Engineering Design Handbook - Computer Aided Design of Mechanical Systems

349 Pages·1973·40.197 MB·English

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mC PAMPHLET AMCP 706-192
oENGT ERlJh~,LNG DESIGN
H11ANDBOOK
I
COPUTRAl DED DESIG'N
OF
M EC-H A".N ICAL, SYSTEMS
JAUQUARYERS, U.S. AP'lY MATERIEL COMMAND JULY 1973
R-,.v.od bw
N'ATIONAL TEC.WX"AL
114FORMATiON StRVICE
f,41 VA -
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HEADQUARTERS
US ARMY HAIEBIEL COM10ND
5001 EISENHOWER AVENUE
ALEXANDRIA, VA 22304
AHC PAIU'HLET 15 July 1973
NO, 706-192
ENGINEERING DESIGN HANDBOOK
COMPUTER AIDED DESIGN
OF O ECHANICAL SYSTEMS
TABLE OF CONTENTS
Paragraph Page
LIST OF ILLUSTRATIONS ................. ix
LIST OF TABLES ......................... xiii
PREFACE . .............................. xv
CHAPTER I ELEMENTS OF COMPUTER AIDED DESIGN
S-1 Synthesis vs Analysis in Engineering Design .... I - I
--2 The Philosophy of System Engineering ......... 1-2
1-3 Computer Aided Design in the
Mechanical Sciences ....................... --4
1-4 Mathematical Preliminaries ................... 1-5
!-5 Illustrative Military Computer Aided
Design Problems .. ........ ... .......... ! -8
1-5.1 Optimal Design of Structures ............ ... .1-8
1-5.2 Application of the Steepest Descent Method
in !nteractive Computer Aided Design .......... -- 13
1- 5.3 Dc,ign of Artillery Recoil Mechanisms ......... I - 15
Rc'.erences ................ ............ 1 - 17
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CHAPTER 2. FINITE DIMENSIONAL
UNCONSTRAINED OPTIMIZATION
2-1! Introduction........ ...................... 2-
2-2 Necessary Conditions for Extrema ............... 2-2
2--3 One-dimensional Minimization .................. 2-6
2--3.1 Quadratic Interpolation .................. ... 2-6
2-3 2 Fibonacci Search (or Golden Se~ction Search) .. 2-7
2.4 The Method of Steepest Descent (or Gradient) ... 2
2-5 A Generalized Newton Method ............ .. 'I
2-6 Methods of Conjugate Directions ..............- 1 2
2-6.1 The Conjugate Gradient Method .......... 1
2-6.2 The Method of Fletcher and Powell .... 2-16
2-6.3 A Conjugate Direction Method
Without Derivatives ................ 2-17
2-7 Comparison of the Various Methods 2-18
2-7.1 Method of Steepest Descent ............ 2 - 18
2-7.1.1 Cost Function f (X). ............. 2-18
2-7.1.2 Cost Function f "X) ................
2
2-7.1.3 Cost Function f (Y) .......
3
2-7 2 Generalized Newton Method ... . 19
2 -7.2. 1 Cost Function f,(W)....... .. ..
2-7.2.2 Cost Function f(x) ...............
2-7.2.3 Cost Functionf (x) ........ 2
3
2-7.3 Conjugate Gradient Method .. ................
2-7.3.1 Cost Functionf, Wx.( .... .. 2- 1
2-7.3.2 Cost Function f (X)......................
2
2--7.3.3 Cost Function f3 (r) ...........-
2-7.4 Fletcher-Po'vcll Method .. ......... . 2-21
2-7.4.1 Cost Functionfj(x) ...................... 2-21
2-7.4.2 Cost Function f (x) ..... ................. 2-22
2
2-7.4.3 Cost Functionf (X) ...................... 2-221
3
2-7.5 Conjuigate Directions Without Deriva'ives ........ 2- 22
2-7.5.1 Cost Functionf (Wx... ..... .............. 2--22
1
2- 7.5.2 Cost Function f (W)............... .... 2-23
2
-7.5.3 Cost Functionf (x)............2- 23
3
28 An Application of Ur~constiained
Optriization to Stru,-ural Anadysis.........2- 24
References.... .... ...... 2 25
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Paragaph? Faie
CHAPTER 3. LINEAR PROGRAMMING
3-1 lntrodv~ction .............................. 3-1
12 Properties of Lineat Programs .................. 3-2
3-3 The Simplex Algorithm .......... ............ 3-8
3- 3.1 Deterziination of a Basic Feasible Point .......... 3-8
3-3.2 Solution of LP........... ................ 3-9
3--3.3 1Iihe Degenerate Cae ...................... 3-10
3-4 Mininum Weight Truss Design ................. 3-12
.3-5 An Applicat~on of Linear Programming
to Analysis .......................... ... 3-i4
References.............................. 3-17
CHAPTER 4. NONLINEAR PROGRAMMING AND
FINITE DIMENSIONAL OPTIMAL DESIGN
t.-1 Introduction to tlie Theory of
Nonlinear Programming (NLP) ....... ......... 4-1
412 Global Theory ............... ............. 4-4$
42 Theory of Finite Dimensional Optimsl Design . ... 48
421 Finite Dimensional Optimal Design
Problems................ .............. 4-8
4-23.2 LIToah eoryto ..........................4-10
-3.2 ecxtierio Method.........................4- 14
4-3.1 Iii1In tirEterior Method.................. 4 H1
4- 3. FxVterinatMonho an.I.t..or..oin....... ...... 4 -18
4--.3 ted estDescrent erios Metor NL.... .. ........- 16
4 -4.1 The Dircetion of Steepest Descent .............. 4-20
4- 4.2 Step Siz~e Determination................... .4- 23
4 -4.2, 1 Rosen's Method for Linear Constrints ......... 4 -23
4-.42-~2 Fixed Step With Variable Weigtihig .... 4 -24
'7'4-4.2.3 Steepest D)escent With Constraint
Tolerances ..... ................... ... 4 25
4 4.3 A Steepest Desect Alethod With
Constraint E'.ror Compensation. .. .. .. .... 4 2
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4-5 Steepest Descent Solution of the Finite
Dimensional Optimal Design Problem ......... 4-27
4-5.1 An Approximation of the Problem OD ........ 4-28
4-5.2 Solution of the Approximate Problem ......... 4-30
4-5.3 Steepest Desceit A gorithm ................. 4-33
4-5.4 Use of the Computat'onal Algorithm .......... 4-34
References ............................... 4-35
CHAPTER 5. FINITE DIMENSIONAL
OPTIMAL STRUCTURAL DESIGN
5- 1 Introduction ......... ................... 5-1
5-1.1 Lightweight vs Structural Performance
Trade-offs ....... ................ .... . 5-2
5- 1.2 Weapon Development P A"'., ociated
With Lightweight Require,.%. .......... ... 5-2
5-1.2.1 Aircraft Armament ......... ............ 5-2
5-1.2.2 Gun Barrel Design .................. ..... 5-3
k
5-1.2.., Towed Artillery ......................... 5-3
5-1.2.4 Other Weapon Problems ................... 5-4
5-I .3 Plan for Technique Development ............. 5-4
5-2 Elements of the Elastic Structural
Design Problem ... ...................... 5-4
5-2.1 The Optimality Criterion .................... 5- S
5-2.2 Stress and Displacement Due to
Static Loading .......................... 5 -5
5-2.3 Natural Frequency and Buckling ............. 5-6
5-2.4 Method of Solution ....................... 5-7
5-3 Steepest Descent Programming for
Optimal Structural Design ...... ... ....... 5-7
5-3.1 Linearized Cost and Constraint Functions ..... 5-8
5-3.2 Stcepest Descent Algorithnm foi
Optimal Structural Design. ................. 5-10
5-3.3 Computation. Consideration, ... ......... 5- 10
5-4 Optimization of Special Purpose Structures ...... 5- 1I
5-4.) A Minimum Weight Column ................. 5-- I!
5-4.2 A Minimum Weight Vibratirg Bean- ........ ... 5-- 14
5- 4.3 A Minimum Wei2.t Pcrtal Frame With a
Natural 'requeicy ('c.:istral .t 5 -16
.5 -4.4 / Minimum Weight Frame With Multiple
Failure Criteria . .. .5 18
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~-4.5 A Minimum Weight Plate With Frequency
Constiaints............................. 5-20
5-5 General Treatment of Truss Design..............5-22
5-5.1 Special Problem Formulation ................. 5-22
5 -5.1. 1 Frequency Constraints..................... 5-24
5-5.1.2 Stress Constraints ........................ 5-25
5-5. 1.3 Buckling Constraints...................... 5-25,
5-5.1.4 Displactment Consiraints................... 5-263
5-5.1.5 Buund.,.on Design Vari ihles.................5-26
5-5.2 Multiple Loading Condtions.................. 5-28
S-5 .3 Example Problems ............... ......... 5-28
5-6 A General Treatment of Plane Frame Design ... 5-36
5-6.1 Proble.m Formulation .................. . 5-
5 --6.2 Stress Constraint Calcu~ations ................. 5-39
5-0.3 Example Problems ........................ 5-46
5-7 Interactive Computing in Structural
Optimization ............................ 5-51
5--7.1 The Interactive Approach.................... 5-51
5-7.2 Interactive Structural Design Using
Sensitivity Data ......................... 5-52
5 -7.3 Example Problems ................. ....... 5-54
5-7.4 Interactive Computing Conclusions ............. 5-61
References ... ........................... 5-62
CHAPTER 6. THE CALCULUS OF VARIATIONS AND
OPTIMAL PROCESS THEORY
6-1 Introduction.......................... ... 6-I
6-2 The Fundamental Pcoblein of the Calculus
of Variations............................. 6-4
6-2.1I Ntcessary Conditions for the Fundamental
Problem..................... .......... 6-5
6- 2.2 Special Ca~ses and Exampies ................... 6-8
f- 2.3 Vaiiat~on,: Notation and Second-order
Conditions ............................ 6-10
6'-2.4 Di~rect Methods ... .............. ........ 6-13
6 -2.4.1 The Ritz Method ......... ............. 6-14
6- 2 4.2, Method of Finite Differences. .............- 15
6--3 A Problem of Bolza..............6-16
6 3 I Statement of the Problem..................6- 16
4-3.2 A Multiplim Rale.. ............ .. ... 6- 17
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o-3.3 Ntvcessary Conditions for the Bolza
Problem.............................. 6-18
6- 3.4 Application of the Boiza Problem .............. 6-22
6-4 Fhoblems of Optimal Design and Control .......... 6-27
6-4.1 Design Variable Inequality Constrants ........... 6-28
5-4.2 State V'ariable Inequality Constraints ............ 6-31
6-4.3 Application of the Theory of Optimal
Design............................... 6-33
6-5 Methods of Satisfying Necessary Coin(titions ....... 6-40
6-5,1I Initial Value Methods (or Shooting
Teclh-niques) ............................ 6-,*0
6-5.2 A Generalized Newton Method ................ 6-42
References .... .......................... 6-45
CHAPTER 7. OPTIMAL STRUCTURAL DESIGN
BY THE INDIRECT METrHOD
7-1I Introduction .............................. 7-1
7- 1.1 The Class of Problems Considered ............. 7-1
7-1.2 H-istorical Development .................. ... 7-I
7-1.3 Methods Employed .................... -
7-2 A Minimum Weight Column ................ 2
7-3 A Minimum Weight Structure With
Angular Deflection Requirements .............. 7- 5
7-3.1I Statement of the Problem .......... ......... 7-S
7--3.2 Tower With One Design Variable ............... 7- -7
7- .2.1 Method 1. Tower With Base Rigidly
Fastened to the Earth ............. ..... 7
7-3.2.2 Method 2. Trower With Case Pinned to
Earth and With Top Supported by
Guy L~ines ............................ 7-
7-3.3 Tower With f~wo Design Variables.....- ... .. T7 -3
7--3.3.1 Method 1. Trower With Base Rigidly
Fastened to the Earth ............ ...... 7-14
7 -3.3.2 Method 2. Tower With Base Pinned to
Earth arid With Top SuiM,.irted by
Guy Lines .. .. .. .. .. .. .. .. .. .. .. .. ... 7- 14
7 3.4 Discussion of Results ....... .......... ..... 7-15
7 -4 Minimum Weight Design of IReans With
Inequality ('onstrmints onl Stress and
Deflection 7 16
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7-4.1I Statement of the Problem ......... .......... 7-16
7-4.2 Necessary Conditions for the Beam
Des~gn Problem ......................... 7-19
7-4.3 Statically Indeterminate Problems ... .......... 7-23
7-4.4 Solutions of the Equations of Theorem 7-1I.. 7-24
7-4.5 Beams With Rectangular Cross Section
of Variable Depth ........................ 7-25
7-4.5.1 A Problem Which Can Be Solved
Analytically ............ ........ ....... 7-29
t 7-4.5.2 Simpiy Supported Beam With Positive
Distributed Load........................ 7-32
7-4.5.3 A Problem of a More General Typ-. .... ....... 7-38
7 -4.5.4 Conclusions ............................ 7-43
References ......................... ..... 7-44
FOR OPTIMAL DESIGN PROBLEMS
8-1 Introduction ............................. 8-I
3-2 A Steepest Descent Method for the Basic
R ~Optimal Design Problem..................... 8-2
8-211 The Problem Considered..................... 8-2
t5- 2. 2 Effects of Small Changes in Design
Variables and Parameters ........ ........... 8-2
8-- 2..3 A Steepcst Descent Approach .............. 8-5
8-3 A Stcepest Descent Method for a General
Optimal D~esign Problem ......... ...... .... 8-15
8-3.1I The Problem Considered....................8-15
8-3.2 The Effect of Small Changes in Design
Variables and Parameters ............... ...-
8--3.3 A Steepest Descent Computational
rth..-.-.-............... .......... 8-23
8 -4 Steepest Descent Programming to, a Class Systemn
Described by Faitial Diffezwna~al iEq''rns.......8-25
8-4.1 The Class oi Pioblems Considered ............... 8-25
n 4.2 Effeut of Silall Changes in Design
Variables and idaer......... ........ 8-27
8 -4.3 A Steepest Descent Comnputational
Akor~R- 10
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8-5 Optimal Desigii of an Ariller R"coi
Mecharism ..................... ........ 8-35
8-.1 Formnulation of the Problem.................. 8-37
8-5.2 Equations of Motion for the XM 164
Howitzer............................. 8-39
8-5.3 Steepest Descent Formulation................ 8-43
8-5.3. 1 Determination of the Adjoint Equations .... 8-44
8-5.3.2 Determination of the B'undar Conditions
for the Adjoint Equations. ................ 8-45
8-5.3.3 Computation of Design Improvemjents .......... 8-47
8-5.4 Results and Conclusions .................... 3-48
References.............................. 8-49
CHAPTER 9. APPLICATION OF STEEPEST DESCENT
METHODS TO OPTIMAL STRUCTURAL DESIGN
9-1 Introduction .............................. 9-1
9.2 Steepest Descent Method for Optimal
Structural Design.......................... 9-2
9-3 A Minimum Weight Column................... 9-8
9-4 A Minimum Weight Vibrating Beam .............. 9-9
9-5 A Minimum Weight Vibrating Frame ............. 9-10
9-6 A Minimum Weight Frame With Multiple
Failure Criteria........................... 9-14
9-7 A Minimum Weight Vibrating Plate .............. 9-16
References .................. ..... ....... 9- 18
APPENDIX A CONVEXITY ....... ........... A-I
APPENDIX B ANALYSIS OF I3EAM-T\ PE
STRUCTURES ........ .. ...... B-1
INDEX........................ .... ..... I1-I
Vil
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LIST OF ILLUSTRATIONS
Fig. No. Title Page
1-.1 A System Engineering Model ................. 1-2
1-2 Structural Requirement .................... . 1-8
1-3 Conceptual Designs ....... ................ 1-9
1-4 Uniform Initial Design ...................... 1-11
1-5 Direction of Steepest Descent ................. 1-12
1-6 Tower With Base Rigidly Fitcned
to the Earth ......... ................... 1- 12
1-7 Tower With Base Simply Supported and
Top Supported With Guy Lines .............. .1 13
1-8 Sensitivity to Design Variations ............... 1-14
1-9 Sensitivity to Two Performance Indicators ....... 1-14
> 1
1-10 Howitzer, Towed, 105 mmi, XM164 ............. 1-16
1-11 Traditional Recoil Design Goal ............... 1-16
1-12 Recoil Distribution in Time .................. .1 -16
1-13 Sensitivity to Gun Hop ...................... 1-i7
1-14 Optimum Reioil Ctuve ...................... 1-17
2-1 f(x) = (x - 2)2 ........................... 2-2
2-2 A Cost Function ......................... . . 2-3
2-3 Function of Single Variable .................. 2-7
2-4 Interval Partition ................... ...... 2-7
2-5 Descent Steps ............................ 2-10
3-1 Graphical Solution of Example 3-1 ........... 3-2
3-2 Polyhedral Constraint Set ...... ............. 3-5
3-3 Admissible Joints for Bridge Truss ............. 3-14
3-4 Optimum Bridge Trus;es ............ ...... 3-14
3-5 Boundary Conditi.n for Example 3-4 .......... 3-16
j 4-1 Graphical Solution of Example 4-1 ............ 4-3
4--2 First-order Con~strairt Qualification ............ 4-6
4-3 Penalty Function . ......................... 4-11
5-1 Column ................................. 5- 12
5-2 Column Eleament .................. ....... 5--12
5-3 Profiles ot Optimal Columns ................. 5- 14
5-4 Stepped Beam ............. .............. 5-14
5-5 Tyrical Elem-nt .......................... 5 15
5-6 Profile of Optimum Beam ................... 5 -16
5-7 Portal Frame ... . .. ............... .... 5 -16
5-8 Typical Elements .............. .......... 5 16
5- 9 Optimum Portal Frame for = 3000
rad/sec ................... 5 18
Ix