Force and torque measurement

Force and torque measurement

Strain gauges are an important basis for the electrical measurement of mechanical quantities. Strain gauges measure strain, distinguishing expansion or compression merely by sign. The strain itself is, however, rarely the ultimate result sought from the measurement. Strain gauges are much more useful when applied to modelling techniques, bio-mechanics and as the basic elements in sensing devices.

When a mechanical force is applied to a component it expands or contracts in proportion to the applied force. Any other quantities that are related to this expansion can thus be measured using strain gauges. These include mass, pressure, force, torque, displacement, torsional angle etc.

A strain gauge does not measure the force itself, only the deformation at the surface of the component, but this deformation is related to the forces extending the element. If the laws that relate to the quantities are known, a corresponding conclusion regarding the actual forces can be reached. Nevertheless, this can only be achieved if the strain gauges are placed technically in the correct arrangement.

Strain gauges allow the forces on a component to be measured without causing any damage to it, as long as certain broad limits are not exceeded. This is highly important when the components have a shape that makes calculating the forces by mathematical means rather complicated. The actual forces can be determined by experiment. The methods used for this are generalised as "experimental strain analysis".

When constructing sensing equipment, strain gauges are used in many applications as pressure or force sensors. The changes in the strain are particularly minimal in this instance. Special measures must be taken to ensure that very small changes in resistance can be measured with sufficient accuracy.

Strain gauge detectors thus form part of a chain of measurement that also includes an amplifier and a display unit. Depending on how the amplifier is constructed, the output it either a voltage or a current. The subsequent display unit shows what the measured signal is. The type of display equipment depends n the measuring method. Plotting, printing, digital or analog devices are all suitable.

Strain

Strain is defined as the change in a length L with respect to an initial length L0 (see graphic below):

The change in length L is the difference between an original length L0 and the length L when the measurement is taken:

If elongation occurs L is positive and if the length becomes shorter (compression) the sign is negative.

The strain is a ratio of two lengths. Lengths are measured in metres. The unit of strain is thus m/m but this can be disregarded although it may result in confusion.

The following are recommended for defining a numerical value for strain:

1 m/m or 10-6 m/m

1 mm/m or 10-3 m/m

1 cm/m or 10-2 m/m

Stress

Stress on a material refers its being acted on by forces. There is a distinction between normal stresses () and transverse stresses ().

Stress occurs when forces of the same magnitude act in different directions on a material. This causes there to be tension forces (z) or compression forces (d) in certain directions (see graphic below).

The stresses in a material become greater when the forces are large and the cross section of the stressed material is small. is the ratio of the acting force to the cross-sectional area:

The unit of stress is therefore N/mm2.

Modulus of elasticity

Where a material under test is in its zone of elasticity, there is a proportional relationship between stress and strain . The constant of proportionality is the modulus of elasticity E. This is a measure of the stiffness of the material and is the ratio of normal stress to strain:

The unit is the same as that for stress N/mm2. The relationship

is called Hooke's law.

Every material has a different modulus of elasticity. The modulus for steel is approximately 20 104 N/mm2 and that for aluminium is 4 104 N/mm2.

The modulus of elasticity is only applicable within the so-called "zone of elasticity", i.e. the region in the stress-strain diagram (see graphic below) where the relationship is linear. It is only possible to draw conclusions on the stresses within the material from the strain within this region.

Stress-strain diagram: (1) Hooke's gradient, (2) Limit of proportionality Bending stresses

Bending stresses are in the nature of normal stresses but they arise in a different way. During bending, forces of tension and compression occer at the same time.

The following is an example of a beam with a force acting at one end. Positive and negative stresses occur on either side of the block. Between these theere are unstressed or neutral grains.

The distribution of stress over the cross section at any given point along the stressed beam is not constant. The stress actually rises linearly from zero in the neutral grains towards a maximum value in the surface at the edges. The following graphic shows that more clearly.

Distribution of bending stress over a coross section, (1) Neutral grains Consider a stressed beam of constant cross section with a force F acting at one end. The force gives rise to a bending moment, that increases from the point where the forces acts towards the anchoring point (see graphic below).

A bending moment is the product of the force and the distance from the anchor. A bending moment at a point x is thus

and for the maximum bending moment at that point x = L

For a known section modulus Wb of the beam cross section, the bending moment can thus be used to calculate the stresses in the beam. For grains at the edge where y = h/2 the bending moment is

The section modulus depends on the nature of the cross section. For the cross section shown it is

The 'Force and torque measurement' card SO4203-5T

General

The Force and Torque Measurement UNI-TRAIN card SO4203-5T has a bending rod for measuring force and a torsion rod for measuring torque. The bending rod has four wire strain gauges that can be connected to a quarter bridge, half-bridge or full bridge. The torsion rod has two strain gauges that can be connected in a half bridge. The operating voltage for the two measuring bridges comes from a DC supply that can taken from one of the constant current sources on the measuring amplifier card SO4203-5N and maintained with the help of a Zener diode.

Move the mouse over the info icons on the photograph of the card to read the details of each section

The following graphic shows the arrangement of the individual strain gauges on the bending rod and the corresponding full bridge for measurement.

When the bridge is in use as a half bridge, the two strain gauges DMS2 and DMS4 are replaced by fixed resistors R5 and R6. For a quarter bridge only DMS1 is utilised. In this version of the circuit potentiometer P1 is used as the second leg of the circuit. The tosion rod can only be used in a half-bridge arrangement.

A range of different weights from 2g up to 200 g is provided for loading the beam or the torsion rod (see the following photograph). The torsion rod has two balance pans so that torques and moments of differing sign can be produced.

Principle of the strain gauge-beam

Principle of the strain gauge

Strain gauges are metallic sensors that convert quantities such as force pressure or torque into an extension which is reflected in changes in resistance of the gauge. The formula for the resistance R of a conductor is

where is the specific resistivity of the conductor, l the length of the conductor and A its surface area. If a force of tension F is then applied, as in the the following animation, this causes the conductor to increase in length (this is exaggerated in the animation). At the same time the conductor becomes more tapered, i.e. the cross sectional area of the conductor decreases.

It can be shown that the relative change in resistance of metallic strain gauges is approximately given by the following equation:

The constant k is called the k factor and is close to 2 for metal conductors. is the extension of the conductor due to the force acting upon it.

Designs of strain gauge

Industrial strain gauges are usually etched onto foil. Depending on the application, there are a wide variety of strain-gauge types available. The graphic alongside illustrates some typical designs. In use they are firmly attached to the surface of the object to be measured so that any deformation of the object is reflected identically by the strain gauge.

How the beam works

Most force sensors using strain gauges involve a beam (see graphic below). If a force F acts downward on one end of the beam, the top of the beam (the side under tension) increases in length and the bottom (the side under compression) shortens by the same amount.

The extension of the top of the beam is proportional to the magnitude of the force acting on it as given in the following equation

where E is the modulus of elasticity for the materialof the beam and WB is the resisting moment of the beam. For a beam of rectangular cross section this works out to be

In order to measure the force two strain gauges are attached to the top of the beam and two more to the bottom. The force causes the resistance of the strain gauges at the top to increase and that of the two underneath to decrease by the same amount. By connecting up the strain gauges in a full bridge, the force can be measured directly. The following animation elucidates the relationships involved.

Principle of the torque rod

Generation of shearing stresses

If a moment is applied to a shaft, torsional stresses appear within it. These are in the nature of shearing stresses. To understand the principle for the measurement of these forces, imagine an infinitesimal square element of area on the surface of a component. Assume that the element is fixed at one edge and that a force F acts parallel to that along the other. The animation below illustrates the situation. A force of reaction FR acts at the fixed edge and the two forces combine to distort the element.

It can clearly be seen that the lengths of the diagonal (shown here as dotted lines) change due to the force acting on the element. Diagonal D1 becomes longer and diagonal D2 gets shorter. The degree of deformation of the surface element can thus be determined by measuring the compression or extension of the diagonals using strain gauges. The change in length of the diagonals is more than along any other axis of the element. The changes measured in any other direction are smaller. The lines around the perimeter of the square are not changed if the degree of deformation is small.

Relationships between variables for the torque rod

The following diagram indicates a typical case of torsional loading of a shaft. The shaft (a) is loaded so that there is a pair of moments MD1 and MD2. Torsional stresses thus appear in the cross sections. These reach a maximum at the outer edge and fall to zero at the centre.

Diagram (b) shows a cross section. The length of the vectors drawn alongside it represent how the torsional stress increases from 0 in the centre to a maximum of max at the outer edge. The stress changes from one of the concentric rings shown in the diagram to the next but the around the circumference of each circle it is constant. This is described by the following equation:

If we consider the bounding surface of a chaft under torque (the surface of a cylinder), the following may be observed: The line along the surface runs parallel to the central axis when there is no load but when a load is applied, it spirals sharply. Taking an infinitesimal surface element on the outer surface, the same deformation as in the above animation can be seen except that above the vectors representing the force are horizontal whereas, in this case, they are vertical. This is unimportant in terms of the actual effect. By turning either of the diagrams 90, the similarity becomes obvious. The largest amount of torsional deformation also occurs along the diagonals of the surface element, i.e. at +45 or -45 to the surface line parallel to the shaft axis.

Move the mouse over the info icons on the graphic to read the details

Arrangement of strain gauges

When a shaft twists (under a torque) the maximum degee of deformation occurs along a line at +45 or -45 to the axis of the shaft. Parallel and perpendicular to the axis there is no extension or compression. Therefore the strain gauges should be set up as shown in the following diagram so that they can be suitable connected up to form a full bridge. Alternatively, it is possible to use just two strain gauges and connect them in a half-bridge configuration.

BENDING ROD 1

What is the shape of the characteristic? What is the gradient of the line in mV/g if the gain of the amplifier is eliminated? Enter your answers in the box below.BENDING ROD 2

What is the shape of the characteristic? What is the gradient of the line in mV/g if the gain of the amplifier is eliminated? How do the results compare with those for the full bridge? Enter your answers in the box below.

BENDING ROD 3

What is the shape of the characteristic? What is the gradient of the line in mV/g if the gain of the amplifier is eliminated? How do the results compare with those for the full bridge and half bridge? Enter your answers in the box below.

TORSION ROD

What is the shape of the characteristic? What is the gradient in mV/g when the gain of the amplifier is eliminated? Enter your answers in the box below.