Kinematic Differential Geometry and Saddle Synthesis of Linkages

Wiley (Verlag)
  • erschienen am 8. Mai 2015
  • |
  • 496 Seiten
E-Book | ePUB mit Adobe DRM | Systemvoraussetzungen
978-1-118-25507-0 (ISBN)
With a pioneering methodology, the book covers the fundamental aspects of kinematic analysis and synthesis of linkage, and provides a theoretical foundation for engineers and researchers in mechanisms design.
* The first book to propose a complete curvature theory for planar, spherical and spatial motion
* Treatment of the synthesis of linkages with a novel approach
* Well-structured format with chapters introducing clearly distinguishable concepts following in a logical sequence dealing with planar, spherical and spatial motion
* Presents a pioneering methodology by a recognized expert in the field and brought up to date with the latest research and findings
* Fundamental theory and application examples are supplied fully illustrated throughout
weitere Ausgaben werden ermittelt
DELUN WANG, Dalian University of Technology, China
WEI WANG, Dalian University of Technology, China
Preface ix
Acknowledgments xi
1 Planar Kinematic Differential Geometry 1
1.1 Plane Curves 2
1.1.1 Vector Curve 2
1.1.2 Frenet Frame 6
1.1.3 Adjoint Approach 10
1.2 Planar Continuous Kinematics 14
1.2.1 Displacement 14
1.2.2 Centrodes 18
1.2.3 Euler-Savary Equation 26
1.2.4 Curvatures in Higher Order 33
1.2.5 Line Path 42
1.3 Plane Coupler Curves 49
1.3.1 Local Characteristics 49
1.3.2 Double Points 51
1.3.3 Four-bar Linkage I 55
1.3.4 Four-bar Linkage II 61
1.3.5 Oval Coupler Curves 67
1.3.6 Symmetrical Coupler Curves 73
1.3.7 Distribution of Coupler Curves 75
1.4 Discussion 78
References 80
2 Discrete Kinematic Geometry and Saddle Synthesis of Planar Linkages 83
2.1 Matrix Representation 84
2.2 Saddle Point Programming 85
2.3 Saddle Circle Point 88
2.3.1 Saddle Circle Fitting 89
2.3.2 Saddle Circle 92
2.3.3 Four Positions 95
2.3.4 Five Positions 97
2.3.5 Multiple Positions 100
2.3.6 Saddle Circle Point 101
2.4 Saddle Sliding Point 106
2.4.1 Saddle Line Fitting 108
2.4.2 Saddle Line 109
2.4.3 Three Positions 111
2.4.4 Four Positions 114
2.4.5 Multiple Positions 116
2.4.6 Saddle Sliding Point 116
2.5 The Saddle Kinematic Synthesis of Planar Four-bar Linkages 120
2.5.1 Kinematic Synthesis 122
2.5.2 Crank-rocker Linkage 129
2.5.3 Crank-slider Linkage 139
2.6 The Saddle Kinematic Synthesis of Planar Six-bar Linkages with Dwell Function 145
2.6.1 Six-bar Linkages 146
2.6.2 Local Saddle Curve Fitting 149
2.6.3 Dwell Function Synthesis 150
2.7 Discussion 163
References 167
3 Differential Geometry of the Constraint Curves and Surfaces 171
3.1 Space Curves 171
3.1.1 Vector Representations 171
3.1.2 Frenet Trihedron 175
3.2 Surfaces 177
3.2.1 Elements of Surfaces 177
3.2.2 Ruled Surfaces 183
3.2.3 Adjoint Approach 186
3.3 Constraint Curves and Surfaces 192
3.4 Spherical and Cylindrical Curves 195
3.4.1 Spherical Curves (S-S) 195
3.4.2 Cylindrical Curves (C-S) 197
3.5 Constraint Ruled Surfaces 201
3.5.1 Constant Inclination Ruled Surfaces (C'-P'-C) 201
3.5.2 Constant Axis Ruled Surfaces (C'-C) 204
3.5.3 Constant Parameter Ruled Surfaces (H-C, R-C) 208
3.5.4 Constant Distance Ruled Surfaces (S'-C) 212
3.6 Generalized Curvature of Curves 214
3.6.1 Generalized Curvature of Space Curves 215
3.6.2 Spherical Curvature and Cylindrical Curvature 218
3.7 Generalized Curvature of Ruled Surfaces 224
3.7.1 Tangent Conditions 224
3.7.2 Generalized Curvature 225
3.7.3 Constant Inclination Curvature 227
3.7.4 Constant Axis Curvature 228
3.8 Discussion 228
References 230
4 Spherical Kinematic Differential Geometry 233
4.1 Spherical Displacement 233
4.1.1 General Expression 233
4.1.2 Adjoint Expression 235
4.2 Spherical Differential Kinematics 240
4.2.1 Spherical Centrodes (Axodes) 240
4.2.2 Curvature and Euler-Savary Formula 245
4.3 Spherical Coupler Curves 257
4.3.1 Basic Equation 257
4.3.2 Double Point 257
4.3.3 Distribution 262
4.4 Discussion 263
References 266
5 Discrete Kinematic Geometry and Saddle Synthesis of Spherical Linkages 267
5.1 Matrix Representation 267
5.2 Saddle Spherical Circle Point 269
5.2.1 Saddle Spherical Circle Fitting 269
5.2.2 Saddle Spherical Circle 272
5.2.3 Four Positions 274
5.2.4 Five Positions 275
5.2.5 Multiple Positions 278
5.2.6 Saddle Spherical Circle Point 279
5.3 The Saddle Kinematic Synthesis of Spherical Four-bar Linkages 282
5.3.1 Kinematic Synthesis 283
5.3.2 Saddle Kinematic Synthesis of Spherical Four-bar Linkages 289
5.4 Discussion 298
References 300
6 Spatial Kinematic Differential Geometry 303
6.1 Displacement Equation 303
6.1.1 General Description 304
6.1.2 Adjoint Description 306
6.2 Axodes 310
6.2.1 Fixed Axode 310
6.2.2 Moving Axode 312
6.3 Differential Kinematics of Points 314
6.3.1 Point Trajectory 315
6.3.2 Darboux Frame 319
6.3.3 Euler-Savary Analogue 320
6.3.4 Generalized Curvature 323
6.4 Differential Kinematics of Lines 326
6.4.1 Frenet Frame 326
6.4.2 Striction Curve 330
6.4.3 Spherical Image Curve 332
6.4.4 Connecting Kinematic Pairs 334
6.4.5 Constant Axis Curvature 338
6.4.6 Constant Parameter Curvature 349
6.5 Differential Kinematics of Spatial Four-Bar Linkage RCCC 355
6.5.1 Adjoint Expression 355
6.5.2 Axodes 358
6.5.3 Point Trajectory 361
6.5.4 Line Trajectory 368
6.6 Discussion 378
References 380
7 Discrete Kinematic Geometry and Saddle Synthesis of Spatial Linkages 383
7.1 The Displacement Matrix 384
7.2 Sphere Point PSS 386
7.2.1 Spherical Surface Fitting 386
7.2.2 Saddle Spherical Surface 390
7.2.3 Five Positions 391
7.2.4 Six Positions 393
7.2.5 Multiple Positions 396
7.2.6 Saddle Sphere Point 396
7.3 Cylinder Point PCS 401
7.3.1 Cylindrical Surface Fitting 402
7.3.2 Saddle Cylindrical Surface 404
7.3.3 Six Positions 406
7.3.4 Seven Positions 407
7.3.5 Multiple Positions 410
7.3.6 Saddle Cylinder Point 410
7.3.7 The Degeneration of the Saddle Cylinder Point (R-S, H-S) 412
7.4 Constant Axis Line LCC 417
7.4.1 Ruled Surface Fitting 417
7.4.2 Saddle Spherical Image Circle Point 418
7.4.3 Saddle Striction Cylinder Point 420
7.4.4 Saddle Constant Axis Line 425
7.5 Degenerate Constant Axis Lines LRC and LHC 426
7.5.1 Saddle Characteristic Line LRC (R-C, R-R) 426
7.5.2 Saddle Characteristic Line LHC (H-C, H-R, H-H) 428
7.6 The Saddle Kinematic Synthesis of Spatial Four-Bar Linkages 444
7.6.1 A Brief Introduction 445
7.6.2 The Spatial Linkage RCCC 450
7.6.3 The Spatial Linkage RRSS 454
7.6.4 The Spatial Linkage RRSC 458
7.7 Discussion 461
References 464
Appendix A Displacement Solutions of Spatial Linkages RCCC 467
Appendix B Displacement Solutions of the Spatial RRSS Linkage 473
Index 477

Chapter 1
Planar Kinematic Differential Geometry

Kinematics, a branch of dynamics, deals with displacements, velocities, accelerations, jerks, etc. of a system of bodies, without consideration of the forces that cause them, while kinematic geometry deals with displacements or changes in position of a particle, a lamina, or a rigid body without consideration of time and the way that the displacements are achieved. As a combination of kinematic geometry and differential geometry both in content and approach, kinematic differential geometry describes and studies the geometrical properties of displacements.

There are a number of articles and books on kinematic geometry. Pioneers such as Euler (1765), Savary (1830), Burmester (1876), Ball (1871), Bobillier (1880), and Müller (1892) established the theoretical foundation and developed the classical geometrical and algebraic approaches for studying kinematic geometry in two dimensions some hundred years ago. The classical geometric and algebraic approaches are still in use today. Differential geometry is favored by many researchers studying the geometrical properties of positions of a planar object, changes in its positions, and their relationships. Invariants, independent of coordinate systems, are introduced to describe the geometric properties concisely. Thanks to the moving Frenet frame for describing infinitesimally small variations of successive positions, the positional geometry can be naturally and conveniently connected to the time-independent differential movement of a planar object.

This chapter deals with the kinematic characteristics of a two-dimensional object (a point, a line) in a plane without consideration of time by means of differential geometry. Though abstract, the explanation is judiciously presented step by step for ease of understanding and will be a necessary foundation for studying the kinematic characteristics of a three-dimensional object by means of differential geometry in later chapters.

1.1 Plane Curves

1.1.1 Vector Curve

A plane curve is represented in rectangular coordinates as


where is a parameter. The above equation can be rewritten in the following way by eliminating the parameter :


or in implicit form as


In a fixed coordinate frame , the vector equation of curve can be written as




Obviously, both the magnitude and direction of in equation (1.5) vary.

To describe a curve in the vector form, a real vector function, represented by a unit vector with an azimuthal angle with respect to axis , measured counterclockwise, is defined as a vector function of a unit circle (see Fig. 1.1). A plane curve can be denoted by the following vector function:


In the above equation, the magnitude and direction of vector depend on the scalar function and the vector function of a unit circle .

Figure 1.1 Vector function of a unit circle

Another vector function of a unit circle can be obtained by rotating counterclockwise about by p/2 (in Chapters 1 and 2, is the unit vector normal to the paper and directed toward the reader).

The vector function of a unit circle has the following properties:

  1. Expansion 1.7
  2. Orthogonality

    For a unit orthogonal right-handed coordinate system consisting of , , and , we have the following identities:

  3. Transformation 1.9
  4. Differentiation 1.10

The descriptive form of a curve depends on the chosen parameters and coordinates. A curve may have many descriptive forms, which differ in complexity if the parameters and reference coordinates are chosen differently. Below are three examples.

Example 1.1

A circle with radius and center point C is shown in Fig. 1.2. Write its equation in both vector and parameter forms.

Figure 1.2 A circle


The parameter equation of a circle in rectangular coordinates can be written as


where are the coordinates of the center of the circle in the reference frame .

Alternatively, the same circle can be represented as a vector function of a unit circle:


Example 1.2

An involute is shown in Figs 1.3 and 1.4. Write its equation in both vector and parameter forms.

Figure 1.3 An involute

Figure 1.4 An involute with a unit circle vector function


The equation of an involute can be written in three different forms using polar coordinates, rectangular coordinates, and a vector function of a unit circle, where is the radius of the base circle.

  1. Polar coordinates: E1-2.1
  2. Rectangular coordinates: E1-2.2
  3. Vector function of a unit circle: E1-2.3

Example 1.3

A planar four-bar linkage is shown in Fig. 1.5. Write the equation of the coupler curve in both parameter and vector forms.

Figure 1.5 A planar four-bar linkage


As shown in Fig. 1.5, links and , in a planar four-bar linkage with link lengths , form an inclination angle and with respect to the fixed link. A moving rectangular coordinate system attached to link BC and a fixed coordinate system attached to the fixed link are established. Point in the coupler with polar coordinates can be represented in the coordinate system as

  1. The parameter equation of coupler curves

    A coupler curve traced by point can also be expressed in the fixed frame as


    A sextic algebraic equation can be deduced for a coupler curve if parameters and are replaced by function in the displacement solution of a four-bar linkage.

  2. The vector equation of coupler curves

    Link AB rotates about joint A of the fixed link AD, and link BC rotates about joint B of link AB. Since a circle can be expressed by a vector function of a unit circle, a coupler curve of a four-bar linkage can be written as


    A point in link AB traces a circle vector in the fixed frame . A point in coupler link BC produces a circle vector in the reference frame of link AB. The subscripts inside the brackets are independent variables. Here, we deal with the coupler point relative to the coordinate system by the vector function of a unit circle.

Based on the above three examples, we observe that the description of a plane curve in terms of a vector function of a unit circle is simpler than the traditional algebraic equation. Moreover, since a vector function of a unit circle has intrinsic properties, its successive derivatives with respect to the chosen parameters can be conveniently obtained.

Invariants of a curve, independent of the coordinate system used, can be used to simplify the equation of the curve, which is considered a general rule in differential geometry. The arc length of a curve, which is also termed a natural parameter, is an invariant. Other invariants will be introduced in the later of this chapter and other chapters of the book. For equation (1.4), can be replaced by . The differential relationship between and can be written as


Then, the vector equation of curve is expressed in terms of as


It is recognized that . Using the Taylor expansion, curve can be expressed in the neighborhood of point by


where .

1.1.2 Frenet Frame

In a fixed frame, a curve is traced by a point of a moving body. There exists a connection between the point path and the moving body. A frame that moves along the curve can be employed to study the intrinsic geometrical properties of the curve.

Assume that the unit tangent vector of a plane curve is always in the direction of increasing arc length. Adopting the right-handed rule, as in the case of equation (1.8), the unit normal vector of a curve may be defined as . A unit orthogonal right-handed coordinate system may be uniquely established for each point on the curve. This moving Cartesian reference frame is called the Frenet frame, or the moving frame of a plane curve (see Fig. 1.6). The Frenet frame for a plane curve may be defined as


where , an invariant of the curve, is the curvature. Performing a dot product of both sides of the second equation in (1.14) with vector , we obtain


If a vector equation with a general parameter is given, as in equation (1.4) for a plane curve , the unit tangent vector can be expressed as


Utilizing the identity equation , the unit normal vector is obtained as


According to...

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