Controlling Degrees of Freedom From a kinematic viewpoint, excessive constraints must be avoided to produce a rationally designed guide. Controlling DESIGN PRINCIPLE: To achieve a highly precise and smooth Degrees of Freedom action which requires minimal force to operate, avoid excessive constraints. (The principle of kinematic design) DESIGN OF PRECISION MACHINES Kinematic design thus means employing optimum constraints when designing a machine. 3 DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Excessive constraints/ Over-constrained Eliminating excessive constraints Smooth action in x-direction is difficult to achieve unless the parts have been perfectly machined and assembled so that interference among the redundant constraints does not interfere. DESIGN OF PRECISION MACHINES 2 DESIGN OF PRECISION MACHINES 4
Controlling Degrees of Freedom Controlling Degrees of Freedom Four column press Table top and table legs Constrained DOFs = 8 x 4 DOFs = 32 DOFs Much of the machining effort is intended to prevent positional over constraint being manifested in the design. Kinematic mount of table top on table Bending moments transferred to table top structure 5 7 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Thermal centre Alternative concept Exact constrained design For a high level of predictable behaviour and a minimum number of accurate dimensions, exact constrained design is a necessary condition. Without preload With preload DESIGN OF PRECISION MACHINES 6 DESIGN OF PRECISION MACHINES 8
Controlling Degrees of Freedom Controlling Degrees of Freedom Elements to constrain degrees of freedom Contact stress and stiffness with ball support F Steel ball : 1 � � F 3 � � � � � 6 2 1360 10 N/m Hz , max � 2 � r � contact stiffness 1 1 dF � � � � 6 3 3 4 . 3 10 r F N/m � Symbol for ONE translation Symbol for ONE rotation d Constrained DOF 9 11 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining one translation by contact Contact stress and stiffness with ball support F F � � Constrained DOF Constrained DOF A sphere against a flat surface DESIGN OF PRECISION MACHINES 10 DESIGN OF PRECISION MACHINES 12
Controlling Degrees of Freedom Controlling Degrees of Freedom Damaging steel balls is very easy! Poor position accuracy in z-direction due to spherical contact area approximat ion z - error : � � � � � � z ( r_h r) (r_h r) cos( ) � 2 � � 2 3000 N/mm ! � � � g cos( ) 1 Hz 2 20 mm � � � � 2 � � 2 � � z r_h - r 13 15 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Poor position accuracy in z-direction due to spherical Constraining ONE translation contact area approximat ion z - error : � 2 � � � � z r_h - r 2 � � � � 2 � � � cos( ) 1 2 DESIGN OF PRECISION MACHINES 14 DESIGN OF PRECISION MACHINES 16
Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining ONE translation Demo – Folded sheet flexure 17 19 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom The principle of the “Folded Sheet Flexure” � � � 1 � � 3 3 F a b 6 a b � � � � z � � c � z � � u 3 EI 5 G A z Only one location and direction: - no high bending compliance - no torsion DESIGN OF PRECISION MACHINES 18 DESIGN OF PRECISION MACHINES 20
Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining ONE rotation Constraining ONE rotation 21 23 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining ONE rotation Constraining ONE rotation Be careful with intermediate elements! torsionally stiff body - buckling - resonances - fatigue torsionally stiff body torsionall DESIGN OF PRECISION MACHINES 22 DESIGN OF PRECISION MACHINES 24
Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining ONE rotation Constraining TWO translations More robust embodiment!!! 25 27 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining TWO translations Constraining one translation and one rotation about perpendicular axes... about the same axis…. DESIGN OF PRECISION MACHINES 26 DESIGN OF PRECISION MACHINES 28
Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining two translations and one rotation in one plane Constraining three translations 29 31 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining two rotations Constraining three translations Example: elastic ball joint with coinciding poles DESIGN OF PRECISION MACHINES 30 DESIGN OF PRECISION MACHINES 32
Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining three DOFs Constraining three DOFs using folded sheet flexures 33 35 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining three DOFs using folded sheet flexures Constraining three rotations DESIGN OF PRECISION MACHINES 34 DESIGN OF PRECISION MACHINES 36
Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining two translations and two rotations Free one DOF: translation Correct parallel guiding 37 39 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Two degrees of freedom (translations) in one plane Free one DOF: rotation DESIGN OF PRECISION MACHINES 38 DESIGN OF PRECISION MACHINES 40
Controlling Degrees of Freedom Controlling Degrees of Freedom Free one DOF: rotation Constraining 6 DOFs with 2 sheet flexures? centre line leaf spring Example: cross spring pivot 41 43 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining 6 DOF: kinematic mount Constraining 6 DOFs with 2 sheet flexures? translation is over-constrained, rotation free rotation free DESIGN OF PRECISION MACHINES 42 DESIGN OF PRECISION MACHINES 44
Controlling Degrees of Freedom Controlling Degrees of Freedom Ball joint with intermediate body Examples of controlling DOFs Play-free mounting of the ball hinge cup 45 47 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Ball joint with intermediate body Ball hinge based on “wire flexures” Flexure ball hinge lacks hysteresis Friction release is uncontrolled DESIGN OF PRECISION MACHINES 46 DESIGN OF PRECISION MACHINES 48
Controlling Degrees of Freedom Controlling Degrees of Freedom Linking shafts which are not aligned precisely Linking shafts which are not aligned precisely ◦ Transferring ONE DOF: a rotation ◦ Transferring ONE DOF: a rotation Angular precision dependent on alignment errors: 49 51 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Linking shafts which are not aligned precisely Flexible coupling with 3 thin bars ◦ Transferring ONE DOF: a rotation (universal joint) Coupling with single thin bar DESIGN OF PRECISION MACHINES 50 DESIGN OF PRECISION MACHINES 52
Controlling Degrees of Freedom Controlling Degrees of Freedom Flexible coupling with 2 times 3 thin bars Detailed design “Koster-coupling” 53 55 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Torsionally stiff coupling based on elastic elements Torsionally stiff coupling based on folded sheet flexures equivalent to “Oldham coupling” � out out Courtesy Huco � in DESIGN OF PRECISION MACHINES 54 DESIGN OF PRECISION MACHINES 56
Controlling Degrees of Freedom Controlling Degrees of Freedom Point of support should be in neutral plane of membrane Version suitable for high rotational velocities Transmitter of information on one angle 57 59 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom Constraining axial displacement without using material at Self-centering driver the centre line - shafts remain centered in relation to another Compare to a splined-shaft... - every point of contact has equal load DESIGN OF PRECISION MACHINES 58 DESIGN OF PRECISION MACHINES 60
Controlling Degrees of Freedom The body is free to expand without affecting the centre position Concept of Thermal Centre (TC) 61 63 DESIGN OF PRECISION MACHINES DESIGN OF PRECISION MACHINES Controlling Degrees of Freedom Controlling Degrees of Freedom The Concept of Thermal Centre (TC) Constraining a centre line Instant centre of rotation does not coincide with the thermal centre Using symmetry to position the thermal centre in the object centre DESIGN OF PRECISION MACHINES 62 DESIGN OF PRECISION MACHINES 64
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