TOLERANCE DESIGN OF MECHANICAL ASSEMBLY OF A PISTON

TOLERANCE DESIGN OF MECHANICAL ASSEMBLY OF A PISTON CYLINDER USING NSGA II AND FINITE ELEMENT ANALYSIS

Under the guidance of Dr. Kent Lawrence

BY SANDESH VARMA BUDDARAJU

1000849357

INTRODUCTION

• The allocation of design tolerances to components of mechanical assemblies and tolerances to machining processes used in the components fabrication plays an important role in cost reduction and quality improvement.

• Conventional optimization methods such as iteration method ,sequential quadratic programming ,adaptive branch and bound method ,heuristic method ,LaGrange multiplier algorithm ,interval analysis and non linear integer programming method are used.

• Potential of finite element simulation has not been fully utilized in tolerance design of mechanical assembly ,and the benefits of integrating finite element simulation with tolerance design has not been realized.

In this we are analyzing the piston cylinder and the design considerations are as follows

Ref . Tolerance design of mechanical assembly using nsga II and fea by Jayaprakash Govindarajulu, Sivakumar Karuppan and Thilak Manoharan


Dimensions Name Tolerance code Nominal dimensions(mm)

X1 Distance between cylinder top surface and centerline of cylinder spindle T1 292.2

X2 Eccentricity between centerline of cylinder spindle and axis diameter T2 0

X3 Centre distance between crankshaft spindle and crankshaft axle T3 50

X4 Eccentricity between centre of big hole and centre line of crankshaft axle T4 0

X5 Centre distance between centre line of big hole and centre line pinhole T5 174

X6 Eccentricity between centre line of pine hole and centre line of piston pinhole T6 0

X7 Distance between piston top surface and centre line piston pinhole T7 69

X21 Diameter of the crank shaft spindle T21 70

X22 Diameter of the cylinder spindle T22 70

X41 Diameter of the crank shaft axle T41 57

X42 Diameter of the connecting rod big hole T42 57

X61 Diameter of connecting rod pin hole T61 32

X62 Diameter of piston pin hole T62 32

X∑ Distance between piston top surface and cylinder top surface T∑ 0.7

Table 1: The component parts and code name of the diesel engine

Properties Piston (Ni Cr Al ALLOY) Connecting rod & Crankshaft (35 Mn 2M028 alloy steel)

Mass density (kg/mm3) 0.027 E -3 0.0784 E -3

Modulus of rigidity (E) (N/mm2) 0.7 E 5 2.1 E 5

Thermal conductivity (K) w/mm 204.2 E -3 51.9 E -3

Co-eff of thermal expansion (α) mm/0c 23.8 E -6 11.1 E -6

Specific heat (c) J/Kg K 896 446

Poisson ratio (1/m) 0.34 0.3

Table 2 : Material properties for various components of assembly


THE INERTIA EFFECT

• The inertia effect like gravity ,velocity results in deformation of a component .

• The amount of deformation produced by the gravity effect is directly proportional to density of the material.

• In this study, the deformation due to gravity effect is determined using finite element analysis and it is suitably incorporated in the tolerance stack up equation of tolerance design ,thereby loosening tolerance requirement of critical components.

•Piston top and cylinder top must fall with in the limits 0.7±0.27mm.

•Resultant tolerance as per statistical tolerance analysis method is T1

2+T22+T3

2+T42+T5

2+T62+T7

2 ≤0.272

•Total cost is equal to sum of manufacturing cost and quality losses and it is expresses as TC=Manufacturing cost + Quality losses= M+L = and CM(t)= a+be-ct

•Reciprocal and exponential expressions are commonly used. Here exponential expression is used.

•The average loss function for each product L is Kat2r where k is determined by

2

1

)( ri

n

i

i kattC

2AK

PART NOMINAL(MM)

PROCESS b a c

X1 11.5 TURNING 0.15997103 4$ 0.4389869

X21 2.75 TURNING 0.11800302 4$ 0.4389868X22 2.75 TURNING 0.11800302 4$ 0.4389868X3 1.968 TURNING 0.11800302 4$ 0.4389868X41 2.24 GRINDING 0.0133561 4$ 0.7827624X42 2.24 GRINDING 0.0133561 4$ 0.7827624X5 6.85 GRINDING 0.12576137 4$ 0.46536684X61 1.25 GRINDING 0.10123338 4$ 0.44723008X62 1.25 GRINDING 0.10123338 4$ 0.44723008X7 1.25 TURNING 0.11800302 4$ 0.4389869

Table 3 : Manufacturing cost data for each component


Type A

Normal 0.111

Triangle 0.165

Sine 0.191

Moving normal 0.218

Semi circle 0.250

Rectangle 0.333


Table 4 : Conversion constants for different distributions

•A is the overall loss and Δ is deviation of product dimensions from the target value.

•The constant K = 2058.

•The resultant dimensions of bath products have a constant density function(rectangle),the constant A equals 0.333 from table 4.

•Therefore ,the average loss for each product will be L=0.679x104xt2r

•Once the deformation is determined ,then optimal values of the component dimensions and tolerances are determined by using NSGA II.

•The piston cylinder assembly case can be found by solving the following mathematical model Minimize TC =

2

1

)( ri

n

i

i kattC

Subjected to:Statistical model: T1

2+T22+T3

2+T42+T5

2+T62+T7

2 ≤ (0.27+δ)2

Where δ = deformation due to gravity =0.044521mm

• The first constraint equation is the technical requirement and the remaining three constraints are constraints of eccentricity.

•Since the compression ratio should be kept constant ,a clearance of 0.27 mm has to be maintained.

•The deformation determined is included in the functional constraint equation.

2262

261

2242

241

2222

221

05.0

05.0

05.0

TT

TT

TT

The Tolerance design model

The objective functionTo minimize

Total cost

Manufacturing cost=

Quality loss function=KAtr2

(Deformation)

Machine capability

Functional requirement

Tolerance stack up

Vector assembly Model

(The assembly functions)

m/c tool database

2

1

)( ri

n

i

i kattC

Schematic of tolerance design model


RESULTSTable 5 : Comparison of optimal values of tolerance and total cost


S.NO VARIABLES

Optimal values of tolerance in (mm) Total cost in $

% savings in costSimulated annealing NSGA II Stimulated

annealing NSGA II

1 T1 0.0210 0.0119

48.2619 46.5260 3.6%

2 T2 0.0041 0.0023

3 T3 0.0062 0.0025

4 T4 0.0102 0.0119

5 T5 0.0031 0.0020

6 T6 0.0075 0.0080

7 T7 0.0103 0.0101

8 T8 0.0064 0.0031

9 T9 0.0041 0.0027

10 T10 0.0123 0.0083

Method Parameter Rigid body approach (mm)

The proposed method(mm) Cost savings in %

Statistical method

T1 0.0119 0.01013

5.22%

T21 0.0023 0.0045

T22 0.0025 0.0057

T3 0.0119 0.0041

T41 0.0020 0.0010

T42 0.0080 0.0010

T5 0.0101 0.0075

T61 0.0031 0.0011

T62 0.0027 0.0015

T7 0.0083 0.0059

Manufacturing cost 42.7877 43.4571

Quality loss 3.7383 1.8479

Quality loss due to deformation 1.2744 Nil

TC 47.8004 45.3050

Table 6: Comparison of optimal values obtained by rigid body approach and the proposed method


CONCLUSION•Tolerance allocation is done based on a hypothesis that the assembly process deals with infinitely rigid bodies.

•In reality every body undergoes deformation due to inertia effect and temperature effect.

•Through finite element simulation ,the value is suitably incorporated in the constraint equation of tolerance design problem.

•The component tolerance values found are the most robust to variation of operating conditions during the products application.

•Due to this ,the tolerance requirements of the given assembly are relaxed to certain extent for critical components ,resulting in reduced manufacturing cost and high reliability.

REFERENCESY. M. Huang and C. S. Shiau, Optimal tolerance allocation for a sliding van compressor, J Mech Des, 128 (2006) 98-107.

J. Deng and S. Deng, The adaptive branch and bound method of tolerance synthesis based on the reliability index, Int J Adv Manuf Tech, 20 (2002) 190-200.

M. F. Huang, Y. R. Zhong and Z. G. Xu, Concurrent process tolerance design on minimum product manufacturing cost and quality loss, Int J Adv Manuf Tech, 25 (2005) 174-722.

Tolerance design of mechanical assembly using NSGA II and finite elementAnalysis Jayaprakash Govindarajalu1, Sivakumar Karuppan and Thilak Manoharan

P. Wang and M. Liang, An integrated approach to tolerance synthesis, process selection and machining parameter optimization problems, Int J Prod Res, 43 (11) (2005) 2237-2262.

S. S. Rao and W. Wu, Optimum tolerance allocation in mechanical assemblies using an interval method, Eng Optim, 37 (3) (2005) 237-257.

C. Kao, C. C. Li and S. P. Chen, Tolerance allocation via simulation embedded sequential quadratic programming, Int J Prod Res, 38 (17) (2000) 4345-4355.

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TOLERANCE DESIGN OF MECHANICAL ASSEMBLY OF A PISTON