Принципы нанометрологии (1027506), страница 14
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Kinematics is a branch of mechanics that deals with relationshipsbetween the position, velocity and acceleration of a body. Kinematic designaims to impart the required movements on a body by means of constraints [6].A rigid body possesses six degrees of freedom in motion - three linear andthree rotational. In Cartesian coordinates the degrees of freedom are in the x,y and z directions plus rotations about each of the axes. A constraint is thatwhich prevents minimally motion in just one of the degrees of freedom.There are two lemmas of kinematic design [3]:Kinematic design-any unconstrained rigid body has six degrees of freedom;-the number of contact points between any two perfectly rigid bodies isequal to the number of constraints.This means thatNumber of constraints þ remaining number of degrees of freedom ¼ 6:There are often many assumptions applied when carrying out kinematicdesign.
Real bodies are not perfectly rigid and will experience both elastic andpossibly plastic deformations under a load. Such deformations will excludeperfect point contacts and cause unwanted motions. For this reason it isoften important to choose with care the materials, shapes and surface textureof a given part. Despite this, kinematic design is an extremely importantconcept that the designer must master.
Two examples of kinematic designwill be considered here – the Kelvin clamp and a single degree of freedommotion system. These are, essentially, the only two kinematic designs usedon the majority of MNT measuring instruments.3.2.1 The Kelvin clampsThe Type I and Type II Kelvin clamps are examples of fully constrainedsystems, i.e. ones with six constraints.
When designed properly these clampsare very effective where accurate re-positioning is required and are stable towithin nanometres [7].Both clamps have a top-plate (on which, for example, the object to bemeasured is placed) that has three rigid spheres spaced on a diameter. Thethree spheres then contact on a flat and in a vee and a trihedral hole, as inFigure 3.1a, or in three vee-grooves, as in Figure 3.1b. In the Type II clamp itis easy to see where the six points of contact, i.e. constraints are – two in eachvee-groove. In the Type I clamp one contact point is on the flat, two more arein the vee-groove and the final three are in the trihedral hole.
The Type Iclamp has the advantage of a well-defined translational location based on theposition of the trihedral hole, but it is more difficult to manufacture. Atrihedral hole is produced by pressing three spheres together in a flatbottomed hole (the contacting sphere will then touch at a common tangent)or by complex angled machining techniques. For miniature structures ananisotropic etchant can be used on a single crystalline material [8]. The TypeII clamp is more symmetrical and less influenced by thermal variations.
Notethat the symmetrical groove pattern confers its own advantages but is nota kinematic requirement; any set of grooves will do provided that they are notall parallel.3738C H A P T ER 3 : Precision measurement instrumentation – some design principlesFIGURE 3.1 (a) A Type I Kelvin clamp, (b) a Type II Kelvin clamp.3.2.2 A single degree of freedom motion deviceThere are many methods for producing single degree of freedom motion (seefor example [9]). One method that directly uses the idea of single pointcontacts is the prismatic slideway [3].
The contact points are distributed ontwo non-parallel flat surfaces as shown in Figure 3.2. In practice the sphereswould be attached to the carriage. The degrees of freedom in the system canbe deduced by considering the loading necessary to keep all five spheres incontact. Firstly, the three-point support could be positioned onto the horizontal plane, resulting in a linear constraint in the z axis and rotaryconstraints about the x and y axes.
A carriage placed on this plane is free toslide in the x direction until either of the two remaining spheres contacts thevertical face. The x axis linear degree of freedom is then constrained. Furtherhorizontal force would cause the carriage to rotate until the fifth spherecomes into contact, removing the rotary degree of freedom about the z axis.This gives a single degree of freedom linear motion along the y axis.3.3 DynamicsMost precision instruments used for MNT metrology involve some form ofmoving part. This is especially true of surface texture measuring instrumentsand CMMs. Motion usually requires some form of guideway, this being twoor more elements that move relative to each other with fixed degrees offreedom. For accurate positioning, the play and the friction between the partsin the guideway must be reduced (unless the friction characteristics are beingused to impart damping on the guideway).
To avoid sticking and slipping ofDynamicsFIGURE 3.2 A single degree of freedom motion device.the guideway the friction should normally be minimised and kept ata constant value even when there are velocity or acceleration changes. It isalso important that a guideway has a smooth motion profile to avoid highaccelerations and forces.The symmetry of a dynamic system plays an important role. Witha rotating part the unbalance and mass moment of inertia must bereduced. A linear guideway should be driven through an axis that minimizes any angular motion in its travel (its axis of reaction). Stiffness isanother important factor; there must be a trade-off between minimizingthe forces on a guideway and maximizing its stiffness. As with themetrology frame the environment in which the instrument is housedaffects its dynamic characteristics.Guideways can be produced using many techniques, but the most popularthree are:-flexures – usually used only over a small range owing to the elasticlimit and parasitic motion [3,10,11];3940C H A P T ER 3 : Precision measurement instrumentation – some design principles-dry or roller-bearing linear slideways – as used on surface profilemeasuring instruments, for example [12];-hydrostatic bearings (air bearings) [4].Many of the most advanced guideways use active feedback control systems[13,14].3.4 The Abbe PrincipleThe Abbe Principle was first described by Ernst Abbe (1890) of Zeiss andstates:If errors of parallax are to be avoided, the measuring system mustbe placed co-axially (in line with) the line in which displacement(giving length) is to be measured on the work-piece.Abbe error occurs when the measuring point of interest is displacedlaterally from the actual measuring scale location (reference line or axis ofmeasurement), and when angular errors exist in the positioning system.Abbe error causes the measured displacement to appear longer or shorterthan the true position, depending on the angular offset.
The spatial separation between the measured point and reference line is known as the Abbeoffset. Figure 3.3 shows the effect of Abbe error on an interferometricmeasurement of length. To ensure zero Abbe error, the reflector axis ofmovement should be co-linear with the axis of measurement. To account forthe Abbe error in an uncertainty analysis relies on knowing the magnitude ofthe Abbe offset and the magnitude of the errors in motion of the positioningsystem (for example, straightness).FIGURE 3.3 Effects of Abbe error on an optical length measurement.Elastic compressionThe Abbe Principle is, perhaps, the most important principle in precisioninstrument design and is also one that is commonly misunderstood – Bryan[14] described it as ‘the first principle of machine design and dimensionalmetrology’.
Abbe’s original paper concentrated on one-dimensional measuringinstruments. Bryan re-stated the Abbe Principle for multi-dimensionalsystems as:The displacement measuring system should be in line with thefunctional point whose displacement is to be measured. If this is notpossible, either the slideways that transfer the displacement must befree of angular motion or angular motion data must be used tocalculate the consequences of the offset.Many three-axis instruments, especially coordinate measuring machines(CMMs), attempt to minimize the Abbe error through good design principles(see chapter 8). Two good examples of this are the Zeiss F25 CMM [16] andan elastically guided CMM developed at the Eindhoven University of Technology [17].3.5 Elastic compressionWhen any instrument uses mechanical contact, or when different parts of aninstrument are in mechanical contact, there will be some form of compression due to any applied forces. With good design such compression will beminimal and can be considered negligible, but when micrometre or nanometre tolerances or measurement uncertainties are required, elasticcompression must be accounted for, either by making appropriate correctionsor taking account of the compression in an uncertainty analysis.
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