Consider a perfect kinematic version of the triskelion in which the suspension arms are rigid rods carrying a revolute and a universal joint. Any ’vertical’ motion of their ends (i.e., of the platform support points) shortens the projection of their length into the original plane, implying that the platform must move laterally to accommodate it. But, the three arms require platform motions in different directions. Thus, the system would ideally lock (through kinematic over-constraint) in its null position. (Of course, real devices could not be perfectly rigid and some motion is likely to occur). For a beam, the axial effect equivalent to shorting length of the rods projection into its original plan is know as curvature of shorting [198]. For a beam truly fixed at both ends, any transverse deflection must be associated with the build up of axial (tensile) forces, so stretching it to a length that fits the curved shape into the required space. Note that such axial forces will tend to pull the beam into a straighter shape, so an alternative reading is that complete constraint increases the lateral stiffness beyond the predicted value.
The axial shortening of the curved beam is calculated formally by integrating the projection of the local slope along the beam. It is completely ignored in basic texts, but is covered briefly in advanced textbooks on structures or strength of materials [198]. For a pinned rod of length ` having an end deflection δ, the effective displacement along the original axial direction is (`−(`2 −δ2)), which for small δ reduces to (δ2/2L). For a transversely-loaded cantilever and closely-related forms such as the double-built-in shallow-S leaf-spring, the equivalent axial shift - sometimes known as curvature shortening is (0.6δ/`). The linear model ignores such effects.
If the triskelion platform moves in az-translation without any externally imposed tilts, then the symmetry means that ideally it will have no in-plane translations. However, the movement causes each suspension arm to bend, resulting in them having equal curvature shortening. Maintaining a continuous physical path from the base to the platform z-axis then requires a combination of other relative motions. The elbow angle between the platform arm and the beam axis could expand through a small degree of (elastic) rotary freedom or, effectively a variant, the beam could bend in-plane to open the projected angle. Az-rotation of the platform will generate a displacement at the end of its arms that directly compensates for a component of the axial shortening; this would still require slight rotary freedom at the arm-beam joint.
If tilt is imposed on the platform, with no overall z-motion, the arm ends will not necessarily displace the same amount and the component of the twist accommodated by end y-rotation of each of the beams will generally differ, as, therefore, will their in-plane curvature shortening. Since any bending causes a shortening from the default (’zero’) posi- tion, there will still be a tendency for platform rotation. The overall effect is considerably more complex than for simplez-motion and is likely to lead to small in-plane shifts of the platform centre from thez-axis. On at least some of the suspension beams a component of the platform twist must be matched by both an end-slope in the design bending axis and a twist about the longitudinal axis. The latter is relatively compliant in typical designs , but generally the overall stiffness will vary according to the relative orientation of the beam to the imposed twist. Under tilt, the in-plane projections of the arms will also shorten, generally requiring additional bending of the suspension beams in their stiff planes, and so increasing overall device stiffness. Recall here that, unlike the microprobe, the force artefact will ideally respond only to az-stimulus: this point might be an indicator of need for divergent design strategies.
A simple way to visualize to how the second-order shortening effects will influence things is to consider thexy-projection of the path through the beam and platform arm. It forms a triangle with the straight line from the beam base to thez-axis. The length of the latter is fixed. The arm-projection will vary if there is platform twist, but remains at its original
length for pure translation. The beam-projection shortens always for any deflection. For any specific initial case, the triangle is solved for the fixed ‘pseudo-hypotenuse’ by the cosine rule, using some combination of the fix base-line,the arm and beam lengths and their included angle. As the beam shortens, we need small changes in this and other angles to maintain the triangle.
An unpublished design study by Chetwynd (2009) included an exploration of this pro- jection triangle by use of spreadsheet calculations. It suggested that there is a slow-moving minimum in the need for the angle to change, dependent on the ratio, a/`, of arm length. Hence it tentatively suggested that a ratio of around 2 is best for the 60◦ elbow used traditionally in a triskelion. However, most published designs have a ratio around 0.5. This raised, but could not adequately answer, the question of whether different elbow angles might give more linear stiffness characteristics in physically compact devices.
In terms of accommodating beam shortening by platform rotation, at least for platform translation, the beams should clearly be tangential, i.e., the included elbow angle, should be 90◦. Also, longer platform arms reduce the rotation needed for a specific shortening. Smaller rotation leads to less need for lateral deflection of the suspension beam in order to maintain continuity (a square law relationship so valid even if the arm length is increased). Longer suspension beams suffer less curvature shortening for a given end deflection, again leading to less platform rotation.
If inherent platform rotations are likely to occur at magnitudes broadly similar to other second-order deflections (as is the case for dimensions and geometries being considered here), a criterion based on stabilizing the included angle is not necessarily best. While the cosine model effectively treats the projected lengths as rigid rods, a real device will presumably relax elastically to an intermediate value between all deflections, governed by the relative local stiffness. Over-constraint stiffening caused by platform rotation will arise from work done in a combination of expanding this angle and laterally deflecting (slope and linear) the suspension beam.
Depending upon the exact form of the arms projecting from the central platform, the best simple model for the junction to the suspension beam is probably the slightly conservative one that the arm and connection are locally rigid. Then there can be no local change to the included angle and all accommodation must be by in-plane bending of the suspension beam. For this model, we would impose in the plane of the device an end-slope equal to the platform in-plane rotation and an end-deflection equal to the change in projected radial length of the arm caused by that rotation.
From the reasons discussed earlier, there is no point in perusing these ideas further unless they are shown experimentally to be of practical importance. This emphasizes further need for experimental investigations of non-linear stiffness region of a variety of triskelion design; no such data exist in the public domain at the time of writing.
3.12
Conclusion
A novel contribution has been presented in this chapter for the development of analyt- ical linear elastic model for triskelion force artefacts. The previously published linear models for micro-probe have some inconsistencies in notation, but build in some common approximations and force the use of highly pre-ordained design geometries. The new approach retains good features of earlier ones while introducing consistent, systematic use of conventions generally used in mechanism analysis. More importantly, it makes direct use of direct vector-matrix tools so that the model can deal with all potential design parameters; most important here are variation in ‘elbow angle’ and the number and or distribution of suspension legs. Extensive use of pseudo-kinematic concepts highlights the like types of non-linear stiffness behaviour, which is of great importance for applications to force transfer artefacts than to micro-probe. This leads to the proposal for ‘angle-beam’ design and second linear model is developed to analysis this family. Considerable discussion of the general needs for force transfer artefacts includes exploration of like patterns for non-linearity and some of outline strategies for modelling it. However, it is conclude that detailed work is not justified in the absence of practical data on the behaviour of real device (triskelion force artefacts). Hence this chapter provides means for imposed modelling of
designs and highlights the importance of gaining experimental data.
The implementation of an enhanced linear elastic model has been discussed in the next chapter for development of triskelion software program.
Chapter 4
Implementation of enhanced linear
elastic model:
Numerical
experiments and data analysis
4.1
Introduction
This chapter presents the development of a triskelion software program that is purely dependant on the mathematical equations of the analytical linear model for a triskelion force artefacts. The newly developed program is not limited, as previous ones have been to studying the fixed geometry of the planar flexure artefact with a 60◦ elbow angle [120], [14], [118], [188], [148]. Because force artefacts have different requirements from and so many benefit from different designs, to micro-probe suspensions, research with new system is intended. The new search work is intended to explore the relative sizes, orientation of angles, etc. it is an important first stage, in combination with experimental studies in chapter 6, 7, and 8 towards discovering ‘best designs’ and minimizing the non-linearity of the triskelion force artefacts and micro-probe artefacts. The new concept of variable elbow angle has been incorporated into an enhanced linear elastic model to provide a flexible for triskelion software program for computing the behaviour of the suspension beams, forces Fz, moments Mx & My, stress σn, strain εn, stiffness constant kz, and torsional stiffnessλx of triskelion force artefact and the micro probe suspension artefact of
any elbow angle prior to their fabrication processes.
This approach has never previously been seen in the public domain or any scientific published paper for predicting the stiffness kz and torsional stiffens λx or λy of triskelion force artefact or micro probe suspension artefact.