7/28/2019 Uncertainty Exercise - Material Test System
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ME 4600:483 Uncertainty Exercise Material Test System
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Problem Description: Estimate the uncertainty in determining the
Youngs modulusfor a material test system. The test system applies a
compressive forceFto a test
cylinder made of an elastic material and the cylinder
deformation L is measured. Fromthis data the linear material
response (Youngs modulusE) of the material is calculated.
The forceFis measured with a load cell and generates a voltage
signal V, which is
captured by a DAQ board. The block compression L is measured by
a displacementsensor with a digital readout.
Assumptions:
The cylinder deforms uniformly without barreling (the cylinder
surfaces are lubricated). The material is incompressible, therefore
the cylinder volume is constant. Use this fact to calculate D1
andA1 (cross
section area) based on L1.
Test Block:Nominal block dimensions: L0=20mm,D0=10mm. After
deformation theblock dimensions areL1 andD1.A1 is the cylinder
cross section area afterdeformation.
Instrumentation:
Load Cell to measure the applied loadFhas a Gain factorG=2004
N/Volt.
DAQ card measures the load cell voltage V, has a 12 bit A/D
converter andis configured for a voltage range of 0-5V. It also has
a hysteresis error of
0.5% at full scale. Displacement sensor forL has a digital
display with a resolution of 0.1mm.
Other errors for this sensor can be neglected. Ruler to check
undeformed block dimensions (L0 andD0) has a 0.5mm resolution.
Test Data:
The max. applied loadFis 100N. At that load the nominal
deflection L is about 1.5mm.
The experimental data process consists of increasing and
decreasing the load and measuring the displacement 14times at
different loads (up to the max of 100N). The raw data of block
displ. vs. load cell voltage and the result afterconversion to
strain and stress are shown below . A linear regression statistics
of the strain / stress data is calculated
Equations:
VGFD
AALALLLLL
L
A
FE
;
4;;;;;
2
001101
01
Solution Approach:1. Determine the nominal values of the
different variables in the test system. Nominal in this case is at
the max load
expected on the test system.2. Write an equation, which
determines the test result (the modulus) from the test variables.3.
Determine uncertainties for the test variables.4. Calculate, how
uncertainties propagate from the test variables to the test
result.5. Apply the known uncertainties to the propagation equation
to calculate the total design stage uncertainty (or bias
error); also determine, which variables contribute the most to
the design stage uncertainty of the test system.6. Use the test
data regression statistics to determineE7. From the standard
deviation of the slope Sa1, determine its uncertainty (= precision
error forE)
8. Combine design state uncertainty and precision error forEto
get the total test system uncertainty for this test.9. State the
Youngs modulusEof the elastomer including its uncertainty
interval.
Regression Statistics
Strain to Stress:
N: 14
a0: 0.0318 [N/mm2]
a1: 15.216 [N/mm2]
r: 0.992
Syx: 0.0504 [N/mm2]
Sy: 0.399 [N/mm2]
Sa0: 0.0242 [N/mm2]
Sa1: 0.538 [N/mm2]
L0
L1
D0
D1
F
L
