4.2.2 Torque measurement using strain gauges

A strain gauge is an element whose resistance changes if the surface it is mounted to experiences mechanical strain. Silicon strain gauges are often used in force sensors due to their extremely high sensitivity. Their constitutive relationship is given by (21).

(21)

where R is the nominal resistance, is the change in resistance due to some externally imposed strain, , and is the gain factor. Silicon strain gauges have gain factors around 130, as compared to metal foil gauges which have a gain factor of about 2. The Wheatstone Bridge with four active elements is often used in strain gauge applications, due to its low sensitivity to temperature changes and its ability to measure strains in specific directions. In our case, we used the Analog Devices 1B31 Signal Conditioner to drive the bridge input, amplify the output signal, and provide an anti-aliasing filter.

The damper cup has four flexure elements (designed by Paul Millman), and maintains high stiffness while still allowing the flexure elements to undergo substantial strain. (Figure 20).

Figure 20. Damper cup with four flexures and close-up of flexure with strain gauge.

In the initial design, we placed all four strain gauges on the same flexure element, mounted vertically as shown in Figure 21.

Figure 21. Top view of a flexure element with four active strain gauges. The gauge pairs

and

experience the same strain under torsional loading of the damper.

The electrical configuration of the strain gauges is as shown in Figure 22.

Figure 22. Electrical diagram of a Wheatstone Bridge with four active elements.

is the bridge excitation and

is the output voltage.

The output voltage, , can be expressed in terms of the input voltage and resistances of the four bridge elements :

(22)

Further simplification can be made if the nominal resistances are all R, and if each element changes resistance by an amount . For example, suppose and increase by , while and decrease by . The output voltage is then given by :

(23)

Using the constitutive relationship for strain gauges (21), we express output voltage as a function of flexure strain :

(23)

The next step in this development is to find the strain, , as a function of damper velocity, U. Since the top of the cup is bolted to the 1 DOF box, it can be considered mechanical ground. Each flexure can be modeled as a cantilever beam, as shown in Figure 23. Note that the end of the each flexure is constrained to zero slope due to the high stiffness of the cup for rotations around a radial axis.

Figure 23. Model of a single flexure element. P is the load on the flexure due to shear in the damper, h is the length of the flexure, and t is the thickness.

Since the load is shared equally among four flexure elements, P is calculated using (20) from 4.2.1 :

(24)

Assuming small displacements, the strain experienced by each gauge is given by :

(25)

where E is Young's Modulus of Elasticity, and I is the area moment of the flexure element, and x is the position of the center of the strain gauge (notice that the length of the gauge doesn't affect the signal in this case). In this case, we set x=(1/4)h, so that (25) simplifies to (26).

(26)

where the area moment of the flexure element is given by :

(27)

where w is the width of the flexure and t is the thickness. Substituting (26) and (27) into (23), we finally get an equation that relates the output voltage of the strain gauge circuitry to the velocity of the damper :

(28)

It is important to remember at this point that we are not relying on this result to ensure performance of the system, but rather as a tool to evaluate and redesign it. In the following subsections, this will become clear, as we highlight a specific problem with our damping cancellation technique, along with an effective solution.

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