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###### 日期：2020-10-13 11:39

Homework 4

Mechanical Engineering Lab I - MECE E3018 - Fall 2020 - Professor Yevgeniy Yesilevskiy

Due: October 15, 2020 at 11:59pm

1. You are given the task to measure the number of moles of a gas. To do so you fill a cylindrical

pressure vessel with a gas and wait for the system to come to thermal equilibrium with the

room before measuring the temperature and pressure of the system. The Ideal Gas Law, ???? =

??????, can be used to describe the system, where P is absolute pressure, V is volume, n is number

of moles, R is the universal gas constant, and T is the absolute temperature. What follows are

some questions which will guide you through the process to measure the number of moles and

also to estimate the uncertainty in the measurement.

a. You need to calculate the best estimate of the volume and the uncertainty in the

volume. The cylindrical pressure vessel has internal diameter, d, and internal length, L,

so the internal volume of the cylinder is ?? = (??4) ??2??. To calculate the best estimate

and the uncertainty in volume, you must therefore first calculate the best estimates and

the uncertainties of d and L.

i. The file `diameter.txt' contains measurements (units of mm) of the inside

diameter of the cylinder. Calculate the sample mean, ???, which is the best

estimate for the diameter.

ii. Calculate the precision (or random) absolute uncertainly in the measurements

as:

and also the precision (or random) relative uncertainty as

where the d subscripts indicate diameter.

iii. The specification sheet for the caliper used to measure the inside diameter

states that the accuracy of the instrument is ±0.01 mm, which is determined by

calibration against a standard from the National Institute of Standards and

Technology (NIST). The accuracy quantifies the bias (or systematic) absolute

uncertainty that we denote as ????. Calculate the bias relative uncertainty

defined as:

where again the d subscripts indicate diameter.

iv. The total absolute uncertainty, ????,

defines the total relative uncertainty. Calculate both ???? and ??????

and show that

??????

can also be calculated directly from ??????

and ??????

v. The file ‘length.txt' contains measurements (units of mm) of the inside length of

the cylinder. A different caliper with accuracy 0.02mm was used to make the

measurements. Repeat steps (i) through (iv) to calculate the best estimate, ???,

and the absolute and relative total uncertainties denoted as ????and ??????

,

respectively.

vi. Calculate the best estimate of the volume denoted as ???.

vii. As we discussed in class, the uncertainty in a result (in this case in volume) can

be calculated in terms of how the standard deviations associated with the

measurements (in this case of ???? and ????

) combine to determine the standard

deviation of the result (in this case ????). Specifically

for the case when the measurements of d and L are uncorrelated. Upon

normalizing this expression with the equation for the result (or volume) one

obtains

Show that for this case, and determine the total

uncertainty in volume as ??????

by calculating ??????.

b. You need to calculate the best estimate of the temperature and the uncertainty in the

temperature. The temperature measurements (with units of Kelvin) contained in the le

`temperature.txt' were made with a Hart Scientic 5665 thermistor.

i. Calculate the best estimate of the temperature denoted as ???.

ii. Calculate the absolute and relative precision uncertainties denoted as ???? and

??????

for the temperature.

iii. Refer to the specification sheet (DataSheets2.pdf) to determine the absolute

bias uncertainty for the temperature measurement denoted as ????. Calculate

the relative bias uncertainty denoted ??????

.

iv. Calculate the absolute and relative total uncertainties denoted as ???? and ??????

for the temperature.

c. You need to calculate the best estimate of the pressure and the uncertainty in the

pressure. The pressure in the cylinder is measured with a Honeywell PPTR pressure

transducer with a full-scale (FS) of 500 psi in analog mode with a maximum voltage of

5V. The output voltage of the pressure transducer is measured by a data acquisition

card (DAQ) and the measured voltage is then numerically converted to a pressure with

the calibration constant of 100 psi/V after which it is converted to MPa. The file

‘pressure.txt' contains the results of the absolute pressure in units of MPa.

i. Calculate the best estimate of the pressure denoted as ???.

ii. Calculate the absolute and relative precision uncertainties denoted as ???? and

??????

for the pressure.

iii. Refer to the specification sheet (DataSheets2.pdf) to determine the absolute

bias uncertainty for the pressure measurement denoted as ???? . Calculate the

relative bias uncertainty denoted ??????

.

iv. Calculate the absolute and relative total uncertainties denoted as ???? and ??????

for the pressure.

d. You are now prepared to calculate the best estimate of the number of moles as well as

the related total uncertainty, based upon the best estimates and uncertainties in

pressure, temperature and volume.

i. From the Ideal Gas Law, calculate the best estimate, ??? of the number of moles

of gas contained in the pressure vessel.

ii. Calculate the total absolute uncertainty of the measurement of moles, ???? and

the total relative uncertainty of the measurement of moles, ??????

2. The shear modulus, G, of a metal can be determined by measuring the angular twist, ??, resulting

from a torque applied to a cylindrical rod made of the metal. For a rod of radius, ??, and a

torque, ??, applied at a length ?? from a fixed end, the modulus can be calculated to be

In this problem you will examine the effect of the relative uncertainty of each measured variable

on the shear modulus to assist with the design of an experimental setup. With the equipment

you have in the laboratory, you can measure ??, ??, and ?? each with relative total uncertainty of

±1.0 %. However you need to purchase an instrument to measure ??. The goal of your

measurements is to have a total relative uncertainty in ?? of ?2.0% or less. What is the

maximum relative uncertainty allowable in the measurement for R?

3. The tip deflection of a cantilever beam with rectangular cross-section subjected to a point load

at the tip is given by the formula.

where ?? is the force, ?? is the length of the beam, ?? is the Young's modulus of the material, ?? is

the width of the cross-section, and ? is the height of the cross-section. The instrument

uncertainties in ??, ??, ??, ??, ? are each 2 %.

a. Estimate the fractional (i.e. relative) uncertainty in ??.

b. This beam is used in an experiment to determine the value of an unknown load, ????, by

performing a series of four repeated measurements of ?? at that load under the same

controlled conditions. The resulting sample standard deviation is 8 μm and the average

deflection is 20 μm. Determine the overall uncertainty in the deflection measurements

estimated at 90 % confidence assuming that the resolution of the instrument used to

measure ?? is so small that it produces negligible uncertainty.

4. A hand-held velocimeter uses a heated wire and, when air blows over the wire, correlates the

change in temperature to the air speed. The reading on the velocimeter, ????????, is relative to

standard conditions defined as ???????? = 70 °F and ???????? = 14.7 psia. To determine the actual

velocity, ????????, in units of feet per minute, the equation

???????? = ???????? [460+??460+????????][??????????]

must be applied, where ?? is in °F and ?? is in psia. The accuracy of the reading on the velocimeter

is 5% or 5 ft/min, whichever is greater. The velocimeter also measures air temperature with an

accuracy of 1°F. During an experiment, the measured air velocity is 400 ft/min and the

temperature is 80°F. The air pressure can be assumed to be at standard conditions. Assuming

that the relative precision uncertainties are much smaller than the relative bias uncertainties,

determine

a. the best estimate of the actual air velocity

b. the absolute uncertainty in the actual air velocity

c. the percent uncertainty in the actual air velocity.

5. Seven identical model truss bridges are constructed. Each is loaded with increasing masses until

it collapses. The masses used are in 50g increments. The loads at failure are 1.40 kg, 1.40 kg,

1.45 kg, 1.50 kg, 1.55 kg, 1.60 kg, and 1.60 kg. If an additional truss is constructed, estimate with

95 % confidence the loaded mass at which the truss will fail.