Minimizing Distortion in Precision Devices

• Introduction
• The Model
• The Selection
• Postscript
• Further reading
• Related case studies

Introduction

The precision of a measuring device, like a sub-micrometer displacement gauge, is limited by its stiffness, and by the dimensional change caused by temperature gradients. Compensation for elastic deflection can be arranged; and corrections to cope with thermal expansion are possible too — provided the device is at a uniform temperature. Thermal gradients are the real problem: they cause a change of shape — that is, a distortion of the device, for which compensation is not possible. Sensitivity to vibration is also a problem: natural excitation introduces noise, and thus imprecision, into the measurement. So, in precision instrument design it is permissible to allow expansion, provided distortion does not occur (Chetwynd, 1987). Elastic deflection is allowed, provided natural vibration frequencies are high.

What, then, are good materials for precision devices? Table 27.1 lists the requirements.

Figure 27.1 A precision device. All such devices have a force loop ; the precision depends on its dimensional stability.

Table 27.1 The design requirements

The Model

Figure 27.1 shows, schematically, the features of such a device. It consists of a force loop, an actuator and a sensor. We aim to choose a material for the force loop. It will, in general, carry electrical components for actuation and sensing, and these generate heat. The heat flows into the force loop, setting up temperature gradients, and these in turn generate strain-gradients, or distortion. The relevant performance index is found by considering the simple case of one-dimensional heat flow through a beam with one surface exposed to a heat source (Figure 27.2).

Figure 27.2 The distortion of a beam caused by a heat flux on one of its faces.

In the steady state, Fourier's law for one-dimensional steady-state heat flow states:

,

(M27.1)

where q is heat input per unit area, λ is the thermal conductivity and dT/dy is the resulting temperature gradient. The thermal strain ε is related to temperature by

,

(M27.2)

where α is the thermal expansion coefficient and T_o is ambient temperature.

A temperature gradient creates a strain gradient dε/dy in the beam, causing it, if unconstrained to take up a constant curvature K, such that:

,

(M27.3)

where u is the transverse deflection of the beam. Integrating along the beam, accounting for the boundary conditions, gives an equation for the central deflection (distortion) δ:

,

(M27.4)

where C is a constant.

Thus for a given geometry and heat flux q, the distortion δ is minimized by selecting materials with large values of the index

.

(M27.5)

The second problem is that of vibration. The sensitivity to external excitation is minimized by making the natural frequencies of the device as high as possible (Chetwynd, 1987; Cebon and Ashby, 1994). In general, it is the flexural vibrations which have the lowest frequencies; for a beam, their frequencies are proportional to

.

(M27.6)

A high value of this index will minimize the problem.

The Selection

Here is an example in which the use of compound properties as axes is helpful. Figure 27.3 shows one way of tackling the problem, starting with the Generic record subset. The vertical axes shows the thermal-distortion index M1 ; the horizontal axis is the stiffness index M₂.

Figure 27.3 A chart of M₂ plotted against M₁

Figure 27.4 The same chart as Figure 27.3, but this time we concentrate on the light alloys branch of the materials tree

Figure 27.3 is a close-up view of the part of the chart which is interesting — the part with high M₁ and M₂. Steels, nickel and copper alloys are relatively poor by both criteria. The innovative choices lie at the top right. Diamond is outstanding, but practical only for the smallest devices (precision bearings, for example). Silicon carbide and aluminum nitride are excellent, but difficult to form to complex shapes. Silicon, an unexpected finding, is almost as good as the other fine ceramics and it is a practical choice: silicon is available cheaply, in large sections, and with high purity (and thus reproducibility). Silicon carbide is only slightly less good. The resulting short-list of candidates is provided in Table 27.2.

Light alloys feature in Table 27.2. It is worth examining them more closely. Figure 27.4 shows the results of plotting the light-alloy branch of the materials tree on the same axes. Among the light alloys, beryllium excels. But the Al-SiC metal-matrix composites are nearly as good; the composite Al-70% SiC(p) particularly so.

Table 27.2 Materials to minimize thermal distortion

Postscript

Nano-scale measuring and imaging systems present the problem analyzed here. The atomic-force microscope and the scanning-tunnelling microscope both support a probe on a force loop, typically with a piezo-electric actuator and electronics to sense the proximity of the probe to the test surface. Closer to home, the mechanism of a video recorder and that of a hard disk drive qualify as precision instruments; both have an actuator moving a sensor (the read head) attached, with associated electronics, to a force loop. The materials identified in this case study are the best choice for force loop.

Further Reading

Chetwynd, DG (1987) Precision Engineering, 9, (1), 3.

Cebon, D and Ashby, MF (1994) Meas. Sci. and Technol., 5, 296.

Related Case Studies

1. Mirrors for Large Telescopes

2. Materials for Shaker Tables

3. Insulation for Isothermal Containers

4. Heat Exchanger Tubes

5. Aperture Grills for Cathode Ray Tubes

6. Heat Sinks for Microchips