me21 final exam

in general, all experimental error derives from one of 3 sources:

human, systematic, random

the most difficult type of error to guard against

human

(type of error) transposing two digits of a reading, misinterpreting data

human

(type of error) voltmeter reading below correct value, thermocouple that reads temperature too high

systematic

cure for systematic error

calibration

comparing the reading of a given instrument to a known standard (preferably traceable to the National Institute of Standards and Technology - formerly National Bureau of Standards)

calibration

cure for random error

average a sufficiently large number of measurements

when plotted as frequency of observation (f) vs. measured value (x), random experimental errors follow

a bell-shaped curve

bell-shaped curve random error follows

Gaussian or normal distribution

Gaussian distributions are characterized by their

mean and standard deviation

(sample) standard deviation equation

sigma(^^x)=

population standard deviation

replace n-1 to n in denominator

small sigma in bell curve

thinner peak

if data is Gaussian/normal distribution, ______% of measured x values will fall within interval (mean + stdv, mean - stdv)

68%

if data is Gaussian/normal distribution, ______% of measured x values will fall within interval (mean + 2*stdv, mean - 2*stdv)

95

if data is Gaussian/normal distribution, ______% of measured x values will fall within interval (mean + 3*stdv, mean - 3*stdv)

99

percentage confidence level (defined by stdv) of if a particular x value falls within a particular interval

confidence intervals

confidence interval widely used for engineering measurements

95% (mean + 2*stdv, mean - 2*stdv)

propagation of experimental error: mean (true value)

mean value = f(mean of each individual variable)

propagation of experimental error: st. dev (uncertainty)

uncertainty value = sqrt(sum of (derivative of f in respect to each individual variable)squared))

fractional uncertainty

uncertainty / value

resistance to penetration of an indenter under a prescribed load

hardness

prescribed combination of load and indenter

Rockwell hardness tests

linear regression aka

linear least-squares fit

linear least squares equations

(sumxi^2)a+(sumxi)b=(sumxiyi)

(sumxi)a+nb=(sumyi)

in general, an n-th order polynomial fit will yield ______ linear equations for the constant in the polynomial

n+1

how to fit a power-law relation of the form y=cx^a

take logs of both sides so that

lny=alnx+lnc

where b=lnc

(so lny=alnx+b)

fit curve fit of form y=ce^ax

take logs of both sides so that

lny=ax+lnc

where b=lnc

(so lny=ax+b)

types of graph paper

linear

semi-log (lny vs x)

log-log

data falls close to a straight line on linear paper

y=ax+b (linear fit)

data falls close to a straight line on semi-log paper

y=ce^ax (exponential fit)

data falls close to straight line on log-log paper

y=cx^a (power-law fit)

**may be typo in notes?

indicates how good a curve fit is

correlation coefficient (r)

solve for correlation coefficient

ym=(1/n)(sumyi) -> mean of values

oy=sqrt([sum(yi-ym)^2]/n-1) -> stdv

ybari=axi+b

oxy=sqrt([sum(yi-ybari)^2]/n-2)

r=sqrt(1-[oxy^2/oy^2])

values for a perfect fit

oxy=0

r=1

the closer "r" is to 1...

the better the fit

length of time a system takes to make one complete oscillation

period of oscillation

before being converted to a digital signal, the measured quantity must be converted to an ______ signal (usually a voltage)

analog

a signal that can take on any value and can vary continuously with time

analog

representations of analog signals that can take on only certain discrete values and are not continuous functions of time

digital

usually we measure experimental quantities using ___

transducers

a device that converts the quantity of interest into an electrical voltage

transducer

usually the output of a transducer will be ____ in the quantity of interest (at least over a certain range)

linear

v=kx, what is k?

sensitivity of the transducer (constant of proportionality)

v(t) represents the _______ we wish to convert to digital form

analog signal (v is the output of transducer)

take the value of the analog signal at certain times and convert this value to digital form

sample

in order to convert the analog signal to digital form, we _____ the analog signal at fixed intervals of time

sample

take continuous analog signal and replace it with ______ digital representation

discrete (value of the signal known only at the same instants of time we sampled the analog signal)

if signal is sampled often enough, easy to estimate the value of the signal at a time other than sampling times using....

simple interpolation

rate at which we sample the signal (usually samples/sec, or hertz)

sampling rate

the higher the sampling rate

the more accurate our representation of the analog signal

device used to convert analog signal to digital form

analog-to-digital converter (A/D converter)

how does an A/D converter work

integrated circuit (IC) board takes 1+ analog voltages and outputs their digital representations into one of the ports of a personal computer (PC)

determines what the analog signal can be representated as digitally by an A/D converter

bits

(1 bit = can only represent as 0 or 1

2 bits = can represent as 4 options, (0,0), (0,1) etc

n bits = can represent as n^2 options)

value that A/D converter distinguishes between

resolution

(1 bit = divide range of signal voltages by 2

2 bit = divide by 4...

n bit = divide by n^2)

bit value of A/D converter we used

16

slope of linear portion of output voltage of A/D converter vs displacement curve

sensitivity of LVDT (LABVIEW data acquisition system)

how does LVDT (measures linear displacement) operate

senses difference in voltage induced in 2 adjacent coils by a moving inner rod

LVDT output cannot be used in raw form; must be passed through a _______ to be converted to usable form

signal conditioner

output of signal conditioner is fed directly to

A/D converter

how to calibrate LVDT transducer

take the slope of linear portion of voltage vs displacement curve (displacement=(change in voltage)/k)

slope k depends on the _______ to the LVDT

voltage supplied by the power supply (if input voltage is changed, k will change and LVDT must be recalibrated)

LVDT initially clamped in a fixture in series with...

micrometer head (LVDT core may be precisely adjusted by moving micrometer head)

micrometer used for LVDT has a range of ___ and may be adjusted to within ____

25 mm, +/- 0.01 mm

calibrate LVDT by taking __ data points

5

as long as yield stress is not exceeded anywhere in the LVDT beam, the beam will obey Hooke's Law

we can expect linear relation of form F=k& between load F applied at end of beam and tip deflection &

Hooke's Law: F=k& what is k

stiffness (spring constant of Hooke's Law)

stiffness ratio for 2 aluminum beams of different radii

(k1/k2)=(r1^3/r2^3)

stiffness for steel and aluminum beams of equal diameter

(ks/kal)=(Es/Eal)

(E is Young's modulus)

Esteel

200 GPa

Ealuminum

70 GPa

used as force transducers and commonly referred to as "load cells"

elastic members

output of an elastic member (load cell) transducer

electrical voltage that is proportional to the force that is transmitted through the elastic element

as long as yield stress is not exceeded in elastic member (and we are operating at sufficiently low temperature so creep is not relevant),

behavior of member in response to applied force or displacement will be completely linear

response of elastic member is measured by measuring the ____ at a point on the member

strain

using elastic member, strain at a point is ______ of either applied displacement or applied load

linear function

device commonly used to make strain measurements

strain gage

limitations of strain gage

environmental conditions (high temp, corrosive environment, etc)

how are strain gages applied

normally bonded (glued or welded) to location at which we wish to measure strain

strain gages function by

changing their electrical characteristics (resistance, capacitance) as a (hopefully linear) function of the applied strain; we measure the change to get measure of applied strain form of electrical signal

most strain gages experience change in _____ as they are strained

resistance

strain gage factor F formula

E = (1/F)(change in R / R)

where R is gage resistance

the larger the value of F, the larger the value of ______ for given strain level E

change in R (gage resistance)

are larger or smaller values of F preferred for strain gage?

larger because larger values of (change in R) can be measured with greater precision

most common type of resistance-type strain gage

foil gage

how does foil gage work

thin strip of foil laid in rows along axis of gage; as foil is stretched along axis its resistance changes linearly with strain

examples of resistance-type strain gages

foil gage, semiconductor gage

how does semiconductor gage work

same as foil but uses strip of silicon; has gage factor of F=100, so good for applications where accuracy is important (like load cells)

pros of foil gage

inexpensive, easy application, flexible so can be applied to curved surfaces

cons of semiconductor gage

higher cost, brittle so difficult to apply to curved surface

counteract thermal strains due to changes in temperature (which would add to measured strain value)

strain gages attached to loaded members too often used w dummy gages attached to unloaded material (both loaded and unloaded members experience same thermal strain, so can subtract)

simple force transducer explanation

elastic ring w/ 4 strain gages wired as the 4 resistors of a Wheatstone Bridge & connected to signal conditioning device; when ring is extended, outside gages are in tension* and inside in compression; as long as yield stress not exceeded, force required to elongate ring will be linear function of measured strain values;

simple force transducer input and output

input: 6 volt d.c. battery

output: fed into A/D board inside a computer (voltage proportional to the strain at the gour gages and the input voltage)

using simple force transducer

calibration: under static conditions, keep adding weights for 5 weight/voltage points, then record slope k of linear fit

experiment: series of measurements of friction force required to pull a weight across a flat surface

how universal testing machine works

applies load through motion of a crosshead driven by two screwjacks; as screwjacks rotate, crosshead moves up and down

behavior of movement friction experiment

"slip and stick" - cable slackens as weight moves forward, reducing applied force and causing weight to come to temporary halt because force is no longer great enough to overcome friction force

a material element in a state of pure shear (such as exists in a torsion test) undergoes no...

change in volume

easier to interpret results of a stress-strain plot obtained in ______ than it is for more common tensile test

torsion

to obtain stress-strain behavior we need situation where the stress is ____ as is the case in a uniaxial stress test

statically determinate

if tube is sufficiently thin (relative to mean radius), we can assume the the shear stress is...

approximately constant across the thickness