Force - I

• Calibration • Force Measurement • Strain Measurement • Torque Cells Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 0

Force - I

• Strain gages are widely used for measurement such as force, pressure, torque, and strain.

• This is done by converting these forms of input into mechanical strain using an elastic member, which is then converted into resistance change.

• Finally the resistance change is converted into voltage using a bridge circuit.

• A combination of SGs and an elastic body are also available commercially and come in different sizes and shapes.

• Before these instruments can be used for measurement, however, they must be properly calibrated first.

Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 1

Calibration

• A measurement instrument must be calibrated by applying the inputs of known values (standards) and measuring its output. This process is called

calibration

.

• It involves a comparison of the instrument with a higher standard and, thus, reduces bias errors.

• Once this relationship is established or verified, the input values can be inferred from the measured values (often voltage) accurately.

Force Bridge Circuit + Power Source Load Cell SG Load Cell (Cantilever Beam) Output Voltage Strain Gage(s) The voltage vs force relationship must be determined experimentally.

(Calibration) Force (F=mg) (Standard brass-weight) Bridge Circuit Excitation Voltage Voltage Voltage Force Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 2

Calibration

• There is standards.

a For hierarchy instance, of the following figure is one such example for weight standards.

• A primary standard defines the value of a unit and provides the means to describe the unit with a unique number.

Secondary standards (inter-laboratory transfer standard, local standard, etc.) are only reasonable approximations to the primary standard but can be accessed more readily.

The international kilogram (Paris) National primary standard kg (Ottawa) National secondary standard weight (Ottawa) National transfer standard weight (portable) Local standard weight (weight and measures dept) Laboratory standard weight (Meas. lab. ?) Your weight instrument Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 3

Force Measurement

• Transducers which measure force, torque, or pressure usually contain an elastic member that converts the mechanical quantity into a deflection or strain.

• Elastic members employed in these transducers include link, column, rings, beams, cylinders, tubes, washers, diaphragms, etc. For force measurement, such members are called load cells.

• Two main types of load cell are considered in the following (actually we have seen them already): 1) Bending-beam load-cell and 2) Axial load cell.

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Bending-Beam Load Cell

• A bending-beam load-cell is one of the most popular types due to its simple design and low cost.

To measure an applied force F, strain gages are mounted on the beam.

Support Beam

L l

F H b • Beam theory predicts that the strain at the SG location is proportional to the applied force.

In fact,   at SG location  6 

L

   

bh

2  geometry of beam 1 property of material 

F

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Force Measurement

• Calibration gives the true (nonlinear) relationship between the measured voltage and the actual force, even for beams with a non-uniform shape, for which a theoretical relationship may be difficult to find or does not exist.

2) Axial Load-Cell

• The axial load (force) can be measured using strain gages as shown below: Beam theory predicts that   1  1 geometry of beam property of material 

F F h b

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Strain Measurement

• Ideally, calibration of measurement instrument is performed using accurate samples of input. While making such samples is easily done for many quantities such force, torque, and pressure, this may not be so for others such as strain. For strain measurement, the bridge circuit can be calibrated against force instead and the measurements (

E AC

) taken.

Then, the strain can be calculated from the measured data using a theoretical equation. The bridge output, which we have obtained earlier, can be expressed as 

R E AC



k R E ex

where k is the bridge constant which depends on a particular configuration. For instance, k=1/4 for one-arm-active bridge, ½ or 1.3/2 for two-arm-active bridge, and 1 for four-arm-active bridge, etc. Since the gage factor is defined as (GF)=  R/R/  , the strain can be written as   1

k

(GF)

E AC E ex

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Strain Measurement

• Strain gages are manufactured under carefully controlled conditions and the gage factor is provided by the manufacturer within an indicated tolerance of about  0.2%. Thus, accurate measurement of

E AC

can give a reasonably accurate strain. Of course, there will be a propagation error.

The following figure summarizes these theoretical equations.

Measurement Lab 19 Feb 2003 

E ex



R R

A

R

 

R

C 

E ex

 

R

 A

R

 

R

2 b g

E E AC ex



E ex

   4 (GF)

E AC E ex R

 

R R

  A

R R

 

R R

  C

R



E ex

   2 (GF)

E AC E ex R

  

R R

 A  

R R

 

R R

C  

R

  1 (GF)

E AC E ex

  (GF) 2  1 +  

E AC E ex

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Strain Gage Rosettes

• In the previous section of strain measurement, it was implicitly assumed that the orientation of the axis of principal (maximum) stress was known. In a more general case, this axis is not known and must be calculated using a SG rosettes, which consists of three SGs. This is not discussed further this year.

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/strain _gage_rosette.cfm

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Torque Cell

• Torque transmission through a rotating shaft can be measured using a 1) Torque cell or 2) Dynamometer. Torque cells require the measurement of angular twist or strain of the shaft, while dynamometers require that of reaction force with an arm.

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Torque Cell

Using a cylindrical elastic member, either the deflection or the strain measurement may be used to determine the applied torque.

1

.

Torque measurement from angular deflection.

A (static) torque may be measured by observing the angular deflection of a bar or hollow cylinder.

 The torque is related to the deflection angle by

T

 

G



r o

4 2

L



r i

4   No torque applied Torque T applied

r i T

where,

T

: applied torque (Nm)

r o L

G : Shear Modulus of elasticity(N/m 2 )

G

 2  1

E

   E = Young’s Modulus(N/m 2 ) μ = Poisson’s ratio r o = Outside radius (m) r i = Inside radius (m) L = Length of cylinder (m) φ = angular deflection (rad) Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 11

Strain Gage Torque Cell

The above method may be suitable for non-rotating cylinder but not for rotating one. A strain gage torque cell can be used for both cases.

The torque T applied to a solid cylindrical shaft produces tensile and compressive strains on the shaft surface.

SG 1 is mounted with its sensing the (active) tensile axis strain at 45 degrees to the shaft axis, where has a maximum value of + compressive strain  . Similarly, SG2 is mounted at -45 degrees to the shaft axis where the has a maximum value of  . SG 3 and SG4 are mounted at similar angles on the other side of the shaft and experience strains +  and  , respectively.

T r

45  45  SG3 and SG4 on opposite side 3 4   1 2   Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 12

Strain Gage Torque Cell

• Theory predicts that this maximum strain is given by   1  3  1 

T

geometry property • This four-arm-active bridge arrangement is temperature-compensated and known to be insensitive to bending and axial strains.

• When SGs are used on rotating members, slip rings are often used for signal transmission between a rotating body and a stationary instrument.

• Four SGs are connected on the shaft to form a four-arm-active bridge and the slip-rings are used to connect the bridge, which is rotating, to the power supply and recording instrument, which are stationary.

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Strain Gage Torque Cell

• The slip-ring assembly consists of a shaft on which rotating conductor rings are mounted and the outer shell which houses stationary brushes.

It is attached to the rotating member whose torque is to be measured, such that the axes of rotation of the slip-ring shaft and the rotating member coincide. Complete assemblies (some with speed sensors for power calculation) are available commercially for various torque ranges.

• Brush contacts cause wear, attract dirt, and tend to produce electrical noises. A better, but more expensive, way is to use telemetry, which employs radio frequencies and operates in a similar manner to wireless microphones Measurement Lab 19 Feb 2003 Note: this material may be copyright protected and may only be used for personal use. 14