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10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
1
SAMPLE
ONLY
Pressure TransientFourier Analysis
Experimentby
Student XGroup Y
ME 435L Winter 2007
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
2
SAMPLE
ONLY Objectives
• Calibrate a strain gage pressure transducer and compare to manufacturer’s calibration data
• Study transient response pressure fluctuations generated by rapid release of water from a raised tank
• Create a computer generated curve fit of analog pressure transient curve using a Discrete Fourier Transform
• Compare physical system to computer generated model
• Find pressures at t=120° and t=180°
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
3
SAMPLE
ONLY Background Theory
• Strain Gage Pressure Transducer– Strain gages bonded to diaphragm in Wheatstone Bridge
configuration.– Pressure gradient causes deflection in diaphragm– Resistance in strain gages is proportional to diaphragm
deflection
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
4
SAMPLE
ONLY Background Theory
• Strain Gage Pressure Transducer (Continued)– Low mass and relative stiffness of diaphragm lead to a high
natural frequency and quick response time– Well suited to transient measurements
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
5
SAMPLE
ONLY Background Theory• Fourier Analysis
– Infinite expression of coefficients multiplied by sines and cosines to approximate a continuous, complex function
• Fourier Transform– Method for decomposition of a measured signal (y(t))
into its amplitude-frequency components– Discrete Fourier Transform (DFT)
• Approximation of the Fourier Transform for use with finite data sets
– Fast Fourier Transform (FFT)• Algorithm to compute DFT quickly• Uses N log2 N operations as opposed to N2 in the DFT
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
6
SAMPLE
ONLYFourier Analysis Theory
2n
2nn BAC
40
1nnno ) tsin(nCAF(t)
t)2
Ncos(
2
A t)sin(nB t)cos(nA
2
AF(t) 2
N12
N
1nnn
o
4096
VA
4096
1ii
o
n
n1n A
Btan
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
7
SAMPLE
ONLY Equipment
• Viatran Corp. Model 119 Pressure Transducer– FSO Range:
• 0-40” WCD
– Static Sensitivity:• K = 100.54 +/- 15.3% mVDC / in WCD
• Agilent Technologies HP34970A Data Acquisition / Switch Unit – Operating Range:
• 0-10 VDC
– Bias Error:• 0.0035% of Reading + 0.0005% of Range
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
8
SAMPLE
ONLY Equipment
• Agilent Technologies HP35670A Dynamic Signal Analyzer– Range:
• 90 dB
– Accuracy• +/- 0.15dB
• Operational Amplifier Bridge
Signal Conditioning board– Gain potentiometer set to obtain 0.993
VDC at 10” WCD– Gain, G=120
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
9
SAMPLE
ONLYEquipment
• Water Supply Tank– Hole in bottom plugged by
stopper
• Ruler– Accuracy
• +/- 0.0625”
• PC with LABVIEW installed
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
10
SAMPLE
ONLYPressure Transducer Calibration Curve
Figure G2: Calibration curve calculated after shifting the y-intercept.
y = 87.193x - 44
R2 = 1
y = 100.54x - 44
R2 = 0.9882
-100
100
300
500
700
900
0 1 2 3 4 5 6 7 8 9 10
Water Column Depth (in. WCD)
Tra
nu
cer
Ou
tpu
t V
olt
age
(m
VD
C)
Experimental Cal Curve
Mfg. Cal Curve
Linear (Mfg. Cal Curve)
Linear (Experimental Cal Curve)
Graph of the Experimental and Manufacturer's Calibration Curves for Viatran Model 119 Pressure TransducerStatic Sensitiviy: Experimental = 100.54 mVDC / in WCD Manufacturer = 87.193 mVDC / in WCDResulting % difference: % difference = [(100.54 - 87.193) / 87.193] * 100 = 15.3%
Mechanical Engineering DepartmentCal Poly Pomona UniversityMeasurements Lab9/30/02
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
11
SAMPLE
ONLYUncertainty Analysis
2FAE
2SA
2DAQ
2Tape
2PTFAE PBBBBu
2ZB
2REP
2HYS
2LIN
2SEPT BBBBBB
WCD 0.740" BPT
BSE= 0.04” WCD BLIN= 0.16” WCD
BHYS= 0.08” WCD BREP= 0.004” WCD
BZB= 0.716” WCD
eFAE 1.96 P
WCD 0.03125" BTAPE
WCD "10x 8.54 B -4DAQ
WCD 0.0667" BSA
WCD 0.0995" e
WCD .1949"0 PFAE WCD "766.0uFAE
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
12
SAMPLE
ONLY
Figure 6: Actual Pressure Transient Curve superimposed on the Fourier Pressure Transient Curve
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0 0.5 1 1.5 2 2.5
Time (s)
Head
(in
. W
CD
)
Actual Pressure TransientFourier Analysis Pressure Transient
Graph of Pressure Transient
Mechanical Engineering DepartmentCal Poly PomonaMeasurments Lab
Fourier Analysis ofPressure Transient Curve
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
13
SAMPLE
ONLYFigure 7: Chart of the frequency spectrum for the full pressure transient curve
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10
.5
11
.5
12
.5
13
.5
14
.5
15
.5
16
.5
17
.5
18
.5
19
.5
Frequency
He
ad
(in
. WC
D)
Frequency Spectrum ChartPressure Transient Curve
Mechanical Engineering DepartmentCal Poly PomonaMeasurements Lab
Fourier Analysis ofPressure Transient Curve
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
14
SAMPLE
ONLYFourier Analysis of
Pressure Transient Curve
P(t) = 2.986 + 11.936 sin (πt ) + 3.282 sin (2πt + 1.512) + 1.482 sin (3πt - 1.557) + 1.064 sin (4πt - 1.408) + 0.577 sin (5πt - 1.164) + 0.577 sin (6πt + 1.152) + 0.656 sin (7πt - 1.198) + 0.318 sin (8πt - 1.222) + 0.239 sin (9πt - 0.246) + 0.328 sin (10πt + 0.653) + 0.537 sin (11πt + 1.241) + 0.477 sin (12πt - 0.931) + 0.129 sin (13πt + 0.766) + 0.368 sin (14πt + 0.436) + 0.517 sin (15πt + 1.57) + 0.338 sin (16πt - 0.349) + 0.169 sin (17πt - 0.733) + 0.338 sin (18πt + 0.960) + 0.348 sin (19πt - 1.132) + 0.129 sin (20πt + 0.069) + 0.149 sin (21πt + 0.155) + 0.288 sin (22πt - 1.247) + 0.239 sin (23πt - 0.944) + 0.050 sin (24πt + 0.724) + 0.199 sin (25πt + 0.91) + 0.159 sin (26πt + 1.479) + 0.119 sin (27πt - 1.136) + 0.030 sin (28πt - 1.217) + 0.099 sin (29πt + 0.973) + 0.139 sin (30πt + 1.533) + 0.099 sin (31πt - 1.139) + 0.050 sin (32πt + 1.529) + 0.090 sin (33πt + 1.187) + 0.109 sin (34πt + 1.546) + 0.080 sin (35πt + 1.299) + 0.050 sin (36πt + 1.57) + 0.070 sin (37πt + 1.294) + 0.090 sin (38πt + 1.516) + 0.070 sin (39πt -1.391) +0.050 sin (40πt -1.528) [in. WCD]
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
15
SAMPLE
ONLYFigure 8: Actual static noise curve superimposed on the Fourier static noise curve
-0.3000000
-0.2000000
-0.1000000
0.0000000
0.1000000
0.2000000
0.3000000
0.4000000
0.5000000
0.6000000
0 0.01 0.02 0.03 0.04 0.05 0.06
Time (s)
Hea
d (in
. WC
D)
Actual Static Noise
Fourier Equation Static Noise
Graph of Static Noise Curves
James Beale and Tyler HaraMechanical Engineering DepartmentCal Poly PomonaSystem Dynamics Lab 9/30/02
Fourier Analysis of Noise in Static Region
10/31/06 FOR ME 435L DEMOSTRATION PURPOSES ONLY
16
SAMPLE
ONLY
Fourier Analysis of Noise in Static Region
Figure 9: Chart of frequency spectrum for the Static Noise Curve
0.0000000
0.0200000
0.0400000
0.0600000
0.0800000
0.1000000
0.1200000
0.1400000
0.1600000
0.1800000
0.2000000
50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
Frequency
Frequency Spectrum ChartStatic Noise Curve
Mechanical Engineering DepartmentCal Poly PomonaMeasurements Lab
P(t) = 0.039 + 0.149 sin (100πt ) + 0.020 sin (200πt - 0.233) + 0.030 sin (300πt - 1.238) + 0.189 sin (400πt + 1.477) + 0.010 sin (500πt - 0.406) + 0.020 sin (600πt - 0.098) + 0.109 sin (700πt + 0.871) + 0.030 sin (800πt + 1.216) + 0.010 sin (900πt + 1.431) +
0.010 sin (1000πt + 1.045) + 0.010 sin (1100πt - 1.161) + 0.010 sin (1200πt + 0.139)+ 0.040 sin (1300πt + 0.507) + 0.020 sin (1400πt + 0.314) + 0.020 sin (1500πt + 0.015)+
0.020 sin (1600πt - 1.327) + 0.010 sin (1700πt + 1.121) + 0.010 sin (1800πt - 0.81) + 0.010 sin (1900πt + 0.458) + 0.010 sin (2000πt + 0.078) [in. WCD]