ME 322: InstrumentationLecture 40

April 29, 2015

Professor Miles Greiner

Review Labs 10 and 11

Announcements/Reminders• This week: Lab 12 Feedback Control• Working with TSI to get the Lab 11 CTA to work

• Remember you are not graded on how well the CTA’s work

• HW 13 due now • HW 14 due Monday (may drop)

• Review Labs 10, 11, and 12; Today and Friday

• Supervised Open-Lab Periods • Saturday and Sunday

• Drop Extra-Credit Lab 12.1 and LabVIEW Workshop• 6 points will be added to everyone’s Midterm II score

• Lab Practicum Finals (May 6-14)– Guidelines, Schedule

• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Tests/Index.htm

• College of Engineering Innovation Day: Friday, May 1, 2015• http://www.unr.edu/engineering/news-and-events/special-e

vents/innovation-day

Lab 10 Vibration of a Weighted Cantilever Beam

• Accelerometer Calibration – i.e. C = 616.7 mV/g

– Use the calibration constant for the accelerometer you are issued

– Inverted Transfer function: a = V/C• During Final you will be given values and 3 s

uncertainties of E, W, T• You will measure (3s uncertainty)

• MT, MW: Analytical balance, = 0.1 g

– LE, LT: Tape measure, = 1/16 in

– Two values of LB

W

LT MT

T

LBLE

Accelerometer

ClampMW

E (given)

Table 1 Measured and Calculated Beam Properties

• Use the units you are given, or the ones you used to measure

W

LT MT

T

LBLE

Accelerometer

ClampMW

E (given)

Units Value3 s

UncertaintyElastic Modulus, E GPa 206.3 7.5

Beam Width, W inch 0.9968 0.0015Beam Thickness, T inch 0.1220 0.0015

Beam Total Length, LT inch 23.813 0.063End Length, LE inch 0.469 0.063

Beam Mass, MT g 377.2 0.1Combined Mass, MW g 333.9 0.1

Figure 2 VI Block Diagram

Convert to Dynamic Data Convert to Dynamic Data Converts numeric, Boolean, waveform and array data types to the dynamic data type for use with Express VIs.

Statistics This Express VI produces the following measurements: Time of Maximum

Figure 1 VI Front Panel

• Use fS ~600 Hz > 2fM to capture the peaks • When plotting a versus t, use time increment Dt = 1/fS

• The oscillatory frequency is clearly higher for the shorter LB.

• Looks like ( )=𝑎 𝑡 𝐴𝑒𝑏𝑡sin (2 + )𝜋𝑓𝑡 𝜙– Measure f from spectral analysis ( fM )

– Find b from exponential fit to acceleration peaks

Disturb beam and measure a(t) for two LB’s

Warning: Be careful to check (view) you’re a(t) data before processing

• For example, see oscillations between 2 and 4 seconds

Figure 4 Oscillatory Amplitude Versus Frequency

• The sampling time and frequency were T1 = 10 sec and fS = 600 Hz, so the system is capable of detecting frequencies between 0.1 and 300 Hz, with a resolution of 0.1 Hz. – To plot aRMS vs t, use frequency increment Df = 1/T1

• For beam lengths of LB = 13 and 7 inches, the peak frequencies are, respectively, fM = 7.50 ± 0.1 and 18.60 ± 0.1 Hz. – These frequencies are easily detected from this plot.

• Can we predict these measured oscillatory frequency from length, mass and elastic modulus measurements?

Fig. 5 Peak Acceleration versus Time

• The average exponential decay constants for the beam lengths of LB = 13 and 7 inches, are b = -0.176 and -0.192, respectively

• The magnitude of these “constants” (slope of the curves) decreased slightly with time.

Equivalent Endpoint Mass

• Beam is not massless, – Its mass affects its motion and natural frequency

• (linear sum)– mass of weight, accelerometer, pin, nut– (contribution form beam mass)

MEQBeam Mass MB

LT MT

LBLEClamp MW

Beam Equivalent Spring Constant, KEQ

– Power product?

F

d

LB

Predicted Frequencies• Undamped

– – Power Product?

• Damped

– Power product?– If (it is), then , and

• Measured Damping Coefficient

Table 2 Calculated Values and Uncertainties

• The intermediate mass is small compared to the equivalent mass.

• For both beam lengths, the damping is sufficiently low so that the predicted undamped and damped frequencies, fOP and fP, are nearly the same

• The predicted damped frequencies are roughly 12 to 17% higher than the measured values, and their confidence intervals do not overlap.

Value3s

UncertaintyValue

3s

Uncertainty

Beam Length, LB [m] 0.1778 0.0016 0.3302 0.0016

Intermedate Mass, MI [kg] 0.033 0.001 0.055 0.001

Equivalent Mass, MEQ [kg] 0.367 0.001 0.389 0.001

Equivalent Beam Spring

Constant, kEQ[N/m] 6912 402 1079 58

Predicted Undamped

Frequency, foP[Hz] 21.8 0.6 8.4 0.2

Measured Damped

Frequency, fM [Hz] 18.60 0.05 7.50 0.05

Decay Constant, b [1/sec] -0.192 - -0.176 -Damping Coefficient, lM [Ns/m] 0.1409 - 0.1368 -

Damped Frequency, fp [Hz] 21.8 0.6 8.4 0.2Percent Difference

(fP/fM-1)*100%17% - 12% -

LB = 7 inch LB = 13 inch

Units

Time and Frequency Dependent Data• http

://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Lab%20Index.htm

• Plot a versus t – Time increment Dt = 1/fS

• Plot aRMS versus f

– Frequency increment Df = 1/T1

• Measured Damped (natural) Frequency, fM

– Frequency with peak aRMS

– Uncertainty

• Exponential Decay Constant b – Show how to find acceleration peaks versus time

• Use AND statements to find accelerations that are larger than the ones before and after it

• Use If statements to select those accelerations and times• Sort the results by time and Plot

– Fit data to y = Aebx to find b

Lab 11 Unsteady Speed in a Karman

Vortex Street

• Nomenclature– U = air speed

– VCTA = Constant temperature anemometer voltage

• Two steps– Statically-calibrate hot film CTA using a Pitot probe (this part worked)

– Find frequency, fP with largest URMS downstream from a cylinder of diameter D for a range of air speeds U (working on this part)

• Compare to expectations (StD = DfP /U = 0.2-0.21)

Setup

• Measure PATM, TATM, and cylinder D

• Find air m from text– A.J. Wheeler and A. R. Ganji,

Introduction to Engineering Experimentation, 2nd Edition, Pearson Prentice Hall, 2004, p. 430

• Tunnel Air Density

DTube

PP

Static

Total+ -

IP

Variable SpeedBlower Plexiglas

Tube

Pitot-Static Probe

3 in WC

BarometerPATM TATM

CTA

myDAQ

Cylinder

VCTAHot Film Probe

Calibration Calculations• Based on analysis we expect

IP

[mA]VCTA

[V]U

[m/s]

U1/2

[m1/2/s1/2]

VCTA2

[V2]4.00 2.140 0.0 0.00 4.585.70 3.670 12.4 3.52 13.477.40 3.930 17.5 4.18 15.449.40 4.070 22.0 4.70 16.56

11.60 4.130 26.2 5.11 17.0616.80 4.460 33.9 5.83 19.8914.40 4.340 30.6 5.53 18.8413.30 4.290 28.9 5.38 18.4011.00 4.160 25.1 5.01 17.318.50 4.000 20.1 4.49 16.006.30 3.820 14.4 3.79 14.594.00 2.140 0.0 0.00 4.58

𝑆𝑉𝐶𝑇𝐴2 , √𝑈

𝑆√𝑈 ,𝑉 𝐶𝑇𝐴2

Hot Film System Calibration

• The fit equation VCTA2 = aU0.5+b appears to be appropriate

for these data.

How to Construct VI (Block Diagram)

Spectral Measurements Selected Measurements: Magnitude (RMS) View Phase: Wrapped and in Radians Windowing: Hanning Averaging: None

Formula Formula: ((v**2-b)/a)**2

• Use for both static-calibration and unsteady measurements• Don’t need to store speed vs time

Front Panel

Unsteady Speed Downstream of a Cylinder

• When the cylinder is removed the speed is relatively constant• When the cylinder is installed, downstream of the cylinder

– The average speed is lower compared to no cylinder– There are oscillations with a broadband of frequencies

0

0.1

0.2

0.3

0.4

0 500 1000 1500 2000 2500 3000

f [Hz]

Srm

s [m

/s]

Fig. 4 Spectral Content in Wake for Highest and Lowest Wind Speed

• The sampling frequency and period are fS = 48,000 Hz and TT = 1 sec.

• The minimum and maximum detectable finite frequencies are 1 and 24,000 Hz (not all are shown).

• It is not straightforward to distinguish fP from this data. Its uncertainty is wfp ~ 50 Hz.

(a) Lowest Speed

0

0.1

0.2

0.3

0.4

0.5

0 500 1000 1500 2000 2500 3000f [Hz]

Srm

s [m

/s]

(b) Highest Speed

fp = 2600 Hz

fp = 751 HzURMS [m/s]

URMS [m/s]

Dimensionless Frequency and Uncertainty

• UA from LabVIEW VI

• fP from LabVIEW VI plot

– ½(1/tT) or eyeball uncertainty

• Re = UADr/ m (power product)

• StD = DfP/UA (power product)

UA [m/s] WUa [m/s] fP [Hz] wfp [Hz] Re WRe St WSt

37.8 1.3 2600 50 7084 236 0.218 0.00834.1 1.2 2427 50 6385 224 0.226 0.00927.3 1.1 1892 50 5121 201 0.220 0.01023.0 1.0 1596 50 4312 184 0.220 0.01216.5 0.8 1218 50 3081 156 0.235 0.01511.8 0.7 751 50 2214 132 0.202 0.018

Fig. 5 Strouhal versus Reynolds

• The reference value is from A.J. Wheeler and A.R. Ganji, Introduction to Engineering Experimentation, 2nd Edition, Pearson Prentice Hall, 2004, p. 337.

• Four of the six Strouhal numbers are within the expected range.

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 1000 2000 3000 4000 5000 6000 7000 8000Re

St

Expected St Range

Process Sample Data

• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2011%20Karmon%20Vortex/Lab%20Index.htm

Lab 9 Transient TC Response in Water and Air

• Start with TC in room-temperature air• Measure its time-dependent temperature when it is plunged into boiling

water, then room-temperature air, then room-temperature water• Determine the heat transfer coefficients in the three environments,

hBoiling, hAir, and hRTWater

• Compare each h to the thermal conductivity of those environments (kAir or kWater)

• Also calculate Biot number (dimensionless thermal size) and delay time for center to respond

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8

Time, t [sec]

Tem

pe

ratu

re,

T [

oC

]

tB = 0.78 sIn Boiling Water

tA = 3.36 sIn Air

tR = 5.78 sIn Room Temperature Water

LabVIEW VI

Dimensionless Temperature Error

• At time t = t0 a thermocouple at temperature TI is put into a fluid at temperature TF.

• Theory for a uniform-temperature TC predicts:– Dimensionless Error: – Time Constant for a spherical thermocouple

T

tt = t0

TI

TF

Error = E = TF – T ≠ 0

T(t)

TI

TF

Environment Temperature

Initial Error EI = TF – TI

𝜌 ,𝑐 ,𝐷

h

• From this chart, find– Times when TC is placed in Boiling Water, Air and RT Air (tB, tA, tR)

– Temperatures of Boiling water (maximum) and Room (minimum) (TB, TR)

• Thermocouple temperature responds more quickly in water than in air• Slope does not exhibit a step change in each environment

– Temperature of TC center does not response immediately • Transient time for TC center: tT ~ D2rc/kTC

Measured Thermocouple Temperature versus Time

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8

Time, t [sec]

Te

mp

era

ture

, T

[oC

]

tB = 0.78 sIn Boiling Water

tA = 3.36 sIn Air

tR = 5.78 sIn Room Temperature Water

• State estimated diameter uncertainty, 10% or 20% of D

• Thermocouple material properties (next slide) – Citation: A.J. Wheeler and A.R. Gangi, Introduction to Engineering

Experimentation, 2nd Ed., Pearson Education Inc., 2004, page 431. – Best estimate: – Uncertainty:

• tT ~ D2rc/kTC; = ?

Type J Thermocouple Properties

Effective Diameter D

[in]Density ρ [kg/m3]

Thermal Conductivity kTC [W/mK]

Specific Heat c [J/kgK]

Initial Transient

Time tT [sec]

Value 0.059 8400 45 421 0.183s Uncertainty 0.006 530 24 26 0.10

TC Wire Properties (App. B)

Dimensionless Temperature Error,

– For boiling water environment, TF = TBoil, TI = TRoom

– For room-temperature air and water, TF = TRoom, TI = TBoil

• How can we find the time range t1 < t < t2 when decays exponentially with time?

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8

Time, t [sec]

Tem

pe

ratu

re,

T [

oC

]

tB = 0.78 sIn Boiling Water

tA = 3.36 sIn Air

tR = 5.78 sIn Room Temperature Water

Data Transformation (trick)

– Where , and b = -1/ t are constants

• Take natural log of both sides

• Instead of plotting versus t, plot ln() versus t– Or, use log-scale on y-axis– During the time period when decays exponentially, this

transformed data will look like a straight line

Find decay constant b using Excel

• Use curser to find beginning and end times for straight-line period– q exhibits random variation when it is less than q < 0.05

• Add a new data set using those data• Use Excel to fit a y = Aebx to the selected data

– For this data b = -13.65 1/s – Since b = -1/ , t and t , – Calculate (power product?), ?

• Assume uncertainty in b is small compared to other components

For t = 1.14 to 1.27 sq = 1.867E+06e-1.365E+01t

0.01

0.1

1

0.8 0.9 1 1.1 1.2 1.3 1.4

Time, t [sec]

q BO

IL =

(T

B-T

(t))

/(T

B-T

R)

Dimensionless Temperature Error versus Time for Room-Temperature Air and Water

• Decays exponentially during two time periods:– In air:

• t = 3.83 to 5.74 sec, b = -0.3697 1/s

– In water: • t = 5.86 to 6.00 sec, b = -7.856 1/s.

In AirFor t = 3.83 to 5.74 sec

q = 2.8268e-0.3697t

In Room Temp WaterFor t = 5.86 to 6.00 sec

q = 2E+19e-7.856t

0.01

0.1

1

3 3.5 4 4.5 5 5.5 6 6.5 7

Time t [sec]

q Ro

om

Lab 9 Results

• Water environments have orders of magnitude higher h (and b) than air– Similar to kFluid

• Nusselt numbers (power product) are more dependent on flow conditions (steady versus moving) than environment composition

• Biot number (dimensionless size)

Environment b [1/s]

h

[W/m2C]

Wh

[W/m2C]kFluid

[W/mC]NuD

hD/kFluid

Bi hD/kTC

Lumped (Bi < 0.1?)

Boiling Water -13.7 12016 1603 0.680 26 0.403 noAir -0.37 325 43 0.026 19 0.011 yes

Room Temperature Water -7.86 6915 923 0.600 17 0.232 no

Air and Water Properties (bookmark)

Lab 9 Sample Data• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/L

ab%2009%20TransientTCResponse/LabIndex.htm

• Plot T vs t– Find TB, TR, tB, tA, and tR

• Calculate q and plot vs time on log scale– In Boiling Water, TI = TR, TF = TB

– In Room Temperature air and water, TI = TB, TF = TR

– Select regions that exhibit exponential decay• Find decay constant for those regions

• Calculate and wh for each environment

• For each environment calculate – NuD

– BiD

Thermal Boundary Layer for Warm Sphere in Cool Fluid

•

– h increases as k increase and object sized decreases

– = Dimensionless Nusselt Number (power product?)

T

D r

TF

𝛿

Conduction in Fluid

Thermal BoundaryLayer

Lab 9 Transient TC Response in Air & Water

Wire yourself

TC → Conditioner(+)TC → White Wire(-)TC → Red StripeConditioner to → MyDAQ

Com → (-)

Vout → (+)

Write VI Easy Fig 1 & 2 will not be given

Acquire Data

Fs = 1000 Hz T i = 8 sec

At least 2 seconds in each environment.

•Room temp water

•Boiling water

•Room temp air

•Room temp water

Fig 3 Plot T Vs. t

ID time tB , tA , tR

ID Temp

TRoom = Tmin

TBoil = Tmax

Fig 4 For boiling water

vs. tIdentify: Start & end times of exponential decay period (looks linear)•Select exp decay data y•Add data to plot•Fit to that data

Show results on the plot•Find b

Units s-1

Fig 5 Room Temp Air & Water

vs. t

Find

Find

Table 2

Lab 10: Vibrating Beam

You will be given beam and its E and WE

VI fig 1 &2

Table 2

Undamped Predicted Frequency if b = 0, λ = 0

Measured Damping Coeff

If

Then Wfp ≈ Wfop

Is