Freedom in Machinery

Cover......Page 1
Frontmatter......Page 2
Contents......Page 10
Preface......Page 13
Some observations......Page 14
On the history of screws......Page 15
Guide and synopsis......Page 17
Notes and references......Page 18
1 - Mechanism and the mobility of mechanism......Page 19
Some opening remarks......Page 20
Links and joints......Page 21
Loops and the paths of points......Page 22
Constraint and the idea of mobility......Page 23
The freedom of joints......Page 24
A criterion for mobility......Page 26
An example mechanism of mobility unity......Page 27
Zero mobility and modes of assembly......Page 29
Spots of contact and the perfection of joints......Page 31
Some wider considerations and a summary......Page 34
Notes and references......Page 36
2 - Overconstraint and the nature of mechanical motion......Page 39
Overconstraint and the idea of apparent mobility......Page 40
Full cycle and transitory, apparent mobilities......Page 41
The locking in of passive freedoms......Page 42
Some examples of overconstraint in working machinery......Page 43
The question of completed chains......Page 44
More transition comment......Page 45
Spherical and planar mechanism......Page 46
More about passive freedoms......Page 47
On the question of complex joints......Page 48
Worked exercise on figures 2.03 and 2.04......Page 49
Worked exercise on figures 2.05 and 2.06......Page 51
Worked exercise on the figures 2.07......Page 52
Worked exercise on figure 2.08......Page 54
Manipulators and robots......Page 55
Notes and references......Page 56
3 - Some of the various lines in a moving body......Page 57
Right lines and n-lines in a body with 1 °F......Page 58
An important theorem......Page 59
The whole multitude of right lines in a moving body......Page 60
The nature of the relative motion at an n-line......Page 61
Mechanical substitutions at existing n-lines......Page 62
Revolute axes, Mines, and the actional right lines......Page 64
The various lines in a body with 2 °F......Page 67
The central, symmetrical case of 3 °F......Page 69
Another important theorem......Page 70
Some concluding remarks......Page 71
4 - Enumerative geometry and the powers of infinity......Page 73
Some first thoughts......Page 74
The regulus and the linear complex of lines......Page 75
5 - Rigidity and the instantaneous screw axis......Page 77
Retreat for simplicity to a point with zero velocity......Page 78
The locus of other points of velocity zero......Page 80
A consideration of three points in a body......Page 81
Relevance of the velocity polygon......Page 82
Some implications of the unique direction......Page 83
The ISA and a definition for angular velocity......Page 84
Curvature and the paths of points......Page 85
Closure......Page 86
Notes and references......Page 87
6 - Irregularity and the freedoms within a joint......Page 89
On regularity, and the irregularity at surfaces......Page 90
Various other preliminary remarks......Page 91
Consider first two irregular bodies free in space......Page 92
Consider next a single point of contact......Page 93
Two points of contact between the bodies......Page 94
Three points of contact between the bodies......Page 96
One way into a first study of the 3-system......Page 97
Four points of contact between the bodies......Page 101
Five points of contact between the bodies......Page 102
On the question of the immobilisation of a body......Page 103
Appendix 6A. Notes on the quasi-general 5-system......Page 104
Appendix 6B. Notes on the quasi-general 4-system......Page 106
Appendix 6C. Notes on the quasi-general 3-system......Page 112
7 - The possibilities in reality for practical joints......Page 117
The possibility and practicality of joints......Page 118
Remarks in passing......Page 120
On trying to set up an intermediate f4 joint......Page 121
On the projective (or linear) dependence of lines......Page 122
A reversal of the argument: the idea of contact patch......Page 123
A practical intermediate f 4 joint......Page 124
Other intermediate, simple, f 4 joints......Page 125
On the projective dependence of lines at an f 3 joint......Page 126
Towards a practical f 3 joint......Page 127
Other practical, simple, f 3 joints......Page 129
Some of the simple, f 3 joints that can exist......Page 131
The reciprocal and the complex f 3 joints......Page 134
The population of the simple f 2 joints......Page 135
The population of the simple f 1 joints......Page 138
The six (the only six) lower, simple joints......Page 140
Notes and references......Page 141
8 - Some elementary aspects of two degrees of freedom......Page 143
The setting up of the cases for consideration......Page 144
Some further remarks about the bases for argument......Page 145
The third case (c) in figure 8.03......Page 146
The fourth case (d) in figure 8.04......Page 147
Intersecting complexes of right lines......Page 148
Towards a wider generality......Page 149
9 - The linear complex of right lines in a moving body......Page 153
The helicoidal field of velocity vectors......Page 154
The wheel of right lines at a point......Page 155
The total number of right lines......Page 156
All of the binormals to all of the helices......Page 157
Towards a visualisation of the set......Page 158
Other reguli of the complex......Page 159
When the pitch of the complex is zero or infinity......Page 162
Notes and references......Page 163
10 - Line systems and the dual vectors in mechanics......Page 165
Opening remarks......Page 166
The linear line systems......Page 167
Some experiments with the balancing of forces......Page 168
Interlude......Page 170
The cases of five, six, and more than six forces......Page 171
The concept of couple......Page 172
Some striking analogies......Page 173
Parallel assembly and the natures of the duals......Page 175
The general, oblique assembly of the duals......Page 176
The addition to rest of rates of twisting......Page 177
When one or both of the duals or the power is zero......Page 179
The rolling of a wheel......Page 181
The reciprocal systems at an f 3 joint......Page 182
Notes and references......Page 184