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14ME703/ME413 Hall Ticket Number:

IV/IV B.Tech (Regular/Supplementary) DEGREE EXAMINATION

November, 2017 Mechanical Engineering

Seventh Semester Engineering Metrology &Mechanical Measurements Time: Three Hours Maximum : 60 Marks

Answer Question No.1 compulsorily. (1X12 = 12 Marks)

Answer ONE question from each unit. (4X12=48 Marks)

1. Answer all questions (1X12=12 Marks)

a) Differentiate between unilateral tolerance and bilateral tolerance with examples.

Unilateral System

In this system, the dimension of a part is allowed to vary only on one side of the basic size. i.e., tolerance lies

wholly on one side of the basic size either above or below it.

Examples of unilateral tolerance are:

25+0.02+0.01 , 25

+0.02−0.00 , 25

−0.01−0.02 , 25

+0.00−0.02 etc

Bilateral System

In this system, the dimension of the part is allowed to vary on both the sides of the basic size. i.e., the limits

of tolerance lie on either side of the basic size; but may not be necessarily equally disposed about it.

Examples of bilateral tolerance are; 25±0.02 , 25+0.02−0.01

b) Sketch the different limit gauges used in industry for quality checking. Limit gauges are very widely used in industries. As there are two permissible limits of the dimension of a part,

high and low, two gauges are needed to check each dimension of the part, one corresponding to low limit of

size and other to the high limit of size of that dimension. These are known as GO and NOGO gauges.

The difference between the sizes of these two gauges is equal to the tolerance on the work piece. GO gauges check the Maximum Metal Limit (MML) and NO-GO gauge checks the Least or Minimum Metal Limit (LML).

In the case of a hole, maximum metal limit is when the hole is as small as possible, that is, it is the low limit of size. In case of hole, therefore, GO gauge corresponds to the low limit of size, while NO-GO gauge corresponds to high limit of size.

For a shaft, the maximum metal limit is when the shaft is on the high limit of size. Thus in the case of a shaft GO gauge corresponds to the high limit of size and NO-GO gauge corresponds to the low limit of size.

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c) Describe the working principle of Autocollimator.

Auto collimator is an optical instrument used for the measurement of small angular differences, changes or

deflection, plane surface inspection. It is also used to determine straightness and flatness.

The behaviour of a reflected light can be reviewed here to understand the working principle of Auto

collimator. If a beam of light strikes a flat reflecting surface it is reflected and if the surface is perpendicular

to the ray, it is turned back along its original path.

When the surface is tilted at any other angle, (say θ) the total angle through which the light deflected is twice

the angle, 2θ.

Assume a converging lens with a point of source of light ‘O’ at its principal focus. When a beam of light

strikes a flat reflecting surface it is reflected and if the surface is perpendicular to the ray it is turned back

along its original path. When the surface is tilted at an angle, the total angle through which the light is

deflected is twice the angle through which the reflector is tilted and is brought to a focus in the same plane

as the light source but shifted through some distance as shown in figure.

On examination of the triangle formed by the ray passing through the geometric centre of the lenses and the

focal length ‘f’ shows that;

tan 2𝜃 =𝑂𝑂′

𝑓=

𝑑

𝑓

𝑑 = 𝑓 tan 2𝜃

𝑑 ≈ 𝑓 2𝜃 = 2𝜃𝑓

Where, ‘f’ is the focal length of the lens.

d) What are the requirements of good comparators? 1. Robust design and construction: The design and construction of the comparator should be robust so

that it can withstand the effects of ordinary uses without affecting its measuring accuracy.

2. Linear characteristics of scale: Recording or measuring scale should be linear and uniform (straight

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line characteristic) and its indications should be clear.

3. High magnification: The magnification of the comparator should be such that a smallest deviation in

size of components can be easily detected.

4. Quick in results: The indicating system should be such that the readings are obtained in least

possible time.

5. Versatility: Instruments should be designed that it can be used for wide range of measurements.

6. Minimum wear of contact point. The measuring plunger should have hardened steel contact or

diamond to minimize wear effects. Further the contact pressure should be low and uniform.

7. Free from oscillations: The pointer should come rapidly to rest and should be free from oscillations.

8. Free from back lash: System should be free from back lash and unnecessary friction and it should

have minimum inertia.

9. Quick insertion of work piece: Means should be provided for lifting the plunger for quick insertion of

work.

10. Adjustable Table: The table of the instrument should, preferably, be adjustable in a vertical sense.

11. Compensation from temperature effects: The indicator should be provided with maximum

compensation for temperature effects.

12. Means to prevent damage: Suitable means should be provided for preventing damage of the

instrument in the event of the plunger moving through a greater distance than that corresponding

to the range of its measuring scale.

e) What are the various features to be measured on threaded components?

Major Diameter, Minor diameter, Effective diameter, Pitch, Flank angle and Thread form

f) Name the various instruments required for performing the alignment tests on machine tools.

Dial gauges, Test mandrels, Straight edges and squares, Spirit levels, Autocollimator and Waviness meter

g) List out the elements of measuring system.

Detector transducer element, signal conditioning element and output or readout element.

h) How to measure the velocity of flow using pitot tube?

Pitot tube used to measure the velocity of flow at a point is a tube bent at right angles and placed facing the

direction of flow as shown in figure.

At the tip, the fluid is brought to rest. That is velocity becomes zero and kinetic energy gets converted in to

pressure energy.

Applying Bernoulli’s equation between a point in the free stream and another at the tip of the Pitot tube,

𝑝

𝜌𝑔+

𝑣2

2𝑔=

𝑝1

𝜌𝑔+ 0 𝑜𝑟

𝑝1

𝜌𝑔=

𝑝

𝜌𝑔+

𝑣2

2𝑔

Where ‘p’ is the static pressure, v is the free stream velocity and ρ the density of flowing fluid. P1/ρg is the

total head or impact pressure head or stagnation head. Thus the Pitot tube (1) measures the sum of static

pressure head and velocity head.

Above equation assumes that the flow is steady, incompressible and frictionless.

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If another tube (2) is fixed normal to the direction of flow, it will measure only a static pressure head p/ρg

and is called a Piezometer tube.

A U-tube manometer connected across the 1 and 2 will directly give the velocity head ‘h’.

𝑕 =𝑝2

𝜌𝑔−

𝑝1

𝜌𝑔=

𝑣2

2𝑔

𝑣 = 2𝑔𝑕

Above expression gives theoretical velocity.

Actual velocity = 𝐶𝑣 2𝑔𝑕

Where Cv = Coefficient of pitot tube. ‘h’ in the above relation is expressed in terms of the column of flowing

fluid.

i) Differentiate between sensitivity and resolution.

Sensitivity:

It is defined as the ratio of the magnitude of output signal to the magnitude of input signal.

𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 =𝐶𝑕𝑎𝑛𝑔𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡 𝑠𝑖𝑔𝑛𝑎𝑙

𝐶𝑕𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑝𝑢𝑡 𝑠𝑖𝑔𝑛𝑎𝑙

A 1mv recorder might have a 10 cm scale length. Assuming a linear scale, its sensitivity would be 10 cm/mv.

The sensitivity is constant in a linear instrument and usually it is required to be high.

Resolution:

This is the smallest change in input signal or measured value which can be detected by the instrument. The

least count of an instrument can be taken as the resolution of an instrument.

j) List out different types of mechanical pressure gauges. Dead weight tester, piezometer, manometer, McLeod gauge, Bourdon tube, Elastic diaphragms, Bellows and Bridgman gauge.

k) How to measure the temperature using Bi-metallic thermometers?

Principle When two metal strips having different coefficients of expansion are bonded together, an increase in temperature causes the deflection of the free end of the strip as shown in figure.

The deflection with temperature is nearly linear. Invar (iron, nickel alloy) is commonly used as a low expansion material. Brass or other materials are used as high expansion material.

Bimetal strip is commonly used as temperature sensing and control device called as thermostat (on-off type) in home applications such as geysers and ovens, etc.

The bimetallic strip has the advantages of low-cost, negligible maintenance expense, and stable operation over extended periods of time.

For temperature measurement, the sensitivity of bimetal is increased by coiling it in a helical form (see figure). As the temperature increases, the bimetal expands and the helical bimetal rotates at its free end.

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l) What is a load cell?

Strain gauge load cells are most often constructed of a metal, and have a shape such that the range of forces

to be measured results in a measurable output voltage over the desired operating range. The shape of the

linearly elastic member is designed to meet the following goals: (1) provide an appropriate range of force-

measuring capability with necessary accuracy, (2) provide sensitivity to forces in a particular direction, and

(3) have low sensitivity to force components in other directions.

A load cell comprises four strain gauges; two of these are used for measuring the longitudinal strain while the

other two for measuring the transverse strain. The four strain gauges are mounted at 90° to each other, as

shown in below figure.

Two gauges experience tensile stresses while the other two are subjected to compressive stresses. At the no-

load condition, resistance in all the four gauges will be same. The potential across the two terminals B and D

are same.

The Wheatstone bridge is now balanced and hence output voltage is zero. When the specimen is stressed

due to the applied force, the strain induced is measured by the gauges.

UNIT I

2. a) Sketch three main types of fits and name them.

Clearance Fit: In this type of fit shaft is always smaller than the hole. i.e., the largest permissible shaft

diameter is smaller than the diameter of the smallest hole. So that, the shaft can rotates or slide through

with different degrees of freedom according to the purpose of mating part.

Interference Fit: In this of type of fit the minimum permissible diameter of the shaft is larger than the

maximum allowable diameter of the hole. Thus the shaft and the hole members are intended to be attached

permanently and used as a solid component.

Elastic strains developed on the mating surfaces during the process of assembly prevent relative movement

of the mating parts. For example steel tyres on railway car wheels, gears on intermediate shafts of trucks,

bearing in the gear of a lathe head stock, drill bush in jig plate, cylinder linear in block, steel ring on a wooden

bullock cart wheels etc.

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Transition Fit: Transition fit lies mid way between clearance and interference fit. In this type the size limits of

mating parts (shaft and hole) are so selected that either clearance or indifference may occur depending upon

the actual sizes of the parts. Push fit and wringing fit are the examples of this type of fit.

b) Calculate the limits of hole and shaft in the hole and shaft pair designated by 40H7d 8.

Assume: (i) 40 lies in the diameter step of 30 and 50mm (ii) The standard tolerance unit, i

(microns) = 0.45D1/3

+ 0.001D where D is the geometric mean of the lower and upper limits

if diameter step in which the diameter consideration lies, D is in mm, (iii) The fundamental

deviation for shaft d= -16D0.44

microns (iv) The standard tolerance grade IT7 = 16 i , (v)The

standard tolerance grade IT8 = 25 i

Solution:

𝐷 = 30 × 50 = 38.73𝑚𝑚 Fundamental deviation for hole H is zero. Fundamental deviation for shaft = −16𝐷0.44

= −16 × 38.730.44 = −79.957 𝑚𝑖𝑐𝑟𝑜𝑛𝑠 ≅ −80 𝑚𝑖𝑐𝑟𝑜𝑛𝑠 𝑖. 𝑒 − 0.080 𝑚𝑚

The tolerance unit

𝑖 = 0.45 𝐷3

+ 0.001𝐷 = 0.45 38.733

+ 0.001 × 38.73 = 1.5612 𝑚𝑖𝑐𝑟𝑜𝑛𝑠 Tolerance grade IT7=16i

= 16 × 1.5612 = 24.9782 𝑚𝑖𝑐𝑟𝑜𝑛 = 0.025𝑚𝑚 Tolerance grade IT8=25i

= 25 × 1.5612 = 39.03 𝑚𝑖𝑐𝑟𝑜𝑛 = 0.039𝑚𝑚 HOLE Low limit=Basic size fundamental deviation=40.00+0.000=40.00 mm High limit= Low limit tolerance =40.00+0.025= 40.025 mm. SHAFT High limit=Basic size-fundamental deviation of shaft=40.00-0.080=39.92 mm Low limit= High limit-tolerance=39.92-0.039=39.881 mm.

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(OR) 3. a) Discuss the design procedure for plug and ring gauges as per Taylor’s principle of gauging.

Taylor’s Principle of Gauge Design

(1) It states that GO gauges should be designed to check the maximum material limit (MML), while the NO-GO gauges should be designed to check the minimum or Lowest Material Limit (LML).

Figure: GO gauge to check MML and NOGO to check LML

The difference in size between the GO and NO-GO plug gauges & Snap gauges is approximately equal to the tolerance of the tested hole or shaft as may be the case.

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(2) GO gauges should check all the related dimensions (roundness, size, location etc) simultaneously, where as NO-GO gauge should check only one element of the dimension at a time. According to this rule, GO plug gauge should have a full circular section and be of full length of the hole it has to check. This ensures that any lack of straightness, or roundness of the hole will prevent the entry of full length of GO plug gauge.

For example, suppose the bush to be inspected has a curved axis and a short ‘GO’ plug gauge is used to check it. The short plug gauge will pass through all the curves of the bent bushing. This will lead to a wrong result that the work piece (hole) is within the prescribed limits. Actually such a bushing with a curved hole will not mate properly with its mating part and thus defective. A GO plug gauge with adequate length will not pass through a curved bushing and the error will be detected. A long plug gauge will thus check the cylindrical surface not in one direction, but in a number of sections simultaneously. The length of the GO plug gauge should not be less than 1.5 times the diameter of the hole to be checked.

Now suppose, the hole to be checked has an oval shape. While checking it with the cylindrical ‘NOT-GO’ gauge the hole under inspection will overlap (hatched portion) the plug and thus will not enter the hole. This will again lead to wrong conclusion that the part is within the prescribed limits. It will be therefore more appropriate to make the ‘NOT-GO’ gauge in the form of a pin as shown in below figure.

Figure : checking the oval shaped hole with NOT-GO gauge

b) (i) What are slip gauges? What are their uses? Slip gauges or gauge blocks are universally accepted end standard of length in industry. These were

introduced by Johanson, a Swedish engineer and are also called Johanson gauges.

Slip gauges are rectangular blocks of high grade steel with exceptionally close tolerances. These blocks are

suitably hardened through out to ensure maximum resistance to wear. They are then stabilized by heating

and cooling successively in stages so that hardening stresses are removed. After being hardened they are

carefully finished by high grade lapping to a high degree of finish, flatness and accuracy.

For successful use of slip gauges their working faces are made truly flat and parallel. Slip gauges are also

made from tungsten carbide which is extremely hard and wear resistance.

The cross sections of these gauges are 9mm x 30mm for sizes up to 10mm and 9mm x 35mm for larger sizes.

Any two slips when perfectly clean may be wrung together. The dimensions are permanently marked on one

of the measuring faces of gauge blocks.

Gauge blocks are used for:

(i) Direct precise measurement, where the accuracy of the work-piece demands it.

(ii) For checking accuracy of vernier calipers, micrometers, and such other measuring instruments.

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(iii) Setting up a comparator to a specific dimension.

(iv) For measuring angle of work piece and also for angular setting in conjunction with a sine bar.

(v) The distances of plugs, spigots, etc on fixture are often best measured with the slip gauges or

end bars for large dimensions.

(vi) To check gap between parallel locations such as in gap gauges or between two mating parts.

There are many measurements which can be made with slip gauges either alone or in conjunction with other

simple apparatus such as straight edges, rollers, balls, sine bars etc.

(ii) Briefly illustrate the steps involved in manufacture of slip gauges. 1. The high grade steel gauge blocks are hardened and after rough grinding they are subjected to a cyclic

low temperature heat treatment with the purpose of a balancing the internal stresses produced by hardening thus leaving gauges in a stable condition.

2. Batches of eight blanks of similar nominal size are mounted on eight co-planar faces of a magnetic chuck.

3. While mounted on the magnetic chuck one set of faces are lapped truly flat. 4. The gauges are then removed and their lapped faces wrung to the eight faces of solid chuck which are

also accurately in one plane. 5. The exposed faces of gauges are now lapped flat and in one plane. The opposing faces of each gauge are

now truly flat but not necessarily parallel. The required degree of accuracy in parallelism and equality of size is achieved by interchanging four of the eight gauges diagonally and turned end for end.

6. To determine whether the gauges are of the required sizes they are removed from the chuck, wrung together in combination and their aggregate size compared with an appropriate sized master in a suitable comparator.

UNIT II

4. a) Explain the utility of straight edge and surface plate in laboratories.

STRAIGHT EDGES

These are used to check the straightness and flatness of parts.

Figure 2.13 Classifications of Straight Edges

They are broadly classified in to:

(a) Tool makers straight edges

(b) Wide edge straight edges

Tool makers’ straight edges have the highest accuracy. They are available in lengths from 75 to 175 mm and

with one to four working edges. With these edges the straightness and flatness are checked by sight test.

Straight edges with a single edge are mainly used for checking straightness. For checking the straightness of

an element of a cylindrical or taper surface, the straight edge is applied along the full length of the surface

and is held before a bright background. The absence of light between the straight edge and work surface

indicates the straightness of the element and vice versa. For checking the flatness, single edged as well as

three and four-edged straight edges may be used. With single edge straight edge, it is applied in different

directions at different places on the surface to be tested. The flatness of the surface is judged by the light

showing through. Two typical types of straight edges (i) Bow shaped (ii) I-section straight edge are shown in

figures below.

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SURFACE PLATE Surface plate forms the basis of measurement. They are extensively used in workshops and metrological laboratories where inspection is carried out. They are used as:

(i) A reference or datum surface for testing flatness of surfaces. (ii) Reference surface for all other measuring instruments having flat bases. E.g., for mounting V-

blocks, angle plates, sine bars, height gauges, dial gauges, comparators etc. Surface plates are massive and highly rigid in design. They have truly flat level planes. They are generally made up of C.I free from blow holes, inclusions and other surface defects and are heat treated to relieve internal stresses.

b) Discuss with a neat sketch working principle of Taylor Hobson Talysurf.

Above figure shows the schematic of the Talysurf surface roughness tester which comprises the following

elements:

(i) E-shaped soft iron head

(ii) Armature

(iii) Stylus

(iv) Cylindrical skid

The outer limbs of the E-shaped soft iron head are provided with induction coils and an armature is provided

at its central limb. There exists a small air gap between the armature and other limbs. The armature carries

the stylus (very fine needle) at its one end. When the stylus engages with the valley of the surface irregularity

(moves downward) the air gap at the primary coil increases and there occurs corresponding decrease of air

gap at the secondary coil. This variation in air gap results in to a change in the impedance of the coils. An

additional alternating current, proportional to the movement of the stylus, then flows in the secondary coil.

As such, the position of stylus controls or modulates the carrier.

The cylindrical skid acts as the datum while the motor and gear box unit is provided to move the pickup unit

across the test surface. The skid length is about twice the pitch of waviness. The stylus is of four sided

pyramid shape and has the following features.

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Pyramid angle = 90 degree

Nose radius = 2µm,

Measuring force = 0.1 gm.

The maximum length that can be traced is 10 mm, and the vertical magnification of the system can be varied

in steps from 1000 to 50000 times.

(OR)

5. a) Explain the working of sigma comparator with a neat sketch

SIGMA COMPARATOR

This is a mechanical comparator providing magnification in the range of 300 to 5000. It consists of a plunger

mounted on two flat steel strings. It consists of a plunger mounted on two flat steel strings (slit diaphragms).

This provides a frictionless linear movement for the plunger.

The plunger carries a knife edge, which bears upon the face of the mounting block of a cross-strip hinge. The

cross strip hinge is formed by pieces of flat steel springs arranged at right angles and is a very efficient pivot

for smaller angular movements. The moving block carries a light metal Y-forked arm. A thin phosphor bronze

ribbon is fastened to the ends of the forked arms and wrapped around a small drum, mounted on a spindle

carrying the pointer.

Any vertical displacement of the measuring plunger and hence that of the knife edge makes the moving block

of the cross strip liver to pivot. This causes the rotation of the Y-arms. The metallic band attached to the arms

makes the driving drum and hence the pointer to rotate.

The ratio of the effective length (L) of the arm and the distance (a) of the knife edge from the pivot gives the

first stage magnification and the ratio of the pointer length (l) and radius (r) of the driving drum gives second

stage magnification of the instrument.

Total magnification of the instrument is thus 𝐿

𝑎×

𝑙

𝑟 .

The magnification of the instrument can be varied by changing the distance (a) of Knife edge of tightening or

slackening of the adjusting screws: The range of instruments available provides magnifications of X 300 to X

5000, the most sensitive models allowing scale estimation of the order of 0.0001 mm to be made.

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b) Discuss with suitable sketches on alignment tests on lathe: i) True running of locating

cylinder of main spindle, ii) True running of taper socket in main spindle, iii) True running

of headstock center. True running of locating cylinder of main spindle:

Locating cylinder is provided to locate in the chuck or face plate. The locating surface is cylindrical and this

must run truly, then only the face plate can run truly. The dial indicator is fixed to the carriage and the feeler

of the indicator touches the locating surface. The surface is then rotated on its axis and indicator should not

show any movement of needle. The arrangement is shown in above figure. The permissible error is 0.01 mm.

By rotating the cylinder for one complete revolution, the deflection of the dial gauge pointer is noted down.

This gives the actual error of the locating cylinder. The actual error must be less than the permissible error.

True running of taper socket in main spindle:

If the axis of the tapered hole of the socket is not concentric with the main spindle axis, eccentric and

tapered jobs will be produced. To test it, a mandrel is fitted in to the tapered hole and readings at two

extremes of the mandrel (i.e., nearest to the spindle nose and at a distance of 300 mm) are taken by means

of a dial indicator, by rotating the spindle (see above figure). Permissible error near the spindle is 0.01 mm.

Permissible error at a distance of 300 mm is 0.03 mm.

True running of headstock centre:

Head stock centre is live centre and the work-piece has to rotate with this centre. If it is not true with the axis

of rotation of the spindle, eccentricity will be caused while turning a work, as the job axis would not coincide

with the axis of rotation of main spindle. For testing this error, the feeler of the dial indicator is pressed

perpendicular to the taper surface of the centre (see figure) and the spindle is rotated. The deviation

indicated by the dial gauge gives the error in trueness of the centre. Allowable error is 0.01 mm.

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UNIT III 6. a) Describe the various elements of mechanical measurement system with a neat block

diagram.

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b) List out different types of strain measurement systems and explain about strain gauge

rosettes.

Strain Measurement

In general, any strain measurement is to be made over a finite length of a work piece or test piece. The

measurement will approximate to strain at a point, if the length over which strain is measured (called gauge

length) is as small as possible. Various strain measuring devices are classified as below:

1. Mechanical: Huggenbeger tensometer (It uses mechanical levers for the magnification of displacement).

2. Optical: Tuckerman optical strain gauge (In this ray of light is substituted for mechanical levers for the

magnification of displacement).

3. Electrical: (i) Inductance strain gauges

(ii) Capacitance strain gauges

(iii) Resistance strain gauges

The choice of strain measuring device depends upon many factors such as sensitivity, range, nature if strain,

whether static or dynamic. Among the above Electrical resistance strain gauges are very convenient and

popular.

Above methods are point by point methods of measuring strain. There are other methods, termed as Whole

field methods of measuring strain. By these methods, strain can be measured in the whole domain of

interest at a time. E.g., Moiré technique and holographic Interferometry.

Stress-Strain Relationship and Rosettes

Electrical resistance strain gauges are generally mounted on the free surface of a test specimen to determine

the stress at a particular point. Three strains at point define completely either the strain or stress field.

Alternatively two principal strains ϵ1, ϵ2 are to be measured along with the direction of ϵ1 relative to the x-

axis as given by the principal angle ф.

Conversion of strains in to stresses requires additional information about the elastic constants of the

material of the specimen. e.g., Modulus of elasticity (E) and Poisson’s ratio (ν).

An arrangement of a number of strain gauges (two to four) bonded in the same backing material and

oriented in definite relative positions is called rosette.

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The different types of rosettes are:

1. Two element rosette

2. Three element rosette (Rectangular rosette)

3. Three element rosette (Delta or equiangular rosette)

4. Four element rosette (T-delta rosette )

Three element rectangular and delta rosettes are quite popular for strain analysis.

To find the principal strains and stresses from the strains recorded by a rosette, following steps can be

followed.

1) Choose two orthogonal directions, say (x and y). Express strains recorded by the gauges fixed in

various directions (ϵ1, ϵ2 and ϵ3) in terms of strains ϵxx, ϵyy and γxy.

2) Obtain the strains x & y directions in terms of strains as recorded by different gauges.

3) Find the principal strains using the above values.

4) Find the principal stresses.

(OR)

7. a) Explain the working principle of hot wire anemometer with a neat sketch.

This device is mainly used in research applications to study rapidly varying flows. E.g., for turbulence studies

to measure mean and fluctuating components of velocities. The principle of hot wire anemometer is rather

simple. A thin wire (5µm diameter, approximately) of resistance Rw (usually platinum, nickel or tungsten) is

heated by passing a current 𝑖 and placed in the flowing fluid. The convective heat transfer characteristics of

the heated wire become a measure of fluid velocity.

Consider a heated wire in a flowing fluid as shown below.

For equilibrium condition, the 𝑖2𝑅𝑤 heat generated is equal to the convective heat transfer to the fluid.

𝑖2𝑅𝑤 = 𝐾𝑐𝑕𝐴 𝑇𝑤 − 𝑇𝑓 --------------- (1)

Where, 𝑖 = current flowing in hot wire 𝑅𝑤 = resistance of the hot wire

𝑇𝑤 = temperature of the hot wire 𝐴 = heat transfer area 𝑕 = convective film coefficient of heat transfer 𝑣 = velocity of the fluid 𝑇𝑓 = temperature of the flowing fluid

𝑅𝑓 = resistance of water at fluid temperature

𝐾𝑐 = conversion factor, electrical to thermal power The functional relationship between 𝑕 and flow velocity 𝑣 for a given range can be expressed as,

𝑕 = 𝑎 + 𝑏 𝑣 --------------------- (2) (a & b are constants) From substituting for 𝑕 in (1) we have,

𝑖2𝑅𝑤 = 𝐾𝑐𝐴 𝑇𝑤 − 𝑇𝑓 𝑎 + 𝑏 𝑣 ------------------- (3)

From the above expression it can be seen that for a given wire the velocity of the flowing fluid is related

to (i) the current flowing through hot wire, (ii) resistance of the wire and (iii) temperature difference

between hot wire and flowing fluid. In addition, convective heat transfer also depends upon the size and

shape of the wire and the physical properties of flowing fluid.

In actual practice, one of the variables ( 𝑖, 𝑅𝑤 ,𝑇𝑤 ) is kept constant and a change in the other becomes a

measure of the velocity of flowing fluid. Further, the resistance of the wire depends upon its

temperature,

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𝑅𝑤 − 𝑅𝑓 = 𝑐(𝑇𝑤 − 𝑇𝑓) -------------------------------- (4) (c is a constant)

There are therefore, two basic modes of using hot wire anemometer as described below.

Constant Current Mode

It uses an electrical circuit to feed a constant current to the wire. Variations in the flow result in a

changed temperature of the hot wire and hence its resistance changes, which becomes a measure of

flow velocity. If the wire is to sense a fluctuating flow its response time will lag behind the actual

fluctuations in the flow because of the heat capacity of the hot wire. A compensating network (an

electrical circuit) compensates for this time lag and improves the dynamic response.

Constant Temperature Mode

In this arrangement, hot wire resistance hence its temperature is kept constant by continuously

adjusting the current through the hot wire using a suitable servo system. The current (or the voltage

across the hot wire) becomes a measure of flow velocity. Compensation forms an inherent part of the

basic system. The constant temperature design has the advantage that wire is protected against burn

out.

The hot wire probe can be calibrated against a pitot static tube in a wind tunnel. Accumulation of dirt on

the wire can lead to serious heat transfer error. Hot wire probe is suitable only for gas flow

measurements.

Hot film anemometer probe

For measuring the flow of liquids, hot film probe is employed, the electric circuitry for which is the same

as used for gases. The sensing element or wire is called as hot film probe. It consists of a film of gold or

platinum of about 5 µm thickness coated over glass and covered with an insulating layer of epoxy to

avoid short circuiting.

Hot Wire Anemometer

Figure Schematic Circuit for Constant Temperature

15

b) Explain different methods used to fixing and bridge circuits for measuring strain changes. 6M

The following circuits are used for measuring change in resistance for electrical resistance strain gauges.

1. Ballast circuit (voltage sensitive potentiometric circuit)

2. Wheatstone bridge circuit.

(i) Balanced (null) condition

(ii) Unbalanced (Deflection) condition

- Quarter bridge

- Half bridge

- Full bridge

Ballast Circuit (Voltage-sensitive potentiometric circuit)

𝑉𝑖 = 𝐼𝑛𝑝𝑢𝑡 𝑠𝑢𝑝𝑝𝑙𝑦 𝑣𝑜𝑙𝑡𝑎𝑔𝑒

𝑉𝑜 = 𝑂𝑢𝑡𝑝𝑢𝑡 𝑣𝑜𝑙𝑡𝑎𝑔𝑒

𝑅𝑏 = 𝐵𝑎𝑙𝑙𝑎𝑠𝑡 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑅𝑔 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡𝑕𝑒 𝑢𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑔𝑎𝑢𝑔𝑒

The output voltage, when no stress is applied to the strain gauge is given by,

𝑉𝑜 = 𝑅𝑔

𝑅𝑔 + 𝑅𝑏

𝑉𝑖

When the gauge is strained, the gauge resistance changes to (Rg+dRg) and the output voltage becomes;

𝑉𝑜 + 𝑑𝑉𝑜 = 𝑅𝑔 + 𝑑𝑅𝑔

𝑅𝑔 + 𝑑𝑅𝑔 + 𝑅𝑏

𝑉𝑖

∴ The change in output voltage,

𝑑𝑉𝑜 = 𝑅𝑔 + 𝑑𝑅𝑔

𝑅𝑔 + 𝑑𝑅𝑔 + 𝑅𝑏

−𝑅𝑔

𝑅𝑔 + 𝑅𝑏

𝑉𝑖

= 𝑅𝑔 + 𝑑𝑅𝑔 𝑅𝑔 + 𝑅𝑏 − 𝑅𝑔

𝑅𝑔 + 𝑑𝑅𝑔 + 𝑅𝑏

𝑅𝑔 + 𝑑𝑅𝑔 + 𝑅𝑏 𝑅𝑔 + 𝑅𝑏

𝑉𝑖

= 𝑅𝑔

2 + 𝑅𝑔𝑅𝑏 + 𝑑𝑅𝑔𝑅𝑔 + 𝑑𝑅𝑔𝑅𝑏 − 𝑅𝑔2 − 𝑑𝑅𝑔𝑅𝑔 − 𝑅𝑔𝑅𝑏

𝑅𝑔 + 𝑅𝑏 2 𝑉𝑖 ……… . . (∵ 𝑑𝑅𝑔 < 𝑅𝑔)

= 𝑑𝑅𝑔 .𝑅𝑏

𝑅𝑔+𝑅𝑏 2 𝑉𝑖 =

𝑅𝑔 .𝑅𝑏

𝑅𝑔+𝑅𝑏 2⋅𝑑𝑅𝑔

𝑅𝑔𝑉𝑖 ……𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑖𝑛𝑔 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑎𝑛𝑑 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑏𝑦 𝑅𝑔

Also, condition of maximum sensitivity is given by; Rg=Rb

Hence, 𝑑𝑉𝑜 =𝑉𝑖

4∙𝑑𝑅𝑔

𝑅𝑔

Also, 𝑑𝑅𝑔

𝑅𝑔= 𝐺𝐹 ∙ 𝜖

16

∴ 𝑑𝑉𝑜 = 𝐺𝐹

4 ∙ 𝜖 ∙ 𝑉𝑖 Where, GF denotes Gauge Factor.

From the above equation it is evident that change in output voltage when gauge is strained is directly proportional to strain.

The ballast circuit is used for dynamic strain measurements, where static strain components are ignored.

Wheatstone bridge Circuit

The Wheatstone bridge technique can be used in the following two ways;

1. Null mode

2. Deflection mode

1. Null mode

Refer to figure 3.14.The resistance, with no straining, are so arranged that VB=VD and the galvanometer gives zero deflection.

Then, 𝑅1

𝑅3=

𝑅2

𝑅4 ------------------ (1)

Where, R1 = Rg = Unstrained resistance of the gauge.

In measurement of strains, generally R1 is the strain gauge, R2 and R4 are the fixed resistance and R3 is a variable resistor.

When the gauge is strained, its resistance R1 changes by an amount dR1. This change unbalances the bridge resulting in to the

deflection of the galvanometer. The balance is then regained by adjusting R3 by an amount dR3. The rebalanced condition

gives:

𝑅1 + 𝑑𝑅1

𝑅3 + 𝑑𝑅3

=𝑅2

𝑅4

𝑜𝑟 𝑅1 + 𝑑𝑅1 = 𝑅3 + 𝑑𝑅3 𝑅2

𝑅4

𝑜𝑟 𝑅1 + 𝑑𝑅1 = 𝑅3 ×𝑅2

𝑅4

+ 𝑑𝑅3 ×𝑅2

𝑅4

𝑜𝑟 𝑅1 + 𝑑𝑅1 = 𝑅1 + 𝑑𝑅3 ×𝑅2

𝑅4

𝑎𝑠 ∵ 𝑅1 =𝑅3

𝑅4

× 𝑅2

𝒐𝒓 𝒅𝑹𝟏 = 𝒅𝑹𝟑 ×𝑹𝟐

𝑹𝟒 ----------------------------- (2)

If the resistances of all the limbs of the Wheatstone bridge are equal, then

𝑅1 = 𝑅2 = 𝑅3 = 𝑅4 = 𝑅𝑔

𝑎𝑛𝑑 𝑑𝑅1 = 𝑑𝑅3 ------------------------------- (3)

The change in resistance dR1, in terms of strain is given by

17

𝑑𝑅1 = 𝐺𝐹 ∙ ∈ ∙ 𝑅𝑔 𝑤𝑕𝑒𝑟𝑒, 𝐺𝐹 𝑖𝑠 𝑔𝑎𝑢𝑔𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑎𝑛𝑑 𝜖 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑡𝑟𝑎𝑖𝑛.

∴ 𝑑𝑅3 = 𝐺𝐹 ∙ ∈ ∙ 𝑅𝑔 --------------------------- (4)

Equation (4) indicates that the change in the value of resistance R3 is direct measurement of strain.

2. Deflection mode

Initially, the bridge resistance are so adjusted that the bridge is in balanced. The equilibrium gets disturbed when the gauges

are strained. Then, the voltage Vo is measured under this unbalanced condition.

(i)Quarter- Bridge

Single gauge used for strain measurement (Quarter bridge)

Figure shows single gauge used for strain measurement (quarter-bridge). In this arrangement only one strain gauge is used and

the other three elements of the bridge are fixed resistors.

Let us assume that the galvanometer (measuring instrument) has infinite impedance and therefore no current flows through it.

Then,

Current flowing through the limbs AB and BC, 𝐼1 =𝑉𝑖

𝑅𝑔1 +𝑅3 ------------------ (5)

Voltage drop in limb AB (voltage at terminal B), 𝑉𝐴𝐵 = 𝐼1𝑅𝑔1 =𝑅𝑔1

𝑅𝑔1 +𝑅3∙ 𝑉𝑖 ----------------- (6)

Similarly, 𝐼2 =𝑉𝑖

𝑅2 +𝑅4 --------------------- (7)

And 𝑉𝐴𝐷 = 𝐼2𝑅2 =𝑅2

𝑅2 +𝑅4∙ 𝑉𝑖 ---------------------- (8)

Initially, Rg1 = R2 = R3 = R4 = R

∴ 𝑉𝐴𝐵 = 𝑉𝐴𝐷 =𝑉𝑖

2

And, Vo = Voltage across the terminals B and D.

𝑉𝑜 = 𝑉𝐴𝐵 − 𝑉𝐴𝐷 = 0

Obviously, the bridge is balanced under unstrained conditions.

When the gauge is strained, the resistance Rg1 changes by an amount dRg1. Then,

𝑉𝐴𝐵 = 𝑅𝑔1 + 𝑑𝑅𝑔1

𝑅𝑔1 + 𝑑𝑅𝑔1 + 𝑅3

𝑉𝑖 = 𝑅 + 𝑑𝑅

2𝑅 + 𝑑𝑅 𝑉𝑖 [∵ 𝑅𝑔1 = 𝑅3 = 𝑅 𝑎𝑛𝑑 𝑑𝑅𝑔1 = 𝑑𝑅]

𝑉𝐴𝐷 = 𝑅2

𝑅2 + 𝑅4

𝑉𝑖 =𝑉𝑖

2 [∵ 𝑅2 = 𝑅4 = 𝑅]

The changed output voltage, is given by

𝑉𝑜 + 𝑑𝑉𝑜 = 𝑅 + 𝑑𝑅

2𝑅 + 𝑑𝑅−

1

2 𝑉𝑖

18

= 2𝑅 + 2𝑑𝑅 − 2𝑅 − 𝑑𝑅

2(2𝑅 + 𝑑𝑅) 𝑉𝑖 =

𝑑𝑅

4𝑅 + 2𝑑𝑅 𝑉𝑖

Since, dR<< R and Vi=0 (under unstrained conditions), therefore

𝑑𝑉𝑜 =𝑉𝑖

4∙𝑑𝑅

𝑅

𝑜𝑟 𝑑𝑉𝑜 = 𝐺𝐹

4 ∙ ∈ ∙ 𝑉𝑖 ---------------- (9)

From equation (9) it is clear that the output voltage is directly proportional to the applied strain.

(ii)Half-bridge

Figure shows two gauges used for strain measurement (Half bridge). In this arrangement two of the bridge elements are strain

gauges and the other two are fixed resistors. The strain gauge 1 is bonded to the upper surface of the cantilever beam and a

second strain gauge 3 is bonded to the lower surface and located precisely underneath the gauge 1. These gauges are

connected electrically to form adjacent limbs of the Wheatstone bridge.

Two gauges used for strain measurement (Half bridge)

The temperature effects are cancelled out by having R2 = R4 and using two identical gauges in the opposite arms of the bridge.

Suppose Rg1 = Rg3 = R2 = R4 = R

Under no strain condition: 𝑉𝐴𝐵 = 𝑉𝐴𝐷 =𝑉𝑖

2 ; 𝑉𝐵 = 𝑉𝐷 𝑎𝑛𝑑 𝑉𝑜 = 0

On the application of load to the cantilever beam, the resistance of the gauge Rg1 increases due to tensile load whilst Rg3

decreases due to equal compressive strain so that,

Resistance of gauge Rg1 = Rg1 + dRg1

Resistance of gauge Rg3 = Rg3 - dRg3

𝑁𝑜𝑤, 𝑉𝐴𝐵 =𝑅𝑔1

𝑅𝑔1 + 𝑅𝑔3

∙ 𝑉𝑖

=𝑅 + 𝑑𝑅

𝑅 + 𝑑𝑅 + (𝑅 − 𝑑𝑅)𝑉𝑖 =

𝑅 + 𝑑𝑅

2𝑅∙ 𝑉𝑖

𝑎𝑛𝑑 𝑉𝐴𝐷 =𝑅2

𝑅2 + 𝑅4

∙ 𝑉𝑖 =𝑉𝑖

2

The changed output voltage, is given by

𝑉𝑜 + 𝑑𝑉𝑜 = 𝑅 + 𝑑𝑅

2𝑅∙ 𝑉𝑖 −

𝑉𝑖

2

= 𝑉𝑖 𝑅 + 𝑑𝑅

2𝑅−

1

2 = 𝑉𝑖

2𝑅 + 2𝑑𝑅 − 2𝑅

4𝑅

𝑜𝑟 𝑉𝑜 + 𝑑𝑉𝑜 =𝑉𝑖

2∙𝑑𝑅

𝑅

19

Since under unstrained conditions Vo=0, therefore change in output voltage due to applied strain becomes;

𝑑𝑉𝑜 =𝑉𝑖

2∙𝑑𝑅

𝑅

𝑑𝑉𝑜 = 𝐺𝐹

2 ∙ ∈ ∙ 𝑉𝑖

This is twice the output of Wheatstone bridge using one gauge only.

The above equation can be rewritten as

𝑑𝑉𝑜 =𝑉𝑖

4 𝑑𝑅

𝑅 −

−𝑑𝑅

𝑅

𝑑𝑉𝑜 =𝑉𝑖

4

𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑕𝑎𝑛𝑔𝑒 𝑖𝑛 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑔𝑎𝑢𝑔𝑒 𝑖𝑛 𝑙𝑖𝑚𝑏 𝐴𝐵

− 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑕𝑎𝑛𝑔𝑒 𝑖𝑛

𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑔𝑎𝑢𝑔𝑒 𝑖𝑛 𝑙𝑖𝑚𝑏 𝐵𝐶

The –ve sign with fractional change in resistance of the gauge in limb BC is due to the fact that compressive and tensile strain

are of opposite signs.

In general, for two gauges connected in the adjacent limbs of a bridge circuit, we have:

𝑑𝑉𝑜 =𝑉𝑖

4 𝑑𝑅𝑔1

𝑅−

𝑑𝑅𝑔3

𝑅

Thus when both the gauges are mounted on the top of the cantilever beam, the two effects cancel each other and the output

voltage is zero.

(iii) Full-Bridge

Figure belowshows four gauges used for strain measurement (Full Bridge). In this arrangement all the four elements of the

bridge are strain gauges.

All the four gauges are similar and have equal resistances when unstrained. i.e.,

Rg1 = Rg2 = Rg3 = Rg4 = R

Under no strain condition: 𝑉𝐴𝐵 = 𝑉𝐴𝐷 =𝑉𝑖

2 ; 𝑉𝐵 = 𝑉𝐷 𝑎𝑛𝑑 𝑉𝑜 = 0

When the load is applied to the cantilever beam, the resistance Rg1 and Rg4 increase due to tensile load whilst resistances Rg2

and Rg3 decrease due to equal compressive strain. When strained, the resistances of the various gauges are

Rg1 = Rg4 = R+dR (tension)

Rg2 = Rg3 = R-dR (compression)

Four gauges used for strain measurement (Full Bridge)

𝑎𝑛𝑑, 𝑉𝐴𝐵 =𝑅𝑔1

𝑅𝑔1 + 𝑅𝑔3

∙ 𝑉𝑖

𝑉𝐴𝐵 =𝑅 + 𝑑𝑅

𝑅 + 𝑑𝑅 + (𝑅 − 𝑑𝑅)𝑉𝑖 =

𝑅 + 𝑑𝑅

2𝑅∙ 𝑉𝑖

20

𝑎𝑛𝑑, 𝑉𝐴𝐷 =𝑅𝑔2

𝑅𝑔2 + 𝑅𝑔4

∙ 𝑉𝑖

𝑉𝐴𝐷 =𝑅 − 𝑑𝑅

𝑅 − 𝑑𝑅 + (𝑅 + 𝑑𝑅)𝑉𝑖 =

𝑅 − 𝑑𝑅

2𝑅∙ 𝑉𝑖

The changed output voltage

𝑉𝑜 + 𝑑𝑉𝑜 =𝑅 + 𝑑𝑅

2𝑅∙ 𝑉𝑖 −

𝑅 − 𝑑𝑅

2𝑅∙ 𝑉𝑖

= 𝑉𝑖 𝑅 + 𝑑𝑅

2𝑅−

𝑅 − 𝑑𝑅

2𝑅 = 𝑉𝑖

𝑑𝑅

𝑅

Since the output voltage under unstrained conditions, Vo = 0, therefore, change in output voltage due to applied strain

becomes,

𝑑𝑉𝑜 = 𝑉𝑖

𝑑𝑅

𝑅

𝑖. 𝑒. , 𝑑𝑉𝑜 = 𝐺𝐹 ∙ ∈ ∙ 𝑉𝑖

Which is the four times the output of Wheatstone bridge using one gauge only.

UNIT IV 8. a) Explain the working principle of McLeod gauge with a neat sketch.

It is used to measure low pressure up to about 10-4 torr.

It is based on the principle of compressing a sample of known volume of low pressure gas to a known volume

and to a pressure sufficiently high which can be read with a simple manometer. The unknown low pressure is

then calculated by using Boyle’s law.

The unknown pressure (Pi) is connected to the point A. The plunger is then withdrawn lowering the mercury

level in the capillary (C), bulb (B) and reference column (R) thus admitting the gas at unknown pressure (Pi).

Next, the plunger is pushed and the mercury level goes up. When the level reaches a cut-off point F, a known

volume (V) of the gas is trapped in the bulb and capillary. The pressure (P) of the gas trapped in capillary (C)

is read as difference in height ‘h’.

6M

21

Applying Boyle’s law

𝑝𝑖𝑉 = 𝑝(𝐴𝑕)

𝑝 = 𝑝𝑖 + 𝑕𝛾

𝑝𝑖𝑉 = 𝑝𝑖 + 𝑕𝛾 𝐴𝑕

𝑝𝑖 =𝛾𝐴𝑕2

𝑉 − 𝐴𝑕

Pressure in terms of mercury column 𝑝𝑖

𝛾=

𝐴𝑕2

𝑉−𝐴𝑕

Assuming, V >> Ah,

𝑝𝑖

𝛾=

𝐴𝑕2

𝑉

𝑝𝑖 = 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

𝐴 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦

𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑓 𝑔𝑎𝑠 𝑖𝑛 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑎𝑓𝑡𝑒𝑟 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛

𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡𝑕𝑒 𝑏𝑢𝑙𝑏 𝑎𝑛𝑑 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑡𝑖𝑙𝑙 𝑝𝑜𝑖𝑛𝑡 𝐹

𝛾 = 𝜌𝑔 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔𝑕𝑡 𝑜𝑓 𝑚𝑒𝑟𝑐𝑢𝑟𝑦

The limitations of this gauge are: (i) lack of continuous measurement, (ii) if the measured gas contains any

vapor that are condensed by compression, there will be an error.

McLeod gauge is also used calibrate other gauges.

b) Brief notes on (i) Elastic force meters (ii)Pyrometers

Spring

A simple spring is a force-displacement transducer, often used for weighing.

Force and displacement are related as,

𝐹 = 𝑘𝑥

Where, 𝐹= the applied force

𝑥= displacement

𝑘= stiffness or spring constant (force per unit displacement)

Axially Loaded Member

It is a bar or a rod axially loaded either in tension or compression (Figure 4.36).

Displacement of the free end 𝑥 =𝐹𝐿

𝐴𝐸=

𝐹

𝑘

Stiffness, 𝑘 =𝐴𝐸

𝐿

22

Strain in axial direction 𝜀1 =𝐹

𝐴𝐸

Strain in lateral direction 𝜀2 = −𝜈𝜀

Where, F= force

A= area of cross-section

L=length

E= Young’s modulus of the material

𝜈 = Poisson’s ratio

Cantilever Beam

Figure shows a cantilever, where a force F is applied at a distance L from the fixed end. The force causes a

bending moment and the deflection of the free end strains, 𝜀1 , 𝑎𝑛𝑑 𝜀2 induced are equal in magnitude but

opposite in sign.

cantilever elastic member

Deflection of the free end 𝑦 =1

3

𝐹𝐿3

𝐸𝐼=

4𝐹𝐿3

𝐸𝑏𝑡 3

Stiffness 𝑘 =3𝐸𝐼

𝐿3 =𝐸𝑏𝑡 3

4𝐿3

Where, 𝐼 =𝑏𝑡 3

12 is the moment of inertia of the section about the centroidal axis.

Strain, 𝜀1 =6𝐹𝐿

𝐸𝑏𝑡 2 (𝑡𝑒𝑛𝑠𝑖𝑜𝑛)

𝜀2 =6𝐹𝐿

𝐸𝑏𝑡2 (𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛)

The opposite sign of strains ε1 and ε2 (in above case) is of advantage while using strain gauges employing a

bridge circuit.

Ring (Proving Ring)

This is a very useful and important device for measurement of load with a range of 1.5 kN to 1.5 MN. This

device has been the standard for calibrating tensile testing machines. Figure (b) shows that when the ring is

subjected to diametrical compression by a force F it deflects by amount ‘y’. Deflection can be measured by a

dial gauge or micrometer with a vibrating reed [Figure (c)].

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Bending moment 𝑀 =𝐹𝑅

2 cos 𝜃 −

2

𝜋

M (at θ=0), is equal to 0.182 FR

Deflection 𝑦 =1

16 𝜋

2−

4

𝜋

𝐹𝐷3

𝐸𝐼

𝑦 =1.79𝐹𝑅3

𝐸𝑏𝑡3

Where, F= load

D= diameter of ring

E= Young’s modulus

I= moment of inertia of the section about centroidal axis of bending.

Above relations are derived on the assumption that radial thickness of the ring is small as compared to the

radius. Under the force applied, as shown in the figure (a), ring develops compressive strains on the inner

surfaces (2, 3) and tensile strains on the outer surfaces (1, 4) with maximum stresses occurring at θ=0o and

both compressive and tensile strains are equal. This fact is used when employing strain gauges to get a

higher output from the bridge circuit.

PYROMETERS

The temperature measuring devices require that the device must be brought in to physical contact with the

body whose temperature is to be measured. This may not be possible under following situations:

(i) The body is too hot and a contact may damage or melt the device or thermometer.

(ii) The body is not accessible for actual physical contact as it may be moving or is too far off.

(iii) When temperature distribution over the surface of the body to be determined.

In the situations, non contact type temperature measuring devices are to be used.

All bodies above absolute zero temperature radiate energy. Heat radiations emitted from the body above

temperatures 650oC are of sufficient intensity and are used for temperature measurements. The instruments

based on sampling and measuring radiant energy from the hot body are called Pyrometers.

(OR)

9. a) Explain the working principle of Bourdon tube pressure gauge with a neat sketch

The bourdon tube is an elastic element which is closed at its one end the other end is open to receive the applied pressure. The cross section of the tube is elliptical and will tend to become circular when pressure is applied. This gives rise to deflection at its free end and is used to move a pointer on a calibrated scale.

4M

24

b) List out different types of dynamometers used for torque measurement. Explain electrical

type dynamometer with a neat sketch.

8M

Torque measuring devices are commonly known as dynamometers and are classified as,

1. Absorption dynamometers

2. Driving dynamometers

3. Transmission dynamometers.

Cradling

The concept of cradling is widely used in dynamometers and is explained before describing various types of dynamometers.

These dynamometers consist of a rotor and a stator (or housing). The stator is not fixed but can turn relative to the rotor.

The rotor is connected by a shaft to the machine to be tested. The rotor and stator are coupled electromagnetically or

hydraulically. The torque is measured as the reaction torque acting on the stator.

Schematic of a Cradled Dynamometer

An arm is attached on the stator and the force acting on this arm is determined. The torque is then given by the product of this

force and the arm of the dynamometer because the restraining torque equals the reaction torque. The rotor and stator can be

an electric motor (source of power) or a hydraulic dynamometer (sink). Figure shows the basic elements of a cradling type

dynamometer.

Electrical Generator type Dynamometer

It is a cradle type dynamometer and is essentially a generator driven by the test machine. The mechanical energy is converted

into the electrical energy by the electrical generator. The electrical energy is then dissipated through a variable resistance grid,

which also serves to vary the load. Cradling in trunnion bearings permits the determination of reaction torque on the stator.