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ME 564 Final Report

Optimization Design of Variable Stiffness Coil Spring of Vehicle Suspension

Kun Tang Renjie Xie

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Contents Abstract ............................................................................................................................................. 2 Chapter 1 Introduction ...................................................................................................................... 3

Section 1.1 Stiffness Coil Spring .............................................................................................. 3 Section 1.2 Optimization Problem ............................................................................................ 4

Chapter 2 Modeling Assumption ...................................................................................................... 6 Chapter 3: Optimization Problem Statement and Mathematical Formulation .................................. 9

Section 3.1 Optimization Problem Statement ........................................................................... 9 Section 3.2 Selection of Variable ............................................................................................ 10 Section 3.3 The Initial Parameters of Coil Spring ................................................................... 10 Section 3.4 The Objective Function ........................................................................................ 11 Section 3.5 Constraints ........................................................................................................... 11

Chapter 4 Optimization Process ...................................................................................................... 14 Section 4.1 Solution Method ................................................................................................... 14 Section 4.2 Results .................................................................................................................. 14 Section 4.3 Detailed Design .................................................................................................... 16

Chapter 5 Performance Verification Analysis ................................................................................. 17 Chapter 6 Conclusion ...................................................................................................................... 18 Reference ........................................................................................................................................ 19 Appendix ......................................................................................................................................... 20

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Abstract

An optimal design method was implemented for variable stiffness cylinder

coil spring basing on the ideas of dispersion variable optimizing method

and the existing design theory and method of spring. Provided the basic

design process and developed the visual optimizing procedure using

Microsoft Office Excel Solver used the procedure to carry on the optimized

analysis operation to the spring in the instance. The result indicates that the

algorithm is feasible and effective, and makes the design and calculation

of the spring more concise and clear than the conventional design method,

which imp roves work efficiency.

Key words: Vehicle suspension Variable stiffness coil spring Optimal design

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Chapter 1 Introduction

Section 1.1 Stiffness Coil Spring

As an elastic element, stiffness coil springs are used commonly in the

applications of suspension of modern light vehicles especially in the

medium bus because of its simple structure, manufacturing convenient and

a highly specific capacity (Figure.1 Stiffness Coil Spring). Due to the

requirements of better driving, the suspension guide mechanism still has

the ability to maintain the angle of the wheel under big swing. For instance,

with the replacement of stiffness coil springs, they are able to install shock

absorbers, stroke limit device or guide column to make the structure more

compactible. What is more, the spring structure can be used to achieve

variable stiffness characteristics via adopting different kinds of pitches,

winding or both of those. The objective goal to select variable stiffness coil

spring is that Cars at light loads or driving on a flat road, is hoped to have

damping spring stiffness smaller to ensure driving comfort. And when the

vehicle is in heavy load or driving on uneven road, if the spring stiffness is

small, deformation is excessive, it will cause the entire spring and tight to

lose cushioning and damping effect, affecting handling and stability, then

spring stiffness is expected to be larger. This requires the same spring to

have low stiffness at the light load deformation, have high stiffness at

heavy load large deformation. Thus adjustable, pitch and other variable

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stiffness spring arises.

Figure.1 Variable Stiffness Coil Spring

Section 1.2 Optimization Problem

Designing and solving optimization problem is to find the best ways so that

a decision maker or a designer can achieve a maximum benefit from the

available resources and situations. Classified on the basis of nature of

equations with respect to design variables, the optimization problem is of

great importance in structural design. Here are two classifications in

general: If the objective function and the constraints involving the design

variable are linear, the optimization is termed as linear optimization

problem then. If even one of them is non-linear, it is classified as the non-

linear optimization problem. Basically the design variables are real but

sometimes they could be integers such as number of springs, mass, etc.

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These constraints could be equality constraints or inequality constraints

which depend on the nature of the optimization problem.

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Chapter 2 Modeling Assumption

Two inequality parts of coil spring

The structure of stiffness coil spring of suspension in a mini-van is showed

in figure 2. It is a variable coil spring which consists of the following two

parts: each part has different pitch of coil spring. When it comes to

structural analysis, we could consider them as two equal pitch which are in

series. As presented in figure 2, considering part A and part B which are

separated into two part by imaginary plane MN, we think the wire diameter

d and pitch circle Dm are equal in two parts.

Figure.2 Variable Pitch Circle of Coil Spring

Shape of coil spring’s end point

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Typically, the end of spring can be grind fine or tight. It can still not only

be cut into right angle but also be inner curved. Owing to research involved

in variable stiffness coil spring of suspension in a mini-van, we find that

this special structure has great advantages which occupies smaller space

and has flat end points. According to the reference, the relationship

between numbers of circle d and height Hs is that

1.01 ( 1) 2 ( )Hs d n b mm= − + (1)

Where: 1.01 represents compensation coefficient helix angle;

d represents the diameter of the spring wire;

b represents a terminal end portion of the thickness of the grinding

time, / 3b d= mm.

The deformation f of the coil spring in its axial load P is showed as follows:

3 48 / ( )Mf PD n Gd= mm (2)

Where: Dm represents the pitch diameter of coil spring;

d represents diameter of coil spring wire;

n represents working turns of coil spring;

G represents the shear modulus of elasticity of the coil spring

material;

P represents its axial load.

Thus,

Stiffness K:

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4

8 m

P GdKf D n

= = (N/mm) (3)

Shear stressτ :

3 2

8 ' 8 'mPD K PCKd d

τπ π

= = (MPa) (4)

Where: C represents Spring index (winding ratio), mDC d= ;

K’ represents curvature of the Department Number.

4 1 0.615'4 4

CKC C−

= +−

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Chapter 3: Optimization Problem Statement and

Mathematical Formulation

Section 3.1 Optimization Problem Statement

Optimization objective can vary with different working environments and

requirements. For those springs which tend to have fatigue damage because

of the working feature, maximizing the fatigue safety factor should be

primarily considered. For those springs affected by varying loads of high-

speed operation mechanism , first-order natural frequency maximum or

minimum should be considered as the optimization design goal to avoid

resonance. While in this project, this spring is working in the vehicle

suspension system. The objective is reducing the volume and the weight,

considering the cost and the limited space of suspension system of vehicle.

Meanwhile, the expected functions of the spring should be guaranteed.

In this project, spring stiffness as large as possible and spring-mass

minimum are the objective function.

The material of the spring is 50CrV, which is Infiltrated shot peening. So

the density of the spring 37.9t mρ −= ⋅ .

Shear modulus is 81G GPa= . The allowable shear stress is [ ] 810MPaτ = .

Fatigue limit of spring material is 0 2000τ = . Allowable safety coefficient is

[ ] 1.3S = . Spring coefficient: kmin=46N/mm; kmax=75N/mm.

Critical load Pmax=1425kgf; Pmin=408kgf

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Section 3.2 Selection of Variable

According to structure of variable stiffness coil spring, we select the

following four variables:

d: the diameter of coil spring wire;

Dm: the pitch diameter of coil spring;

nA and nB: working turns, n= nA + nB.

Thus, the design variables which affect spring stiffness and spring -mass

are spring wire diameter d, the number of turns n and spring diameter Dm.

Write them in matrix form (Figure.3):

Figure.3 Matrix Form

Where: 12mm≤x1 ≤22mm, 120mm≤x2 ≤180mm, 2≤x3 ≤5, 2≤x4 ≤5.

Section 3.3 The Initial Parameters of Coil Spring

Spring wire diameter d = 17mm;

Spring diameter Dm = 147mm;

The number of turns nA = 3.2 nB = 4.26.

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Section 3.4 The Objective Function

Minimize 2 2 ( )( )

4m A Bd D n nf x π ρ+

= (6)

Or

2 21 2 3 4( )( )

4x x x xf x π ρ +

= (7)

Where: ρ represents density of coil spring.

Section 3.5 Constraints

Selecting constraints is an indexical work which means we have to

consider every part of constraint that may have effect on objective function.

In general constraints include stiffness requirements, spring wire diameter

restrictions, the installation space restrictions on diameter, number of turns

scope of work, range of winding ratio, natural vibration frequency range,

intensity conditions, stable condition, fatigue strength conditions, etc.

And in order to make sure our redesign variable stiffness coil spring can

survive under the high load and low load, we make the following six

constraints:

(1)Stiffness requirement

Before the spring coils compress, the total rigidity should be within a

certain range min maxk k k≤ ≤ : kmin=46N/mm kmax=75N/mm

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44 1

( ) 32 3 48 ( ) 8 ( )A B

m A B

Gd Gxk kD n n x x x+= = =

+ + (8)

(2)Rotation ratio requirement

According to previous experience:

4 18C≤ ≤ (9)

2

1

mD xC d x= = (10)

(3)Strength requirement

When the coil spring is under the largest axial load, shear stress should be

less than the allowable stress ( K` is curvature coefficient):

2ma max

max 3 31

8 x ' 8 ' [ ]mP D K P x Kd x

τ τπ π

= = ≤ (11)

(4) Fatigue strength requirement

According to the requirement of coil spring fatigue strength,

0.84max 2

2.8max1 min

4.23P xxτ τ= . For instance, coil spring fatigue strength safety

factor S should be greater than the allowable safety factor:

0 min

max

0.75 [ ]S Sτ ττ

+= ≥ (12)

Where: [ ]S represents the allowable safety coefficient.

(5)The requirement that spring at the maximum working load does not

touch the ring:

0 max sH Hδ− ≥ (13)

δmax is the amount of deformation at the maximum working load:

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3max 2 3 4

4max1

8 ( )( )

P x x xGxδ += (14)

H0 is the spring free height, Hs is spring compress height:

0 A A B BH n t n t d= + + (15)

At and Bt are pitch diameter of coil spring A and coil spring B, respectively:

13

50 1At x x= + + (16)

150 14Bt x x= + + (17)

Hs is spring compress height:

21.01 [( ) 1] 3s A BdH d n n= + + − + (18)

(6)Stability requirement

For compression coil spring, when the axial load reaches a certain level

will have a greater lateral bending and destabilization. Critical load of

instability is connected with its height diameter ratio of 0

m

HD

λ =

λ≤5.3

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Chapter 4 Optimization Process

Section 4.1 Solution Method

Using Microsoft Office Excel Solver with GRG to optimize the design.

Using excel solver with GRG to optimize the design

Considering the fact that this is nonlinear problem, so the GRG nonlinear

method is chosen to find the optimize solution.

Section 4.2 Results

Solution with excel:

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The results are showed as follow:

Figure.6 Before Optimization Figure.7 After Optimization

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Section 4.3 Detailed Design

Figure.8 Before Optimization

Free hight: 255mm; Compressed hight: 126mm

Spring diameter: 147mm; Spring wired diameter: 17mm

Figure.9 After Optimization

Free hight:171mm; Compressed hight:49mm

Spring diameter:120mm; Spring wired diameter:13mm

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Chapter 5 Performance Verification Analysis

Final parameters of the spring:

(1) Spring wire diameter d = 13.3mm;

(2) Spring diameter Dm = 120mm;

(3) The number of turns nA = 2 nB = 2;

(4) Stiffness: K = 46 ∈ [Kmin,Kmax]=[46,75];

(5) Rotation ratio: C = 9 ∈ [4,18];

(6) Max shear stress: maxτ = 214.5 ≤ [ ] = 810;

(7) Safety factor: S = 9.3 ≥ [S] =1.3;

(8) H0 – δmax= 139.6 ≥ Hs=49.2;

(9) Height diameter ratio: λ = 1.4 ≤ 5.3.

All performance requirement are satisfied.

Using simulation software to verify that the loaded spring after

optimization satisfies the performance requirements.

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Chapter 6 Conclusion

After the optimization, the spring wire diameter and spring diameter are

reduced by 21.8% and 18.4% separately. And the total turn of the spring

are reduced by 48.2%. The average reduction reaches almost thirty percent.

The weight and volume of the spring are reduced to a lower state which

makes it easier to be installed and cheaper, considering the available space

of vehicle`s suspension system and the cost of material. The reduction of

the spring turns improve the riding comfort and stability.

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Reference

[1]Variable Stiffness Coil Spring Design Optimization of Vehicle

Suspension, Peicheng Shi, Jiancheng Gong (Volume11,2006).

[2] Optimization in Structural design, N. G. R. Iyengar.

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Appendix

g1: ((B7*(B1)^4)/((8*(B2)^3)*(B3+B4)))-B14

g2; B13-((B7*(B1)^4)/((8*(B2)^3)*(B3+B4)))

g3: (B2/B1)-18

g4: 4-(B2/B1)

g5: ((8*B12*B2*B15)/(B18*(B1)^3))-B8

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g6: B10-

((B9+0.75*((8*B11*B2*B15)/(B18*(B1)^3)))/((8*B12*B2*B15)/(B18*(

B1)^3)))

g7: B23-(B22-B19)

g8: B17-5.3

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