COMPLIANT SUSPENSION DESIGN OPTIMIZATION

ME-555 W08 DESIGN OPTIMIZATION PROJECT TEAM-5-FINAL REPORT

GIRISH KRISHNAN KARAN GOYAL MUKUND NEMALI ROHAN SINGH

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Lc

Lh

Uaf

thetah

Uar

Laf

Lar

thetala

thetaua

alphauaf

thetab

thetat

betauaf

etauaf

Alphauar

betauar

etauar

alphalaf

betalaf

alphalar

etalaf

etalar

betalar

dc

dc1

thetah

lamdab

lamdat

Basic design of our

compliant A arm

suspension for a

vehicle.

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Contents GIRISH REPORT ................................................................................................................................. 8

(KINEMATIC ANALYSIS) .................................................................................................................... 8

1.1 PROBLEM STATEMENT .............................................................................................................. 8

1.2 NOMENCLATURE ....................................................................................................................... 11

THE VARIABLES:............................................................................................................................... 12

PARAMETERS: .................................................................................................................................. 12

1.3 MATHEMATICAL MODELS: ....................................................................................................... 14

1.4 SUMMARY MODEL: .................................................................................................................... 15

1.5 MODEL ANALYSIS: ..................................................................................................................... 15

MUKUND NEMALI (DYNAMIC ANALYSIS)..................................................................................... 20

2 PROBLEM STATEMENT ............................................................................................................... 20

2.1. MATHEMATICAL MODEL.......................................................................................................... 20

2.2. NOMENCLATURE: ..................................................................................................................... 21

2.3. SUMMARY MODEL: ................................................................................................................... 23

2.4. MODEL ANALYSIS: .................................................................................................................... 24

2.5. NUMERICAL RESULTS: ............................................................................................................ 25

ROHAN SINGH ................................................................................................................................... 29

(STRUCTURAL ANALYSIS) .............................................................................................................. 29

3.2. NOMENCLATURE: ............................................................................................................. 32

3.3. MATHEMATICAL MODEL: ................................................................................................. 33

3.4. CONSTRAINTS: .................................................................................................................. 34

3.5. SUMMARY OF THE MODEL: ............................................................................................ 37

3.6. DESIGN VARIABLES: ........................................................................................................ 38

3.7. DESIGN PARAMETERS: ................................................................................................... 38

3.8. OPTIMIZATION ANALYSIS: .............................................................................................. 39

3.9. ANALYSIS:........................................................................................................................... 46

3.10. SUBSYSTEM TRADEOFFS: .......................................................................................... 47

4. DAMPER PLACEMENT ANALYSIS ............................................................................................. 49

4.1 PROBLEM STATEMENT ............................................................................................................ 49

4.2 NOMENCLATURE: ...................................................................................................................... 51

4.3 MATHEMATICAL MODEL ........................................................................................................... 52

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VARIABLES .................................................................................................................................... 53

CONSTRAINTS .............................................................................................................................. 53

PARAMETERS ............................................................................................................................... 54

4.4 SUMMARY MODEL ..................................................................................................................... 55

4.5 MODEL ANALYSIS ...................................................................................................................... 55

4.6 RESULTS...................................................................................................................................... 56

4.7 CONCLUSION & FUTURE WORK ............................................................................................. 60

FINAL SYSTEM INTEGRATION ....................................................................................................... 61

5.2. CONSTRAINTS: .......................................................................................................................... 61

5.3. FINAL SUMMARY: ...................................................................................................................... 63

5.4. OPTIMIZATION ANALYSIS: ...................................................................................................... 64

5.5. RESULT: ...................................................................................................................................... 68

5.6. FUTURE WORK: ......................................................................................................................... 69

APPENDIX IA...................................................................................................................................... 71

APPENDIX IB...................................................................................................................................... 75

APPENDIX 3A .................................................................................................................................... 77

APPENDIX 4 ....................................................................................................................................... 79

REFERENCES: .................................................................................................................................. 80

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ABSTRACT:

Functions of an automobile suspension are to isolate the occupants from severe levels

of shock and vibration induced by the road surface while ensuring the acceleration and

braking performance and directional response behavior is safe for the occupants at all

operating speeds. Vehicle suspension systems contain several elements, with a main

spring supporting the sprung mass and the damper dissipating energy stored in the

spring during motion. The tire acts a secondary spring system, acting on both the

unsprung mass of the suspension and the sprung mass of the loaded vehicle. The

relatively high number of parts present in traditional suspension systems often

negatively affects the total cost and weight of the system. Minimizing the unsprung

mass will enable a suspension to be more responsive, hence providing better

performance. Compliant mechanisms offer great promise in providing innovative and

efficient solutions to many mechanical design problems. Motivation to use compliant

suspension mechanisms is to design and build ultra light high performance suspension

systems. Compliant or flexible link mechanisms gain some or all of their mobility from

the flexibility of their members rather than from rigid-body joints only. They have many

potential advantages like weight reduction due to reduced number of parts, reduced

wear, manufacturing cost and assembly time. In this project a compliant A-arm

suspension design is chosen for design optimization. It has added advantages of lower

packaging space, lower number of extra joints and rigid links and lower cost to

manufacture. The compliant A-arm suspension design is optimized based on the

kinematics, dynamic response, link and joint stresses and optimal damper placement.

The objective of the optimization study is to find a global optimum compliant A-arm

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suspension design that satisfies all the prescribed constraints and meets all the

cascaded attribute targets.

INTRODUCTION:

The aim of this project is to find an optimal design for a compliant a-arm

suspension mechanism used in automobiles. The problem is decomposed by looking at

what disciplines are required to completely analyze the problem and build a

mathematical model. Such aspect decomposition is shown in Figure 1

Figure 1

Objective of the compliant A-arm topology creation is to provide the kinematics motion

required for the optimum performance of the suspension in a vehicle. The topology will

generate a combination of rigid and flexible links of certain cross sections, which are

intended by design to provide the required motion.

Objective of the dynamics analysis model is to optimize the topology to maximize the

natural frequency of the system, so that it does not degrade the ride of the vehicle. The

Compliant A-Arm

Suspension Mechanism

Design

Kinematics (Input

force vs. output

displacement

relationship for

optimum vehicle

ride and handling

Dynamics (Stiffness

and mass

optimization for

optimal frequency

response)

Component

Stresses

(Minimization of

peak stresses for

maximum system

durability

System Damping

(Placement and

damping force for

maximum damper

performance)

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dynamics analysis will treat the cross section of the complaint A-arm topology as

variables to optimize the mass and stiffness of the topology to match a certain

frequency target. The constraints imposed on this model are based on the geometric

variables required for the kinematics of the suspension mechanism.

Objective of the stress analysis model is to minimize the peak stresses that occur in the

compliant A-arm suspension topology in order to improve the durability of the

suspension system. The stress analysis model will optimize the topology cross section

while satisfying the mass and stiffness constraints imposed due to the dynamic

performance requirement of the compliant A-arm suspension.

Objective of the damping analysis is to optimize the placement of the damper and

determine the optimum damping force required to damp the oscillations of the

suspension system caused due to the vibrations induced by any road input to the

vehicle.

Each of these physics based mathematical models will analyze the system and optimize

it for the required performance attribute. The final step of the optimization problem is to

use an Analytical Target Cascading process to cascade the required design attribute

targets as specifications to each subsystem in a consistent and efficient manner so as

to generate a set of feasible design solutions and use gradient based optimization

algorithms to search and find the optimal design solution to the compliant A-arm design

problem.

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GIRISH REPORT

(KINEMATIC ANALYSIS)

1.1 PROBLEM STATEMENT

The kinematics of the compliant suspension is similar to a normal suspension, with the

compliant members replaced with rigid links. This method of analysis of compliant

systems is known as the pseudo-rigid body model (PRBM). The link lengths that are

used for kinematic analysis is shorter than the actual beam lengths by a factor of 0.8.

1) One of the most important kinematic properties of the mechanism is its ability to

comply with vertical jounce displacements (yv). A mechanism should be able to

do so with relative ease, i.e. without considerable changes in its configuration.

Thus, we need to maximize this quantity, which is the objective function.

2) Another metric which is specific to the compliant suspension mechanism is the

joint rotations (θt - θt’, θa – θa’, γ - γ’, β - β’). This can be explained from the fact

that high stress always occur at regions that undergo significant displacements.

This, when translated to the current kinematic model, would mean low joint

rotations for a specified jounce displacement. In the present problem, this

quantity is expressed as constraints on the four joint rotations that make up the

mechanism.

3) Another important characteristic of a suspension mechanism is the roll-center

(yr). The roll-center distance is the vertical distance above the ground about

where the toppling force acts during cornering. If the roll-center is away from the

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center of gravity of the vehicle, there will be moments, which cause the vehicle to

overturn.

4) The vertical jounce of the mechanism (yv) should move almost in a straight

vertical line. In reality it moves along an arc of the circle. It is of our interest to

increase the radius of travel. The instantaneous center of circle of the arc (xin)

traversed by the wheel needs to be maximized to make it move along a straight

line.

5) The link lengths and the angles ( l1 , l2, l3, l4 and θ1, θt, θh, θa ) which together

form the variables of which define a configuration of the four-bar mechanism are

the variables for the present problem. To put together a four-bar mechanism with

these variables, we need to satisfy the triangle inequality condition with certain

link lengths. These inequalities add a set of three further constraints.

6) Furthermore it should be realized that a four-bar mechanism‟s configuration is

completely dependent on which of the link lengths is shorter than the other. First

of all the link lengths should be such that they satisfy the Grashoff‟s criterion,

which can make a four-bar mechanism possible from a given set of link lengths

and angles. The Grashoff‟s criterion cannot be explicitly put into the model as

constraints because there may be some point in the middle of iterations when the

constraints can get temporarily violated. This is why we need to put bounds on

the link lengths such that Grashoff‟s criterion is always satisfied.

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Figure 1.1.

Different configurations of a four-bar mechanism , ‘s’ is the smallest link, l is the largest

and p and q are intermediate links.

Shown in Fig. 1.1 are all the four configurations that a four-bar mechanism can take. Of

these only certain types of configurations can be suitable for the suspension. Choice a,

where the fixed link is the shortest link, is a good choice because large angles of

rotation with relatively straight line motion for link p is obtained. Option b also allows

large angles of rotation but is limited because the largest link rotates significantly for

large angles of rotations of links s and p. Option c is not suitable because large crank

angles almost intersect p and l. This situation must be avoided, and can be done so by

making sure that the connecting link is not the shortest. Finally, option d is the best

suited since it gives a pure straight line motion, however, it will be seen later that such a

design has a very bad roll-center height.

Ideally, we have to avoid case ‘c’ and parts of case ‘b’ and case ‘a’ where there

are large variations in the link lengths, leading to large rotations of the connecting link to

which the wheel is to be attached.

a b c d

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1.2 NOMENCLATURE

Figure 1.2.

The design variables and the parameters (a) of the whole car, (b) instant and the roll

center, (c) jounce distance when the lower link arm is given a rotation ang.

lw

ltw

l1

l2

Lh or l3

l4

θt

θ1

θa

θh

Lc

Roll center

Instant center xin

Vertical Roll

center height yr

θh’

θa’

θt’

Vertical travel yv

β

γ

β'

γ'

a

b

c

θt’ - θt ≈ ang

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THE VARIABLES:

l2 = (Lar + Laf)/2 = Length of the lower A arm of the suspension

lh = Length of the beam connecting the upper and lower A-arm.

l4 = (Uar + Uaf)/2 = Length of the upper A arm of the suspension.

θt = Angle made by the link with length l2 to the vertical.

θh = Angle between the connecting beam and the vertical.

θa = Angle between the upper A Arm and the vertical.

Note: Though the lengths and the angles are deemed as the variables, it is most

practical to specify the lengths of the other links relative to one link so as to satisfy both

the Grashoff‟s criterion as well as preferring case „a‟, „b‟, and case „d‟ of Fig. 1.1. This is

done by setting

l4 = l4

l2 = l4 + η2

lh = l4 + ηh

Specifying the design variables by choosing nominal values for η2 and ηh will ensure

that we limit our optimum search only within the required design domain.

PARAMETERS:

Lc = Length of the connecting arm of the suspension to the hub of the wheel

ltw = width of the wheel

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lw = Breadth of the car

lt = Length of the wheel..

ang = Angle by which θt is changed to obtain θt’

All other quantities in other subsystems which define the compliance of the mechanism

do not influence the kinematic quantities as defined for the problem.

Quantities that are functions of the variables and the parameters:

l1 = Length of the fixed link of the suspension.

θ1 = Angle made by the fixed link to the vertical.

θh „ = Angle between the connecting beam and the vertical after rotating the lower A arm

by an angle ang.

θa „= Angle between the upper A Arm and the vertical after rotating the lower A arm by

an angle ang.

β and β’ = Angle between links l2 and l3 before and after rotating the lower A arm by an

angle ang.

γ and γ’ = Angle between links l3 and l4 before and after rotating the lower A arm by an

angle ang.

yv = Vertical distance travelled by the end of the suspension for a rotation of the lower A

arm by an angle ang.

xin = Distance of the Instant center from the wheel

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yin = Distance of the roll-center from the vertical.

Note: All the lengths here are normalized with half the width of the car, i.e. lw/2.

1.3 MATHEMATICAL MODELS:

We have mathematical formulation for all the quantities used for the objective functions

and constraints as defined in the problem statement. The detailed expressions of the

model are given in Appendix I.

To evaluate the jounce distance yv, we increase the lower arm angle from θ to θ’, such

that θ - θ’ is ang. All the other angle changes are evaluated in Appendix I. These

angles should not exceed the input angle ang, and if they do, it leads to large stresses

in the flexures that make up the compliant mechanism.

The instant center is the center about which the wheel rotates instantly when it

encounters some jounce. To maintain perfect straight line motion, the instant center

should be at infinity, ideally. Furthermore the roll-center is the point about which the

cornering forces act when the vehicle takes a turn. The roll center is measured here

from the ground and it is given by a very simple relation, which includes the instant

center as

2 in w

r

in

y ly

x Eq 1.1

This shows that once the width of the car with which all the lengths are normalized with

is fixed, the roll center is inversely proportional to the instant center distance from one

end of the wheel and is its value completely determines the roll-center height. Thus, the

roll-center height and the instant center height cannot be independently imposed as two

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independent constraints. Thus, in the present problem only constraint on the roll-center

is imposed.

1.4 SUMMARY MODEL:

Minimize – yv (the vertical jounce displacement)

w.r.t variables l1 , l2, Lh (or l3), l4, θ1 and θt

Subject to the constraints

g1: 3 4 0l l

g2: 3 4 0l l

g3: 4 3 0l l

g4: 3 4 ' 0l l

g5: 3 4' 0l l

g6: 4 3' 0l l

g7: ' 0ang

g8: ' 0angl

g9: ' 0a a ang

g11: * 0r ry y

and bounds on the design variables

1 1 1

2 2 2

3 3 3

1 1 1

lb ub

hlb h hub

lb ub

lb ub

lb ub

tlb t tub

l l l

L L L

l l l

l l l

1.5 MODEL ANALYSIS:

The problem specification was assumed to be well constrained and bounded.

Optimization of this objective function subjected to the set of constraints. Some trade-

offs expected are

Triangle inequality

Constraints on the rotation of

the joints

Constraints on the roll center.

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1) Angles of link rotations and the vertical jounce: These tend to make the links as

close to the configuration in Fig. 1.1c as possible as making the links

perpendicular to each other is the only way we can achieve a high vertical jounce

for limited angles of link rotations.

2) Roll center: Having a high roll-center would require from Eq. 1.1 a small instant

center, which instantly conflicts with the angle and the jounce requirements.

Otherwise, no monotonicity analysis can be directly seen from the expressions as they

are highly convoluted and nonlinear. This means that more about this model could be

revealed from optimization alone. Optimization was carried out using the Sequential

Quadratic Programming algorithm in the MATLAB‟s optimization toolbox. The necessary

parameter values for the problem are shown below.

Lc = Length of the connecting arm of the suspension to the hub of the wheel =0.2 lw

ltw = width of the wheel = 0.1lw

lw = Breadth of the car =1

lt = Length of the wheel.= 0.8 lw

ang = Angle by which θt is changed to obtain θt’ = π/20

yr = The roll-center height = lw/4.

The initial value, upper bound, lower bound and the optimized value of the design

variables are shown in Table 1.

Table 1.2 Specifications and optimized results for the problem mentioned

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l4 η2= l4 - l2 ηh =lh - l4 θt θh θa

Initial value

0.1 0.3 0.3 10*pi/180 10*pi/180 -10*pi/180

Upper bound

1.2 1 0.9 20*pi/180 20*pi/180 20*pi/180

Lower bound

0 0 0 -20*pi/180 -20*pi/180 -20*pi/180

Optimized value

0.1797 0 0.9 20*pi/180 -20*pi/180 -2.61*pi/180

Figure 1.3

The initial and final configuration of the four-bar mechanism

The algorithm convergence details are given in the appendix. Some of the key features

of the result given by the algorithm are

1) The initial jounce travel was 0.060332 units (normalized with respect to the width

of the car). The final jounce travel was around 0.217668 units. This is an

improvement close to 300%.

2) The intermediate link length lh went to the upper bound along with the angle θt.

The angle θh went to the lower bound.

ltw

l1

l2

Lh or l3

l4

θt

θ1

θa

θh

Lc

β

γ

Initial mechanism

configuration

Optimized

mechanism

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3) Two constraints, one bounding the angle of rotation of link l4 and the roll-center

constraints were active. The Lagrange multipliers corresponding to these

constraints were found to be positive.

4) The hessian was found to be positive definite.

5) The optimized results were independent of the initial guess indicating

some sort of a localized convexity within the range of the problem.

The lengthening of the links was evident from the problem specifications because that is

the only way that one can get large displacements for small link rotation angles. Thus

the optimized solution is in sync with intuition.

The constraint history is shown in the next page with the function evaluations

Figure 1.4.

Convergence history of the algorithm

The only parameter worth varying in this problem is the value of the roll-center

constraint. This is shown in Table 1.2.

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Table 1.2 Parametric variation

1) yr = lw/5 Optimized values: l4=0.1797 η2 = 0.0000 ηh=0.9000 θt = 0.3491 θh = -0.3491 θa =-0.0457

2) yr = lw/3

Optimized values: l4=0.0678 η2 = 0.0000 ηh=0.9000 θt = 0.3491 θh = -0.3491 θa =0.2207

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MUKUND NEMALI (DYNAMIC ANALYSIS)

2 PROBLEM STATEMENT: The dynamics analysis model is developed to study the

dynamic performance of the compliant A-arm topology. The objective of this analysis is

to maximize the natural frequency of the suspension mechanism so as not to cause

degradation in ride or cause any unwanted NVH characteristic. The suspension

mechanism will have a fundamental frequency mode and other higher multiple modes of

vibration. The interest here is in the first natural frequency mode.

2.1. MATHEMATICAL MODEL:

Maximize Natural Frequency ( i ) of the suspension topology

Maximize i in 0 ii XMKXF

Subject to:

ul

uil

lonv

latv

vv

G

AAAG

KKG

KKG

KKG

2,1

*

:5

:4

10:3

10:2

:1

Objective function is a matrix equation relating the mass and stiffness of the suspension

topology expressed as an Eigen value equation to solve for the system natural

frequencies. Though the problem solution will have multiple Eigen values (natural

frequencies) that satisfy the equation, the interest in this problem is to maximize the first

mode natural frequency.

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Constraint G1 establishes an upper bound on the vertical stiffness in the suspension.

The vertical stiffness is required to be low in order for the suspension to be flexible in

the vertical direction. This is because in a compliant suspension the kinematics is

achieved by the elasticity in the structure. The vertical stiffness is also required to be

high in order to maximize the natural frequency of the suspension in order to have good

vehicle ride characteristics. The vertical stiffness has conflicting requirements.

Constraint G2 establishes a constraint on the lateral stiffness in terms of the vertical

stiffness. The lateral stiffness has to be greater than 10 times the vertical stiffness for

good handling. The longitudinal stiffness has to be higher for better energy absorption

during acceleration and braking.

Constraint G3 establishes a constraint on the longitudinal stiffness in terms of the

vertical stiffness. The longitudinal stiffness has to be greater than 10 times the vertical

stiffness for better straight line performance during acceleration and braking.

Constraints G4 and G5 establish bounds on the geometric variables of the suspension

design.

2.2. NOMENCLATURE:

K – Stiffness of the suspension mechanism in N/mm

M – Mass of the suspension mechanism in kg

E – Young's Modulus in N/mm2

- Density of the beam material

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i - Natural frequencies of the suspension mechanism in Hz

Kv – Vertical Stiffness in N/mm

Kv* - Target Vertical Stiffness in N/mm

Klat – Lateral Stiffness in N/mm

Klon – Longitudinal Stiffness in N/mm

Al – Lower limit for beam cross section area in mm2

Au – Upper limit for beam cross section area in mm2

Ai – Cross section area of each beam element in mm2

l - Lower limit of the angle between the two legs of the A-arm in Deg

u - Upper limit of the angle between the two legs of the A-arm in Deg

1 - Angle between the two legs of the upper A-arm in Rad

2 - Angle between the two legs of the lower A-arm in Rad

The cross section area Ai and angle are variables in the optimization algorithm and

the stiffness values (Kv, Klat, Klon) and mass values M are parameters, whose values do

not change in a particular run of the optimization algorithm. Parametric study was

conducted for different values of stiffness, mass and elasticity.

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Figure 2.1

Figure 2.1 shows a double a- arm suspension. This is the suspension topology that will

be analyzed.

2.3. SUMMARY MODEL:

The original mathematical programming problem is formulated in the negative null form

Minimize i in 0 ii XMKXF

Subject to:

0:7

0:6

0:5

0:4

010:3

010:2

:1

2,1

2,1

*

u

l

ui

il

v

lon

v

lat

vv

G

G

ddG

ddG

K

KG

K

KG

KKG

Ai

The areas of cross-section Ai

for each beam

1 2

3

4

5

6

7

8

9 10

11

12

13 14

15

α1

α2

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2.4. MODEL ANALYSIS:

Figure 2.2

The topology is assumed to be made of solid tubes of circular cross section. Each beam

is considered to have homogeneous material properties. Each a-arm has two legs and

each leg is divided into three circular beam sections. The two legs are assumed to be

symmetric in the optimization model. The beam cross section diameters of each leg of

the upper arm are: 321 ,, ddd and the beam cross section diameters of each leg of the

lower arm are: 654 ,, ddd . Diameters 987 ,, ddd are kept constant. Figure 2.2 shows the

suspension topology model that will be used for optimization. There are totally 8 design

variables (six diameters and two angles) in the optimization model. All the diameters ( id

) and the angles ( 2,1 ) can be chosen independently. For a given material, density and

Young's Modulus are fixed. Therefore, natural frequency only depends on cross section

area iA of each circular beam section, which in turn is a function of the diameter id only

and the angle between the a-arms . Constraint G1 is active because it is an equality

Ai

The areas of cross-section Ai

for each beam

3 2

1

1

2

3

7

8

9 4

5

6

4 5

6

α1

α2

P a g e | 25

constraint. Constraints G2 and G3 are active because they are given in terms of vK and

bound the maximum value of the stiffness from above. Stiffness values are given as

parameters in the simulation. Constraints G4 and G5are bounds on the diameters of

each beam cross section. The diameters are bounded from above and below.

Constraints G6 and G7 are bounds on the angles between the legs of the upper arm

and the lower arm. The angles are bound from above and below. The objective of the

constrained optimization problem is to optimize the cross section diameters and a-arm

angles to maximize the first mode of the natural frequency.

2.5. NUMERICAL RESULTS:

The optimization was run in MATLAB using the medium scale constrained optimization

algorithm. This algorithm uses Sequential Quadratic Programming with line search to

solve the optimization problem.

Case 1: Kv = 5, Klat = 100, Klon = 100, Material = Steel, 9107850 X , 5102XE

Figure 2.3

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In the first simulation the maximum number of iterations was set to 20. The algorithm

terminated because the maximum number of iterations was reached. Figure 2.3 shows

that the natural frequency keeps increasing and started to stabilize after about 200

iterations. However, since the algorithm terminated no conclusion can be made about

the convergence of the objective function and the optimum value.

Case 2: Kv = 5, Klat = 250, Klon = 250, Material = Steel, 9107850 X , 5102XE

Figure 2.4

In the second simulation the maximum number of iterations was set to 30. The

algorithm terminated because the function value did not converge within the maximum

number of iterations. Figure 2.4 shows the negative of the natural frequency and hence

it is shown as a decreasing function. The algorithm continues to minimize the function

as long as no constraints are violated. If any one of the constraints is violated then the

algorithm searches in a direction that no longer violates the constraints. Therefore, the

above graph shows the values decreasing and as soon as a constraint is violated the

value increases and reaches an optimum where none of the constraints are violated.

However, since the algorithm was terminated because the maximum number of

iterations was reached, the obtained natural frequency value is not the optimum.

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Case 3: Kv = 5, Klat = 250, Klon = 250, Material = Aluminum, 9102730 X , 31069XE

Figure 2.5

4602.53

9202.33

6734.11

4683.2

8919.5

7732.8

2637.19

6244.19

2

1

6

5

4

3

2

1

d

d

d

d

d

d

Figure 2.5 shows the graph of the objective function value at the end of every iteration.

The algorithm terminated after the maximum number of iterations was reached. The

objective function value did not converge within the maximum number of iterations. The

max constraint value did not go to zero. A zero value indicates that all the constraints

were satisfied. The algorithm could not find an optimal point where all the constraints

were satisfied. This is possibly because the constraint values specified are too stringent.

The parameter stiffness has conflicting requirements in trying to minimize the rigidity of

the suspension in order to achieve the required kinematics while simultaneously trying

to maximize the natural frequency. Low rigidity requires low mass and hence low

P a g e | 28

stiffness and high natural frequency requires high stiffness for a given mass of the

suspension. The lack of an optimal solution suggests that it is very difficult to achieve

this trade off in a real vehicle suspension and therefore a compliant suspension is not

feasible in a real vehicle. It is only applicable to small scale and light weight

applications.

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ROHAN SINGH

(STRUCTURAL ANALYSIS)

3.1. PROBLEM STATEMENT:

As stated in the introduction to the report, the A-arm compliant suspension system is

attached to the chassis of the vehicle at four different points. The suspension

mechanism‟s main purpose is to absorb the energy from the shocks and impulses that

act on the vehicle from the road and various driving conditions. In doing so the

suspension is acted upon by a large number of forces, moments and torques in different

directions. The range of loads that act upon the suspension can be very large from a

minor bump to a bump in a large pothole which can create high amount of impact

stresses. These loads will create a large amount of stress in the beams of the A-arm

mechanism. Since this suspension system is a compliant mechanism, made up of semi

rigid compliant beams (Euler beams), the stresses created by these forces can be very

large and may exceed the yield strength of the A-arm beams. This would prove to be

disastrous for the vehicle as it could have fatal results. Thus the main aim of my sub

system optimization is to reduce or minimize the maximum stress that occurs in the

complaint suspension and thus keep the stress levels in the beams below the yield

stress levels and make sure they have a high fatigue life.

The various forces, moments and torques that can act on the suspension from the

various road and driving conditions are listed below:

P a g e | 30

3.1.1 Vertical impulse force: the force that acts on the wheel when it bumps into a

pothole or goes over rough terrain. This causes a large amount of impulse

force, most of which is absorbed by the suspension. In doing so a large

amount of stress is created in the beams to absorb these forces. These

forces act in the vertical direction denoted by the positive Z direction.

3.1.2. Braking/Traction forces: the forces acting on the suspension system by the

traction provided to the wheel by the engine or the braking forces that are

acted upon by the brakes onto the wheel. These forces act in the longitudinal

direction which is denoted by the positive Y direction.

3.1.3. Cornering forces: these are the forces experienced by the wheel/suspension

during the cornering of the vehicle. Cornering leads to large amount of forces

acting in the lateral direction, denoted by the positive X direction.

3.1.4. Aligning torque: aligning torque is caused due to the self aligning property of

the wheel to center itself automatically after a turn. This causes an aligning

moment around the positive Y direction.

The direction of various forces acting on the wheel/suspension is indicated in the

figure on the next page.

P a g e | 31

Figure 3.1

Compliant A-arm suspension is a new area of suspension design. Thus I have not

been able to find a lot of literature on optimization of a compliant suspension system.

Thus A-arms in the suspension as designed as three different elements thus

represent three different nodes. Each element will be having a length which is a ratio

of the total length of the A-arm. This ratio is a variable and will change according to

the optimization. The diameter for the a-arm is also a variable along with the length

of the upper and lower A-arms. Thus the problem has 7 variables which can change.

The stress levels also depend on the material I will be using. Thus the initial material

chosen is steel and thus the elasticity modulus and the density of the material

become parameters in the problem and can be later changed in case of a parametric

optimization.

M aligning

F shock/bump

F cornering

P a g e | 32

Figure 3.2

3.2. NOMENCLATURE:

The nomenclature is given below in table 3.1. The nomenclature was given with a view

of providing a clear understanding of the different parameters and variables.

symbol units description

Lh mm length of beam connecting the upper and lower a arms

thetah radians angle between the connecting beam and Z axis

Uaf mm length of upper a arm front

Uar mm length of upper a arm rear

thetaua radians angle between the upper a arms

alphauaf no units ratio of element 1 of upper a-arm(front) to the length of a arm

etauaf no units ratio of element 2 of upper a-arm(front) to the length of a arm

betauaf no units ratio of element 3 of upper a arm(front) to the length of a arm

alphauar no units ratio of element 1 of upper a-arm(rear) to the length of a arm

etauar no units ratio of element 2 of upper a-arm(rear) to the length of a arm

betauar no units ratio of element 3 of upper a arm(rear) to the length of a arm

Laf mm length of lower a arm front

Lc

Lh

Uaf

thetah

Uar

Laf

Lar

thetala

thetaua

alphauaf

thetab

thetat

betauaf

etauaf

Alphauar

betauar

etauar

alphalaf

betalaf

alphalar

etalaf

etalar

betalar

dc

dc1

thetah

lamdab

lamdat

P a g e | 33

Lar mm length of lower a arm rear

thetala radians angle between the lower a arms

alphalaf no units ratio of element 1 of lower a-arm(front) to the length of a arm

etalaf no units ratio of element 2 of lower a-arm(front) to the length of a arm

betalaf no units ratio of element 3 of lower a-arm(front) to the length of a arm

alphalar no units ratio of element 1 of lower a-arm(rear) to the length of a arm

etalar no units ratio of element 2 of lower a-arm(rear) to the length of a arm

betalar no units ratio of element 3 of lower a-arm(rear) to the length of a arm

Lc mm length of the arm connecting the mechanism to the wheel hub

lambdat no units point at which the rod connecting the damper is placed on the a arms

lambdab no units point at which the rod connecting the damper is placed on the a arms

thetat radians angle made by upper a arms with the X axis

thetab radians angle made by lower a arms with the X axis

d mm diameter of cross section of the beams

dc mm diameter of the plate holding the damper

dc1 mm diameter of the beam defined by Lf Table 3.1

3.3. MATHEMATICAL MODEL:

General ideology about the suspension systems has always been about reducing the

weight of the suspension system in order to reduce the overall mass of the vehicle. This

was one major difference in our project design. Since we were taking compliant beams

into consideration, they were more prone to higher stress because of lack of rigidity.

Thus we had to ensure that the stress never reached a value that was greater than the

yield strength of the material being used.

The problem that we encountered while evaluating the point or node where the stress

was maximum in the system was that, with different geometries the node with the

maximum amount of stress was varying. The maximum stress nodes were all the nodes

which were connected to the chassis of the vehicle. Thus the four nodes 17,18,21 and

22 Were the ones which were experiencing the maximum amount of stress, but

P a g e | 34

depending on the length of the a arms, the angle between them, the different ratios of

the elements the maximum stress was varying from one node to the other. Thus to

overcome this problem I took the maximum value of stress at each of the nodes in every

iteration and then tried to minimize the stress at that particular node. Thus this formed

the main objective function of the problem.

Zav= ([abs (Z18), abs (Z17), abs (Z22), abs (Z21)]) Yav= ([abs (Y18), abs (Y17), abs (Y22), abs (Y21)])

Objective function= Stress=max (Zav+Yav)

3.4. CONSTRAINTS:

Thus there were varying set of constraints that had to be enforced on the system.

Let us begin with the first and most important of constraints.

3.4.1. Jounce of the suspension: the main objective of the function is to absorb the

bumps and provide passenger comfort. This is accomplished by the spring

damper system in the suspension which makes the suspension to move in

the vertical direction and thus absorbs the various road and engine inputs.

Thus the greater distance it moves the better it absorbs the energy of the

bumps. But due to space requirements and vehicle design, it cannot have a

large value of deflection. Thus it has limit on the maximum deflection of the

wheel. This is basically to stop the wheel from bottoming out. Another

important factor that was required for the suspension was that it had to

absorb a certain amount of energy to provide passenger ride comfort. Thus

for the different range of forces that it was subjected to it had to deflect by a

P a g e | 35

certain amount in the Z direction to absorb the energy of the bump. Thus

another constraint had to be added, requiring that it had to deflect by a certain

minimum amount in the vertical or Z direction to provide the required

passenger comfort level. This deflection was determined by assuming certain

stiffness for the vehicle and thus calculating the deflection based on that. The

constraint was thus given by

G1:w1-jmax<=0 G2:=jmin-w1

In the above equation w1 gives the displacement of node 2 in the vertical z direction.

The jounce travel jmax and jmin were determined by the available space and the

minimum stiffness requirement for the suspension.

3.4.2. The various forces as shown in figure 1.1 cause a lot of deflection in various

directions which cause unwanted effects in the suspension mechanism. The

cornering force causes a moment about the X axis which causes a rotation

about the y axis thus changing the camber angle. The camber angle plays an

important role in the handling of the vehicle, thus changes in it causes a

negative effect for the vehicles handling. Thus this camber angle change had

to be kept to a minimum. The constraint for this is given by:

G3: (x4+x1)-cambmin<=0

Here the points x1 and x4 give the deflection in the X direction for nodes 1 and

4. Cambmin indicates the minimum camber angle change allowed for the

vehicle.

3.4.3. The braking force causes a displacement in the longitudinal Y direction and

also causes a moment around the vertical Z axis. The moment by the aligning

P a g e | 36

torque also causes a rotation about the vertical Z axis. This rotation causes

the steering angle to change which is undesirable for the vehicle. Thus a

constraint had to be placed on the allowable rotation for the steering angle.

The constraint is given by:

G4: thz3-stirmin<=0

The thz3 represents the change in angle about the Z direction of node 3. The

stirmin value for the suspension was given as half a degree.

3.4.4. Since there were forces acting in the longitudinal direction, I had to bind the

movement of the node 2 in the longitudinal direction, such that it didn‟t

displace greater than a certain amount, thereby causing slipping of the wheel.

G5: uy2-longdpmax<=0

The term uy2 is the deflection caused in the Y axis at node 2. The

Longdpmax was selected on the basis of the suspension having stiffness

which was very high compared to the stiffness in the vertical(Z) direction but not

being infinitely rigid.

3.4.5. One of the main things in any vehicle is to reduce the mass of the vehicle in

order to improve the efficiency and the performance of the vehicle. The

suspension system has to be light in order for the overall vehicle weight to be

light as well as it reduces the unsprung weight of the vehicle which improves

the ride and handling of the vehicle. Thus a constraint had to be added to

bind the upper weight restriction on the suspension system. The constraint

was given by:

G6: Mass-massmin<=0

P a g e | 37

The mass in the above equation indicates the sum of the individual masses of

the beams. The massmin is given by the current weight of a similar rigid

suspension system.

3.5. SUMMARY OF THE MODEL:

MIN (F): [Zav= ([abs (Z18), abs (Z17), abs (Z22), abs (Z21)])

Yav= ([abs (Y18), abs (Y17), abs (Y22), abs (Y21)])]

Subjected to:

G1: w1-jmax<=0 G2: jmin-w1<=0

G3: (x4+x1)-cambmin<=0 G4: thz3-stirmin<=0

G5: uy2-longdpmax<=0 G6: Mass-massmin<=0

{Jmin=10mm

Jmax=280mm Cambmin=2.5mm

Stirmin=0.5*pi/180radians Longdpmax=2mm

Massmin=20kg} Lower bound: Lb= [400, 400, .1, .55, .1, .55, 10];

Upper bound: Ub= [700, 700, 0.4, 0.9, 0.4, 0.9, 40]; Initial point: D (0) = [450, 450, .3, .6, .3, .6, 20]

P a g e | 38

3.6. DESIGN VARIABLES:

The following are the design variables:

Uaf[D(2)] mm length of upper a arm front

Uar[D(2) mm length of upper a arm rear

alphauaf[D(5)] no

units ratio of element 1 of upper a-arm(front) to the length of a arm

alphauar[D(5)] no

units ratio of element 1 of upper a-arm(rear) to the length of a arm

betauar[D(6)] no

units ratio of element 3 of upper a arm(rear) to the length of a arm

betauaf[D(6)] no

units ratio of element 3 of upper a arm(front) to the length of a arm

Laf[D(1)] mm length of lower a arm front

Lar[D(1)] mm length of lower a arm rear

alphalaf[D(3)] no

units ratio of element 1 of lower a-arm(front) to the length of a arm

betalaf[(D(4)] no

units ratio of element 3 of lower a-arm(front) to the length of a arm

alphalar[D(3)] no

units ratio of element 1 of lower a-arm(rear) to the length of a arm

betalar[D(4)] no

units ratio of element 3 of lower a-arm(rear) to the length of a arm

d[D(7)] mm diameter of cross section of the beams Table 1.2

Table 3.2

D (I) represent that the design variables are similar to each other and will have the

same value. Thus I have 7 design variables in this problem and since I cannot find

out the number of active constraints, the degree of freedom of the system is 7.

3.7. DESIGN PARAMETERS:

The main design parameters in terms of the material constants are the density

(rho) and young‟s modulus of the material. These values change when the

material of the suspension is a changed. The other design parameters which

P a g e | 39

relate the design with the suspension geometry are given by

Table 3.3

These parameters have been fixed for the static analysis but can/will change based

on the other subsystems analysis. This will lead to an interesting tradeoff between

different subsystems.

3.8. OPTIMIZATION ANALYSIS:

The main software used for this optimization was MATLAB. The function used was

FMINCON. On starting out with the optimization of the system, I found that the

system was not converging and the number of iterations crossed the number of

iterations allowed in mat lab.

This happened irrespective of the change in the bounds, constraints etc. the function

value was not converging at all.

On trying to analyze the reasons as to why the function was not converging, I came to a

conclusion that since the stress values were varying at different nodes based on the

thetah radians angle between the connecting beam and Z axis

thetaua radians angle between the upper a arms

thetala radians angle between the lower a arms

thetat radians angle made by upper a arms with the X axis

thetab radians angle made by lower a arms with the X axis

lambdat

no units

point at which the rod connecting the damper is placed on the a arms

lambdab

no units

point at which the rod connecting the damper is placed on the a arms

P a g e | 40

different geometries it came up with using different iterations; it could be that the

objective function was not differentiable at some points in the domain. This could be

because the value of the maximum stress which could change from one node to

another based on the geometry of the suspension at any iteration could make the

function highly discontinuous. This could lead to the function not converging, even

though I had double checked the values of the constraints and the bounds and the

forces that were acting upon it. This was confirmed by the fact that the function was not

converging even when all the constraints were relaxed except constraints 1 and 2. This

made me sure that the function was not continuous at every point in the domain. This

was the result of the optimization for a given initial point. This was the case that

happened with every initial point that was taken into the consideration. Another fact

could also be that I had too less constraints and more variables, thus this could also be

another factor in the optimization not working.

This gave rise to a new problem of how to optimize the structural aspect of the

subsystem. In order for the function to structurally stable and be able to take the various

loads acting on the vehicle, the stress at any point could not exceed the yield stress.

Since I was sure that the maximum stress was occurring at one of the four extreme

nodes, which was confirmed by the initial analysis of the suspension system. I decided

to put a stress constraint at those nodes and came up with four constraints that bound

the stress at those nodes from above.

3.4.6. The maximum stress acting at any node in the system has to be less than the

yield strength of the material. Thus on analyses of the suspension system we

P a g e | 41

found that the maximum stresses were occurring at the four nodes 17,18,22

and 21, thus we have to have an upper bound at those four nodes and limit

the maximum stress at those nodes. Thus the constraints are given by:

G6: abs (Z18) +abs (Y18)-yield1 G7: abs (Z17) +abs (Y17)-yield1 G8: abs (Z22) +abs (Y22)-yield1; G9: abs (Z21) +abs (Y21)-yield1

In the above equations Z18, Z17, Z22and Z21 refers to the stresses in the Z

direction fibers at nodes 18,17,22,21 respectively. Y18, Y17, Y22, Y21 denote the

stresses in the Y direction at nodes 18,17,22,21 respectively. Yield1 denotes the

yield strength of the material which is taken as steel. Thus the yield strength is

250MPA.

This lead to a new objective function, in which I looked at minimizing the mass of the

suspension system while maintaining the stress limits under bounds. Thus I changed

the objective function of the problem to minimize the weight of the suspension subjected

to the stress at the four nodes being within the stress limits. Thus the changed

optimization function was given as:

MIN (F) = Mass=sum (mass)

Subjected to:

Con (1) =w1-jmax<=0 Con (2) =jmin-w1<=0

Con (3) =(x4+x1)-cambmin<=0 Con (4) =thz3-stirmin<=0

Con (5) =uy2-longdpmax<=0 Con (6) =abs (Z18) +abs (Y18)-yield1<=0 Con (7) =abs (Z17) +abs (Y17)-yield1<=0

P a g e | 42

Con (8) =abs (Z22) +abs (Y22)-yield1<=0 Con (9) =abs (Z21) +abs (Y21)-yield1<=0

Jmin =10

Jmax=280 Cambmin=2.5

Stirmin=0.5*pi/180 Longdpmax=2;

yield1=250; Lower bound: Lb= [400, 400, .1, .55, .1, .55, 10];

Upper bound: Ub= [700, 700, 0.4, 0.9, 0.4, 0.9, 40]; Initial point: D (0) = [450, 450, .3, .6, .3, .6, 20]

On running the optimization with the changed function, I came across some new

problems. Irrespective of the start point of the function, at some iteration the value of

one of the function variable was being taken as zero and thus I was getting a singular

matrix as a result. This caused the optimization to stop and thus I couldn‟t continue with

the algorithm. Thus I had to try for different initial points to give me a converging

algorithm and a final answer. Thus I tried out the optimization with various different

initial points, till I got one which gave me a converged value and a final optimized

minimum. From the above analysis, the mathematical model seems to be very sensitive

to the initial point and thereby needs careful selection of the initial point for the system

to converge. This suggests that the algorithm has local properties. The upper bounds

had to be increased to a slightly larger value. Though this increase in upper bound can

be accommodated in the vehicle, this would cause the wheel to stick out a bit from the

vehicle.

At the optimum value, the constraints 7, 8 which were the one‟s bounding the stress

from above were active implying that the stress at those points was equal to the yield

strength. This was at nodes 17 and 22. Thus to have a factor of safety I had to reduce

P a g e | 43

the yield strength values at each of the nodes to 245N/MM2 from 250N/MM2. Running

the optimization again the final value are given below.

The final mass of the system is given below:

(F)= 27.6284Kg

Mass at initial point:

(F) = 17.34

The comparisons of the final and initial values are given below:

variable D(i)

initial

value

Optimal

value

upper a arm length 450 646

lower a arm length 450 636

Alphalaf 0.3 0.4

Betalaf 0.6 0.55

Alphalar 0.3 0.4

Betalar 0.6 0.55

Diameter 20 23.5

P a g e | 44

The graph representing the values of the objective function for all the iteration is given

below:

Graph 3.1

The final geometry of the A arm suspension is given below in figure 3.4

Figure 3.4

P a g e | 45

The deflected shape for the final values of the A arm is given by figure 3.5:

Figure 3.5

The deflected shape at the initial starting point is given by figure 3.6:

Figure 3.6

P a g e | 46

The final results also validate the fact that the constraint #7, 8 are active at the optimum

value, but the value of stress at that value is lesser than the yield strength of the

material. Hence it poses no problems for the yielding of the suspension system.

3.9. ANALYSIS:

Lambda values for the system are given by: 0, 2.5814, 0, 0, 0, 0, .1202, .0866, 0. this

confirms that the constraints 2, 7 and 8 are active at the final optimal point.

The following values of the hessian and the Eigen values are given in the

APPENDIX. The hessian is positive semi definite. This indicates that there might

exist, a valley of minimizers for the objective function above. Thus the optimal

point represents a minimum value for the function.

I have found that the length of the upper and lower A arm is 646,636 mm

respectively. Though this can be incorporated in the vehicle it is a bit too long for

the vehicles requirement and may cause the wheel to jut out by 15mm from the

vehicle‟s side fenders. On analyzing it further I came across the fact that though

the diameter has an upper bound of 40mm, the optimum value is around 23mm.

this could have increased slightly and could have negotiated the length to a

certain aspect, but from a mass point of view this would not be appropriate. The

mass would have increased but would have given us a shorter suspension which

could be easily incorporated. But since the main objective is the mass it has

increased the length and kept the diameter as small as possible.

From the optimum value given in the table above we see that the length of the

middle element in all the 4 „A‟ arms has been reduced from the initial value. This

P a g e | 47

must have happened cause the suspension arms need long flexible arms to

absorb the loads and the short rigid element in between just gives it a bit more

rigidity but doesn‟t help out with the load absorption purposes. This was in

expectation with the results I was expecting initially.

The final weight of the suspension system is 27.684Kg, which is quite heavier

when compared to the mass of the rigid suspension. Thus this increases the

unsprung mass, thereby slowing down the reaction of the suspension system.

At the final optimal solution three of the constraints are active as indicated by the

Lagrange multipliers. This indicates that the optimal doesn‟t lie at the interior

point of the function curve but at the boundary. This makes the problem highly

sensitive that is the solution is not a robust one since any error in the final value

of the variables could lead to the violation of the constraints. This could occur

due to manufacturing tolerances which could offset the optimal point by some

length and hence lead to constraint violation. Since the maximum jounce

constraint is active at the optimal point, this could lead to bottoming of the

suspension due to any defects either manufacturing or design of the part.

3.10. SUBSYSTEM TRADEOFFS:

The main objective function of my system was to reduce the mass of the

system, keeping in mind the stress levels at the nodes of the suspension. For a

compliant system the general mass was found to be higher than the regular

suspension system, which is not a good case when the dynamics of the

suspension is considered. For the compliant mechanism to be able to absorb

the loads and to keep the stress below required levels it needs longer arms and

P a g e | 48

thus the overall length of the system increases. For the dynamic analysis the

required length of the arms should be as minimum, as possible. This creates an

interesting tradeoff for the complete system optimization. In my case I have

considered the top plate connecting the upper a arms and holding the damper,

to be very small. In the damper placement analysis, this plate disappears

altogether and this will lead to a tradeoff in which the plate doesn‟t exist, and

the stress levels might vary. This also leads to an interesting tradeoff in the

sense that, the damper forces are going to act on the bottom plate which has

not been taken into consideration in my analysis. In my analysis I have

considered a fixed topology for the initial design, whereas in the kinematic

analysis the topology is going to be varied. This might create the maximum

stresses at different nodes, which may differ from the nodes taken into

consideration now. The final analysis will focus on evaluating the above

tradeoffs and finding out an optimal design to the solution.

P a g e | 49

4. DAMPER PLACEMENT ANALYSIS - by Karan Goyal

4.1 PROBLEM STATEMENT

The objective of this problem is to find the optimum position of the damper on the

suspension mechanism so as to minimize the displacement/travel in the vertical

direction. The position of the damper refers to the points where the damper is attached

to the mechanism. Minimum displacement/travel in the vertical direction is desired to

restrain violent impacts and to attenuate vibrations.

Compared to other vibration dampers, the fluid viscous damper is a favorable device as

its stiffness and damping coefficient are completely independent. The fluid viscous

damper generates damping by the liquid flowing through orifices or valves. Therefore,

the damping characteristics of the vehicle are analyzed by considering a mono-tube

passive hydraulic shock absorber attached between four links of the suspension

mechanism.

The response of a vehicle depends upon the vehicle suspension which in turn depends

upon the road input conditions. Road is the primary input for the vehicle suspension.

This road input could be nonlinear in nature due to the irregularities in the road surface.

Since the main objective is to find the optimum position of the damper and not design a

damper, a unit force was considered as an input for the vehicle suspension system.

A shock absorber in a car is designed to damp the oscillations of the suspension

springs in the car. Without this damping after a car passes over a bump, it will bounce

P a g e | 50

(oscillate) up and down many times rather than just once. Damping in shock absorbers

is obtained by forcing a piston to move through a liquid-filled cylinder with an

appropriate amount of fluid flow through or around the cylinder. This provides a drag

force that is approximately proportional to the speed with which the piston moves. Since

the damper shows a non linear behavior, it is difficult to construct a mathematical model

to find the optimal placement of the damper while taking into account the oil

characteristics, damper orifice dimensions and the size and length of the damper etc.

Therefore, to facilitate the process of analysis and design, the damping is approximately

taken to be linear. Also to include the effects of the size and length of the damper as

well as the oil characteristics and the effect of the orifice dimensions the damping

coefficient is considered to be one of the variables.

Also for the ease of calculations and to model the damper position as close to the real

conditions as possible, the upper end of the damper is considered to be fixed at a

random point on the chassis. The lower end of the damper is fixed to the mounting plate

that is connected to the two arms of the lower a-arm of the mechanism. The position of

this lower end of the damper is what we are really interested in, as after optimizing for

minimum displacement we will find a point where the damper would be connected to the

suspension mechanism. Thus we get three more variables. Two being the distance of

the two ends of the mounting plates from the fixed end of arms of the lower a-arm. The

other being the position of the point on the mounting plate where the damper is actually

connected. Doing this gives us the ability to position the damper anywhere on the plane

between the two arms of the lower a-arm.

P a g e | 51

4.2 NOMENCLATURE:

F: The external force applied

K : Stiffness of the suspension mechanism in N/mm

M : Mass of the vehicle in kg

C : Damping Coefficient

x1 : Position of the point on the lower link where the damper is attached

x2 : Position of the point on the lower link where the damper is attached

Y : Displacement/Travel in the vertical direction in mm

Ỳ : Velocity in mm/sec

Ÿ : Acceleration in mm/sec2

Laf: Length of the lower a-arm front in mm

Lar: Length of the lower a-arm rear in mm

Lambdat : the distance between the fixed end of the front link of the lower a-arm and the

point where the mounting plate is connected to the link in mm

Lambdab : the distance between the fixed end of the rear link of the lower a-arm and

the point where the mounting plate is connected to the link in mm

Lmp : length of the mounting plate in mm

P a g e | 52

Lambdap : the distance between the end of the mounting plate on the rear lower a-arm

link and the point where the damper is connected

Figure 4.1

4.3 MATHEMATICAL MODEL

In the objective function below, the minimum value of x that satisfies the equation needs

to be determined. This function takes into account the mass of the vehicle while

neglecting the un-sprung mass i.e. the mass of the wheels. It also takes into account

the stiffness of the spring or in the case of compliant suspension mechanism, the total

stiffness of the beams. Also considered is the damping coefficient of the shock

absorber. A linear relation between the damping force and the velocity is considered.

The unit damping force is applied at the point as shown in the accompanying figure.

P a g e | 53

The lesser is the displacement of the piston in the damper, the lesser is the overall

displacement at the wheel. While we are minimizing the maximum displacement of the

damper, we will still be able to find the optimum position for the damper as a function of

the position of the damper‟s lower end on the mounting plate. The mathematical model

for this configuration is obtained using COMSOL and the optimization is performed

using MATLAB.

VARIABLES

Two of the variables are the position of the points on the beams of the lower a-arm

where the mounting plate is connected to the beams. The third variable is the position of

the lower end of the damper on the mounting plate itself. The fourth variable is the

damping coefficient. It was decided to include the damping coefficient itself to account

for the possibility of using a different damper with different working characteristics. The

design of the damper for minimum displacement could be a separate optimization

problem in itself.

CONSTRAINTS

The major constraints are on the position of the lower end of the damper on the

compliant mechanism. Its position could be anywhere in the plane between the two

beams of the lower a-arm. Thus it was decided to include a mounting plate that is

connected between the two beams of the lower a-arm. The damper can be mounted

anywhere on this plate. The constraints and hence the optimization problem would

determine the location of the point in this plane where the displacement would me

minimum. Other constraints were placed on the value of the damping coefficient. In the

P a g e | 54

absence of a mathematical model that would take into account the design

characteristics of a damper such as the oil properties, orifice diameter and size and the

diameter and the length of the damper, it was decided to limit the value of the damping

coefficient. Since only a unit force is being applied for the purpose of analysis, the

damping coefficient was reduced by a factor of hundred in the optimization model. Thus

the position of the damper is dictated by the length of the beams and the length of the

mounting plate. The position of the upper end of the damper is considered to be fixed to

the chassis to generate a model as close to the real solution.

PARAMETERS

For this subsystem the mass and the stiffness, the topology of the four bar mechanism

and the area of the cross section of the beams are the parameters. The fluid used while

finding the damping coefficient for the different dimensions of the damper is assumed to

be the same in all cases. Parametric study can be performed by using different set of

values for each of the parameters listed above. But for this optimization model, the

parametric study was performed only on the position of the fixed upped end of the

damper. The position of the fixed end of the damper on the chassis was changed in

case. This also led to a change in the length of the damper but its effect was assumed

to be included in the determination of the damping coefficient.

Note: This sub-system has a single degree of freedom as the only motion possible is in

the vertical direction.

P a g e | 55

4.4 SUMMARY MODEL

Objective Function is:

Min Y in f(Y) = MŸ + CỲ + KY – F

s.t.

g1: -C ≤ 0

g2: C -2 ≤ 0

g3: - Lambdap ≤0

g4: Lambdap - Lmp ≤0

g5: - Lambdat ≤0

g6: Lambdat - Laf ≤0

g7: - Lambdab ≤0

g8: Lambdab - Lar ≤0

4.5 MODEL ANALYSIS

The main optimization software being used for this subsystem was MATLAB. Initially the

position of the upper end of the damper was also considered to be a variable.

Constraints similar to the lower end of the damper were placed on the upper end too.

But it gave difficulty in determining the difference in the velocities of those two points for

determining the net displacement. Further, the use of two dampers instead of one was

also considered. This system would have one damper each connected to one of the

P a g e | 56

mounting plates. The other end of the dampers would be fixed. In this system too, using

MATLAB, it was difficult to obtain considerable displacement values.

Finally, it was decided to use a single damper with one end fixed to the chassis and the

other end connected to the compliant mechanism using a mounting plate. A quarter car

model was used for performing simulation. The mass was assumed to be constant and

the stiffness value of the compliant system was also assumed to be constant during the

parametric study. Only one variable i.e. is the damping coefficient is included in the

objective function. This variable is well bounded in both the directions. The damping

coefficient directly relates to the displacement if the damper and of the overall system.

The other variable relate directly to the position of the damper. Due to the absence of

any mathematical relation between the position of the damper and its displacement

characteristics, these variables could not be included in the objective function. But all

the constraints are active and none of the four variables is redundant. This was shown

during the parametric study. Varying the position of the fixed end of the damper gave

reasonably different values for the variables.

Using MATLAB, the optimized results for the four variables were obtained. These

values were obtained in a single iteration. After that the parametric study was

performed.

4.6 RESULTS

The position of the damper on the lower mounting plate can be visualized with the help

of the following image. The upper end of the damper is fixed to the chassis.

P a g e | 57

Figure 4.2

Figure 4.3

x

y z

z

y x

P a g e | 58

Sr.

No.

Parameter Design Variables Displacement

Fixed Point Lambdat Lambdab Lambdap C

1 [400 0 300] 0.3 0.3 0.9 2 9.7663e-004

2 [350 0 300] 0.7956 0.8 0.3992 .9991 9.7673e-004

3 [200 0 300] 0.3 0.3 0.8977 1.9998 9.5804e-004

4 [400 0 200] .7903 .8 .4065 .9938 9.7600e-004

5 [400 0 100] .3 .3 .8977 1.9998 9.5354e-004

6 [400 300 300] .3 .3 .9 2 9.9714e-004

7 [400 200 300] .3 .3 .9 2 9.8415e-004

8 [400 100 300] .7919 .8 .4042 .9954 9.7762e-004

Table 4.1 shows the result of the optimization model and the parametric study.

The results indicate that the position of the fixed point has considerable effect on the

values of the different design variables considered in this problem.

1. Changing the x coordinate of the fixed point has a major affect on the position of

the damper on the mounting plate. The position of the mounting plate itself

doesn‟t vary much if only the x coordinate of the fixed point is changed. It shows

the tendency of minimizing the length of the damper. It means the while finding

for the minimum displacement, the length of the damper is also being optimized.

2. The results also show the non linear nature of the model. This problem could be

solved only after simplification by removing some of the design variables and

constraints that were considered in other models.

P a g e | 59

3. The parametric study also shows that, the position of the fixed point would have

a big impact on the overall displacement. It is shown by the fact that for the

different positions of the fixed point, we get same values for the variables but

different value for the overall displacement.

4. The nature of results obtained in the parametric study also highlight that the

length of the damper may also be an important factor in the determining the

placement of the damper. If we take the length of the damper into consideration,

we would also need to consider other design variables of the damper such as the

diameter, oil viscosity, orifice diameter etc. Thus, a more complex model would

be required.

5. While considering the length of the damper, we need to be extra careful with

setting the constraints/bounds on the length of the damper. As shown in the

results, decreasing the length may lead to a lower displacement, but an

extremely small damper may not be feasible. One way of overcoming this could

be the alignment of the damper at certain angles to get the same damping

characteristics.

6. From the parametric study we have been able to establish that the placement of

the damper in an automotive suspension system has considerable effect on the

displacement characteristics of the damper.

P a g e | 60

4.7 CONCLUSION & FUTURE WORK

The future work may involve creation of more complex mathematical models for taking

into consideration different damper characteristics. Taking just both the ends of the

damper as free and not fixed would involve an exhaustive study into the relationship

between the velocity difference of the two free points and the damping force. Design of

the damper could be considered a separate sub-system in itself.

P a g e | 61

FINAL SYSTEM INTEGRATION

For the final system integration we considered various different objectives like the best

kinematic topology, the best natural frequency and the best mass for the compliant A-

arm suspension. On analyzing the different objectives we came to the conclusion that

the main requirement for the suspension would be to reduce the mass of the compliant

A-arm suspension. The mass plays a very important role in the performance of a

suspension and can make a lot of difference. Thus minimizing the mass seemed to

increase the performance by a bigger margin when compared to the other objectives.

Thus minimizing the mass became our final objective function which was subjected to

various constraints derived from other subsystems.

Final Objective: MIN: F = MASS

5.2. CONSTRAINTS:

Since we carried over the structural optimization objective to our final system objective

the constraints that were placed in the structural optimization were all valid constraints

for the final system integration. Those are given by:

G1 =w1-jmax<=0

G2 =jmin-w1<=0

G3 =(x4+x1)-cambmin<=0

G4 =thz3-stirmin<=0

G5 =uy2-longdpmax<=0

G6 =abs (Z18) +abs (Y18)-yield1<=0

G7 =abs (Z17) +abs (Y17)-yield1<=0

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G8 =abs (Z22) +abs (Y22)-yield1<=0

G9 =abs (Z21) +abs (Y21)-yield1<=0 The additional constraints that were added form the different subsystem optimization are given as follows.

G10 =1-nat

G11 = riset-0.8

G12 = 0.1-riset

The constraint g10 places a lower bound on the natural frequency derived from

Mukunds dynamic optimization. The lower bound is given as 1Hz. This is selected

because 1Hz is the maximum natural frequency for a general automobile, Hence the

value.

The constraints g11 and g12 basically place a lower and upper bound on the rise time

of the damper. This is generally taken as an ideal rise time for the vehicle of our class.

Hence it is bound from the top and from the bottom. A very short rise time would disrupt

passenger comfort and a very large rise time will allow the suspension to vibrate for a

longer time. This range basically provides an ideal range for the rise time to be in.

The next sets of constraints are derived from Girish‟s kinematic optimization. It places a

constraint on the maximum angle of rotation for the suspension arms and on the

kinematic jounce of the suspension.

G13 = 0.19-kinob (Gvar)

G14= gcons (7)

G15 = gcons (8)

G16=gcons (9)

G17=gcons (10)

P a g e | 63

5.3. FINAL SUMMARY:

The final problem is

Final Objective: MIN: F = MASS

Subjected to:

G1 =w1-jmax<=0 G2 =jmin-w1<=0

G3 =(x4+x1)-cambmin<=0 G4 =thz3-stirmin<=0

G5 =uy2-longdpmax<=0 G6 =abs (Z18) +abs (Y18)-yield1<=0 G7 =abs (Z17) +abs (Y17)-yield1<=0 G8 =abs (Z22) +abs (Y22)-yield1<=0 G9 =abs (Z21) +abs (Y21)-yield1<=0

G10 =2-nat G11 = riset-0.8 G12 = 0.1-riset

G13 = 0.19-kinob (Gvar) G14= gcons (7) G15 = gcons (8) G16=gcons (9)

G17=gcons (10)

variable Start point Lower bound Upper bound

X1 0 1

X2 0 1.2

X3 .5 1.3

Thetab -20 degree 20 degree

thetah -20 degree 20 degree

thetaa -20 degree 20 degree

thetala 5 degree 80 degree

thetaua 5 degree 80 degree

alphala .15 .45

alphaua .15 .45

betala .7 .95

betaua .7 .95

etala .5 .65

lamdab .1 .95

D1 1 25

D2 1 25

D3 1 40

D4 1 25

D5 1 25

P a g e | 64

D6 1 40

The lengths in the final integration have been normalized with respect to the semi width

of the car. This has been done in order to scale the variables and to try and get decent

convergence of the objective.

The number of variables for the final problem is 20 and there are 17 constraints. The

parameters here are the material properties and the damper properties. A parametric

optimization will be done with respect to the material properties, mainly the elasticity

modulus and the density.

5.4. OPTIMIZATION ANALYSIS:

We used OPTIMUS5.2 for the final optimization at the system level. When we started

the optimization initially it was giving us the warning saying that time stepping has been

exceeded. This indicated a problem with the constraint of the damper analysis. This was

rectified and the optimization was carried out using different algorithms for optimization

like NLPQL, SQP, SELF ADAPTIVE SIMULATION, GRG and SIMULATED

ANNEALING. All these algorithms failed to converge. At the end points we still had

some constraints which were still not satisfied. Hence we tried to diagnose what was

wrong with the formulation. Hence we first scaled all the variables to being in between 0

and 1. Then some of the constraint that could be relaxed by a certain margin were

relaxed. We also found that constraint g13 corresponding to the kinematic jounce was

equivalent to the constraint g2 which was the actual jounce of the vehicle on application

of the forces. This was being dominated by the constraint g2 and hence was removed

P a g e | 65

from the final optimization. Hence the final problem consisted of 20 variables and 16

constraints.

Final problem:

Final Objective: MIN: F = MASS

Subjected to:

G1 =w1-jmax<=0 G2 =jmin-w1<=0

G3 =(x4+x1)-cambmin<=0 G4 =thz3-stirmin<=0

G5 =uy2-longdpmax<=0 G6 =abs (Z18) +abs (Y18)-yield1<=0 G7 =abs (Z17) +abs (Y17)-yield1<=0 G8 =abs (Z22) +abs (Y22)-yield1<=0 G9 =abs (Z21) +abs (Y21)-yield1<=0

G10 =2-nat G11 = riset-0.8 G12 = 0.1-riset G13= gcons (7) G14 = gcons (8) G15=gcons (9)

G16=gcons (10)

For the final optimization run all the algorithms were not terminating. They all ended

after reaching the maximum number of iterations and were still violating some of the

constraints at the final point. Hence what we decided to do was, to run the optimization

for different algorithms and at the end point select the algorithm that gives us the best

end point and then use this end point as the initial point for the next run of the

optimization. This continued iteratively for 4 times till we reached an initial point for

which the sequence ended with all the constraints being satisfied. It still didn‟t give us

an optimal point, but at least gave us a feasible point for the design. On carrying out the

optimization again for different algorithm with the feasible initial point, the optimization

did not reach a termination point. Hence we decided to put in the best result of the

P a g e | 66

entire algorithm as our feasible optimal design. This was done due to lack of time.

Otherwise it could be revised and most probably could have got a optimal value. The

results are mentioned in appendix 4.1. The initial point mentioned is not the actual initial

point that we started out with but the initial point at the final sequence of optimization.

The result here represents the best value of the results obtained from all the different

algorithms it was run on.

Initially we considered steel as a material for the initial structure and the mass

that we got in the structural optimization was 27Kgs. This was a local minima

corresponding to the initial point that was used for the structural optimization.

When we ran the optimization for the final integration using steel as the material,

it was not able to satisfy the constraints for any given start points. We followed

the iterative process for steel and magnesium alloy and high strength steel, but

these materials could not satisfy the requirements for the suspension design.

Hence we used carbon fiber and titanium as the material and they showed us

some promise. Hence we went ahead with these materials for the suspension.

The results shown here are for carbon fiber. We did not have enough time to

carry out the final sequence of optimization for titanium as a material. Hence the

mass has been reduced by such a high percentage when compared to steel as

the material.

FINAL MASS: 4.72177kg

FINAL MASS FOR SUB SYSTEM OPTIMIZATION(STEEL): 27Kg

INITIAL MASS FOR INITIAL POINT DURING FINAL OPTIMIZATION

SEQUENCE: 15Kg

P a g e | 67

Hence this represents an improvement of 306.54% when compared to the initial

point in final sequence of optimization. This is actually a very big improvement

and could lead to a very light weight suspension.

The final suspension geometry is shown in figure 5.1. This represents a feasible

design not an optimal one.

Figure 5.1

P a g e | 68

The deflected shape for the forces applied as shown in figure 5.2

Figure 5.2

5.5. RESULT:

The results mentioned in the appendix 4.1 show that the final optimization runs yielded

a design that is very light. Some of the conclusions that can be arrived from the final

results are:

Compliant suspension design needs highly exotic materials such as carbon fiber,

titanium etc to be able to satisfy the requirement for vehicular suspension for the

given topology mentioned in our problem.

P a g e | 69

Compliant suspension need long link lengths to be able to satisfy the stress

requirements. This was in expectation with what we thought would be the final

results.

The given design requires a very large thetaua to be able to satisfy the

requirements of the suspension. This is not a general design. Most A arm

suspension have a comparatively smaller thetaua when compared to the design

of the compliant suspension.

In our optimization runs we found that sequential quadratic programming (SQP)

kept giving us the best results among all the other algorithms. Even the final best

feasible result is from the run made in SQP.

The final mass is very low for an A arm suspension and if implemented could

improve the ride and handling characteristic of a vehicle as it reduces the

unsprung mass of the vehicle by a large margin. General rigid A arm suspension

made of steel weigh in around 12-15 kilos. We have not been able to find results

on how much would a rigid A-arm suspension made of carbon fiber would weigh

but we estimate that it would be higher than the corresponding compliant

suspension.

5.6. FUTURE WORK:

1. Topology optimizations to be done to determine an optimal topology design for a

complaint A arm suspension.

2. Different materials to be considered when carrying out the optimization process.

P a g e | 70

3. Multiple damper placements can be considered for even better damping results.

MUCHAS GRACIAS (THANKS)

We would like to thank Jarod Kelly for his help and support to us during our

optimization analysis. This would not have been possible without his useful

insights into our projects.

We would like to thank Dr. Michael Kokkolaras for making this a very interesting

Course and giving us a very useful insight about the basics of optimization.

We would also like to thank James Allison for helping us out during the initial

stages of our optimization with the subsystem division.

P a g e | 71

APPENDIX IA

Mathematical formulations for the kinematics

1) Vertical Displacement yv.

For given link lengths, l1, l2, l3 and l4, and two link angles θ1 and θt, the four bar

mechanism configuration is completely determined. We can then determine θh , β , γ,

and θa. These are determined from the formulae below

Let θd = θt – θ1

2 2

1 2 1 22 cos( )dl l l l

2 2 2

1 2

1

cos( )2

l la

l

2 2 2

3 4

3

cos( )2

l la

l

2 2 2

4 3

4

cos( )2

l la

l

2 2 2

4 3

4 3

cos( )2

l la

l l

t h

1

1( )

h

a

θh

θa

θt β

γ

λ

ψ θ1 δ

ξ

P (xp,yp)

P a g e | 72

The Y-coordinate of the point P, is given by

1 2 3cos( ) cos( ) / 2p t hy l l

When the lower A-arm is rotated clockwise by an angle ang, we would get

corresponding quantities θd’(=θt – θ1 -ang ), ξ’, ψ’, λ’, β’, θh’, and θa’. The corresponding

Y-cordinate of point P would be

1 2 3' sin( - ')) cos( ') cos( ') / 2p c t t t hy l l l

The net vertical displacement would be

1 1'v p py y y

2) Joint rotations.

Joint rotations can be obtained from the previous analysis as

'

'

'

'

t t

a a

ang

3) Instant center xin.

This can be got from geometry. The expression can be given below for both the x-

coordinate and the y-cordinate of the instant centers.

P a g e | 73

4) Roll-center distance from the ground

5) Furthermore, since l1 ,l2, θ1 and θt are variables, they determine the fixed link and

one moving link of the mechanism.

P a g e | 74

The link lengths l3 and l4 should be chosen such that the triangle law between them

should be valid

i.e.

3 4

3 4

4 3

l l

l l

l l

These relations should hold good even when l2 is rotated by an angle ang. δ would

then be δ’.

3 4

3 4

4 3

'

'

'

l l

l l

l l

l2

l1

l4

l3

δ

P a g e | 75

APPENDIX IB

Algorithm details for the kinematics part.

max Directional First-order

Iter F-count f(x) constraint Step-size derivative optimality Procedure

0 7 -0.295508 0.6283 Infeasible start point

1 15 -0.359618 0.1806 1 -0.0482 0.468

2 23 -0.210512 0.006013 1 0.147 0.466

3 31 -0.207044 0 1 0.00357 0.0334

4 39 -0.21263 0.002121 1 -0.00396 0.0462 Hessian modified

5 47 -0.214904 1.558e-005 1 -0.00227 0.0293 Hessian modified

6 55 -0.215326 6.928e-005 1 -0.000411 0.0262 Hessian modified

7 63 -0.21731 0.00327 1 -0.00143 0.00394 Hessian modified

8 71 -0.217672 4.544e-006 1 -0.000362 0.0215 Hessian modified

9 79 -0.217668 9.521e-011 1 3.88e-006 4.29e-006 Hessian modified

10 87 -0.217668 6.106e-016 1 8.4e-011 3e-007 Hessian modified

lower upper ineqlin ineqnonlin

2 3 9

5 4 10

X = 0.1797 0.0000 0.9000 0.3491 -0.3491 -0.0457

FVAL = -0.2177

EXITFLAG = 1

OUTPUT =

iterations: 10

funcCount: 87

P a g e | 76

stepsize: 1

algorithm: 'medium-scale: SQP, Quasi-Newton, line-search'

firstorderopt: 2.9967e-007

cgiterations: []

message: [1x144 char]

Lagrange Multipliers: [0 0 0 0 0 0 0 0 0.9903 0.0461]

P a g e | 77

APPENDIX 3A

Structural optimization:

3.1.1. Gradient of the function at the system is given by

0.0217 0.0212

-34.3435 34.3435 -14.8930 2.1136 2.2611

3.1.2. The hessian at the final value is given by the matrix below:

1.0e+006 * [0.0000 -0.0000 -0.0014 0.0011 0.0004 0.0019 0.0001 -0.0000 0.0000 0.0028 -0.0021 -0.0009 -0.0038 -0.0003 -0.0014 0.0028 3.4167 -2.4889 -1.2331 -4.6428 -0.3481 0.0011 -0.0021 -2.4889 4.0732 -0.5477 4.4760 0.2509 0.0004 -0.0009 -1.2331 -0.5477 1.3739 0.9772 0.1276 0.0019 -0.0038 -4.6428 4.4760 0.9772 6.8450 0.4715 0.0001 -0.0003 -0.3481 0.2509 0.1276 0.4715 0.0355]

3.1.3. The Eigen values for the hessian are given by:

1.0e+007 * [0.0000 0.0000 0.0000 0.0002 0.0003 0.2621 1.3119]

3.1.4. The final coordinates of the suspension system are given below: all are in mm

0 0 0 54.0274 0 250.6411 64.0274 0 250.6411 128.0548 0 501.2822 239.3959 -88.4772 42.2120 239.3959 88.4772 42.2120 370.3597 -65.9270 458.5573 370.3597 65.9270 458.5573 430.9360 -82.4087 447.8760

P a g e | 78

430.9360 0 447.8760 430.9360 82.4087 447.8760 299.2449 -110.5964 52.7649 299.2449 0 52.7649 299.2449 110.5964 52.7649 461.2241 -90.6496 442.5354 461.2241 90.6496 442.5354 733.8172 -164.8175 394.4699 733.8172 164.8175 394.4699 329.1694 -121.6561 58.0414 329.1694 121.6561 58.0414 598.4898 -221.192 105.5299 598.4898 221.1929 105.5299

3.1.5. The value of the constraints at the optimum value is given below:

-270.0000 -0.0000 -2.6395 -0.0086 -2.3588 -3.0483 -0.0000 -0.0000 -29.4553

P a g e | 79

APPENDIX 4

4.1.

Start END Lb Ub

Inputs

u1 1.30E-12 0 0 1

u2 0 0 0 1.2

u3 0.5 0.5 0.5 1.3

thetab 0.3489 0.3489 -0.3489 0.3489

thetah -0.3489 -0.3489 -0.3489 0.3489

thetaa -0.33421 -0.20697 -0.3489 0.3489

thetala 0.483695 0.219686 0.0872 1.397

thetaua 0.607457 0.414391 0.0872 1.397

alphala 0.303742 0.305649 0.15 0.45

alphaua 0.288913 0.250401 0.15 0.45

betala 0.742543 0.72771 0.7 0.95

betaua 0.74716 0.739827 0.7 0.95

etala 0.58 0.578426 0.5 0.65

lamdab 0.5 0.5 0.1 0.9

d1 20 1.211838 1 40

d2 20 3.285178 1 40

d3 20 4.828643 1 40

d4 20 5.104398 1 40

d5 20 13.54892 1 40

d6 20 16.04535 1 40

Outputs

Output1 15.42 4.72177

Output2 -261.166 -128.923 0

Output3 -8.83354 -141.077 0

Output4 -1.16984 -15.4181 0

Output5 -0.00917 -0.02488 0

Output6 -2.54052 -4.73377 0

Output7 -1312.79 -1471.47 0

Output8 -1270.87 -1460.39 0

Output9 -1047.52 -107.217 0

Output10 -1042.14 -78.6202 0

Output11 -1.42994 -0.00441 0

Output12 -0.7 -0.5 0

Output13 0 -0.2 0

Output14 0 0

Output15 -0.17184 -0.14311 0

Output16 -0.28274 -0.28274 0

Output17 -0.05661 -0.05309 0

Output18 -0.04353 -0.07409 0

MASS 15.42 4.72177

P a g e | 80

REFERENCES:

1) Allred, T., Howell, L. L., Magleby, S. P., Todd, R. H., “ The Compliant A-Arm

suspension”, Proceedings of IMECE‟ 03, Wahington, D. C., November 15-21,

2003

2) Principles of optimal design. Modeling and computation. Panos.y.Papalambros

and Douglass.J.Wilde

3) “One-piece compliant Automotive suspension mechanism”, an ME 450 report

chaired by Prof. Sridhar Kota, April 22, 1994.

4) Gillespie, Thomas D., Fundamentals of Vehicle Dynamics, SAE, 1992.

5) Howell, Midha and Norton, Evaluation of Equivalent Spring Stiffness for use in a

Pseudo Rigid Body model of Large Deflection Compliant Mechanisms, 1996,

ASME Journal of Mechanisms Design / Vol. 118

6) Kota, Kikuchi and Joo, Topological synthesis of Compliant Mechanisms Using

Linear Beam Elements, 2000, Mech. Struc. & Mach., 28(4)

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PA

8) http://www.fourwheeler.com/techarticles/suspension/129_0601_2005_gmc_duram

ax_jounce_shocks/photo_02.html

9) COMPLIANT MECHANISMS BY LARRY HOWELL

10) Halliday, resnick, and Walter, 5th edition, chapter 16, special attention to section

16-8 (damped simple harmonic motion)

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11) Mechanical characteristics of oil-damping shock absorber for protection of

electronic-packaging components by yang ping, Research Center of Micro-Nano

Science and Technology, Jiangsu University, Zhenjiang, China

12) Development and experimental verification of a parametric model of an

automotive damper, a thesis by Kirk Shawn Rhoades, Texas A&M University

13) Dixon, J. C. “Shock Absorber Handbook”, SAE International, Feb. 1999

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