ESA Module 1 Part-B ME832. by Dr. Mohammed Imran
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This document discusses electrical resistance strain gauges. It begins by introducing the principle behind how they operate, which was discovered by Lord Kelvin in 1856 using a Wheatstone bridge. Electrical resistance strain gauges have advantages like small size, negligible mass, and producing an electrical output. The document then discusses gauge factor and how a gauge's sensitivity is determined by both dimensional changes and changes in resistivity when strained. It also describes different types of gauge construction methods, including bonded wire, foil, and wrap-around styles. Key applications mentioned include experimental stress analysis of vehicles, structures, and machines.
![1
GHOUSIA COLLEGE OF ENGINEERING
RAMANAGARAM-562159
EXPERIMENTAL STRESS ANALYSIS
[15ME832]
Dr. MOHAMMED IMRAN
ASST PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING](https://image.slidesharecdn.com/esamodule-1-part-bme832-200515073538/85/ESA-Module-1-Part-B-ME832-by-Dr-Mohammed-Imran-1-320.jpg)
![27
h = 1 −
1
1 +
1
(1 + m)
D
h = 1 −
1
1 +
1
(1 + m)
. ∈
(4)
It shows that the nonlinear term is depends on magnitude of strain∈, gauge factor ‘F’
and ratio of m = R2 / R1. In most strain measurements the nonlinearity term h can be
neglected & E [Eq (2)] can be determined from
D =
( + )
D
−
D
− −( )
However, if high accuracy in strain measurement is required or larger strains are to be
measured. Then plastic strain determinations in metallic materials the output signal
determined through Eq (5) can be corrected for error due to nonlinearity.
(a) Range and sensitivity of the circuit:
The output signal per unit strain is known as circuit sensitivity Sc of the
potentiometer circuit is given by
D =
m
(1 + m)2
D
−
D
V (1)
E output signal per unit strain = =
∈
D
∈
=
m
(1 + m)2
D
−
D
V
∈
=
m
(1 + m)2
D 1
1
−
D 2
2
V
∈
(2)
With an active strain gauge R1 and fixed-blast resistor R2
i.e. = ; D = D ; = ; D =
=
m
(1 + m)2
D g
g
V
∈](https://image.slidesharecdn.com/esamodule-1-part-bme832-200515073538/85/ESA-Module-1-Part-B-ME832-by-Dr-Mohammed-Imran-27-320.jpg)
![Consider next the voltage
V = I (R
From which
= +
Eq. (IV) can be substituted in to Eq. (v)
I =
(
(
Substituting eqs. (vi) in to (1)
= (
From Eq. (3) It is evident that the balance condition (E = 0) for the constant
Wheatstone bridge is the same as that for the constant
namely.
=
If resistance R1, R2, R3, and R
voltage E+E measured with a very high impedance meter is
+ D = (Ʃ Ʃ )
[(
where ƩR = R + R
Expanding Eq. (vii) and simplifying after assuming the initial balance condition gives
Fig: constant Wheatstone bridge designed to minimize nonlinear effect
37
Consider next the voltage V and note the
(R + R ) = I (R + R ) (iii)
I =
( )
( )
I (iv)
(v)
Eq. (IV) can be substituted in to Eq. (v)
( )
)
I I =
( )
( )
I (vi)
Substituting eqs. (vi) in to (1)
( )
(R R − R R ) (3)
From Eq. (3) It is evident that the balance condition (E = 0) for the constant
Wheatstone bridge is the same as that for the constant-voltage Wheatstone bridge.
= (4)
, and R4 change by the amounts R1, R2, R
E measured with a very high impedance meter is
[( + D )( + D ) − ( + D )( + D
+ R + R ƩR = R + R +
and simplifying after assuming the initial balance condition gives
Fig: constant Wheatstone bridge designed to minimize nonlinear effect
(iii)
From Eq. (3) It is evident that the balance condition (E = 0) for the constant-current
voltage Wheatstone bridge.
R3, and R4, the
)] (vii)
+ R + R
and simplifying after assuming the initial balance condition gives
Fig: constant Wheatstone bridge designed to minimize nonlinear effect](https://image.slidesharecdn.com/esamodule-1-part-bme832-200515073538/85/ESA-Module-1-Part-B-ME832-by-Dr-Mohammed-Imran-37-320.jpg)
ESA Module 1 Part-B ME832. by Dr. Mohammed Imran
- 1. 1 GHOUSIA COLLEGE OF ENGINEERING RAMANAGARAM-562159 EXPERIMENTAL STRESS ANALYSIS [15ME832] Dr. MOHAMMED IMRAN ASST PROFESSOR DEPARTMENT OF MECHANICAL ENGINEERING
- 2. 2 Module-1-Part-B Electrical Resistance Strain Gauges 1. INTRODUCTION The electrical resistance strain gauge is widely used as it has a good measure of all the characteristics a strain gauge should process. Three important advantages of electrical resistance strain gauges are: (i) They are small size and used in situations where other types of gauges cannot be used. (ii) As they have negligible mass, their effect on the quantity being measured is significant. Further, they respond faithfully to rapidly fluctuating strains. (iii) As the output is electrical, remote observation is possible. Further, the output can be displayed, recorded or processed as required. The principle on which the electrical resistance strain gauge operates was discovered in 1856 by Lord Kelvin. Using a Wheatstone bridge, he measured the change in resistance in copper and iron wires due to a tensile strain. He established that the change in resistance is a function of strain and that different materials have different sensitivities, i.e. the ratios of change in strain are different. Lord Kelvin noted that the resistance of a wire increases with increasing strain and decreases with decreasing strain. The question then arises whether this change in resistance is due to the dimensional change in the wire under strain or to the change in resistivity of the wire with strain. It is possible to answer this question by performing a very simple analysis and comparing the results with experimental data which have been compiled on the characteristics of certain metallic alloys. The analysis proceeds in the following manner. The strain gauge can be easily bonded to the test component with a suitable adhesive as shown in fig (1). Any strain compressive or tensile in the test component is faithfully transmitted to the strain gauge, after attaching lead wires to the solder tabs on the gauge by soft soldering, the grid of the gauge, solder tab and base lead wires are covered with a protective in coating to prevent oxidation, electrical shorting and mechanical damage. The electrical circuit required for the measurement of the very small changes in the gauge resistance is a variation of the well-known Wheatstone bridge.
- 3. 3 The potentiometer circuit is used in some applications where purely dynamic stress measurements are involved. Temperature sensitivity, i.e. the error in strain measurement due to temperature variation can be reduced to a minimum either through the use of suitable compensation circuits or by using self-temperature- compensated gauges. However, expensive and complex auxiliary equipment is needed to energize and record the signal from the gauge. With automatic data acquisition and processing systems, output from hundreds of strain gauges bonded to the structure under test can be processed and read out in units of strain or its derived quantities. Typical applications of electrical strain gauges include: (i) Experimental study of stresses in transport vehicles – aircraft, ships, automobiles, trucks, etc.; (ii) Experimental analysis of stresses in structures and machines –apartment buildings, pressure vessels, bridges, dams, transmission towers, engines, steam and gas turbines, machine tools, etc.; (iii) Experimental verification of theoretical analysis; (iv) Aid design and development of machines and structures; (v) Assist failure analysis; and (vi) As a sensing element in transducers for measurement of force, load, pressure, displacement, torque, etc.
- 4. 4 2. STRAIN SENSITIVITY IN METALLIC ALLOYS GAUGE FACTOR: The gauge factor or strain sensitivity of a metal is denoted by FA and is defined as the ratio of the resistance change in a conductor per unit of its initial resistance to applied axial strain. Expression for gauge factor The resistance R of a straight conductor of length L, area of cross section A and resistivity ρ is given by = / -------- (1) If the conductor is stretched its length will increase and area of cross section will decrease. This result in a change of resistance R Taking log on both sides of equation (1) = ( / ) = + – . ( ) Differentiating we get = r r + − − − (b) Now A = cD 2 Where C = a constant D = some dimensions of conductor like width, diameter etc. Therefore = + 2 − − − − − ( ) = 2 − −( ) The term dA represents the change in cross-sectional area of the conductor due to the transverse strain, which is equal to - dL / L . . . = −ν − −( ) put in Eq. (d) Where ν is the Poisson’s ratio for the conductor material we get = −2ν − −( ) . ( ) Therefore Eq. (b) becomes = r r + − −2 = r r + (1 + 2)
- 5. 5 But Longitudinal strain ε = dL / L Hence = r r + ε(1 + 2) Or ε = r r ε + (1 + 2) − − − − − − − (2) The term on the left hand side of equation (2) which represents the change in resistance / unit, resistance / unit strain is defined as the gauge factor and is denoted by FA or SA. Hence FA = SA = r r + ( + n) Where SA or FA is the sensitivity of the metallic alloy used in the conductor and is defined as the resistance change per unit of initial resistance divided by the applied strain.Examination of the above Eq (2) Shows that the strain sensitivity of any alloy is due to two factors, namely, The change in the dimensions of the conductor, as expressed by the 1 + 2 term. The change in specific resistance with respect to ( dρ / ρ) / ϵ. Experimental results show that F A varies from about 2 to 4 for most metallic alloys. If strain sensitivity approaches to 2 when the gauge experiences plastic deformation, this specify that specific resistance is zero (0) and Poisson’s ratio approaches to 0.5 ie., if have strain gauge which has strain sensitivity close to 2 from elastic (E.R) to plastic region (P.R) it do not need any modification it becomes linear as shown in figure. The strain sensitivity is modified slightly by the form of construction and the pattern of the strain gauge then sensitivity of strain gauge is termed the gauge factor F and is given by = The manufacturer gauge factor ‘F’ for strain gauges is determined normally through a calibration test in a uniaxial stress field, Ex. The tensile test it is to be note that the strain gauge bonded to the calibration test is a destructive test 2 E.R P.R R/R % of Strain Fig(2) R/R V.S % of Strain
- 6. 6 3. GAUGE CONSTRUCTION : It is theoretically possible to measure strain with a single length of wire as the sensing element of the strain gauge; however circuit requirements needed to prevent overloading of the power supply and minimum resistance required from instrumentation point of view is 100Ω (i.e. lower limit of resistance) If for example, diameter of the conductor is 0.025mm (0.001in) and resistance per meter is 1000Ω calculate to have a minimum resistance of 100Ω, what is the length of wire is required. As a result a 100Ω strain gauge fabrication from wire having a diameter of 0.025mm (0.001in) & have resistance of 25 Ω / in (1000 Ω) requires a single length of wire 4 in (100mm) long, to make a measurement. But it is too long, obviously one cannot measure strain at a point using a long wire! Hence, the gauge is formed by folded grid etched on metal foil & wire grid. Resistance strain gauges with a metallic-sensing element may be broadly classified into four groups: (i) Un-bonded- wire strain gauges, (ii) Bonded-wire strain gauges, (iii) Foil strain gauges, and (iv) Wieldable strain gauges. Construction of bonded- wire strain gauges: Two methods of construction are generally employed for constructing bonded-wire strain gauges. In the flat-grid type gauge illustrated shown in fig (3), the sensing element or grid is formed by winding the wire around pins on a jig. The grid is then lowered on a backing material and cemented to it before withdrawing the pins. The grid is then covered with a suitable protective material. In the wrap-around type gauge (fig 4), the wire is wound in the form of a helix around a thin walled cylinder of insulating material. This cylinder is then flattened and bonded between two sheets of a suitable insulating material. Alternatively, the wire may be wound on a thin card of appropriate size and bonded between two sheets of insulating material. As the wrap-around type gauge has two layers of wire and three layers of insulating material it is considerably thicker than the flat-grid type gauge. The wrap-around
- 7. 7 gauge is not suitable for use on thin sections subjected to bending as the strain indicated by it is inaccurate due to thickness effects. The performance of the wrap- around gauge is unsatisfactory when transient or rapidly varying strain and/or temperature are involved. Generally, flat-grid gauges are preferred as they are superior to wrap-around gauges in terms of hysteresis, creep, elevated-temperature, performance, stability, especially under hydrostatic pressure fluctuations, and current carrying capacity. Formerly, only wrap-around gauges were available in shorter (less than 6mm) gauge lengths. Today, mainly through improvements in production methods, flat-grid gauges are also available in shorter gauge lengths. Construction of foil gauges: In the foil gauge, the foil grid is made by etching the desired grid pattern in a metal foil only a few microns in thickness. The grid pattern can also be cut from the foil using high-precision dies. The foil grid is carefully bonded to a thin flexible carrier or backing as shown in fig (5). Any conceivable grid configuration can be produced by these processes accurately.
- 8. 8 The width of the section at each turn of the straight filament portions as shown in figure is increased to reduce the resistance in the transverse direction to a relatively low value. This configuration makes the foil gauge quite insensitive to strains in the transverse direction. The large surface area near the ends of the straight filaments also ensures that linear conditions prevail over the complete active length of the grid. As foil gauges have a greater bonding (surface) area to cross-sectional area ratio than wire-gauges, they have enhanced (avoid) heat-dissipation properties. As this permits use of higher voltage levels for gauge excitation, higher sensitivity can be achieved. As the foil gauge has a larger contact area for bonding onto the test component, the stress in the adhesive is lower. Consequently, the stress relaxation and hysteresis are significantly less in foil gauges. For these reasons the performance of the foil gauge is superior to that of the wire gauge. Currently, foil gauges are used extensively. The use of wire gauges is mostly limited to applications such as stress analysis at elevated temperatures, where it still possesses an edge over the foil gauges. Gauge materials: The strain sensitive alloy used in the wire or foil grid determines to a great extent the operating characteristics of a strain gauge. Other factors which influence significantly the performance of a strain gauge are the properties of the backing material and the bonding material. The desirable features or properties in a grid material are: (i) High gauge factor, gauge factor constant over a wide range of strain, (ii) High specific resistance (iii) Low temperature coefficient of resistance (iv) High elastic limit, (v) High fatigue strength, (vi) Good workability, soldering and weldability, (vii) Low mechanical hysteresis, (viii) Low thermal emf when joined with other materials, and (ix) Good corrosion resistance.
- 9. 9 Probably the most serious problem in strain measurement is sensitivity to temperature variations. This effect is minimized through the use of self-temperature-compensated gauges and/or the bridge-compensation method. Thermal emf superimposed on the gauge output must be avoided if dc circuitry is employed. This factor presents no problem in the case of ac circuitry. Corrosion at a junction between the grid and lead wire could possibly result in a miniature rectifier; this would be more serious in an ac circuit. In some applications the influence of strong magnetic fields on gauge performance is of great importance. Grid materials of high nickel content are susceptible to the effect of magnetostriction and magnetoresistivity. The apparent strain caused by these two effects may be significant enough to preclude use of such materials as gauge grid material. Some of the important alloys or its equivalent that ore commonly used as gauge grid material are (i) Constantan or Advance, (ii) annealed Constantan, (iii) iso-elastic, (iv) Nichrome V (v) Karma (Ni-Cr-Al alloy with iron), (vi) iron-chromium aluminium alloy (Armor D), and (vii) Platinum-tungsten alloys. Backing materials or carrier materials: A strain gauge backing material has several functions to perform. It provides support to the grid and ensures dimensional stability of the grid. It also provides mechanical protection to the grid during handling and mounting. The backing material transmits the strain from the test material to the grid and provides electrical insulation between the grid and the test material or component. The backing should be stiffer than the grid to support it and also to ensure that the grid is not disturbed when it is strained. However, the backing should be flexible enough to faithfully follow strain changes in the test component without in any way
- 10. 10 altering it. There are several backing materials which satisfy these requirements. They are i) Paper ii) Polymide iii) Epoxy plastics iv) Epoxy-phenolic resin v) Glass fibre-reinforced epoxy-phenolic vi) Bakelite A Good Carrier Materials should have the following characteristics 1. Minimum thickness 2. High Mechanical strength 3. High dielectric strength 4. Minimum Temperature restrictions 5. Good Adherence to cement used 4. ADHESIVES AND MOUNTING TECHNIQUE Adhesives or bonding cement: The bonded wire or foil gauge should be bonded to the test component with a suitable adhesive. The strain gauge adhesive should be sufficiently elastic to faithfully transfer strain in the test component to the gauge – sensing element or grid. For optimum performance, the adhesive prescribed by the gauge manufacturer should be used and the recommended procedure for mounting the gauge should be followed. Several important factors have to be considered while selecting the adhesive for a particular strain gauge and test component combination. It is very important to ensure that the adhesive is compatible with both the gauge-backing material and test material. The adhesive should not damage either of them. The adhesive should also have long-term stability and high creep resistance at the maximum strain level over the expected temperature range of operation. The adhesive should also have high insulation resistance. The main types of adhesives which are commonly used are i) Nitro-cellulose cement ii) Epoxy cements iii) Cyanoacrylate cement iv) Phenolic adhesives v) Ceramic cements
- 11. 11 Desirable characteristics of the Bonding cement or Adhesive 1. High mechanical strength 2. High creep resistance 3. High dielectric strength 4. Minimum temperature restrictions 5. Good adherence giving shear strength 10.5 to 14 Mpa 6. Minimum moisture absorption 7. Ease of application 8. Low setting time A Typical Method for Bonding Strain Gauges: A strain gauge can only give best results if it is bonded to the test piece in such a manner so that the strain experienced by the gauge grid is precisely the same as the strain of the test specimen. To achieve this the proper installation of gauge is very important. In fact the bonding technique depends upon the type of gauge and the cement to be used, which is supplied by the manufacturer. Hence the instructions as per manufacturers catalogue should be followed. However, the following steps, in general, may be followed while applying the strain gauge to the test specimens: 1. Surface preparation 2. Gauge preparation 3. Adhesive preparation 4. Gauge installation 5. Lead wire connection 6. Environmental protection. A typical method of bonding a strain gauge is described here. 1. Surface preparation The surface to which the gauge is to be bonded should be properly clean, smooth and have the proper chemical affinity to the adhesive. Using emery paper or cloth, any rust or paint on the surface is removed to obtain a smooth but not highly polished surface. The prepared surface is then washed with a solvent to remove metal or dirt particles and grease. Some of the cleaning fluids suitable for this purpose are acetone, trichloroethylene, methyl-ethyl-ketone, chlorethene NU and Freon TF. To achieve the degree of cleanliness required, the surface may have to be washed several times and the washed surface wiped with clean paper towel or lint-free cloth until the wiping cloth or paper no longer picks up dirt. The cleaned surface may be treated with a basic solution to give the surface the proper chemical affinity to the adhesive.
- 12. 12 2. Gauge preparation Fig (6) Gauge Location The strain gauge with its backing (bonding surface) also degreased is bonded to the test surface immediately after it is cleaned. The location of the gauge on the test surface is marked with a pencil of hardness 4H to 8H. The lines are scribed outside the area where the gauge is to be bonded. The gauge is then laid in position. 3. Adhesive preparation Fig (7) Adhesive preparation The supply leads on the gauge are taped so that the gauge can be raised up for applying the adhesive without altering its location. Alternatively, the gauge can be positioned by using a rigid transparent tape shown in figure (7a). After the adhesive is applied to the area marked on the test surface, the gauge is brought into position and is pressed down gently with a thumb as shown in figure. This gentle pressure with the thumb is to squeeze out excessive adhesive and to ensure that there are no air bubbles between the test surface and the backing Figure (7b). Terminal tabs may be cemented along with the gauges.
- 13. 13 4. Gauge installation The gauge installation is then subjected to a proper combination of pressure and temperature for a recommended length of time to ensure complete cure of the adhesive. During the curing process, the adhesive may expand because of heat, undergo a volume reduction during polymerization, and contract upon cooling. Any residual stresses in the adhesive will deform the relatively flexible gauge grid element and influence the output of the strain gauge. Therefore the curing is critical and should be complete particularly when the gauge installation is for long-term strain measurements. 5. Lead wire connection Fig (8) Lead – wire layout Lead wires are needed to transmit the electrical signals from strain gauges to the strain measuring instrument. The lead-wire system connected to the gauges must perform satisfactorily under all environmental conditions. The lead wires should have low resistance and low temperature coefficient of resistance. They should not introduce significant resistance change, or generate or transmit electrical noise. As copper has low specific resistance, it is commonly used as material for lead wires. However, it should be noted that copper has a large temperature coefficient of resistance and has poor corrosion and fatigue resistance. Tinned, plated or metal clad solid copper wires have superior corrosion resistance.
- 14. 14 Stranded copper wires are flexible. Hence they are used when relative motion between the lead wire and the component is expected. Nickel - chromium alloy lead wires are prescribed in high-temperature applications. These wires are suitable for temperatures up to about 370°C. As it has a high specific resistance, only short-length lead wires should be used. 6. Environmental protection Strain-gauge installations are affected by environments containing water or moisture or chemical vapours. They are also susceptible to mechanical damage. Therefore unless tests are to be conducted under laboratory conditions within a short time after installation, protective coatings are essential. The important considerations that influence the selection of a coating are test environment, test duration and the degree of accuracy required. Though several commercial coatings are available, only those proven by tests to be electrically and chemically compatible with a gauge installation should be used in any application. 5. GAGE SENSITIVITIES AND GAGE FACTOR The strain sensitivity of a single, uniform length of a conductor was preciously defined as = e » D e (1) Where ϵ is a uniform strain along the conductor and in the direction of the axis of the conductor, this sensitivity FA is strain sensitivity to the axial strain, whenever the conductor is wound into a strain gauge grid to yield the short gage length required for measuring strain, the gage exhibits sensitivity to both axial and transverse strain. The change is introduced by end loops, which are transverse to the straight portion of the grid. Thus the gauge in addition to measures the strain reading. This is known as the transverse or cross sensitivity of the gauge. Transverse sensitivity of a strain gauge is a measure of its response to strains perpendicular to a primary sensing axis, a-a (fig 9) however, their transverse sensitivity is a small fraction of their axial sensitivity Fig (9) Biaxial Strain Gauge
- 15. 15 The response of a bonded strain gage to a biaxial strain field can be expressed as D = ϵ + ϵ + g (2) Where ϵa = normal strain along axial direction of gage ϵt = normal strain along transverse direction of gage at = shearing strain Fa = sensitivity of gage to axial strain Ft = sensitivity of gage to transverse strain Fs = sensitivity of gage shearing strain In general, the gage sensitivity to shearing strain is small and can be neglected. The response of the gage can then be expressed as D = ϵ + ϵ + 0 D = ϵ + ϵ (2′) D = (ϵ + ϵ ) D = ϵ 1 + ϵ ϵ (3) Where = is defined as the transverse sensitivity factor for the gage. Now it is common practice to calibrate a strain gauge in a uniaxial stress field, i.e. in a biaxial strain field with the ratio of the transverse-to-axial strain equal to the Poisson’s ratio of the specimen material Eq (a). Thus Strain-gage manufacturers provide a calibration constant known as the gage factor ‘F’ for each gage. The gage factor “F” relates the resistance change to the axial strain Eq (4). With this method of calibration, the strain field experienced by the gage is biaxial, with ϵt = -o ϵa (a) D = F . ϵ (4) where o = 0.285 is Poisson’s ratio of the material. If Eq.(a) is substituted into Eq.(3), the resistance change in the calibration process is D = ϵ (1 + (− ))
- 16. 16 D = ϵ (1 − ) (5) Since the resistance changes given by Eqs. (4) and (5) are identical, the gage factor is related to both and by the expression D = ϵ (1 − ) = F . ϵa = ( − n ) (6) It may be noted that even when k t is as high as 10% and o = 0.3 is Poisson’s ratio from Eq (6) Fa is only 1.03 times of ‘F’. It should be realized that Eq (4) is valid for the uniaxial stress field used by the manufacturer to calibrate the strain gauge. 6. PERFORMANCE CAHRECTERISTICS OF FOIL STRAIN GAGES. Foil strain gages are small precision resistors mounted on a flexible carrier that can be bonded to a component part in a typical application. The gage resistance is accurate to ±0.4 percent, and the gage factor, based on a lot calibration, is certified to ±1.5 percent. These specifications indicate that foil-type gages provide a means for making precise measurements of strain. The results actually obtained however, are a function of the installation procedures, the state of strain being measured, and environment conditions during the test. All these factors affect the performance of a strain- gage system, in general. a. Strain- Gage Linearity, Hysteresis, and Zero Shift b. Temperature Compensation c. Elongation Limits d. Stability A. Strain- Gage Linearity, Hysteresis, and Zero Shift One measure of the performance of a strain gage system (system here implies gage, adhesive, and instrumentation) involves considerations of linearity, hysteresis, and zero-shift. If gage output, in terms of measured strain, is plotted as a function of applied strain as the load on the component is cycled, results similar to those shown in fig.4. Will be obtained A slight deviation from linearity is typically observed, and the unloading curve normally falls below the loading curve to form a hysteresis loop. Also, when the applied strain is reduced to zero, the gage output indicates a small negative strain,
- 17. termed zero shift. The magnitude of the deviation from linearity, hysteresis, and zero shift depends upon the strain level, the adequ of the foil material, and the carrier material. Fig 10: A typical strain cycle showing nonlinearity, hysteresis, and zero For properly installed gages, deviations from linearity should be approximately 0.1 percent of the maximum strain for polyimide carriers and 0.05 percent for epoxy carriers. First cycle observed in typical applications If possible, shift cycling to 125 percent of the maximum test strain is recommended since the amount of hysteresis and zero shift will decrease to less than 0.2 percent of the maximum strain after 4 or 5 cycles. B. Temperature Compensation A very important problem in strain measurement is the effect of temperature on the performance. Strain gauge installation is subjected to temperature changes during the test period, and careful consideration must be given for determining whether the change in resistance is due to applied strain or temperature change. When the ambient temperature changes four effects occur which may alter the performance characteristics of the gauge 1. The strain sensitivity of the metal alloy used for the grid changes 2. The gage grid either elongates or contracts ( 17 termed zero shift. The magnitude of the deviation from linearity, hysteresis, and zero shift depends upon the strain level, the adequacy of the bond, the degree of cold work of the foil material, and the carrier material. A typical strain cycle showing nonlinearity, hysteresis, and zero shift (scale exaggeration) For properly installed gages, deviations from linearity should be approximately 0.1 percent of the maximum strain for polyimide carriers and 0.05 percent for epoxy carriers. First cycle hysteresis and zero shifts are more frequently lications If possible, shift cycling to 125 percent of the maximum test strain is recommended since the amount of hysteresis and zero shift will decrease to less than 0.2 percent of the maximum strain after 4 or 5 cycles. Temperature Compensation ry important problem in strain measurement is the effect of temperature on the performance. Strain gauge installation is subjected to temperature changes during the test period, and careful consideration must be given for determining resistance is due to applied strain or temperature change. When the ambient temperature changes four effects occur which may alter the performance characteristics of the gauge 1. The strain sensitivity of the metal alloy used for the grid changes ge grid either elongates or contracts (∆L/L = α∆T) termed zero shift. The magnitude of the deviation from linearity, hysteresis, and zero acy of the bond, the degree of cold work A typical strain cycle showing nonlinearity, hysteresis, and zero For properly installed gages, deviations from linearity should be approximately 0.1 percent of the maximum strain for polyimide carriers and 0.05 and zero shifts are more frequently lications If possible, shift cycling to 125 percent of the maximum test strain is recommended since the amount of hysteresis and zero shift will decrease to less than 0.2 percent of the maximum strain after 4 or 5 cycles. ry important problem in strain measurement is the effect of temperature on the performance. Strain gauge installation is subjected to temperature changes during the test period, and careful consideration must be given for determining resistance is due to applied strain or temperature change. When the ambient temperature changes four effects occur which may alter the performance 1. The strain sensitivity of the metal alloy used for the grid changes
- 18. 18 3. The base material upon which the gage is mounted either elongates or contracts (∆L/L = β∆T). 4. The resistance of the gage changes because of the influence of the temperature Co - efficient of resistivity of the gage material (∆R/R = υ∆T) Where α = Thermal Co efficient of expansion of gage material β = Thermal Co efficient of expansion of base material ρ = Thermal Co efficient of resistivity of gage material The combined effect of these three factors will produced a temperature induced change in resistance of the gauge (∆R/R) ∆T which may be expressed as (∆R/R) ∆T = (β – α) ∆T.F + υ∆T -------- (1) Where F = gauge factor and ∆T is rise in temperature In order to prevent significance errors due to this effect some form of temperature compensation is usually employed when strain gauges are used in applications where steady state strain must be measured. There are two methods are available for effecting temperature compensation in a gauge installation. In the first method, the gauge alloy coefficients α & β are adjested to minimize the range when bonded to a test material with a matching coefficient for linear expansion α. Such gauge is called a Self – temperature compensated gauge The second method involves in general, the use of a compensating (dummy) gauge in the electrical system to eliminate the error due to the apparent strain. In theis method for zero error, the apparent strains be exactly equal i.e. in turns to temperature effect is approximated. C. Elongation Limits The maximum strain that can be measured with a foil strain gage depends on the gage length, the foil alloy, the carrier material, and the adhesive. The Advance and Karma alloys with polyimide carriers, used for general-purpose strain gages, can be employed to strain limits of ±5 and ±1.5 percent strain, respectively. This strain gage is adequate for elastic analyses on metallic and ceramic components, where yield or fracture strains rarely exceed 1 percent; however, these limits can easily be exceeded in plastic analyses, where strains in the post yield range can become large. In these instances, a special post-yield gage is normally employed, it is fabricated using a double annealed Advance foil grid with a high-elongation polyimide carrier. Urethane-modified epoxy adhesives are generally used to bond post-yield gages to the
- 19. 19 structure. If proper care is exercised in preparing the surface of the specimen, roughening the back of the gage, formulating a high-elongation plasticized adhesive system, and attaching the lead wires without significant stress raisers, it is possible to approach strain levels of 20 percent before cracks begin to occur in the solder tabs or at the ends of the grid loops. Special purpose strain gage alloys are not applicable for the measurements of large strains. The Isoelastic alloy will withstand ±2 percent strain; however, it undergoes a change of sensitivity at strains larger than 0.75 percent (see fig. below). Armour D and Ni chrome V are primarily used for high temperature measurements and are limited to maximum strain levels of approximately ±1 percent. Fig 11: A liquid metal electrical resistance strain gage. For very large strains, where specimen elongations of 100 percent may be encountered, liquid-metal strain gages can be used. The liquid-metal strain gage is simply a rubber tube filled with mercury or a gallium-indium-tin alloy, as indicated in fig. 12. When the specimen to which the gage is attached is strained, the volume of the tube cavity remains constant since Poisson’s ratio of the rubber is approximately 0.5. Thus the length of the tube increases (l = ϵl) while the diameter of the tube decreases (d = -ϵd). The resistance of such a gage increases with strain, and it can be shown that the gage factor is given by F = 2 + ϵ (1) Performance characteristics response of a liquid metal gauge shows lightly nonlinear with increasing strain due to the increase in gauge factor with strain and due to change in resistance of gauge. Rubber capillary tube Gallium Indium tin Lead wire
- 20. 20 D. Stability: In some applications strains in the test components may have to be recorded over a long period of time running into months and years with no freedom to unloading the component for checking the zero reading in such applications maximum stability of the strain gauge insulation is very important requirement. Moisture and humidity effects on the backing, stress relaxation in the adhesive, backing and grid material and instabilities in the resistance in the inactive arm of the strain measuring bridge are the primary causes for drift in the zero reading of a strain gauge installation. 7. ENVIRONMENTAL EFFECTS The performance of resistance strain gages is markedly affected by the environment. Moisture, temperature extremes, hydrostatic pressure, nuclear radiation, and cyclic loading produce changes in gage behavior which must be accounted for in the installation of the gage and in the analysis of the data to obtain meaningful results. Each of these parameters is discussed in the following subsections. (a) Moisture and humidity: Absorption of moisture by the backing material and the adhesive causes significant degradation of short term and long term performance of the strain gauge installation. Moisture absorption causes are A decrease in insulation resistance of the gauge which manifests itself in the form of an apparent strain due to change in the effective resistance of the gauge. Variation in the gauge factor due to decrease in the strength and rigidity of the bond Apparent strain due to strain in the grid induced by stress in the adhesive Apparent tensile strain resulting from the thing of the grid elements caused by gauge cussed induced electrolysis in the moisture laden adhesive. Effective protection of the gauge installation from moisture will prevent the absorption of moisture and avoid the consequent degradation in the performance by using protective coatings.
- 21. 21 Protective coatings: Strain-gauge installations are affected by environments containing water or moisture or chemical vapours. They are also susceptible to mechanical damage. Therefore unless tests are to be conducted under laboratory conditions within a short time after installation, protective coatings are essential. The important considerations that influence the selection of a coating are test environment, test duration and the degree of accuracy required. Though several commercial coatings are available, only those proven by tests to be electrically and chemically compatible with a gauge installation should be used in any application. Fig (12) Waterproofing a strain gauge in severe environment Wax coatings like microcrystalline wax, are melted and poured directly over the strain gauge installation to form a coating of thickness in the range 0.5 to 1.5 mm. These are good barriers to water or moisture but provide little mechanical protection. It is well-suited for general-purpose laboratory work and field applications. It can be used at temperatures up to 60°C. An air-drying polyurethane coating or acrylic coating can also be used in the laboratory for protection against moisture over a temperature range of -40° to +100°C. Both single-component and two-component epoxy plastics are available for use as protective coatings. For example, a solvent-thinned single – component epoxy resin compound can be brushed on the gauge and cured to provide an excellent chemical and mechanical protection. It is an excellent coating for transducer applications up to 120°C. Synthetic rubber coatings such as nitrile rubber, silicone rubber, poly- sulphides, butyl polymers, etc. provide good protection against moisture, fresh and
- 22. 22 salt water immersion or spray. The operating temperature range varies according to the polymer selected over the range of -150° to +300°C. A two-part 100% solids polysulphide modified epoxy compound is available as a general-purpose coating. It gives good protection against oils, greases, gasolines, most acids, alkalis and solvents. However, the operating temperature range for this compound is only 0° to 100°C. Metal-foil tapes when used with teflon or nylon padding and a sealing material provide protection against mechanical damage and moisture. In case of gauge installations subjected to long-term exposure to adverse environment, such as sea water, several layers of different types of coatings are needed. A cross-sectional view of a gauge instal1ation protected for long-term use is shown in Fig (12). (b) Effects of Hydrostatic Pressure: In the stress analysis of pressure vessels and piping systems, strain gages are frequently employed on interior surfaces where they are exposed to a gas or fluid pressure which acts directly on the sensing element of the gage. Under such conditions, pressure-induced resistance changes occur which must be accounted for in the analysis of the strain gage data. Milligan and Brace independently studied this effect of pressure by mounting a gage on a small specimen, placing the specimen in a special high-pressure vessel, and monitoring the strain as the pressure was increased to 140,000Ib/in2 (965MPa). In this type of experiment, the hydrostatic pressure p produces a strain in the specimen which is given by W.N.T Hook’s law ϵ = 1 − n + (1) Hydrostatic pressure p produces a strain in the specimen which is given by = = = −p ϵ = − n (−p) + (−p) = − n = K . p (2) Where KT = - (1-2)/E is often referred to as the compressibility constant for a material. The strain gages were monitored during the pressure cycle, and it was observed that the indicated strains were less than the true strains predicted by Eq. (2).
- 23. 23 The difference between the true strains and the indicated strains was attributed to the pressure effect. (c) Effects of High Temperature Resistance-type strain-gages can be employed at elevated temperatures for both static and dynamic stress analyses; however, the measurements require many special precautions which depend primarily on the temperature and the time of observation. At elevated temperatures, the resistance R of a strain gage must be considered to be a function of temperature t and time t in addition to strain ϵ Thus R=f (ϵ, T, t) (2) The resistance change R/R is then given by D = ¶ ¶ ϵ + ¶ ¶ T + ¶ ¶ t (3) Where ¶ ¶ =F =gage sensitivity to strain (gage factor) ¶ ¶ =F = gage sensitivity to temperature ¶ ¶ =F = gage sensitivity to time The combined effect of this three factor will produced a temperature induced change in strain with respect to time; this induced change in resistance of the gauge D (∈, , ) can then be expressed in terms of the three sensitivity factor as D = F ϵ + F T + F t (4) The discussion of performance characteristics of foil strain gages. It was shown that sensitivity of the gages t p temperature and time was minimized at normal operating temperatures of 0 to 1500 F (-18 to 650 ) by proper selection of the strain-gage alloy and carrier materials. As the test temperature increases above this level, however, the performance of the gage changes, and FT and Ft are not usually negligible.
- 24. 24 8. STRAIN GAUGE CIRCUITS The change in the resistance due to variation in applied strain is extremely small. Two electrical circuits – the potentiometer circuit and Wheatstone bridge – are used to measure such small changes in resistance. Mostly, some variation of the Wheatstone bridge is used for this purpose. I. The potentiometer and its applications to strain measurements: The potentiometer circuit is well suited for dynamic measurements. An attractive feature of the circuit is its extreme simplicity. The potentiometer circuit, which is often employed in dynamic strain-gage applications to convert the gage output R/R to a voltage signal E, is shown in Fig.7. Small increment in the open circuit voltage E of the potentiometer circuit can be derived as follows. Fig.13: potentiometer circuit. Consider the resistance in the circuit R1 & R2 Apply krickoff’s-voltage law to circuit = I ( + ) Consider the resistance in the circuit R1 & R2 the open-circuit voltage E across AB is = I = + R V = ( + R ) V = 1 1 + r V (1) Where V is the input (excitation) voltage and r = R2 / R1 is the resistance ratio for the circuit. If incremental change R1 and R2 occur in the value of the resistors R1 and R2, the change E of the output voltage E can be computed by using Eq. (1) as follows; + D = + D + D + R + D V (a) Solving Eq. (a) for E gives
- 25. 25 D = + D + D + R + D V − D = + D + D + R + D . V − + R . V D = + D + D + R + D − + R . V D = ( + D )( + R ) − ( + D + R + D ) ( + D + R + D )( + R ) . V (b) D = . V (b) Numerator= = ( + D )( + R ) − ( + D + R + D ) = ( + D )( + R ) − ( + R ) − (D + D ) = ( + R )( + D − ) − (D + D ) = ( + R )D − (D + D ) = D + R D − D − D = R D − D = D − D = = ( + D + R + D )( + R ) = ( + R ) 1 + D ( ) + D ( ) ( + R ) = ( + R ) 1 + D 1 + R + D R R + 1 Where = resistance ratio = (1 + m) 1 + D (1 + m) + D R 1 m + 1 = ( + ) + ( + ) D + D
- 26. 26 Substituting A & B in Eq (b) Which can be expressed in the following form by introducing r = R2/R1 D = ⎝ ⎜ ⎛ D − D ( + ) + ( + ) D + D ⎠ ⎟ ⎞ V D = ⎝ ⎜ ⎛ ( + ) D − D 1 + ( + ) D + D ⎠ ⎟ ⎞ V D = ⎝ ⎜ ⎛ ( + ) D − D 1 + ( + ) D + D ⎠ ⎟ ⎞ V (2) Examination of Eq.(2) shows that the voltage signal E from the potentiometer Circuit is a nonlinear function of R1/R1 and R2/R2.To inspect the nonlinear aspects of this circuit further, it is possible to rewrite Eq. (2) in the form D = (1 + m) D − D (1 − h) (3a) Where nonlinear term h is expressed as (1 − h) = 1 1 + 1 (1 + m) D + m D R h = 1 − 1 1 + 1 (1 + m) D 1 1 − D 2 2 (3b) Equations (3) are the basic relationships which govern the behavior of the potentiometer circuit, and as such they can be used to establish the applicability of this circuit for strain-gage measurements. Error due to nonlinearity of the circuit can be estimated with R1 as the resistance due to string gauge, R2 as a resistor of fixed resistance and R1 as the change in the resistance of the gauge due to a strain . Where R2 as a resistor of fixed R2 = 0 Equation 3b becomes
- 27. 27 h = 1 − 1 1 + 1 (1 + m) D h = 1 − 1 1 + 1 (1 + m) . ∈ (4) It shows that the nonlinear term is depends on magnitude of strain∈, gauge factor ‘F’ and ratio of m = R2 / R1. In most strain measurements the nonlinearity term h can be neglected & E [Eq (2)] can be determined from D = ( + ) D − D − −( ) However, if high accuracy in strain measurement is required or larger strains are to be measured. Then plastic strain determinations in metallic materials the output signal determined through Eq (5) can be corrected for error due to nonlinearity. (a) Range and sensitivity of the circuit: The output signal per unit strain is known as circuit sensitivity Sc of the potentiometer circuit is given by D = m (1 + m)2 D − D V (1) E output signal per unit strain = = ∈ D ∈ = m (1 + m)2 D − D V ∈ = m (1 + m)2 D 1 1 − D 2 2 V ∈ (2) With an active strain gauge R1 and fixed-blast resistor R2 i.e. = ; D = D ; = ; D = = m (1 + m)2 D g g V ∈
- 28. 28 = m (1 + m)2 V ⎣ ⎢ ⎢ ⎡ D g g ∈ ⎦ ⎥ ⎥ ⎤ = m (1 + m)2 . V . F (3) Where gauge factor = D ∈ for strain gauge, thus the circuit sensitivity of the potentiometer circuit is depends on the voltage V and ratio m = R2/R1, the Sc is limited by the maximum power Pg that can be dissipated by the gauge without unfavorable effect of performance. As the power dissipated in the gauge is equal to I2 g Rg i.e. Pg = I2 g Rg input voltage. = ( + ) = (1 + ) Where m = R2/R1 = (1 + ) = (1 + ) = (1 + ) ( ) Substituting Eq (a) in Eq (3) we get = m (1 + m)2 . (1 + ) . F = ( + ) . . Circuit sensitivity of the potentiometer circuit is depends on m ( ) and . F the term m ( ) is completely depends on m = R2/R1 resistance ratio limited value is 9. Therefore, m ( ) ≈ 0.9, the term . F is depends on the characteristics of the strain gauge range between 3 to 700 and Sc range 5 to 10µV per micro-strain.
- 29. 29 II. WHEATSTONE BRIDGE: Wheatstone bridge is 2nd circuit which can be employed to determine the change in resistance to a strain. Wheatstone bridge can be used to determine both dynamic and static strain gauge readings. The bridge as a direct readout device where the output voltage E is measured & related to strain. The bridge may be used as a null balance system, where the output voltage E is adjusted to zero value by adjusting the resistive balance of the bridge. There are two types of Wheatstone bridge circuits are used for the strain measurement. o Null balance type (balanced E = 0) o Out of balance type (unbalance E ≠ 0) A dc Wheatstone bridge consisting of four resistance arms with a battery and a meter is shown in figure (14). Fig (14) Wheatstone bridge In this bridge the resistance shown in each of the four arms of the bridge can represent a strain gauge. A voltage V is applied to the bridge. Some measuring instrument or meter such as a galvanometer is used to measure the output of the bridge.
- 30. 30 (a) Null Balance Type (balanced E = 0): Condition for balancing of Wheatstone bridge. The requirement for balance, i.e. zero potential difference E between points B and D for the bridge shown in fig (8) can be determined as follows: The voltage drop VAB across R1 is VAB = i1R1= V / (R1 + R2) . R1 --------------------- (1) Similarly the voltage drop VAD across R4 is VAD = i4R4= V / (R3 + R4) R4 ---------- (2) The potential difference between B and D, VBD, is VBD = VAB - VAD = E ------------ (3) Substituting of equations. (1) and (2) in (3), we get = ( + ) – ( + ) = ( + ) − ( + ) ( + )( + ) = − ( + )( + ) ( ) The condition for balance is that the voltage E should be zero i.e. the numerator in Eq. (4) should be zero: i.e − = 0 = Or = (5) Or = (6) Equation (5) or (6) gives the condition for the Wheatstone bridge to balance, that is the ratio of resistances of any two adjacent arms of the bridge must be equal to the ratio of the resistances of the remaining two arms taken in the same order.
- 31. 31 Balancing Wheatstone Bridge: Consider an initially balanced bridge i.e. consider equation for condition of balancing R1 / R2 = R4 / R3 satisfied, and then change R1 and/or R2 by a small increment. Any imbalance that may result from these changes can be removed and the bridge can be again balanced by changing, say the ratio R4 / R3 in such a way that the above condition for balancing is again satisfied. For example, assume that the resistance R1 is a strain gauge mounted on a specimen. The bridge can be balanced first under no load by altering the ratio of resistances R4 / R3 suitably. After the specimen is loaded, the bridge can be balanced again by adjusting the ratio of resistances R4 / R3. if the change in this ratio is known, then the change in the strain gauge resistance ∆R1 due to the load can be determined. The corresponding strain can be calculated from equation ε = ∆ R/R / F (b) Out of Balance Type: (unbalance E ≠ 0) It is this feature balancing which permits the Wheatstone bridge to he employed for static strain measurements. The bridge is initially balanced before strains are applied to the gages in the bridge; thus the voltage E is initially zero, and the strain-induced voltage E can be measured directly for both static and dynamic applications. Consider an initially balanced bridge with R1R3 = R2R4 so that E = 0 and then change each value of resistance R1 R2 R3and R4 by an incremental amount R1 D R2 R3and R4.The voltage output E of the bridge can be obtained from Eq. (4) which becomes D = D D D D D D D D D = . (a) Where A is the determinant in the numerator and B is the determinant in the denominator. By expanding each of these determinants, neglecting second-order terms, and noting R1R3 = R2 R4 it is possible to show that
- 32. 32 Numerator A = + D + D + D + D = ( + D )( + D ) − ( + D )( + D ) = . + .D + . D + D D − − D − D − D D = .D + . D − D − D = .D + . D − D − D = . D + D − . D + D = ∆ − ∆ + ∆ − ∆ (b) = + D + + D 0 0 + D + + D = ( + D + + D )( + D + + D ) = + + + = ( + ) + ( + ) = ( + )( + ) = ( + ) 1 + = ( + ) 1 + = ( + ) + = ( + ) = ( ) (c) Substituting Eqs. (b) to (d) yields D = . ∆ − ∆ + ∆ − ∆ 1 3( 1 + 2)2 1 2 D = ( ) ∆ − ∆ + ∆ − ∆ (7) By letting R2/R1 = r it is possible to rewrite Eq.(8.19) as D = ( ) ∆ − ∆ + ∆ − ∆ (8) In reality, Eqs, (7) and (8) both carry a nonlinear term 1 - h, However, the influence of the nonlinear term is quite small and can be neglected, provided the strains being measured are less than 5 percent Equation(8) thus represents the basic equation which governs the behavior of the Wheatstone bridge in strain measurement.
- 33. 33 B. Wheatstone-Bridge Sensitivity: The sensitivity of the Wheatstone bridge must be considered from two points of view: (1) With a fixed voltage applied to the bridge regardless of gage current (a condition which exists in most commercially available instrumentation) and (2) With a variable voltage whose upper limit is determined by the power dissipated the particular arm of the bridge which contains the strain gage. By recalling the definition for the circuit sensitivity given, and using the basic bridge relationship given in Eq. (8), it is clear that the circuit sensitivity is = D = D = ( ) ∆ − ∆ + ∆ − ∆ (9) = ϵ (1 + ) (∈ −∈ +∈ −∈ ) If all strain have equal value ∈ = −∈ =∈ = −∈ =∈ = ϵ (1 + ) (4 ∈) = 4. (1 + ) If a multiple-gage circuit is considered with n gages (where n = I. 2. 3, or 4) whose out puts sum when placed in the bridge circuit. it is possible to write ∆ = ∆ (10) Which by Eq. using ∆ = . . ϵ (11) Substituting Eq. (11) into Eq. (9) gives the circuit sensitivity as = ( ) (12) This sensitivity equation is applicable in those cases where the bridge voltage V is fixed and independent of gage current. The equation shows that the sensitivity of the bridge depends upon the number n of active arms employed, the gage factor F , the
- 34. 34 input voltage, and the ratio of the resistances R1 /R2 . A plot of r versus (I + r)2 (the circuit efficiency) that maximum efficiency and hence maximum circuit sensitivity occur when r = 1. With four active arms in this bridge a circuit sensitivity of F×V can be achieved, whereas with one active gage a circuit sensitivity of only F × (V/4) can be obtained. 9. CONSTANT-CURRENT CIRCUITS: The potentiometer and Wheatstone bridge circuits driven with a voltage source which ideally remains constant with changes in the resistance of the circuit. These voltage- driven circuits exhibit nonlinear output whenever DR/R is large. This nonlinear behavior limits their applicability to semiconductor strain gages. It is possible to replace the constant- voltage source with a constant-current source, and it can be shown that improvements in both linearity and sensitivity result. Constant-current power supplies with sufficient regulation for strain-gage applications are relatively new and have been made possible by advances in solid-state electronics, basically the constant-current power supply is a high impedance (1 to 10M) device which changes output voltage with changing resistive load to maintain a constant current. (a) Constant-Current Potentiometer Circuit: Consider the constant-current potentiometer circuit shown in Fig, 34a. When a very high impedance meter is placed across resistance R1 the measured output voltage E is E = I R (1) When resistances R 1 =R and R2 =R b change by DR1 =DR and DR2 = DR b =0 the output voltage becomes E + DE = I(R+ DR) (a) DE = I(R+ DR) – E (a’) Thus from Eq’s, (1) and (a) DE = I(R+ DR) - IR = I DR = I R D (2)
- 35. 35 Fig (15): constant current potentiometer circuits. It should be noted that DR, does not affect the signal output. Indeed. Even R2 is not involved in the output voltage, and hence it can be eliminated to give the very simple potentiometer circuit shown in Fig. 15. If R= Rg is the resistance of a strain gauge with gauge factor F and DRg is the change in resistance corresponding to strain ϵ, Substituting Eq. D = F . ϵ into Eq. (2) yields D = R . F . ϵ (3) By increasing the gauge current Ig to the maximum value dictated by power dissipation considerations, the circuit sensitivity can be maximized thus, The circuit sensitivity Sc = DE/ ϵ reduces to = . R . (4) If the constant-current Source is adjustable, so that the current I can be increased to the power-dissipation limit of the strain gage, then I = I g and Eq (4) can be rewritten as = P R (5) Thus, the circuit sensitivity is totally dependent on the strain-gage parameters Pg and R g and S g and is totally independent of circuit parameters except for the capability to adjust the current source. Comparison of Eqs Sc = (1+r) F. Pg. Rg and (5) shows that the sensitivities differ by the r / (1+ r) multiplier for the constant-voltage potentiometer: thus. Sc will always be higher for the constant-current potentiometer.
- 36. 36 It was noted in deriving Eq. (2) that R2 and DR2 did not affect the signal output of the constant-current potentiometer. This indicates that temperature compensation by signal cancellation in the strain-gage circuit or signal addition cannot be performed. It is possible to maintain the advantages of high sensitivity and perfect linearity of this circuit and to obtain the capability of signal addition or subtraction by using a double constant-current potentiometer circuit. (b) Constant-Current Wheatstone bridge Circuits: To consider a bridge driven by a constant current supply as shown in fig below. The current I delivered by the supply divides at point A or the bridge into currents I1 and I2 where I=I1+I2 the voltage drop between points A and B of the bridge is = + = (i) And the voltage drop between points A and D is = (ii) Thus the output voltage E from the bridge can be expressed as = V = V − V = I R − I R (1) Fig: Constant Wheatstone bridge with constant current supply. Far the bridge to be in balance (E=0) under no-load conditions, = (2)
- 37. Consider next the voltage V = I (R From which = + Eq. (IV) can be substituted in to Eq. (v) I = ( ( Substituting eqs. (vi) in to (1) = ( From Eq. (3) It is evident that the balance condition (E = 0) for the constant Wheatstone bridge is the same as that for the constant namely. = If resistance R1, R2, R3, and R voltage E+E measured with a very high impedance meter is + D = (Ʃ Ʃ ) [( where ƩR = R + R Expanding Eq. (vii) and simplifying after assuming the initial balance condition gives Fig: constant Wheatstone bridge designed to minimize nonlinear effect 37 Consider next the voltage V and note the (R + R ) = I (R + R ) (iii) I = ( ) ( ) I (iv) (v) Eq. (IV) can be substituted in to Eq. (v) ( ) ) I I = ( ) ( ) I (vi) Substituting eqs. (vi) in to (1) ( ) (R R − R R ) (3) From Eq. (3) It is evident that the balance condition (E = 0) for the constant Wheatstone bridge is the same as that for the constant-voltage Wheatstone bridge. = (4) , and R4 change by the amounts R1, R2, R E measured with a very high impedance meter is [( + D )( + D ) − ( + D )( + D + R + R ƩR = R + R + and simplifying after assuming the initial balance condition gives Fig: constant Wheatstone bridge designed to minimize nonlinear effect (iii) From Eq. (3) It is evident that the balance condition (E = 0) for the constant-current voltage Wheatstone bridge. R3, and R4, the )] (vii) + R + R and simplifying after assuming the initial balance condition gives Fig: constant Wheatstone bridge designed to minimize nonlinear effect
- 38. 38 + D = I (ƩR + ƩR) ∆ − ∆ + ∆ − ∆ + ∆ ∆ − ∆ ∆ (5) Inspection of Eq. (5) shows that the output signal E, is nonlinear with respect to DR because of the term ƩDR in the denominator and because of the second-order terms ∆ ∆ and ∆ ∆ within the bracketed quantity The nonlinear, of the constant-current Wheatstone bridge, however is less than that with the constant- voltage bridge Indeed, if the constant-current Wheatstone bridge is properly designed, the nonlinear terms can he made insignificant even for the large ∆ ′ encountered with semiconductor strain gages. The nonlinear effects in a typical situation call he evaluated by considering the constant-current Wheatstone bridge shown in Fig below. A single active gage is employed in arm R1, and it temperature-compensating dummy gage is employed in arm R4 Fixed resistors arc employed in arms R2 and R3. Thus = = = = D = D = 0 Under stable thermal environments,D = D D = 0 Equation (5) then reduces to D = I 2(1 + m) + ∆ ⁄ ∆ (6) Again it is evident that. Eq. (6) is nonlinear due to the presence of the term ∆ ⁄ in the denominator to determine the degree of the nonlinearity let I 2(1 + m) + ∆ ⁄ ∆ = I 2(1 + m) ∆ (1 − h) h = ∆ ⁄ 2(1 + m) + ∆ ⁄ = ∈ 2(1 + m) + ∈ (7) Inspection of Eq. (7) shows that the nonlinear term h can be minimized by increasing m in this case; nonlinear term h will depend on the gage factor F and on the magnitude of the strain ∈.