapter 3). The layout constraints were same as in the Performance Index Optimization (Ref. Chapter 3).

Design variables and parameters for Force Optimization were defined in the Figure 5 and Table 3 and 4.

Symbol Comment Unit Picture Notation

Lba_x Lower arm ball joint (x coordinate value) mm

Lba_y Lower arm ball joint (y coordinate value) mm

Lba_z Lower arm ball joint (z coordinate value) mm

A

uba_x Upper arm ball joint (x coordinate value) mm

uba_y Upper arm ball joint (y coordinate value) mm

uba_z Upper arm ball joint (z coordinate value) mm

B

Udt_x Damper upper joint (x coordinate value) mm

Udt_y Damper upper joint (y coordinate value) mm

Udt_z Damper upper joint (z coordinate value) mm

C

fkbt_x Damper lower joint (x coordinate value) mm

fkbt_y Damper lower joint (y coordinate value) mm

fkbt_z Damper lower joint (z coordinate value) mm

D

Lbf_x Lower arm bush front joint (x coordinate value) mm

Lbf_y Lower arm bush front joint (y coordinate value) mm

Lbf_z Lower arm bush front joint (z coordinate value) mm

E

Lbr_x Lower arm bush rear joint (x coordinate value) mm

Lbr_y Lower arm bush rear joint (y coordinate value) mm

Lbr_z Lower arm bush rear joint (z coordinate value) mm

F

Ubf_x Upper arm bush front joint (x coordinate value) mm G

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Ubf_y Upper arm bush front joint (y coordinate value) mm

Ubf_z Upper arm bush front joint (z coordinate value) mm

Ubr_x Upper arm bush rear joint (x coordinate value) mm

Ubr_y Upper arm bush rear joint (y coordinate va lue) mm

Ubr_z Upper arm bush rear joint (z coordinate value) mm

H

Fdu_z Damper upper point force ( z direction) N

Flbf_y Lower arm bush front point force ( y direction) N

Flbr_y Lower arm bush rear point force ( y direction) N

Table 3

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lba_x

lba_y

lba_z

uba_x

uba_y

uba_z

udt_x

udt_y

udt_z

fkbt_x

fkbt_y

fkbt_z

lbf_x

lbf_y

lbf_z

lbr_x

lbr_y

lbr_z

ubf_x

ubf_y

ubf_z

ubr_x

ubr_y

ubr_z

- All given as the Design Variables and

Parameters are only position vectors for

Performance Index Optimization.

- Unimportant parameters were omitted in

this table such as bush stiffness.

- Selection of the design variables and

parameters was based on the sensitivity

analysis (Ref. Chapter 4)

- Fdu_z : Damper upper point force

( z direction)

- Flbf_y : Lower arm bush front point force

( y direction)

- Flbr_y : Lower arm bush rear point force

( y direction)

- Fdu_z, Flbf_y, and Flbr_y are used only

in objective function for Force

Optimization to minimize these reaction

forces (Ref. Chapter 3)

- Fdu_z, Flbf_y, and Flbr_y are the function

of Design Parameters and Variables.

Table 4

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2.3 Suspension Component Size Optimization


To design the components of the suspension in the view point of meeting the allowable maximum

stress level and weight reduction, the optimization of the components’ size should be carried out. In this

project we considered only lower arm component to present simply the procedure of this optimization. Even

though only this component was considered, it could not be represented as a simple bar or beam, and was

configured in 3-D space. Therefore, our optimization procedure included the structural simulation using

MSC/NASTRAN for the stress analysis because it was difficult to get the explicit objective function in

terms of design variables. Also we assumed that this component was made of the homogeneous material and

the link shape seen as an annular rod. We minimized the total weight of the lower arm and the constraints

were allowable maximum stress and layout space.

Design variables and parameters are given in Table 5,6 and Figure 6 illustrates the representative

lower arm component.

Figure 6

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Symbol Comment Unit Ref. Figure 5

lba_x Lower arm ball joint (x coordinate value) mm

lba_y Lower arm ball joint (y coordinate value) mm

lba_z Lower arm ball joint (z coordinate value) mm

A

lbf_x Lower arm bush front joint (x coordinate value) mm

lbf_y Lower arm bush front joint (y coordinate value) mm

lbf_z Lower arm bush front joint (z coordinate value) mm

E

lbr_x Lower arm bush rear joint (x coordinate value) mm

lbr_y Lower arm bush rear joint (y coordinate value) mm

lbr_z Lower arm bush rear joint (z coordinate value) mm

F

di Inner Diameter of the Lower Arm Section mm

do Outer Diameter of the Lower Arm Section mm

S Allowable Maximum Stress MPa


lba_x

lba_y

lba_z

lbf_x

lbf_y

lbf_z

lbr_x

lbr_y

lbr_z

di

do

S

- S (Allowable Maximum Stress) is used in

the one of the constraint

- di < do

- Position vector components were decided

through Performance Index and Force

optimization in advance.

Table 5

Table 6

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3. Mathematical Model for Optimization

(Object Function and Constraints)

3.1 Performance Index Optimization

3.1.1 Objective Function

Minimize average( Scale * | Tt(i) – Ts(i) | )

Tt(i) : Target Toe Curve at bump position i

Ts(i) : Toe Curve which is achieved from the dynamic simulation at bump position i

Scale : 100000.0 à Scaling factor ( because the toe value is very small usually)

3.2.2 Constraints

- Design Variables

Minimum

(relative

difference)

Design Variable /

initial layout value (mm)

Maximum

(relative

difference)

Negative Null

Form 1

Negative Null

Form 2

-10 X2 = lba_y / 655.74 0 -10-X2 ≤ 0 X2 ≤ 0

-10 X3 = lba_z / 466.11 10 -10-X3 ≤ 0 X3-10 ≤ 0

-10 X6 = uba_z / 749.23 5 -10-X6 ≤ 0 X6-5 ≤ 0

-30 X11 = lbf_y / 295.83 30 -30-X11 ≤ 0 X11-30 ≤ 0

Table 7

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- Design Parameters ( This range information was used in sensitivity analysis (Ref. Chapter 4))

Minimum

(relative

difference)

Design Variable /


Maximum

(relative

difference)

Negative Null

Form 1

Negative Null

Form 2

-20 X1 = lba_x / 793.9 20 -20-X1 ≤ 0 X1-20 ≤ 0

0 X4 = uba_x / 814.2 20 -X4 ≤ 0 X4-20 ≤ 0

-20 X5 = uba_y / 595.3 20 -20-X5 ≤ 0 X5-20 ≤ 0

-30 X7 = fkbt_x / 793.9 30 -30-X7 ≤ 0 X7-30 ≤ 0

-15 X8 = fkbt_y / 544.5 15 -15-X8 ≤ 0 X8-15 ≤ 0

-10 X9 = fkbt_z / 457.11 0 -10-X9 ≤ 0 X9 ≤ 0

0 X10 = lbf_x / 590 20 -X10 ≤ 0 X10-20 ≤ 0

-20 X12 = lbf_z / 475.7 20 -20-X12 ≤ 0 X12-20 ≤ 0

-20 X13 = lbr_x / 997.91 0 -20-X13 ≤ 0 X13 ≤ 0

-30 X14 = lbr_y / 295.83 30 -30-X14 ≤ 0 X14-30 ≤ 0

-20 X15 = lbr_z / 484.31 20 -20-X15 ≤ 0 X15-20 ≤ 0

-30 X16 = ubf_x / 653.95 0 -30-X16 ≤ 0 X16 ≤ 0

-20 X17 = ubf_y / 380.92 0 -20-X17 ≤ 0 X17 ≤ 0

0 X18 = ubf_z / 732.43 20 -X18 ≤ 0 X18-20 ≤ 0

-30 X19 = ubr_x / 971.0 30 -30-X19 ≤ 0 X19-30 ≤ 0

-30 X20 = ubr_y / 380.92 30 -30-X20 ≤ 0 X20-30 ≤ 0

0 X21 = ubr_z / 712.08 20 -X6 ≤ 0 X6-20 ≤ 0

Table 8

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Minimize f = 100 * ( Weight1 * | Fdu_z | / Scale1 + Weight2 * | Flbf_y + Flbr_y | / Scale2 )

Fdu_z : Damper upper point force(z-direction)

Flbf_y : Lower arm bush front point force ( y-direction)

Flbr_y : Lower arm bush rear point force ( y-direction)

Scale1 : 10000

Scale2 : 8500

Weight1 : 0.6

Weight2 : 0.4

3.2.2 Constraints

- Design Variables

Minimum

(relative

difference)

Design Variable /


Maximum

(relative

difference)

Negative Null

Form 1

Negative Null

Form 2

-20 X1 = lba_x / 793.9 20 -20-X1 ≤ 0 X1-20 ≤ 0

0 X4 = uba_x / 814.2 20 -X4 ≤ 0 X4-20 ≤ 0

-15 X8 = fkbt_y / 544.5 15 -15-X8 ≤ 0 X8-15 ≤ 0

-20 X12 = lbf_z / 475.7 20 -20-X12 ≤ 0 X12-20 ≤ 0

-20 X15 = lbr_z / 484.31 20 -20-X15 ≤ 0 X15-20 ≤ 0

Table 9

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- Design Parameters ( This range information was used in sensitivity analysis (Ref. Chapter 4))

Minimum

(relative

difference)

Design Variable /


Maximum

(relative

difference)

Negative Null

Form 1

Negative Null

Form 2

-10 X2 = lba_y / 655.74 0 -10-X2 ≤ 0 X2 ≤ 0

-10 X3 = lba_z / 466.11 10 -10-X3 ≤ 0 X3-10 ≤ 0

-20 X5 = uba_y / 595.3 20 -20-X5 ≤ 0 X5-20 ≤ 0

-10 X6 = uba_z / 749.23 5 -10-X6 ≤ 0 X6-5 ≤ 0

-30 X7 = fkbt_x / 793.9 30 -30-X7 ≤ 0 X7-30 ≤ 0

-10 X9 = fkbt_z / 457.11 0 -10-X9 ≤ 0 X9 ≤ 0

0 X10 = lbf_x / 590 20 -X10 ≤ 0 X10-20 ≤ 0

-30 X11 = lbf_y / 295.83 30 -30-X11 ≤ 0 X11-30 ≤ 0

-20 X13 = lbr_x / 997.91 0 -20-X13 ≤ 0 X13 ≤ 0

-30 X14 = lbr_y / 295.83 30 -30-X14 ≤ 0 X14-30 ≤ 0

-30 X16 = ubf_x / 653.95 0 -30-X16 ≤ 0 X16 ≤ 0

-20 X17 = ubf_y / 380.92 0 -20-X17 ≤ 0 X17 ≤ 0

0 X18 = ubf_z / 732.43 20 -X18 ≤ 0 X18-20 ≤ 0

-30 X19 = ubr_x / 971.0 30 -30-X19 ≤ 0 X19-30 ≤ 0

-30 X20 = ubr_y / 380.92 30 -30-X20 ≤ 0 X20-30 ≤ 0

0 X21 = ubr_z / 712.08 20 -X6 ≤ 0 X6-20 ≤ 0

Table 10

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3.3 Suspension Component Size Optimization


Minimize f = weight of the lower arm

Notes : weight of the lower arm component is calculated internally in NASTRAN

3.3.2 Constraints

- Constraints Table

Minimum

(relative

difference)

Design

Variable and

Parameters

initial

Maximum

(relative

difference)

Negative Null

Form 1

Negative Null

Form 2

0 XS1 = di 20 20.1 -XS1 ≤ 0 XS1-20 ≤ 0

20.2 XS2 = do 21 100 20.2-XS2 ≤ 0 X4-100 ≤ 0

0 XS3 = S 200MPa N/A N/A N/A

Auxiliary

di – do < 0

Table 11

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4. Sensitivity Analysis for Performance Index

and Force Optimization 4.1 Performance Index Optimization

In this sensitivity, 18 among 21 position vector components were selected as variables to decide which

variable influence the toe curve characteristic severely. The damper lower points were not considered

because it is obvious that they are not sensitive to the toe characteristic in the general suspension structure.

In 21 position vector components, the tie rod point was not included. The tie rod point was assumed as

fixed, and the position parameters related with this point were eliminated from entire model because tie rod

point is too sensitive to the toe characteristic. Therefore changing this point is the last method in the

suspension design. It is no meaning to change this point in order to get optimal toe curve. That is, if the tie

rod point is moved, the optimal toe curve can be obtained easily without considering other hard points

effects. Normally, position of tie rod point is modified in the final tuning the handling performance. Because

it is related with the exterior shape and the engine room layout, suspension engineer alone cannot change it

at will.

Toe curve was achieved from the bump and rebound simulation. As seen in Figure 6, two wheels were

forced to move upward and downward fixing the body, respectively 80mm. From this analysis, the toe

curve could be obtained.

Figure 7

80 mm

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The sensitivity analysis of the 18 variables– lba_x, lba_y, lba_z, lbf_x, lbf_y, lbf_z, lbr_x, lbr_y, lbr_z,

uba_x, uba_y, uba_z, ubf_x, ubf_y, ubf_z, ubr_x, ubr_y, ubr_z - were performed. We changed only one

variable value remaining the other variables as the same. The influence of the variable on the toe curve

could be estimated through the result graph and the dominant variables could be found. Using ADAMS’s

DOE (Design of Experiment) routine and variables for the toe characteristic, this analysis was performed.

Three levels of DOE were selected, which select 3 points – first, middle, and last points - in the constraint

range of the given variable and acquired the results of DOE. For example, if the following variable is

considered,

-30 ≤ lba_x ≤ 30

-30, 0, and 30 points are selected. Figure 8 shows that the analysis results.

4.2 Force Optimization In this analysis, all 21 position vector components were selected as variables. Damper lower point was

included additionally. The analysis was comprised of two load situations. One was vertical 3g force and the

Figure 8

1. Lower Arm B/J Point Y 2. Lower Arm B/J Point Z

3. Lower Arm Frt Bushing Point Y 4. Upper Arm B/J Point Z

24

other was lateral 2g force. 1g is the weight of each wheel. Vertical 3g and lateral 2g were the normal force

condition in vehicle load simulation. For convenience, two load cases were simulated at once. (Figure 9)

The sensitivity analysis of the 21 variables– lba_x, lba_y, lba_z, lbf_x, lbf_y, lbf_z, lbr_x, lbr_y, lbr_z,

uba_x, uba_y, uba_z, ubf_x, ubf_y, ubf_z, ubr_x, ubr_y, ubr_z, fkbt_z, fkbt_y, fkbt_z - was performed. We

changed only one variable value letting the other variables unchanged. The influence of the variable on the

force curve could be estimated. After checking the reaction force at the damper top in the z-direction and

lower arm front/rear bushing points in the y-direction, the dominant variables could be found. - as before

mentioned, in the conventional suspension structure, maximum force normally occurs at those location and

direction. Figure 10 shows that the analysis results.

2g for lateral direction

3g for vertical direction

Force

Time Figure 9

Vertical 3g

Lateral 2g 2. Lower Arm B/J Point X

5. Lower Arm Frt. Bushing Point Z

3. Upper Arm B/J Point X

4. Lower Arm Rr. Bushing Point Z

1. Damper Lower Point Y

Figure 10

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5. Procedures for Performance Index and

Force Optimization 5.1 Performance Index Optimization

Chart 2 shows overall procedure for Performance Index Optimization. Using ADAMS optimization

technique (DOT2à Design Optimization Technique 2), the optimal point for the target toe curve could be

obtained. DOT2 uses the optimization algorithm provided by ADAMS/View with Fletcher-Reeves for

unconstrained problems and Sequential Linear Programming (SLP) for constrained problems. In this

project, SLP routine was used.

5.2 Force Optimization Chart 3 shows overall procedure for Force Optimization. Using ADAMS optimization technique,

the optimal point for the target toe curve could be obtained. In this project, SLP routine was used.

Chart 2

Initial Hard Point

+/- 80 mm Bump/Rebound static

Check the objective function.

Change Hard Constraints (layout and

Optimal Hard Point

Chart 3

Initial Hard Point

Force Simulation ( applying at the wheel tip, considering 3 axis

directions)

Check the objective function.

Change Hard Constraints (layout)

Optimal Hard Point

26

6. Results and Trade-off for Performance Index and

Force Optimization 6.1 Performance Index Optimization

We achieved the following results in this optimization. Table 12 shows the final optimized values of

the design variables in the Performance Index Optimization. The comparison of the toe curve is shown in

the Figure 11. As seen in this figure, toe curve with optimized results traces well target toe curve.

Design Variable /


Initial layout

value (mm)

Final value

(mm) Hard Point and Direction

X2 = lba_y / 655.74 655.74 645.217 Lower Arm B/J Point Y

X3 = lba_z / 466.11 466.11 465.94 Lower Arm B/J Point Z

X6 = uba_z / 749.23 749.23 747.76 Upper Arm B/J Point Z X11 = lbf_y / 295.83 295.83 295.38 Lower Arm Frt. Bushing Point Y

Toe Curve

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-80 -64 -48 -32 -16 0 16 32 48 64 80

Bump Distance

To

e(+:

in, -

:ou

t)

Target_toe

Org_toe

Optimization Result

Table 12

Figure 11

27


We achieved the following results in this optimization. Table 13 shows the final values of the design

variables in Force Optimization. The comparison of the toe curves is shown in the Figure 12 for the damper

top point in the z-direction, in the Figure13 for the lower arm front bushing point in the y-direction, and in

the Figure14 for the lower arm rear bushing point in the y-direction. As seen in those figures, reaction force

levels with optimized results were reduced from original (initial) values.

Design Variable /


Initial layout

value (mm)

Final value

(mm) Hard Point and Direction

X1 = lba_x / 793.9 793.9 813.9 Lower Arm B/J Point X

X4 = uba_x / 814.2 814.2 794.2 Upper Arm B/J Point X

X8 = fkbt_y / 544.5 544.5 559.5 Damper Lower Point Y X12 = lbf_z / 475.7 475.7 455.7 Lower Arm Frt. Bushing Point Z X15 = lbr_z / 484.31 484.31 464.31 Lower Arm Rr. Bushing Point Z

Damper Z Force

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

0.00 0.19 0.38 0.57 0.76 0.95 1.15 1.34 1.53 1.72 1.91 2.10

Time

Fo

rce

(N)

ORG Optimization Result

Table 13

Figure 12

28

Lower Arm Front Bushing Y Force

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

8.00E+03

9.00E+03

0.00 0.19 0.38 0.57 0.76 0.95 1.15 1.34 1.53 1.72 1.91 2.10

Time

Fo

rce

(N)


Figure 13

Lower Arm Rear Bushing Y Force

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

8.00E+03

9.00E+03

0.00 0.19 0.38 0.57 0.76 0.95 1.15 1.34 1.53 1.72 1.91 2.10

Time

Fo

rce

(N)


Figure 14

29

6.3 Trade-off

After Performance Index Optimization and Force Optimization, a sort of the trade-off design was

performed because the hard points that set from the each optimization were different. In this project, we

focus more on the Performance Index (toe only) than the reaction force. The Performance Index is only

affected by the position of the hard points – Those kind of characteristics are only kinematically decided.

Once the position of the hard points is set, it is impossible to change the characteristic. However, for the

reaction force, it can be covered with changing the component size to reduce the maximum stress of that

component even though slightly large reaction forces exist. Someone may think that the Force Optimization

is redundant, but it has still significant meaning because the smaller reaction force can induce the smaller

maximum stress.

When the design variables of the Force Optimization had the negative effect greatly for tracing the

target toe curve, they ware omitted in the design variables. The design variables of the Force Optimization

were fkbt_y (Damper Lower Point Y), lba_x (Lower Arm B/J Point X), uba_x (Upper Arm B/J Point X),

lbr_z (Lower Arm Rr. Bushing Point Z), and lbf_z (Lower Arm Frt. Bushing Point Z). Among them, lba_x,

lbr_z, and lbf_z had a great effect negatively for tracing the toe curve, which could be seen in the result of

the toe sensitivity analysis (Ref. Figure 16).

The trade-off procedure is the following. First, we conducted sensitivity analysis to toe curve with the

design variables in the Force Optimization, and the design variables that had the great effect negatively for

tracing the target toe curve were omitted and restored to the original (initial) values. Second, we carried out

the Force Optimization again with the remaining design variables of the previous Force Optimization.

Figure 15 shows the procedure of the trade-off.

Retry Force Optimization

Sensitivity Analysis for the Performance Index (Toe Curve) with the design variables of the Force Optimization

The remaining design variables of the Force Optimization

Figure 15

30

As mentioned before, the Force Optimization was performed again using remaining two design

variables (fkbt_y and uba_x). Finally, we could get the trade-off result of the hard points. Even if there is a

little discrepancy with target toe curve at the edge, we could estimate that the reasonable optimized toe

curve was achieved when compared with the target curve. (Ref. Figure 17)

lba_x lbr_z lbf_z

fkbt_x uba_x

Figure 16 : Sensitivity Analysis Results for toe curve with the design variables of the Force Optimization

31

Figure 18,19, and 20 shows the reaction forces of the damper top point in the z-direction, the lower

arm front bushing point in the y-direction, the lower arm rear bushing point in the y-direction with the final

optimized design variables. For the lower arm rear bushing reaction force, it is shown that optimized

reaction forces are higher than original one because of the lower weighting factor in the objective function.

For the lower arm front bushing reaction force, it is shown that the final result is similar with the original

one. From this result, we estimated that overall optimization procedure was not applied well to the lower

arm component for the reaction force optimization. In that time, we had to choose the curing method to

solve this situation such as modifying objective function, But since we originally planned to optimize the

size of the lower arm component and we thought that the size optimization could solve this problem because

stress in the individual component is more important than the reaction force of the components, we continue

to proceed the next step, structural optimization.

Figure 17

Toe Curve

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-80 -64 -48 -32 -16 0 16 32 48 64 80

Time or Bump Distance

To

e(+:

in, -

:ou

t)Target_toe

Org_toe

Optimization Result

Final Result(considering Force)

32

Figure 18

Figure 19

Lower Arm Rear Bushing Y Force

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

8.00E+03

9.00E+03

0.00 0.19 0.38 0.57 0.76 0.95 1.15 1.34 1.53 1.72 1.91 2.10

Time

Fo

rce

(N)

Original

Optimization Result (only forceoptimization)Final Optimization

Lower Arm Front Bushing Y Force

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

8.00E+03

9.00E+03

0.00 0.19 0.38 0.57 0.76 0.95 1.15 1.34 1.53 1.72 1.91 2.10

Time

Fo

rce

(N)

Original

Optimization Result (only force optimization)

Final Optimization

33

Damper Z Force

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

0.00 0.19 0.38 0.57 0.76 0.95 1.15 1.34 1.53 1.72 1.91 2.10

Time

Fo

rce

(N)

Original

Optimization Result (only forceoptimization)Final Optimization

Figure 20

34

7. Procedures and Results for Suspension Component

Size Optimization 7.1 Procedure

Initial Diameter do : 21 di : 20

Reaction Force from the Force Optimization

Ground Spring Elements for Boundary Condition

Figure 22

Figure 21 : Lower Arm Component à Original Model and Simplified Model

35

Figure 21 and 22 illustrate the lower arm component model in the suspension system. As

mentioned before, the real component was simplified as annular rod shape to reduce the simulation

time. The lower arm front and rear bushing points were constrained using the ground spring elements

in x, y and z direction for imposing proper boundary condition and the reaction force which had been

achieved from the Force Optimization was given at the lower arm ball joint in x, y, and z direction as

force inputs. The distance between the lower arm front and rear bushing points was 400mm and the

normal distance from the lower arm ball joint to the line between above two points was 300 mm

approximately. The finite elements model was constructed using the total 65 number of the ‘Cbar’

elements in the NASTRAN, which has three degree of freedom in x, y, and ϑ at each node. In the

optimization procedure, Initial design variables were set as following; outer diameter was 21mm and

inner diameter was 20mm. Using an optimization method in NASTRAN (SQP), we conducted the

diameter size optimization of the lower arm component. Objective function was minimum weight of

the component. After 2 iterations of the linear static analysis, we achieved the minimum objective

function value satisfying the constraints.

36

7.2 Results

As seen in Figure 23, initial points was not in the feasible region because maximum stress was over

the allowable stress(200 N/mm2 ). Optimal objective function value was 426g when the maximum

stress was 190 N/mm2. Outer diameter and Inner diameter in the design variables was 21.11mm and

19.86mm respectively at the minimum point. We achieved the minimum weight far from the

conventional lower arm component’s one in the sports utility vehicle (SUV) (normally about 3 kg)

because we oversimplified the model and the loading condition was much less severe than realistic

situation. To get the realistic result, we should have made the model more refined including shapes,

loading and boundary condition. However, in this project, our aim was only to demonstrate the

procedure how to optimize the size of the components, therefore we thought that it would be

sufficient for this purpose.

*************************************************************** S U M M A R Y O F D E S I G N C Y C L E H I S T O R Y *************************************************************** (HARD CONVERGENCE ACHIEVED) (SOFT CONVERGENCE ACHIEVED) NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 3 NUMBER OF OPTIMIZATIONS W.R.T. APPROXIMATE MODELS 2 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY --------------------------------------------------------------------------------------------------------------- OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT --------------------------------------------------------------------------------------------------------------- INITIAL 3.324234E-04 2.446335E+02 1 3.991801E-04 3.989822E-04 4.960272E-04 2.036552E+02 2 4.263013E-04 4.262615E-04 9.347114E-05 1.905573E+02 --------------------------------------------------------------------------------------------------------------- 1 LOWER ARM OPTIMIZATION APRIL 17, 2002 MSC/NASTRAN 10/23/98 PAGE 51 DESIGN VARIABLE HISTORY ---------------------------------------------------------------------------------------------------------------------------------- INTERNAL | EXTERNAL | | DV. ID. | DV. ID. | LABEL | INITIAL : 1 : 2 : 3 : 4 : 5 : ---------------------------------------------------------------------------------------------------------------------------------- 1 | 1 | B | 2.1000E+01 : 2.1098E+01 : 2.1141E+01 : 2 | 2 | H | 2.0000E+01 : 1.9898E+01 : 1.9858E+01 :

Figure 23

37

8. Conclusion In this term project, we applied the optimization method to the design process of the vehicle

suspension system. Overall optimization procedure was divided into two parts; One was to decide the hard

point position and the other was to decide the size of the suspension component. In the decision of the hard

point position, we divided the optimization problem into two sub-problems, which were called Performance

Index Optimization and Force Optimization. At first, two sub-problems were separately optimized and then

trade-off was conducted. In the size optimization of the component, we selected only the lower arm

component in the suspension system. The reasons were that one was to reduce the problem size and the

other was for this component to undergo the largest reaction force by the design experiences. Another

reason was found during the project by product because in the force optimization, we set lower weighting

factor on the force term related with the lower arm points than any other, therefore it was hard to minimize

the reaction force at the point of this component when the trade-off was done.

To get the more realistic result, we have to consider other performance indices including toe, and

more analytical trade-off method should be developed. Also shape optimization should be included and

more realistic component model should be used not annular bar model. Last, we have to impose more

realistic and diverse loading condition on the suspension system.

Through this project, we knew how to apply the optimization skill to the design process using

commercial S/W, and that the well-defined optimization procedure including the trade-off method is needed

for the competitive practical design.

38

References

1. Murali M.R. Krishna and Scott V. Anderson, “Shape Optimization Application in Upper Control

Arm Design”, SAE 2000-01-3445

2. Taeoh Tak, Sunghun Chung and Hyungho Chun, “An Optimal Design Software for Vehicle

Suspension Systems”, SAE 2000-01-1618

3. Kwon-Hee Suh, Yoon-Ki Lee and Hi-Seak Yoon, “Optimization of Front Bump Steer Using Design

of Experiments”, SAE 2000-01-1617

4. Keiichi Motoyama and Takashi Yamanaka, “A Study of Suspension Design Using Optimization

Technique and DOE”, 2000 International ADAMS User Conference

5. Minen & M. Montenegro, “New Strategy for Choosing Optimal Design Parameters via Xmath and

ADAMS Interconnected Simulations”, 1994 International ADAMS User Conference

6. G. Stevens, D. Peterson, StatDesign, and U. Eichhorn, "Optimization of Vehicle Dynamics Through

Statistically Designed Experiments on Analytical Vehicle Models”, 1997 MDI User Conference

7. E. Ekert, “The Integration of Designed Experiments and ADAMS”, 1994 International ADAMS

User Conference

8. P. McNally, “Applying DOE to Mechanical Designs”, 1995 MDI User Conference

39

Appendix

- NASTRAN INPUT FILE $*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $* $* EXECUTIVE CONTROL $* $*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $* SOL 200 TIME 900 CEND $* $*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $* $* CASE CONTROL $* $*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $* TITLE = LOWER ARM OPTIMIZATION $ECHO = ALL $* GLOBAL CASE SPC = 1 LOAD = 10 $* DISPLACEMENT = ALL STRESS = ALL SPCFORCES = ALL $* $ OPTIMIZATION $ $ just write, modify for this application!!! $ ANALYSIS = STATICS $ OBJECTIVE EXPRESSION DESOBJ(MIN) = 20 $ CONSTRAINT EXPRESSION DESSUB = 10 $* $*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $* $* BULK DATA $* $*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $*

40

BEGIN BULK $* $* PARAM CARDS $* PARAM GRDPNT -1 PARAM AUTOSPC YES PARAM POST -2 PARAM BAILOUT -1 $* LOAD 10 1.00000 1.00000 1 $* $* COORDINATE SYSTEM CARDS $* CORD2R 2 0 0.0 0.0 0.0 0.0 0.0 1.00000+ + 1.00000 0.0 0.0 $* $* GRID CARDS $* GRID 1 0 997.910 295.830 484.310 0 GRID 2 0 590.000 295.380 475.700 0 GRID 5 0 977.515 295.808 483.880 0 GRID 6 0 957.119 295.785 483.449 0 GRID 7 0 936.724 295.763 483.019 0 GRID 8 0 916.328 295.740 482.588 0 GRID 9 0 895.933 295.718 482.158 0 GRID 10 0 875.537 295.695 481.727 0 GRID 11 0 855.142 295.673 481.297 0 GRID 12 0 834.746 295.650 480.866 0 GRID 13 0 814.351 295.628 480.436 0 GRID 14 0 793.955 295.605 480.005 0 GRID 15 0 773.560 295.583 479.575 0 GRID 16 0 753.164 295.560 479.144 0 GRID 17 0 732.769 295.538 478.714 0 GRID 18 0 712.373 295.515 478.283 0 GRID 19 0 691.978 295.493 477.853 0 GRID 20 0 671.582 295.470 477.422 0 GRID 21 0 651.187 295.448 476.992 0 GRID 22 0 630.791 295.425 476.561 0 GRID 23 0 610.396 295.403 476.131 0 GRID 24 0 595.344 313.708 475.076 0 GRID 25 0 600.842 332.541 474.434 0 GRID 26 0 606.493 351.877 473.776 0 GRID 27 0 612.297 371.717 473.101 0 GRID 28 0 618.255 392.061 472.408 0 GRID 29 0 624.365 412.910 471.699 0 GRID 30 0 630.630 434.262 470.973 0 GRID 31 0 637.047 456.118 470.229 0 GRID 32 0 643.618 478.478 469.469 0 GRID 33 0 650.342 501.342 468.691 0 GRID 34 0 657.249 524.627 467.901 0

41

GRID 35 0 664.996 546.485 467.185 0 GRID 36 0 673.851 566.158 466.581 0 GRID 37 0 683.815 583.649 466.088 0 GRID 38 0 694.888 598.955 465.707 0 GRID 39 0 707.069 612.077 465.436 0 GRID 40 0 720.359 623.016 465.278 0 GRID 41 0 734.758 631.771 465.230 0 GRID 42 0 750.265 638.343 465.294 0 GRID 43 0 766.881 642.730 465.469 0 GRID 44 0 784.605 644.934 465.755 0 GRID 45 0 803.162 644.954 466.146 0 GRID 46 0 820.889 642.788 466.609 0 GRID 47 0 837.508 638.436 467.135 0 GRID 48 0 853.019 631.899 467.726 0 GRID 49 0 867.423 623.175 468.382 0 GRID 50 0 880.718 612.266 469.102 0 GRID 51 0 892.906 599.171 469.886 0 GRID 52 0 903.985 583.890 470.736 0 GRID 53 0 913.957 566.423 471.649 0 GRID 54 0 922.820 546.771 472.627 0 GRID 55 0 930.576 524.932 473.670 0 GRID 56 0 937.492 501.664 474.752 0 GRID 57 0 944.224 478.814 475.814 0 GRID 58 0 950.803 456.468 476.852 0 GRID 59 0 957.229 434.626 477.866 0 GRID 60 0 963.501 413.287 478.857 0 GRID 61 0 969.619 392.452 479.825 0 GRID 62 0 975.584 372.120 480.769 0 GRID 63 0 981.396 352.292 481.689 0 GRID 64 0 987.054 332.968 482.586 0 GRID 65 0 992.559 314.147 483.460 0 GRID 68 0 569.604 295.358 475.270 0 GRID 69 0 1018.31 295.853 484.740 0 GRID 70 0 793.900 645.217 465.940 0 $* $* ELEMENT CARDS $* CBAR 1 1 1 5 1.10E-3-1.00000 0.0 CBAR 2 1 5 6 1.10E-3-1.00000 0.0 CBAR 3 1 6 7 1.10E-3-1.00000 0.0 CBAR 4 1 7 8 1.10E-3-1.00000 0.0 CBAR 5 1 8 9 1.10E-3-1.00000 0.0 CBAR 6 1 9 10 1.10E-3-1.00000 0.0 CBAR 7 1 10 11 1.10E-3-1.00000 0.0 CBAR 8 1 11 12 1.10E-3-1.00000 0.0 CBAR 9 1 12 13 1.10E-3-1.00000 0.0 CBAR 10 1 13 14 1.10E-3-1.00000 0.0 CBAR 11 1 14 15 1.10E-3-1.00000 0.0 CBAR 12 1 15 16 1.10E-3-1.00000 0.0 CBAR 13 1 16 17 1.10E-3-1.00000 0.0

42

CBAR 14 1 17 18 1.10E-3-1.00000 0.0 CBAR 15 1 18 19 1.10E-3-1.00000 0.0 CBAR 16 1 19 20 1.10E-3-1.00000 0.0 CBAR 17 1 20 21 1.10E-3-1.00000 0.0 CBAR 18 1 21 22 1.10E-3-1.00000 0.0 CBAR 19 1 22 23 1.10E-3-1.00000 0.0 CBAR 20 1 23 2 1.10E-3-1.00000 0.0 CBAR 21 1 2 24-0.96002 0.27993 0.0 CBAR 22 1 24 25-0.95993 0.28023 0.0 CBAR 23 1 25 26-0.95985 0.28051 0.0 CBAR 24 1 26 27-0.95977 0.28078 0.0 CBAR 25 1 27 28-0.95970 0.28104 0.0 CBAR 26 1 28 29-0.95963 0.28128 0.0 CBAR 27 1 29 30-0.95956 0.28151 0.0 CBAR 28 1 30 31-0.95949 0.28173 0.0 CBAR 29 1 31 32-0.95943 0.28194 0.0 CBAR 30 1 32 33-0.95937 0.28214 0.0 CBAR 31 1 33 34-0.95871 0.28439 0.0 CBAR 32 1 34 35-0.94255 0.33406 0.0 CBAR 33 1 35 36-0.91188 0.41045 0.0 CBAR 34 1 36 37-0.86889 0.49500 0.0 CBAR 35 1 37 38-0.81023 0.58612 0.0 CBAR 36 1 38 39-0.73291 0.68033 0.0 CBAR 37 1 39 40-0.63551 0.77210 0.0 CBAR 38 1 40 41-0.51954 0.85444 0.0 CBAR 39 1 41 42-0.39017 0.92074 0.0 CBAR 40 1 42 43-0.25531 0.96686 0.0 CBAR 41 1 43 44-0.12339 0.99236 0.0 CBAR 42 1 44 45-1.07E-3 1.00000 0.0 CBAR 43 1 45 46 0.12128 0.99262 0.0 CBAR 44 1 46 47 0.25330 0.96739 0.0 CBAR 45 1 47 48 0.38839 0.92150 0.0 CBAR 46 1 48 49 0.51804 0.85535 0.0 CBAR 47 1 49 50 0.63433 0.77307 0.0 CBAR 48 1 50 51 0.73202 0.68128 0.0 CBAR 49 1 51 52 0.80959 0.58700 0.0 CBAR 50 1 52 53 0.86845 0.49579 0.0 CBAR 51 1 53 54 0.91158 0.41113 0.0 CBAR 52 1 54 55 0.94234 0.33466 0.0 CBAR 53 1 55 56 0.95856 0.28490 0.0 CBAR 54 1 56 57 0.95923 0.28263 0.0 CBAR 55 1 57 58 0.95929 0.28243 0.0 CBAR 56 1 58 59 0.95935 0.28222 0.0 CBAR 57 1 59 60 0.95942 0.28200 0.0 CBAR 58 1 60 61 0.95948 0.28177 0.0 CBAR 59 1 61 62 0.95955 0.28153 0.0 CBAR 60 1 62 63 0.95963 0.28127 0.0 CBAR 61 1 63 64 0.95971 0.28100 0.0 CBAR 62 1 64 65 0.95979 0.28072 0.0 CBAR 63 1 65 1 0.95988 0.28042 0.0

43

CBAR 64 1 2 68 1.10E-3-1.00000 0.0 CBAR 65 1 69 1 1.10E-3-1.00000 0.0 RBE2 66 70 123456 44 45 $ $ GROUND SPRING ELEMENT $ CELAS2 67 690.9 1 1 CELAS2 68 6425.55 1 2 CELAS2 69 6425.55 1 3 CELAS2 70 3111.35 1 4 CELAS2 71 1.5e4 1 5 CELAS2 72 1.5e4 1 6 CELAS2 73 690.9 2 1 CELAS2 74 6425.55 2 2 CELAS2 75 6425.55 2 3 CELAS2 76 3111.35 2 4 CELAS2 77 1.5e4 2 5 CELAS2 78 1.5e4 2 6 $* $* MATERIAL CARDS $* I-DEAS Material: 1 name: GENERIC_ISOTROPIC_STEEL $* MAT1 1 2.07E+5 0.29000 7.82E-9 $* $* PROPERTY CARDS $* $* I-DEAS property: 1 name: LINEAR BEAM1 $* Fore Section : 2 name: PIPE 45.0 X 15.0 PBAR 1 1 32.185 27081.3 27081.3 54162.6 $* $* LOAD CARDS $* FORCE 1 70 0 1.00000 40.3540-14349.0 4258.40 $ $ $ DESIGN MODEL $ $ $DESVAR, ID, LABEL, XINIT, XLB, XUB, DELXV DESVAR, 1, B, 21., 20.2, 100., 0.01 DESVAR, 2, H, 20., 0., 20., 0.01 $ $ $ EQUATION FOR CROSS SECTION DEQATN 1301 AREA(B,H) = 3.14*(B**2-H**2)/4. DEQATN 2302 I(B,H) = 3.14*(B**4-H**4)/4. DEQATN 3303 J(B,H) = 3.14*(B**4-H**4)/2. $ $ $ BAR PROPERTY RELATION

44

DVPREL2, 10, PBAR, 1, 4, , , 1301, , +DP2 +DP2, DESVAR, 1, 2 DVPREL2, 11, PBAR, 1, 5, , , 2302, , +DP3 +DP3, DESVAR, 1, 2 DVPREL2, 12, PBAR, 1, 6, , , 2302, , +DP4 +DP4, DESVAR, 1, 2 DVPREL2, 13, PBAR, 1, 7, , , 3303, , +DP5 +DP5, DESVAR, 1, 2 $ $ $ DESIGN RESPONSE IDENTIFICATION DRESP1, 20, W, WEIGHT DRESP1, 21, SBARA, STRESS, PBAR, , 8, , 1 DRESP1, 22, SBARB, STRESS, PBAR, , 15, , 1 $ $ $ CONSTRAINT BOUND DCONSTR, 10, 21, -1., 2.e2 DCONSTR, 10, 22, -1., 2.e2 $ DOPTPRM, APRCOD, 3, CONV1, 0.1, CONV2, 0.1, CONVDV, 0.1, +1 +1 , CONVPR, 0.1, METHOD, 3, CTMIN, 2.e2 ENDDATA

Documents

final project suspension+designode.engin.umich.edu/me555reports/2002/555-02-09.pdf · Term Project Final Report OF OPTIMIZATION OF SUSPENSION DESIGN ... In the process of developing