Experimental MethodsExperimental Methods

MEC3045FMEC3045F

Force, Torque and Strain Force, Torque and Strain MeasurementMeasurement

Force BasicsForce Basics

Newton’s second law of motion:Newton’s second law of motion:– For a particle of constant mass, force is proportional to the For a particle of constant mass, force is proportional to the

product of mass and accelerationproduct of mass and acceleration

What is weight?What is weight?– The weight of a body is the force exerted on the body by the The weight of a body is the force exerted on the body by the

acceleration of gravity at sea levelacceleration of gravity at sea level

Weight Weight ≠ mass (eg, what about the moon)≠ mass (eg, what about the moon)Mass represents a quantity of matter and is related to Mass represents a quantity of matter and is related to force through F = maforce through F = maIt is not a physical property that is measured directly. It is It is not a physical property that is measured directly. It is determined by force measurements or by comparing to determined by force measurements or by comparing to mass standardsmass standards

maF

mgWF

Mass balance measurementsMass balance measurements

An analytical balance rotates aboutAn analytical balance rotates aboutfulcrum O.fulcrum O.

Unbalanced position indicated by Unbalanced position indicated by φφ

Centre of gravity of arm given by GCentre of gravity of arm given by G

ddGG is the distance from O to G is the distance from O to G

WWBB is the weight of balance arm is the weight of balance arm

and the pointerand the pointer

When WWhen W1 1 = W= W22 φφ = 0 and weight = 0 and weight

of balance arms has no influenceof balance arms has no influence

Sensitivity of the balance is a measure of the angular Sensitivity of the balance is a measure of the angular displacement, displacement, φφ, per unit unbalance, per unit unbalancein the two weights Win the two weights W11 and W and W22

WWWS

21

Sensitivity of the balanceSensitivity of the balance

How does the sensitivity of the balance depend on the How does the sensitivity of the balance depend on the size and mass of the balance itself?size and mass of the balance itself?

W1(cosφ)

W1 (sinφ)

φ

+ve

sinsincossincos 2211 GBBB dWdWLWdWLW

sin)sincos()sincos( 21 GBBB dWdLWdLW For small deflection angles, sinφ ≈ φ; cosφ ≈ 1

GBBB dWdLWdLW )()( 21

GBBB dWdWLWdWLW 2211

GBBB dWdWdWLWLW 2121

)()( 2121 GBBB dWdWdWWWL

GBBGBBB dWWWd

L

dWdWdW

L

W

)( 2121

Near equilibrium, W1 ≈ W2

GBB dWWd

L

WS

2

If constructed so that dB = 0

GBdW

LS Independent of

the loads!!!

0M

Accounting for buoyancy forcesAccounting for buoyancy forces

Assume we measure a mass against known brass weightsAssume we measure a mass against known brass weights

Archimedes's principle: When a rigid object is submerged Archimedes's principle: When a rigid object is submerged in a fluid (completely or partially), there exits an upward in a fluid (completely or partially), there exits an upward force on the object that is equal to the weight of the fluid force on the object that is equal to the weight of the fluid that is displaced by the objectthat is displaced by the object

We are not measuring the weight forces of the unknown We are not measuring the weight forces of the unknown mass, but the weight forces less the buoyancy forces mass, but the weight forces less the buoyancy forces acting on each massacting on each mass

If we measure in a vacuum or if the brass weights and If we measure in a vacuum or if the brass weights and unknown mass have the same volume, the buoyancy unknown mass have the same volume, the buoyancy forces will cancel outforces will cancel out

Accounting for buoyancy forcesAccounting for buoyancy forces

The true weights of the unknown and standard weights areThe true weights of the unknown and standard weights are

The forces exerted on the balance arms areThe forces exerted on the balance arms are– wherewhere

ρρuu = density of the unknown = density of the unknown

ρρss = density of the standard weights = density of the standard weights

ρρaa = density of the surrounding air = density of the surrounding air

VVuu, V, Vss = volumes of the unknown and standard weights = volumes of the unknown and standard weights

At equilibriumAt equilibrium

sas

uau

VW

VW

)(

)(

2

1

sssuuu VWVW ;

assu

aususu WW

ExampleExample

An analytical balance like that An analytical balance like that alongside is used to weigh a alongside is used to weigh a 15-oz (troy) quantity of 15-oz (troy) quantity of platinum in air at 1atm and platinum in air at 1atm and 20°C. The density of platinum 20°C. The density of platinum is 21 380 kg/mis 21 380 kg/m33 and that of the and that of the standard brass weights is 8 standard brass weights is 8 490 kg/m490 kg/m33. If the price of . If the price of platinum is R5373.61 per oz, platinum is R5373.61 per oz, calculate the Rand value of the calculate the Rand value of the buoyancy error in weight buoyancy error in weight determination.determination.

Using Elastic elements for NUsing Elastic elements for N

We can get an indication of the magnitude of an applied We can get an indication of the magnitude of an applied force by measuring a displacementforce by measuring a displacement

Simple springSimple spring Bar elementBar element Cantilever elementCantilever element

yL

AEF y

L

EIF

3

3ykF

ExampleExample

A farmer decides to build a crude weighing device to A farmer decides to build a crude weighing device to weigh bags of grain up to 68kg. For this purpose he weigh bags of grain up to 68kg. For this purpose he employs a section of 25mm steel pipe (24.3mm OD, employs a section of 25mm steel pipe (24.3mm OD, 33.4mm ID) as a cantilever beam. He intends to 33.4mm ID) as a cantilever beam. He intends to measure the deflection of the beam with a metal measure the deflection of the beam with a metal carpenter's scale having graduations of 1mm. He feels carpenter's scale having graduations of 1mm. He feels that an error of 1% at the maximum deflection would that an error of 1% at the maximum deflection would be acceptable. On the basis of these data, what length be acceptable. On the basis of these data, what length of pipe would you recommend for the farmer's of pipe would you recommend for the farmer's application?application?

Torque measurementsTorque measurements

Angular deformation of a bar or hollow cylinderAngular deformation of a bar or hollow cylinder

φφ = angular deflection = angular deflection

G = shear modulus of elasticityG = shear modulus of elasticity

μμ = Poisson’s ratio (≈ 0.3) = Poisson’s ratio (≈ 0.3)

L

rrGM i

2

440

12

EG

Torque measurementsTorque measurements

The Prony brakeThe Prony brake

DynamometerDynamometer

TNPower 2T is the torqueN is the rotational speed

ExampleExample

A model airplane engine develops 750W at A model airplane engine develops 750W at 10000 rpm. Calculate the torque produced at 10000 rpm. Calculate the torque produced at these conditions. Suppose this torque acts at a these conditions. Suppose this torque acts at a moment arm of 150mm and is to be measured moment arm of 150mm and is to be measured with the deflection of a cantilever beam having a with the deflection of a cantilever beam having a nominal deflection of 10mm. Specify suitable nominal deflection of 10mm. Specify suitable design dimensions for a beam to accomplish this design dimensions for a beam to accomplish this measurement.measurement.

Stress (Stress (σσ) ) and strain (and strain (εε))

Elastic deformation vs plastic deformationElastic deformation vs plastic deformation– Elastic deformation: area reduces for + Elastic deformation: area reduces for + ΔΔLL– Plastic deformation: area constant for + Plastic deformation: area constant for + ΔΔLL

εεaa= axial strain= axial strain

T = axial load T = axial load A = cross-sectional area A = cross-sectional area σσaa= axial stress= axial stress

Unit axial strain Unit axial strain εεaa = dL/L = dL/L

Ratio of unit strain in transverse direction to unit strain in Ratio of unit strain in transverse direction to unit strain in axial direction called axial direction called Poisson’s ratioPoisson’s ratio 3.0

/

/

LdL

DdD

a

t

EE

AT aa

/

Strain basicsStrain basics

We are aiming to get the unit strain at a pointWe are aiming to get the unit strain at a point

Strain measurements must be made over a finite lengthStrain measurements must be made over a finite length

As L As L 0, 0, measurement approaches unit strainmeasurement approaches unit strain

Length over which average stain measurement is taken Length over which average stain measurement is taken called the base lengthcalled the base length

Deformation sensitivity is the minimum deformation that Deformation sensitivity is the minimum deformation that can be indicated by an appropriate gaugecan be indicated by an appropriate gauge

Strain sensitivity is the minimum deformation that can be Strain sensitivity is the minimum deformation that can be indicated per unit base lengthindicated per unit base length

So, we need to be able to measure deformation:So, we need to be able to measure deformation:But how?But how?

Electrical resistance strain gaugesElectrical resistance strain gauges

Operation based on the principle that a conductor’s Operation based on the principle that a conductor’s electrical resistance changes when it is mechanically electrical resistance changes when it is mechanically deformed.deformed.

Bond the conductor to the specimenBond the conductor to the specimen

Apply a loadApply a load

The specimen and the resistance element deformThe specimen and the resistance element deform

Deformation indicated by a change in resistanceDeformation indicated by a change in resistance

Wire gauge (12-25μm)Foil gauge (<25μm)Semiconductor gauge

Basic strain gauge relationshipBasic strain gauge relationship

The resistance of the conductor isThe resistance of the conductor is– ρρ = resistivity of the material (assume constant) = resistivity of the material (assume constant)

Differentiating givesDifferentiating gives

The area can be related to the diameter (D)The area can be related to the diameter (D)

Substituting for axial strain and Poisson’s ratioSubstituting for axial strain and Poisson’s ratio– εεaa= dL/L;= dL/L; εεtt= dD/D;= dD/D; μμ = - = -εεtt/ / εεaa

A

LR

A

dA

L

dL

R

dR

D

dD

A

dADA 2

42

21 aR

dR

Calculating strainCalculating strain

A gauge factor can be defined asA gauge factor can be defined asandand

We can now express local strain in terms of:We can now express local strain in terms of:– The gauge factor (F)The gauge factor (F)– The resistance of the gauge (R)The resistance of the gauge (R)– The change in resistance (The change in resistance (ΔΔR)R)

The gauge factor (-140<F<175) and the resistance of the The gauge factor (-140<F<175) and the resistance of the gauge is given by the manufacturergauge is given by the manufacturerA high gauge factor is desirable in practice because a A high gauge factor is desirable in practice because a larger change in larger change in ΔΔR is produced for a given strain inputR is produced for a given strain input– This requires a less sensitive readoutThis requires a less sensitive readout

R

R

F

1

a

RdRF

/

21F

Temperature compensationTemperature compensation

Compensation done experimentallyCompensation done experimentally

Functional layoutsFunctional layouts

Functional layoutsFunctional layouts

Strain-gauge rosettesStrain-gauge rosettes

We have been measuring a principle strain onlyWe have been measuring a principle strain only

Normally more than one strain presentNormally more than one strain presentAND we don’t know the orientation of theAND we don’t know the orientation of theprinciple strain axesprinciple strain axes

Rectangular rosette Delta rosette

ExampleExample

A strip of steel sheet 1.6 x 50 x 500 mm, is available for A strip of steel sheet 1.6 x 50 x 500 mm, is available for use as a force measuring elastic element. The strip is to use as a force measuring elastic element. The strip is to be used by cementing strain gauge to its flat surfaces and be used by cementing strain gauge to its flat surfaces and measuring the deformation under load. The strain gauges measuring the deformation under load. The strain gauges have a maximum strain of 2000 µm/m and a gage factor of have a maximum strain of 2000 µm/m and a gage factor of 1.90. Calculate the force for which the measurement 1.90. Calculate the force for which the measurement system is applicable.system is applicable.An electrical-resistance strain gauge records a strain of An electrical-resistance strain gauge records a strain of 400 µm/m on a steel tension member. What is the axial 400 µm/m on a steel tension member. What is the axial stress?stress?A rectangular rosette is mounted on a steel plate having E A rectangular rosette is mounted on a steel plate having E = 29x10= 29x1066 psi and µ= 0.3. The three strains are measured psi and µ= 0.3. The three strains are measured asas

ε1 = +500 µin/in , ε2 = +400 µin/in, ε3 = -100 µin/inε1 = +500 µin/in , ε2 = +400 µin/in, ε3 = -100 µin/in

Calculate the principal strains and stresses and the Calculate the principal strains and stresses and the maximum shear stress. Locate the axis of the principal maximum shear stress. Locate the axis of the principal stress.stress.