Measurement of Force and Torque

One of the most effective devices for measuring force is a coil spring. Strictly speaking, this is not a transducer as such because it converts mechanical force into proportional mechanical movement; thus, it's an all-mechanical system rather than an energy converter. It works on the same principle as a spring balance.

You can easily make such an instrument by mounting a spring in a tube, together with a rod passing through the center. One end of the spring is fixed to the tube, the other end to the rod (see ill. 20-1). End A of the rod can be bent at right angles to lock onto something to measure pull or tensile force. End B is plain and pressed against something to measure compressive force. Thus, regardless of where tensile force or compressive force is being measured the spring is extended in the same way.

ill. 20-1. A simple spring device for measuring force.

The amount of spring extension is a measure of the force involved. This can be indicated directly by extending the tube length to accommodate the full length of the spring when fully extended and adding a pointer attached to the rod at the point at which the spring is attached to it. The tube is slotted to accommodate movement of the pointer, and a graduated scale can be added to indicate actual pointer movement (ill. 20-2).

ill. 20-2. A calibrated scale indicates the relative force in the device of ill. 20-1.

It now remains to design a suitable spring size for the range of force to be measured. The basic formula involved is:

spring deflection = (8FD^3 / Gd^4) x 4

...where:

F is the force

D is the mean diameter of the spring

d is the diameter of the wire

G is the modulus of rigidity of the spring material

N is the number of active coils in the spring

The limiting force the spring can accept is related to the torsional stress developed in the spring when extended, so another formula is involved here:

torsional stress (S) = 8FD / π x d^3

As a starting point, let’s look at spring materials first. Music wire is a logical choice (hardened spring steel wire), followed by oil-tempered steel wire or hard-drawn steel wire. These together with stainless steel wire are the strongest spring materials. For completeness we could also consider materials such as phosphor bronze and hard brass wire. The material properties we are interested in are shown in Table 20-1.

Table 20-1. Properties of Spring Materials.

Once the material is chosen, the geometry of the spring needs to be worked out: we need values for D, d, and N. Points to remember are that the stiffness of a spring (resistance to deflection) is proportional to the fourth power of the wire diameter d and varies inversely as the cube of the mean diameter D.

Both d and D thus have a marked effect on spring performance. Using a wire size only one gauge up can appreciably reduce the deflection, and vice versa. Similarly, only a small increase in spring diameter D can considerably increase the deflection; or a small decrease in D can make the spring much stiffer.

The latter effect, particularly, should be borne in mind when making a helical spring by wrapping around a mandrel. There will be an inevitable “spring back” resulting in a spring inner diameter size greater than that of the mandrel. An undersize mandrel is thus required to form a spring of required diameter. The degree of under size can only be estimated from experience because it will vary with the quality of the spring material used and with the coiling technique.

The number of active coils is those actually “working” as a spring. The usual practice is to allow three fourths of a turn (or one complete turn) at each end in the case of a plain compression spring to produce parallel ends. Thus, geometrically the spring has a total number of coils equal to N + 1 1/2 (or N + 2), the number of active coils being calculated for the required deflection performance. Extension springs, on the other hand, commonly have all the coils “active,” the ends being made off at right angles to the main coil.

Another important parameter is the spring rate (or load rate), which is simply the load divided by the deflection.

spring rate = P / deflection

= (8FD^3 / Gd^4)

When the spring is of constant diameter and the coils are evenly pitched, the spring rate is constant.

Basically, spring design involves calculating the spring diameter and wire size required to give a safe material stress for the load to be carried. it's then simply a matter of deciding how many coils are required (how many active turns) to give the necessary spring rate or “stiffness” in pounds per inch of movement.

Although the working formulas are straightforward, spring design is complicated by the fact that three variables are involved in the spring geometry: diameter (D), wire diameter (d), and number of active coils (N). However, only D and d appear in the stress formula, which is the one to start with. So here it's a case of “guesstimating” one figure and calculating the other on that basis.

DESIGNING THE SPRING GEOMETRY

The most direct way of deciding the spring geometry is to fix a value for wire diameter d (that is, guesstimate a likely standard gauge size) and calculate the required value of D from

D = (π Sd^3 / 8F)

Here F is the maximum force to be measured, and S is determined from the wire material value for maximum safe torsional stress.

Alternatively, you may need to start with a fixed value for the spring according to diameter D to fit inside a suitable tube. In this case you calculate the wire diameter size required from

The snag here is that the calculated value for d is unlikely to correspond exactly to an available gauge size. That means you will have to select the nearest available gauge size and recalculate the actual value of D required from the first formula.

Having finally arrived at suitable values for D and d, the number of active turns required is

Deflection in this case refers to the amount of spring movement you want to accommodate, that is, the length of the slot in the tube.

TORSION SPRINGS

ill. 20-3. A torsion spring may be used to measure force.

Another way of measuring force is with a torsion spring. Here one end of the spring is fixed, and the other terminates in an ex tended arm (ill. 20-3). A force applied to the end of this arm wilt produce a deflection related to the amount of applied force and the length of the spring arm. In this case the stiffness of the spring (its resistance to deflection) will be directly proportional to the fourth power of the wire diameter (d) and usually proportional to the mean spring diameter (D). The design formulas involved are

where E is the elastic modulus of the spring material, all spring steel wire E = 30,000,000

stainless steel wire E = 28,000,000

phosphor bronze E = 15,000,000

hard brass E = 9,000,000

Design calculations are again based on working the spring material within acceptable limits of stress. The force (P) acting on the spring is applied over a radius (R) equal to the effective length of the free arm of the spring. Design calculations can proceed as follows:

- Knowing the force to be accommodated and the spring arm leverage required (R), use the stress formula to calculate a suitable wire size:

where S is the maximum permissible material stress.

- Adjust to a standard wire gauge as necessary.

- Calculate the angular deflection of such a spring, using a specified value of diameter D from the deflection formula and ignoring the factor N. This will give the deflection per coil. Then simply find out how many coils are needed to produce the required deflection.

This stage may, of course, be varied. The load moment FR may be the critical factor: the spring is required to exert (or resist) a certain force (F) at a radius R with a specific deflection. In this case, having adopted a specific value for D, the deflection formula can be used to find a solution for the number of turns required. If the spring is to be fitted over a shaft or spindle, you should check that in its tightened position it does not bind on the shaft. Incidentally all springs when deflected have a store of energy (mechanical force has been connected into mechanical energy).

MEASURING TORQUE

Springs can also be used to measure torque. In this case they take the form of a flat helix or clock spring, best wound from flat- strip spring material (ill. 20-4). Any load applied in such a way as to wind the spring up (clockwork fashion) generates a torque in the spring proportional to the applied turning movement and the (unwound) spring diameter.

ill. 20-4. A spiral-wound, clock-type, spring device for measuring torque.

This principle is utilized in simple, mechanical, static, torque-measuring instruments incorporating a calibrated spring and attached pointer movement, the spring being rotated by a protruding shaft or check or similar device for attachment to a workpiece. Rotating the workpiece or instrument then gives a torque indication. and , allied to a slipping brake mechanism, measurement of dynamic torque is possible.

A slipping brake system is, in fact, the basis of mechanical dynamometers. The brake applies a load to the machine on test, the corresponding load force being determined over a specified moment arm; the product of brake load and moment arm is the torque developed.

TRUE TRANSDUCERS

Devices that are true transducers are also widely used for measuring force and torque, normally generating an electrical signal that is suitable for interfacing with instrumentation. Load cells, e.g., are widely used for measuring tensile and compressive loads, not just for weighing applications. Force measurement is, after all, only another form of weighing. Strain gauges, too, are widely used for both static and dynamic torque measurements.

ELECTRIC CURRENT INTO TORQUE

Just as specific types of transducers convert mechanical movement into an electrical signal, the same process can be used in re verse. A simple example is the movement of readout instruments (ammeters and voltmeters). The signal circuit they receive is passed through a pivoted coil to which a needle is attached. The coil faces a fixed permanent magnet. Current passing through the coil generates an electromagnetic field in opposition to that of the magnet, causing the coil to rotate away from its initial position by an amount proportional to the electromagnetic field strength. This in turn is directly proportional to the signal current flowing through the coil. A compass can be surrounded by a coil, producing deflection of the needle in a similar way (ill. 20-5).

ill. 20-5. Electric current is converted into torque by means of a compass galvanometer device.

A moving-coil instrument movement thus works as a transducer, connecting an electrical signal into a proportional movement indicating the strength of that signal. The actual value of torque (Q) generated is given by

where Q is the torque in gram-cm

A is the area of the coil in cm

B is the flux density in lines per cm or the air gap

n is the number of turns in the coil

I is the current in milliamps

With typical small meter movements the torque generated is of the order of 0.01 gram-cm per milliamp. Thus, a 0-5-mA milliammeter movement develops a torque of about 5 x 0.01 = 0.05 gram- cm. A far more robust coil would be used, developing a torque of about 5000 x 0.01 = 50 gram-cm torque. At the other end of the scale, lighter movements with very low friction bearings are needed in microammeters, with rather higher pro rata torque developed by using more turns in the coil.

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